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So bear with me if you can. I will try to type my understanding (which is bad) to this subject (regarding simultaneity), what's exactly the question and why it has came up.

Why

Reading some papers on numerical relativity (Any search of "Numerical Relativity" or "Magnetically Arrested Disk" will result in papers, not posting certain papers for obvious reason. They study a closed system, and the observer, which is us humans, are at inf distance) they refer to "simulation time".

Let's take Kerr Spacetime geometry for example. They relate that simulation time by multiplying it to $\dfrac{GM}{c^3}$, which is just the "unit time" to go from simulation to "real" time, and that's the proper time that passes for a ZAMO observer on infinity (us). Basically its the same quantity multiplied by unity ($\delta \tau = \delta t$)

That got me thinking because I'm not really that experienced with General Relativity (have studied some books, mostly recently, but still my intuition is not good) how can we correlate the coordinate time to an observer time for events that happened so far away (not referring to information travel speed) so easily.

Me, trying to understand it

So what I will try and elaborate here, are things that I have mostly read on 3+1 Formalism in General Relativity which is actually a free e-book from Éric Gourgoulhon and I only refer to it because one can check Chapter 2,3,4 and find these things for himself (hopefully you won't need for this question as I'll try to explain them to my understanding).

If we consider Hypersurfaces of constant coordinate $t$, then, at each point, we can define a unit vector which is prepandicular to that Hypersurface. An example of this is the following picture:

Hypersurfaces of a spacetime, where $t$ is the coordinate time and $\mathbf{n}$ is the ZAMO/Eulerian timeline at a certain point. There are infinite points, so there are infinite ZAMO observers.

Now, I will post a "part" of text that comments on simultaneity of that observer:

Since $\mathbf{n}$ is a unit timelike vector, it can be regarded as the 4-velocity of some observer. We call such observer an Eulerian observer. The worldlines of the Eulerian observers are thus orthogonal to the hypersurfaces $\Sigma _t$. Physically, this means that the hypersurface $\Sigma _t$ is locally the set of events that are simultaneous from the point of view of the Eulerian observer, according to Einstein-Poincaré simultaneity convention

From this mark, only the local events that are orthogonal to the ZAMO time-like vector are simultaneous.

So from the "Why" paragraph before, our purpose is to correlate the coordinate time $t$ with the proper time $\tau$ of a ZAMO/Eulerian Observer.

Again, quoting from the book (Italic are comments), we got this:

Let us consider two close events $p$ and $p'$ on the worldline of some Eulerian observer (Meaning that both of these points are on the "line" characterized by n vector).

Let $t$ be the “coordinate time” of the event $p$ and $t + \delta t$ $(\delta t > 0)$ that of $p'$ , in the sense that $p \in \Sigma _t$ and $p' \in \Sigma _{t+\delta t}$. Then $p' = p + \delta t \mathbf{m}$ (consider $\mathbf{m}$ the vector that travels from one hypersurface to the other, and as one can intuitively guess, it's linear to $\mathbf{n}$. No need to over-analyze this I feel like).

The proper time $\delta \tau$ between the events $p$ and $p'$, as measured the Eulerian observer, is given by the metric length of the vector linking $p$ and $p'$:

$$\delta \tau = \sqrt{\mathbf{-g(m,m)}} \delta t = N\delta t$$ (Where $N$ is the lapse function $g^{00} = -\tfrac{1}{N^2}$)

Notice that the lapse function $N$ goes to unity $N_{r \rightarrow \infty} = 1$, so from this equation, the time parameter is actually the proper time, for events on the worldline n.

Of course, the "area" which is perpendicular to the $\mathbf{n}$ as we go to $\infty$ (meaning that the point on Hypersurface of the $\mathbf{n}$ is at infinite) is "the whole" local space. As we can see from this gif, the events on the same parameter $t$ are indeed simultaneous.

The question

Since in bibliography they translate the parameter time $t$ immediately to the proper time $\tau$ of our observer, the "right" answer should be that all events on hypersurface $\Sigma _t$ are simultaneous to the observer in $\infty$.

But how can we prove it? Since from the second quote, there is no immediate result regarding 2 events at "curved" spacetime in relation to the Lorentzian Spacetime in $\infty$.

Really thank you for even reading to this point. Wasn't sure if I should just "throw" the question or try to include some information for it to be more complete. Since in the future, one might have similar questions, I decided to do the last. Sorry for the long text :)

Any input is helpful!

EDIT:

To be a bit more clear, from the above 3+1 General Relativity theory, there is an observer who is called Eulerian/ZAMO and his 4-velocity is always perpendicular to the $\Sigma _t$ Hyper-surfaces.

