Worldline congruence and general covariance

[PeterDonis]

When you say "once you've stablished..." , I guess you don't even realize that the way you stablish that in FRW manifolds is thru the Weyl's principle, now if you argue this, you need to go back to read some cosmological relativity texts.

Would you mind pointing me to a reference that describes how the Weyl postulate is used to establish which half of the light cone is the "future" half? The Weyl postulate deals with the assumption of homogeneity and isotropy, and the "comoving" worldlines of fluid elements being hypersurface orthogonal. It says nothing about which direction of time is "future" vs."past". If you are saying that the Weyl postulate somehow decrees that the "expanding" direction of time is the future, that may be the convention in cosmology because we observe the actual universe as a whole to be expanding; however, there are perfectly valid collapsing FRW models that obey the Weyl postulate, in the sense of having a congruence of "comoving" timelike worldlines that are hypersurface orthogonal. They are just converging instead of diverging. So physically, I don't see how the condition of hypersurface orthogonality picks out a preferred direction of time, even in a non-stationary spacetime; both the "expanding" and "contracting" versions of the FRW spacetimes are valid, physically speaking.

Please, we know there are very physically weird spacetime solutions of GR so let's keep the discssion strictly within the scope of spacetimes compatible with what we observe in our universe, the OP was about our spacetime and the models of our own spacetime.

As I said before, if you're going to argue that the Weyl postulate is *required* to establish causality, you need to show that it is *necessary*, which means you need to consider models where it doesn't hold and see if causality is still there. Nobody is disputing that the Weyl postulate is *sufficient* to establish causality. If we're going to restrict discussion to spacetimes compatible with what we actually observe, then there's nothing to be discussed, because we actually observe that the Weyl postulate holds to a certain approximation.

If youvread more about cosmology you'd see you're wrong here, a Cauchy surface is basically a spacelike hypersurface that acts as cosmic time and intersected by worldlines just once, sound familiar?

Yes. But remember that the presence of a Cauchy surface is a stronger condition than just the presence of a global time function.

Now put that in an expanding spacetime and guess what you get: timelike geodesics diverging. Cool, ain't it?

Here's what you said in the previous post of yours that I was responding to:

The global time function (or the cosmic time, to use the term Hawking used in his 1968 paper about stably causal spacetimes) assumes an agreement about a future-directed timelike which is precisely what you don't necessarily have in a curved non-stationary manifold like ours unless you slice it according to the condition that the ’particles’ in the universe lie on a congruence of time-like geodesics, that is the perfect fluid condition is a necessary assumption for the "global time function" .
When to the previous condition you add that the time-like geodesics diverge from from a point in the finite (or infinite) past, you get the globally hyperbolic manifold.

I understood you to be arguing that (a) you need the Weyl postulate with a perfect fluid to have a "global time function", and (b) you need that plus geodesics diverging to get global hyperbolicity. Both of those claims are false. (Even if we restrict attention *only* to non-stationary "expanding" spacetimes, they're false. If we restrict attention to only spacetimes that meet the Weyl postulate requirements, then as I said above, I don't see the point of this whole discussion.) If I misunderstood you and those claims aren't what you were saying, then what exactly were you saying? If you were only saying that the Weyl postulate with an expanding universe is *consistent* with a global time function and global hyperbolicity, of course I agree; but you appeared to be making a much stronger claim than that.

Very true, but since we want to apply the equation to FRW universes, guess what you find:a a timelike geodesic congruence, a.k.a the Weyl's postulate

For the Weyl postulate to hold the congruence has to be hypersurface orthogonal, i.e., vorticity-free. The Raychaudhuri equation is not limited to that case, even in FRW spacetimes; there are plenty of timelike worldline congruences in such spacetimes that are non-geodesic and/or not hypersurface orthogonal. See next comment.

Read carefully, I said the absence of vorticity, not the vorticity.
Wrong again. There is a very interesting explanation by John Baez in the web, I'll try to find the link, but basically the symmetric connection forces geodesic in GR to not twist.

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

I'm quite familiar with that web page (and I agree it's a very good one). In particular, I read the part where it says:

Relatively few people understand why in GR we assume the connection --- the gadget we use to do parallel translation --- is torsion-free.

Do you understand what the bolded phrase means? It means that in GR, there is no twisting of a vector when you parallel transport it along a worldline. It says nothing about twisting of a congruence of worldlines relative to one another, which is what the vorticity in the Raychaudhuri equation refers to. Again, from the Baez page:

If, no matter how we choose P and Q and v, the time derivative of the distance between C(t) and D(t) at t = 0 is ZERO, up to terms proportional to epsilon^2, then the torsion is zero!

Again, the bolded phrase is crucial. Parallel transport deals with the first time derivative; but the vorticity in the Raychaudhuri equation, which is a particular piece of the curvature tensor, deals with the *second* time derivative, the part that would be proportional to epsilon squared, and which is *not* constrained by the torsion-free connection. So it is perfectly possible, as I said above, to have a congruence of timelike worldlines in an expanding FRW spacetime that has non-zero vorticity; the torsion-free nature of the connection used in GR does not prohibit that.

 

[PeterDonis]

...the vorticity in the Raychaudhuri equation, which is a particular piece of the curvature tensor, deals with the *second* time derivative, the part that would be proportional to epsilon squared, and which is *not* constrained by the torsion-free connection.

On re-reading, I should clarify the above. The vorticity is a property of a congruence of worldlines, not of the spacetime itself, so I shouldn't have said it was a "piece of the curvature tensor". What I should have said is that the vorticity is related to the curvature tensor; or perhaps a better way of stating it would be that the vorticity of a congruence of worldlines can be used to deduce properties of the curvature tensor. The key point, that the vorticity is not constrained by the torsion-free connection in GR, still stands.

 

[TrickyDicky]

Would you mind pointing me to a reference that describes how the Weyl postulate is used to establish which half of the light cone is the "future" half?

It turns out it does establish it. I gave you a reference with the explicit original wording of the postulate: "The particles of the substratum (representing the nebulae) lie in spacetime on a bundle of geodesics diverging from a point in the (finite or infinite) past". Remember this was 1923 without notion of expanding universe (Friedman had published his paper a few months earlier but at the time Weyl wrote his postulate he had not read it).
So the "future" half is established by the diverging direction.

