Conditions for spacetime to have flat spatial slices

[JDoolin]

I'm a little troubled that the Robertson Walker chart is either mapping coordinate time or proper time, depending on who gives me an answer. If it is supposed to be proper time, then I understand what you mean by events "not lining up" The problem is that the intersection of two world-lines on a space versus proper time graph do not indicate co-location in space (a collision).

If the diagram represents space vs. proper time, also, this casts into doubt the presence of the light-cones *anywhere* in the diagram. Light does not age, and therefore it should not move forward in proper time. All the light rays in a space vs. proper time diagram should be horizontal. And the intersection of light rays and world-lines on the space versus proper time diagram are all but meaningless. (since as mentioned before, they don't "line up")

PeterDonis told me that Lewis Epstein was encouraging the confusion of proper time and coordinate time, but it appears to me that Robertson Walker have already successfully muddled the two. Epstein, in fact, made great strides in clarifying the difference.

 

[Mentz114]

I'm a little troubled that the Robertson Walker chart is either mapping coordinate time or proper time, depending on who gives me an answer.

I'm not sure what you mean by 'chart' or 'mapping'. Care to elaborate ?

The problem is that the intersection of two world-lines on a space versus proper time graph do not indicate co-location in space (a collision).

The coincidence of worldlines always means collision. Same place, same time. No transformation can change that.

Coordinate time is a parameter in the model, proper time is the integral of the Lorentzian interval along a worldline.

 

[JDoolin]

I'm not sure what you mean by 'chart' or 'mapping'. Care to elaborate ?


The coincidence of worldlines always means collision. Same place, same time. No transformation can change that.

Coordinate time is a parameter in the model, proper time is the integral of the Lorentzian interval along a worldline.

It's my question in post #42. You're sometimes saying that the vertical coordinate in the Robertson-Walker diagram represents the proper time of particles. Other times, you're acting like it is the actual time passed by the central observer. You can either have one or the other. Not both.

The only place where those two definitions can be shared is along the single line representing the worldline of the "stationary" particle.

As I've pointed out before, the parameters of \tau and t are very different. In an x vs t diagram, you are correct. The coincidence of worldlines always mean a collision, and no legitimate transformation can change that...EXCEPT FOR a transformation into an x vs \tau coordinate system, which is totally NOT a legitimate transformation, because \tau is not a coordinate. It is a property of matter.

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.

 

[Mentz114]

OK.

On a spacetime diagram, the vetical axis is t, coordinate time. For the vertical worldline we can calculate the proper time on the clock,

<br /> d\tau^2=dt^2-dx^2=dt^2 \Rightarrow d\tau=dt<br />

because dx=0. For the tilted worldlines we get a different proper time. So every observer thinks his clock is showing coordinate time. Can you see why people say that the vertical axis shows proper time ? Because for the stationary observer it does. None of this is of the least significance. Assume that in a spacetime diagram it is t vs x.

 

[JDoolin]

OK.

On a spacetime diagram, the vetical axis is t, coordinate time. For the vertical worldline we can calculate the proper time on the clock,

<br /> d\tau^2=dt^2-dx^2=dt^2 \Rightarrow d\tau=dt<br />

because dx=0. For the tilted worldlines we get a different proper time. So every observer thinks his clock is showing coordinate time. Can you see why people say that the vertical axis shows proper time ? Because for the stationary observer it does. None of this is of the least significance. Assume that in a spacetime diagram it is t vs x.

This is why I'm saying I get different answers depending on who answers. http://www.astro.ucla.edu/~wright/cosmo_03.htm" "But in cosmological models we do have cosmic time, which is the proper time since the Big Bang measured by comoving observers, and it can be used to set up a deck of cards."

It has to be proper time that he's talking about, because, he then proceeds to do a Galilean Transformation on the diagram. The reason that it's valid to do this transformation is that the horizontal lines in the diagram represent lines of constant PROPER time. (Specifically, proper time for particular observers, who are following particular worldlines.)

