Discuss events which are simultaneous in one frame?

[neopolitan]

Forgive me chipping in here, but you've gone a bit off track from simultaneity.

<snip>

I think you're making heavy weather of this simultaneity thing. What does it matter if two observers disagree about two events being simultaneous or not ?

I refer you to post #143. Hopefully that will make clear what mattered and you can see that I personally wasn't wandering away from simultaneity.

Here are the relevant paragraphs, but the whole lot in glorious context lies below.

I know that you could select any inertial frame and use that as what could be called an "ether frame". But what I am asking is, is there anything preventing a "one true rest frame" or an "absolute (at) rest frame"?

Then going further, if there is nothing preventing such a frame (even if we cannot distinguish it), would not that frame's hypersurface of simultaneity be the boundary of the universe? It seems to me that the boundary of the universe is more of a "when" question than a "where" question.

This has not been addressed.

I agree that the discussion about spacetime is a little off the track but in post #147 I did again try to bring it back -

My understanding, which may be wrong, is that the universe doesn't care what you do in it (as equally as it doesn't care about simultaneity). I can choose whatever frame of reference I like, I can consider ourselves to be at rest, and I can subsequently work out the spacetime locations of events around me, in terms of my selected frame of reference. Other observers can follow the same process, choosing whatever frame of reference they like, considering themselves to be at rest and they can subsequently work out the spacetime locations of events around them, in terms of their own selected frames of reference.

If we all then consider the same event, we can use transformations between our frames of reference and work out that we are indeed considering the same event. Is this correct?

If it is correct, then I don't see what relevance there is to "a rotating spacetime". I don't see how it relates to what I was initially pondering, the possibility of a "one true rest frame" (or "absolute at rest frame", or AAR frame) which is indistinguishable from any "nominally in motion" NIM frame. I don't think there is any reason to assume that an AAR frame would be, or should be, rotating. It could be, I guess. You are just left with the question "what is it rotating in reference to?" Since this is a conceptual "absolute at rest" frame, it is at rest. So really what you would be saying is that all other frames, which appear to be inertial, are for some reason actually rotating relative to the AAR frame (actually, they would be orbiting - the vast majority of them impossibly, since their rotation velocity would be greater than the speed of light).

You effectively answered the question in the second paragraph, thanks. Now if I could get someone to address the other questions in #143 without spiralling off into what is really another topic, it would great!

cheers,

neopolitan

 

[Mentz114]

But what I am asking is, is there anything preventing a "one true rest frame" or an "absolute (at) rest frame"?

Hmm. A frame of reference is not a physical thing. It has to be associated with some observer who is using coordinates of her choice. I'm not sure what an absolute frame could mean. If the universe was filled with some kind of fluid that was flow free, one could use it a frame of reference, and we could define absolute velocity. But the universe is not filled in such a way. So probably the notion of an absolute frame is not useful.

Then going further, if there is nothing preventing such a frame (even if we cannot distinguish it), would not that frame's hypersurface of simultaneity be the boundary of the universe? It seems to me that the boundary of the universe is more of a "when" question than a "where" question.

The entirety of the universe has no boundary by definition. What can be outside the universe to make a boundary ? But your question makes sense if we consider the limits of our observable universe. You are aware that the further light travels to us the more red-shifted it is. So there is a distance from us at which this light becomes undetectable, and this is a boundary in the sense that we cannot see past it. Because the boundary is defined in terms of light propagation, the distance is also a time.
If we pointed our two telescopes in opposite directions and received a signal with identical red-shift ( say an H2 Lyman line) in both telescopes at the same time on our clock, I'm not sure if we could say that the light had been emitted 'simultaneously'. It doesn't have much physical import.

I may have repeated stuff from earlier posts, I've read through but I don't remember it all.

 

[cyberdyno]

But what I am asking is, is there anything preventing a "one true rest frame" or an "absolute (at) rest frame"?

You don't need to make an absolute frame out of empty space. There need to be landmarks before we can have a preferred frame, there needs to be a mass in it. You can make flat space into a fixed frame, but there is no need for it. Not in GR. You need to use frames only in curved space.

 

[Dale]

Accepting what you have to say about "rotating spacetimes", in reference to what is this spacetime rotating?

Rotation is intrinsic. You don't have to be rotating wrt anything to be rotating. If you are in a closed laboratory you can do all sorts of experiments to determine if you are rotating or not. You don't need to refer to anything outside the laboratory.

Don't get too sidetracked about the details of a rotating spacetime. The point is simply that in GR there is no concept of simultaneity that makes sense universally. I think it is problematic at best to talk about "boundaries" and "center" of the universe in terms of a concept that doesn't apply universally.

 

[JesseM]

None of this answers my questions. I understand what you mean by "rotating spacetime" and I accept that it may only be possible to define simultaneity "locally in a rotating spacetime" but I fail to see the relevance.

If anything, it gives me reason to wonder if the idea of an AAR has more relevance than I initially though, since your explanation forces me to ask this:

Accepting what you have to say about "rotating spacetimes", in reference to what is this spacetime rotating?

Are you sure you're familiar with the notion of "rotating spacetime"? Note that it has nothing to do with what coordinate system you choose, it is a physical feature of the spacetime itself. The rotating universe is a GR solution that was originally discovered by Kurt Godel (of Godel's incompleteness theorem in logic) to allow "closed timelike curves", i.e. time travel into the past; I think this may be the key reason it is not possible to come up with a global definition of simultaneity in such a spacetime. Physicists refer to the idea of dividing up a 4D spacetime into a stack of 3D spacelike hypersurfaces as a "foliation" of that spacetime, and if I'm remembering correctly I think it's only possible to foliate a spacetime which is globally hyperbolic (and the third paragraph here seems to confirm my memory), with part of the definition of globally hyperbolic spacetimes being that they do not contain closed timelike curves.

Here is a nice essay on the Godel rotating universe solution, written by someone who wrote a Ph.D. thesis on the subject. The author gives a good explanation of what it means physically for the universe as a whole to be rotating, and why it does not require a center of rotation:

When I tell people about the possibility of a rotating universe, their reaction is usually either a silly smile, or the very well motivated question: With respect to what would the universe rotate? I viciously reply: With respect to something that does not rotate, that is, something that does not experience any centrifugal forces. OK, this is correct, but it needs some elaboration.

First of all, don't try to imagine the universe as rotating as a whole. That way of thinking is misleading. I'll come back to rotation as a whole later.

Second, don't think that this implies some center of rotation. According to the Copernican principle, all places in the universe are equivalent. This is a simplifying assumption adopted by most cosmologists; whether it holds in reality is an open question. On smaller scale the universe is badly inhomogeneous, but there is still hope that the large scale structure is homogeneous.

Third, study carefully the following attempt to visualize a rotating universe.

Imagine you are in a laboratory without windows floating around somewhere in the universe. If you and the other objects in the laboratory get pressed against the walls, you would say that the laboratory is rotating, and centrifugal forces are responsible for the effects. Now, the laboratory happens to be equipped with small engines that can be used to control the rotation. Use the engines until you have totally eliminated the centrifugal forces, and thereby the rotation. When done, drill some peepholes in the laboratory (but please make sure you don't lose your air supply). Observe the galaxies. If you find that the galaxies rotate around you, then the universe is said to be rotating.

You have actually only seen that the universe rotates around the point where you are, but if the Copernican principle holds, then it rotates around any point. That's a rotating universe.

So keep in mind that when I talk about a rotating universe, I mean that the matter of the universe rotates around the non-rotating observer. There is a better word for it: vorticity. In classical hydrodynamics, the vorticity w of a velocity field v is defined using the rot operator:

click for image of equation

In general relativity, there is a similar definition. One expresses the vorticity four-vector in terms of matter four-velocity field (a four-vector is a vector with one 'time' and three 'space' components).

 

[JesseM]

This is not quite what I am asking. I know that you could select any inertial frame and use that as what could be called an "ether frame". But what I am asking is, is there anything preventing a "one true rest frame" or an "absolute (at) rest frame"?

As a metaphysical belief this is possible, but it's meaningless as a physical theory. You're free to imagine that one coordinate system is "metaphysically preferred" in the sense that its judgments about simultaneity (or about other frame-dependent questions, like which of two objects has a higher speed) represent the "real truth" of the matter. But relativity says that no coordinate system is physically preferred over any other, so you can never have any empirical evidence to justify the idea that one frame's judgments are better than any other's. And as far as metaphysics goes, if you like the principle of Occam's razor than the fact that a metaphysically preferred frame would have no empirical consequences whatsoever would be a pretty good argument for dispensing with such a notion, although nothing forces you to accept the Occam's razor argument.

This would mean that, in the terms that we normally use for thinking about such things, there would be no 3 dimensional edge to the universe and no 3 dimensional centre. Instead there would be a 4 dimensional edge and a 4 dimensional centre. (It may help to remove one dimension and think of a sphere. The two dimensional surface of the sphere has no centre and no edge. The three dimensional sphere itself, however, has the two dimensional surface as the boundary and the centre of the sphere is surrounded by and separated from the surface - so two dimensional people living on the sphere would never be able to reach the centre of the universe. The centre of our universe would, therefore, be in the past - in a big bang event, or some equivalent to a big bang event.)

The standard cosmological model has the universe expanding like the surface of a balloon. Is what I have expressed above just saying the same thing, perhaps in another way?

The standard cosmological model actually allows for three possible "shapes" of an expanding universe, depending on the density of matter and energy throughout space. Also, when you talk about a "4 dimensional center", it seems like you're imagining the universe as the 3D surface of a 4D hypersphere which is sitting in a larger 4D space--this is what would be called an "embedding space", but the mathematics of differential geometry actually allows you to describe the curvature of a 3D surface without the need for it to be curved in a higher-dimensional space. I discussed both these points in post #4 here:

According to the Big Bang theory, the Big Bang was not an explosion in a preexisting 3-dimensional space, with matter and light expanding out into empty space from some central point--instead, matter and energy are understood to fill all of 3D space, and what's expanding is space itself. The key is to understand that the Big Bang theory is based on Einstein's theory of general relativity, which explains gravity in terms of matter/energy causing spacetime to become curved--depending on the average density of matter/energy throughout the universe, a consequence of this is that the universe as a whole can be curved, with either positive curvature, zero curvature, or negative curvature. For a closed universe with positive curvature, you can visualize it if you drop the dimensions by one--instead of curved 3-dimensional space, which is impossible for us to visualize, picture a 2D universe a la Flatland in which 2D space is actually curved into a sphere, and "expanding space" means that the sphere is blowing up like a balloon while the bits of 2D matter on the surface do not change in size. You can see that if you pasted a bunch of bits of paper on a balloon and then blew it up, each bit would see the other bits receding from it, just like what we see with other galaxies. If you play the movie backwards so that the size of the sphere approaches zero, you can seen that all the bits of matter throughout the universe get more and more squished together, approaching infinite density as the size approaches zero--this is what the big bang is supposed to be. Of course, this analogy forces you to picture the 2-dimensional surface of the sphere expanding in a higher 3rd dimension, and while it is possible that our curved 3D space is expanding in some kind of higher 4D space, mathematically there is no need for such a thing--instead of describing the curvature of a surface with reference to a higher-dimensional "embedding space", it is possible to describe curvature using purely intrinsic features that could be observed by a being confined to the surface (like whether the sum of angles of a triangle drawn on the surface is more, less, or equal to 180 degrees), and general relativity uses only such intrinsic features to describe what it means for space to be curved (see this page on differential geometry, the mathematical basis for general relativity, which talks about the difference between intrinsic and extrinsic descriptions of curvature).

For a universe with zero curvature, picture an infinite chessboard in which all the squares are growing at the same rate, while the pieces at the center of each square remain unchanged in size. If you play the movie backwards, the distance between any two squares approaches zero as you approach the moment of the big bang, which means the density of the matter on the squares (represented by the chess pieces) approaches infinity as it gets smushed together more and more tightly. A universe with negative curvature would be something like an infinite saddle-shape which is a little harder to picture expanding, but if you can picture the other two you get the basic idea. From Ned Wright's Cosmology Tutorial, a graphic showing the 2D analogues of the three types of spatial curvature, negative, zero, and positive:

One other thing to point out is that even if you want to embed curved 3D space in a higher-dimensional euclidean space, or curved 4D spacetime in a higher-dimensional flat spacetime, one additional dimension may not be enough (as an analogy you might think of a 1D line curved into a corkscrew shape, which can't be embedded in 2D space). As discussed in this thread, it has been proven that any curved 4D spacetime could be embedded in a flat spacetime with 90 dimensions, 87 spacelike and 3 timelike. I don't know if anyone has come up with an example of a spacetime that would require this many dimensions to embed, but this is the upper bound.

 

[Mentz114]

You don't need to make an absolute frame out of empty space. There need to be landmarks before we can have a preferred frame, there needs to be a mass in it. You can make flat space into a fixed frame, but there is no need for it. Not in GR. You need to use frames only in curved space.

Cyberdyno - please correct post #153. I did not ask the question you attribute to me.

You can make flat space into a fixed frame, but there is no need for it.

Wrong. You just said that matter is required. Matter is always required to define a frame.

 

[neopolitan]

The entirety of the universe has no boundary by definition. What can be outside the universe to make a boundary ? But your question makes sense if we consider the limits of our observable universe. You are aware that the further light travels to us the more red-shifted it is. So there is a distance from us at which this light becomes undetectable, and this is a boundary in the sense that we cannot see past it. Because the boundary is defined in terms of light propagation, the distance is also a time.
If we pointed our two telescopes in opposite directions and received a signal with identical red-shift ( say an H2 Lyman line) in both telescopes at the same time on our clock, I'm not sure if we could say that the light had been emitted 'simultaneously'. It doesn't have much physical import.