The Weyl postulate deals with the assumption of homogeneity and isotropy, and the "comoving" worldlines of fluid elements being hypersurface orthogonal. It says nothing about which direction of time is "future" vs."past". If you are saying that the Weyl postulate somehow decrees that the "expanding" direction of time is the future, that may be the convention in cosmology because we observe the actual universe as a whole to be expanding; however, there are perfectly valid collapsing FRW models that obey the Weyl postulate, in the sense of having a congruence of "comoving" timelike worldlines that are hypersurface orthogonal. They are just converging instead of diverging. So physically, I don't see how the condition of hypersurface orthogonality picks out a preferred direction of time, even in a non-stationary spacetime; both the "expanding" and "contracting" versions of the FRW spacetimes are valid, physically speaking.

I've explained to you earlier that it doesn't deal with that assumption, is totally independent of it, it's just that in the FRW cosmology acts as a necessary precondition to the cosmology principle assumption.
In fact hypersurface orthogonality was an addition to the original Weyl's postulate made by Robertson when introducing the FRW metric, it is just a logic outcome of using the original postulate in an expanding FRW metric context.

As I said before, if you're going to argue that the Weyl postulate is *required* to establish causality, you need to show that it is *necessary*, which means you need to consider models where it doesn't hold and see if causality is still there. Nobody is disputing that the Weyl postulate is *sufficient* to establish causality. If we're going to restrict discussion to spacetimes compatible with what we actually observe, then there's nothing to be discussed, because we actually observe that the Weyl postulate holds to a certain approximation.

You have a confusion about what I argue and what I don't (and I admit it can be due to my sloppy way of argumenting). I'll try to clarify:I say that Weyl's postulate establish causality only in the case of the FRW cosmology.
You seemed to be arguing that Weyl postulate was not "sufficient" to establish causality above.

For the Weyl postulate to hold the congruence has to be hypersurface orthogonal, i.e., vorticity-free.

See above comment.

The Raychaudhuri equation is not limited to that case, even in FRW spacetimes; there are plenty of timelike worldline congruences in such spacetimes that are non-geodesic and/or not hypersurface orthogonal.

I don't agree with you here, as I explained the very "constructor" of the FRW metric, Robertson, used the Weyl p. as precondition and added the hypersurface orthogonality bit to the postulate.
I already agreed that the Raychaudhuri equation refers to a more general congruence than the used in the Weyl's postulate. But I explained that within torsion-free GR it amounts to the same one.

Do you understand what the bolded phrase means? It means that in GR, there is no twisting of a vector when you parallel transport it along a worldline. It says nothing about twisting of a congruence of worldlines relative to one another, which is what the vorticity in the Raychaudhuri equation refers to.

Let's see if we can reach some mutual understanding. Do you agree that due to torsion-free timelike geodesics are not allowed to twist in GR (rotate around their axis)?
Now let's quote the wikipedia page on the Raychaudhuri equation:"let \vec{X} be a timelike geodesic unit vector field with vanishing vorticity, or equivalently, which is hypersurface orthogonal. For example, this situation can arise in studying the world lines of the dust particles in cosmological models which are exact dust solutions of the Einstein field equation (provided that these world lines are not twisting about one another, in which case the congruence would have nonzero vorticity)."
I understand this last phrase to mean that worldlines twisting around each other would have nonzero vorticity, even if the wording is a bit confusing.
I infer from this that you are not correct when you say that vorticity is totally unrelated to torsion-free GR.
Also according to the quoted wiki paragraph I'd say it is not possible to have a congruence of timelike worldlines in an expanding FRW spacetime that has non-zero vorticity as you claim, that is precisely what the Weyl's postulate and hypersurface orthogonality in expanding FRW metric prohibit.

 

[PeterDonis]

I gave you a reference with the explicit original wording of the postulate: "The particles of the substratum (representing the nebulae) lie in spacetime on a bundle of geodesics diverging from a point in the (finite or infinite) past". Remember this was 1923 without notion of expanding universe (Friedman had published his paper a few months earlier but at the time Weyl wrote his postulate he had not read it).
So the "future" half is established by the diverging direction.

I agree that this reference establishes that Weyl, when he proposed the postulate, *claimed* that the "future" direction of time was established by the diverging direction. I'm not sure I agree that that claim is still physically valid, in the light of what we know today. Weyl was not only unaware of the expanding universe and the FRW models of same; he was also unaware of the "time reversed" versions of those models, the collapsing FRW models, for example the one used in the classic Oppenheimer-Snyder paper in 1939.

You have a confusion about what I argue and what I don't (and I admit it can be due to my sloppy way of argumenting). I'll try to clarify:I say that Weyl's postulate establish causality only in the case of the FRW cosmology.
You seemed to be arguing that Weyl postulate was not "sufficient" to establish causality above.

No, I am arguing that the Weyl postulate is not *necessary* to establish causality in the case of "expanding universe" cosmologies. I say "expanding universe" since it's more general than "FRW cosmology", which could be taken to restrict attention only to spacetimes that satisfy the Weyl postulate; and as I've said several times now, the whole question is whether such a restriction is *necessary* to establish causality, which means to answer the question you have to consider models that don't meet the restriction, and see whether causality still holds; if, as I claim, it does, then the Weyl postulate is not necessary for causality. I explicitly said in previous posts that the fact that the Weyl postulate is *sufficient* to establish causality is not in question.

I already agreed that the Raychaudhuri equation refers to a more general congruence than the used in the Weyl's postulate. But I explained that within torsion-free GR it amounts to the same one.

No, it doesn't. See below.

Let's see if we can reach some mutual understanding. Do you agree that due to torsion-free timelike geodesics are not allowed to twist in GR (rotate around their axis)?

No. See below.

Now let's quote the wikipedia page on the Raychaudhuri equation:"let \vec{X} be a timelike geodesic unit vector field with vanishing vorticity, or equivalently, which is hypersurface orthogonal. For example, this situation can arise in studying the world lines of the dust particles in cosmological models which are exact dust solutions of the Einstein field equation (provided that these world lines are not twisting about one another, in which case the congruence would have nonzero vorticity)."

Just to clarify, this part of the Wiki page is discussing a particular application of the Raychaudhuri equation, not the equation in general.

I understand this last phrase to mean that worldlines twisting around each other would have nonzero vorticity, even if the wording is a bit confusing.

I understand it the same way, provided that "worldlines twisting around each other" is interpreted correctly; see below. I agree the wording is not optimal (which is often the case with Wikipedia).

I infer from this that you are not correct when you say that vorticity is totally unrelated to torsion-free GR.

This is because you are confusing vorticity with the torsion of the connection; as I said in my last post, they are two different things. To see why, look again at that John Baez web page on torsion in GR that you linked to. It describes a thought experiment (unfortunately I don't know how to make Baez' ASCII art look the same here as it does on his page, so I'll leave out the drawings):

Take a tangent vector v at P. Parallel translate it along a very short curve from P to Q, a curve of length epsilon. We get a new tangent vector w at Q. Now let two particles free-fall with velocities v and w starting at the points P and Q. They trace out two geodesics...

