Conditions for spacetime to have flat spatial slices

[JDoolin]

I think some more clarification of terminology is in order; I should have clarified this before since we've been using the term "proper time" in more than one sense.

(1) Proper time *along a particular worldline between two particular events* is an invariant; geometrically, it's the analogue in spacetime of the invariant "length" of a particular line segment in a Euclidean space.

(2) Proper time *along a particular worldline*, without specifying events on the worldline, is a *parameter*: a range of real numbers you can use to label events on the worldline, by arbitrarily assigning some particular event the value 0 and then labeling every other event by its invariant proper time along the worldline from the event with the value 0 (with earlier events having negative proper time and later events having positive proper time).

(3) Once you have a labeling of events on a worldline by the proper time parameter, you can then look for a coordinate system that uses that same event labeling as its time coordinate. If you're really lucky, you can find a coordinate system that does this, not just for one worldline, but for a whole family of worldlines that are picked out by some symmetry property of the spacetime. This is what is meant by "coordinate time directly represents proper time" for a particular family of observers (in the case I've been discussing, the "comoving" observers).

When things are proven mathematically, there is a certain inevitability of the next step in the process. You recognize your axioms and state them clearly, and then those axioms lead inevitably to certain conclusions. Even then, you acknowledge that if your assumptions are false, then your conclusions would also be false.

I find Special Relativity to be an axiomatically sound theory. Namely because the Lorentz Transformations leave the speed of light constant, but they allow for acceleration. But it seems to me that you've kind of nailed it here with General Relativity. "If you're really lucky, you can find a coordinate system that does this"

We start by making an assumption; I don't know what it is--there's no axiom behind General Relativity. If you ask "Are you assuming that the density is the same throughout," the answer is no. If you ask, "Are you assuming that the proper time is some universal parameter," the answer is no. There's no starting point.

You just say, let's assume the coordinate time is equal to the proper time, and then you run with it. "If you're lucky" you find a coordinate system that does this. And hey, you got lucky:

http://www.astro.ucla.edu/~wright/photons_outrun.html

You just need the coordinate system to stretch over time, and you need to have the particles to be appearing to move apart, but it's just an illusion formed by the stretching of space. And then, voila, you've created a system where Special Relativity no longer works. Yay!

So when I ask, why do Robertson Walker think they can set proper time to be coordinate time, I'm also asking, is there anything axiomatic that FORCES them to throw away the results of Special Relativity? Is there some assumption that they made that made the proper-time = coordinate time assumption inevitable?

I don't care how LUCKY they got in coming up with a system that throws away Special Relativity theory. I want to know the assumption they made that requires them to throw away the Special Theory of Relativity.

 

[PAllen]

I don't care how LUCKY they got in coming up with a system that throws away Special Relativity theory. I want to know the assumption they made that requires them to throw away the Special Theory of Relativity.

Not sure if this is really what you're asking, but gravity is what killed Special Relativity. Before, and after General Relativity, there were researchers attempting to come up with consistent gravity+SR, without accepting GR (or other theories with spacetime curvature). Mathematically consistent theories were set up, but they all conflicted with experiment. There are proofs of impossiblity of an SR based theory matching known experiments, but, as you note, proofs always assume something you might find a way to remove.

Anyway since a pretty common view among physiscists is that GR is 'next theory to fall', and given that an SR based gravity would provide a great fit for Quantum Gravity, you could be a hero if you were to produce one.

 

[PeterDonis]

So when I ask, why do Robertson Walker think they can set proper time to be coordinate time, I'm also asking, is there anything axiomatic that FORCES them to throw away the results of Special Relativity? Is there some assumption that they made that made the proper-time = coordinate time assumption inevitable?

I don't care how LUCKY they got in coming up with a system that throws away Special Relativity theory. I want to know the assumption they made that requires them to throw away the Special Theory of Relativity.

I think you're misunderstanding the process of arriving at a model in general relativity. First of all, the FRW solution (like any solution of the Einstein Field Equation) does not "throw away special relativity". At any event in the spacetime, you can set up a local inertial frame in which the laws of SR hold locally. You can't set up a *global* Minkowski coordinate system in which the laws of SR hold that covers the entire spacetime, because the spacetime is curved, and SR assumes a flat spacetime. This is not something unique to the FRW solution; it's true of any solution in GR (except, of course, the trivial solution of Minkowski spacetime itself, a spacetime that's globally flat, zero curvature everywhere).

Second, as I said before, the FRW model does not *assume* that proper time is equal to coordinate time. Let me lay out the steps in the process more explicitly:

(1) We are looking for a solution to the Einstein Field Equation that describes (to some appropriate level of approximation) the universe as a whole.

(2) We observe that, to some appropriate level of approximation (and after correcting for our own peculiar velocity), the universe appears homogeneous and isotropic. Therefore, in looking for a solution to the EFE, we decide to try imposing the condition that the universe be homoegenous and isotropic (meaning, spatially).

(3) The condition of homogeneity and isotropy picks out a certain particular subset of solutions to the EFE. After some mathematical work, we find that that subset of solutions has the property that the metric, by appropriate choice of coordinates, can always be written in the following form:

$$d\tau^{2} = dt^{2} - a(t) \left[\frac{1}{1 - k r^{2}} dr^{2} + r^{2} \left( d\theta^{2} + sin^{2} \theta d\phi^{2} \right) \right]$$

where $t$, $r$, $\theta$, and $\phi$ are the coordinates, and $k$ is a constant that can take one of three values: +1, 0, or -1. (The case I've been discussing up to now, where the spatial slices are flat, is the case k = 0, which makes the metric look like the one I wrote in an earlier post.)

(4) We notice that the metric as written above has the property that, for any worldline with constant values for all the space coordinates, $d\tau^{2} = dt^{2}$. That means three things: first, those worldlines are also integral curves of the t coordinate (since the $dt^{2}$ term is the only term on the RHS); second, the t coordinate directly represents the proper time parameter along those worldlines (since there are no other coefficients on either side of the equation); and third, since the worldlines are timelike, we can define a family of observers moving along those worldlines, who we call "comoving" observers, and we can they say that the t coordinate directly represents their proper time.

So we never *assumed* that coordinate time was equal to proper time; we *discovered* that, if we look for solutions to the EFE that are spatially homogeneous and isotropic, those solutions can be described by a coordinate system in which coordinate time directly represents proper time for "comoving" observers. (That last qualifier, by the way, is key: some of your comments seem to imply that you think the FRW coordinate system somehow has coordinate time equal to proper time period, for *all* observers, which is false. A non-comoving observer--one whose spatial coordinates in the FRW coordinate system change with time--will find that their proper time is *not* represented directly by coordinate time.) Of course this coordinate system is *not* a Minkowski coordinate system such as we would use in SR; it can't be, because it's a global coordinate system covering the entire spacetime, and the spacetime it is describing is curved, and Minkowski coordinates can't globally represent a curved spacetime. That's not "throwing away SR"; it's recognizing that SR has a limited domain of applicability.