Extremely red-shifted light will still be detectable. It's called infrared radiation. IR radiation is still the same sort of thing as light, it's just a "colour" that we can't see. What we normally refer to as light is just the spectrum of EMR that we can see unaided but we can use different telescopes and "see" any frequency in the EMR spectrum we want to.

I will get to the boundary issue in a moment.

The standard cosmological model actually allows for three possible "shapes" of an expanding universe, depending on the density of matter and energy throughout space. Also, when you talk about a "4 dimensional center", it seems like you're imagining the universe as the 3D surface of a 4D hypersphere which is sitting in a larger 4D space--this is what would be called an "embedding space", but the mathematics of differential geometry actually allows you to describe the curvature of a 3D surface without the need for it to be curved in a higher-dimensional space. I discussed both these points in post #4 here:

One other thing to point out is that even if you want to embed curved 3D space in a higher-dimensional euclidean space, or curved 4D spacetime in a higher-dimensional flat spacetime, one additional dimension may not be enough (as an analogy you might think of a 1D line curved into a corkscrew shape, which can't be embedded in 2D space). As discussed in this thread, it has been proven that any curved 4D spacetime could be embedded in a flat spacetime with 90 dimensions, 87 spacelike and 3 timelike. I don't know if anyone has come up with an example of a spacetime that would require this many dimensions to embed, but this is the upper bound.

I actually have thought that one extra dimension is not sufficient. It has been a concern, since it seemed to lead to an infinite progression of dimensions.

It is interesting to see that 90 dimensions appears to be an upper limit. Personally, I cannot fathom why, once you get up to 90 dimensions, you should suddenly stop there. What is special about 90? If it were a more natural number I might feel slightly more comfortable about it: say 81, as either three to the power of six or 9 squared, or 128, as two to the power of seven, or 91 as the factorial of thirteen, or 100, as 10 squared, or 89, either as the nearest prime number or as the 12th number in the fibonacci sequence, or 85 as the seventh in a sequence of summed squares. It's difficult to see any significance to 90.

As for boundaries, my perception is that a 90-dimensional universe is unbounded (in terms of 90 dimensions). In terms of 89 dimensions, however, the universe would be bounded and so on, all the way down to the "useful" dimensions, if I can call them that.

(Note, I do see that this is an 89+1 dimensional universe, which would make more sense if there were 89 spacelike dimensions and 1 timelike. But you specified that it was 87 spacelike dimensions and 3 timelike. Finding something that makes sense of 87 and three is equally difficult. (29+1)*3? But then we are looking for a meaning for 29 and 3. I can live with 3 since we perceive our universe to have three spacelike dimensions. 29 as the tenth prime number? Ok, so we are left looking for a meaning for 10. The factorial of four? Why four and why a factorial, and why a prime number before? Anyway, I think you see the point, where is the physical significance of 90 dimensions?)

I agree with Mentz in so much as a four dimensional universe has no meaningful boundary in terms of four dimensions. However, a three dimensional universe does have a meaningful boundary in terms of four dimensions.

If you think about the surface of sphere, it is not bounded in two dimensions (although that two dimensional space is actually curved in terms of three dimensions). It is, however, "limited" or bounded in terms of three dimensions, when the radius of the sphere becomes apparent.

Similarly, the hypersurface of a hypersphere is not bounded in three dimensions (although that three dimensional space is actually curved in terms of four dimensions). Similarly, the hypersurface of the hypersphere is bounded in terms of four dimensions, when the "hyper-radius" of the hypersphere becomes apparent.

While mathematically you might be able to express this in terms where there is no need for a hypersphere or any other 4-D shape, but I am trying to interpret this in useful terms. So, when we don't observe any open surfaces in our perceived 3-D universe, other than in the world of mathematics, is there any reason to assume an open hypersurface in what we know to be (at the very least) a 4-D universe?

cheers,

neopolitan

 

[Mentz114]

Extremely red-shifted light will still be detectable. It's called infrared radiation. IR radiation is still the same sort of thing as light, it's just a "colour" that we can't see. What we normally refer to as light is just the spectrum of EMR that we can see unaided but we can use different telescopes and "see" any frequency in the EMR spectrum we want to.

Are you saying there is no limit below which we cannot detect light ? That's just plain wrong. Our instruments are not infinitly sensitive. Already we have to cool the detectors to very low temperatures.

This thead has got ridiculous. From the relativity of simultaneity, which is simple and easy to grasp, we now have multi-dimensional cosmologies and all sorts of weird stuff that has nothing to do with the thread topic.

I'm unsubscribing from this farago.

 

[neopolitan]

Are you saying there is no limit below which we cannot detect light ? That's just plain wrong. Our instruments are not infinitly sensitive. Already we have to cool the detectors to very low temperatures.

It is largely irrelevant, but I think it is also largely wrong. I am pretty damn sure that we can detect all frequencies below the light spectrum at least down to the ELF radio spectrum - we might have problems with weak signals but not lowish frequencies. Admittedly you need a landmass as the detector (like a peninsular or a subcontinent), but it is technically feasible to detect an ELF signal.

Problems with detecting frequencies below ELF (below 1 hertz for example) would have nothing to do with the temperature of the detectors and more to do with the size of the detectors.

I don't think there is any transmitter in the universe that is moving fast enough to cause doppler shift down to below ELF, is there?

Note, I know this is entirely off topic. I will start a new thread on it - please respond to it there rather than here.

I don't know how to make a proper link! :(

But just above this is the link to the new thread

thanks,

neopolitan

 

[Dale]

Neopolitan, I have to agree with Mentz, this tangent of yours is rather absurd.

First, as I already pointed out, there is no universal concept of synchronization with which to easily extract your 3D universal hypersurface. Synchronization is only generally clearly defined locally over small regions where spacetime is essentially flat.

Second, you are rather ignorant about the concept of embedding (not a criticism, I am too) so it is unwise to blindly assume that it is possible to use a fourth timelike dimension to embed even a simple 3D spacelike geometry and further unwise to claim that the result of such an embedding would be that "a three dimensional universe does have a meaningful boundary in terms of four dimensions". If you really wish to pursue this line of thought I would highly recommend that you study the embedding concept for a while and then actually do the math.

And third, so what? If we were 2D beings living on the surface of a 3D sphere what benefit would we get from projecting our space up into 3 dimensions. We would find that the "boundary" of our universe is everywhere and that the "center" of our universe is nowhere and that our universe is curved at a certain curvature. All of which we could have deduced intrinsically. I don't see the value added by the embedding.

PS Light can be red-shifted below zero frequency. That is what an event horizon is.

 

[neopolitan]

And third, so what? If we were 2D beings living on the surface of a 3D sphere what benefit would we get from projecting our space up into 3 dimensions. We would find that the "boundary" of our universe is everywhere and that the "center" of our universe is nowhere and that our universe is curved at a certain curvature. All of which we could have deduced intrinsically. I don't see the value added by the embedding.

Is it generally agreed that 'the "boundary" of our universe is everywhere and that the "center" of our universe is nowhere and that our universe is curved at a certain curvature'? (Noting that I don't say the centre of the universe is nowhere, I say it is in the past.)

If that's the case, taking into account my note, then I am happy.

If it is also the case that the universe expands in such a way that that expansion can be interpreted as the passage of time, then I am also happy. - Note, I am not saying that the universe is expanding with time, or over time, but effectively that very expansion is time. If that is the generally accepted case, then I am very happy.

Is that the case? If it is then it seems from what you are saying that I have somehow come to this via an unorthodox route, and it involves the idea of what is effectively a hypersurface of simultaneity - one which constitutes the boundary of the universe in terms of four dimensions.

I am sorry that the conversation spins off into weird directions, it is certainly not my intention that it should.

cheers,

neopolitan

PS And as for "embedding", I don't think I used that term at all. It's a bit like reading some of what I typed earlier and writing it off as the "Lorentzian ether interpretation". Giving what I muse about a label doesn't make it what the label says it is, and it doesn't mean that I go along with all the baggage normally associated with the label.

 

[JesseM]

Is it generally agreed that 'the "boundary" of our universe is everywhere and that the "center" of our universe is nowhere and that our universe is curved at a certain curvature'?

It is generally accepted that the center is nowhere (or at least nowhere within the 3D space of the universe), but I'm not sure what the difference is between saying "the boundary of the universe is everywhere" and saying "the universe has no boundary". If we look at a 2D surface without an edge like a flat plane, would you say that the entire surface is its own boundary? Would you say that even if the surface is not embedded in a higher-dimensional space?

(Noting that I don't say the centre of the universe is nowhere, I say it is in the past.)

What do you mean "the universe is curved at a certain curvature"? As I said in a previous post, the curvature of 3D space depends on the density of mass and energy--if the density is higher than a certain critical value it has positive spatial curvature, which is analogous to the 2D surface of a sphere, but if it's right at that critical value it'd be flat (zero spatial curvature) like an infinite 2D plane, and if it's below that critical value it'd have negative curvature which is similar to the 2D surface of an infinite saddle. Again, just have a look at the diagrams and explanation here.

If it is also the case that the universe expands in such a way that that expansion can be interpreted as the passage of time, then I am also happy. - Note, I am not saying that the universe is expanding with time, or over time, but effectively that very expansion is time. If that is the generally accepted case, then I am very happy.

You should not imagine that time is just the radial dimension of the hypersphere representing a positively-curved space, so that successive moments would be like layers of an onion in a higher-dimensional space, if that's what you're suggesting; this would imply that time has to reverse if the universe begins to contract again (which positively-curved universes naturally do unless the cosmological constant is high enough), but that isn't a prediction of general relativity. If you want to imagine a positively-curved universe that expands from a big bang and then collapses back into a big crunch, it's better to drop the dimensions in your analogy down by one again, so that a hypersurface of simultaneity is represented by a 1D line curved into a circle; then spacetime as a whole would look like the 2D surface of an American football, with one pointy end being the big bang and the other pointy end being the big crunch, and each cross-section of the football surface would give a different-sized circle representing the size of the universe at a particular moment; as you moved from the big bang to the big crunch, the circles would grow bigger for a while, then shrink again. The fact that simultaneity is relative could be represented by the fact that you are free to slice the football at different angles in order to produce your stack of expanding and contracting circles.

Of course this analogy still requires us to imagine the surface embedded in a higher-dimensional space, which should not be taken seriously as anything physical--as I said, the mathematics of differential geometry allows you to describe curvature in purely intrinsic terms without reference to an embedding space, so the notion of space (or spacetime) sitting in some higher-dimensional space becomes physically irrelevant, another "metaphysical" notion like absolute simultaneity (although some variants of superstring theory do imagine the universe as a sort of membrane in a higher dimension, and in this theory the additional dimension does have physical consequences).

PS And as for "embedding", I don't think I used that term at all.

But that's the standard term for a higher-dimensional space in which a lower-dimensional curved surface is "sitting", like a curved 3D hypersurface of simultaneity sitting in a 4D (or higher) space. Is this not what you were talking about when you referred to the universe having a "4 dimensional centre" in post #143? The center of a 3D surface which is curved into a hypersphere cannot lie anywhere on the surface itself.

 

[neopolitan]

It is generally accepted that the center is nowhere (or at least nowhere within the 3D space of the universe), but I'm not sure what the difference is between saying "the boundary of the universe is everywhere" and saying "the universe has no boundary". If we look at a 2D surface without an edge like a flat plane, would you say that the entire surface is its own boundary? Would you say that even if the surface is not embedded in a higher-dimensional space?

No, I said that the surface of a sphere ( a 2D surface ) represents the boundary of the sphere ( a 3D volume ). Only when you think in terms of a 3D hypersurface and a 4D "hypervolume" can you consider that the apparent 3D universe is its own boundary - noting that this phrasing is yours, not mine. I don't disagree with the phrasing, so I am not 100% sure that I didn't use it, but checking back I can't see anywhere where I did.

What do you mean "the universe is curved at a certain curvature"? As I said in a previous post, the curvature of 3D space depends on the density of mass and energy--if the density is higher than a certain critical value it has positive spatial curvature, which is analogous to the 2D surface of a sphere, but if it's right at that critical value it'd be flat (zero spatial curvature) like an infinite 2D plane, and if it's below that critical value it'd have negative curvature which is similar to the 2D surface of an infinite saddle. Again, just have a look at the diagrams and explanation here.

Actually that phrasing is DaleSpam's.

You should not imagine that time is just the radial dimension of the hypersphere representing a positively-curved space, so that successive moments would be like layers of an onion in a higher-dimensional space, if that's what you're suggesting; this would imply that time has to reverse if the universe begins to contract again (which positively-curved universes naturally do unless the cosmological constant is high enough) <snip>

I disagree. It does not imply that time has to reverse in order to reach another big-bang event. There is another possibility. I don't have time to go into it right now, but perhaps you can work it out yourself.

Think about the fact that 1) the universe is expanding and 2) the universe is not expanding uniformly. If it were expanding uniformly, we would never notice it because we would expand with it. What is expanding is the space between masses (masses being concentrations of energy).

What happens when the universe is at maximum entropy? I am thinking here about "http://en.wikipedia.org/wiki/Heat_death" ".

If at this point the universe continues to expand but expands uniformly, and all the energy in the universe is homogenously distributed, it is basically indistinguishable from the entirety of the universe compacted homogenously into a very small volume (in the past). As long as the heat distribution is homogenous nothing will happen, but if the distribution becomes hetrogenous and gap opens up, this gap will expand faster than where the relative concentration of energy is. Maybe this will be overcome and a new equilibrium will be reached, but eventually the gap will open up enough to effectively flip the universe inside out, so that rather than having a small empty space in large otherwise homogenous heat energy distribution, you now have a relatively large empty space surrounding a highly compact concentration of energy. And that then explodes.

Maybe.

