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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!)

Qmechanic
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JakaQuan
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3 Answers3

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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."

soulphysics
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11

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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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.

benrg
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