Worldline congruence and general covariance

[PeterDonis]

I still have problems with the bolded phrase. This seems like a coordinate condition.

It's true that you can define coordinates with either direction of time. For example, I could define "inverted" FRW coordinates where the sign of t was reversed and nothing else was changed; in those coordinates, the expansion would be negative and the universe would be "contracting". However, there would be no way to do a Lorentz transformation at any event between those "inverted" coordinates and standard FRW coordinates (more precisely, between a local patch of one and a local patch of the other). So local Lorentz invariance is enough to ensure that, if we pick a direction of time in one coordinate system, any others that we relate to it must have the same direction of time. I guess if we wanted to be really careful, we would have to say that local Lorentz invariance is part of "general covariance", so general covariance does require you to pick a time orientation. In principle you could pick either one (since the expanding and contracting FRW models are both valid solutions of the EFE), but since we actually observe the universe to be expanding in the same direction of time as we feel ourselves to be "moving", we pick that direction of time as the "future".

 

[TrickyDicky]

It's true that you can define coordinates with either direction of time. For example, I could define "inverted" FRW coordinates where the sign of t was reversed and nothing else was changed; in those coordinates, the expansion would be negative and the universe would be "contracting". However, there would be no way to do a Lorentz transformation at any event between those "inverted" coordinates and standard FRW coordinates (more precisely, between a local patch of one and a local patch of the other). So local Lorentz invariance is enough to ensure that, if we pick a direction of time in one coordinate system, any others that we relate to it must have the same direction of time. I guess if we wanted to be really careful, we would have to say that local Lorentz invariance is part of "general covariance", so general covariance does require you to pick a time orientation. In principle you could pick either one (since the expanding and contracting FRW models are both valid solutions of the EFE), but since we actually observe the universe to be expanding in the same direction of time as we feel ourselves to be "moving", we pick that direction of time as the "future".

Sure, but you are ultimately agreeing here that in the FRW metric, expansion/contraction is coordinate dependent, in a similar way that entropy effects would be (as you say "we" are the ones that pick the correct time coordinate afterwards), and we all know this dependency is generally considered the hallmark of an unphysical effect. (See Ben's post)

 

[PeterDonis]

Sure, but you are ultimately agreeing here that in the FRW metric, expansion/contraction is coordinate dependent

I think you are using the term "coordinate dependent" here in a different sense than Ben was using it in his post. I would use the term "direction of time dependent", which is just another way of saying "time asymmetric". But the time asymmetry can be described in entirely coordinate independent terms, except for the choice of which side of the asymmetry is to be called the "future" side, i.e., except for picking a direction of time. (For example, in the FRW spacetime--we'll assume we're talking about the case where it doesn't recollapse, to avoid any issues with the closed, recollapsing version being "time symmetric"--the time asymmetry can be described in coordinate-independent terms simply by saying that there is a curvature singularity at one "end" of time, but not the other. The only question then is whether we call that end the "past" or the "future" end. See below.)

But as I've said several times already, the only thing required to pick a direction of time is to pick which half of the light cone is the "future" half, and in any spacetime meeting a very general set of conditions, which I outlined earlier (and FRW spacetime, as well as any other spacetime that's been considered as physically reasonable, as far as I know, meets those conditions), once you've made that choice at any particular event, you can continuously extend it throughout the spacetime. And since the light cones are invariant geometric features of the spacetime (i.e., they are not coordinate dependent), the choice of which half of the light cones is the "future" half will also be coordinate independent, except for the (trivial, in my view--but see below) fixing of the sign of the time coordinate.

So I guess what this boils down to is: the fact that a particular spacetime is time asymmetric is not coordinate dependent. And given a choice of which half of the light cones is the "future" half, the physics in such a spacetime is not coordinate dependent. That choice itself could be considered "coordinate dependent" in the sense that it fixes the sign of the time coordinate; but I don't think this is a big issue, because it's inherent in the very fact that the spacetime is time asymmetric.