I don't know, since I wasn't around when it happened before, and I doubt that I will be here when it happens again. But this does allow me to have my cake and eat it to. The universe continues to expand and I effectively get a heat death and I get a big crunch followed by a big bang.

But that's the standard term for a higher-dimensional space in which a lower-dimensional curved surface is "sitting", like a curved 3D hypersurface of simultaneity sitting in a 4D (or higher) space. Is this not what you were talking about when you referred to the universe having a "4 dimensional centre" in post #143? The center of a 3D surface which is curved into a hypersphere cannot lie anywhere on the surface itself.

It may be the standard term, but I didn't use it. If I use the standard term, I might get tied to things that I don't intend. Embedding sounds contrived to me. If we can say the two dimensional surface of the Earth is embedded in the apparent 3D universe then I probably can go along with it, but DaleSpam indicated that there is much to study with the concept, so I worry that using the term "embedded" will sign me up for ideas and concepts that I am not aware of.

Yes, the centre of a 3D surface which is curved into a hypersphere cannot lie anywhere on the surface itself.

cheers,

neopolitan

 

[JesseM]

No, I said that the surface of a sphere ( a 2D surface ) represents the boundary of the sphere ( a 3D volume ). Only when you think in terms of a 3D hypersurface and a 4D "hypervolume" can you consider that the apparent 3D universe is its own boundary - noting that this phrasing is yours, not mine.

OK, but once again you are assuming that curved 3D space is embedded in a higher-dimensional space, so that we can talk about the "volume" enclosed by 3D space in this higher dimension. This is not an assumption of general relativity--again, general relativity uses differential geometry to describe curved space and curved spacetime without the notion that they are embedded in a higher dimensional space.

I disagree. It does not imply that time has to reverse in order to reach another big-bang event.

What does not imply it? Are you indeed imagining that time is just the radial dimension of a hypersphere? If so, then besides the fact that this relies on unphysical notions about embedding space in a higher dimension, and doesn't make sense in the case of a universe with negative or zero curvature (in which case space is not shaped like a hypersphere), I don't see how you could believe this and yet not believe that as the universe shrinks it is returning to earlier times, since the you're now moving towards the center on the radial dimension rather than away from it. But maybe you're not really thinking of time as just the radial dimension in this way, please clarify what you meant by "effectively that very expansion is time".

There is another possibility. I don't have time to go into it right now, but perhaps you can work it out yourself.

Think about the fact that 1) the universe is expanding and 2) the universe is not expanding uniformly. If it were expanding uniformly, we would never notice it because we would expand with it. What is expanding is the space between masses (masses being concentrations of energy).

I don't know what you're hinting at, but suffice to say that trying to understand the meaning of time through intuitive embeddings of space is likely to lead only to weird ideas which have nothing to do with the mathematical predictions of general relativity.

What happens when the universe is at maximum entropy? I am thinking here about "http://en.wikipedia.org/wiki/Heat_death" ".

If at this point the universe continues to expand but expands uniformly, and all the energy in the universe is homogenously distributed

That's not what maximum entropy would necessarily look like--for gravitating systems, greater entropy often leads to more clumpiness, not more homogeneity (the amount of clumpiness in the equilibrium distribution will depend on the temperature).

As long as the heat distribution is homogenous nothing will happen, but if the distribution becomes hetrogenous and gap opens up, this gap will expand faster than where the relative concentration of energy is.

Why do you assume GR would predict that a gap would expand faster?

Maybe this will be overcome and a new equilibrium will be reached, but eventually the gap will open up enough to effectively flip the universe inside out, so that rather than having a small empty space in large otherwise homogenous heat energy distribution, you now have a relatively large empty space surrounding a highly compact concentration of energy. And that then explodes.

What would cause the energy to become "highly compact"? Self-gravity? If so, why would it "explode" rather than just becoming more compact and perhaps forming a black hole?

I don't know, since I wasn't around when it happened before, and I doubt that I will be here when it happens again. But this does allow me to have my cake and eat it to. The universe continues to expand and I effectively get a heat death and I get a big crunch followed by a big bang.

Just in a way that follows from your own vague imaginings, not in a way that follows from any well-defined theory of physics, as far as I can tell.

It may be the standard term, but I didn't use it. If I use the standard term, I might get tied to things that I don't intend. Embedding sounds contrived to me.

"Contrived" in what way? All that embedding means in this context is having a curved lower-dimensional surface sitting in a noncurved higher-dimensional space or spacetime.

If we can say the two dimensional surface of the Earth is embedded in the apparent 3D universe then I probably can go along with it

Yes, of course.

but DaleSpam indicated that there is much to study with the concept, so I worry that using the term "embedded" will sign me up for ideas and concepts that I am not aware of.

The "much to study" is just geometry--pure math, not any new physics. For example, it's a nontrivial mathematical result that for any possible curved 4D spacetime (with the curvature defined in terms of differential geometry), it's guaranteed to be embeddable in a flat spacetime with 87 space dimensions and 3 time dimensions.

Yes, the centre of a 3D surface which is curved into a hypersphere cannot lie anywhere on the surface itself.

And do you agree it is possible to describe the curvature of a 3D surface with no reference whatsoever to any higher-dimensional space, so the idea that such a higher-dimensional space exists at all is physically superfluous?

 

[Dale]

Is it generally agreed that 'the "boundary" of our universe is everywhere and that the "center" of our universe is nowhere and that our universe is curved at a certain curvature'? (Noting that I don't say the centre of the universe is nowhere, I say it is in the past.)

What I am referring to is the widely accepted "Copernican principle" which basically states that we are probably do not occupy a special point in the universe. My understanding is that, based on what we observe astronomically and on the Copernican principle the general agreement is that at every point in the universe it would look like the universe is expanding in all directions. So there is no center and there is no edge in terms of 3D space. As JesseM mentioned saying that the boundary is everywhere is not much different from saying that there is no boundary.

PS And as for "embedding", I don't think I used that term at all. It's a bit like reading some of what I typed earlier and writing it off as the "Lorentzian ether interpretation". Giving what I muse about a label doesn't make it what the label says it is, and it doesn't mean that I go along with all the baggage normally associated with the label.

Objecting to the use of standard terms is pointless. The Lorentz ether is an undetectable but still real absolute reference frame. The concept of a lower-dimensional curved space described from the point of view of a higher-dimensional flat space is embedding. Sorry you don't like the terms, but they are useful for communication, and your ideas are substantially described by those terms.

I don't know why you think using an appropriate label is "writing it off", that is certainly not my intent. I think the Lorentz ether concept is useful for explaining relativistic Doppler effects, and I think embedding is a useful way to understand basic curvature concepts. I see nothing dismissive or otherwise wrong with using the appropriate standard terminology.

 

[neopolitan]

What I am referring to is the widely accepted "Copernican principle" which basically states that we are probably do not occupy a special point in the universe. My understanding is that, based on what we observe astronomically and on the Copernican principle the general agreement is that at every point in the universe it would look like the universe is expanding in all directions. So there is no center and there is no edge in terms of 3D space. As JesseM mentioned saying that the boundary is everywhere is not much different from saying that there is no boundary.

Well, yes and no.

It's the difference between an infinitely large universe with no boundary and a closed universe with no boundary. The Copernican principle could be applied to both, as far as I can work out. I think you might run into problems with an infinitely large universe in that such a universe would then have to have infinite mass in order to avoid there being vantage points from which the mass in the universe seems to be approaching you from one direction and the universe in the other direction is empty.

Are there people who hold to both the Copernical principle and the idea of an open universe? It seems not to work for me.

I agree with JesseM that "saying that the boundary is everywhere is not much different from saying that there is no boundary". If we are happy that it is the same thing from different perspectives there is no problem.

Objecting to the use of standard terms is pointless. The Lorentz ether is an undetectable but still real absolute reference frame. The concept of a lower-dimensional curved space described from the point of view of a higher-dimensional flat space is embedding. Sorry you don't like the terms, but they are useful for communication, and your ideas are substantially described by those terms.

I don't know why you think using an appropriate label is "writing it off", that is certainly not my intent. I think the Lorentz ether concept is useful for explaining relativistic Doppler effects, and I think embedding is a useful way to understand basic curvature concepts. I see nothing dismissive or otherwise wrong with using the appropriate standard terminology.

Perhaps I am too touchy, but I would prefer "what you are describing has much in common with Lorentzian ether theory" rather than "this is essentially Lorentzian ether theory". I am familiar with argumentation techniques which involve a form of guilt by association, and since I have been accused before of looking at things from (too say the least) a slightly different perspective, I don't want to be labelled - especially when there is potential for someone to read some of these posts and go off thinking that I believe in ether (which is usually thought of as the luminiferous stuff that Michelson and Morley were looking for, not something almost entirely conceptual as in the theoretical interpretation that you refer to).

I accept that embedding "substantially describes" my ideas, but just be aware that if you or anyone else make the step "those who espouse embedding also say this", then that step is not valid. Note also that pretty much every single reference to "embedding" that I can find on the internet is linked to brane theory. So, using the term may inadvertently link me to that school of thought - and by extension to M-theory and thence to superstring theory. And as things currently stand, I am sceptical about all sorts of string theory which form a theoretical frame that seems pre-Copernican to me.

cheers,

neopolitan

 

[neopolitan]

You should not imagine that time is just the radial dimension of the hypersphere representing a positively-curved space, so that successive moments would be like layers of an onion in a higher-dimensional space, if that's what you're suggesting; this would imply that time has to reverse if the universe begins to contract again (which positively-curved universes naturally do unless the cosmological constant is high enough<snip>.

I disagree. It does not imply that time has to reverse in order to reach another big-bang event.

What does not imply it? Are you indeed imagining that time is just the radial dimension of a hypersphere? If so, then besides the fact that this relies on unphysical notions about embedding space in a higher dimension, and doesn't make sense in the case of a universe with negative or zero curvature (in which case space is not shaped like a hypersphere), I don't see how you could believe this and yet not believe that as the universe shrinks it is returning to earlier times, since the you're now moving towards the center on the radial dimension rather than away from it. But maybe you're not really thinking of time as just the radial dimension in this way, please clarify what you meant by "effectively that very expansion is time".

If it is also the case that the universe expands in such a way that that expansion can be interpreted as the passage of time, then I am also happy. - Note, I am not saying that the universe is expanding with time, or over time, but effectively that very expansion is time. If that is the generally accepted case, then I am very happy.

Just trying to get this in context, if you pull out bits and ask for explanations out of context, it doesn't really help.

We know that the universe is expanding. We also know that we have this phenomenon which we perceive as the passage of time (we can possibly say that as "we also know that there is this thing called time" or however you prefer to state it).

How do we know that? We can only tell when there is change. Velocities tell us (because things which have unlike velocities experience change in their relative positions).

Imagine for a moment that everything in the universe was stationary (yes, I know it is not possible). Nothing changes and there is, effectively, no passage of time (your clocks don't move, everything is stationary remember).

Now, add in universal expansion. You have time passing again because things are moving apart from each other, there is change. This is sort of what I mean by the very expansion of the universe is time but this is illustrative not prescriptive.

Without considering the whole of what I have said, you won't grasp what I am trying to say. If you think of one moment in time (a "one true simultaneity", one that we cannot distinguish, but which might have some importance), and then the moment after that, you have two moments which you can not only use to observe change, but between which change can occur.

Now if this "when", this one moment in time, is the surface of the universe, then is it subsequently surrounded by the next moment, and is slightly larger. This is universal expansion as passage of time. It will not be observable unless there is something which counteracts or resists this expansion, since even the space between our constituent atoms will expand and our rulers would expand. However, we do notice that the universe doesn't expand uniformly.

That's (referring to Heat Death) not what maximum entropy would necessarily look like--for gravitating systems, greater entropy often leads to more clumpiness, not more homogeneity (the amount of clumpiness in the equilibrium distribution will depend on the temperature).

Some people seem to think that heat death is where the universe is going. Even some people with far more letters behind their name than I have. Now I am not appealing to authority, but just saying that while I cannot disagree that there are others who think differently, like yourself, I am not alone in thinking that heat death is possible.

Why do you assume GR would predict that a gap would expand faster?

Which bits of the universe are expanding fastest? Where there is a concentration of mass or where there isn't? I am not using GR to predict this, I am looking at the universe and noting

the fact that 1) the universe is expanding and 2) the universe is not expanding uniformly. If it were expanding uniformly, we would never notice it because we would expand with it.

As you can see from the quote, I had already said that.

What would cause the energy to become "highly compact"? Self-gravity? If so, why would it "explode" rather than just becoming more compact and perhaps forming a black hole?

Well, actually, gravity is possibly a symptom of other factors rather than a cause in itself. In reality the mass of the universe wouldn't be anymore compact than it was previously, but relative to the expanding empty space around it would be highly compact.

As to forming a black hole, there are arguments that the mass of the universe already is a black hole, but on the inside. And anway, your argument here is "why the big bang, why didn't it just turn into a black hole"? It apparently happened once, I can't see why it couldn't happen again.

"Contrived" in what way? All that embedding means in this context is having a curved lower-dimensional surface sitting in a noncurved higher-dimensional space or spacetime.

It's more a personal thing, embedding sounds active: "General Disorder embedded the journalist Ms Tellall into the troop of soldiers". If it is not meant this way, it is entirely a descriptive term and means something like "the 2D surface of a 3D object is intrinsic to our mathematical description of the object" then I can agree that it isn't contrived.

And do you agree it is possible to describe the curvature of a 3D surface with no reference whatsoever to any higher-dimensional space, so the idea that such a higher-dimensional space exists at all is physically superfluous?

Not really sure what you are saying here. Is our universe this curved 3D surface? Why do we now ignore 4D space, which is the basis of 4-vector notation which most GR devotees are so fond of? Or are you talking about the surface of a 3D object (which I, perhaps erroneously, refer to as 2D) and want to not mention a fourth dimension? That would be fine by me.

