Simultaneity in general relativity

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Hi guys,

in most relativity articles on the internet, and in books, relative simultaneity is always mentioned in the context of special relativity, and that it depends on the state of motion of the intertial reference frame (or observer). But curiosity has brought me another question, which is related to general relativity, and its take on the relativity of simultaneity.

The most known effect on time which is related to GR is gravitational time dilation which is mostly discussed when people talk about GR, but does GR change anything (or add anything) to the definition of relative simultaneity in special relativity. By this, I mean any new criteria or conditions when observers disagree about simultaneity or something like that. I hope somebody could answer me this with a concrete example, similar to the train example by Einstein which describes relative simultaneity of moving observers. I apologize in advance if my language is not very scientific, I'm pretty new to this topic so I hope you don't mind.

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Hi guys,

in most relativity articles on the internet, and in books, relative simultaneity is always mentioned in the context of special relativity, and that it depends on the state of motion of the intertial reference frame (or observer). But curiosity has brought me another question, which is related to general relativity, and its take on the relativity of simultaneity.

The most known effect on time which is related to GR is gravitational time dilation which is mostly discussed when people talk about GR, but does GR change anything (or add anything) to the definition of relative simultaneity in special relativity. By this, I mean any new criteria or conditions when observers disagree about simultaneity or something like that. I hope somebody could answer me this with a concrete example, similar to the train example by Einstein which describes relative simultaneity of moving observers. I apologize in advance if my language is not very scientific, I'm pretty new to this topic so I hope you don't mind.

In GR , there are no frames of reference covering the whole space, so, the notion of simultaneity can only be discussed locally. For a small enough region, the spacetime is locally Minkowski, so the GR notion of simultaneity reduces to the SR notion of simultaneity. For large® extents of space, nothing can be said about simultaneity. There is not much else that can be said on this subject.

Edited by xyzt
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In GR , there are no frames of reference covering the whole space, so, the notion of simultaneity can only be discussed locally. For a small enough region, the spacetime is locally Minkowski, so the GR notion of simultaneity reduces to the SR notion of simultaneity. For large® extents of space, nothing can be said about simultaneity. There is not much else that can be said on this subject.

How large the mentioned portion of space is, is there some kind of barrier which limits the local notion of simultaneity?

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How large the mentioned portion of space is, is there some kind of barrier which limits the local notion of simultaneity?

This is an excellent question. The answer can only be given via experimentation, in the presence of strong gravitational fields, violations of Lorentz covariance start showing up over large "enough" extents. This is due to the effects of the gravitational fields, it is the presence of the gravitational field that makes SR no longer applicable. For example, if we were trying to run a Michelson-Morley experiment very close to the Sun, we might get a non-null result due to the fact that the Sun has a much stronger gravitational field than the Earth.

Edited by xyzt
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How large the mentioned portion of space is, is there some kind of barrier which limits the local notion of simultaneity?

You can speak of the simultaneity of events across all of space, but the property of simultaneity of those events only applies locally.

"A local definition of simultaneity is an assignment to each spacetime event of a spacelike hyperplane that, roughly speaking, determines the events that are locally simultaneous." [E. Minguzzi. Simultaneity in special and general relativity. http://arxiv.org/abs/gr-qc/0506127]

In other words, you can imagine a hyperplane through all of space being like an instant in time, and all events on that hyperplane can be considered locally simultaneous according to some observer, but that hyperplane might not apply to some other observer, for whom the same events couldn't meaningfully be considered simultaneous.

However, if you keep reading the paper I cited and understand it better than I do, you'll see that the way that you'd define these hyperplanes is not unique but depends on a convention of simultaneity. The author selects such a convention, defining "$\bar{C}$-simultaneity, as the most natural and useful in the week field limit." But that doesn't mean that there's any absolute physical meaning to the simultaneity of events defined by that.

Edited by md65536
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How large the mentioned portion of space is, is there some kind of barrier which limits the local notion of simultaneity?