A final note on my use of the word "lucky": the "luck" comes in finding a particular symmetry property that allows a particular solution (or set of solutions) of the EFE to be written in a form that looks simple. In the general case, this is not possible. It just so happens that a number of spacetimes of interest in physics *do* happen to have particular symmetry properties that allow us to write them in a form that looks simple.

 

[PeterDonis]

I find Special Relativity to be an axiomatically sound theory. Namely because the Lorentz Transformations leave the speed of light constant, but they allow for acceleration. But it seems to me that you've kind of nailed it here with General Relativity. "If you're really lucky, you can find a coordinate system that does this"

We start by making an assumption; I don't know what it is--there's no axiom behind General Relativity. If you ask "Are you assuming that the density is the same throughout," the answer is no. If you ask, "Are you assuming that the proper time is some universal parameter," the answer is no. There's no starting point.

Roughly, the general assumptions of GR are

(1) That we describe physics by means of invariant geometric objects in spacetime, which is itself a geometric object (a 4-dimensional pseudo-Riemannian manifold with a locally Minkowskian metric);

(2) That if we can express a given physical law in terms of invariant geometric objects in flat spacetime (i.e., in SR terms), that law will continue to hold if we allow the spacetime to be curved (which it must be in the presence of "gravity"), as long as we replace ordinary derivatives by covariant derivatives, which account for the curvature of spacetime.

These general assumptions are enough to get pretty close to the Einstein Field Equation. As far as I know, the shortest route the rest of the way is the one Hilbert discovered, about the same time Einstein was finishing his 1915 paper that announced GR:

(3) We assume that the dynamics of spacetime (or spacetime coupled to matter-energy, if the latter is present) is determined by a principle of least action.

(4) We assume that the appropriate action for spacetime is the unique action (which Hilbert found) that depends only on the metric and its first and second derivatives.

(5) We assume that, if matter-energy is present, we are given its action as well. I'm not sure if there are any general conditions on the matter action other than it being in appropriate Lagrangian form.

This is enough to get us to the Einstein Field Equation.

These are general assumptions, though, not assumptions particular to any specific physical problem. For a particular problem, of course, you will need additional specifications, such as the specific form of the matter action, or equivalently its stress-energy tensor (e.g., a perfect fluid with a particular equation of state--or the specification that you're looking for a vacuum solution, with zero stress-energy tensor), and any particular properties the solution must have (e.g., isotropy or spherical symmetry).

 

[PeterDonis]

And when two pieces of matter cross the same coordinate space, when they reach the same PROPER AGE, that does NOT mean that they are colliding. It means they just happened to reach that same spot in space when they happened to be the same proper age. That does NOT mean they reached the same spot at the same TIME.

So, I repeat, (more emphatically, this time,) that an intersection on a space vs proper time graph, is all but physically meaningless.

Reading back through the thread, I saw this post, which helped me understand better what you meant by "a graph of space vs. proper time". This is definitely *not* what is being shown in any of the spacetime diagrams we have been discussing; the "time" is always *coordinate* time, which in some cases happens to also represent the proper time of a particular family of observers, as I've described.

However, your statement here does bring up a further issue, which is: suppose we have a coordinate system where coordinate time directly represents the proper time of some family of observers, and two of those observers happen to pass through the same spatial point? Wouldn't this raise the question you raise above, about how the coordinate system can possibly represent events accurately, if it's possible for two observers to be at the same point in space at the same *proper* time, but *not* necessarily at the same "time"?

I believe (but see one caveat at the end of this post) the answer is that no such question can ever arise, because whenever you have a coordinate system where coordinate time directly represents the proper time of a family of observers, each such observer has a unique worldline that can never intersect the worldline of any other such observer. This follows from the way the family of observers is picked out: their worldlines are the set of integral curves of the time coordinate. None of those integral curves can ever intersect: each event in the spacetime lies on one and only one such integral curve. So if, for some reason, the spatial coordinates were set up such that two different observers' worldlines both passed through the same spatial point (i.e., the same set of values for the spatial coordinates), they would *have* to do so at different proper times.

Going further along this line, note that we can use these integral curves to label the spatial points, such that each observer in the family of observers, for whom coordinate time directly represents proper time, has their own unique space point (meaning, label for their particular integral curve) at which they remain for all time. If there are three spatial dimensions in the spacetime, then the labels for the integral curves will need to contain three numbers to uniquely specify each curve. This amounts to finding a set of spatial coordinates that "matches up" with the time coordinate, in the sense that each set of unique values for the spatial coordinates is paired up one-to-one with a unique worldline in the family of integral curves of the time coordinate (and thus with a unique observer in the family of observers). So we can always find a coordinate system that not only has coordinate time directly representing proper time for a particular family of observers, but also has each observer "at rest" at his own unique spatial point for all time. The Robertson-Walker coordinates we've been discussing are such a coordinate system, with respect to the "comoving" observers.

Note: The one caveat I referred to above is that there may be singular "events" at which integral curves of the time coordinate can "intersect". For example, the initial singularity (the Big Bang) in FRW spacetime is such an event--all timelike worldlines originate there. However, the word "singularity" is key: technically, the initial singularity is not actually "part of the spacetime", because the curvature is infinite there (also, the spatial part of the metric collapses to zero, meaning that "space" at the initial singularity is no longer three-dimensional, but a single point). Physically, this means GR breaks down at the singularity; we need new physics (e.g., a quantum theory of gravity) to understand what's going on there. But leaving out singularities, I believe what I said above is correct.

 

[PAllen]

Note: The one caveat I referred to above is that there may be singular "events" at which integral curves of the time coordinate can "intersect". For example, the initial singularity (the Big Bang) in FRW spacetime is such an event--all timelike worldlines originate there. However, the word "singularity" is key: technically, the initial singularity is not actually "part of the spacetime", because the curvature is infinite there (also, the spatial part of the metric collapses to zero, meaning that "space" at the initial singularity is no longer three-dimensional, but a single point). Physically, this means GR breaks down at the singularity; we need new physics (e.g., a quantum theory of gravity) to understand what's going on there. But leaving out singularities, I believe what I said above is correct.

Curious about this. Are you saying if you remove singular points, you can construct a single coordinate patch (not just any, but a nice one!) covering any solution of GR? That there are no GR solutions analagous to a sphere, which cannot be covered in one patch but has no singularity of any kind? Or were you implicitly referring to some class of cosmologic solutions that have this property?