If it is the former, I can't imagine that 4-space is physically superfluous, so sorry, no.

cheers,

neopolitan

 

[JesseM]

Just trying to get this in context, if you pull out bits and ask for explanations out of context, it doesn't really help.

We know that the universe is expanding. We also know that we have this phenomenon which we perceive as the passage of time (we can possibly say that as "we also know that there is this thing called time" or however you prefer to state it).

How do we know that? We can only tell when there is change. Velocities tell us (because things which have unlike velocities experience change in their relative positions).

Imagine for a moment that everything in the universe was stationary (yes, I know it is not possible).

It's possible to have a flat SR spacetime (which is also allowed in GR) where all particles are stationary relative to one another, if that's what you mean.

Nothing changes and there is, effectively, no passage of time (your clocks don't move, everything is stationary remember).

Now, add in universal expansion. You have time passing again because things are moving apart from each other, there is change. This is sort of what I mean by the very expansion of the universe is time but this is illustrative not prescriptive.

But do you agree there would equally be change in a non-expanding flat spacetime if particles were simply moving relative to one another? If so, then I still don't get why you would say the expansion "is" time.

Without considering the whole of what I have said, you won't grasp what I am trying to say. If you think of one moment in time (a "one true simultaneity", one that we cannot distinguish, but which might have some importance), and then the moment after that, you have two moments which you can not only use to observe change, but between which change can occur.

Now if this "when", this one moment in time, is the surface of the universe, then is it subsequently surrounded by the next moment, and is slightly larger.

But why do you say "surrounded by", if you're not picturing time as the radial axis in a 4D space where the 3D universe at a single instant is embedded? Did you read my analogy where if we picture the universe at any given instant (according to some definition of simultaneity) as a 1D line curved into a circle rather than a 2D surface curved into a sphere, then spacetime as a whole can be pictured as an upright American football, with the bottom point as the big bang and the top point as the big crunch, and each successive cross-section from top to bottom giving a circle that represents the universe at a given instant? In this case, a later moment would lie "above" a previous one in our visualization, it wouldn't surround the previous one like layers of an onion. You can see a fuzzy illustration of this sort of visualization http://www.fortunecity.com/emachines/e11/86/space.html .

This is universal expansion as passage of time.

You still haven't explained why you see expansion as passage of time, rather than just one of the many examples of things in the universe which change over time.

It will not be observable unless there is something which counteracts or resists this expansion, since even the space between our constituent atoms will expand and our rulers would expand. However, we do notice that the universe doesn't expand uniformly.

The fact that small bound systems don't expand can be understood in the context of GR (a ruler is held together by non-gravitational forces, but you can also look at gravitationally bound systems like the solar system, which isn't expected to expand with the universe either)--see this section of the Usenet Physics FAQ.

That's (referring to Heat Death) not what maximum entropy would necessarily look like--for gravitating systems, greater entropy often leads to more clumpiness, not more homogeneity (the amount of clumpiness in the equilibrium distribution will depend on the temperature).

Some people seem to think that heat death is where the universe is going. Even some people with far more letters behind their name than I have. Now I am not appealing to authority, but just saying that while I cannot disagree that there are others who think differently, like yourself, I am not alone in thinking that heat death is possible.

You misread what I said there. You inserted the parentheses "(referring to Heat Death)" in my sentence, but I wasn't referring to Heat Death, I was referring to your statement "all the energy in the universe is homogenously distributed". In pure GR, a state of maximum entropy (and 'Heat Death' is the idea that the universe will go to a maximum entropy state, if you didn't know) will not be a homogenous distribution of matter, it will actually be very clumpy, with matter collected into black holes. Of course, if you try to incorporate quantum effects, particularly Hawking radiation which is expected to cause black holes to evaporate into mostly photons, then things get more complicated; in this case, it might again be true that the maximum-entropy state would be pretty homogenous spatially, just a universe filled with photons left over from Hawking radiation (see the photon age from the wikipedia article on heat death).

Which bits of the universe are expanding fastest? Where there is a concentration of mass or where there isn't? I am not using GR to predict this, I am looking at the universe and noting

Yes, and as I said, GR can explain this observation. But it's not clear that this is equivalent to the idea that if you have a mostly homogeneous distribution of matter/energy throughout the universe and then a small empty or almost empty region forms in one spot, then this region will begin to expand faster than the rest of the universe (and even if it expands a little faster the difference might not be very significant, I highly doubt that it would be so much faster that 'the gap will open up enough to effectively flip the universe inside out' and force all the matter to occupy a small region as you suggested).

As to forming a black hole, there are arguments that the mass of the universe already is a black hole, but on the inside. And anway, your argument here is "why the big bang, why didn't it just turn into a black hole"? It apparently happened once, I can't see why it couldn't happen again.

Look, just making vague speculative arguments based on isolated facts you have read about cosmology is a very bad way to achieve any understanding of physics, I highly discourage this approach as it will tend to lead you into crackpot-land. All cosmological predictions are based on GR, and GR has a perfectly good answer to why concentrating a bunch of matter in one place in a larger space will cause a black hole to form, while the dense but fairly homogenous distribution of matter and energy throughout space in the first moments after the big bang did not form a black hole--read over this section of the Usenet Physics FAQ.

It's more a personal thing, embedding sounds active: "General Disorder embedded the journalist Ms Tellall into the troop of soldiers".

That's a pretty recent usage of the word. How about something like "a large emerald was embedded in the king's crown?"

If it is not meant this way, it is entirely a descriptive term and means something like "the 2D surface of a 3D object is intrinsic to our mathematical description of the object" then I can agree that it isn't contrived.

The embedding space/spacetime is always assumed to have zero curvature, so regardless of the number of dimensions it's easy to set up a coordinate system with straight orthogonal axes, like a Cartesian coordinate system where all the axes are straight lines that meet at right angles at a single origin. So, the curved surface that is embedded in this embedding space can be completely described in terms of this coordinate system, in the same way that a 2D spherical surface of radius one can be described in a 3D embedding space using the equation x^2 + y^2 + z^2 = 1 (any x,y,z coordinates that lie on the surface of the sphere will satisfy this equation, while x,y,z coordinates that don't lie on the sphere won't satisfy it). That's basically all that embedding space implies, that it's possible to completely describe the curved surface in terms of what points it occupies in a coordinate system in the embedding space.

Not really sure what you are saying here. Is our universe this curved 3D surface? Why do we now ignore 4D space, which is the basis of 4-vector notation which most GR devotees are so fond of?

No! There is no 4th dimension of space in GR, only a 4th dimension of time. And the 4th dimension is not an uncurved "embedding space" for curved 3D space; rather, GR describes the curved 4D surface of spacetime in purely intrinsic terms, without the need for a 5th dimension for this curved 4D surface to be embedded in. Just think back to the visualization where I pictured a big bang/big crunch spacetime as the surface of an American football; here I have dropped the number of spatial dimensions by 2, so that spacetime is a curved 2D surface with one space dimension and one time dimension. To visualize this curved 2D surface we intuitively have to picture it sitting in an uncurved (Euclidean) 3D space, but GR could describe its curvature in intrinsic terms, with no need for a higher dimension for the spacetime to be embedded in.

 

[Dale]

I am familiar with argumentation techniques which involve a form of guilt by association, and since I have been accused before of looking at things from (too say the least) a slightly different perspective, I don't want to be labelled - especially when there is potential for someone to read some of these posts and go off thinking that I believe in ether (which is usually thought of as the luminiferous stuff that Michelson and Morley were looking for, not something almost entirely conceptual as in the theoretical interpretation that you refer to).

Fair enough. You are correct, the Lorentz ether theory is championed by some real crackpots that I have encountered on other forums. Unfortunately, they generally are pretty ignorant about its predictions and implications (particularly experimental implications) and therefore tend to apply it incorrectly or just generally spout unrelated nonsense and call it "Lorentz ether". So it is not unreasonable of you to be concerned about guilt by association.

However, there are two differences in this case: 1) generally the crackpots themselves claim to agree with Lorentz in order to lend their idea some authority (which you did not do here) and 2) they misapply it to reach erroneous conclusions (which I did not do here). As mentioned before, I am completely comfortable with people applying and using the Lorentz ether approach as long as they do so correctly. It is not a perjorative in my mind as I use it to understand relativistic Doppler effects. Your idea was essentially the Lorentz ether as correctly applied, not a crackpot bastardization.

I accept that embedding "substantially describes" my ideas, but just be aware that if you or anyone else make the step "those who espouse embedding also say this", then that step is not valid.

Also fair enough. Another "guilt by association" argument. If you think I make such a logical fallacy please point it out. In the meantime, my use of the labels makes communication easier and is not intended dismissively.

 

[neopolitan]

Hi DaleSpam

Also fair enough. Another "guilt by association" argument. If you think I make such a logical fallacy please point it out. In the meantime, my use of the labels makes communication easier and is not intended dismissively.

I don't think you were making any logical fallacy yourself, and with the understanding of the risks involved, I accept the use of the labels as aids to communication.

Note the following post, in which I point out that it seems to me that JesseM is misusing your label to associate what I am saying with the assertions of others.

cheers,

neopolitan

 

[neopolitan]

But do you agree there would equally be change in a non-expanding flat spacetime if particles were simply moving relative to one another? If so, then I still don't get why you would say the expansion "is" time.

Actually I do agree that would equally be change in a non-expanding flat spacetime if particles were simply moving relative to one another, which is why I was careful to say

This is sort of what I mean by the very expansion of the universe is time but this is illustrative not prescriptive.

It's also what I pointed to the need to think of the argument as a whole. Without thinking of the argument as a whole, I don't think it is possible to understand what I am getting at. I am happy to have my argument as a whole shot down, but if someone considers one leg of the table I have made and says "your table won't stand if it only has one leg", I have to agree with them on that specific level even if I know that in a grander scale, my table may well still stand. It's possible that it won't, but the reason won't be that "my table won't stand if it only has one leg".

But why do you say "surrounded by", if you're not picturing time as the radial axis in a 4D space where the 3D universe at a single instant is embedded?

I do though, with reservations about the term "embedded" as noted in a previous post replying to DaleSpam.

Did you read my analogy where if we picture the universe at any given instant (according to some definition of simultaneity) as a 1D line curved into a circle rather than a 2D surface curved into a sphere, then spacetime as a whole can be pictured as an upright American football, with the bottom point as the big bang and the top point as the big crunch, and each successive cross-section from top to bottom giving a circle that represents the universe at a given instant? <snip>

Yes I did. It's a piecemeal thing though.

Since I "(picture) time as the radial axis in a 4D space where the 3D universe at a single instant is embedded" then the American Football model doesn't work for me. Additionally, I am not sure that the American Football model produces such a neat explanation for length contraction and it's temporal equivalent (effectively the inverse of time dilation) and the invariance of c that my visualisation does. It may, but I doubt it.

You misread what I said there. You inserted the parentheses "(referring to Heat Death)" in my sentence, but I wasn't referring to Heat Death, I was referring to your statement "all the energy in the universe is homogenously distributed". In pure GR, a state of maximum entropy (and 'Heat Death' is the idea that the universe will go to a maximum entropy state, if you didn't know) will not be a homogenous distribution of matter, it will actually be very clumpy, with matter collected into black holes. Of course, if you try to incorporate quantum effects, particularly Hawking radiation which is expected to cause black holes to evaporate into mostly photons, then things get more complicated; in this case, it might again be true that the maximum-entropy state would be pretty homogenous spatially, just a universe filled with photons left over from Hawking radiation <snip>

Or all the matter in the universe finally collects into one black hole. Note this is part of the whole, the universe is bounded and unless the black holes are absolutely stationary with respect to each other (ignoring the relative velocity due to expansion) these black holes will eventually collide, if Hawking radiation doesn't make them evapourate. I understand that while Hawking radiation is generally accepted, it is still speculative and has not been inequivocably observed.

All the mass of the universe being in one black hole is not really an issue from my perspective, since I think there is something to the idea that we are already inside the effective event horizon of an extremely supermassive black hole.

I note that you said

Look, just making vague speculative arguments based on isolated facts you have read about cosmology is a very bad way to achieve any understanding of physics, I highly discourage this approach as it will tend to lead you into crackpot-land.

I guess I should be happy that you make that a prediction, rather than a diagnosis. However, this is not something that I took from isolated readings of cosmology and patched into my visualisation. When I thought about it, I came to the conclusion that what I was thinking was perhaps totally impossible since it implies that we would be inside a black hole. It was after that that I got to hear that someone else had done the maths and that showed that the Schwartzschild radius of the universe's mass matched the universe's radius. The equation is there, the figures are there, you can do the maths yourself. It works.

What you may need to do is explain why the equation doesn't apply in this instance.

Yes, and as I said, GR can explain this observation. But it's not clear that this is equivalent to the idea that if you have a mostly homogeneous distribution of matter/energy throughout the universe and then a small empty or almost empty region forms in one spot, then this region will begin to expand faster than the rest of the universe (and even if it expands a little faster the difference might not be very significant, I highly doubt that it would be so much faster that 'the gap will open up enough to effectively flip the universe inside out' and force all the matter to occupy a small region as you suggested).

There's no forcing the matter to occupy the small region. It is just that the matter doesn't expand like the (relatively) empty space. Relative to this (relatively) empty space, the matter becomes more and more compact. But it is only a relative thing.

The real issue, one for which I have no explanation is why relative compactness of the energy (which may be in the form of photons) should change its form to the stuff of a big bang. It's possible that your link hints at an answer to that question (only possibly, I am not hinting that does).

That's a pretty recent usage of the word. How about something like "a large emerald was embedded in the king's crown?"

