Discuss events which are simultaneous in one frame?

[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

 

[JesseM]

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

In light of which part of that quote? Are you referring to the fact that the SR line element no longer works in curved spacetime? In that sense, GR itself is incompatible with SR, although it reduces to SR locally.

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?

As long as any visualization doesn't lead you to conclusions that are inconsistent with GR, I don't have a problem. But your visualization seemed to lead to the conclusion that "time is expansion", which doesn't seem compatible with GR, since GR does allow for non-expanding universes where time can still be measured perfectly well. And I also had a problem with some of your other statements that may not have been based on your visualization, like your claim that the universe could be a black hole, and the fact that you were taking isolated cosmological facts or out-of-context statements from physicists and taking them to be confirmation of your own notions.

 

[neopolitan]

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

In light of which part of that quote? Are you referring to the fact that the SR line element no longer works in curved spacetime? In that sense, GR itself is incompatible with SR, although it reduces to SR locally.

Let's make the question simpler.

Is my model inconsistent with SR?

cheers,

neopolitan

 

[JesseM]

Let's make the question simpler.

Is my model inconsistent with SR?

I don't know enough about what your model is supposed to consist of to answer this question definitely, although it seems likely it would be inconsistent based on what you've said. Does your model allow for the theoretical possibility that each inertial observer could be accompanied by an infinite lattice of clocks and rigid rulers filling all of space at all times, with all the clocks and rulers moving inertially and at rest with respect to one another and the observer, such that if each observer assigns the coordinates of events using local readings on this lattice, then each observer will see the same laws of physics respected everywhere, including the law that all other observers are moving slower than light in their coordinate system, and that light is always moving at exactly c? And also including the law that if two straight rigid rods meet at right angles, and a third rod is laid out to meet the ends of each one, then the angles where they meet will always add up to 180? (aren't you allowing for the possibility of curved space in your model, or have I misunderstood?) If not, then your model isn't consistent with SR globally, although I suppose you might still try to make it consistent with SR locally as in GR.

 

[neopolitan]

I don't know enough about what your model is supposed to consist of to answer this question definitely, although it seems likely it would be inconsistent based on what you've said. Does your model allow for the theoretical possibility that each inertial observer could be accompanied by an infinite lattice of clocks and rigid rulers filling all of space at all times, with all the clocks and rulers moving inertially and at rest with respect to one another and the observer,

Yes (assuming the clocks and rigid rulers are massless, otherwise you run into problems - but I am assuming these clocks and rulers are also theoretical)

such that if each observer assigns the coordinates of events using local readings on this lattice, then each observer will see the same laws of physics respected everywhere, including the law that all other observers are moving slower than light in their coordinate system,

Yes, nothing moves faster than light relative to any rectilinearly inertial observer who is not rotating.

If I spin on the spot and assume that I am stationary (which means I have to ignore the tidal effects in my own body), then in my reference frame I will have distant objects moving at faster than the speed of light according to me - but this requires that I have to make what I feel is an invalid assumption by ignoring important cues. Similarly, I feel that we err if we assume to be stationary a reference frame which does not have rectilinear motion like my house on the Earth (effectively in a very low geostationary orbit around the centre of the Earth), or the Earth itself (in orbit around the sun) - I have to ignore a lot of other things going on around me (tides, currents, atmospheric effects, the molten core, the totality of the moon's orbit, etc).

and that light is always moving at exactly c?

Yes, in fact this is what sparked the model in the first place.

And also including the law that if two straight rigid rods meet at right angles, and a third rod is laid out to meet the ends of each one, then the angles where they meet will always add up to 180? (aren't you allowing for the possibility of curved space in your model, or have I misunderstood?)

Well, if you have an absolutely huge set of rods, which span a substantial portion of the universe, no ...

If not, then your model isn't consistent with SR globally, although I suppose you might still try to make it consistent with SR locally as in GR.

... but locally, yes.

The same thing happens on the surface of a sphere. A little triangle with one corner where the international date line (IDL) meets the equator and the two adjacent sides running one metre parallel to the IDL and equator respectively will have a sum of internal angles (SIA) of 180+9x10^-13 degrees (give or take a little). This is close enough to 180 degrees for most people. Make a larger triangle, with the adjacent sides 2000 kilometers long, and you will no longer be able to ignore the effects of curvature (your SIA is now 183.6 degrees). But this represents a lot of surface area, about one percent of the total surface area of the Earth, or 5 million square kilometers - a bit over one half the area of continental USA.

A correspondingly large triangle in the observable universe would have sides which are not quite half a billion (10^9) light-years long. I think the chances of us manipulating rods of that magnitude are rather remote. For entertainment's sake, let's say we could manipulate rods which are half a million light years long, then we could create a triangle which has a SIA of 180+3x10^-6 degrees assuming the universe is hyperspherical. I think this is close enough to 180 degrees for the majority of us.

cheers,

neopolitan

 

[Dale]

inertial observer who is not rotating

An inertial observer is not rotating by definition.

