Conditions for spacetime to have flat spatial slices

[PeterDonis]

When you say " the actual, physical spacetime (meaning, the actual physical metric that determines proper times and proper distances) is curved." It is curved with respect to what? The answer is, that it is curved with respect to the Minkowski coordinates.

Even if we don't know what they are, can't we at least say the Minkowski Coordinates exist?

The answer is really simple. Don't apply the metric. Just use the original unmodified event coordinates, and you'll have Minkowski spacetime.

On re-reading, I realized that I may not have fully addressed these points. Let's go back to the example I gave, of the two objects freely falling towards the Earth, starting at coordinate time t = 0 at radial coordinates R and R + r, with dr/dt = 0 for both at t = 0. I said that the curves these objects follow are geodesics, and that if we insist on using Minkowski coordinates, we will find that we need to modify the metric to accurately represent distances and times, because physically the distance between these objects increases with time, even though Minkowski coordinates would assign them a constant coordinate separation (since their initial velocities are equal).

First of all, let's look at your suggestion to "Don't apply the metric. Just use the original unmodified event coordinates, and you'll have Minkowski spacetime." You seem to have a misconception here that the metric somehow modifies the coordinate values that we assign to events. It doesn't; the metric just tells us, given a certain system of coordinates, how to calculate actual physical distances and times from the coordinate differentials between events. So, as I said, you can use "unmodified" Minkowski coordinates if you want, but you won't be able to use the unmodified Minkowski metric, because it will give the wrong answers for physical distances and times.

Second, I said that the curves the two falling objects were following were geodesics, and then I proceeded to infer curvature from the fact that these geodesics were initially parallel and then separated. You could respond by saying, basically, so what? Maybe it's just that these curves are *curved*--that is, that they aren't "straight lines", and so the fact that they are initially parallel but later separate isn't an issue. I would respond by asking how, then, would you determine what a "straight line" is? You answer, by using the "unmodified" Minkowski coordinates.

The problem with this is that "use the unmodified Minkowski coordinates" is not a *physical* condition, because there is nothing, physically, that picks out the Minkowski "straight lines" from among all the other possible curves in spacetime. No physical phenomenon propagates along such lines--not even light, since gravity affects the path of light. So if you tell me that a particular curve--a particular object's worldline--is or is not a Minkowski straight line, I have no way of telling whether or not you're correct; no physical test I can run will tell me what the Minkowski "straight lines" are.

These "geodesics" I refer to, on the other hand, *do* have a definite physical meaning: they're the worldlines of freely falling objects, and it's easy to test, physically, whether an object is freely falling: just attach an accelerometer to it and make sure it reads zero. Geometrically, this condition corresponds to the condition that a geodesic has zero "proper acceleration": the covariant derivative of its 4-velocity with respect to proper time along it is zero. Now, you may say, that's the *same* condition that picks out Minkowski straight lines! In special relativity, you're correct; but one of the key points of GR is that, in the presence of gravity, freely falling worldlines no longer satisfy the *geometric* requirements for Minkowski straight lines--for example, initially parallel freely falling worldlines can increase separation over time, which can't happen with Minkowski straight lines.

So another way of stating my statement that "the background flat spacetime is unobservable", when I was talking about the "flat spacetime" interpretation of GR, is that the Minkowski spacetime of SR is based on an assumption that "straight lines", in the sense of freely falling worldlines, are *also* "straight lines" in the sense of having certain geometric properties, such as parallel lines staying parallel. In the presence of gravity, that assumption no longer holds.

 

[PeterDonis]

If you are applying rotation to surface of a planet, you can do it in steps. Take whatever coordinates you have and map them, one-to-one into \mathbb{R}^2, the mapping that you CAN apply the rotation. Then do the rotation, and convert back.

It should be the same with Lorentz Transformation; simply map whatever coordinates you have into \mathbb{R}^4, apply the Lorentz Transformation, and then convert back.

There's one other issue involved with this that I didn't mention. If the surface of the planet is a sphere, you can't map it one to one into \mathbb{R}^2, because the topology is different. You have to include a "point at infinity" as well. Similar remarks might apply when trying to map a curved spacetime into \mathbb{R}^4; for example, I don't know if you can do it with any of the FRW spacetimes without running into this type of difficulty. (This is all in addition to the point I already made, that whatever the final transformation turns out to be, if you *can* do the "map-transform-map back" procedure, it must preserve the intrinsic geometry of the original spacetime; you can't make the surface of the Earth flat by applying a rotation in this way, even assuming you find a way to deal with the "point at infinity" issue.)

 

[JDoolin]

There's one other issue involved with this that I didn't mention. If the surface of the planet is a sphere, you can't map it one to one into \mathbb{R}^2, because the topology is different. You have to include a "point at infinity" as well. Similar remarks might apply when trying to map a curved spacetime into \mathbb{R}^4; for example, I don't know if you can do it with any of the FRW spacetimes without running into this type of difficulty. (This is all in addition to the point I already made, that whatever the final transformation turns out to be, if you *can* do the "map-transform-map back" procedure, it must preserve the intrinsic geometry of the original spacetime; you can't make the surface of the Earth flat by applying a rotation in this way, even assuming you find a way to deal with the "point at infinity" issue.)

You can map all the points on the surface of the Earth into \mathbb{R}^2, but there are two points on the Earth mapped to each point. Sorry, I didn't make that clear. I'm not talking about a one-to-one mapping. I'm just saying that if you are rotating, you can draw a line through the center of the earth, and each point on the Earth has an r and theta with respect to that line around which you are rotating.

No, you're right, once you map all the points this way, you had better have kept track of the third coordinate which will tell you whether the point is on your hemisphere, or the opposite hemisphere.

Every point on the Earth's surface has a coordinate relative to your axis of rotation, but every coordinate relative to your axis of rotation is associated with an infinite number of points on the earth.

I may add more if I remember what my point actually was.

Yes, you're right. I said you could convert to \mathbb{R}^2 and convert back. That is not quite true. You can convert fairly easily into cylindrical coordinates; though. Not sure if that was quite my point either.

What it comes down to is that I wouldn't be entirely satisfied with a rotation, unless we had the whole business defined in \mathbb{R}^4. If you take the Earth speeding by at 99% of the speed of light and rotate around the coordinates around the point speeding by, you get quite a different result than if you rotate the Earth around a point sitting on the earth.

I'm running into difficulty, though, because even in \mathbb{R}^4 there is some difficulty. From the observer's perspective at an instant, we regard the Earth as a set of simultaneous events. If there were no such thing as slowing of time in a gravity well, then it would be a simple matter to claim that we have a set of simultaneous events. But with gravity, we cannot merely take a clock far enough out into space and say, it measures the "real time" for that observer.

Yet, I feel that the "real time" does exist for that observer. What I mean is that there is an objective meaning of universal simultaneity, which is NOT related to the proper time of a particle following a geodesic. At each instant, every observer has a momentarily comoving universal reference frame, even though, in general, there is no other particle in that reference frame in the entire universe.

I feel like you're saying "Ah, we have no way of determining a universal reference frame because everything is moving." And then, in frustration, you say "Let's just use all the moving stuff as our universal reference frame."

What I'm hearing is "even light bends in gravity, and therefore there is no such thing as a straight line, therefore, define geodesics as straight lines." To me, these things simply don't follow one another.

To me, it seems like, yes, light bends in gravity, but the vast majority of light that reaches us is coming straight. And we can use those straight lines to define what we mean by straight. I honestly don't even think I can begin to fathom how anyone could think a geodesic is a straight line. I honestly don't even think I can begin to fathom how anyone could see a "great circle" on a sphere as a straight line. To me, these ideas are very very distinct.

In regards to a(t) the scale factor which is a function of cosmological time, I do not think it is appropriate for the cosmos to have an age independent of the matter that is passing through it. I certainly think that the universe is less dense now than it was before, and that means things are moving apart. But that is not a matter of scale. That is a matter of movement.

If all of what I am saying means that I don't "believe" the Einstein Field Equations, I really don't know. I do know that these graduate texts on Riemannian Geometry are beyond me, and I ordered some undergraduate texts from the library. But I think there might be some more fundamental disagreement if the Einstein Field Equations make the assumption a priori that distant galaxies are comoving. And I'm afraid that might be the case, given that a large chunk of "Relativity, Gravitation, and World Structure" were devoted to criticizing Eddington's flawed assumptions about homogeneity, and I do, definitely agree with Milne.

 

[PeterDonis]

But with gravity, we cannot merely take a clock far enough out into space and say, it measures the "real time" for that observer.

Yet, I feel that the "real time" does exist for that observer. What I mean is that there is an objective meaning of universal simultaneity, which is NOT related to the proper time of a particle following a geodesic. At each instant, every observer has a momentarily comoving universal reference frame, even though, in general, there is no other particle in that reference frame in the entire universe.