These observers are unique at each point of the Hyper-surface. For these observers, we can consider what is "simultaneous".

The relation $\delta \tau = N \delta t$ shows how much proper time has passed for these events, and the gif shows what can be considered "simultaneous" for an observer. (Only one can be stattic with respect to the Kerr coordinates though, so one of those 2 observers, would not be considered ZAMO).

So the final question:

Can we prove that the hypersurface of $\Sigma _t$ is the surface of the events that happen at the same proper time of an Eulerian Observer/ ZAMO at $\infty$ ?

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  • $\begingroup$ It is unclear to me, what exactly you are trying to ask. I'm sure you are aware that one of the central "points" of general relativity is that there is no notion of simultaneity that can be globally agreed upon. Any two events that are no each others causal past/future can be considered "simultaneous". Are you asking how to prove that no two events on $\Sigma_t$ are in each others causal past or future? $\endgroup$
    – TimRias
    Aug 17, 2021 at 15:14
  • $\begingroup$ Hi @mmeent. Really sorry if I didn't make any sense. I edited the question ( last paragraph). I'm specifically referring to the "family" of observers that are Eulerian/ZAMO and are at $\infty$ distance, so there can be an agreement about them. I "presented" the analysis for events happening on an Eulerian Time line and the proper time / Coordinate time relationship. But I lack the knowledge/skill to do that for an observer at $\infty$ while all papers consider it trivia and just take $\tau = t$ $\endgroup$ Aug 17, 2021 at 17:15
  • $\begingroup$ What does ZAMO mean? $\endgroup$
    – D. Halsey
    Aug 17, 2021 at 18:48
  • $\begingroup$ If you take the limit of the Kerr metric as $r\rightarrow\infty$, you get the Minkowski metric. All observers at spatial infinity are in flat space. $\endgroup$
    – Paul T.
    Aug 17, 2021 at 18:50
  • $\begingroup$ @D.Halsey Zero Angular Momentum Observer. $\endgroup$ Aug 17, 2021 at 19:22

2 Answers 2

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Can we prove that the hypersurface of $\Sigma_t$ is the surface of the events that happen at the same proper time of an Eulerian Observer/ ZAMO at $\infty$?

It is extremely common for people to pick a coordinate system, show that at $r=\infty$ their $t$-coordinate (say) coincides with proper time for a given observer, then conclude the $t = \textrm{const}$ hypersurfaces define simultaneity for that observer. By itself, this is not justified! Compare for example the Kerr and Boyer-Linquist time coordinates. These concur at infinity, but are distinct elsewhere.

Another fairly obvious choice is to take a local reference frame, and extend it via geodesics. But various ambiguities and problems can arise, and I would want to see clear physical justifications, especially for extending a frame an infinite distance! Suppose your "observer" is a single point in spacetime, with an orthonormal frame given there. You could use Riemann normal coordinates to extend this frame into a coordinate system, at least over some neighbourhood of the point, and then interpret level sets $T = \textrm{const}$ of your new time coordinate as simultaneity hypersurfaces. Or maybe you use Fermi normal coordinates: but then, what is the entire worldline of the observer -- are they stationary, having always been at $r = \infty$, or a "raindrop", freely falling to or from finite $r$? (I need to do more concrete calculations here, but in the very least this concern applies to the next paragraph, if not here.)

I prefer to take a field of observers, and consider hypersurfaces orthogonal to their 4-velocities, if possible. This keeps things local. ("Orthogonal" implies the Einstein-Poincare conventionality of simultaneity, which I am impressed to see Gourgoulhon mentions.) The idea is to patch together these local choices of simultaneity, into a global choice. Hence, consider the field of Zero Angular Momentum Observers everywhere they are defined, not just a single ZAMO at infinity. (For ZAMOs see e.g. Frolov & Novikov 1998 $\S3.3.3$). This velocity field has zero vorticity, hence is orthogonal to a family of hypersurfaces $t = \textrm{const}$ say. Physically, this means we can extend (the most natural choice of) simultaneity over a spacetime region. Boyer-Lindquist time is sufficient. However the observers have nonzero 4-acceleration, hence no parametrisation $t$ exists which also coincides with the proper time of all our ZAMO observers. (This is related to the Frobenius theorem, see e.g. Ellis+ 2012 $\S4.6.2$.) There is nontrivial lapse.