It's less amusing (no opportunity for a character called General Disorder, for instance), but still equally describing something being done to a pre-existing crown - the crown would still be a crown without the emerald (although, it is possible that without the emerald it is not a king's crown, depending on your definitions and cultural expectations). I can't see that a sphere could exist without its surface. So, I don't think there is an equivalent process by which the surface of the sphere is embedded in the sphere.

The embedding space/spacetime is always assumed to have zero curvature, so regardless of the number of dimensions it's easy to set up a coordinate system with straight orthogonal axes, like a Cartesian coordinate system where all the axes are straight lines that meet at right angles at a single origin. So, the curved surface that is embedded in this embedding space can be completely described in terms of this coordinate system, in the same way that a 2D spherical surface of radius one can be described in a 3D embedding space using the equation x^2 + y^2 + z^2 = 1 (any x,y,z coordinates that lie on the surface of the sphere will satisfy this equation, while x,y,z coordinates that don't lie on the sphere won't satisfy it). That's basically all that embedding space implies, that it's possible to completely describe the curved surface in terms of what points it occupies in a coordinate system in the embedding space.

I direct your attention to the discussion with DaleSpam, and I direct DaleSpam's attention to this paragraph above. For JesseM, this is relevant:

I accept that embedding "substantially describes" my ideas, but just be aware that if you or anyone else make the step "those who espouse embedding also say this", then that step is not valid.

The coloured section in your paragraph is your equivalent of saying "those who espouse embedding also say this".

No! There is no 4th dimension of space in GR, only a 4th dimension of time. And the 4th dimension is not an uncurved "embedding space" for curved 3D space; rather, GR describes the curved 4D surface of spacetime in purely intrinsic terms, without the need for a 5th dimension for this curved 4D surface to be embedded in. Just think back to the visualization where I pictured a big bang/big crunch spacetime as the surface of an American football; here I have dropped the number of spatial dimensions by 2, so that spacetime is a curved 2D surface with one space dimension and one time dimension. To visualize this curved 2D surface we intuitively have to picture it sitting in an uncurved (Euclidean) 3D space, but GR could describe its curvature in intrinsic terms, with no need for a higher dimension for the spacetime to be embedded in.

I am pretty sure this is another case of taking the use of the term "embedding" and trying to tie me to something that I don't subscribe to. I should have written 4-space, rather than 4D space. Is that better?

I never mentioned a 5th dimension, except by association when responding to your post where you talked about there being 90 dimensions. I didn't call for another higher dimension for spacetime (3+1) to be embedded in.

I think you have misunderstood me somewhere (or we have misunderstood each other somewhere).

cheers,

neopolitan

 

[JesseM]

It's also what I pointed to the need to think of the argument as a whole. Without thinking of the argument as a whole, I don't think it is possible to understand what I am getting at.

But that's my point, it doesn't seem to me that you've presented any argument as to why we should consider the expansion of space to be time, rather than just one of many things that are changing with time.

Since I "(picture) time as the radial axis in a 4D space where the 3D universe at a single instant is embedded" then the American Football model doesn't work for me. Additionally, I am not sure that the American Football model produces such a neat explanation for length contraction and it's temporal equivalent (effectively the inverse of time dilation) and the invariance of c that my visualisation does. It may, but I doubt it.

Just to be clear, are you claiming that your way of thinking about time might just be a good way of visualizing the consequences of general relativity, or are you proposing that general relativity might need to be replaced by a new theory which explains thing differently? If the latter then your speculations don't really belong in this forum.

GR does in fact deal with spacetime as a single curved surface, so it's natural to visualize this by dropping the number of space dimensions by two so we can think about a curved 2D surface like the football (we 3-dimensional creatures can't really visualize a curved 3D surface directly, much less a curved 4D surface).

Or all the matter in the universe finally collects into one black hole. Note this is part of the whole, the universe is bounded and unless the black holes are absolutely stationary with respect to each other (ignoring the relative velocity due to expansion) these black holes will eventually collide, if Hawking radiation doesn't make them evapourate. I understand that while Hawking radiation is generally accepted, it is still speculative and has not been inequivocably observed.

True, but then again few of the defining features of black holes that are predicted by GR, like the presence of an event horizon or a singularity, have anything in the way of observational evidence. Very dense, non light-emitting objects have been observed, but without GR there'd be no reason to think they have these features.

All the mass of the universe being in one black hole is not really an issue from my perspective, since I think there is something to the idea that we are already inside the effective event horizon of an extremely supermassive black hole.

I'm not aware of any model that says we could be in a black hole, since this would seem incompatible with expansion--the page I linked to suggested the possibility we might be in a giant white hole (the temporal reverse of a black hole, it only spits out matter and nothing can enter its horizon), but even this model would probably be considered fairly far-fetched by most physicists.

However, this is not something that I took from isolated readings of cosmology and patched into my visualisation. When I thought about it, I came to the conclusion that what I was thinking was perhaps totally impossible since it implies that we would be inside a black hole. It was after that that I got to hear that someone else had done the maths and that showed that the Schwartzschild radius of the universe's mass matched the universe's radius. The equation is there, the figures are there, you can do the maths yourself. It works.

Unless you have a GR-based model which puts us in a black hole and yet gives the same observational predictions about things like the redshifts of distant galaxies, then this is indeed just spinning ungrounded speculations based on a few isolated facts you have read (namely, your claim that the Schwarzschild radius mass matches the universe's radius--though I'd like to see the source of this, estimates of the radius of the observable universe have changed over the years, and no one claims to know the size of the entire universe beyond the distance that light has had time to get to us, if the curvature of space is flat or negative then mainstream models typically treat it as infinite).

What you may need to do is explain why the equation doesn't apply in this instance.

I already pointed out the well-understood fact that in GR, the "Schwarzschild radius" only applies to a non-expanding space, in an expanding universe you can have more mass in a space of that size without a black hole being formed. Did you not read the page I linked to?

That's a pretty recent usage of the word. How about something like "a large emerald was embedded in the king's crown?"

It's less amusing (no opportunity for a character called General Disorder, for instance), but still equally describing something being done to a pre-existing crown - the crown would still be a crown without the emerald (although, it is possible that without the emerald it is not a king's crown, depending on your definitions and cultural expectations). I can't see that a sphere could exist without its surface. So, I don't think there is an equivalent process by which the surface of the sphere is embedded in the sphere.

Who said anything about the surface of the sphere being embedded in the sphere? I was talking about "embedding" the curved 2D surface in 3D euclidean (noncurved) space.

I direct your attention to the discussion with DaleSpam, and I direct DaleSpam's attention to this paragraph above. For JesseM, this is relevant:

The embedding space/spacetime is always assumed to have zero curvature[/color], so regardless of the number of dimensions it's easy to set up a coordinate system with straight orthogonal axes, like a Cartesian coordinate system where all the axes are straight lines that meet at right angles at a single origin. So, the curved surface that is embedded in this embedding space can be completely described in terms of this coordinate system, in the same way that a 2D spherical surface of radius one can be described in a 3D embedding space using the equation x^2 + y^2 + z^2 = 1 (any x,y,z coordinates that lie on the surface of the sphere will satisfy this equation, while x,y,z coordinates that don't lie on the sphere won't satisfy it). That's basically all that embedding space implies, that it's possible to completely describe the curved surface in terms of what points it occupies in a coordinate system in the embedding space.

The coloured section in your paragraph is your equivalent of saying "those who espouse embedding also say this".

So you are imagining the curved 3D surface of space at a particular moment as being part of a 4D spacetime which is also curved? This would not be "embedding", and while I suppose it's true that these 3D surfaces of simultaneity are in some sense "contained in" the curved 4D spacetime (in the same sense that the 1D circles are 'contained in' the 2D football surface), all attempts to make arguments based on how you visualize such things are likely to go badly wrong since we can really only visualize shapes in the context of 3D euclidean geometry. And your arguments about earlier surfaces of simultaneity being "surrounded by" later ones seems to be based on such a concrete Euclidean visualization. Can you say what it means for one 2D surface to "surround" another one if we aren't picturing them in ordinary Euclidean 3D space, but are instead trying to imagine them as being contained in a curved 3D space or spacetime which is impossible for us to visualize directly? Do you think that successive 3D surfaces of simultaneity "surround" each other in a sense that successive 1D circular cross-sections of the curved football surface do not surround each other?

I am pretty sure this is another case of taking the use of the term "embedding" and trying to tie me to something that I don't subscribe to.

No, "embedding" can apply either to space or spacetime (all that matters is that the embedding space/spacetime has zero curvature, like Euclidean space or minkowski spacetime), I was just responding to the fact that you seemed to be visualizing the curved surface of space using a geometric picture of a 2D spherical surface sitting in ordinary 3D space.

I never mentioned a 5th dimension, except by association when responding to your post where you talked about there being 90 dimensions. I didn't call for another higher dimension for spacetime (3+1) to be embedded in.

I didn't say you had mentioned a 5th dimension--but I had thought you were assuming the higher dimension that curved space was sitting in would itself be uncurved since you seemed to be visualizing it in terms of ideas taken from ordinary Euclidean geometry (like the notion of one surface 'surrounding' another), so a natural extension of this would be that if GR describes curved 4D spacetime, it must be sitting in a higher-dimensional flat space or spacetime. If you never meant to suggest the higher-dimensional space should be uncurved then I misunderstood, but then see my points above about how any attempts at visualizations (which necessarily involve uncurved Euclidean space, since we can't picture curved 3D space directly) should be regarded with suspicion. The specific idea that one surface "surrounds" another doesn't necessarily make sense if the higher-dimensional space/spacetime they are both contained in is itself curved, as in the case of the football where we can't really say which of two circular cross-sections is surrounding the other.

 

[neopolitan]

But that's my point, it doesn't seem to me that you've presented any argument as to why we should consider the expansion of space to be time, rather than just one of many things that are changing with time.

Because I am heading off to yet another meeting, I can't respond in depth. However, I think it may be worth the couple of minutes it needs to drag this back into perspective.

And third, so what? If we were 2D beings living on the surface of a 3D sphere what benefit would we get from projecting our space up into 3 dimensions. We would find that the "boundary" of our universe is everywhere and that the "center" of our universe is nowhere and that our universe is curved at a certain curvature. All of which we could have deduced intrinsically. I don't see the value added by the embedding.

Is it generally agreed that 'the "boundary" of our universe is everywhere and that the "center" of our universe is nowhere and that our universe is curved at a certain curvature'? (Noting that I don't say the centre of the universe is nowhere, I say it is in the past.)

If that's the case, taking into account my note, then I am happy.

If it is also the case that the universe expands in such a way that that expansion can be interpreted as the passage of time, then I am also happy. - Note, I am not saying that the universe is expanding with time, or over time, but effectively that very expansion is time. If that is the generally accepted case, then I am very happy.

Is that the case? If it is then it seems from what you are saying that I have somehow come to this via an unorthodox route, and it involves the idea of what is effectively a hypersurface of simultaneity - one which constitutes the boundary of the universe in terms of four dimensions.

I am sorry that the conversation spins off into weird directions, it is certainly not my intention that it should.

This is where this strand comes from.

I said I would be very happy if the expansion of the universe could be interpreted as the passage of time. I see that, in my visualisation, it works.

As far as I can tell, you haven't actually said that this is not possible. I think you have said that it is not necessary and that it is not something in your (and perhaps the standard) interpretation of GR. So I could make a slight modification to the original and say:

If it were also the case that it is not impossible that the universe could be expanding in such a way that that expansion could be interpreted as the passage of time, then I would be vaguely happy. - Note, I am suggesting the possibility not so much that the universe is expanding with time, or over time, but that the passage of time we experience is a symptom of that very expansion.

I did subsequently try to explain how I visualise things, but there's no real need to convince you. I would much prefer to have you understand and disagree with what I am trying to express than have you agree without understanding. So, there is no should intended from my side (check back to the quote at the beginning, there was a should in there).

cheers,

neopolitan

 

[JesseM]

I said I would be very happy if the expansion of the universe could be interpreted as the passage of time. I see that, in my visualisation, it works.

As far as I can tell, you haven't actually said that this is not possible. I think you have said that it is not necessary and that it is not something in your (and perhaps the standard) interpretation of GR.

Because I still don't understand what you mean when you say the expansion is the passage of time. Put it this way, do you accept that GR allows for the possibility of a non-expanding universe (either static or contracting) where time still passes? If you accept that GR allows for the possibility, is your idea of interpreting the expansion as the passage of time meant to say that GR is wrong that this is a possibility?

 

[neopolitan]

Because I still don't understand what you mean when you say the expansion is the passage of time. Put it this way, do you accept that GR allows for the possibility of a non-expanding universe (either static or contracting) where time still passes? If you accept that GR allows for the possibility, is your idea of interpreting the expansion as the passage of time meant to say that GR is wrong that this is a possibility?

This is maybe a point where we need to be clear about our terms.

How would the passage of time be experienced in a non-expanding universe? Would it be experienced?

Would the passage of time be reversed in a contract so that from inside such a universe it still seems to expand?

Wasn't the idea of a static universe http://www.astrophysicsspectator.com/commentary/commentary20041020.html" (not my words, apparently his, but as far as I can tell it is hearsay)?

A http://www.americanscientist.org/template/AssetDetail/assetid/18638/" directed at string theory is that it explains too much, to the extent that (perhaps exaggerating a little) even if the universe was completely different, with laws of physics quite unlike ours, string theory could explain it. Such (interpretations of) theories lose predictive power.

I would have thought that a beauty of (stardard interpretations of) GR would be that GR explains the universe as it is. What you seem to be saying is that GP would also explain the universe if it were different, if it were static or contracting.

It seems that there is http://www-theory.phys.utas.edu.au/theory/qftfest/public.pdf" that universal expansion is actually increasing, rather than slowing down. If, as far as we can tell, the universe has always expanded and there are no indications that this expansion is on its way to stopping - what benefit is there in an interpretation of GR which indicates that it's ok to have a static or contracting universe?