Okay. Now, let's call our two geodesics C(t) and D(t), respectively. Here we use as the parameter t the proper time: the time ticked out by stopwatches falling along the geodesics. (We set the stopwatches to zero at the points P and Q, respectively.)

Now we ask: what's the time derivative of the distance between C(t) and D(t)? Note this "distance" makes sense because C(t) and D(t) are really close, so we can define the distance between them to be the arclength along the shortest geodesic between them.

If, no matter how we choose P and Q and v, the time derivative of the distance between C(t) and D(t) at t = 0 is ZERO, up to terms proportional to epsilon2, then the torsion is zero! And conversely! (One can derive this from the definition of torsion, assuming our recipe for parallel transport is metric preserving.)

If v got "rotated" a bit when we dragged it over to Q...then the time derivative of the distance would not be zero (it'd be proportional to epsilon). In this case the torsion would not be zero.

This thought experiment gives us a recipe for generating a congruence of timelike worldlines: start with some chosen worldline V, and pick a spacelike curve S that intersects V at point P, and call V's tangent vector at P, v. We also specify that V is a geodesic, so that its tangent vector at P is sufficient to specify it throughout the spacetime.

Now parallel transport v along curve S. Take any point Q of S, and call the parallel transported version of v at Q, w. Now find the timelike geodesic intersecting S at Q whose tangent vector at Q is w. The set of all such timelike geodesics, intersecting S, will form a congruence (with one caveat: I haven't worked out exactly what conditions the spacetime as a whole has to satisfy for this to be true, in the sense that the worldlines don't intersect unless the spacetime as a whole has a singularity, such as the initial singularity in FRW spacetime; see further comments below). And the torsion-free nature of the connection in GR does guarantee that this particular congruence will have vanishing vorticity.

However, the congruence I've just described is not necessarily the *only* congruence that might have a worldline intersecting spacelike surface S at point V with tangent vector v. There might be other such congruences, either because worldline V itself belongs to more than one congruence, or because there are other congruences that are non-geodesic but contain worldlines intersecting S at P with tangent vector v (for non-geodesic worldlines, the tangent vector at a point is not sufficient to specify a single worldline). The torsion-free connection does *not* prevent this. What the torsion-free connection does allow us to say is this: consider point Q on spacelike surface S, where the parallel transported tangent vector of worldline V is w. There may be another worldline passing through Q, call it Z, whose tangent vector z at Q is *different* from w; and Z may be part of some *other* congruence of timelike worldlines that includes either V, or some other worldline with the same tangent vector v at P. If that is the case, then this second congruence will have *non-zero* vorticity.

Now for the caveat: as I said above, I have not worked out specifically what conditions the spacetime as a whole has to satisfy for the recipe given above to produce a congruence of non-intersecting timelike geodesics. I believe that global hyperbolicity is sufficient; I suspect that even stable causality might be sufficient. If either of those is correct, then what I've said above will hold in a far more general set of spacetimes, even "expanding" non-stationary ones, than those which satisfy the Weyl postulate. (In fact, even in spacetimes which do satisfy the Weyl postulate, such as expanding FRW spacetimes, the torsion-free connection does not force all congruences of timelike geodesics to be vorticity-free; see next comment below.)

Also according to the quoted wiki paragraph I'd say it is not possible to have a congruence of timelike worldlines in an expanding FRW spacetime that has non-zero vorticity as you claim, that is precisely what the Weyl's postulate and hypersurface orthogonality in expanding FRW metric prohibit.

No, they don't. The postulate does not claim that *all* congruences of timelike worldlines in expanding FRW spacetime must be hypersurface orthogonal; it only claims that there *exists* such a congruence (the congruence of worldlines of "comoving" observers), and that that congruence describes the worldlines of the "particles" of the cosmological fluid. In other words, it claims that the cosmological fluid has vanishing vorticity; but there are plenty of other congruences of worldlines, which could describe families of observers who are *not* comoving with the fluid, and which could have non-zero vorticity.

 

[PeterDonis]

There may be another worldline passing through Q, call it Z, whose tangent vector z at Q is *different* from w; and Z may be part of some *other* congruence of timelike worldlines that includes either V, or some other worldline with the same tangent vector v at P. If that is the case, then this second congruence will have *non-zero* vorticity.

I should expand on this a little more. In fact, it could even be the case that there is another worldline passing through Q, call it Y, whose tangent vector at Q *is* w (the same as the geodesic W passing through Q which is part of the first congruence), but which is not a geodesic and therefore is not the same as W. Even in *this* case, the congruence containing V and Y can have non-zero vorticity. This possibility is what I was thinking of when I said that vorticity is related to curvature: if points P and Q are separated by distance epsilon, as in Baez' scenario, then even though worldlines V and Y have "the same" tangent vectors along surface S (i.e., one is the parallel transported version of the other), so the first time derivative of the "distance" between V and Y is zero at surface S, the *second* time derivative of that distance (the term proportional to epsilon squared instead of epsilon) can be non-zero, because worldlines V and Y curve differently, and so they might twist around each other taken as a whole, even though they are "parallel" for an instant as they cross surface S. Again, the torsion-free connection in GR does not prevent this. (And the different curvature of V and Y might tell us something about the curvature of the spacetime as well.)

 

[TrickyDicky]

This is because you are confusing vorticity with the torsion of the connection

I don't claim that vorticity and the torsion of the connection are the same thing

This thought experiment gives us a recipe for generating a congruence of timelike worldlines: start with some chosen worldline V, and pick a spacelike curve S that intersects V at point P, and call V's tangent vector at P, v. We also specify that V is a geodesic, so that its tangent vector at P is sufficient to specify it throughout the spacetime.

Now parallel transport v along curve S. Take any point Q of S, and call the parallel transported version of v at Q, w. Now find the timelike geodesic intersecting S at Q whose tangent vector at Q is w. The set of all such timelike geodesics, intersecting S, will form a congruence (with one caveat: I haven't worked out exactly what conditions the spacetime as a whole has to satisfy for this to be true, in the sense that the worldlines don't intersect unless the spacetime as a whole has a singularity, such as the initial singularity in FRW spacetime; see further comments below). And the torsion-free nature of the connection in GR does guarantee that this particular congruence will have vanishing vorticity.

This is what I'm saying, no more.