 

[PeterDonis]

Curious about this. Are you saying if you remove singular points, you can construct a single coordinate patch (not just any, but a nice one!) covering any solution of GR? That there are no GR solutions analagous to a sphere, which cannot be covered in one patch but has no singularity of any kind?

No, I wasn't saying that. I was only saying that within a given coordinate patch, leaving out singularities, what I said was (as far as I know) true. However, on thinking it over, I realize that there are some subtleties, which are worth mentioning, if I may be permitted to "think out loud" for a little.

The specific example I used, that of FRW spacetime, *does* have the property that a single coordinate patch can be used to cover the entire spacetime, and what I said in my previous post *is* true for FRW spacetime, without doubt (I pretty much gave the explicit construction).

Consider another simple example, Schwarzschild spacetime. The "obvious" coordinate patch in which what I said in my last post is true is the Schwarzschild exterior coordinates, for which the Schwarzschild time coordinate fulfills the requirements I gave. However, this patch does not cover the horizon or the region inside the horizon. A coordinate patch which does cover the interior region is the Schwarzschild interior coordinates; in this coordinate patch, the t coordinate is spacelike (so it can't be used as a "time" coordinate there) and the r coordinate is timelike. I think, but am not certain, that the r coordinate meets the requirements in the interior region; to be certain, I'd have to think some more about what the surfaces of constant r look like and whether a set of integral curves orthogonal to those surfaces can be used to uniquely label spatial points in the surfaces. I think they can, but I'm not certain.

There are, of course, other coordinate systems that can be used to describe Schwarzschild spacetime. The Painleve coordinates cover both the exterior and the "future interior" region, and the Painleve time coordinate satisfies the conditions (as long as we include the bit about leaving out singularities, since each integral curve of Painleve time ends at the central singularity at r = 0--this is also true for the r coordinate of the Schwarzschild interior coordinates). If we restrict ourselves to talking about black holes that form as a result of the collapse of a massive object (such as a star), then the two regions I just mentioned cover the entire spacetime in question.

However, if we allow ourselves to consider mathematical solutions that may or may not be "acceptable" physically, there is a "maximal analytic extension" of Schwarzschild spacetime which includes two more regions, the "past interior" region and the "second exterior" region. This maximal analytic extension is described by Kruskal coordinates, and one version of these (the version that does not use null coordinates) has a timelike coordinate that I believe meets the requirements I gave, although again I'm not certain. Assuming it does, the integral curves of this time coordinate all have finite endpoints both past and future, on the past and future singularities, so again the bit about leaving out singularities is needed.

Assuming that what I've said so far is correct, I believe similar comments would apply to the other "black hole" spacetimes, the most general of which is the Kerr-Newman spacetime (which includes all other "black hole" spacetimes, including Schwarzschild, as special cases).

However, I realize that there are a *lot* of other possible spacetimes that are solutions to the EFE. From the examples above, I think there may be at least two other conditions that would need to be satisfied for what I said to be true: first, the spacetime would need to be time orientable (no closed timelike curves, so something like the Godel solution, which has CTCs, would not work); second, I think the spacetime would need to be simply connected (no "wormholes" or other topological anomalies).

 

[PeterDonis]

The Painleve coordinates cover both the exterior and the "future interior" region, and the Painleve time coordinate satisfies the conditions (as long as we include the bit about leaving out singularities, since each integral curve of Painleve time ends at the central singularity at r = 0--this is also true for the r coordinate of the Schwarzschild interior coordinates).

On further consideration, there's another subtlety here. The integral curves of the Painleve time coordinate are non-intersecting, so that's all right; but they do *not* have constant values of the Painleve radial coordinate r. So if we wanted to use these curves to uniquely label "spatial points", we would have to use some other labeling that didn't involve the Painleve r coordinate. There are such labelings: the simplest one I can think of is to label each integral curve of Painleve time by the event at which that curve crosses the horizon at r = 2M; we can use, for example, the Kruskal spatial coordinate X to uniquely label each such event.

So I do have to add another qualification to what I said: the spatial coordinates that "match up" with the time coordinate, so that the integral curves of the time coordinate have constant values of the spatial coordinates for all time, may *not* be the same as the spatial coordinates of the coordinate system that gave rise to the time coordinate!

 

[PAllen]

Consider another simple example, Schwarzschild spacetime. The "obvious" coordinate patch in which what I said in my last post is true is the Schwarzschild exterior coordinates, for which the Schwarzschild time coordinate fulfills the requirements I gave. However, this patch does not cover the horizon or the region inside the horizon.

Of course, coordinate time in these coordinates is not actual proper time for any observer except at r=infinity. You used an interesting phrasing earlier:

"suppose we have a coordinate system where coordinate time directly represents the proper time of some family of observers"

Do you mean something other than equals for "directly represents"? Or are you thinking of some simple transform of the standard Schwarzschild coordinates that normalized t to equal tau ?

 

[George Jones]

Let $T$ and $R$ be Painleve coordinates.

On further consideration, there's another subtlety here. The integral curves of the Painleve time coordinate are non-intersecting, so that's all right; but they do *not* have constant values of the Painleve radial coordinate r.

Other subtleties:

1) even though $R = r$, $\partial / \partial R \neq \partial / \partial r$, so their integral curves are quite different;

2) even though $T \neq t$, $\partial / \partial T = \partial / \partial t$, so their integral curves are the same;

3) the worldline of an observer who falls freely from infinity is not an integral curve of $\partial / \partial T$;

4) the integral curves of $\partial / \partial R$ intersect orthognally the worldline of an observer who falls freely from infinity.

Here, $t$ and $r$ are standard Schwarzschild coordinates.

1) is an example of what Penrose calls Woodhouse's Second Fundamental Confusion of Calculus. Even though $R = r,$ $\partial / \partial R$ is not the same as $\partial / \partial r$ because lines of constant $(T,\theta,\phi)$ are not the same as lines of constant $(t,\theta,\phi)$. For lines of constant $(R = r,\theta,\phi)$, $T$ and $t$ differ by a constant, and hence 2). Because $\tau = T$ on the worldline, 3) might seem a little odd. However, $g \left( \partial / \partial T , \partial / \partial T \right) = 1 - 2M/R \neq 1$, so integral curves of $\partial / \partial T$ cannot be worldlines of observers. The 4-velocity of an observer who falls freely from infinity,

$$\bf{u} = \frac{\partial}{\partial T} - \sqrt{\frac{2M}{R}} \frac{\partial}{\partial R},$$

is used to derive 4).

If anyone wants, I can elaborate mathematically on the above.