While I have no objection to graphing proper time vs. position, this way, I think we deserve a certain amount of clarity about which is being graphed, because it can be either one or the other, but not both. If we have a space vs. proper-time graph, the speed of light should be represented as horizontal lines; not light cones. If we have a space vs TIME graph, then it is not valid to do a galilean transformation, because the relativity of simultaneity has to come into play.

(I don't know what you do, when the Lorentz Transformations are only valid "locally" but you need to have some kind of transformation that looks like the Lorentz Tranformations locally, and whatever they do with the rest of the stretching space, I really cannot say.)

 

[Mentz114]

I can see why you're confused. The FRW model is special in this way - because of the homogenity and isotropy, every worldline is the same.

While this diagram is drawn from our point-of-view, the Universe is homogeneous so the diagram drawn from the point-of-view of any of the galaxies on the diagram would be identical.

From my previous post you can see that for the stationary observer proper time and coordinate time coincide. Now combine that with the quote above, and one concludes that all the observers' clocks are showing coordinate time.

Ned Wright says 'we do not have absolute time', which I find puzzling, but we do have 'cosmic time', which coincides with any observer if we choose them to be at rest ( vertical wl). But coordinate time also has the property that it coincides with the stationary clock time. Therefore cosmic time = coordinate time, as far as I can see.

Looking at the spacetime diagram that shows all the worldines as straight and radiating from t=0, r=0. In a Minkowski diagram, those lines should all be curving away from the central stationary observer, because they are accelerating. They are straight because there's been some serious deformation of the axes, so this is not a Minkowski ST diagram. Therefore it's not odd that to change frames we use Gallilean transformations. If we plotted the worldlines in a Minkowski diagram, the LT would connect instants on the worldlines.

There are lots of coordinate transformations going on in those notes that are not explicitly stated. But he does say how the conformal coordinates are found

Sometimes it is convenient to "divide out" the expansion of the Universe, and the space-time diagram shows the result of dividing the spatial coordinate by a(t). Now the worldlines of galaxies are all vertical lines.

(my emphasis)

 

[PeterDonis]

This is why I'm saying I get different answers depending on who answers. http://www.astro.ucla.edu/~wright/cosmo_03.htm" "But in cosmological models we do have cosmic time, which is the proper time since the Big Bang measured by comoving observers, and it can be used to set up a deck of cards."

It has to be proper time that he's talking about, because, he then proceeds to do a Galilean Transformation on the diagram. The reason that it's valid to do this transformation is that the horizontal lines in the diagram represent lines of constant PROPER time. (Specifically, proper time for particular observers, who are following particular worldlines.)

While I have no objection to graphing proper time vs. position, this way, I think we deserve a certain amount of clarity about which is being graphed, because it can be either one or the other, but not both.

I agree that some clarification of terms is in order:

(1) The FRW *spacetime* is a geometric object; it can be described using a number of different coordinate systems or charts (or metrics--see next item), but it's the same geometric object no matter what chart/metric is used to describe it. So statements about invariant quantities, like the proper time experienced by a given observer with a given worldline between two events on that worldline (e.g., the big bang and the Earth "now", events O and N in your terminology from an earlier post), are independent of the specific coordinate chart/metric being used.

(2) What I've been calling the Robertson-Walker *metric* is a specific coordinate system used to describe the FRW spacetime, along with the expression for the metric using that coordinate system. (The expression "coordinate chart" is also sometimes used to refer to a coordinate system.) This coordinate system is useful because it has the property I described, that in this particular coordinate system, the "time" coordinate t happens to directly represent the proper time experienced by "comoving observers", observers who remain at the same spatial coordinates (r, theta, phi) for all time. But this is a property of the particular coordinate system; a different coordinate system may not have it (see next item).

(3) What Mentz114 is calling the Painleve chart for the FRW spacetime is a different coordinate system used to describe that spacetime, in which the metric looks quite different than it does in the Robertson-Walker coordinate system. In this coordinate system, the "time" coordinate t does *not* directly represent the proper time of "comoving" observers (at least, I don't think it does based on looking at the metric--Mentz114, please correct me if I'm wrong). But again, this is a property of the specific coordinate system.