However, the congruence I've just described is not necessarily the *only* congruence that might have a worldline intersecting spacelike surface S at point V with tangent vector v. There might be other such congruences, either because worldline V itself belongs to more than one congruence, or because there are other congruences that are non-geodesic but contain worldlines intersecting S at P with tangent vector v (for non-geodesic worldlines, the tangent vector at a point is not sufficient to specify a single worldline). The torsion-free connection does *not* prevent this. What the torsion-free connection does allow us to say is this: consider point Q on spacelike surface S, where the parallel transported tangent vector of worldline V is w. There may be another worldline passing through Q, call it Z, whose tangent vector z at Q is *different* from w; and Z may be part of some *other* congruence of timelike worldlines that includes either V, or some other worldline with the same tangent vector v at P. If that is the case, then this second congruence will have *non-zero* vorticity

Once again this is trivial and I have said anything contrary to this, let's not distract from the OP.
It is quite obvious that a slicing of the manifold that models our spacetime that has cross terms of the type dxdt, dydt, dzdt, doesn?t guarantee the presence of a synchronous cosmic time and therefore doesn't guarantee an agreement on the second law for observers using such asynchronous time coordinate.
It doesn't matter at all whether you consider the universe is contracting or expanding as long as everybody agrees on which one is the case, since choosing the Wey¡s slicing what guarantees is the agreement on that not the particular direction one chooses -this said, I found very few people that like you seems ready to argue that our universe is contracting ;)

Not choosing this particular slicing of spacetime allows the disagreement among differently located observers.
I found this on the web that actually suits well part of what I'm trying to clarify in the OP
"Weyl's cosmic time thus becomes a global, standard clock time that applies to every observer in the universe, making possible simultaneity of events. Unfortunately, this kind of cosmic time flies in the face of relativity, where time is always relative, depending on things like particle velocity and gravitational effects. Consequently, Weyl's postulate appears to prevent a completely covariant treatment of the simple cosmological models that utilize his postulate "

No, they don't. The postulate does not claim that *all* congruences of timelike worldlines in expanding FRW spacetime must be hypersurface orthogonal; it only claims that there *exists* such a congruence (the congruence of worldlines of "comoving" observers), and that that congruence describes the worldlines of the "particles" of the cosmological fluid. In other words, it claims that the cosmological fluid has vanishing vorticity; but there are plenty of other congruences of worldlines, which could describe families of observers who are *not* comoving with the fluid, and which could have non-zero vorticity.

Sure, those happen not to be geodesics,wich are the type of worldlines that I'm referring to from the start.

 

[PeterDonis]

I don't claim that vorticity and the torsion of the connection are the same thing

Ok, good. I wasn't sure based on your previous posts, but now I understand better where you were coming from.

It is quite obvious that a slicing of the manifold that models our spacetime that has cross terms of the type dxdt, dydt, dzdt, doesn?t guarantee the presence of a synchronous cosmic time...

True, in the sense that such a slicing will not correspond to a global "comoving" frame.

...and therefore doesn't guarantee an agreement on the second law for observers using such asynchronous time coordinate.

False. One does not need to be at rest in a global "comoving" frame in order to agree on the second law; the spacetime does not even have to *admit* a global "comoving" frame. All that needs to be true is that all observers agree on the direction of time, in the sense of agreeing on which half of each local light cone is the "future" half, and on the definition of that direction being continuous throughout the spacetime. That is guaranteed by a much weaker set of conditions than the presence of a global "comoving" frame, as I showed in previous posts.

It doesn't matter at all whether you consider the universe is contracting or expanding as long as everybody agrees on which one is the case, since choosing the Wey¡s slicing what guarantees is the agreement on that not the particular direction one chooses -this said, I found very few people that like you seems ready to argue that our universe is contracting ;)

I wasn't arguing that our actual universe is contracting, just that there are valid FRW-type models in which the future direction of time is the contracting direction. I agree that the important point is global agreement on the direction of time, as I said above.

I found this on the web that actually suits well part of what I'm trying to clarify in the OP
"Weyl's cosmic time thus becomes a global, standard clock time that applies to every observer in the universe, making possible simultaneity of events. Unfortunately, this kind of cosmic time flies in the face of relativity, where time is always relative, depending on things like particle velocity and gravitational effects. Consequently, Weyl's postulate appears to prevent a completely covariant treatment of the simple cosmological models that utilize his postulate "

This looks to me like an equivocation on the word "simultaneity". It is true that the time coordinate of a global "comoving" frame can be used to set up a global sense of simultaneity. However, it is *not* true that this sense of simultaneity will coincide with the *local* sense of simultaneity (meaning the simultaneity of the local Lorentz frame) of *every* observer in the universe, whatever their state of motion. And the claim that having the global "simultaneity" somehow contradicts relativistic covariance requires the latter to be true, not the former. So the claim is false.

Another way to put this is to imagine an observer who is not at rest in the "comoving" frame of the universe, and suppose that he wants to set his clock by the global "cosmic time". He will find that he has to build in a correction to the clock's rate; the "natural" rate of ticking of his clock, which is determined by his proper time, will *not* be the same as the rate of ticking of cosmic time (which he could check by exchanging light signals with another observer who *is* at rest in the "comoving" frame, and whose proper time is the same as cosmic time). In other words, "cosmic" time is *not* the same as proper time for any observer who is not at rest in the "comoving" frame. And that means that the presence of the "comoving" frame, and the decision to adopt its time as the global "cosmic" time, does *not* contradict relativistic covariance; that would only be contradicted if observers not at rest in the comoving frame somehow found that their proper time *was* the same as cosmic time, and they won't.

For example: the worldline of the Earth is *not* a "comoving" worldline; we see a large dipole anisotropy in the CMBR, for example. Therefore, the global sense of simultaneity that is provided by the global "comoving" frame for our actual universe is *not* the same as the local sense of simultaneity here on Earth. That is, a pair of events which are simultaneous according to the global "cosmic time" of the "comoving" frame are *not* simultaneous to us here on Earth. The difference is small, and it is normally not an issue in cosmology because we don't need a level of accuracy where the difference would be significant, but it's there. Our proper time here on Earth is *not* the same as cosmic time. We could, if we chose, decide to adopt a "cosmic time standard", so that we recorded the times of events, for the record, as their "cosmic" times instead of according to our local Earth proper time; but we would then have to build corrections into all our clocks, precisely because relativistic covariance works. Only if we found that our clocks somehow kept "cosmic time" *without* needing correction would we have any reason to doubt relativistic covariance.

(And it's also worth noting, as I've said before, that we here on Earth observe the second law to hold and the universe to be expanding, even though we are not at rest in the "comoving" frame.)

Can you provide a link to the full article you quoted?

 

[TrickyDicky]

True, in the sense that such a slicing will not correspond to a global "comoving" frame.