Technically "locally" actually means "at an infinitely small scale." There is, in general no notion of simultaneity in GR for two different points. For short distances you can talk approximately about simultaneity, and at everyday scales/speeds/gravity "short" covers a lot. There's no cutoff or "barrier" as you say, it just depends on how accurate you're trying to be in your calculations and measurements.

Edited by elfmotat
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• 3 years later...

In GR , there are no frames of reference covering the whole space, so, the notion of simultaneity can only be discussed locally. For a small enough region, the spacetime is locally Minkowski, so the GR notion of simultaneity reduces to the SR notion of simultaneity. For large® extents of space, nothing can be said about simultaneity. There is not much else that can be said on this subject.

Do I understand it correctly by saying that from Newtonian physics to SR, simultaneity went from being absolute to relative, and from SR to GR, it more or less "disappeared" (since it can only be discussed locally as you say)?

Please correct me if I am wrong. I try to wrap my head around this.

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Here is a simple analogy.

But beware analogs are never exact, but they can offer enough correspondence to promote understanding.

Right so SR is 'flat or euclidian' GR is not.

Going down to 2 dimensions for the analog consider a plane and a sphere, both of which have two dimensions, X and Y for the plane, Latitude and Longitude for the sphere. Remember a sphere is mathematically a surface, not a solid.

The plane is flat like SR.

The sphere corresponds to GR.

The plane only approximates a small portion of the surface of the sphere with any pretention to accuracy.

The smaller the curvature ( ie the larger the sphere) the greater is this 'patch'.

A function or transformation that maps points on the plane to points on the sphere is called a chart.

Clearly a chart good at the equator is not good for mapping the polar regions.

This is what is meant by local.

In fact no single transformation will put every point of any given sphere into one-to one correspondence with a point on the plane.

There will always be at least one sphere point left out, when every available point on the plane has been used up.

Proving this is the subject of topology.

So the planar grid can never be a global coordinate system for the surface of a sphere.

You have to 'cover' the sphere with reference to at least two different planes.

Of course the sphere is the simplest most regular 2 D surface in 3D.

The real universe is quite irregular so the only global grid is the universe itself, which is effectively saying there is no global grid.

That is enough for now, but does it help?

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Here is a simple analogy.

But beware analogs are never exact, but they can offer enough correspondence to promote understanding.

Right so SR is 'flat or euclidian' GR is not.

Going down to 2 dimensions for the analog consider a plane and a sphere, both of which have two dimensions, X and Y for the plane, Latitude and Longitude for the sphere. Remember a sphere is mathematically a surface, not a solid.

The plane is flat like SR.

The sphere corresponds to GR.

The plane only approximates a small portion of the surface of the sphere with any pretention to accuracy.

The smaller the curvature ( ie the larger the sphere) the greater is this 'patch'.

A function or transformation that maps points on the plane to points on the sphere is called a chart.

Clearly a chart good at the equator is not good for mapping the polar regions.

This is what is meant by local.

In fact no single transformation will put every point of any given sphere into one-to one correspondence with a point on the plane.

There will always be at least one sphere point left out, when every available point on the plane has been used up.

Proving this is the subject of topology.

So the planar grid can never be a global coordinate system for the surface of a sphere.

You have to 'cover' the sphere with reference to at least two different planes.

Of course the sphere is the simplest most regular 2 D surface in 3D.

The real universe is quite irregular so the only global grid is the universe itself, which is effectively saying there is no global grid.

That is enough for now, but does it help?

I think I got the point.

It is easy to draw a straight line (e.g. the x-axis of a specific frame of reference) if you have a flat surface (and SR talks about flat spacetime), however, GR talks about a curved spacetime, hence you cannot describe a straight x-axis in the same way due to the curvation of spacetime. So simultaneity (e.g. the x-axis of a specific frame of reference) is relative in SR but not clearly described in GR.

Is this correct?

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