It is entirely possible that I have misinterpreted JesseM and so I look forward to a clarification.

cheers,

neopolitan

 

[neopolitan]

<snip>I'm not aware of any model that says we could be in a black hole, since this would seem incompatible with expansion <snip>

I posted about this https://www.physicsforums.com/showpost.php?p=1622966&postcount=9". It's not a model per se and it seems that JesseM's counterargument will be that the relevant equation only applies to flat space so I am misusing the equation by applying it to the universe as a whole.

I do wonder why is it not paradoxical to apply that equation to flat space when the existence of the very thing the equation applies to makes space not flat.

Again, I may be misinterpreting (and putting words into other's mouths). In any event, this discussion of black holes is not really central to this thread. So perhaps those who want to respond can do so in the thread linked, not this one.

cheers,

neopolitan

Edit: JesseM, you were worried that I had not read that page you linked to. It is actually on my desk and has been read, but this just caught my eye

The Schwarzschild limit does not apply to rapidly expanding matter.

My emphasis.

In any event, I have gone through the process of deriving the Schwarzschild solution from "first principles" so I don't thing I am using it out of context. (I put "first principles" in inverted commas because you can do it from Lorentz Transformations onwards pretty simply. I have also gone through the process of deriving the Lorentz Transformations from "first principles".) If there are limitations to application of this solution, which were not a necessity in my derivation, perhaps I am looking at something different and am doing what I asked DaleSpam not to do - ie labelling myself with something that doesn't necessarily apply.

For me, the Schwartzschild radius is an expression of the minimum separation from the centre of a mass that a photon must have to escape that mass (with the implication that the mass must be contained by the volume described by that radius). Usually that radius describes a volume which is smaller than the volume which the mass inhabits, so the volume described by the Schwartzschild radius for the Earth's mass is smaller than the the volume of the Earth. When the Schwartzschild radius is equal to or larger than the volume inhabited by the mass in question, then you can talk about a black hole.

If this is more than, or not all that, is implied, then perhaps I should not use the term "Schwarzschild radius".

 

[JesseM]

How would the passage of time be experienced in a non-expanding universe? Would it be experienced?

Interactions between objects, cyclic processes like orbits, radioactive decay, changes in entropy...all the things that people noted as changing with time prior to the discovery of the universe's expansion could still happen in a non-expanding universe.

Would the passage of time be reversed in a contract so that from inside such a universe it still seems to expand?

Not unless the laws of thermodynamics reversed so that there was a low-entropy boundary condition at the big crunch just like the one at the big bang (it is entropy which is thought to be responsible for the arrow of time, since the fundamental laws of nature are all time-symmetric or CPT-symmetric)...there's nothing in current physics to indicate this would happen, although it was once proposed by a physicist named Thomas Gold as a speculation.

Wasn't the idea of a static universe http://www.astrophysicsspectator.com/commentary/commentary20041020.html" (not my words, apparently his, but as far as I can tell it is hearsay)?

Einstein had to adjust the "cosmological constant" in his equations to a precise value to keep the universe static--he later saw this as a blunder because if he had made the equations simpler by dropping the cosmological constant entirely, he could have predicted the expansion of space which was discovered by Hubble with his observation of galactic redshifts. But nowadays physicists think there is actually a cosmological constant, although it's not thought to have the precise value to keep the universe static. Still, there is nothing invalid about Einstein's static solution, it's a valid solution to the equations of GR even if it doesn't describe our universe (and it was also discovered later that it was highly unstable--a slight perturbation could cause the universe to begin to expand or contract).

Also, Einstein was assuming a homogenous universe with matter evenly distributed throughout space, but you can also have a static universe if you assume the average density is zero. For example, a flat SR spacetime filled with "test particles" of infinitesimally small mass is a GR solution, as is an "asymptotically flat" universe where all the matter is concentrated in a finite region and the universe is empty elsewhere, so as you get farther and farther away from the matter-containing region, space approaches perfect flatness. The Schwarzschild black hole solution assumes an asymptotically flat universe, although this is just meant to be an approximation.

A http://www.americanscientist.org/template/AssetDetail/assetid/18638/" directed at string theory is that it explains too much, to the extent that (perhaps exaggerating a little) even if the universe was completely different, with laws of physics quite unlike ours, string theory could explain it. Such (interpretations of) theories lose predictive power.

I would have thought that a beauty of (stardard interpretations of) GR would be that GR explains the universe as it is. What you seem to be saying is that GP would also explain the universe if it were different, if it were static or contracting.

When making predictions in physics, you have to know both the general dynamical laws which govern the system, and also the initial "boundary conditions" like the specific initial positions and momentum of the particles that make up the system in classical physics, or the initial state of the system's wavefunction in quantum physics. GR is the "general dynamical laws" here, but how the universe behaves depends on the initial conditions you start with just like in any other area of physics; GR can't tell you whether the universe is going to be homogenous on large scales or whether all the matter will be concentrated in one region, and for an approximately homogenous universe it can't tell you whether the density of matter will be low enough for the universe to expand forever or whether it will be high enough for the universe to reach a maximum size and then begin to contract again in a big crunch (and in the latter case, it certainly can't predict whether a given observer will find himself in the expanding phase or the contracting phase). This is really no more problematic than the fact that Newtonian physics can't predict the specific sizes of the planets in our solar system, and that the same Newtonian laws could be used to describe an infinite range of physically distinct solar systems.

I guess it's sort of ambiguous whether the cosmological constant is part of the basic laws of physics or if it's part of the distribution of matter and energy--no one really knows the source of the cosmological constant, although it's referred to as "dark energy"--so if it is seen as part of the basic laws, then I suppose we might way that once we have determined the cosmological constant experimentally, we should assume that all universes governed by GR would have that same value for the cosmological constant. But I would guess that no matter what the value of the cosmological constant, by choosing the right density for matter/energy one could come up with a universe that expanded for a while and then contracted again into a big crunch, or a static universe where the density of matter/energy perfectly balanced the cosmological constant.

Anyway, you still haven't given me a clear answer to whether your speculations that expansion is time is supposed to be something you think is a consequence of GR, or if you are suggesting some kind of alternate theory. If the argument is supposed to be based on GR, does the precise value of the cosmological constant play a critical role in your reasoning? If not then your reasoning must be flawed in some way, since without specifying a value of the cosmological constant GR is quite trivially compatible with static universes like the asymptotically flat one.

 

[JesseM]

I posted about this https://www.physicsforums.com/showpost.php?p=1622966&postcount=9". It's not a model per se and it seems that JesseM's counterargument will be that the relevant equation only applies to flat space so I am misusing the equation by applying it to the universe as a whole.

I do wonder why is it not paradoxical to apply that equation to flat space when the existence of the very thing the equation applies to makes space not flat.

Again, the Schwarzschild solution assumes asymptotically flat space, which approaches perfect flatness as your distance from the black hole approaches infinity.

In any event, I have gone through the process of deriving the Schwarzschild solution from "first principles" so I don't thing I am using it out of context. (I put "first principles" in inverted commas because you can do it from Lorentz Transformations onwards pretty simply. I have also gone through the process of deriving the Lorentz Transformations from "first principles".)

The Schwarzschild solution is inherently a GR matter, so I highly doubt any derivation based only on the Lorentz transformations (which apply to inertial coordinate systems in the flat spacetime of SR) could be valid. Would you mind sharing it, perhaps along with your derivation of the Lorentz tranformation itself?

For me, the Schwartzschild radius is an expression of the minimum separation from the centre of a mass that a photon must have to escape that mass (with the implication that the mass must be contained by the volume described by that radius). Usually that radius describes a volume which is smaller than the volume which the mass inhabits, so the volume described by the Schwartzschild radius for the Earth's mass is smaller than the the volume of the Earth. When the Schwartzschild radius is equal to or larger than the volume inhabited by the mass in question, then you can talk about a black hole.

Just to be clear, would you agree that in a case like the Earth where the actual radius is larger than the Schwarzschild radius, a photon will always be able to escape no matter how close it is to the center?

 

[neopolitan]

First an admission, I misspoke when I wrote (paranthetically) about deriving the Schwartzschild solution:

(I put "first principles" in inverted commas because you can do it from Lorentz Transformations onwards pretty simply. I have also gone through the process of deriving the Lorentz Transformations from "first principles".)

I did the derivation a long long time ago, in the same timeframe as my derivation of the Lorentz Transformations. However, the Lorentz transformations are not required (see the following).

The Schwarzschild solution is inherently a GR matter, so I highly doubt any derivation based only on the Lorentz transformations (which apply to inertial coordinate systems in the flat spacetime of SR) could be valid. Would you mind sharing it, perhaps along with your derivation of the Lorentz tranformation itself?

You have already seen a derivation of Lorentz Transformations from me Jesse. We discussed it via email, you probably have the relevant document on your computer. Since, while the results are the same, my method is not exactly standard, this is possibly not the right place to discuss it - given the rules of the forum. In any case, I am not sure that specifically my derivation of Lorentz Transformations is pertinent to this thread.

You wanted my non-GR derivation of Schwartzschild radius. Since what I am talking about is possibly not precisely the Schwartzschild radius I had better explain what my derivation provides you with.

My understanding is that a black hole is a concentration of mass of such proportions that not even photons can escape. There is a limit to the consequences of this concentration of mass, in so much as you could approach a black hole and escape so put more precisely-

A black hole is a concentration of mass of such proportions that, once they are within a certain boundary, not even photons can escape.

My equation is for that certain boundary. I believe this is the event horizon and that, strictly speaking, the black hole itself (the singularity) lies within this boundary. I do think that I have interpreted http://en.wikipedia.org/wiki/Schwartzschild_radius" correctly, but I accept that wikipedia is not authoritative.

That said, this certain boundary can be thought of as just inside the closest point of approach that a photon could make to a black hole without falling in (more accurately perhaps, the radius at which a tangentially moving photon would be captured by the mass and locked into a circular orbit, assuming a non-rotating mass). This is, in other words, the radius at which the escape velocity is infinitesimally greater than the speed of light.

The equation for escape velocity (per Newton) is:

v= sqrt (2GM/r)

Since we are seeking the radius and have the velocity (c), then we rearrange.

r= 2GM/c^2

This is the equation for Schwartzschild radius (according to wikipedia which, I point out again, is not authoritative).

No GR involved.

Just to be clear, would you agree that in a case like the Earth where the actual radius is larger than the Schwarzschild radius, a photon will always be able to escape no matter how close it is to the center?

Of course. I don't understand why you could have thought that I thought otherwise.

cheers,

neopolitan

 

[neopolitan]

Anyway, you still haven't given me a clear answer to whether your speculations that expansion is time is supposed to be something you think is a consequence of GR, or if you are suggesting some kind of alternate theory. If the argument is supposed to be based on GR, does the precise value of the cosmological constant play a critical role in your reasoning? If not then your reasoning must be flawed in some way, since without specifying a value of the cosmological constant GR is quite trivially compatible with static universes like the asymptotically flat one.

Neither really. I think that the equations of GR would apply irrespective of my interpretation but I do also think that the universe works perfectly well without our theories. All we achieve with our theories (and our interpretations, I guess) is a description of the universe.

So, in a sense, no observed behaviour of the universe is a consequence of GR.

cheers,

neopolitan

 

[yuiop]

Just for info: The radius that a photon can orbit a black hole is generally accepted to be 3GM/R while the Shwarzchild radius is 2GM/R. Google "photon orbit".

 

[neopolitan]

I already pointed out the well-understood fact that in GR, the "Schwarzschild radius" only applies to a non-expanding space, in an expanding universe you can have more mass in a space of that size without a black hole being formed. Did you not read the page I linked to?

Yes I did. I did want time to absorb it. Note that I didn't say the big bang was a black hole, per se, I said there are arguments supporting the concept that we are inside a black hole. So, the article is not entirely relevant. It is not entirely irrelevant either.

Something that struck me was this:

The first clear difference is that the big bang singularity of the FRW models lies in the past of all events in the universe, whereas the singularity of a black hole lies in the future.

If the universe is in a black hole, and boundary of the universe is a "when" issue and not a "where" issue, then yes, the black hole that contains the universe is in the future, since we are surrounded by that future. It's saying the same thing.

And this:

A white hole has an event horizon which is the reverse of a black hole event horizon.

Now this is probably a misreading, but I also recall seeing diagram supporting the idea that on the "other side" of a black hole is a white hole. Note http://en.wikipedia.org/wiki/White_hole#Recent_speculations" which is conceptually equivalent to what I have been pondering.

Being "in a black hole" is the same thing as having had the universe created in a white hole style big bang.

But you did say that this is highly speculative stuff, so I won't take it further.

The author of the article you linked did though:

A black hole in thermal equilibrium with surrounding radiation might have to be time symmetric in which case it would be the same as a white hole. This idea is controversial, but if true it would mean that the universe could be both a white hole and a black hole at the same time. Perhaps the truth is even stranger. In other words, who knows?

cheers,

neopolitan

 

[neopolitan]

Just for info: The radius that a photon can orbit a black hole is generally accepted to be 3GM/R while the Shwarzchild radius is 2GM/R. Google "photon orbit".

That's a relatively stable orbit, but still an effectively unstable orbit, I think. In reality, a photon which gets as close as the Schwartzschild radius will be pulled in pretty much immediately, while those between the Schwartzchild radius and the photon orbit radius will spiral in. The photon orbit radius is where the photon can either escape or spiral in.

Photons don't seem to want to get into orbits. I blame it on their indecisiveness vis á vis wave and/or particle status.

But you are right, unless I am mistaken myself.

cheers,

neopolitan

 

[neopolitan]

JesseM,

I don't like your American football model so it is quite possible that I have misunderstood.