False. One does not need to be at rest in a global "comoving" frame in order to agree on the second law; the spacetime does not even have to *admit* a global "comoving" frame. All that needs to be true is that all observers agree on the direction of time, in the sense of agreeing on which half of each local light cone is the "future" half, and on the definition of that direction being continuous throughout the spacetime. That is guaranteed by a much weaker set of conditions than the presence of a global "comoving" frame, as I showed in previous posts.

I see I can't manage to make you understand what I mean. I never said anything about needing to be at rest in the comoving "frame". If a coordinate system with cross terms is used there is not even a defined comoving frame to have the possibility wrt which be at rest.
It's about using different coordinates, not about frames in the sense of state of motion.
You say: "All that needs to be true is that all observers agree on the direction of time..." and yet you don't realize that using a different coordinate system with crossed terms is what precisely would prevent you from having that agreement.

Another way to put this is to imagine an observer who is not at rest in the "comoving" frame of the universe, and suppose that he wants to set his clock by the global "cosmic time". He will find that he has to build in a correction to the clock's rate; the "natural" rate of ticking of his clock, which is determined by his proper time, will *not* be the same as the rate of ticking of cosmic time (which he could check by exchanging light signals with another observer who *is* at rest in the "comoving" frame, and whose proper time is the same as cosmic time). In other words, "cosmic" time is *not* the same as proper time for any observer who is not at rest in the "comoving" frame. And that means that the presence of the "comoving" frame, and the decision to adopt its time as the global "cosmic" time, does *not* contradict relativistic covariance; that would only be contradicted if observers not at rest in the comoving frame somehow found that their proper time *was* the same as cosmic time, and they won't.

For example: the worldline of the Earth is *not* a "comoving" worldline; we see a large dipole anisotropy in the CMBR, for example. Therefore, the global sense of simultaneity that is provided by the global "comoving" frame for our actual universe is *not* the same as the local sense of simultaneity here on Earth. That is, a pair of events which are simultaneous according to the global "cosmic time" of the "comoving" frame are *not* simultaneous to us here on Earth. The difference is small, and it is normally not an issue in cosmology because we don't need a level of accuracy where the difference would be significant, but it's there. Our proper time here on Earth is *not* the same as cosmic time. We could, if we chose, decide to adopt a "cosmic time standard", so that we recorded the times of events, for the record, as their "cosmic" times instead of according to our local Earth proper time; but we would then have to build corrections into all our clocks, precisely because relativistic covariance works. Only if we found that our clocks somehow kept "cosmic time" *without* needing correction would we have any reason to doubt relativistic covariance.

You keep using this example as if it were relevant to the discussion. It is not, it doesn't matter at all that we may use a different local time as long as it is still calculated in terms of the global cosmic time, if it is referenced to cosmic time it means we are using the comoving observers slicing. The coreect example would be using a coordinate system that doesn't allow to be referenced to comoving observers, that is one with the cross terms, in this coordinate system's metric there can't be no agreement between certain separate observers as to what direction time goes.

Can you provide a link to the full article you quoted?

It's not an article, it's just some guy with a blog on the web, I only included it to see if using someone else's words helped, Obviously, it didn't:

 

[PeterDonis]

You say: "All that needs to be true is that all observers agree on the direction of time..." and yet you don't realize that using a different coordinate system with crossed terms is what precisely would prevent you from having that agreement

You're right, I don't "realize" why that would have to be true. It amounts to saying that no two observers in relative motion can agree on the direction of time. That's obviously absurd. See following comments.

You keep using this example as if it were relevant to the discussion. It is not, it doesn't matter at all that we may use a different local time as long as it is still calculated in terms of the global cosmic time, if it is referenced to cosmic time it means we are using the comoving observers slicing.

No, it doesn't. Consider the Earth example again. Our local proper time on Earth, and the simultaneity associated with it, automatically implies a slicing of spacetime that is different from the "comoving" one. That has to be the case because we are not at rest in the "comoving" frame. Relative motion, and the consequent change in the local surfaces of simultaneity, is all that is required to change the slicing that "local" time is based on. But relative motion, by itself (and even if it includes non-zero vorticity and consequent "cross terms" in the metric--see next comment), is *not* enough to change the perceived direction of time.

The coreect example would be using a coordinate system that doesn't allow to be referenced to comoving observers, that is one with the cross terms, in this coordinate system's metric there can't be no agreement between certain separate observers as to what direction time goes.

Yes, there can. You keep confusing agreement on the *direction* of time, which only requires agreement on which half of the light cones is the "future" half, with agreement on the *surfaces of simultaneity*, which is a much stronger restriction, and is *not* required for agreement on causality, the second law, etc.

 

[TrickyDicky]

Once againg you are confusing frames, motion and coordinates.
Quote:"You keep confusing agreement on the *direction* of time, which only requires agreement on which half of the light cones is the "future" half, with agreement on the *surfaces of simultaneity*, which is a much stronger restriction, and is *not* required for agreement on causality, the second law, etc."
In your opinion how exactly is agreement on which half of the light cones is the "future" half achieved in the FRW metric?

 

[PeterDonis]

Once againg you are confusing frames, motion and coordinates.

It seems to me that you are often using language which invites confusion as to what you are trying to say. For example, consider the quote of yours that I was responding to that prompted your latest post:

The coreect example would be using a coordinate system that doesn't allow to be referenced to comoving observers, that is one with the cross terms, in this coordinate system's metric there can't be no agreement between certain separate observers as to what direction time goes.

You speak of "a coordinate system that doesn't allow to be referenced to comoving observers", but the worldlines of such comoving observers are coordinate-independent, geometric objects in the spacetime; they don't somehow disappear or become non-describable when I adopt coordinates in which the comoving observers are not at rest. The comoving worldlines are still there, and they can still be described even in a different coordinate systems; their description just won't look as simple. But all their invariant features (such as, for example, the fact that they are orthogonal to a particular set of spacelike hypersurfaces) can still be calculated and verified in any coordinate system.

You also speak of "that coordinate system's metric", but the metric, as a geometric object, is coordinate-independent; what changes from one coordinate system to another is only the *expression* of the metric in terms of the coordinate differentials. Changing coordinates certainly doesn't change the invariants that depend on the metric, and those invariants include the causal structure of the spacetime, i.e., the light cones and their sense of orientation. So if I have agreement, with reference to one coordinate system, as to which half of the light cones is the "future" half, and that sense of the light cones is continuous throughout the spacetime (which it must be if the spacetime is stably causal--see below), then since that is part of the causal structure, it is coordinate-independent; changing coordinate systems doesn't change it, so if I have such agreement in one coordinate system, I have it in any coordinate system, including one with "cross terms" in its expression for the metric.