There's nothing stopping you from setting up such a reference frame "at each instant", but to have a complete description of the spacetime, you have to have one at *every* instant, and they need to match up with each other smoothly and continuously. Depending on your state of motion, the rest of the universe may look simple in such a description, or it may not. As long as your frame meets the continuity requirement above, and as long as it allows you to calculate the invariant quantities of the spacetime correctly, your frame is just as "valid" as any other. Whether or not you can assign a reasonable physical meaning to all the frame-dependent quantities in such a frame is a separate question.

I feel like you're saying "Ah, we have no way of determining a universal reference frame because everything is moving." And then, in frustration, you say "Let's just use all the moving stuff as our universal reference frame."

As *one possible* universal reference frame--the one that happens to match up well with the symmetry we imposed as a condition on the solution, the symmetry of isotropy, and which therefore makes the universe look simple to any observer that happens to be moving in a way that matches up with that symmetry. Once again, if you're in a different state of motion, there's nothing stopping you from setting up your own personal reference frame and assigning coordinates to every event in the entire spacetime in that frame; but in such a frame, the universe probably won't look as simple (it won't look isotropic, for example, because of your state of motion).

What I'm hearing is "even light bends in gravity, and therefore there is no such thing as a straight line, therefore, define geodesics as straight lines." To me, these things simply don't follow one another.

To me, it seems like, yes, light bends in gravity, but the vast majority of light that reaches us is coming straight. And we can use those straight lines to define what we mean by straight. I honestly don't even think I can begin to fathom how anyone could think a geodesic is a straight line.

But these "straight" lines that light travels on *are* geodesics! And geodesics like these *are* what we use to define what we mean by "straight". But you have to remember that this is "straight" in *spacetime*, not just in space. The paths of light rays grazing the Sun look bent in *space*, but in *spacetime* they are as straight as it's possible for a line to be. In regions of spacetime that are flat to within some accuracy of measurement, the paths of light rays will be "straight" in the sense you're thinking (the Minkowski frame sense of "straight") to that same accuracy. The "vast majority of light that reaches us" is coming from such regions of spacetime, so those paths do appear "straight" to us in the everyday sense. But they are still geodesics of spacetime.

I honestly don't even think I can begin to fathom how anyone could see a "great circle" on a sphere as a straight line. To me, these ideas are very very distinct.

You're right, they are. And the question is, which idea is the better idea for use in modeling a particular geometry that we're interested in? If the geometry you want to model is a Euclidean plane (or Euclidean 3-space), then obviously you need to use the Euclidean notion of "straight line". But if the geometry is that of a 2-sphere (not a 2-sphere embedded in Euclidean 3-space, but the intrinsic geometry of the 2-sphere itself), then you need to use great circles as your "straight lines" if you want those straight lines to meet the axioms and postulates of geometry. The only exception is the parallel postulate, of course, but the whole point of studying the intrinsic geometry of the 2-sphere in the first place is that in that geometry the parallel postulate doesn't hold--on a 2-sphere there simply are no "straight lines" in the Euclidean sense.

Similarly, in spacetime, if the geometry you are studying is the actual flat Minkowski spacetime, then of course you want to use Minkowski straight lines as your "straight lines". But if the spacetime geometry you are studying is not flat, then you need to find the right notion of "straight line" for that geometry if you want your straight lines to satisfy the appropriate axioms and postulates. If you really don't like calling such objects "straight lines", well, that's why the word "geodesic" was invented.

In regards to a(t) the scale factor which is a function of cosmological time, I do not think it is appropriate for the cosmos to have an age independent of the matter that is passing through it.

I'm not sure what you mean by this; the FRW solution to the EFE is certainly not "independent of the matter"--the actual dynamics of the scale factor a(t) depend crucially on the specific equation of state for the matter-energy that is present.

I certainly think that the universe is less dense now than it was before, and that means things are moving apart. But that is not a matter of scale. That is a matter of movement.

I would say that these are two different ways of saying the same thing--or maybe two different ways of describing the same physical reality. It's a matter of "scale" if you look at it one way, and it's a matter of "movement" if you look at it another way. The two viewpoints are not mutually exclusive; they're like looking at the same object from different vantage points.

If all of what I am saying means that I don't "believe" the Einstein Field Equations, I really don't know. I do know that these graduate texts on Riemannian Geometry are beyond me, and I ordered some undergraduate texts from the library.

I can't remember if this has been linked to before, but you might try John Baez' article on "The Meaning of Einstein's Equation" here:

http://math.ucr.edu/home/baez/einstein/

He goes into the sorts of things we've been talking about at a pretty basic level--he mentions differential geometry but only in passing, more or less. I think this page does a pretty good job of describing the basic physical content of the EFE without requiring you to dig into the heavy mathematical machinery.

You might also try the pages linked to at Baez' relativity tutorial index page here:

http://math.ucr.edu/home/baez/gr/

Also, as I mentioned before, I recommend Kip Thorne's Black Holes and Time Warps if you want a good non-technical book on relativity (as well as giving a lot of interesting historical background). Thorne also does a good job of getting across the physical content without requiring you to dig into heavy math.

But I think there might be some more fundamental disagreement if the Einstein Field Equations make the assumption a priori that distant galaxies are comoving. And I'm afraid that might be the case, given that a large chunk of "Relativity, Gravitation, and World Structure" were devoted to criticizing Eddington's flawed assumptions about homogeneity, and I do, definitely agree with Milne.

I haven't read the book you refer to, but there appears to be a free ebook download available so I'll look through it. One thing to bear in mind, though, is that the EFE itself is separate from particular solutions such as those used as cosmological models; obviously the latter will be dependent on the reasonableness of the conditions imposed (such as homogeneity and isotropy), but that doesn't invalidate the EFE itself, it just means we need to investigate other possible conditions and see how the various models, with various different conditions imposed, match up with experimental data. The FRW models are the front runners right now because they appear to do the best job at that (more precisely, the more detailed models that include perturbations about the FRW "baseline" do the best job to date).

One should also bear in mind that what look like different solutions to the EFE may actually be describing the same spacetime geometry, just in different coordinates. The Milne model, for example (on Wikipedia here: http://en.wikipedia.org/wiki/Milne_model), looks at first glance like a distinct solution that might address some of the potential issues with the FRW models, but it turns out to be a special case of the FRW models.

 

[JDoolin]

I can't remember if this has been linked to before, but you might try John Baez' article on "The Meaning of Einstein's Equation" here:

http://math.ucr.edu/home/baez/einstein/

He goes into the sorts of things we've been talking about at a pretty basic level--he mentions differential geometry but only in passing, more or less. I think this page does a pretty good job of describing the basic physical content of the EFE without requiring you to dig into the heavy mathematical machinery.

This is helpful, and no, I had not seen it before. I wonder if you could verify what Baez says, that the following is a good representation of the Einstein Field Equations in Plain English

http://math.ucr.edu/home/baez/einstein/node3.html]We[/PLAIN] promised to state Einstein's equation in plain English, but have not done so yet. Here it is:

Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the x direction at that point, plus the pressure in the y direction, plus the pressure in the z direction.

I find this to be comforting, because it means I do not have to conflict with the Einstein Field Equations.

The key is that to apply the EFE, we begin by assuming we have a small ball of freely falling particles initially at rest with respect to one another. I am starting with a small ball of freely falling particles that have relative velocity with one another of v= d/t, where t is the time since the big bang. When we consider any group of particles which follow geodesics from the big bang, there can be no such "small ball" of comoving particles.

So the case I'm considering does not "conflict" with the EFE's, but it does lie outside the scope of the EFE's. In other words, I think I do "believe" the EFE's are true, but I'm pretty sure they don't apply in this case.

We have to start from scratch, considering a ball of particles with not v=0, but v=r/t.

 

[JDoolin]

One should also bear in mind that what look like different solutions to the EFE may actually be describing the same spacetime geometry, just in different coordinates. The Milne model, for example (on Wikipedia here: http://en.wikipedia.org/wiki/Milne_model), looks at first glance like a distinct solution that might address some of the potential issues with the FRW models, but it turns out to be a special case of the FRW models.

Check out the discussion page for the Milne Model, because there are some things there that came from the actual book. When I tried to put actual quotes from Milne in the main article, they were removed.

Wikipedia's policy is to use secondary; not primary sources. You must go back to "Relativity, Gravitation, and World Structure" to actually get any idea of what Milne actually wrote.

 

[PeterDonis]

The key is that to apply the EFE, we begin by assuming we have a small ball of freely falling particles initially at rest with respect to one another. I am starting with a small ball of freely falling particles that have relative velocity with one another of v= d/t, where t is the time since the big bang. When we consider any group of particles which follow geodesics from the big bang, there can be no such "small ball" of comoving particles.