 

[JDoolin]

I think you're misunderstanding the process of arriving at a model in general relativity. First of all, the FRW solution (like any solution of the Einstein Field Equation) does not "throw away special relativity". At any event in the spacetime, you can set up a local inertial frame in which the laws of SR hold locally. You can't set up a *global* Minkowski coordinate system in which the laws of SR hold that covers the entire spacetime, because the spacetime is curved, and SR assumes a flat spacetime. This is not something unique to the FRW solution; it's true of any solution in GR (except, of course, the trivial solution of Minkowski spacetime itself, a spacetime that's globally flat, zero curvature everywhere).

This is a major part of what I can't understand. The Effect of the Lorentz Transformations are essentially directly proportianal to distance in space and time. i.e. If you Lorentz Transform an event that is 2 light years away, the effect will be roughly twice as much as if you Lorentz Transform an event that is 1 light-year away.

If you do a Lorentz Transform on an event that is a billion light years away, the effect is roughly a billion times as much as if you do an LT on an event that is 1 light year away. Maybe I'm misinterpreting what you're saying when you say "the laws of SR hold locally." The way I'm taking your meaning is that you can Lorentz Transform events within a certain radius in spacetime, but events beyond that radius are not Lorentz Transformed.

Maybe you mean something different by "the laws of SR hold locally."

Second, as I said before, the FRW model does not *assume* that proper time is equal to coordinate time. Let me lay out the steps in the process more explicitly:

(1) We are looking for a solution to the Einstein Field Equation that describes (to some appropriate level of approximation) the universe as a whole.

(2) We observe that, to some appropriate level of approximation (and after correcting for our own peculiar velocity), the universe appears homogeneous and isotropic. Therefore, in looking for a solution to the EFE, we decide to try imposing the condition that the universe be homoegenous and isotropic (meaning, spatially).

But you reject the possibility that maybe space ISN'T stretching, and you reject the possibility that perhaps there was an era of non-uniform acceleration, and you reject the possibility that the galaxies might actually be moving apart. And you reject the possibility that Lorentz Transformations might actually work on the long range.

(3) The condition of homogeneity and isotropy picks out a certain particular subset of solutions to the EFE. After some mathematical work, we find that that subset of solutions has the property that the metric, by appropriate choice of coordinates, can always be written in the following form:

$$d\tau^{2} = dt^{2} - a(t) \left[\frac{1}{1 - k r^{2}} dr^{2} + r^{2} \left( d\theta^{2} + sin^{2} \theta d\phi^{2} \right) \right]$$

where $t$, $r$, $\theta$, and $\phi$ are the coordinates, and $k$ is a constant that can take one of three values: +1, 0, or -1. (The case I've been discussing up to now, where the spatial slices are flat, is the case k = 0, which makes the metric look like the one I wrote in an earlier post.)

So it is an available option to just set a(t)=1 and k=0, right? So Minkowski spacetime actually is a possible solution to the Einstein Field Equations? And then we don't have to throw away Special Relativity.

(4) We notice that the metric as written above has the property that, for any worldline with constant values for all the space coordinates, $d\tau^{2} = dt^{2}$. That means three things: first, those worldlines are also integral curves of the t coordinate (since the $dt^{2}$ term is the only term on the RHS); second, the t coordinate directly represents the proper time parameter along those worldlines (since there are no other coefficients on either side of the equation); and third, since the worldlines are timelike, we can define a family of observers moving along those worldlines, who we call "comoving" observers, and we can they say that the t coordinate directly represents their proper time.

Ah, I see. Yes, of course, if we have comoving observers, then of course, they would all share the same proper time. But it is one thing to define a family of observers moving along those parallel worldlines. It is quite another to claim that the galaxies in the real universe are a family of observers moving along those worldlines.

So we never *assumed* that coordinate time was equal to proper time; we *discovered* that, if we look for solutions to the EFE that are spatially homogeneous and isotropic, those solutions can be described by a coordinate system in which coordinate time directly represents proper time for "comoving" observers. (That last qualifier, by the way, is key: some of your comments seem to imply that you think the FRW coordinate system somehow has coordinate time equal to proper time period, for *all* observers, which is false. A non-comoving observer--one whose spatial coordinates in the FRW coordinate system change with time--will find that their proper time is *not* represented directly by coordinate time.) Of course this coordinate system is *not* a Minkowski coordinate system such as we would use in SR; it can't be, because it's a global coordinate system covering the entire spacetime, and the spacetime it is describing is curved, and Minkowski coordinates can't globally represent a curved spacetime. That's not "throwing away SR"; it's recognizing that SR has a limited domain of applicability.

I'm sorry. Unintentionally, I've been switching back and forth between two ideas, and now there are three. The third idea is what you are explaining, that a set of comoving observers share a proper time, and that proper time is the same as their coordinate time. That's valid. The first idea is that those comoving observers are the galaxies in the real universe, who only appear to be moving apart because of the stretching of space. (That's wierd, but not where my argument was coming from.) What I was thinking about were galaxies moving apart from each other, with real recessional velocities, whose proper times were all different. In this environment, it would be ridiculous to simply set proper time equal to coordinate time, because the galaxies wouldn't be comoving.

A "limited domain" of applicability for SR seems to me, the same as "throwing it away." If I told you that "rotation" had a limited domain of applicability, it would mean that if you turn to the left or right, only nearby objects respond. Things in your room might change positions relative to your facing, but distant stars would not cooperate; they would remain in the same place; stubbornly remaining in front of you as you spin around, because rotation is "only valid locally".

If you say "SR is valid only locally" you're saying that only nearby objects are affected by the Lorentz Transformations. It is absurd. Either SR is valid or it's not.

A final note on my use of the word "lucky": the "luck" comes in finding a particular symmetry property that allows a particular solution (or set of solutions) of the EFE to be written in a form that looks simple. In the general case, this is not possible. It just so happens that a number of spacetimes of interest in physics *do* happen to have particular symmetry properties that allow us to write them in a form that looks simple.

We have discussed General Relativity on the small scale, (say 0 to 1000 Astronomical Units across) and the large scale (say, beyond 1 billion light years) and recognized that there is a curvature associated with each one. The curvature at the local level is caused by gravitating bodies such as the earth, moon, sun, etc. The curvature at the global level is caused by making the mathematical generalization that

$$d\tau^{2} = dt^{2} - a(t) \left[\frac{1}{1 - k r^{2}} dr^{2} + r^{2} \left( d\theta^{2} + sin^{2} \theta d\phi^{2} \right) \right]$$

where $t$, $r$, $\theta$, and $\phi$ are the coordinates, and $k$ is a constant that can take one of three values: +1, 0, or -1.

One thing that I think we have established is that the large scale curvature, if it occurs at all, occurs at a level that is almost imperceptible up to a scale of at least a billion light years, and a billion years. Yet no one will entertain the idea that the large scale curvature actually is null; that a(t)=1 and k=0; that is, that the universe actually is, on the large scale, Minkowski.