Reading your quote above as it stands, it appears to deny what I just wrote. If you did not intend to do that, then what did you mean when you wrote what I quoted above?

In your opinion how exactly is agreement on which half of the light cones is the "future" half achieved in the FRW metric?

To answer the question as you stated it, obviously in our actual universe we experience time to flow in the direction that the universe is expanding. So the half of the light cones in which the universe is larger is obviously the "future" half. It's worth noting once more, though, that as I said above, obtaining such agreement does *not* require adopting the "comoving" coordinate system; agreement on the sense of direction of the light cones can be obtained in any coordinate system, and since it is invariant once obtained, it will hold in any coordinate system. For example, we can choose, here on Earth, which half of our light cones is the "future" half based on which direction of time has the universe expanding, and if we compared our choice with that of a "comoving" observer, made using the same criterion, we would find agreement.

However, as I've pointed out repeatedly now, the question you asked in the quote above is the wrong question to ask. The question you should be asking is:

Is there a way to get agreement on which half of the light cones is the future half, in a non-stationary spacetime that does *not* meet the conditions of the Weyl postulate?

Answer: yes, as long as it is stably causal, so a global time function can be defined. The gradient of the global time function is everywhere timelike, and we can choose its increasing direction as the future direction of time; or, if we decide that the decreasing direction makes more sense, we can simply invert the sign of the global time function, to get a new global time function whose gradient is likewise timelike, but points in the opposite direction. In other words, any global time function can be used to obtain a global agreement on which half of the light cones is the "future" half. So it can be done in any stably causal spacetime, which includes spacetimes that do not meet the conditions of the Weyl postulate.

You might also ask another question: Does the global time function guarantee that, once we've obtained agreement on which half of the light cones is the "future" half in a stably causal spacetime, that future direction won't "flip over" from one observer to another along a spacelike surface? We know that can't happen in a spacetime that satisfies the Weyl postulate; so the question is, could it happen in a spacetime that is stably causal and has a global time function, but does not satisfy the Weyl postulate?

Answer: no, it can't happen. Here's why: pick a spacelike hypersurface, and suppose that at some point on it, point A, the gradient of the global time function picks out the "future" half of the light cone as pointing one way. Now ask: what would have to happen for the gradient of the global time function to point the other way at some other point, B, on the same spacelike hypersurface? That could only happen in one of two ways: at some point, C, between A and B, the gradient would either have to go to zero, or else it would have to be tangent to the surface. But since the gradient is everywhere timelike, neither of those things can happen: a zero vector is not timelike (because it has a zero norm, and a timelike vector can't have a zero norm); and the surface is spacelike, so a timelike vector can't be tangent to it. So the global time function, or more precisely the fact that its gradient is everywhere timelike, guarantees that the direction of time can't "flip".

One other thing to remember: as I've said before, the existence of a global time function, as defined above (i.e, a scalar with a gradient that is everywhere timelike and future-directed), does *not* guarantee that the spacetime must satisfy the Weyl postulate. It doesn't even guarantee the existence of a Cauchy surface, and even a spacetime with a Cauchy surface may not satisfy the Weyl postulate; a Cauchy surface implies the existence of a global slicing of the spacetime, but it does not, by itself, guarantee that there is a congruence of "comoving" worldlines which are everywhere orthogonal to the slicing. The phrase "time function" by itself is ambiguous, and the quotes you've given have shown that some authors use it to imply a much tighter constraint than the standard definition does; in this post (and indeed in all my posts in this thread), I am using the term only to refer to its standard definition.

 

[TrickyDicky]

To answer the question as you stated it, obviously in our actual universe we experience time to flow in the direction that the universe is expanding. So the half of the light cones in which the universe is larger is obviously the "future" half.

This is the right question because apparently is the one you can't answer.
How is that so obvious to you, that we "experience", built into the FRW metric, mathematically?

 

[PeterDonis]

This is the right question because apparently is the one you can't answer.
How is that so obvious to you, that we "experience", built into the FRW metric, mathematically?

It isn't. The FRW metric is equally valid, mathematically, for either direction of time (expanding or contracting), as I've said several times. We have to make a choice, based on actual observation, that the expanding model better fits the data for cosmology. If we observed the universe to be contracting in the "future" direction of time, we would choose a contracting FRW spacetime as our model, and the math would work just as well. The math alone can't make the choice.

Having said that, I noted in my last post that none of the questions about the direction of time depend on the use, or even the existence, of a set of "comoving" worldlines that are hypersurface orthogonal. So what does the question you just asked, and I just answered, have to do with the Weyl postulate? Wouldn't the same question apply just as well if the actual data showed an expanding universe that wasn't homogeneous and isotropic (so the Weyl postulate was not satisfied and we had to adopt a somewhat different expanding spacetime model, one that didn't have a congruence of "comoving" worldlines that was hypersurface orthogonal)?

 

[TrickyDicky]

It isn't. The FRW metric is equally valid, mathematically, for either direction of time (expanding or contracting), as I've said several times. We have to make a choice, based on actual observation, that the expanding model better fits the data for cosmology. If we observed the universe to be contracting in the "future" direction of time, we would choose a contracting FRW spacetime as our model, and the math would work just as well. The math alone can't make the choice.

Having said that, I noted in my last post that none of the questions about the direction of time depend on the use, or even the existence, of a set of "comoving" worldlines that are hypersurface orthogonal. So what does the question you just asked, and I just answered, have to do with the Weyl postulate? Wouldn't the same question apply just as well if the actual data showed an expanding universe that wasn't homogeneous and isotropic (so the Weyl postulate was not satisfied and we had to adopt a somewhat different expanding spacetime model, one that didn't have a congruence of "comoving" worldlines that was hypersurface orthogonal)?

Once again the postulate was written before there was observations leading to think of expansion. And it already provided a cosmic time and a way to agree about time direction, not about how to label that agreement. The key word here is agreement, not whether we call it future or past.

If you don't know what my question has to do with the Weyl's postulate, I'm afraid we need to leave it here until you do.