 

[neopolitan]

An inertial observer is not rotating by definition.

True.

I do note that there are some references which say "non rotating inertial observers" and that inertial refers to an absence of acceleration. This is what I meant, an observer who is neither rotating nor undergoing acceleration. I do think it is possible to be both rotating and undergoing acceleration, so as to not be inertial in any sense.

I can't easily find a reference which states that inertial means non rotating, but I accept without hesitation that it is the case (my reasoning is that centipetal forces are involved with a rotating frame).

 

[neopolitan]

JesseM (and any other interested parties),

Can you take a look at post #204 and see if it is sufficient for you to answer post #202?

Posts #205 and #206 were an unnecessary detour (probably due to both DaleSpam and myself being pathologically finicky).

thanks,

neopolitan

 

[JesseM]

JesseM (and any other interested parties),

Can you take a look at post #204 and see if it is sufficient for you to answer post #202?

Yes, based on what you said your model is incompatible with SR in the same sense that GR is incompatible with SR (which I mentioned in #201), although like GR it might still reduce to SR locally (and haven't you said that you see your 'model' just as a way of visualizing GR rather than something with new physical content?) But as I said in #201, your conclusion about expansion = time doesn't really fit with GR, so if that's a consequence of your model (and I still don't understand the logic behind that conclusion, even in the context of your model) then that suggests a problem with it. Also, it still seems to me that your model ignores the fact that it is fundamentally spacetime that is curved in GR, not space. On that last subject, do you still object to my football visualization of the spacetime for a closed universe that starts in a big bang and ends in a big crunch, or did my explanations resolve your problems with this visualization?

Finally, minor note on this comment:

A correspondingly large triangle in the observable universe would have sides which are not quite half a billion (10^9) light-years long. I think the chances of us manipulating rods of that magnitude are rather remote. For entertainment's sake, let's say we could manipulate rods which are half a million light years long, then we could create a triangle which has a SIA of 180+3x10^-6 degrees assuming the universe is hyperspherical.

Here you also seem to be making the additional assumption that the universe is a hypersphere whose size is equal to the observable universe. As I've said before in the context of your "the universe's radius is close to the schwarzschild radius for its mass" argument, that assumption doesn't really make any sense at all, since the boundaries of the observable universe just have to do with how far light has had time to travel since the Big Bang (an observer in the Andromeda galaxy would have a slightly different 'observable universe' that would include some regions that lie outside our own observable universe).

 

[neopolitan]

Yes, based on what you said your model is incompatible with SR in the same sense that GR is incompatible with SR (which I mentioned in #201), although like GR it might still reduce to SR locally (and haven't you said that you see your 'model' just as a way of visualizing GR rather than something with new physical content?) But as I said in #201, your conclusion about expansion = time doesn't really fit with GR, so if that's a consequence of your model (and I still don't understand the logic behind that conclusion, even in the context of your model) then that suggests a problem with it. Also, it still seems to me that your model ignores the fact that it is fundamentally spacetime that is curved in GR, not space. On that last subject, do you still object to my football visualization of the spacetime for a closed universe that starts in a big bang and ends in a big crunch, or did my explanations resolve your problems with this visualization?

I will get back to this.

Finally, minor note on this comment:

Here you also seem to be making the additional assumption that the universe is a hypersphere whose size is equal to the observable universe. As I've said before in the context of your "the universe's radius is close to the schwarzschild radius for its mass" argument, that assumption doesn't really make any sense at all, since the boundaries of the observable universe just have to do with how far light has had time to travel since the Big Bang (an observer in the Andromeda galaxy would have a slightly different 'observable universe' that would include some regions that lie outside our own observable universe).

Not really, if the universe is bigger (and curved) you just need longer rods for your triangle to make a noticeable difference in the sum of internal angles. I took an optimistic case, that the observable universe is all there is. There is an even more optimistic case, one which would require shorter rods to make a noticeable difference in the sum of internal angles - that is if the observable universe is such that some of what we see in one direction we can also see in another direction:

http://en.wikipedia.org/wiki/Observable_universe]It[/PLAIN] is also possible that the universe is smaller than the observable universe. In this case, what we take to be very distant galaxies may actually be duplicate images of nearby galaxies, formed by light that has circumnavigated the universe. It is difficult to test this hypothesis experimentally because different images of a galaxy would show different eras in its history, and consequently might appear quite different. A 2004 paper [2] claims to establish a lower bound of 24 gigaparsecs (78 billion[3] light-years) on the diameter of the universe, based on matching-circle analysis of the WMAP data.