It seems to me like this should be a starting point. That we should be willing to explore this simplest of possible options, and see what the actual expectations would be.

 

[JDoolin]

You could do a "velocity change" in the FRW spacetime, but the resulting coordinate system would no longer respect the symmetries of the spacetime--the metric would look different (because space would no longer look isotropic in the "moving" frame--Earth itself is an example of such a "moving frame", since the CMBR does not look isotropic to us). That makes it a different case from Minkowski spacetime, where a Lorentz transformation leaves the metric looking the same in the transformed coordinates as it does in the original coordinates.

If you consider an observer that changes velocity at an event after t=0, he will not see a metric looking the same. He will se a dipole anisotropy. Possibly you are considering the Lorentz Transformation without thinking about the path the accelerating observer takes through it.

 

[PeterDonis]

Or are you thinking of some simple transform of the standard Schwarzschild coordinates that normalized t to equal tau ?

I was thinking of this, but you're correct, this is another subtlety. (I see George Jones has pointed out further subtleties as well.) Probably I need to step back and re-think what I was saying and come up with a better formulation.

 

[PeterDonis]

1) is an example of what Penrose calls Woodhouse's Second Fundamental Confusion of Calculus.

Heh, good phrase. Can you give a reference? I've read a fair amount of Penrose's writing (at least his writing for the lay reader) and I haven't come across this one.

If anyone wants, I can elaborate mathematically on the above.

I can follow what you've written, but I'll have to digest it some more; I may have questions after I do.

 

[PeterDonis]

If you consider an observer that changes velocity at an event after t=0, he will not see a metric looking the same. He will se a dipole anisotropy.

Yes, this is what I was saying.

 

[PeterDonis]

This is a major part of what I can't understand. The Effect of the Lorentz Transformations are essentially directly proportianal to distance in space and time. i.e. If you Lorentz Transform an event that is 2 light years away, the effect will be roughly twice as much as if you Lorentz Transform an event that is 1 light-year away.

...Maybe I'm misinterpreting what you're saying when you say "the laws of SR hold locally." The way I'm taking your meaning is that you can Lorentz Transform events within a certain radius in spacetime, but events beyond that radius are not Lorentz Transformed.

That's not what I mean; see below.

A "limited domain" of applicability for SR seems to me, the same as "throwing it away." If I told you that "rotation" had a limited domain of applicability, it would mean that if you turn to the left or right, only nearby objects respond. Things in your room might change positions relative to your facing, but distant stars would not cooperate; they would remain in the same place; stubbornly remaining in front of you as you spin around, because rotation is "only valid locally".

If you say "SR is valid only locally" you're saying that only nearby objects are affected by the Lorentz Transformations. It is absurd. Either SR is valid or it's not.

There are several issues conflated here, which I'll try to disentangle; hopefully this will also clarify some of the terms (e.g., "domain of applicability") I'm using.

First, the general issue of the "validity" of theories: If you absolutely must have a "go/no go" decision on SR, so to speak, then SR is *not* valid, just as Newtonian mechanics is not valid. Both are approximate theories that have known limitations. Newtonian mechanics can't handle objects moving at speeds large enough relative to the speed of light. SR can't handle situations where gravity must be taken into account.

GR is also an approximate theory with known limitations, but its domain of applicability is wider than both Newtonian mechanics and SR, since it includes both as special cases. If we specialize to weak gravity and slow speeds, we get Newtonian mechanics; if we specialize to negligible gravity (but allow relativistic speeds) we get SR.

GR's known limitations are: (1) It predicts spacetime singularities in certain situations, which basically amounts to saying that it admits it can't cover those particular situations and new physics is needed; (2) It isn't a quantum theory, and the general belief is that a quantum theory of gravity is needed (for example, to cover those situations where GR predicts singularities).

Second, there's the issue of what, given the above, it means to say that SR holds "locally". In the standard interpretation of GR (where gravity = spacetime curvature), SR holds "locally" in a curved spacetime in the same sense that Euclidean geometry holds "locally" on a curved surface, such as the surface of the Earth. The Earth's surface is not Euclidean, but I don't have to worry about its curvature when I'm measuring the square footage of my house; the curvature is too small to matter. But if I try to measure the area of the state of Alaska, for example, I'd better take the Earth's curvature into account or I'll get the wrong answer; in other words, Euclidean geometry does *not* hold on the Earth's surface when you get to that large a scale.

Does that mean that, for example, if my house is in the middle of the state of Alaska, and I stand in the middle of my house and spin around, my house spins but the state of Alaska as a whole doesn't? Of course not. But it does mean, for example, that my spinning around doesn't change the area of the state of Alaska; it's still different than it would be if the Earth's surface was flat. So whatever coordinate transformation is being induced on the entire surface of the Earth by my rotation, it must preserve the non-Euclidean geometry of that surface. If that means that such a transformation is different in some way than a "standard" rotational transformation in flat Euclidean space, then okay, it's different. But locally (within my house), I can still treat the transformation as a standard rotation in flat Euclidean space, as long as I remember that I can only make that approximation over a small enough distance.

Similarly, if I'm in a curved spacetime and I change my velocity, locally (i.e., over a small enough patch of spacetime that the effects of curvature are negligible--same basic criterion as I used above for my house vs. Alaska) I can model this by a standard Lorentz transformation, provided I set up local Minkowski coordinates around the event of the velocity change (just as I can set up local Euclidean coordinates inside my house, even though they don't do a good job of representing the entire state of Alaska). It may well be that the transformation induced on distant parts of spacetime will *not* be a standard Lorentz transformation, because it will have to preserve the global curvature (i.e., non-Minkowskian geometry) of the spacetime. But certainly *some* transformation will be induced; the entire universe will look different after the velocity change, not just a local patch, just as it's not only my house that spins around me when I spin.

Third, there's the issue of "interpretations" of GR. I said above that gravity = spacetime curvature is the standard interpretation. However, it is true that it is not the *only* interpretation. (One good discussion of this is Kip Thorne's, in his book Black Holes and Time Warps: Einstein's Outrageous Legacy, which I highly recommend, and not just for this specific issue but as a generally very good presentation of relativity for the lay reader.) Another way to interpret GR is by treating the metric as a field on a background spacetime that is flat--i.e., Minkowski. Basically, you start by writing the metric as

$$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$

where $\eta_{\mu \nu}$ is the standard Minkowski metric and $h_{\mu \nu}$ is the extra field that accounts for the effects of gravity. Then you try to figure out what $h_{\mu \nu}$ is by doing an expansion in powers of some parameter; when this method was first investigated in the 1950's and 1960's, by Feynman among many others, the motivation was to look for a quantum theory of gravity, so $h_{\mu \nu}$ was taken to be (sorry for the bit of jargon here) a massless spin-two quantum field, the "graviton", on the background spacetime, and the expansion was just a standard perturbation expansion in powers of the graviton's quantum coupling constant (which is related to, but not necessarily the same as, Newton's gravitational constant), adding more and more different Feynman diagrams for different possible virtual graviton exchanges, similar to the methods that had worked so well for quantum electrodynamics. The end result of this process would be an expression for the action of spacetime, to some level of approximation anyway, which could be used, in the classical limit (i.e., letting Planck's constant go to zero), to derive a field equation by the same route that Hilbert had used in 1915 (which I referred to in an earlier post).