 

[PeterDonis]

Once again the postulate was written before there was observations leading to think of expansion. And it already provided a cosmic time and a way to agree about time direction, not about how to label that agreement

I agree that it provided a cosmic time. I don't know that I agree that it provided a way to agree about time direction. I saw that you quoted Weyl as saying that in his model, the geodesics diverged from a single point a finite time in the past. The problem is that mathematically, the time reverse of that model, which has the geodesics converging on a point a finite time in the future, is just as valid. Hypersurface orthogonality alone doesn't pick out a direction of time; both models have comoving worldlines that are hypersurface orthogonal. And hypersurface orthogonality is the only condition I see in the Weyl postulate. Did Weyl give any reason for preferring the expanding model over the contracting one, given that hypersurface orthogonality does not pick out either one over the other? Or, whether Weyl gave an argument or not, do *you* have an argument that somehow gets from hypersurface orthogonality to expansion being preferred over contraction? Or is there some other reason for picking the expanding model if one doesn't already know, by observation, that the universe is expanding?

One other question: how does any of this relate to general covariance? I see the question about the direction of time, but I don't see how it has anything to do with general covariance. Even if general covariance holds (which it does), the question about the direction of time is still there.

 

[TrickyDicky]

Ok, let's forget about picking a direction, what the W. P. gives is a way for anybody to have a reference, this reference being the comoving observers, without the hypersurface orthogonal condition, there is no way to choose a comoving observer time coordinate reference; is this a clearer way to put it?
Now back to general covariance, this was the whole point of the OP, it seems like there are physical laws, those that are not time translation invariant (as I said only a few but usually considered fundamental) that fail to behave in a generally covariant way in the FRW geometry.
How does this happen? Maybe it is best understood with an example that can be agreed by anyone, we all accept that spatial homogeneity is a feature of a certain spacetime slicing, it should be like this to comply with the cosmological principle, any coordinate system of the FRW metric that is not time hypersurface orthogonal should not find homogeneity in the matter distribution. This doesn't affect general covariance because the distribution of matter is not considered a physical law.
However, it is easy to check that those same coordinates that make the FRW spacetime lose its spatial homogeneity produce an observer disagreement about those physical laws that are not time invariant.

 

[PeterDonis]

We seem to be going around in circles.

Ok, let's forget about picking a direction,

But, as I've said for many, many posts now, the physical laws you are concerned about, such as the second law, *only* depend on agreement on the direction of time (meaning which half of the light cones is the future half). Given that agreement, these laws are generally covariant: all observers in whatever state of motion will agree on them. See next comment.

However, it is easy to check that those same coordinates that make the FRW spacetime lose its spatial homogeneity produce an observer disagreement about those physical laws that are not time invariant.

Really? I have said a number of times during this thread that this is not the case: the physical laws you speak of (e.g., the second law) *are* generally covariant, and will be observed to be true by any observers who agree with "comoving" observers on the direction of time. I even gave an example: we, here on Earth, observe the second law to hold, even though we are not at rest relative to "comoving" observers, and our proper time is *not* the same as "cosmic time"; the slicing of spacetime implied by our sense of simultaneity is *not* the same as the "comoving" slicing, in which spacetime appears homogeneous. Spacetime does *not* appear homogeneous to us, even "on average"; as I said before, we see a large dipole anisotropy in the CMBR. (There are other effects as well, such as anisotropy in redshifts of galaxies due to our "proper motion" relative to the comoving frame.) You have not addressed any of these arguments.

Also, the way you state your assertion above invites confusion: you say "coordinates that make the FRW spacetime lose its homogeneity", but we agreed many posts ago that homogeneity is a property of the spacetime, not of a coordinate system; the FRW spacetime is homogeneous and isotropic even if we adopt coordinates that do not make those properties manifest. I think what you meant to say is "coordinates that make the spatial slices no longer *appear* homogeneous". See further comments below on "matter distribution".

what the W. P. gives is a way for anybody to have a reference, this reference being the comoving observers, without the hypersurface orthogonal condition, there is no way to choose a comoving observer time coordinate reference; is this a clearer way to put it?

Yes, but I only agree if it is understood that the "cosmic time" is not the same as proper time for any observer not at rest in the comoving frame, and that this does not in any way contradict general covariance. See my comments above on that.

we all accept that spatial homogeneity is a feature of a certain spacetime slicing, it should be like this to comply with the cosmological principle, any coordinate system of the FRW metric that is not time hypersurface orthogonal should not find homogeneity in the matter distribution. This doesn't affect general covariance because the distribution of matter is not considered a physical law.

Again, I think this way of putting it invites confusion. First of all, as I noted above, we agreed many posts ago that homogeneity is a property of the *spacetime*, not of a particular slicing; a better way to state what I think you meant to say above is that spatial homogeneity is only *manifest* in a particular spacetime slicing. Second, a better term for what I think you meant by "distribution of matter" is "stress-energy tensor" (SET), but the SET is a generally covariant geometric object that appears on the right-hand side of the Einstein Field Equation, so it's not quite correct to say that it is not considered a physical law. A better way to say it would be that the way we typically arrive at a solution to the EFE in cosmology is to make a certain *assumption* about the stress-energy tensor (for example, that it is spatially homogeneous and isotropic, so there will be some coordinate system in which it takes a particular simple form), and then plug into the EFE and solve for the dynamics of the spacetime. Once we have such a solution, both the spacetime curvature and the "matter distribution" throughout the entire spacetime are fixed, and all physical predictions based on them are also fixed, and can't be changed by changing coordinate systems. I think you agree on this, but do you realize that it implies that there *cannot* be observer disagreement on physical laws, once the overall spacetime solution is determined? The only way to change the physical predictions is to change the starting assumptions about the SET, and thereby change the solution of the EFE you are using; that changes which spacetime you are working with, and of course in a different spacetime, with different properties, you will get different physical predictions. (But they will still be generally covariant *with respect to that spacetime*.)

 

[TrickyDicky]

I got distracted by the naughty neutrinos, let's finish this discussion properly.

Really? I have said a number of times during this thread that this is not the case: the physical laws you speak of (e.g., the second law) *are* generally covariant,

Ok, would you explain to me geometrically how can physical laws that involve time asymmetry (like the second law) be generally covariant in a manifold like the FRW universe that is isotropic but not spherically symmetric?, you see spherical symmetry is demanded for any spacetime manifold that is isotropic and that is supposed to be generally covariant for laws that involve time and that are not themselves time symmetric, since in such a spacetime all time derivatives of the metric tensor are set to zero.
As you probably know this spherical symmetry is only seen (in the context of valid solutions of the EFE) in vacuum solutions like Schwarzschild's where it implies a static spacetime by the Birkhoff theorem.

 

[PeterDonis]

I got distracted by the naughty neutrinos, let's finish this discussion properly.

So did I.

Ok, would you explain to me geometrically how can physical laws that involve time asymmetry (like the second law) be generally covariant in a manifold like the FRW universe that is isotropic but not spherically symmetric?