So the case I'm considering does not "conflict" with the EFE's, but it does lie outside the scope of the EFE's. In other words, I think I do "believe" the EFE's are true, but I'm pretty sure they don't apply in this case.

We have to start from scratch, considering a ball of particles with not v=0, but v=r/t.

And if you read further to Baez' page on the Big Bang, in the same series of pages, you'll find that he covers this case, which *is* covered by the EFE.

 

[PeterDonis]

Check out the discussion page for the Milne Model, because there are some things there that came from the actual book. When I tried to put actual quotes from Milne in the main article, they were removed.

I'm shocked, yes, shocked that such a thing could possibly happen on Wikipedia!

You're right, one shouldn't offer Wikipedia as an authoritative source. But see below.

You must go back to "Relativity, Gravitation, and World Structure" to actually get any idea of what Milne actually wrote.

I've downloaded the ebook and am working my way through it. From what I've read so far, I see some justice on both sides of the argument on the discussion page you referred to. However, a key point that I didn't really see brought up in that discussion is that we've learned a *lot* about both the theoretical aspects of relativity and the experimental facts about cosmology (not just the discovery of the CMBR, which is mentioned in the discussion) since Milne wrote his book. For example, as the last paragraph of the actual Wiki article notes, we have a lot of evidence now that we didn't have in the 1930's concerning how exactly the conditions of homogeneity and isotropy actually apply in the universe (i.e, pretty exactly--to one part in a few hundred thousand in the CMBR, for example). I agree with the position you took in the discussion that the Wiki article should fairly represent what Milne actually wrote at the time, not what secondary sources say; but it looks to me like Milne's model itself (apart from the more general remarks he makes about what constitutes the actual physical observations we make) is not a good fit to the data based on our current knowledge.

 

[JDoolin]

And if you read further to Baez' page on the Big Bang, in the same series of pages, you'll find that he covers this case, which *is* covered by the EFE.

Are you talking about this page?

http://math.ucr.edu/home/baez/einstein/node7.html

The case I'm describing is definitely not there. What do you think you see that sounds like it has anything to do with what I am talking about?

I said that the particles were moving apart and d=v*t; or likewise, v=d/t. No comoving particles ANYWHERE.

 

[JDoolin]

I'm shocked, yes, shocked that such a thing could possibly happen on Wikipedia!

You're right, one shouldn't offer Wikipedia as an authoritative source. But see below.



I've downloaded the ebook and am working my way through it. From what I've read so far, I see some justice on both sides of the argument on the discussion page you referred to. However, a key point that I didn't really see brought up in that discussion is that we've learned a *lot* about both the theoretical aspects of relativity and the experimental facts about cosmology (not just the discovery of the CMBR, which is mentioned in the discussion) since Milne wrote his book. For example, as the last paragraph of the actual Wiki article notes, we have a lot of evidence now that we didn't have in the 1930's concerning how exactly the conditions of homogeneity and isotropy actually apply in the universe (i.e, pretty exactly--to one part in a few hundred thousand in the CMBR, for example). I agree with the position you took in the discussion that the Wiki article should fairly represent what Milne actually wrote at the time, not what secondary sources say; but it looks to me like Milne's model itself (apart from the more general remarks he makes about what constitutes the actual physical observations we make) is not a good fit to the data based on our current knowledge.

Pardon me, but does the current model really "FIT" that well? We have no real explanation for inflation. We have no dark energy. We have no dark matter. We have a theory that is inconsistent with quantum mechanics. But we have an equation that matches up really well.

The thing is, I can show you a conformal mapping between the Milne model and the Robertson Walker Metric, which preserves the speed of light and has exactly the same events, if you're interested. All the same variables are there. All the same events are there, except the singularity is transformed into a lightcone in Milne's version, and it's turned into a plane in the "comoving matter" version.

The only difference is that in Milne's version, the transformation makes sense, because you're converting from proper time into coordinate time, whereas in the "comoving matter" version, your converting from proper time to meaningless arbitrary cosmological time coordinates, chosen arbitrarily to make it "look like" all the particles are comoving.

 

[JDoolin]

The thing is, I can show you a conformal mapping between the Milne model and the Robertson Walker Metric, which preserves the speed of light and has exactly the same events, if you're interested. All the same variables are there. All the same events are there, except the singularity is transformed into a lightcone in Milne's version, and it's turned into a plane in the "comoving matter" version.

I have all the images I need, so I will go ahead and do it.

This image is correct as a mapping of proper time vs. distance, or more meaningfully, proper time vs. rapidity.

What goal do you have, then? Do you want to map it into a coordinate system where all of the worldlines are vertical? Then it should look like this:

As for why anyone should want to do such a thing, that is a matter of preference and opinion, only! It was an a priori decision made by Einstein which has only been confirmed by circular reasoning.

On the other hand, would you like to map it into space vs. coordinate time?

Then it should look like the lower diagram below:

[URL]http://www.wiu.edu/users/jdd109/stuff/img/milnemetric.jpg[/URL]

This diagram has been on my blog since September. It represents two different conformal mappings of the Robertson-Walker Diagram.

All the difference is in how you parameterize your variables.

(*Correct Variables:
	r=rapidity, relative to central particle;
	t=proper time of particles on unaccelerated paths from big bang.;
             (both are "invariant" properties of matter.)

Incorrect variables:
	r=distance in "real universe coordinates";
	t=time in "real universe coordinates";
             (both are contravariant properties of space.)
*)

t=1;
e0 = Table[{r, 0}, {r, -10, 10}];
e1 = Table[{r, t}, {r, -10, 10}];
comovingWorldLines = Transpose[{e0, e1}];
ListLinePlot[comovingWorldLines]
e0 = Table[{0 Sinh[r], 0 Cosh[r]}, {r, -1.5, 1.5, .1}];
e1 = Table[{t Sinh[r], t Cosh[r]}, {r, -1.5, 1.5, .1}];
milneWorldLines = Transpose[{e0, e1}];

 

[JDoolin]

I didn't have the light-cone in the earlier diagram.

I put it in, and realized there is also a subtle mistake in the "comoving particle" conformal mapping that doesn't happen in this milne mappping.

In the Friedmann-Walker diagram, the light "from the big bang" crosses every single worldline. But in the "comoving particles" diagram, the light just passes a finite number of worldlines.

In the milne diagram, you have that subtle error fixed, and the light "from the big bang" crosses every worldline, just as it is in the Friedmann Walker diagram.

 

[PeterDonis]

The case I'm describing is definitely not there. What do you think you see that sounds like it has anything to do with what I am talking about?

I said that the particles were moving apart and d=v*t; or likewise, v=d/t. No comoving particles ANYWHERE.

The page I linked to describes a ball B which is "expanding at t = 0". That ball corresponds to your particles moving apart with v = d/t. Baez says that we can't directly apply his equation (2) (which is his statement of the EFE) to ball B because its particles aren't at rest relative to each other at time t = 0. He then defines a second ball, B', which *does* have all its particles at rest relative to each other at time t = 0, and which has the same radius and the same acceleration as ball B at time t = 0. This allows him to apply his equation (2) to ball B', and then show that the equation he derives for the radius r of ball B' vs. time, *also* holds for the radius R of ball B vs. time. So he's showing that the EFE *does* apply to the type of ball you defined, where the particles are moving apart at time t = 0.

 

[PeterDonis]

Pardon me, but does the current model really "FIT" that well? We have no real explanation for inflation. We have no dark energy. We have no dark matter. We have a theory that is inconsistent with quantum mechanics. But we have an equation that matches up really well.

And the Milne model, as far as I can see, suffers from the same problems. See below.

The thing is, I can show you a conformal mapping between the Milne model and the Robertson Walker Metric, which preserves the speed of light and has exactly the same events, if you're interested. All the same variables are there. All the same events are there, except the singularity is transformed into a lightcone in Milne's version, and it's turned into a plane in the "comoving matter" version.

I already know about this; it's what I was referring to before when I said that the Milne model is describing the same spacetime geometry as the FRW models, just in different coordinates. See next comment.

The only difference is that in Milne's version, the transformation makes sense, because you're converting from proper time into coordinate time, whereas in the "comoving matter" version, your converting from proper time to meaningless arbitrary cosmological time coordinates, chosen arbitrarily to make it "look like" all the particles are comoving.

What you are saying amounts to this: you like the coordinate system in Milne's model better than you like the FRW coordinate system. That's fine; particular coordinate systems don't matter. What matters is the spacetime geometry. That's the same either way. It's like saying that you prefer to use a Mercator projection rather than a stereographic projection to map the surface of the Earth.

You apparently don't believe this; you appear to think that the Milne model and the FRW models are describing fundamentally different objects. From what I've read so far, I would have to disagree; it looks to me, so far, like what I said above is valid--both models are describing the same geometry, just in different coordinates. The diagrams you posted give me no reason to change that conclusion; for example, your statement that "All the differences is in how you parametrize your variables" indicates to me that what you're illustrating are simple coordinate transformations that don't change the geometry, and hence, don't change the physics.