I should also stress that I am not saying there are no interesting physical questions left once we've done everything I describe above. It is definitely an interesting physical question *why* we find that the direction of time we experience in our consciousness is the same direction of time in which the universe is expanding. (There is also the question of why it's the same direction of time in which the second law holds, but I answered that in an earlier post: our conscious perception of time depends on the formation of memories, and the formation of memories requires entropy increase.) But that question has nothing to do with general covariance, precisely because we can formulate it in coordinate-independent terms (I pretty much just did; if someone insists on pedantic exactitude, just rephrase what I said above in terms of light cones), so it will arise regardless of how we assign coordinates. Even if we adopt coordinates in which the sign of time is reversed (so we say the universe is "contracting" instead of "expanding"), the same question still arises: we just phrase it as "why is the universe contracting in the same direction of time in which we remember things?" instead of "why is the universe expanding in the same direction of time in which we anticipate things?" There's no physical difference between these versions of the question; it's just a difference in wording.

 

[TrickyDicky]

I think you are using the term "coordinate dependent" here in a different sense than Ben was using it in his post. I would use the term "direction of time dependent", which is just another way of saying "time asymmetric". But the time asymmetry can be described in entirely coordinate independent terms, except for the choice of which side of the asymmetry is to be called the "future" side, i.e., except for picking a direction of time. (For example, in the FRW spacetime--we'll assume we're talking about the case where it doesn't recollapse, to avoid any issues with the closed, recollapsing version being "time symmetric"--the time asymmetry can be described in coordinate-independent terms simply by saying that there is a curvature singularity at one "end" of time, but not the other. The only question then is whether we call that end the "past" or the "future" end. See below.)

But as I've said several times already, the only thing required to pick a direction of time is to pick which half of the light cone is the "future" half, and in any spacetime meeting a very general set of conditions, which I outlined earlier (and FRW spacetime, as well as any other spacetime that's been considered as physically reasonable, as far as I know, meets those conditions), once you've made that choice at any particular event, you can continuously extend it throughout the spacetime. And since the light cones are invariant geometric features of the spacetime (i.e., they are not coordinate dependent), the choice of which half of the light cones is the "future" half will also be coordinate independent, except for the (trivial, in my view--but see below) fixing of the sign of the time coordinate.

So I guess what this boils down to is: the fact that a particular spacetime is time asymmetric is not coordinate dependent. And given a choice of which half of the light cones is the "future" half, the physics in such a spacetime is not coordinate dependent. That choice itself could be considered "coordinate dependent" in the sense that it fixes the sign of the time coordinate; but I don't think this is a big issue, because it's inherent in the very fact that the spacetime is time asymmetric.

I should also stress that I am not saying there are no interesting physical questions left once we've done everything I describe above. It is definitely an interesting physical question *why* we find that the direction of time we experience in our consciousness is the same direction of time in which the universe is expanding. (There is also the question of why it's the same direction of time in which the second law holds, but I answered that in an earlier post: our conscious perception of time depends on the formation of memories, and the formation of memories requires entropy increase.) But that question has nothing to do with general covariance, precisely because we can formulate it in coordinate-independent terms (I pretty much just did; if someone insists on pedantic exactitude, just rephrase what I said above in terms of light cones), so it will arise regardless of how we assign coordinates. Even if we adopt coordinates in which the sign of time is reversed (so we say the universe is "contracting" instead of "expanding"), the same question still arises: we just phrase it as "why is the universe contracting in the same direction of time in which we remember things?" instead of "why is the universe expanding in the same direction of time in which we anticipate things?" There's no physical difference between these versions of the question; it's just a difference in wording.

I find your position reasonable.
I just find the "time direction dependency" an awkward point. This direction property seems defining for the time coordinate.

 

[PeterDonis]

This direction property seems defining for the time coordinate.

I'm curious why it seems this way. There's no corresponding requirement for the space coordinates; nobody objects if I flip the direction of the x axis, let's say. Why should the time axis, as a coordinate, be any different? Is it just because our conscious experience picks out a direction of time and it seems "wrong" to pick a sign for the time coordinate that doesn't match that direction?

 

[TrickyDicky]

I'm curious why it seems this way. There's no corresponding requirement for the space coordinates; nobody objects if I flip the direction of the x axis, let's say. Why should the time axis, as a coordinate, be any different? Is it just because our conscious experience picks out a direction of time and it seems "wrong" to pick a sign for the time coordinate that doesn't match that direction?

Yeah, this is one of the old good questions that doesn't seem to have an answer within physics yet.