Of course there are an infinite number of terms in the perturbation expansion, but remarkably, in the case of gravity, it turned out that there was a way to calculate the sum of all of them, which converged to a finite answer, which, remarkably, turned out to be the *same* action that Hilbert had calculated in 1915! So basically, the theory of a massless spin-two field on flat Minkowski spacetime turns out to be GR, at least in the classical limit. But there are two key points about this:

(1) The flat "background" spacetime is unobservable: the actual, physical spacetime (meaning, the actual physical metric that determines proper times and proper distances) is curved, just as in the standard interpretation of GR. This is why this "field on flat spacetime" model is called an "interpretation" of GR instead of a different theory: it makes exactly the same predictions for all experiments as the "curved spacetime" model.

(2) The assumption of a flat background spacetime restricts the possible solutions in a way that the standard curved spacetime model of GR does not. For example, asymptotically flat solutions, such as the Schwarzschild spacetime, are allowed. But it does not, as I understand it, allow solutions such as the FRW spacetime, at least in the k = 1 and k = -1 cases (I'm not sure whether the k = 0 case would be allowed--it does have flat spatial slices, but conformally it doesn't look the same as Minkowski spacetime). So even though the field equation is the same, the underlying assumptions of the flat spacetime model are more restrictive and exclude solutions which are certainly relevant in physics, and to our discussion here.

In this thread I've been talking entirely within the curved spacetime model, since that's the standard interpretation, and since the flat spacetime interpretation ends up making the same predictions anyway for experiments. But see below for some further comments specifically about Minkowski spacetime as a solution of the EFE.

But you reject the possibility that maybe space ISN'T stretching, and you reject the possibility that perhaps there was an era of non-uniform acceleration, and you reject the possibility that the galaxies might actually be moving apart. And you reject the possibility that Lorentz Transformations might actually work on the long range.

The comoving observers (not galaxies, necessarily--see next comment) *are* moving apart, in the sense that the proper distance between them is increasing with time. I'm not sure what you mean by "an era of non-uniform acceleration"; if you mean that the rate of expansion of the universe (the rate of change of the scale factor a(t) with t) may have changed in the past, it has--we know that by the curvature in the Hubble diagram that you mentioned in an earlier post. If you mean that various individual pieces of matter may have accelerated non-uniformly, against the background of FRW spacetime overall, I agree that certainly may have happened, but as I said before, these details are averaged out in the overall FRW models we've been discussing (though they are treated numerically in more detailed models). I talked about how transformations would work long range above.

So it is an available option to just set a(t)=1 and k=0, right? So Minkowski spacetime actually is a possible solution to the Einstein Field Equations? And then we don't have to throw away Special Relativity.

Minkowski spacetime *is* a solution to the EFE, but only if there is no matter-energy present--i.e., the stress-energy tensor is zero identically. That isn't true of the universe, and the FRW solutions to the EFE are valid in the presence of matter-energy (non-zero stress-energy tensor). In the presence of matter, we can't set a(t) = 1 and k = 0 by fiat; we have to work out the dynamics and see. When we do that, we find that a(t) must change with time, and that there are three possible values for k, and which one actually holds for our universe is something we have to determine by measuring the overall density of matter-energy in the universe, the curvature of the Hubble diagram, etc.

Ah, I see. Yes, of course, if we have comoving observers, then of course, they would all share the same proper time. But it is one thing to define a family of observers moving along those parallel worldlines. It is quite another to claim that the galaxies in the real universe are a family of observers moving along those worldlines.

I agree, and I don't think I've claimed the latter, only the former. Individual galaxies, galaxy clusters, etc. may be moving with respect to the cosmological coordinates. The assumption of a perfect fluid on a cosmological scale allows that, as long as the motions average out to zero, just as with the molecules in an ordinary fluid.

I'm sorry. Unintentionally, I've been switching back and forth between two ideas, and now there are three. The third idea is what you are explaining, that a set of comoving observers share a proper time, and that proper time is the same as their coordinate time. That's valid. The first idea is that those comoving observers are the galaxies in the real universe, who only appear to be moving apart because of the stretching of space. (That's wierd, but not where my argument was coming from.) What I was thinking about were galaxies moving apart from each other, with real recessional velocities, whose proper times were all different. In this environment, it would be ridiculous to simply set proper time equal to coordinate time, because the galaxies wouldn't be comoving.

If individual galaxies are not "comoving" (if they are changing their spatial "location" in cosmological coordinates with time), then their proper times will *not* be directly related to coordinate time. That's quite true. The "comoving" observers are abstractions, and there may not be any actual observers in the actual universe who are exactly "comoving" in this sense. However, the condition for determining whether an observer is "comoving" (do they see the universe, for example the CMBR, as isotropic) is quite clear and physically realizable.

One thing that I think we have established is that the large scale curvature, if it occurs at all, occurs at a level that is almost imperceptible up to a scale of at least a billion light years, and a billion years. Yet no one will entertain the idea that the large scale curvature actually is null; that a(t)=1 and k=0; that is, that the universe actually is, on the large scale, Minkowski.

It seems to me like this should be a starting point. That we should be willing to explore this simplest of possible options, and see what the actual expectations would be.

As I noted above, Minkowski spacetime is only a solution of the EFE if there is no matter-energy present--if the stress-energy tensor is zero. So the actual universe *cannot* be a Minkowski spacetime. That's why we are forced to consider models that are more complicated than Minkowski spacetime. As Einstein said, "Make everything as simple as possible--but not simpler."

 

[PeterDonis]

Other subtleties:

Just to check that I've got this right, after some digestion, here's my grasp of these four items, slightly out of order:

2) The transformation between $t$ and $T$ doesn't "change the direction" of the integral curves; it just reparametrizes them. This means that, in both coordinate systems, we can use the integral curves of "time" to uniquely label spatial points (each curve has constant values of $r$ = $R$, $\theta$, and $\varphi$).

For the rest of this, I'll leave out the angular coordinates (assume them held constant) and only talk about "time" and "radius".