Can you check the following diagrams to make sure that 1) I have understood you correctly and 2) that you have understood me correctly.

cheers,

neopolitan

 

[JesseM]

You have already seen a derivation of Lorentz Transformations from me Jesse. We discussed it via email, you probably have the relevant document on your computer.

I don't think we discussed deriving the Lorentz transform--do you mean the document sr.doc which you mailed to me?

In any case, I am not sure that specifically my derivation of Lorentz Transformations is pertinent to this thread.

OK, let's just stick to the Schwarzschild derivation.

My understanding is that a black hole is a concentration of mass of such proportions that not even photons can escape. There is a limit to the consequences of this concentration of mass, in so much as you could approach a black hole and escape so put more precisely-

A black hole is a concentration of mass of such proportions that, once they are within a certain boundary, not even photons can escape.

My equation is for that certain boundary. I believe this is the event horizon and that, strictly speaking, the black hole itself (the singularity) lies within this boundary. I do think that I have interpreted http://en.wikipedia.org/wiki/Schwartzschild_radius" correctly, but I accept that wikipedia is not authoritative.

You're correct, the event horizon marks the point at which a photon emitted by an infalling object cannot escape (although to make things more complicated, the Schwarzschild solution actually describes an object which can act as both a white hole and a black hole, so objects that fall in can't escape, but the hole can also spit objects and light out of the horizon; this isn't a realistic solution though, since it requires the hole to have existed for an infinite time in the past).

That said, this certain boundary can be thought of as just inside the closest point of approach that a photon could make to a black hole without falling in (more accurately perhaps, the radius at which a tangentially moving photon would be captured by the mass and locked into a circular orbit, assuming a non-rotating mass). This is, in other words, the radius at which the escape velocity is infinitesimally greater than the speed of light.

The equation for escape velocity (per Newton) is:

v= sqrt (2GM/r)

Since we are seeking the radius and have the velocity (c), then we rearrange.

r= 2GM/c^2

This is the equation for Schwartzschild radius (according to wikipedia which, I point out again, is not authoritative).

No GR involved.

OK, this derivation is not really rigorous since Newtonian calculations will often give incorrect answers in GR (for example, if you try to calculate how much light is deflected by a gravitational field using Newtonian assumptions, the deflection you'll calculate is only half what it should be in GR), but in this case you do get the same equation.

Of course. I don't understand why you could have thought that I thought otherwise.

I didn't think you were clearly saying otherwise, but I thought the paragraph was a little ambiguous since you were talking about the Schwarzschild radius for the Earth...but looking over it again, I think I didn't read carefully enough the part where you said "(with the implication that the mass must be contained by the volume described by that radius)".

 

[JesseM]

Neither really. I think that the equations of GR would apply irrespective of my interpretation but I do also think that the universe works perfectly well without our theories. All we achieve with our theories (and our interpretations, I guess) is a description of the universe.

So, in a sense, no observed behaviour of the universe is a consequence of GR.

First of all, when physicists say something is a "consequence of theory X" they aren't making some crazy metaphysical statement about the theory existing beyond the universe and dictating how the universe behaves, they just mean you can derive certain predictions from the theory, and if the behavior of the universe is consistently agreeing with the predictions of the theory (as we hope is true for the most successful theories, at least within the domain that we have reason to think they're trustworthy, like far away from the Planck scale in the case of GR), then we should expect that prediction to be correct in the real world.

Second, you don't get a statement like "time is a consequence of expansion" by going out and observing the real universe with your telescope, such abstractions can only be a consequence of some theory or at least a simple toy model (like the visual models you seem to be using) you have in your head. Whatever model led you to that statement, I'm basically wondering whether you would discard your model if you could be convinced it doesn't match what GR is predicting, or whether you consider your model to be something independent so that if it came into conflict with GR, you wouldn't necessarily side with GR.

 

[JesseM]

Yes I did. I did want time to absorb it. Note that I didn't say the big bang was a black hole, per se, I said there are arguments supporting the concept that we are inside a black hole.

But the only "arguments" are the types I warned you about before--taking some isolated facts about cosmology and putting them together in a suggestive way without any overarching theory to ground these speculations. In this case I haven't seen you present any arguments beyond the notion that the radius of the observable universe is within an order of magnitude or two of the Schwarzschild radius for its mass. Keep in mind that no one believes the observable universe is all there is, its boundaries just depend on the maximum distance that light has been able to travel to reach our eyes since the big bang!

Something that struck me was this:

The first clear difference is that the big bang singularity of the FRW models lies in the past of all events in the universe, whereas the singularity of a black hole lies in the future.

If the universe is in a black hole, and boundary of the universe is a "when" issue and not a "where" issue

Then why are you using the boundaries of the observable universe in your argument, which is a "where" issue?

then yes, the black hole that contains the universe is in the future, since we are surrounded by that future. It's saying the same thing.

It's not saying anything that's related to your onion visualization, if that's what you mean. Your visualization suggests the future is larger in volume then the past, but for observers in a black hole, the singularity is a point in the future where their universe collapses and all the matter they see around them is compressed into zero volume and infinite density.

Now this is probably a misreading, but I also recall seeing diagram supporting the idea that on the "other side" of a black hole is a white hole. Note http://en.wikipedia.org/wiki/White_hole#Recent_speculations" which is conceptually equivalent to what I have been pondering.

It's true that for a Schwarzschild black hole (which again must have existed for an infinite time), it contains both an internal black hole region and an internal white hole region leading to a different region of spacetime (see the Kruskal-Szekeres diagram http://casa.colorado.edu/~ajsh/schww.html#kruskal, with 'our' universe on the right, the black hole region above the pink horizon, the white hole region below the red horizon, and the 'other' universe on the left). But to say this is "conceptually equivalent" to anything in your model, just because a vague verbal summary of this idea may sound similar to something you think might be true in your own speculations, is totally ludicrous. Again, if you continue on this path of trying to understand isolated statements in GR without any attempt to understand the theoretical arguments behind them, connecting them to your own ideas and pictures in a totally whimsical way, then you're going to end up in crackpot-land, if you aren't there already.

Being "in a black hole" is the same thing as having had the universe created in a white hole style big bang.

I don't see how this statement could make any sense. In the normal interpretation of a Schwarzschild black hole, the internal black hole region and the internal white hole region are totally discontinuous, an observer in one would have no access to anything in the other. It's possible to map the two regions to each other but this leads to new problems (see the diagram http://casa.colorado.edu/~ajsh/schwm.html#kruskal and the 'Objections' section below).

 

[JesseM]

JesseM,

I don't like your American football model so it is quite possible that I have misunderstood.

Can you check the following diagrams to make sure that 1) I have understood you correctly and 2) that you have understood me correctly.

cheers,

neopolitan

The diagram of the football visualization seems right, provided you understand that the "cross sections" represent curved 1D space, and thus that the blue dots you drew at the center of each one don't lie anywhere within the spacetime itself, and so are only meaningful in terms of the 3D Euclidean "embedding space" which we use in the visualization (again, the mathematics of GR does not require any embedding space, it can describe the curvature of a surface without reference to any points outside the surface).

As I've said before, your "onion" visualization depends critically on the fact that you are imagining the surfaces of simultaneity as sitting in an uncurved 3D Euclidean space/spacetime (an 'embedding'). In particular, the notion of one surface "surrounding" another depends on Euclidean intuitions. Think about it in terms of 2D Euclidean space. On a flat plane, it's unambiguous whether one circle "surrounds" another or vice versa--every circle divides the plane into two regions, a finite "inside" and an infinite "outside", and if circle A lies in the "inside" of circle B, then circle B lies in the "outside" of circle A, so clearly circle B is surrounding A rather than the other way around. But if we now think of circles drawn on a curved 2D surface like the surface of a sphere, there isn't any ambiguous way to picture which of two circles surrounds the other. For example, on a globe, take two lines of latitude (which look like circles on the globe of course), one to the north of the equator and one to the south of it--can you say that either of these "surrounds" the other?

Basically, the larger problem is that by appealing to your ordinary Euclidean intuitions, you make it impossible to understand what it would mean for spacetime to be curved, as opposed to just having curved spatial surfaces of simultaneity. And GR is fundamentally a theory of curved spacetime, not curved space. That's why my football analogy is less likely to be misleading, because it explicitly shows spacetime as a curved surface.

 

[neopolitan]

Whatever model led you to that statement, I'm basically wondering whether you would discard your model if you could be convinced it doesn't match what GR is predicting, or whether you consider your model to be something independent so that if it came into conflict with GR, you wouldn't necessarily side with GR.

I am pretty sure that the model I have in mind isn't inconsistent with the equations of GR. Certainly, if I am convinced that it doesn't match then I would have to put it on ice.

I say I would have to put it on ice because there have been times when I came across things that I found were inconsistent with the model, so I put the model aside thinking it didn't work. Then later I found that actually the model did still work, I had merely been imagining the consequences incorrectly.

A very very long time ago (more than 20 years), when I first had it mind as a way of explaining to myself why the two spaceships/two flashlights scenario works (spaceships approach each other at ½c and shine lights at each other, etc etc). I am beyond that now of course - so please don't go into an explanation unnecessarily.

Anyway, I put this model aside because it would imply that the entire universe would be expanding in such a way that everything was moving apart from everything else and things that are further away would be moving away from us faster than things that were close. You can see that that is a problem, since at the time I had the concept of a big bang in which there was a defined centre to the universe. Then one day I had a bit of time at a library and looked things up (20 years ago remember, no internet). Hubble had something interesting for me. So I took another look at my model.

The same sort of thing has happened a number of times as I reach what seems to be a flaw, I set the model aside and then later I find that there was something I didn't know or didn't think of. The equations for Time dilation and Length contraction screwed me around for quite a while, I put the model aside for about 10 years because of them. I could find no-one who could immediately tell me that Time dilation and Length contraction are not supposed to be temporal and spatial equivalents of each other, but rather just two non-equivalent equations which have great utility. Even our ultimately successful discussions, JesseM, took months to arrive at the conclusion that what I was saying and what you were saying were concerned two different, but related concepts, and that in reality we fundamentally agreed with each other.

So, yes, if I really have reached the point where my model is no longer of any utility, then I will put it aside again. But I do hope that you can understand that I do want to check that I am putting it aside with good reason. None of us so many decades left that we can afford to put potentially interesting ideas aside for decades at a time!

cheers,

neopolitan

 

[neopolitan]

GR is fundamentally a theory of curved spacetime, not curved space. That's why my football analogy is less likely to be misleading, because it explicitly shows spacetime as a curved surface.

In an attempt to prevent my descent into "crack-pot land" (thank you very much), could you please explain the meaning of "GR is fundamentally a theory of curved spacetime, not curved space".

The American football model may be less misleading, but in part that because it doesn't make any sense (to me) so it doesn't lead me anywhere. One dimensional circles don't make sense to me and what seems to be an implication that the model has a two dimensional football doesn't make sense.

So, can you possibly try another approach. For instance, in what sense is space-time curved? Curved relative to what?

Note that the diagrams I provided assume effectively empty universes since mass in them will perturb the nice smooth surfaces. It is this curvature which you think is missing?

cheers,

neopolitan

 

[DrGreg]

The American football model may be less misleading, but in part that because it doesn't make any sense (to me) so it doesn't lead me anywhere. One dimensional circles don't make sense to me and what seems to be an implication that the model has a two dimensional football doesn't make sense.

It is the one-dimensional circumference of a two-dimensional disk. We call it 1-D because you need only one number to measure the position of any point within it. It looks 2D only when you embed it in a 2D (or more) space.

In the model, 4D spacetime is represented by the 2D surface of the 3D football. We say the surface is 2D because you need only two numbers to measure the position of any point on it.

Our minds can't picture enough dimensions to depict the 4D surface of a 5D football, which would be a more accurate model.

So, can you possibly try another approach. For instance, in what sense is space-time curved? Curved relative to what?

You can measure curvature without reference to anything.

Forget relativity for a moment and just consider a large triangle drawn on the 2D surface of the earth, with vertices at the north pole, the equator at 0 degrees and the equator at 90 degrees longitude. Within the 2D geometry of the Earth's surface, this triangle has straight sides but its angles add up to 270 degrees, not 180. That tells us the 2D space is curved without having to mention a third dimension.

But you need at least two dimensions within the space (the surface) to detect curvature. A 1-D space can never be (intrinsically) curved (meaning that, considered embedded in a higher dimension space, you could always straighten it out without distortion). Even the curved surface of a cylinder is not considered "curved" in this sense, because you could always cut it open and flatten it without distortion. But orange peel cannot be flattened and therefore is intrinsically curved.

 

[neopolitan]

It is the one-dimensional circumference of a two-dimensional disk. We call it 1-D because you need only one number to measure the position of any point within it. It looks 2D only when you embed it in a 2D (or more) space.

Ok, happy with the 1D circumference.

In the model, 4D spacetime is represented by the 2D surface of the 3D football. We say the surface is 2D because you need only two numbers to measure the position of any point on it.

Our minds can't picture enough dimensions to depict the 4D surface of a 5D football, which would be a more accurate model.

Then my attempt at drawing it was wrong, since I had time indicated. But JesseM said it seemed right. I specified that it was of a 3+1 universe, so by implication the surface of the football was 3D space with time being along the length of the football. It seems that is wrong. However, the cross sections can therefore not be instants, and so the argument that JesseM had originally doesn't apply.

Note that my "universe as onion" is intentionally 3+1 dimensional. The surface of the sphere represents curved 3D space. Time is another dimension but it has no specific direction other than "perpendicular to space" wherever there is an observer considering it. (We could say the direction is also "towards the future", "in the same direction as increasing entropy" or "in the same direction of decreasing causal index". By "decreasing causal index" I am referring to causality, in that the vast majority of causes lie in one direction, the past. The future has a reduced capacity to be the cause of events we will experience. I don't know if it is a standard concept, but I have been told often enough in this thread that the universe doesn't care about simultaneity, only causality.)