 

[PeterDonis]

In the Friedmann-Walker diagram, the light "from the big bang" crosses every single worldline. But in the "comoving particles" diagram, the light just passes a finite number of worldlines.

In the milne diagram, you have that subtle error fixed, and the light "from the big bang" crosses every worldline, just as it is in the Friedmann Walker diagram.

I'm not sure that what you're saying about the Friedmann-Walker diagram is correct. Physically, in a spatially infinite universe (which is what's required for an "infinite number of worldlines" in the sense you're using the term), I would *not* expect light reaching us now from the Big Bang event to have crossed *all* of that infinite number of worldlines, because only a finite amount of proper time has elapsed, and light can only cover a finite distance in a finite time--what you're suggesting would require the light to cover an infinite distance in a finite time. The "comoving particles" diagram (a more standard name for it would be a "conformal" diagram, as it is called on the Ned Wright page I linked to earlier) has the advantage that it makes the reasoning I just gave obvious.

As far as I can tell, your "Milne" diagram corresponds to the diagram on the Ned Wright page here...

http://www.astro.ucla.edu/~wright/cosmo_02.htm

...in what Wright calls "special relativistic" coordinates. This means to me that you have raised an interesting question about the light cones, since it certainly appears in your "Milne" diagram and Wright's "special relativistic" diagram that the light from the "Big Bang" *does* cross all the "infinite number" of worldlines. I'll have to think about this one some more.

 

[George Jones]

The Milne universe is an FRW universe that is empty of matter and energy. See

https://www.physicsforums.com/showthread.php?p=1757634&highlight=milne#post1757634

https://www.physicsforums.com/showthread.php?p=2872975#post2872975.

 

[JDoolin]

I'm not sure that what you're saying about the Friedmann-Walker diagram is correct. Physically, in a spatially infinite universe (which is what's required for an "infinite number of worldlines" in the sense you're using the term), I would *not* expect light reaching us now from the Big Bang event to have crossed *all* of that infinite number of worldlines, because only a finite amount of proper time has elapsed, and light can only cover a finite distance in a finite time--what you're suggesting would require the light to cover an infinite distance in a finite time. The "comoving particles" diagram (a more standard name for it would be a "conformal" diagram, as it is called on the Ned Wright page I linked to earlier) has the advantage that it makes the reasoning I just gave obvious.

As far as I can tell, your "Milne" diagram corresponds to the diagram on the Ned Wright page here...

http://www.astro.ucla.edu/~wright/cosmo_02.htm

...in what Wright calls "special relativistic" coordinates. This means to me that you have raised an interesting question about the light cones, since it certainly appears in your "Milne" diagram and Wright's "special relativistic" diagram that the light from the "Big Bang" *does* cross all the "infinite number" of worldlines. I'll have to think about this one some more.

At least on this page, I can see that I'm talking about the same thing as Ned Wright. He correctly notes that the observable universe is the entire universe. He also correctly notes that our past lightcone passes the most distant galaxies at x=c t_0/2

(He does not mention that the proper age of those asymptotically most distant galaxies is zero, so that past light-cone does cross the singularity. I imagine the era of hydrogen recombination to be a big hyperbolic arc in the diagram, and that represents the furthest distance we can actually see.)

Technically, the Friedmann Walker Diagram is ambiguous at the singularity. The question is what is the slope of the past light-cone at the point (0,0). If that slope is zero, then the light should cross all of the past light-cones. If the slope is nonzero, then it must cross only a finite number of light-cones.

(Sorry--in Wright's diagram (at http://www.astro.ucla.edu/~wright/cosmo_02.htm ), I looked again, and the speed of light, as drawn, is definitely a non-zero slope at the singularity. That means that it is not a conformal mapping with the image below it, because it does not cross every worldline.) As for me, I think that the speed of light at that singularity point should be faster than the speed of light for any particle at that point. So I think the line should be horizontal there.

 

[JDoolin]

I'm not sure that what you're saying about the Friedmann-Walker diagram is correct. Physically, in a spatially infinite universe (which is what's required for an "infinite number of worldlines" in the sense you're using the term), I would *not* expect light reaching us now from the Big Bang event to have crossed *all* of that infinite number of worldlines, because only a finite amount of proper time has elapsed, and light can only cover a finite distance in a finite time--what you're suggesting would require the light to cover an infinite distance in a finite time.

No, no. The light wouldn't have to cross an infinite distance; it would only need to cross an infinite number of particles. That's the problem Milne (and Epstein and I) have with the assumption of homogeneity. You're assuming homogeneity throughout the universe. You should be allowing for the particles to approach an infinite density, asymptotically as you go out to the edge of the universe.

We're not suggesting that the light has to cover an infinite distance, but that an infinite number of particles lie within a finite distance.

 

[JDoolin]

The Milne universe is an FRW universe that is empty of matter and energy. See

https://www.physicsforums.com/showthread.php?p=1757634&highlight=milne#post1757634

https://www.physicsforums.com/showthread.php?p=2872975#post2872975.

You are linking to links of you asserting that this is true, but as I told PeterDonis, this is one occasion, where you should really go back to the original text, because the secondary sources do not justly represent Milne's ideas.

 

[PeterDonis]

We're not suggesting that the light has to cover an infinite distance, but that an infinite number of particles lie within a finite distance.

A finite *coordinate* distance, yes. But have you integrated the metric over that finite coordinate distance to confirm that it covers a finite *proper* distance? The fact that the model indicates density going to infinity as you approach that finite coordinate distance is a big red flag to me that there's actually an infinite proper distance (meaning the actual physical density remains finite), and that the apparent infinite density is an artifact of the peculiar coordinate system you are using. Actual infinite physical density is not physically reasonable.

 

[JDoolin]

A finite *coordinate* distance, yes. But have you integrated the metric over that finite coordinate distance to confirm that it covers a finite *proper* distance? The fact that the model indicates density going to infinity as you approach that finite coordinate distance is a big red flag to me that there's actually an infinite proper distance (meaning the actual physical density remains finite), and that the apparent infinite density is an artifact of the peculiar coordinate system you are using. Actual infinite physical density is not physically reasonable.

Infinite physical density is physically reasonable, I think. For an infinite physical density, what is required is that several particles occupy the same place at the same time.

Certainly, you can't have any two particles sharing the same quantum numbers. But these particles have different momenta, and hence different quantum numbers. This situation can only exist for an instant.

However, since the proper time experienced by a particle traveling asymptotically approaching the speed of light is zero, that "instant" is effectively, the whole lightcone; it lasts forever in the frame of the central observer. Infinite physical density for an instant (a single event) is physically reasonable, and due to time dilation, that instant is effectively forever.

As for integrating the proper time, I hope you'll forgive me. As I mentioned before, the Einstein Field Equations need to be done over from scratch assuming this case where particles are diverging as v=d/t. So, no, I have not completed this task. That, I think, is something for better minds than mine. I just want it to be realized that the Einstein Field Equations were derived under the assumption of a finite number of comoving particles, and it does not cover the case of an infinite number of non-comoving particles. Then perhaps someone; (maybe you!) could re-do the EFE's from scratch, develop the theory and see what comes out of it.

 

[PeterDonis]

Infinite physical density is physically reasonable, I think. For an infinite physical density, what is required is that several particles occupy the same place at the same time.

To be precise, what's required is that an *infinite* number of particles occupy the same place at the same time. But I think you realize that--see below.

Certainly, you can't have any two particles sharing the same quantum numbers. But these particles have different momenta, and hence different quantum numbers. This situation can only exist for an instant.

However, since the proper time experienced by a particle traveling asymptotically approaching the speed of light is zero, that "instant" is effectively, the whole lightcone; it lasts forever in the frame of the central observer. Infinite physical density for an instant (a single event) is physically reasonable, and due to time dilation, that instant is effectively forever.

I can't be sure here whether or not you're committing a common confusion in relativity. An instant (event) is a single point. A null worldline--the path of a light ray--has "zero lapse of proper time", but it is *not* an instant--it is *not* a single point. It consists of a series of distinct events. Your language in the quote above is ambiguous; you need to be clear and explicit about whether the infinite density in your model occurs at a single point, or along a whole null worldline.

I suspect that what you mean is the former (the infinite density only occurs at a single point), in which case I would strongly discourage using the language "due to time dilation, that instant is effectively forever", since that language promotes exactly the confusion that I described. The instant of infinite density in your model, I suspect, is supposed to be a single point; it's only the particular coordinates you've chosen that distort that single point into what appears to be a null cone. Physically, it's a single point.