1) The lines of constant $T$, which are integral curves of $\partial_{R}$, "cut at a different angle" from the lines of constant $t$, which are integral curves of $\partial_{r}$. So even though the integral curves of "time" stay the same, the "spatial slices" cut through them can be different if they're cut at different angles.

3) It looks to me like the 4-velocity you gave, which gives us 4), also gives us 3), since it makes it obvious that the 4-velocity is *not* just $\partial_{T}$, so the integral curves of the 4-velocity can't be the same as the integral curves of $\partial_{T}$. The integral curves of the 4-velocity are "tilted inward", while the integral curves of $\partial_{T}$ (and thus, of course, $\partial_{t}$) are "vertical".

4) Since the 4-velocity "tilts inward", the integral curves of $\partial_{R}$, to be orthogonal to them, must "tilt downward" relative to the integral curves of $\partial_{r}$, which are "horizontal".

 

[PeterDonis]

4) Since the 4-velocity "tilts inward", the integral curves of $\partial_{R}$, to be orthogonal to them, must "tilt downward" relative to the integral curves of $\partial_{r}$, which are "horizontal".

Oops, I think this should be "tilt upward", since this is spacetime, not space, so "orthogonal" works differently.

 

[JDoolin]

(1) The flat "background" spacetime is unobservable: the actual, physical spacetime (meaning, the actual physical metric that determines proper times and proper distances) is curved, just as in the standard interpretation of GR. This is why this "field on flat spacetime" model is called an "interpretation" of GR instead of a different theory: it makes exactly the same predictions for all experiments as the "curved spacetime" model.

(2) The assumption of a flat background spacetime restricts the possible solutions in a way that the standard curved spacetime model of GR does not. For example, asymptotically flat solutions, such as the Schwarzschild spacetime, are allowed. But it does not, as I understand it, allow solutions such as the FRW spacetime, at least in the k = 1 and k = -1 cases (I'm not sure whether the k = 0 case would be allowed--it does have flat spatial slices, but conformally it doesn't look the same as Minkowski spacetime). So even though the field equation is the same, the underlying assumptions of the flat spacetime model are more restrictive and exclude solutions which are certainly relevant in physics, and to our discussion here.

In this thread I've been talking entirely within the curved spacetime model, since that's the standard interpretation, and since the flat spacetime interpretation ends up making the same predictions anyway for experiments. But see below for some further comments specifically about Minkowski spacetime as a solution of the EFE.

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.

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. Maybe it's not possible to go backwards, because we've got the transformed coordinates, but we don't know what the transformation actually was.

But those Minkowski coordinates within which the Lorentz Transformations still exist, whether or not we know how to map to them.

The comoving observers (not galaxies, necessarily--see next comment) *are* moving apart, in the sense that the proper distance between them is increasing with time. I'm not sure what you mean by "an era of non-uniform acceleration"; if you mean that the rate of expansion of the universe (the rate of change of the scale factor a(t) with t) may have changed in the past, it has--we know that by the curvature in the Hubble diagram that you mentioned in an earlier post. If you mean that various individual pieces of matter may have accelerated non-uniformly, against the background of FRW spacetime overall, I agree that certainly may have happened, but as I said before, these details are averaged out in the overall FRW models we've been discussing (though they are treated numerically in more detailed models). I talked about how transformations would work long range above.



Minkowski spacetime *is* a solution to the EFE, but only if there is no matter-energy present--i.e., the stress-energy tensor is zero identically. That isn't true of the universe, and the FRW solutions to the EFE are valid in the presence of matter-energy (non-zero stress-energy tensor). In the presence of matter, we can't set a(t) = 1 and k = 0 by fiat; we have to work out the dynamics and see. When we do that, we find that a(t) must change with time, and that there are three possible values for k, and which one actually holds for our universe is something we have to determine by measuring the overall density of matter-energy in the universe, the curvature of the Hubble diagram, etc.

If a(t) is not equal to 1, then I must ask you for an extraordinary level of clarity in what t defines. Is t the proper time at x,y,z, or is t the proper time at 0,0,0, or are these two assumed to be the same thing? Are you assuming that t at x,y,z is the age of particles that have traveled from here to x,y,z over the age of the universe, or are you assuming that the particles at x,y,z have always been at x,y,z?

If the universe is Minkowski, then I would say t represents the proper time at 0,0,0, and a(t)=0. But FRW says that the universe is stretching over time, which means a(t)~t and t represents the proper time of the actual galaxies at x,y,z, but somehow that proper time of the galaxies at x,y,z is the same as the proper time at 0,0,0. Meaning, they've taken as a-priori that the galaxies at x,y,z are comoving with the galaxies at 0,0,0.

If somehow, it can be shown that the stress energy tensor causes the scale of space to stretch over time, that's the sort of rigour that I'm looking for. I'm certainly no expert, but my impression has been that the stress energy tensor operates on velocity and proper time; i.e. properties of matter; not properties of space.

I agree, and I don't think I've claimed the latter, only the former. Individual galaxies, galaxy clusters, etc. may be moving with respect to the cosmological coordinates. The assumption of a perfect fluid on a cosmological scale allows that, as long as the motions average out to zero, just as with the molecules in an ordinary fluid.



If individual galaxies are not "comoving" (if they are changing their spatial "location" in cosmological coordinates with time), then their proper times will *not* be directly related to coordinate time. That's quite true. The "comoving" observers are abstractions, and there may not be any actual observers in the actual universe who are exactly "comoving" in this sense. However, the condition for determining whether an observer is "comoving" (do they see the universe, for example the CMBR, as isotropic) is quite clear and physically realizable.

As I noted above, Minkowski spacetime is only a solution of the EFE if there is no matter-energy present--if the stress-energy tensor is zero. So the actual universe *cannot* be a Minkowski spacetime. That's why we are forced to consider models that are more complicated than Minkowski spacetime. As Einstein said, "Make everything as simple as possible--but not simpler."

Let's investigate these results, which say, "Minkowski solution" -> "No Matter, No Energy." Let's see whether or not, they make any assumptions that the universe is made up of (approximately) comoving particles, or whether they assume that the amount of matter in the observable universe is finite. If they make either of these assumptions, or equivalent ones, they've already precluded the Minkowski solution.

 

[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.

No, the answer is that curvature is, essentially, an invariant; whether or not a spacetime is curved does not depend on what coordinates we use to describe it. It's a real, physical property that corresponds to real, physical measurements. I say "essentially" because curvature is not quantified by a single invariant quantity; there are a number of them, and often, in solving problems in GR, we don't express curvature solely in terms of those invariants, but in terms of geometric objects like the Riemann curvature tensor that do transform when we change coordinates. That's just for calculational convenience, and doesn't change the physics.