(4) Finally, a note on "coordinate time" vs. "proper time". Coordinate time is just what it says: in a coordinate system, there will be one coordinate which is "timelike" and three which are "spacelike" (unless you're using null coordinates, which I won't go into here). The one timelike coordinate is "coordinate time". It's nice if you can set up the coordinate system so the timelike coordinate has at least some kind of actual physical meaning, but there's nothing that requires you to. Proper time, on the other hand, is an invariant quantity, and as I noted above, it must be the same regardless of what coordinate system is used to calculate it. But, as I've also noted, in order to even talk about proper time you have to specify a worldline and events on that worldline (as in the example I gave, the proper time experienced by a "comoving observer" between events O and N).

With the above clarification of terms, hopefully the meaning of what Ned Wright was saying is clearer. Because he explicitly says that he is using the proper time of comoving observers to set up his "deck of cards", I deduce (though he doesn't say so explicitly) that he is using the Robertson-Walker coordinate system. In that coordinate system, the "Galilean transformation" he does is just a spatial translation, moving the origin of the spatial coordinates to a different comoving observer, without changing anything else.

If we have a space vs. proper-time graph, the speed of light should be represented as horizontal lines; not light cones. If we have a space vs TIME graph, then it is not valid to do a galilean transformation, because the relativity of simultaneity has to come into play.

I don't understand either of these statements. What is the difference between a "space vs. proper time" and "space vs. time" graph? In either case, you have the horizontal dimension representing space and the vertical dimension representing time (the only difference is that in the first case, you're specifying that the time coordinate you're using in the vertical dimension directly represents proper time for observers who stay at constant values of all the space coordinates, as with, for example, the Robertson-Walker coordinates as I defined them above). And in either case, light can't possibly travel on horizontal lines, because light doesn't travel in lines of constant time; it travels on null worldlines, and the lines of constant time (the horizontal lines in the diagrams) are spacelike lines. Also, the "Galilean transformation" Ned Wright does leaves the surfaces of constant time invariant; as I noted above, it's just a spatial translation, without changing anything else. Pure spatial translations don't bring in any of the issues involved with relativity of simultaneity. (This is true even in the standard Minkowski coordinates of special relativity; I can always move the spatial origin to a new spatial location without affecting anything else except the specific space coordinates I assign to specific events.)

 

[JDoolin]

I agree that some clarification of terms is in order:

(1) The FRW *spacetime* is a geometric object; it can be described using a number of different coordinate systems or charts (or metrics--see next item), but it's the same geometric object no matter what chart/metric is used to describe it. So statements about invariant quantities, like the proper time experienced by a given observer with a given worldline between two events on that worldline (e.g., the big bang and the Earth "now", events O and N in your terminology from an earlier post), are independent of the specific coordinate chart/metric being used.

2) What I've been calling the Robertson-Walker *metric* is a specific coordinate system used to describe the FRW spacetime, along with the expression for the metric using that coordinate system. (The expression "coordinate chart" is also sometimes used to refer to a coordinate system.) This coordinate system is useful because it has the property I described, that in this particular coordinate system, the "time" coordinate t happens to directly represent the proper time experienced by "comoving observers", observers who remain at the same spatial coordinates (r, theta, phi) for all time. But this is a property of the particular coordinate system; a different coordinate system may not have it (see next item).

(3) What Mentz114 is calling the Painleve chart for the FRW spacetime is a different coordinate system used to describe that spacetime, in which the metric looks quite different than it does in the Robertson-Walker coordinate system. In this coordinate system, the "time" coordinate t does *not* directly represent the proper time of "comoving" observers (at least, I don't think it does based on looking at the metric--Mentz114, please correct me if I'm wrong). But again, this is a property of the specific coordinate system.

(4) Finally, a note on "coordinate time" vs. "proper time". Coordinate time is just what it says: in a coordinate system, there will be one coordinate which is "timelike" and three which are "spacelike" (unless you're using null coordinates, which I won't go into here). The one timelike coordinate is "coordinate time". It's nice if you can set up the coordinate system so the timelike coordinate has at least some kind of actual physical meaning, but there's nothing that requires you to. Proper time, on the other hand, is an invariant quantity, and as I noted above, it must be the same regardless of what coordinate system is used to calculate it.