However, saying that the infinite density is only for a single point does *not* make it physically reasonable. It only means that your model "predicts its own downfall", just as GR does for spacetimes in which a singularity occurs. Your model is telling you that it can't cover that point--you need some new physics to tell you what happens there. The FRW models in GR handle that, as has been mentioned before, by *not* claiming to cover events all the way to the initial singularity; they only cover things back to (roughly) the beginning of the inflationary phase, where the universe was very small, hot, and dense, but not infinitely so. In your model, that would correspond (as you note) to drawing a hyperbola very *close* to the "null cone" boundary, corresponding to the (very small but not zero) proper time of the beginning of inflation (let's say), and saying that that's the actual physical limit of what your model covers; before that, new physics is needed to say what happened and how the initial state in your model came to be, just as with the FRW models.

As for integrating the proper time, I hope you'll forgive me. As I mentioned before, the Einstein Field Equations need to be done over from scratch assuming this case where particles are diverging as v=d/t.

No, the EFE doesn't need to be reworked, as I noted in a previous post. (Small nitpick: the usual term is "Einstein Field Equation", singular, even though it does have multiple tensor components. The tensors on each side of the EFE are considered single geometric objects, just as we normally write vector equations as relating single objects, even though vectors have multiple components.)

But that's actually not required anyway for what I was asking for. I said "integrate the metric", and you already have an expression for the metric; how that metric arises from the EFE is a separate issue. Integrate that expression over the appropriate range of coordinate values, and see what you come up with.

 

[JDoolin]

To be precise, what's required is that an *infinite* number of particles occupy the same place at the same time. But I think you realize that--see below.



I can't be sure here whether or not you're committing a common confusion in relativity. An instant (event) is a single point. A null worldline--the path of a light ray--has "zero lapse of proper time", but it is *not* an instant--it is *not* a single point. It consists of a series of distinct events. Your language in the quote above is ambiguous; you need to be clear and explicit about whether the infinite density in your model occurs at a single point, or along a whole null worldline.

See if I can make it a little less ambiguous without going into the math right now. If I take a "null world line" that comes straight from the big bang, then that world-line has a proper time of zero. You go out to that distance, and take away one meter, you're going to be in a region of incredibly high density. You go to 1 millimeter away from the null world-line, and you'll get many orders of magnitude higher of density. You go 1 micrometer, it will be may orders higher density still; you go to 1 nanometer, many more orders of magnitude. I could go on forever, but I hope you get the picture. The density is just a function that asymptotically approaches infinity as you approach the light ray. You can't say precisely what the density is AT the singularity.

I suspect that what you mean is the former (the infinite density only occurs at a single point), in which case I would strongly discourage using the language "due to time dilation, that instant is effectively forever", since that language promotes exactly the confusion that I described. The instant of infinite density in your model, I suspect, is supposed to be a single point; it's only the particular coordinates you've chosen that distort that single point into what appears to be a null cone. Physically, it's a single point.

I meant both. Why is it appropriate to map the singularity to a straight line, as is done in the "comoving" conformal mapping, but it is not appropriate to map the singularity to a light cone? The infinite density occurs at a single point in the Friedmann Walker Diagram (one single event), but in both the standard model, and in the Milne model, that single event is stretched out to infinity.

However, saying that the infinite density is only for a single point does *not* make it physically reasonable. It only means that your model "predicts its own downfall", just as GR does for spacetimes in which a singularity occurs. Your model is telling you that it can't cover that point--you need some new physics to tell you what happens there. The FRW models in GR handle that, as has been mentioned before, by *not* claiming to cover events all the way to the initial singularity; they only cover things back to (roughly) the beginning of the inflationary phase, where the universe was very small, hot, and dense, but not infinitely so. In your model, that would correspond (as you note) to drawing a hyperbola very *close* to the "null cone" boundary, corresponding to the (very small but not zero) proper time of the beginning of inflation (let's say), and saying that that's the actual physical limit of what your model covers; before that, new physics is needed to say what happened and how the initial state in your model came to be, just as with the FRW models.

I'm not sure if you're being entirely fair. The model I'm describing leaves out a singularity. The model you're describing leaves out a singularity plus some additional time including the inflationary stage.

You want me to stop at some finite time after the big bang. But I don't want to stop there. You name a time after the big bang, and I will name an earlier time after the big bang. No matter what time I choose, the universe has a density at that time. If you choose ZERO, then, I'm stuck. You're right. That is outside the scope of the theory. But it is also outside the scope of General Relativity, so what is the difference?

No, the EFE doesn't need to be reworked, as I noted in a previous post. (Small nitpick: the usual term is "Einstein Field Equation", singular, even though it does have multiple tensor components. The tensors on each side of the EFE are considered single geometric objects, just as we normally write vector equations as relating single objects, even though vectors have multiple components.)

But that's actually not required anyway for what I was asking for. I said "integrate the metric", and you already have an expression for the metric; how that metric arises from the EFE is a separate issue. Integrate that expression over the appropriate range of coordinate values, and see what you come up with.

You asserted that the EFE doesn't need to be reworked. I asserted that the EFE does need to be reworked, and backed up that statement, by pointing out that the EFE's assume a finite number of comoving particles. You asserted that the case I'm talking about has been tried, but I told you I couldn't find it anywhere in whatever you linked to. (By the way, I've been, off and on, looking since 2001 to try to find one "respectable" text on General Relativity that considered this case, and asking for it, and the only place I have found it is in Milne. So it is not a surprise to me if you can't find it on the Baez site.)

So, actually I don't have the metric. I would guess that the metric is {{-1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}} because of symmetry. And yes, it's quite easy to show that the proper distance in this case is finite.

 

[PeterDonis]

I meant both. Why is it appropriate to map the singularity to a straight line, as is done in the "comoving" conformal mapping, but it is not appropriate to map the singularity to a light cone? The infinite density occurs at a single point in the Friedmann Walker Diagram (one single event), but in both the standard model, and in the Milne model, that single event is stretched out to infinity.

In a Mercator projection of the Earth's surface, the North and South Poles are mapped to infinitely long lines. But physically, they're not lines; they're points. The "comoving" conformal mapping does something similar to the initial singularity: it takes what is physically a single point and maps it into an infinitely long line. As I understand it, your model does something similar, except that instead of taking what is physically a single point (like the Earth's North Pole) and mapping it into a line, it maps it into a null cone. I'm just trying to understand if you agree that that's what your model is doing, or if you intend it to be doing something else.

You want me to stop at some finite time after the big bang. But I don't want to stop there. You name a time after the big bang, and I will name an earlier time after the big bang. No matter what time I choose, the universe has a density at that time. If you choose ZERO, then, I'm stuck. You're right. That is outside the scope of the theory. But it is also outside the scope of General Relativity, so what is the difference?

Again, I'm just trying to get clear about whether or not your model actually claims that infinite density is physically reasonable. You were saying earlier that it was; now your comments here indicate that it is not--that your model can get arbitrarily close to the infinite density point (but at any such arbitrarily close point, the density will still be finite), but does not actually cover the infinite density point. As long as the latter is the case, I have no issue. (How close you actually want to get to the initial singularity in your model based on the data you have is a separate issue--the FRW models *can* be extended back arbitrarily close to the singularity, just as yours can; the reason they usually aren't is only because we estimate that the start of the inflation phase is where the new physics we need will actually end up coming in. We won't know for sure until we know what that new physics is.)

You asserted that the case I'm talking about has been tried, but I told you I couldn't find it anywhere in whatever you linked to.

Read my post #103 again. I described there how the page I linked to, in which you claimed you couldn't find anything relevant, *is* relevant. If you can't respond to the specifics I gave there, other than just saying "I can't find anything that applies", I don't think we can get any further on this particular point.

So, actually I don't have the metric. I would guess that the metric is {{-1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}} because of symmetry. And yes, it's quite easy to show that the proper distance in this case is finite.

The metric you just gave is a representation in matrix form of the line element:

ds^{2} = - dt^{2} + dx^{2} + dy^{2} + dz^{2}

This is the standard Minkowski metric, which covers a range of minus infinity to plus infinity for all four coordinates (t, x, y, z). That doesn't correspond to the diagram you drew, which only covers a limited range of coordinate values; so the above metric can't be the one that applies to your diagram.

 

[JDoolin]

Regarding Density:

You're asking whether infinite density is physically reasonable. Well, density is defined as a number of particles per unit area. Even though the local density at any point is finite by my description, If you take the average density of the universe,

\frac{\int \rho dV}{V}

at any given time, yes, the top will be infinite, and the bottom will be finite. Yes, an infinite density of the universe is reasonable. The average density of the universe is always infinite, because you always have an infinite number of particles within a finite distance.

So there are different meanings of an infinite density. You can have

(1) an infinite number of particles in a finite volume, (All the time in the Milne model)

or (2) a finite number of particles in a zero volume (never happens in the Milne model),

or (3) an infinite number of particles in a zero volume (approached asymptotically at r=c*t)


Regarding Mapping the Singularity to a Straight Line vs a Null Light Cone.