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    $\begingroup$ Really thank you for the input. Although I have some questions regarding your answer.(Will check the books tommorow due to pc access limitation). 1st: You make an argument for an extension of simultaneity regarding "adding up" all those local orthogonal "regions" and create the hypersurface which can be that extension. If we then change the parameterization $t$ then there can't be two hypersurfaces of simultaneity since the Zamo at infinity has the same worldine. 2nd, it doesn't matter if parameterization t coincides with every ZAMO proper time. We have the lapse function for conversions, no? $\endgroup$ Aug 23, 2021 at 10:43
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    $\begingroup$ 1: I'm glad it's useful :) Given a field of ZAMO 4-velocities, the level sets $t = textrm{const}$ of Boyer-Lindquist time are everywhere orthogonal to them. If you define a new parameter $T = f(t)$, this has the same level sets [i.e. hypersurfaces], but the lapse changes. $\endgroup$ Aug 24, 2021 at 6:48
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    $\begingroup$ 2: Indeed, a ZAMO at point A should not conclude the hypersurface at point B is orthogonal to them. Comparing distant points is ambiguous; while someone could use parallel transport or similar, physical justification is needed. Instead, the ZAMO at A should think "I trust my ZAMO friend at B to determine local simultaneity there, and together our entire field of ZAMOs will piece together a global simultaneity definition." I like this interpretation because it is simpler and keeps everything local. But for applications to e.g. post-Newtonian theory or numerical relativity: different context $\endgroup$ Aug 24, 2021 at 7:03
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    $\begingroup$ (1/2) Alrighty, first of all thank you again! After reading a bit more about the family of observers I think I get the point. To my mind its now like this: A ZAMO has his local region as simultaneity surface. So he can synchronize his clock with another ZAMO whose 4-vel lands inside that small region and patch his region and this goes on until whole region $t=\textrm{const}$ constists the hypersurface of simultaneity for the family of ZAMO. Although I get the idea now, and since this is the way people work I guess I'm fine. $\endgroup$ Aug 24, 2021 at 16:35
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    $\begingroup$ Nice summary of my approach! I don't claim it's "the way people work" however; there is a lot of bad conceptual understanding IMO. Also we both agree that for a single observer at infinity, choosing any old hypersurface parametrised by some $T = f(t,r)$ such that $dT/d\tau \rightarrow 1$ as $r\rightarrow\infty$ is not very meaningful. My example was Kerr vs Boyer-Lindquist coordinates. $\endgroup$ Aug 25, 2021 at 2:52
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By definition all events that are assigned the same time coordinate by an observer are simultaneous to that observer.

Now, in Kerr spacetime we can define the surface $\Sigma_t$, which is the collection of events that have time coordinate $t$ (I'm guessing in Boyer-Lindquist coordinates). The question seems to be: who is the observer who says $\Sigma_t$ is a surface of simultaneity?

Whose proper time is equivalent to Boyer-Lindquist coordinate time? You stated the answer in the question: a zero angular momentum observer (ZAMO) at $r\rightarrow\infty$.

A ZAMO at spatial infinity is basically at rest in flat Minkowski space, so their proper time agrees with coordinate time. They observe all events on $\Sigma_t$ to be simultaneous.

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  • $\begingroup$ This seems logical but I'm not able to prove that for 2 events,say $p$ and $p'$, on the Hypersurfaces $\Sigma _t$ and $\Sigma _{ t+\delta t}$ ( but on $r \neq \infty$ ) are respectively at proper time $\tau _p = t$ and $\tau _{p'} = t+ \delta t$ on the ZAMO at $r \rightarrow \infty$. One should be able to prove it with taking the lenght of the vector $p,p'$ ? $\endgroup$ Aug 17, 2021 at 20:33
  • $\begingroup$ The proper time between two (time-like separated) events is defined as the length of the spacetime separation between them. I'm not sure what else to say about it. $\endgroup$
    – Paul T.
    Aug 17, 2021 at 20:41
  • $\begingroup$ Yeah, in order to prove what you say you have to verify that by examining the length of that vector $p,p'$, no? $\endgroup$ Aug 17, 2021 at 21:01
  • $\begingroup$ Meaning that you have to be able to prove that the ZAMO at infinite sees the events on $\Sigma _t$ simultanous (referring to the events where the spacetime is curved) $\endgroup$ Aug 17, 2021 at 21:28
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    $\begingroup$ In your question $p$ and $p'$ are on the worldline of the distant ZAMO and are separated by proper time $\delta t$. This is also their coordinate time separation. If we look at two events $q\in\Sigma_t$ and $q'\in\Sigma_{t+\delta t}$ located elsewhere, the proper time between those events is NOT $\delta t$. We can call it $\delta \tau \neq \delta t$. Their coordinate time separation is $\delta t$ by definition of the hypersurfaces. $\endgroup$
    – Paul T.
    Aug 18, 2021 at 0:48

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