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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 $\infty$ ?

Billy Matlock
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  • 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? – TimRias Aug 17 '21 at 15:14
  • 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 $\tau = t$ – Billy Matlock Aug 17 '21 at 17:15
  • What does ZAMO mean? – D. Halsey Aug 17 '21 at 18:48
  • 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. – Paul T. Aug 17 '21 at 18:50
  • @D.Halsey Zero Angular Momentum Observer. – Billy Matlock Aug 17 '21 at 19:22
  • @PaulT. Correct me if I'm wrong (it's highly likely) but even at $\infty$, the timeline of an observer that is moving, relative to the spacetime coordinates, will have a non-zero Angular Momentum (or with regard to the previous analysis, the 4-velocity timeline $\mathbf{n}$ will not be perpendicular to the Hypersurface $\Sigma _t$ and will indeed have a component on $\Sigma _t$. So even at $r \Rightarrow \infty$ the Eulerian Observer is still uniqe, given a spacetime. – Billy Matlock Aug 17 '21 at 19:27

2 Answers2

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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.

Paul T.
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  • 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 $p,p'$ ?

     – Billy Matlock Aug 17 '21 at 20:33
  • 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. – Paul T. Aug 17 '21 at 20:41
  • Yeah, in order to prove what you say you have to verify that by examining the length of that vector $p,p'$, no? – Billy Matlock Aug 17 '21 at 21:01
  • 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) – Billy Matlock Aug 17 '21 at 21:28
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    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.

     – Paul T. Aug 18 '21 at 00:48
  • Sorry for the delay. You are right, my comment was way off. Indeed what you say is the proper time and that's all you can say. But please if you can take a look at the 1st quote of my question. It says only locally on the hypersurface are the events simultaneous. Thats because you can only localy define a lorentzian metric for the observer. So how can we so easily say that these events ( referring to events on low rs ) are simultaneously on an observer at infinity – Billy Matlock Aug 18 '21 at 13:31
  • Another comment to add. If you consider the coordinate change $t'=t(1+\tfrac{1}{r})$ then the observer at the infinity is again Eulerian, but the hypersurface of constant $t$ changes. Only one of.those Hypersurfaces could be the surface of simultaneity – Billy Matlock Aug 20 '21 at 06:36
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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.

Colin MacLaurin
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    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? – Billy Matlock Aug 23 '21 at 10:43
  • Second note(1/2): I dont get how adding all those local regions that are normal to the corresponding ZAMO is proof that this hypersurface is the "simultaneity hypersurface". For example a Unit Circle is a surface that has infinite vectors (all points with lenght of $1$) that are radially outwards. Each point has an infinitesimal region which is normal to the vector but the whole circle (the surface if you add all those regions) is not. – Billy Matlock Aug 23 '21 at 13:52
  • (2/2) Lastly (and I think more to the point) I would expect the surface of simultaneity to derive from an equation like $$U_\mu(R^\mu - R^\mu _0) = 0 $$ where $U$ is some proper Velocity of an observer in Minkowski metrixc. I think that this equation would result in $t=const$ for a proper Velocity of an Eulerian observer at infinity ( judging by almsot all papers I have read that are actually dont pay any attention to this) – Billy Matlock Aug 23 '21 at 14:02
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    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. – Colin MacLaurin Aug 24 '21 at 06:48
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    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 – Colin MacLaurin Aug 24 '21 at 07:03
  • Thanks for the quick response! I'm gonna look a bit more onto what you said and response, but what I mean before is consider a new parameter $T = t+\tfrac{t_0\cdot r_0}{r}$. The set of hyper surfaces changes changes BUT for a ZAMO at infinity, the hypersurfaces asymptotiacally collide. These 2 hypersurfaces can't be both global definition, right? Might be off, but thats what i had in mind. Will look into it more on what you said and hopefully come back stronger :) – Billy Matlock Aug 24 '21 at 08:39
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    (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. – Billy Matlock Aug 24 '21 at 16:35
  • (2/2) I wanted to note why this seems problematic to me. A ZAMO at a point $p$ considers his infinitesimal region of the Hypersurface $t=\textrm{const}$ as simultaneity region because his 4-vel is normal to that surface. Then is straightforward to prove: the point $p' = p+\epsilon$ that belongs to the surface $\Sigma _t$ also belongs on the surface normal to the ZAMO's 4-Vel at $p$ with accuracy of $\mathcal{O}(\epsilon ^2}$. If you consider the surface $\Sigma _t$ region of simultaneity for ZAMOs, there will be an error of $\sum ^{\infty} _{n=1} \mathcal{O}{\epsilon ^2}$ which is infinite – Billy Matlock Aug 24 '21 at 17:00
  • *error refers for a point/event in non infinite radius and ZAMO at infinity – Billy Matlock Aug 24 '21 at 17:13
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    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. – Colin MacLaurin Aug 25 '21 at 02:52
  • We also agree that extending a single observer's local frame/simultaneity (e.g. via geodesics) won't concur with the observer field approach, in general. Both approaches have validity. The usual emphasis on an observer at infinity is understandable, because it models us humans. You might choose to e.g. focus on only physically measurable effects. I like the observer field approach, as a reaction to extravagant claims like "time stops at the event horizon, for an observer at infinity". (But timelike vectors do exist at the horizon!) You'll like sec 7 of my paper https://arxiv.org/abs/1911.05988 – Colin MacLaurin Aug 25 '21 at 03:24
  • (3): P.S. I appreciate qualitatively your point about the $\epsilon$ deviation. I do wonder if curvature has a similar effect. Also if there were an infinite time discrepancy in some sense, we are also considering an infinite space interval $\Delta r\rightarrow\infty$, so the quotient $\Delta\textrm{time}/\Delta\textrm{space}$ seems more significant – Colin MacLaurin Aug 25 '21 at 03:36
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    Haha I have read that article before the post. Didn't realize it was you. Penrose diagrams were new to me and I'm mostly into plasma as you can imagine. Nice writing! As for the last comment. I thyone can easily argue that the quotient $\Delta t/ \Delta r$ will evolve as $\mathcal{O} (\epsilon) so it would be essentially be close to zero but I think it lacks physical meaning – Billy Matlock Aug 26 '21 at 13:46
  • Glad you liked it :) Your points about the "epsilon" deviations are important, certainly the observer field simultaneity convention does not concur with extending a local observer's frame outwards. I would like to understand this better – Colin MacLaurin Aug 29 '21 at 05:03