I erred, I think, which is leading to some confusion.

We have (1 FWD) the Friedmann Walker Diagram, (2 CPD) the comoving particle conformal mapping Diagram, and (3 MMD) the Milne/Minkowski conformal map.

Earlier I said

http://www.astro.ucla.edu/~wright/cosmo_02.htm[/url] ), I looked again, and the speed of light, as drawn, is definitely a non-zero slope at the singularity. That means that it is not a conformal mapping with the image below it, because it does not cross every worldline.) As for me, I think that the speed of light at that singularity point should be faster than the speed of light for any particle at that point. So I think the line should be horizontal there.
"

My error was in thinking that ONCE that line was horizontal at the origin, that it can turn. In fact, the mapping is not from the singularity to the light cone, but from this horizontal line to the lightcone.

You might well ask why this light does not bend back toward the infinite density behind it, which is a very good question, so my answer is going to seem glib. That lightcone surface is not causally connected to anything within the lightcone. If I send a beam of light to chase after a beam of light, it will never reach it.

You can receive light from any matter that is asymptotically close to the null light-cone, but the light-cone, itself is not "covered" by the model.
In other words, there is no straight line speed-of-light path from the big bang to any event inside the lightcone. Only events immediately following the big-bang can ever be observed within the lightcone.

I think this is actually a pretty reasonable conclusion, though it is contrary to the standard model. We shouldn't be able to see the Big Bang itself, since it already happened in our reference frame. But we can see events immediately after the Big Bang.


Regarding whether Baez considered this topic:

The page I linked to describes a ball B which is "expanding at t = 0". That ball corresponds to your particles moving apart with v = d/t. Baez says that we can't directly apply his equation (2) (which is his statement of the EFE) to ball B because its particles aren't at rest relative to each other at time t = 0. He then defines a second ball, B', which *does* have all its particles at rest relative to each other at time t = 0, and which has the same radius and the same acceleration as ball B at time t = 0. This allows him to apply his equation (2) to ball B', and then show that the equation he derives for the radius r of ball B' vs. time, *also* holds for the radius R of ball B vs. time. So he's showing that the EFE *does* apply to the type of ball you defined, where the particles are moving apart at time t = 0.

I'm sorry. I still don't see it on the page http://math.ucr.edu/home/baez/einstein/node7.html" . Maybe there's something in the equations that I'm missing. I kind of skimmed over them, looking for something in the text. If you see it is there in the math, or I am looking on the wrong page, I need more information.

 

[PeterDonis]

Well, density is defined as a number of particles per unit area. Even though the local density at any point is finite by my description, If you take the average density of the universe,

\frac{\int \rho dV}{V}

at any given time, yes, the top will be infinite, and the bottom will be finite.

I assume you mean "number of particles per unit volume" (not "area"), and that V is a spatial volume. How is the corresponding volume element dV related to a product of coordinate differentials (dx, dy, dz, or whatever spatial coordinates you are using) in your diagram? This question is related to my question about coordinate distance vs. proper distance and what metric you are using (since the metric and the volume element are related). Remember my earlier comment that the metric that applies to your diagram can't be the ordinary Minkowski metric, since whatever coordinates the diagram is drawn in do not have infinite ranges.

My error was in thinking that ONCE that line was horizontal at the origin, that it can turn. In fact, the mapping is not from the singularity to the light cone, but from this horizontal line to the lightcone.

This sounds basically the same as what I said, just with an extra mapping step, so to speak: the mapping from FWD to CPD maps a point (the initial singularity) into a horizontal line, and then you do a mapping from CPD to MMD that maps the horizontal line into a null cone.

Regarding whether Baez considered this topic:

Read the first two sentences from my post (that you quoted) again: The page I linked to describes a ball B which is "expanding at t = 0". That ball corresponds to your particles moving apart with v = d/t. Do you see how ball B here covers the case you were considering?

Now read these sentences again: He then defines a second ball, B', which *does* have all its particles at rest relative to each other at time t = 0, and which has the same radius and the same acceleration as ball B at time t = 0. This allows him to apply his equation (2) to ball B', and then show that the equation he derives for the radius r of ball B' vs. time, *also* holds for the radius R of ball B vs. time. Do you see how this shows that the EFE covers the case you were considering?

 

[JDoolin]

I assume you mean "number of particles per unit volume" (not "area"), and that V is a spatial volume. How is the corresponding volume element dV related to a product of coordinate differentials (dx, dy, dz, or whatever spatial coordinates you are using) in your diagram? This question is related to my question about coordinate distance vs. proper distance and what metric you are using (since the metric and the volume element are related). Remember my earlier comment that the metric that applies to your diagram can't be the ordinary Minkowski metric, since whatever coordinates the diagram is drawn in do not have infinite ranges.

From http://casa.colorado.edu/~ajsh/sr/wheel.html we have the lovely animation:

Andrew Hamilton has a nice diagram of the effect of Lorentz Transformation around the event at the origin.

This diagram is pure Minkowski diagram.

What you see in this diagram is events from the top light-cone being moved to other places in the top lightcone. What you never see is an event passing down from the future lightcone past the null light cone.

The transformation keeps on going and going and going forever, with each of an infinite number of worldlines blissfully thinking that it is at the center of that lightcone.

Now, there may or may not be anything outside that lightcone but if the model is correct, then it does not matter, because in order to get intothe lightcone from outside the lightcone, one would need to push past an infinite number of particles.

Still, this is the "ordinary Minkowski Metric." It's just that the entire observable universe lies within a single lightcone of the Minkowski Spacetime. The universe that we know only occupies a fraction of the Minkowski spacetime.

This sounds basically the same as what I said, just with an extra mapping step, so to speak: the mapping from FWD to CPD maps a point (the initial singularity) into a horizontal line, and then you do a mapping from CPD to MMD that maps the horizontal line into a null cone.

But there are differences in how that initial singularity is mapped.
Namely: The CPD maps the initial singularity (0,0) to a horizontal line, while the MMD maps the horizontal line through the singularity (0,0) to a light-cone. That represents the surface of \tau^2=t^2-r^2=0 The whole lightcone represents a singularity in the \tau variable. In the MMD representation, the (0,0) point remains a single event.

If I'm not mistaken, the CPD simply does not recognize the horizontal line in the FWD. But how would the horizontal line be mapped, if it were? Would it become a vertical line "at infinity?"

Read the first two sentences from my post (that you quoted) again: The page I linked to describes a ball B which is "expanding at t = 0". That ball corresponds to your particles moving apart with v = d/t. Do you see how ball B here covers the case you were considering?

Now read these sentences again: He then defines a second ball, B', which *does* have all its particles at rest relative to each other at time t = 0, and which has the same radius and the same acceleration as ball B at time t = 0. This allows him to apply his equation (2) to ball B', and then show that the equation he derives for the radius r of ball B' vs. time, *also* holds for the radius R of ball B vs. time. Do you see how this shows that the EFE covers the case you were considering?

Thank you for clarifying what you saw.

http://math.ucr.edu/home/baez/einstein/node7.html] [/URL]
Suppose that, at some time t=0, she identifies a small ball B of test particles centered on her. Suppose this ball expands with the universe, remaining spherical as time passes because the universe is isotropic.

Okay, I see this now, but can you understand why I didn't recognize it as the same case? This example says the ball "expands with the universe." When I read that, I interpreted it to mean "It is a ball which stretches as the universe stretches." I see now that Baez actually meant to talk about something else, but I can't see how you get v = d/t out of this. He literally says later there is "nothing special" about time, t=0. If you have v=d/t, then you should have infinite density at t=0.

http://math.ucr.edu/home/baez/einstein/node7.html] [/URL]
Let R(t) stand for the radius of this ball as a function of time. The Einstein equation will give us an equation of motion for R(t). In other words, it will say how the expansion rate of the universe changes with time.

It is tempting to apply equation (2) to the ball , but we must take care.

See, he's already attempting to apply the EFE before he even gives the simple answer. He never says, R(t)= v_{max} t + R_0, so I would say he is not really considering the idea at all; not in any clear mathematical sense.

http://math.ucr.edu/home/baez/einstein/node7.html] [/URL]

This equation applies to a ball of particles that are initially at rest relative to one another -- that is, one whose radius is not changing at . However, the ball B is expanding at t=0. Thus, to apply our formulation of Einstein's equation, we must introduce a second small ball of test particles that are at rest relative to each other at t=0.

What has he done? He has said we can't simply apply the Einstein Field Equations. We have to define our coordinate system, THEN apply the Einstein Field Equations.

To me, nothing seems to have changed. He is still using the EFE's, but he gave a tiny bit of lip-service to the idea of not using them. He sure didn't start over from scratch, assuming d=v/t.