Let me give an example. Suppose we have two objects freely falling towards the Earth. Object 1 is at radius R (from the Earth's center), and object 2 is at radius R + r (slightly further away). Both objects start out at rest with respect to the Earth at time t = 0; here "time" refers to coordinate time in a Schwarzschild coordinate system with the Earth as the central mass. How will these objects move? We know from the physics of tidal gravity that the radial separation between them, which starts out as r, will increase with time, as they both fall toward the Earth.

What does this mean geometrically? Look at the problem in the t-r plane of the Schwarzschild coordinate system. We have two geodesics (two curves) in this plane which are initially parallel: at time t = 0, dr/dt is 0 for both curves. However, as time passes, the curves separate; the distance between them increases. That is a manifestation, geometrically, of curvature (initially parallel geodesics changing separation--that can't happen in a flat Euclidean space or a flat Minkowski spacetime), and it's a physical effect, independent of the coordinates. For example, we could measure it by running a string between the two objects (making sure that it is light enough and elastic enough that it will not affect their motion) and measuring how it stretches as time passes.

Now suppose we decide to adopt the "flat spacetime" model I described in a previous post, and decree that we are going to use Minkowski coordinates come hell or high water. What will we find? We will find that we can't use the simple metric corresponding to those coordinates to determine actual physical distances and times; we will need to add this extra field, $h_{uv}$, to the metric we actually use to calculate distances and times. For example, the Minkowski coordinate system would assign a constant coordinate separation between the two falling objects I just described (since they are both freely falling and their initial velocities are equal), but their *physical* separation, as we've seen, increases with time, so to obtain a metric that accurately represents distances throughout the spacetime, we'll need to modify the Minkowski metric. (Again, this is another way of saying that spacetime is curved, as as *physical* effect, independent of coordinates.)

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

As I just described, we can certainly decide to *adopt* "Minkowski coordinates", because GR allows us to use any coordinate system we want. But as we've just seen, since the actual spacetime is curved, physically, the Minkowski metric cannot accurately represent it. In that sense, no, the Minkowski coordinates do not "exist".

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

What does "don't apply the metric" mean? Without the metric, you have no way to translate the coordinates of events, which are just numbers, into actual physical distances and times. You can't dictate what the spacetime geometry is just by assigning coordinates to events. Suppose I decree that everyone must use Euclidean coordinates to label locations on the Earth's surface. Does that make the Earth's surface flat?

If a(t) is not equal to 1, then I must ask you for an extraordinary level of clarity in what t defines. Is t the proper time at x,y,z, or is t the proper time at 0,0,0, or are these two assumed to be the same thing? Are you assuming that t at x,y,z is the age of particles that have traveled from here to x,y,z over the age of the universe, or are you assuming that the particles at x,y,z have always been at x,y,z?

If the universe is Minkowski, then I would say t represents the proper time at 0,0,0, and a(t)=0. But FRW says that the universe is stretching over time, which means a(t)~t and t represents the proper time of the actual galaxies at x,y,z, but somehow that proper time of the galaxies at x,y,z is the same as the proper time at 0,0,0. Meaning, they've taken as a-priori that the galaxies at x,y,z are comoving with the galaxies at 0,0,0.

You keep on saying "assume" and "define" and so on. None of the things you are saying here are assumed or defined. They are all *discovered* as aspects of the solution to the EFE that we obtain when we impose the condition that the universe is isotropic. I described how the process works in a previous post, but I'll revisit one item here: how we determine what the coordinate "t" defines. Here, again, is the process:

(1) We look at the metric we obtain as a solution of the EFE under the condition that the universe is isotropic, and discover that it can be written in the form I gave.

(2) We look at the coordinate "t" as it appears in this metric (which, so far, is just an arbitrary coordinate, we haven't assumed anything about its physical meaning) and discover that, if we consider a curve with all the spatial coordinates held constant (which is an integral curve of $\partial / \partial t$), the coordinate t is "the same" as the actual lapse of proper time $\tau$ along that curve (as given by the metric). (I elaborated on what "the same" means in my previous post.)

(3) We therefore have *discovered* that, *if* an observer were to move along such a curve (which is a timelike curve, so it can be the worldline of an observer), that observer's proper time would be the same as coordinate time. This is true for any such curve, so there is a whole family of curves, each of which can be the worldline of its own unique "comoving" observer.

(4) Thus, we have *discovered* that the coordinate t "directly represents the proper time of comoving observers" in the sense just given.

We didn't have to *assume* anything here; this is all just logical deduction from the metric, which is a solution of the EFE given the condition of isotropy. But let's go on and consider the scale factor a(t) (which, note, we did *not* have to consider in any of the above--all of the above is true independently of what the scale factor is or what it physically means):

(5) We note that the spatial part of the metric is multiplied by a factor a(t), and ask what this factor means. Since the spatial part of the metric that a(t) multiplies is a "standard" metric for one of three known geometric surfaces (hypersphere for k = 1, Euclidean 3-space for k = 0, or "hyperbolic space" for k = -1), the effect of a(t) is to determine the distance scale of space (because it multiplies all the terms in the spatial metric equally), with "space" being the appropriate geometric object for a given value of k. Since a(t) can vary with time, this means the distance scale of space can vary with time, meaning coordinate time. But since coordinate time directly represents proper time for comoving observers, if a(t) varies with coordinate time, comoving observers will see the same variation, which will appear to them physically as a change in their physical separation with time.

Again, we didn't have to assume anything; we deduced the physical meaning of a(t) by looking at the metric and applying known geometric facts and what we deduced above about time and comoving observers.

If somehow, it can be shown that the stress energy tensor causes the scale of space to stretch over time, that's the sort of rigour that I'm looking for. I'm certainly no expert, but my impression has been that the stress energy tensor operates on velocity and proper time; i.e. properties of matter; not properties of space.

The stress-energy tensor does describe properties of matter, not space, but the Einstein Field Equation tells us that the stress-energy tensor can *affect* the properties of space. So if you believe the EFE, then you ipso facto believe that the stress-energy tensor can affect the properties of space.

Let's investigate these results, which say, "Minkowski solution" -> "No Matter, No Energy." Let's see whether or not, they make any assumptions that the universe is made up of (approximately) comoving particles, or whether they assume that the amount of matter in the observable universe is finite. If they make either of these assumptions, or equivalent ones, they've already precluded the Minkowski solution.

The fact that Minkowski spacetime is only a solution of the EFE if the stress-energy tensor is identically zero is mathematically proven; it doesn't involve any assumptions beyond those required to derive the EFE itself. The fact that the stress-energy tensor can only be identically zero if there is no matter or energy in the universe is part of the physical definition of the stress-energy tensor. There's no wiggle room to "investigate these results" as you suggest; it would be like investigating whether the derivative of x^2 is 2x.

 

[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.