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My question is about the extent to which "now" is defined in GR.

In Minkowski spacetime, it's possible to define a "now" for an inertial observer by finding a spacelike 3-plane such that, in the observer's frame, all 4-vectors in the 3-plane have zero time component (or something like that - apologies, my geometry is a bit rusty - anyway, a 3-plane in which all vectors are orthogonal to the tangent vector of the observer's world line). This plane can be defined globally, so that my "now" is a slice through the whole of Minkowski space.

My question is whether it's possible to define such a thing in general relativity. So, for instance, can I meaningfully speak about what the Andromeda Galaxy is doing "right now"? Or is the "present moment" something that can only be locally defined? I remember seeing something in a Roger Penrose book about this, but I can't find the reference (if anyone knows it, please let me know!)

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since (by strong equivalence principle) in GR one can locally define an inertial frame like SR, yes it is but not same globally (except when the manifold is flat) –  Nikos M. Aug 19 at 11:45
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Somewhat related: what is the closest GR equivalent of a time slice? –  romkyns Aug 20 at 0:04

3 Answers 3

What you're asking about is the existence of surfaces of simultaneity. In SR, surfaces of simultaneity can be defined by measurement procedures such as Einstein synchronization, and they turn out to depend on one's frame of reference.

In GR it gets a lot tougher to do this. We don't even have global frames of reference. It turns out that what you need in order to define a notion of simultaneity is for the spacetime to be static. Staticity means essentially that the spacetime has a timelike Killing vector, and it also isn't rotating.

As an example, the Schwarzschild spacetime (for a black hole) isn't static because the killing vector $\partial_t$ is only timelike outside the event horizon; on the inside, it's spacelike. This means that there are no static observers inside the event horizon, and therefore it doesn't make sense to talk about static observers carrying out Einstein synchronization.

So, for instance, can I meaningfully speak about what the Andromeda Galaxy is doing "right now"? Or is the "present moment" something that can only be locally defined?

In cosmological terms, the Andromeda Galaxy is in our local neighborhood. We can make a frame of reference big enough to encompass both our galaxy and the Andromeda Galaxy. We could certainly carry out Einstein synchronization between ourselves and an observer in the Andromeda Galaxy, if we were in a state of mutual rest. But for a cosmologically distant galaxy, none of this will work.

Cosmological spacetimes do, however, allow us to define a preferred time coordinate, which is the time measured on the clock of an observer who is moving with the Hubble flow. This is a special property of these spacetimes, and should not be interpreted as implying that we could do anything like Einstein synchronization with a cosmologically distant observer.

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I don't see how being static is relevant. If I interpret this to mean that the spacetime has a timelike Killing field, then this just means that it admits a sense of time-translation symmetry. But that just means that there is no preferred "origin" for time. And it certainly doesn't guarantee the existence of a Cauchy surface. –  soulphysics Aug 20 at 11:10
    
However, if in a globally hyperbolic spacetime, you take the timelike Killing fields to distinguish a "preferred" class of Cauchy surfaces (those that are orthogonal to the Killing fields) then I suppose you might call those simultaneity surfaces. But it is not clear to me what motivates this definition. In the "spirit" of general relativity, any Cauchy surfaces should equally do, whether or not it is orthogonal to a timelike Killing field. –  soulphysics Aug 20 at 11:18
    
@soulphysics: if in a globally hyperbolic spacetime, you take the timelike Killing fields to distinguish a "preferred" class of Cauchy surfaces (those that are orthogonal to the Killing fields) Global hyperbolicity is an extremely loose condition. Possession of a timelike Killing field is an extremely restrictive condition. Essentially all spacetimes of any physical interest are globally hyperbolic. Very few have a timelike Killing field. –  Ben Crowell Aug 20 at 17:32
    
Hi Ben -- I agree with everything you're saying; we clearly don't disagree on technical matters. I just don't understand why you take the timelike Killing fields to define preferred surfaces of simultaneity. What's the motivation for that? From the perspective of general relativity, why would this frame be preferred over any arbitrary timelike vector field? –  soulphysics Aug 21 at 8:11

It sounds like you're interested in when a spacetime admits a Cauchy surface. The answer is that every spacetime that is globally hyperbolic has this property. This was proved by Geroch in 1970 (article here, see Section 5). This includes most of the textbook relativistic spacetimes --- Schwarzschild, Kerr, FLRW, and many others.

But there are some spacetimes that do not have this property. For example, Gödel spacetime is an exact solution to Einstein's equations that does not at admit a global Cauchy surface. The mathematician Kurt Gödel that discovered it actually thought this provided evidence for an idealist view of time, in which there is no global "now."

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A Cauchy surface isn't necessarily a surface of simultaneity. For example, in Minkowski space, you can make all kinds of Cauchy surfaces that are not flat and that aren't surfaces of simultaneity for any observer. –  Ben Crowell Aug 19 at 18:48
    
In an arbitrary (globally hyperbolic) spacetime, the best meaning I know to give to "simultaneous points" is just "points on the same Cauchy surface". In special relativity there is of course a preferred class of Cauchy surfaces, namely those that are orthogonal to the inertial timelike vector fields. But there is no such preferred structure for arbitrary general relativistic spacetimes. –  soulphysics Aug 20 at 11:16
    
But there is no such preferred structure for arbitrary general relativistic spacetimes. As explained in my answer, there is a condition for such a preferred structure to exist. The condition is that the spacetime is static. –  Ben Crowell Aug 20 at 17:33
    
Good: we agree on the technical matters then. But I still don't understand why you adopt this condition; more in the comments below your answer. –  soulphysics Aug 21 at 8:13

Planes of simultaneity in special relativity don't really mean much of anything. The real physical structure of spacetime is in the light cones. The takeaway from "relativity of simultaneity" is not that there are "different time orderings for different observers", but rather that there is no meaningful time ordering for spacelike separated events. They happen independently and not in lockstep; physics is local.

Planes of simultaneity in Minkowski spacetime are a sort of retreat to quasi-Newtonian physics; instead of accepting the fact that space and time are not separable, you separate them in a physically meaningless way and pretend that things in Andromeda are really happening now for you and four days from now for the guy walking by you, even though those events can only affect you much later via light-speed signals that will reach the two of you less than a second apart (assuming you're both still on earth at the time).

In general relativity, you still have the physically meaningful worldlines and light cones, but no longer have the physically meaningless global plane normal to a worldline at a point. There's no meaningful "now over there" even in SR, but GR makes it clearer.

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