You can measure curvature without reference to anything.

Forget relativity for a moment and just consider a large triangle drawn on the 2D surface of the earth, with vertices at the north pole, the equator at 0 degrees and the equator at 90 degrees longitude. Within the 2D geometry of the Earth's surface, this triangle has straight sides but its angles add up to 270 degrees, not 180. That tells us the 2D space is curved without having to mention a third dimension.
embedded in a higher dimension space, you could always straighten it out without
But you need at least two dimensions within the space (the surface) to detect curvature. A 1-D space can never be (intrinsically) curved (meaning that, considered distortion). Even the curved surface of a cylinder is not considered "curved" in this sense, because you could always cut it open and flatten it without distortion. But orange peel cannot be flattened and therefore is intrinsically curved.

Ok, yes, I know this. You draw a great circle on the surface of the earth, then a second one. Then pick two locations, one on each of the great circles, neither being common to both great circle. Draw a third great circle and you have what may look like a triangle, if the Earth were flat. But because the Earth is not flat then you don't have straight sides of a triangle, but rather three intersecting arcs. And the sum of the internal angles defined by three intersecting arcs is not going to be 180 degrees but somewhere between a smidgen over 180 degrees (for an extremely thin triangle, or a triangle which is very small relative to the surface of the Earth's surface) and a smidgen under 540 degrees (for an extremely fat triangle, for example with corners at the south pole, at the international date line and a centimetre to the east of the international dateline where the whole length of the equator, minus 1 cm, constitutes the longest side of the "triangle").

Fine, happy with that.

However, note that so long as you don't try to draw triangles bigger than 100 thousand square kilometres or so, the angles will sum to very close to 180. (For instance a rather simple "triangle", spanning 1 degree of longitude and 90 degrees of lattitude will have a total of 181 degrees, and will contain 1.5 million square kilometres of surface area. A similar 100 square kilometre "triangle" will span 0.07 degrees of longitude and give you a sum of 180.07 degrees.) Note further that more than half the countries in the world are smaller than this, so you are talking about a pretty big triangle.

I do think that the inherent curvature that you are discussing will similarly only come into noticeable effect when you are considering relatively large chunks of the universe. Do you agree?

cheers,

neopolitan

 

[JesseM]

In an attempt to prevent my descent into "crack-pot land" (thank you very much), could you please explain the meaning of "GR is fundamentally a theory of curved spacetime, not curved space".

In differential geometry you define the curvature of a surface using a measure of "distance" on the surface. The function that you use to define the "distance" between points on the surface is called the "metric". If you want to talk about the spatial distance between points on a 2D Euclidean plane using a cartesian coordinate system, this distance is just given by the Pythagorean theorem, dL^2 = dx^2 + dy^2. Even if you're not talking about a straight-line path, if you know the function y(x) that describes the path, and therefore know dy/dx, you can integrate the "line element" equation above to get the total length of the path in the plane. But if you laid out a coordinate system on the surface of a 2D globe using coordinates \theta and \phi, with the \theta direction going along lines of latitude and the \phi direction going alone lines of latitude, you'd find that for a given path, integrating dL^2 = d\theta^2 + d\phi^2 would not the correct length for the path; because the surface is curved, distance works differently (the correct metric for the surface is given on this thread).

Similarly, in the uncurved 4D minkowski spacetime of SR we have a notion of a type of "spacetime distance" which can be calculated in any inertial coordinate system using dS^2 = c^2*dt^2 - dx^2 - dy^2 - dz^2. And if we want to calculate the proper time along any non-straight worldline, if we know the worldline's position as a function of time, we can use the above "line element" in an integral along the worldline to get the proper time along it. But in general relativity, matter and energy causes spacetime to become curved; just as the Euclidean line element doesn't work in spherical geometry, so the minkowski line element won't work in curved spacetime. The metric function can give you the line element at every point, and the equations of GR tell you how to calculate the metric based on the distribution of matter and energy in the space (matter and energy 'tells spacetime how to curve').

Here's a page that gives an outline:

http://www.theory.caltech.edu/people/patricia/greltop.html

The American football model may be less misleading, but in part that because it doesn't make any sense (to me) so it doesn't lead me anywhere. One dimensional circles don't make sense to me and what seems to be an implication that the model has a two dimensional football doesn't make sense.

Calling the surface of a 3D sphere a 2D surface makes sense to you, but calling the edge of a 2D circle a 1D surface doesn't make sense to you? The idea is the same in both cases; just as you can imagine a flatlander confined to live on the surface of a sphere who would still believe his universe was 2D, you should be able to imagine a linelander confined to live on the edge of a circle who would still believe his universe was 1D.

For instance, in what sense is space-time curved? Curved relative to what?

Curved in the sense that the proper time along a given worldline can no longer be correctly computed with the line element dS^2 = c^2*dt^2 - dx^2 - dy^2 - dz^2. The point of differential geometry is to describe the curvature of surfaces in terms of some geometric notion of "distance" for paths on the surface; you're describing curvature in terms intrinsic to the surface, you don't need a higher-dimensional space that the surface is curved "relative to".

Note that the diagrams I provided assume effectively empty universes since mass in them will perturb the nice smooth surfaces. It is this curvature which you think is missing?

No, I'm just saying your model is misleading because it assumes only space is curved, while in GR it's fundamentally spacetime that's curved. You can pick different ways of defining simultaneity in a curved spacetime, and you'll get a different set of spatial surfaces depending on your choice of how to do it, so the spatial distance between two points on a given surface is not a very physical notion, since it depends on arbitrary choices about how to draw your coordinate system. On the other hand, the proper time along any given worldline through spacetime is a very physical notion since all coordinate systems must agree on this.

 

[JesseM]

Then my attempt at drawing it was wrong, since I had time indicated. But JesseM said it seemed right. I specified that it was of a 3+1 universe, so by implication the surface of the football was 3D space with time being along the length of the football. It seems that is wrong. However, the cross sections can therefore not be instants, and so the argument that JesseM had originally doesn't apply.

I don't understand, how could you think the entire surface of the football represents 3D space and think that time is along the length of the football? If the surface of the football is 3D space, wouldn't every dimension along it be a spatial dimension? My idea was that each 1D circle--a cross section--represents 3D space at a particular instant, and time goes along the length of the football. I'm pretty sure I said earlier that I was dropping the number of dimensions by two, so there was one spatial dimension and one time dimension. And that's what DrGreg was saying too, so I don't understand why you think your drawing was wrong because you "had time indicated"--that part was entirely correct!

Note that my "universe as onion" is intentionally 3+1 dimensional. The surface of the sphere represents curved 3D space. Time is another dimension but it has no specific direction other than "perpendicular to space" wherever there is an observer considering it. (We could say the direction is also "towards the future", "in the same direction as increasing entropy" or "in the same direction of decreasing causal index". By "decreasing causal index" I am referring to causality, in that the vast majority of causes lie in one direction, the past. The future has a reduced capacity to be the cause of events we will experience. I don't know if it is a standard concept, but I have been told often enough in this thread that the universe doesn't care about simultaneity, only causality.)

Again, if you are imagining the surfaces as nesting in an uncurved 3D space, then your analogy makes it impossible to represent the idea that GR fundamentally deals with spacetime curvature, not spatial curvature at different instants.

Ok, yes, I know this. You draw a great circle on the surface of the earth, then a second one. Then pick two locations, one on each of the great circles, neither being common to both great circle. Draw a third great circle and you have what may look like a triangle, if the Earth were flat. But because the Earth is not flat then you don't have straight sides of a triangle, but rather three intersecting arcs. And the sum of the internal angles defined by three intersecting arcs is not going to be 180 degrees but somewhere between a smidgen over 180 degrees (for an extremely thin triangle, or a triangle which is very small relative to the surface of the Earth's surface) and a smidgen under 540 degrees (for an extremely fat triangle, for example with corners at the south pole, at the international date line and a centimetre to the east of the international dateline where the whole length of the equator, minus 1 cm, constitutes the longest side of the "triangle").

Fine, happy with that.

However, note that so long as you don't try to draw triangles bigger than 100 thousand square kilometres or so, the angles will sum to very close to 180. (For instance a rather simple "triangle", spanning 1 degree of longitude and 90 degrees of lattitude will have a total of 181 degrees, and will contain 1.5 million square kilometres of surface area. A similar 100 square kilometre "triangle" will span 0.07 degrees of longitude and give you a sum of 180.07 degrees.) Note further that more than half the countries in the world are smaller than this, so you are talking about a pretty big triangle.

I do think that the inherent curvature that you are discussing will similarly only come into noticeable effect when you are considering relatively large chunks of the universe. Do you agree?

Again, you're talking about spatial curvature here. On human scales, space does appear pretty Euclidean. But spacetime curvature is a lot more obvious--for example, it's why balls travel on parabolas rather than in straight lines (take a look at the nice illustration http://io.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/spacetime_curvature.htm which I definitely recommend picking up a used copy of, showing how although the paths of balls thrown at different speeds trace different curves in space, they can be visualized as having the same curvature in spacetime)

 

[DrGreg]

But spacetime curvature is a lot more obvious--for example, it's why balls travel on parabolas rather than in straight lines

I agree with everything you've said so far, JesseM, but I have to take issue with this. See this post.

Or have I misunderstood GR curvature? Would you say there is curvature in an accelerating frame in the absence of gravity? GR is not my area of expertise, but I thought that counted as a flat metric.

 

[JesseM]

I agree with everything you've said so far, JesseM, but I have to take issue with this. See this post.

Or have I misunderstood GR curvature? Would you say there is curvature in an accelerating frame in the absence of gravity? GR is not my area of expertise, but I thought that counted as a flat metric.

Yeah, I think you're right actually, this is more along the lines of the "uniform gravitational field" in flat spacetime discussed here, only tidal forces are evidence of genuine spacetime curvature...if you're standing in a windowless room on Earth, nothing you see will be noticeably different than if the same room was accelerating at 1G through deep space. But Wheeler did use http://io.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/spacetime_curvature.htm in his book--I'll have to go back and look at the text and see exactly what he said they're supposed to show.

A better example of spacetime curvature would be the orbit of satellites around the Earth. Space in the neighborhood of Earth is pretty close to Euclidean (if you created a giant triangle surrounding the Earth the sum of the angles would be very very close to 180 degrees), so this path is certainly not a straight line in space, but it is "straight" (i.e. a geodesic which minimizes the proper time between events which lie along it) in spacetime.

 

[neopolitan]

JesseM,

You referred to curvature as a consequence of mass, a gravitation effect. But you seemed to be saying that spacetime is inherently curved. Which is it?

Consequential curvature due to mass is not in the model I gave since it is an SR thing, not a GR thing. So far in this discussion (and hence in my model as shown) I haven't brought in mass to cause curvature.

In any event, if there is curvature which is inherent rather than consequential to mass, effectively this will only manifest over large volumes of the universe - as I alluded to in a https://www.physicsforums.com/showpost.php?p=1648368&postcount=193". If the universe is infinite then it won't manifest. If it is bounded then my model seems more fitting than yours and the curvature will manifest, but only noticeably if you were to take readings which are ridiculously distant from each other.

cheers,

neopolitan

 

[JesseM]

You referred to curvature as a consequence of mass, a gravitation effect. But you seemed to be saying that spacetime is inherently curved. Which is it?

Why do you think I am saying that? No, spacetime curvature is only in the presence of mass/energy. In cosmological models which treat all of space as curved, it's because they're assuming a homogeneous distribution of mass--think of a fluid of uniform density filling all of space (of course, if there's a nonzero cosmological constant, this would also contribute to the curvature--it could be thought of as a type of energy filling all of space even when other types of matter/energy are present. But all the cosmological models I've seen assume that even if there's a cosmological constant, there's also ordinary matter and energy throughout space.)

 

[neopolitan]

In differential geometry you define the curvature of a surface using a measure of "distance" on the surface. The function that you use to define the "distance" between points on the surface is called the "metric". If you want to talk about the spatial distance between points on a 2D Euclidean plane using a cartesian coordinate system, this distance is just given by the Pythagorean theorem, dL^2 = dx^2 + dy^2. Even if you're not talking about a straight-line path, if you know the function y(x) that describes the path, and therefore know dy/dx, you can integrate the "line element" equation above to get the total length of the path in the plane. But if you laid out a coordinate system on the surface of a 2D globe using coordinates \theta and \phi, with the \theta direction going along lines of latitude and the \phi direction going alone lines of latitude, you'd find that for a given path, integrating dL^2 = d\theta^2 + d\phi^2 would not the correct length for the path; because the surface is curved, distance works differently (the correct metric for the surface is given on this thread).

Similarly, in the uncurved 4D minkowski spacetime of SR we have a notion of a type of "spacetime distance" which can be calculated in any inertial coordinate system using dS^2 = c^2*dt^2 - dx^2 - dy^2 - dz^2. And if we want to calculate the proper time along any non-straight worldline, if we know the worldline's position as a function of time, we can use the above "line element" in an integral along the worldline to get the proper time along it. But in general relativity, matter and energy causes spacetime to become curved; just as the Euclidean line element doesn't work in spherical geometry, so the minkowski line element won't work in curved spacetime. The metric function can give you the line element at every point, and the equations of GR tell you how to calculate the metric based on the distribution of matter and energy in the space (matter and energy 'tells spacetime how to curve').

In light of this, is my model inconsistent with SR? Is it a valid way to visualise SR?

And if there was a way to use that model to visualise the curvature of spacetime caused by the distribution of matter and energy in space (because as you say "matter and energy 'tells spacetime how to curve'"), such that is was not inconsistent with GR, would there still be a problem?

(Note that, in effect, you asked me the same sort of question in an earlier post.)

cheers,

neopolitan