 

[PeterDonis]

Andrew Hamilton has a nice diagram of the effect of Lorentz Transformation around the event at the origin.

Yes, I understand how all this works. (I find all of the diagrams on Hamilton's pages to be very helpful, btw.) But the description you give based on this diagram...

It's just that the entire observable universe lies within a single lightcone of the Minkowski Spacetime. The universe that we know only occupies a fraction of the Minkowski spacetime.

...is *not* compatible with the description you give a little later concerning how the initial singularity is mapped:

Namely: The CPD maps the initial singularity (0,0) to a horizontal line, while the MMD maps the horizontal line through the singularity (0,0) to a light-cone. That represents the surface of \tau^2=t^2-r^2=0 The whole lightcone represents a singularity in the \tau variable. In the MMD representation, the (0,0) point remains a single event.

If the observable universe is only a fraction of Minkowski spacetime, as you say above, then when you draw a diagram showing a light cone emerging from the initial singularity (the "big bang" event), the light cone is not a "mapping" of the big bang event, or anything else; it's just showing the limiting case of light rays going outward from the big bang event in opposite directions. In other words, the light cone is the boundary of the portion of Minkowski spacetime that is causally connected to the big bang event, but that certainly doesn't mean the light cone "is" the big bang event, or is a "mapping" of the big bang event into some other set of coordinates. You're using Minkowski coordinates throughout, and you're saying that we, now, on the Earth, lie on some timelike worldline in a Minkowski spacetime that, in the far past, met up with all the other timelike worldlines in the "universe" we observe, at an event called the "big bang". All those worldlines must lie within the future light cone of that big bang event, so you've just drawn in the future light cone to mark the spacetime boundary of "our universe", as opposed to the rest of the infinite Minkowski spacetime.

It is true that all points on the light cone have zero spacetime interval from the big bang event, but as I said in an earlier post, that does *not* make the entire light cone a single event! There are other ways of assigning coordinates in Minkowski spacetime that make it clear that there are distinct events on light cones. This is in contrast to the "mapping" that is done in, for example, the "CPD"--see next comment.

Also, in your model, there is nothing preventing other "big bang-like" events from happening *outside* the "big bang" light cone (after all, there is an infinite Minkowski spacetime for events to happen in), and sending signals (light rays or timelike worldlines) *into* the "big bang" light cone that represents our universe. As far as I can tell, your model doesn't account for that at all.

If I'm not mistaken, the CPD simply does not recognize the horizontal line in the FWD. But how would the horizontal line be mapped, if it were? Would it become a vertical line "at infinity?"

There actually isn't a "horizontal line" at the bottom of the FWD (I assume that's what you were asking about). There are lines going left and right that approach the horizontal as close as you like, but there are none that are exactly horizontal. In the CPD, the lines approaching the horizontal are mapped to vertical lines on the left and right sides, further and further out from the center. In the "special relativistic" diagram on Wright's page, those lines are mapped to "timelike" lines that get closer and closer to the bounding light cone, without ever reaching it.

The point about all three of these mappings is that the "singularities"--the single point at the "bottom" of the FWD, the line at the bottom of the CPD, and the bounding light cone of the "SR" diagram--are true *boundaries* of the entire spacetime: there is *no* spacetime at all outside of these boundaries. These diagrams are not embedded in any larger diagrams (for example, there is no "larger" Minkowski spacetime in which the "SR" diagram is embedded); they show the *entire* spacetime, *everything* that physically exists in the model.

Okay, I see this now, but can you understand why I didn't recognize it as the same case? This example says the ball "expands with the universe." When I read that, I interpreted it to mean "It is a ball which stretches as the universe stretches." I see now that Baez actually meant to talk about something else, but I can't see how you get v = d/t out of this. He literally says later there is "nothing special" about time, t=0. If you have v=d/t, then you should have infinite density at t=0.

First, a small clarification: by "t = 0" Baez did not mean the instant of the big bang, but just some arbitrarily defined "origin" of a time coordinate. He says "at some time, t = 0", not "at the time of the big bang, t = 0".

That said, you are correct that, at the actual "t = 0" of the big bang, the density (and hence the spacetime curvature) is infinite, which is why, as I've said before, General Relativity "predicts its own downfall" whenever a spacetime singularity--a point of infinite density, infinite curvature, etc.--occurs. The EFE becomes mathematically singular at those points, so we can't use it to predict what happens. But we can get as close to the singularity as we like, and everything will still be finite, so the EFE, mathematically, works just fine. (Whether it still gives predictions that are accurate physically is a separate question--we don't know at this point because we can't make any observations of regimes where the density, curvature, etc. gets large enough to be comparable to what the EFE says it would have to have been, for example, before the inflation phase started.)

See, he's already attempting to apply the EFE before he even gives the simple answer. He never says, R(t)= v_{max} t + R_0, so I would say he is not really considering the idea at all; not in any clear mathematical sense.

Yes, he is; you're missing the point of his argument. He is saying that (provided we are not at a spacetime singularity, as I noted above) *the initial velocities of the particles in the ball don't matter*, because the EFE relates the ratio of *acceleration* and *radius* of the ball to the density and pressure at the center--velocity doesn't enter into it at all. So we can apply the EFE to a ball with *any* distribution of initial velocities, using the method he describes.

 

[JDoolin]

When I said "A singularity in the tau variable," I think you over-interpreted my meaning. I meant that the value of tau is zero everywhere on the light-cone, but I did not mean that having the same tau represented the same event. \tau^2=t^2-r^2=0 means every event where t=r. That includes more than the big bang event.

The lightcone, even though tau is zero everywhere on it, still does have a distinct before and after. A photon has a source and a destination; a cause and an effect. (Actually, I don't know if these photons have a destination, but the Big Bang is the Source.)

The light cone is the set of events for whom the space-time-interval between them and the big bang is zero. It IS a mapping: the HORIZONTAL PLANE THROUGH (0,0) in the FWD is mapped to the LIGHT CONE in the MMD. It doesn't map an EVENT to the lightcone, it maps the WHOLE PLANE to the lightcone.

However, in the CPD (Comoving Particle Diagram), according to what (I think) you've been saying, the big bang itself is mapped to the horizontal line, and the horizontal line from the FWD is not mapped anywhere! In the CPD, you have a SINGLE EVENT from the FWD which is mapped to MULTIPLE LOCATIONS in the Comoving Particle Diagram. That means that you'll see light from the same event at multiple times. Can you see the difference? There are no "single event" mapped to multiple places in the MMD. There IS a "single event" mapped to multiple places in the CPD. In the CPD, there is always light on its way from the big bang, because this single event has been mapped to an infinite number of locations. It might be hard to switch gears between these two models. In your model, that horizontal line in the FRW metric simply doesn't exist. It's effectively, the vertical line "at" infinity in the CPD. In your model, the Big Bang wasn't a single event; it was a huge number of events which all occurred simultaneously throughout the universe.

In my model (The MMD) the Big Bang was a single event, that very likely, produced an infinite number of infinite intensity photons which mark the outside edge of the expanding sphere I call our universe. The horizontal line in the FRW metric does exist in my model, and it's that massive sphere of radiant energy exploding into a Minkowski Space.

In the MMD, the light from the big bang is gone... long gone. It might hit something OUTSIDE our universe, but it is not coming back in. What we are seeing is not light from the big bang, but light from matter billions of years after the big bang, (time dilated so that it seems instants after the big bang.) (Technically, we see light from the Hydrogen Recombination era, thousands of years after the big bang).

Also, in your model, there is nothing preventing other "big bang-like" events from happening *outside* the "big bang" light cone (after all, there is an infinite Minkowski spacetime for events to happen in), and sending signals (light rays or timelike worldlines) *into* the "big bang" light cone that represents our universe. As far as I can tell, your model doesn't account for that at all.

You're right, for the most part. Milne does bring this up if you have the e-book. He found it rather troubling, as do I, that there was nothing preventing other objects from being outside. Unless they are other Big Bang's though, there isn't much to worry about. The model really is infinite in mass and energy; it's not "really really big; so big we might as well call it infinite." in other words, before something from outside got to you, it would have to pass an infinite number of particles. And if you think of a particle right at the edge, then it has to pass an infinite number of particles before it gets to that particle at the edge, etc.

The only thing you could ever really touch from the outside would be the photon shell. You wouldn't see it coming, of course; it would just be happy pleasant day, and then BAM; instant annihilation.

That being said, I don't know what would happen if we had two Big Bangs in the same Minkowski Spacetime. It would be the unstoppable object meeting the unstoppable object. Just as in GR, you can't model the infinite curvature at t=0, I'm not sure how to model two planes of infinite density colliding at the speed of light.

