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In special relativity for each event and reference frame we can find a plane of simultaneous events.

I wonder is it possible to do the same in general case in curved space? Is simultaneity even meaningful in GR?

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Simultaneity has absolutely no meaning whatsoever in GR. In SR, we say simultaneity is relative and can't be trusted. In GR, we don't even say that. You might occasionally catch someone using the word "simultaneous", but that doesn't mean what you think. In GR, you can take any event, any reference frame, any set of values and define some arbitrary time-like coordinate that foliates spacetime such that almost anything is "simultaneous". Then, you can change the gauge and the whole notion of time shifts.

What we normally do is choose a metric with some coordinate time. Then we can "say" that all events at the same coordinate time are a "hypersurface of constant time", which is the GR equivalent of "simultaneous". However, you could choose some other time-dependent value. For instance, in inflationary cosmology, we might have a metric with some coordinate time, $t$, and an inflaton field, $\phi$. We might evolve through inflation with respect to constant $t$ hypersurfaces, but then we call the time when $\phi$ reaches some value corresponding to the end of inflation to be simultaneous, even if it isn't a constant $t$ hypersurface. We do this because we want to say that inflation ends everywhere at the same time, which means we are now defining a constant $\phi$ hypersurface as simultaneous. This isn't even a difference in reference frames, all we have to do is choose a different gauge for our metric and we can define a new concept of simultaneity.

So to answer your question, no, simultaneity is a meaningless idea in curved space. What we use is constant hypersurfaces; we choose a parameter that has a desired foliation of spacetime and then slice spacetime such that every point in that slice has the same value of the chosen parameter.

You can arbitrarily define a coordinate time and take its constant hypersurfaces as simultaneous, but that's even less meaningful than in SR.

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    $\begingroup$ An uninformed person with only a cursory understanding of SR and GR (that's me!) might answer this question - Yes! given the premise that SR is just a 'special' case of GR where gravitation and acceleration are not considered. But I guess there's allot more to consider, more to learn. Thanks for an interesting question - and answer. $\endgroup$ – docscience Jun 4 '15 at 20:30
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    $\begingroup$ @docscience: IMO, simultaneity has no real meaning in SR, either. It is only used for pedagogical purposes - basically, to explain SR to someone who is used to thinking in Newtonian terms. By the time you get to explaining GR, the student is expected to be knowledgeable enough to cope without it. $\endgroup$ – Harry Johnston Jun 4 '15 at 21:50
  • $\begingroup$ @Luaan If you can define it in GR, then it exists in GR. Objective simultaneity isn't defined in GR. If you ask a physicist "Define objective simultaneity in the context of GR", that will confuse them. They might respond with "that doesn't make sense" or "There isn't such a thing" or "That question can't be answered". It's like asking someone to define a square root in the context of primary colours. What does that even mean? $\endgroup$ – Jim Jun 8 '15 at 15:39
  • $\begingroup$ Even if we can define our coordinates so arbitrarily, how comes there seems to be a natural choice? E.g. we talk about a supernova happened this many years in the past. This formulation suggests that the earths reference frame was simultaneous to the one of the supernova back then. $\endgroup$ – M. Winter Oct 15 '18 at 20:26
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    $\begingroup$ For instance, the supernova age depends on how much gravity exists along the path the light takes from the supernova to us as well as how fast we move, how fast the source is, how far away it is in the comoving frame, etc. The age of that supernova to us depends on our local gravity well; move outside of that and it is different, even if you move to a similar reference frame. This is what I mean by simultaneity is ill defined. The ages you see given are given in a way we can easily understand and we omit that bit because everyone expects it. $\endgroup$ – Jim Oct 17 '18 at 12:44
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Say we have a congruence of observers with some 4-velocity field $u^{\alpha}$. Locally each observer has a simultaneity plane determined by the Einstein synchronizaton convention just as in SR. Now if these observers can Einstein synchronize their clocks with their infinitesimal neighbors we can "patch" these local simultaneity planes together to make a one-parameter family of global simultaneity surfaces with a global time coordinate that all the observers in the congruence agree on.

This is, as far as I know, the closest one can get to the usual notion of simultaneity in inertial frames in SR. Keep in mind this isn't really special to GR; one can encounter subtleties regarding simultaneity even in rotating frames in SR for example.

Anyways, the condition for such a foliation to exist is the usual Frobenius condition that $u_{[\alpha}\nabla_{\beta}u_{\gamma]} = 0$.

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I believe there are a few candidates notions, but it's not obvious which is the right one.

You can talk about surfaces that are everywhere orthogonal to a timelike Killing field.

If there isn't a timelike Killing field you can always talk about the surface geodesically generated by a timelike vector at $p$: the set of points on geodesics passing through $p$ that are orthogonal to that vector.

I believe both definitions give you planes of simultaneity in a flat Minkowski spacetime, but they can come apart in the more general setting.

(People sometimes talk about Cauchy surfaces -- surfaces that intersect every timelike or null curve exactly once. But these are not equivalent to planes of simultaneity in Minkowski space.)

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  • $\begingroup$ I think you should be clearer what you mean: planes of simultaneity in Minkowski space are Cauchy surfaces. Even in SR it must be said too that there's no natural notion of global simultaneity for non-inertial observers. $\endgroup$ – John Davis Jun 4 '15 at 22:17
  • $\begingroup$ All planes of simultaneity are Cauchy surfaces, but not vice versa. You can have spacelike hypersurfaces in Minkowski spacetime that are Cauchy, but have "bumps" in them for example, and so aren't planes of simultaneity. $\endgroup$ – Andrew Bacon Jun 4 '15 at 22:22
  • $\begingroup$ On the other hand, two equivalent definitions of a plane of simultaneity in Minkowski spacetime are (i) a surface orthogonal to a timelike Killing field, or (ii) a surface geodesically generated from a timelike vector. Both of these definitions make sense in the GR case, although they can in principle come apart. $\endgroup$ – Andrew Bacon Jun 4 '15 at 22:27
  • $\begingroup$ Yes, but nevertheless planes of simultaneity are Cauchy surfaces and this is a key property that can't be brushed aside. What we are essentially doing is to use the symmetry of Minkowski spacetime to map a special class of observers to a special class set of foliations, which gives a natural definition of simultaneity ot those observers. $\endgroup$ – John Davis Jun 4 '15 at 23:11
  • $\begingroup$ I would also say that your definition of a plane of simultaneity generated from a timelike vector fails to make sense in GR unless it generates a Cauchy surface. $\endgroup$ – John Davis Jun 4 '15 at 23:13
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Simultaneity has a perfectly well defined meaning in SR, as well as in GR. The answer of FenderLesPaul says this in a refined mathematical way. Simultaneity in GR is a generalization of the same concept in SR: given an observer $O$, it is possible to define a 3D space like sub-manifold called "present of $O$". Events in the present of $O$ are simultaneous. In GR the present of a single observer is defined "in its vicinity": since the spacetime of GR is locally flat, the same construction of SR is applied in the so-called "tangent space". If you want to extend the concept everywhere you need a family of observers (namely a family of world lines) and to fulfill the Frobenius condition.

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