The FRW spacetime *is* spherically symmetric. See below.

you see spherical symmetry is demanded for any spacetime manifold that is isotropic and that is supposed to be generally covariant for laws that involve time and that are not themselves time symmetric, since in such a spacetime all time derivatives of the metric tensor are set to zero.
As you probably know this spherical symmetry is only seen (in the context of valid solutions of the EFE) in vacuum solutions like Schwarzschild's where it implies a static spacetime by the Birkhoff theorem.

Birkhoff's theorem only applies to vacuum solutions. The FRW spacetime is not a vacuum solution. For a non-vacuum solution, you can have a non-stationary metric and still have spherical symmetry and isotropy.

 

[TrickyDicky]

The FRW spacetime *is* spherically symmetric.

It is spatially isotropic but not spherically symmetric in 4-spacetime. Please show a reference where it is stated the FRW metric is spherically symmetric in spacetime.

If you think about symmetries a moment you'll realize that a time asymmetric spacetime like FRW can't be both spherically symmetric in 3-space and 4-spacetime.

For a non-vacuum solution, you can have a non-stationary metric and still have spherical symmetry and isotropy.

Are you sure? see above.

 

[PeterDonis]

It is spatially isotropic but not spherically symmetric in 4-spacetime. Please show a reference where it is stated the FRW metric is spherically symmetric in spacetime.

What does "spherically symmetric in 4-spacetime" mean? The only definition of "spherically symmetric" that I'm aware of involves isometry with respect to the spatial rotation group. See, for example, the Wiki page:

http://en.wikipedia.org/wiki/Spherically_symmetric_spacetime

 

[TrickyDicky]

What does "spherically symmetric in 4-spacetime" mean? The only definition of "spherically symmetric" that I'm aware of involves isometry with respect to the spatial rotation group. See, for example, the Wiki page:

http://en.wikipedia.org/wiki/Spherically_symmetric_spacetime

Last time I checked spacetime was 4-dimensional, has that changed?
The wiki definition is fine with me.

 

[TrickyDicky]

Actually there's some terminology confusion in the wiki article. SO(3) is a spatial symmetry, not a spacetime symmetry.

 

[TrickyDicky]

I guess spherical symmetry is usually referring to the spatial part of the manifold, but since we are dealing with general covariance of the 4-manifold, I am referring to a 4-dimensional symmetry, that wouldn't be able to accommodate asymmetrical laws of physics wrt time.

 

[PeterDonis]

Actually there's some terminology confusion in the wiki article. SO(3) is a spatial symmetry, not a spacetime symmetry.

I guess spherical symmetry is usually referring to the spatial part of the manifold, but since we are dealing with general covariance of the 4-manifold, I am referring to a 4-dimensional symmetry, that wouldn't be able to accommodate asymmetrical laws of physics wrt time.

I don't understand the distinction you're making here. The spatial part of the manifold is part of the manifold, whether the manifold is 3-D space or 4-D spacetime. A spatial symmetry *is* a spacetime symmetry; it's just not a symmetry that includes the time portion of the metric. Birkhoff's theorem says that, for a vacuum solution, the spatial symmetry under SO(3) *requires* the spacetime to be static, i.e., it implies something about the time portion of the metric; but that only applies to a vacuum solution. I'm not aware of any requirement that a non-vacuum solution have any time symmetry in order to be considered spherically symmetric, and I don't see why it would have to, since the key difference with a non-vacuum solution is that you can fill spacetime with a fluid whose density can be uniform in space, so it can still be spherically symmetric (isotropic), but can vary with time. The "density" of a vacuum can't vary with time, either because it's zero, or (if you include a cosmological constant) because it's a constant multiple of the metric.

 

[TrickyDicky]

I don't understand the distinction you're making here. The spatial part of the manifold is part of the manifold, whether the manifold is 3-D space or 4-D spacetime. A spatial symmetry *is* a spacetime symmetry; it's just not a symmetry that includes the time portion of the metric. Birkhoff's theorem says that, for a vacuum solution, the spatial symmetry under SO(3) *requires* the spacetime to be static, i.e., it implies something about the time portion of the metric; but that only applies to a vacuum solution. I'm not aware of any requirement that a non-vacuum solution have any time symmetry in order to be considered spherically symmetric, and I don't see why it would have to, since the key difference with a non-vacuum solution is that you can fill spacetime with a fluid whose density can be uniform in space, so it can still be spherically symmetric (isotropic), but can vary with time. The "density" of a vacuum can't vary with time, either because it's zero, or (if you include a cosmological constant) because it's a constant multiple of the metric.

This is fine but general covariance involves 4-D so it should imply something about the time portion of the metric, don't you think? So how can a manifold that is spherically symmetric in the standard terminology meaning that is isotropic, be also invariant for the form of physical laws under arbitrary coordinate transformations (this includes coordinate transformations that involve the time coordinate) without being also time symmetric?

 

[PeterDonis]

This is fine but general covariance involves 4-D so it should imply something about the time portion of the metric, don't you think? So how can a manifold that is spherically symmetric in the standard terminology meaning that is isotropic, be also invariant for the form of physical laws under arbitrary coordinate transformations (this includes coordinate transformations that involve the time coordinate) without being also time symmetric?

Sure, coordinate transformations include the time coordinate, but why should that have anything to do with time symmetry? General covariance doesn't say that the metric has to look identical in any coordinate system; it just says the laws of physics have to be the same in any coordinate system. Nor does general covariance say that the metric must have the same symmetry in every coordinate system; obviously a metric that looks isotropic in one coordinate system, will not look isotropic in a coordinate system that's in relative motion to the first. That's true even for a static metric; the metric of Schwarzschild spacetime won't look isotropic in a coordinate system that's moving relative to the black hole (nor will it look time-independent). But the Einstein Field Equation will still hold.

 

[TrickyDicky]

Peter, I guess we've reached a blind spot you are not able to get rid of.


Can some actual physicist look at what I'm saying in my previous post and give me an answer?

 

[TrickyDicky]

Is it not demanded in GR that the metric tensor and thus the line element must be generally covariant?

 

[PAllen]

Is it not demanded in GR that the metric tensor and thus the line element must be generally covariant?

That is basically achieved by definition. Given a completely arbitrary metric in one coordinates, its expression in all other coordinates are specified by the transformation rule for covariance. The definition of the this transform quite trivially guarantees that any computation of an invariant based on the metric tensor comes out the same. All observable physics is supposed to be defined in GR as some flavor of invariant or coordinate independent geometric quantity (generally involving the world line of the measuring instrument, so there is observer dependence but not coordinate dependence). Conversely, anything that can only be expressed in a coordinate dependent way cannot be a legitimate observable in GR.