With regard to the Schwartzschild radius argument, it is not just the radius that matters, it is the density. The argument goes a little like this:

1. The Copernican Principle states that wherever we are in the universe it looks pretty much the same (which means there is no big empty space around us into which the mass of the universe is expanding) and leads to the cosmological principle.

http://en.wikipedia.org/wiki/Copernican_principle]In[/PLAIN] cosmology, if one assumes the Copernican principle and observes that the universe appears isotropic from our vantage-point on Earth, then one can prove that the Universe is generally homogeneous (at any given time) and is also isotropic about any given point. These two conditions comprise the cosmological principle.

2. This means that our observable universe is not essentially different from the observable universe as observed from the most distant reaches of our observable universe (or the Andromeda galaxy, to use your example.)

3. The more mass within a Schwartzshild radius the less dense it is. This is because the density is related to the volume which increases with the cube of the radius. The Schwartzschild radius increases in a simple relationship with the mass.

(Figures from http://csep10.phys.utk.edu/astr162/lect/active/smblack.html.)

For example the volume defined by the Schwartzschild radius for the Earth's mass (5.98*10^24 kg giving a radius of 9mm) is:

V= 4/3.pi.(9*10^-3)^3=3.05*10^-6 cubic metres

The volume defined by the Schwartzschild radius for the Sun's mass (1.989x10*30 kg giving a radius of 2.9km) is:

V= 4/3.pi.(2.9x10^3)^3=1.02*10^11 cubic metres

The density of the mass with the Schwartzschild radius associated with the mass of the Earth is therefore:

5.98*10^24 kg / 3.05*10^-6 cubic metres = 1.96*10^30 kg/cubic metre

and the density of the mass with the Schwartzschild radius associated with the mass of the Sun is:

1.989*10^30 kg / 1.02*10^11 cubic metres = 1.95*10^19 kg/cubic metre

4. If the radius of the observable universe and the mass/density of the observable universe matches that for an event horizon, then the universe being bigger and isotropic just means that the mass/density of the universe will be greater than that required to constitute an event horizon. All that remains is to calculate whether, given accepted figures for the radius of the observable universe and the mass of the observable universe (or average density) is sufficient to suggest that we lie within an event horizon.

(Figures from http://en.wikipedia.org/wiki/Observable_universe.)

The observable universe has a radius of about 4.65 billion light years, or 4.65 billion times9,460,730,472,580.8 km = 4.3*10^25 m.

This gives a volume of 3.6*10^77 cubic metres.

The observable universe is calculated to have a mass of 3*10^52 kg - this is taken from the measured stellar density of 3*10^-27 kg/cubic metre (wikipedia contributers did this calculation, not me).

The Schwartzschild radius for the mass of the observable universe is:

r=2Gm/c^2=2*6.67*10^-11*3*10^52/(3*10^8)^2=4.4*10^25 m

Remember if the radius is greater, then the density of the universe has to be lower than has been measured and I have only ever heard arguments for the reverse, that the density of the universe is greater than measured because of "dark matter".

I know you have said that this equation does not apply to expanding matter, but equally, can you see that its a pretty nice match between figures here? (Note that a number of the figures I used were cosmological in nature, so being in the right order of magnitude is as close as you can get. My figure is 2.3% higher than the given figure for the radius of the universe, but how accurate is the 3*10^52 kg figure? and how accurate is 4.3*10^25 m?)

I think you are right about the non-applicability of the equation in so much as it is the expansion that prevents the prediction of the equation coming to fruition - ie:

http://en.wikipedia.org/wiki/Schwarzschild_radius]The[/PLAIN] Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a characteristic radius associated with every mass. It is the radius for a given mass where, if that mass could be compressed to fit within that radius, no known force or degeneracy pressure could stop it from continuing to collapse into a gravitational singularity

The universe is patently not collapsing into a gravitational singularity.

cheers,

neopolitan

 

[neopolitan]

<snip>do you still object to my football visualization of the spacetime for a closed universe that starts in a big bang and ends in a big crunch, or did my explanations resolve your problems with this visualization?

Yes.

This is precisely where you are saying that spacetime is inherently curved. Elsewhere you say that spacetime is curved due to mass and energy.

I have no problem with spacetime being curved due to mass and energy, locally this will be very noticeable. But in a grander scale, curvature of the entire universe due to the mass and energy it contains will only be noticeable if the universe is closed and you are playing around with extraordinarily long rods. I have no problems with that. But elsewhere you said that the universe is not considered to be closed by most people.

So, I am a bit confused.

As an aside, your football model seems to rely on the density of the universe being greater than it is (see my previous post). Enough to overcome the expansion of the universe. Such slowing of expansion should be observable but, as mentioned before, there is evidence of the reverse. I would be happy to read any links you have to reputable sources reporting evidence that the rate of expansion of the universe is slowing. (You might then also want to inform http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/990210c.html"!)

Anyway, once you have cleared up my confusion about what you are trying to say, I have some diagrams to demonstrate my model a little more clearly.

For those joining us, the promised model does clearly mention simultaneity which, although might seem to have disappeared from the thread, has never been far from my mind.

cheers,

neopolitan