But, as I've also noted, in order to even talk about proper time you have to specify a worldline and events on that worldline (as in the example I gave, the proper time experienced by a "comoving observer" between events O and N).

Yes, proper time (OF A WORLDLINE) is an invariant quantity, but proper time is NOT A COORDINATE. Coordinates are contravariant; not invariant. Proper time is a property of a worldline. By itself, proper time does not tell you when the event occurs, unless you also know the world-line-path that the particle has followed.

A good coordinate time will tell you when the event occurred, even if you don't know the world-line-path taken to arrive at that event.

With the above clarification of terms, hopefully the meaning of what Ned Wright was saying is clearer. Because he explicitly says that he is using the proper time of comoving observers to set up his "deck of cards", I deduce (though he doesn't say so explicitly) that he is using the Robertson-Walker coordinate system. In that coordinate system, the "Galilean transformation" he does is just a spatial translation, moving the origin of the spatial coordinates to a different comoving observer, without changing anything else.



I don't understand either of these statements. What is the difference between a "space vs. proper time" and "space vs. time" graph? In either case, you have the horizontal dimension representing space and the vertical dimension representing time (the only difference is that in the first case, you're specifying that the time coordinate you're using in the vertical dimension directly represents proper time for observers who stay at constant values of all the space coordinates, as with, for example, the Robertson-Walker coordinates as I defined them above). And in either case, light can't possibly travel on horizontal lines, because light doesn't travel in lines of constant time; it travels on null worldlines, and the lines of constant time (the horizontal lines in the diagrams) are spacelike lines. Also, the "Galilean transformation" Ned Wright does leaves the surfaces of constant time invariant; as I noted above, it's just a spatial translation, without changing anything else. Pure spatial translations don't bring in any of the issues involved with relativity of simultaneity. (This is true even in the standard Minkowski coordinates of special relativity; I can always move the spatial origin to a new spatial location without affecting anything else except the specific space coordinates I assign to specific events.)

In Minkowski Spacetime, a spatial translation is done by taking the paper your graph is drawn on, and moving it, to the left or right. A velocity change is done by Lorentz Transformation.

But in Friedmann-Walker, (Let me see if I've got this right) a pure spatial translation is represented by a Galilean Transformation, and an actual velocity change is simply beyond the scope of General Relativity.

What makes up a geometric object? Are you sure you can graph an invariant vs. a contravariant quantity, and that qualifies as a geometric object? It seems to me that the thing that Robertson Walker did was just set the contravariant coordinate-time equal to the invariant proper time of a bunch of particles that don't even exist, except by statistical average.

I'd like to know why; what was their point? What made this necessary?

I'd also like to know whether, once you set your coordinate time and proper time equal, is it even possible to determine the effect of a large acceleration?

And if I take\tau=t and plug it into d\tau^2=dt^2-dx^2 I get dx=0, suggesting that nothing can ever change its position. (correction: This is to be expected since the \tau represents properties of particles for whom dx is equal to zero. However, I still think that saying \tau is "timelike" is an overgeneralization. \tau and t are really quantities of a fundamentally different sort.)

 

[PeterDonis]

Yes, proper time (OF A WORLDLINE) is an invariant quantity, but proper time is NOT A COORDINATE. Coordinates are contravariant; not invariant. Proper time is a property of a worldline. By itself, proper time does not tell you when the event occurs, unless you also know the world-line-path that the particle has followed.

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).

(Actually, technically *any* definition of a coordinate system with a time coordinate implicitly specifies a "family of observers", in a sense I'll define below; but that family may not make much sense physically, depending on how the coordinate system is defined. See next comment.)

A good coordinate time will tell you when the event occurred, even if you don't know the world-line-path taken to arrive at that event.

But when the event occurred is frame-dependent (i.e., coordinate system dependent); it's not an invariant. That means that specifying when an event occurs requires specifying a coordinate system; and specifying a coordinate system requires specifying a time coordinate; and specifying a time coordinate implicitly specifies a family of worldlines, the integral curves of the vector field defined by that time coordinate (more precisely, defined by the partial derivative operator with respect to that time coordinate).