(By the way, outside this sphere, there may well be infinite curvature, because you have the infinite density coming AT you from ONE direction, instead of the infinite density going AWAY from you in ALL directions. But inside, due to symmetry, I still argue that there is NO curvature.)

Yes, he is; you're missing the point of his argument. He is saying that (provided we are not at a spacetime singularity, as I noted above) *the initial velocities of the particles in the ball don't matter*, because the EFE relates the ratio of *acceleration* and *radius* of the ball to the density and pressure at the center--velocity doesn't enter into it at all. So we can apply the EFE to a ball with *any* distribution of initial velocities, using the method he describes.

Let me tell you what I think the point of his argument is, and then you can tell me I'm missing the point again. We can have a ball of particles that are moving apart. The point of his argument is that this ball he's talking about are the test particles. It doesn't matter whether these test particles are all comoving or if they are all moving apart. They are all going to follow the laws of physics. In regions where the Einstein Field Equations apply, they're going to follow the Einstein Field Equations. So, this should certainly cover situations like the Schwartszchild metric, or the Kerr Metric, or anyplace where we know the general matter distribution in the space around us; we can use that general matter distribution to figure out how to solve for the motions of the matter.

However, if we disagree on the general distribution of the matter around us, for instance, if I think the matter around us approaches an infinite density within a finite distance in all directions, while you think that the density of the universe is the same in all directions, but the majority of it is not yet causally connected to us, then we probably aren't going to agree on what the Einstein Field Equations say, or necessarily on whether they should even be applied.

 

[PeterDonis]

When I said "A singularity in the tau variable," I think you over-interpreted my meaning. I meant that the value of tau is zero everywhere on the light-cone, but I did not mean that having the same tau represented the same event. \tau^2=t^2-r^2=0 means every event where t=r. That includes more than the big bang event.

Ok, good, that makes it clear what the lightcone in your model means--except that this...

The light cone is the set of events for whom the space-time-interval between them and the big bang is zero. It IS a mapping: the HORIZONTAL PLANE THROUGH (0,0) in the FWD is mapped to the LIGHT CONE in the MMD. It doesn't map an EVENT to the lightcone, it maps the WHOLE PLANE to the lightcone.

...is *not* correct if what you said above is true. There is no "horizontal plane through (0, 0)" in the FWD. See my comments further below.

However, in the CPD (Comoving Particle Diagram), according to what (I think) you've been saying, the big bang itself is mapped to the horizontal line, and the horizontal line from the FWD is not mapped anywhere!

This is correct, because, as I just noted, there is *no* "horizontal line" in the FWD (I said this in a previous post as well). The FWD includes lines that get as close to "horizontal" as you like, but none that are exactly horizontal. Yes, that means that this...

In the CPD, you have a SINGLE EVENT from the FWD which is mapped to MULTIPLE LOCATIONS in the Comoving Particle Diagram.

...is true; that's usually how "conformal" diagrams work. (A Mercator projection of the Earth's surface, for example, is "conformal" in this sense--it maps the North and South poles to horizontal lines, not points.) But that does *not* mean that this...

That means that you'll see light from the same event at multiple times.

is true. Consider the Mercator projection again: the lines of longitude (great circles through the poles) are mapped to vertical lines, which *appear* to meet the poles at "different places". But that's an artifact of the "infinite distortion" that the projection makes at the poles. In the same way, the *apparent* "multiple light rays" from the initial singularity in the CPD are an artifact of the "infinite distortion" that this diagram makes at the initial singularity.

How do we deal with this? The only really consistent way is to accept that these "conformal" diagrams *cannot* actually represent the singularities (just as we don't actually use the Mercator projection at the poles). What the "multiple light rays" in the conformal diagram are actually indicating is that, at very short times after the initial singularity, worldlines which emerged from that singularity "in different directions" will be causally disconnected; the more "different" the initial directions are, the longer it will take for the worldlines to become causally connected again. The light that is reaching is now from "close to the big bang" is coming from worldlines that emerged from the big bang in a direction that was "more different" from ours than light that reached us from close to the big bang some time ago.

I realize the above is a somewhat vague description; when I have more time I can try to make it more precise if needed.

In your model, the Big Bang wasn't a single event; it was a huge number of events which all occurred simultaneously throughout the universe.

Strictly speaking, this is false, although it's sometimes used colloquially to describe what I was describing above, that events very close to the big bang happened throughout the universe (in the sense of causal disconnection I gave above).

In the MMD, the light from the big bang is gone... long gone. It might hit something OUTSIDE our universe, but it is not coming back in. What we are seeing is not light from the big bang, but light from matter billions of years after the big bang, (time dilated so that it seems instants after the big bang.) (Technically, we see light from the Hydrogen Recombination era, thousands of years after the big bang).

As I noted above, the FRW models do not claim that we can see light (or any other signal) "from the big bang" itself. The earliest *photons* we can see in the FRW models are, as you say, those from the time of recombination. However, the FRW model would predict that we could see other radiation from earlier times (e.g., neutrinos from the electroweak phase transition, or gravitational waves from even earlier times). We don't currently have any way of testing such predictions because of our poor ability to detect any kind of radiation other than electromagnetic.

You're right, for the most part. Milne does bring this up if you have the e-book. He found it rather troubling, as do I, that there was nothing preventing other objects from being outside. Unless they are other Big Bang's though, there isn't much to worry about. The model really is infinite in mass and energy; it's not "really really big; so big we might as well call it infinite." in other words, before something from outside got to you, it would have to pass an infinite number of particles. And if you think of a particle right at the edge, then it has to pass an infinite number of particles before it gets to that particle at the edge, etc.

I reached this point in the e-book today, as it happens. I still find the "infinite density" part of the model physically unreasonable, but I agree that *if* you stipulate that the density goes to infinity at the "photon shell", there would be no possibility of anything coming in from outside the shell.

That being said, I don't know what would happen if we had two Big Bangs in the same Minkowski Spacetime. It would be the unstoppable object meeting the unstoppable object. Just as in GR, you can't model the infinite curvature at t=0, I'm not sure how to model two planes of infinite density colliding at the speed of light.

This is one reason (but hardly the only reason) that I find the infinite density physically unreasonable.

(By the way, outside this sphere, there may well be infinite curvature, because you have the infinite density coming AT you from ONE direction, instead of the infinite density going AWAY from you in ALL directions. But inside, due to symmetry, I still argue that there is NO curvature.)

Another reason I find the infinite density physically unreasonable is that it should result in infinite curvature at the "photon shell", which, aside from any other objections, would contradict the initial assumption of a flat background Minkowski spacetime. Even if spacetime *inside* the shell were flat (which it could be since that's a general result for inside a symmetrical spherical shell even in Newtonian gravity), the *complete* spacetime in which everything is embedded could not be.

Let me tell you what I think the point of his argument is, and then you can tell me I'm missing the point again. We can have a ball of particles that are moving apart. The point of his argument is that this ball he's talking about are the test particles. It doesn't matter whether these test particles are all comoving or if they are all moving apart. They are all going to follow the laws of physics. In regions where the Einstein Field Equations apply, they're going to follow the Einstein Field Equations. So, this should certainly cover situations like the Schwartszchild metric, or the Kerr Metric, or anyplace where we know the general matter distribution in the space around us; we can use that general matter distribution to figure out how to solve for the motions of the matter.

However, if we disagree on the general distribution of the matter around us, for instance, if I think the matter around us approaches an infinite density within a finite distance in all directions, while you think that the density of the universe is the same in all directions, but the majority of it is not yet causally connected to us, then we probably aren't going to agree on what the Einstein Field Equations say, or necessarily on whether they should even be applied.

This is much closer, I think, but I still have a couple of comments:

(1) According to General Relativity, there are no situations where the EFE does not apply. It always does. There are certainly a wide variety of particular *solutions* to the EFE, among which are the various spacetimes we've been discussing (Schwarzschild, Kerr, FRW, etc.), and which specific solution applies in a particular case will depend on the distribution of matter. But the EFE, as the equation to be solved, applies in every case. So if we disagree on the distribution of matter, we may well disagree on which specific solution to the EFE to apply, but if we accept GR, we *have* to agree that the EFE applies. If you don't accept that, you don't accept GR.

(The only caveat to the above is what I've said before about spacetime singularities: there the EFE itself tells us it can't apply. But we can get as close to the singularities as we like and still apply the EFE.)

(2) Specifying a "matter distribution" in order to solve the EFE can be done in a variety of ways; it can, as you say, be "general", but you may not be appreciating just how general it can be. For example, to obtain the FRW solutions, we specify: "The matter distribution is a perfect fluid, and we'll write the solution in coordinates in which that fluid is isotropic." That's all. Similarly, to obtain the Schwarzschild solution, we specify: "There is no matter--the stress-energy tensor is identically zero--and the solution must be spherically symmetric." (As you can see, often our "specification" takes the form of symmetry properties that the solution must satisfy.)