Depending on the particular coordinate system, this family of curves may or may not make much sense physically, interpreted as the worldlines of a family of observers; but in spacetimes with particular symmetries, those symmetries will pick out certain families of curves, and therefore certain coordinate systems that have those curves as the integral curves of their time coordinates; and those curves will (at least in the cases we're discussing) make sense as the worldlines of a family of observers. The time coordinate of the coordinate system then directly represents the proper time of those observers, in the sense I described above; and therefore specifying when events occur according to this time coordinate *is* specifying the proper time of those events with reference to that family of observers, without having to know the specific worldline path taken to arrive at the event.

In all these cases, though, when events occur is still frame-dependent, and as I think I've said before, I think it's a mistake to insist on thinking about relativistic physics in terms of frame-dependent quantities. Coordinate systems can be a calculational convenience, but they are not *necessary* for doing physics; you can write all the actual physics, if necessary, solely in terms of invariant quantities, without ever specifying a coordinate system.

In Minkowski Spacetime, a spatial translation is done by taking the paper your graph is drawn on, and moving it, to the left or right. A velocity change is done by Lorentz Transformation.

But in Friedmann-Walker, (Let me see if I've got this right) a pure spatial translation is represented by a Galilean Transformation, and an actual velocity change is simply beyond the scope of General Relativity.

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.

What makes up a geometric object? Are you sure you can graph an invariant vs. a contravariant quantity, and that qualifies as a geometric object? It seems to me that the thing that Robertson Walker did was just set the contravariant coordinate-time equal to the invariant proper time of a bunch of particles that don't even exist, except by statistical average.

I'd like to know why; what was their point? What made this necessary?

The FRW metric is determined by the condition that the universe is homogeneous and isotropic (it looks the same everywhere and in all directions). We know this condition isn't exactly fulfilled by our universe, but it's close, and the condition makes the mathematics tractable for expressing solutions in closed form. More detailed models take the FRW solution as a starting point and do an expansion about it in powers of small perturbations from exact isotropy, which gives more precise answers but requires numerical solutions.

The existence of a family of observers whose proper time is directly represented by the time coordinate in the FRW metric is something that *appears* in the solution, not something that is put in at the start. Basically, it amounts to looking at the integral curves of the FRW time coordinate, as I described above.

I still think that saying \tau is "timelike" is an overgeneralization. \tau and t are really quantities of a fundamentally different sort.)

As you note in your correction, saying that coordinate time directly represents proper time is not the same as setting tau equal to t in the metric to begin with; you first impose a condition on the metric (such as your dx = 0, which I stated in an earlier post as dr = dtheta = dphi = 0), and then see what the metric looks like with the condition imposed. But that also means that even though tau and t *are* different kinds of quantities, there can still be a relationship between them under certain conditions. For example, we impose the condition that all the space coordinates are constant on the FRW metric and obtain d\tau^{2} = dt^{2}. Since there are no other coefficients on either side, that means that tau and t must be measured in the same units, so it makes sense to talk about t "representing" tau for the particular family of observers that meets the condition we imposed. Also, since the signs are the same on both sides, tau must represent a timelike interval (positive squared length, using the sign convention I've been using), so it makes sense to call tau "timelike". (And in fact, this latter property holds in general: for *any* observer moving on a timelike worldline, tau along that worldline will have a positive square.)

 

[PeterDonis]

What makes up a geometric object? Are you sure you can graph an invariant vs. a contravariant quantity, and that qualifies as a geometric object?

I realized on reading over my last post that I didn't fully respond to this point. A geometric object is, by definition, invariant; the object itself remains the same regardless of what coordinate system you use to label points in it. You don't "make" a geometric object by graphing anything; you draw graphs to illustrate how a particular coordinate system represents points, curves, etc. in a geometric object (or a portion of one). What you graph are always coordinates, but sometimes, as I noted in my last post, a particular coordinate happens to represent something with direct physical meaning, such as coordinate time representing proper time (in the sense I described in my last post).

 

[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,

<br /> \bf{u} = \frac{\partial}{\partial T} - \sqrt{\frac{2M}{R}} \frac{\partial}{\partial R},<br />

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.