Hypersurfaces of constant proper time

Difficulty level:   ★ ★ ★ ★

Suppose a given spacetime has a region filled with worldlines (a timelike congruence), and a foliation defined by hypersurfaces of constant proper time along these worldlines. As a boundary condition, all times can be set to zero on a given initial hypersurface. The question is, will the proper time hypersurfaces remain spacelike? I investigate this for two straightforward examples: static observers in Schwarzschild spacetime, and the rotating disc in Minkowski spacetime.

George Ellis mentions the possibility of the hypersurfaces becoming timelike, in a 2014 paper  on his “evolving block universe” interpretation. The context is cosmology, and the worldlines are (in principle) some coarse-grained flow of matter:

The flow lines are not necessarily orthogonal to the surfaces of constant time. This does not matter: no physical phenomena are directly determined by simultaneity in the usual sense. More than that, the surfaces determined in this way are not even necessarily space-like in an inhomogeneous spacetime. In that case, the implied initial value problem will locally be time-like, and the way it works will need to be rethought.

Perhaps the possibility of proper time hypersurfaces becoming timelike has not been investigated in detail. Presumably Ellis’ superb earlier publication Relativistic Cosmology (2012), coauthored with Maartens and MacCallum, would not discuss it either.

Recall a congruence is proper time synchronisable if and only if it is geodesic and vorticity-free, by Frobenius’ theorem. If so, the gradient of a proper time coordinate T say, is given by the dual vector to the 4-velocity:

    \[dT := -\mathbf u^\flat\]

where the minus sign compensates for the metric signature choice -+++. Now consider the congruence of static observers in Schwarzschild spacetime. These have 4-velocity parallel to the Killing vector field which is timelike at infinity, hence are defined for all r>2M. The dual-velocity is

    \[-\mathbf u^\flat = \sqrt{1-\frac{2M}{r}}\,dt\]

in terms of the Schwarzschild t-coordinate. This is clearly not integrable, as expected because the static observers are accelerating. But it still suggests the proper time coordinate T := t\sqrt{1-2M/r}. Then the gradient covector is

    \[dT = \sqrt{1-\frac{2M}{r}}\,dt + \frac{Mt}{r^2}\Big(1-\frac{2M}{r}\Big)^{-1/2}dr\]

In general this is not orthogonal to the worldlines, and one interpretation is a non-standard simultaneity convention, as discussed in my forthcoming paper  “Time, black holes, and infinity”. However it is still proper time, because dT(\mathbf u) = 1. Using the inverse metric,

    \[dT\cdot dT = -1+\frac{M^2t^2}{r^4}\]

so dT is timelike for |t|/M < (r/M)^2, and since dT is a normal to the hypersurfaces they are spacelike in the same region. The figure below shows three examples on a Penrose diagram. The hypersurfaces are spacelike for sufficiently large r, but become null at |t|/M = (r/M)^2 which is the dotted red line in the diagram below.

Hypersurfaces of constant proper time for static observers in Schwarzschild spacetime
Hypersurfaces of constant proper time for static observers in Schwarzschild spacetime

This makes sense intuitively. Near the horizon, the static observers are heavily gravitationally time-dilated, so for a proper time of e.g. T = 1 occurs well into the “future”. This is seen from the curves bending upwards in the diagram for t>0, and bending downwards for t<0, near the horizon. The claim of being in the “future” has some dependence on one’s choice of simultaneity convention, however once the red line is crossed it is an unambiguous statement because the T = \textrm{const} events are timelike separated. Incidentally T=0 at t=0, but this is just an initial condition, and in general one could define T := t\sqrt{1-2M/r} + h(r,\theta,\phi) for any function h, which is also proper time along the worldlines.

Now consider a rigidly rotating disc in 2+1-dimensional Minkowski spacetime. Using polar coordinates (t,r,\phi), the “4”-velocity of each particle on the disc is

    \[u^\mu = \Big(\frac{1}{\sqrt{1-r^2\Omega^2}},0,\frac{\Omega}{\sqrt{1-r^2\Omega^2}}\Big)\]

where \Omega := d\phi/dt \ge 0 parametrises rotation speed. The previous procedure of deriving a time coordinate wasn’t fully general. Here we expect a proper time coordinate to depend on r and t but not \phi. The proper time runs more slowly (compared to t) towards the edge of the disc, note the disc is bounded by r < 1/\Omega for timelike motion. Also it is well known there is a “time-lag” when trying to define simultaneity around a circle r=\textrm{const}. However one can use non-standard simultaneity (i.e. constant “time” hypersurfaces not orthogonal to the worldlines) to avoid this problem: see Relativity in Rotating Frames (2004), particularly the chapter by Rizzi & Serafini.

Based on u^t \equiv dt/d\tau above, define

    \[\bar t := t\sqrt{1-r^2\Omega^2}\]

This deliberately avoids any angular dependence. The gradient is

    \[d\bar t = \sqrt{1-r^2\Omega^2}\,dt -\frac{\Omega^2 tr}{\sqrt{1-r^2\Omega^2}}\]

One can check d\bar t(\mathbf u) = 1, so this is a proper time coordinate. From the above expression one can quantify an implied non-standard simultaneity convention, but I will avoid this here. The hypersurfaces turn null at |t| = (1-r^2\Omega^2)/r\Omega^2.

The spacetime diagram below represents hypersurfaces of constant proper time \bar t, and is independent of rotation rate due to scaling of the coordinates. The dotted red line is where the hypersurfaces are null; to the left of it they are spacelike.

Hypersurfaces of constant proper time for particles on a rotating disc in Minkowski spacetime
Hypersurfaces of constant proper time for particles on a rotating disc in Minkowski spacetime

As \Omega r\rightarrow 1 the hypersurfaces quickly turn null. For small \Omega r, they turn null at \Omega|t|\approx 1/r\Omega. Thus for small rotation / acceleration, the desynchronisation is slow but cumulative.

The disc particles are accelerated, so for variety let’s choose an example with vorticity but no acceleration. Take Schwarzschild spacetime, with circular orbits on the coordinate equator \theta = \pi/2. These are valid anywhere outside the photon sphere at r = 3M, not merely the ISCO at r = 6M. The 4-velocity is:

    \[u^\mu = \Bigg( \frac{1}{\sqrt{1-\frac{3M}{r}}},0,0,\frac{\sqrt{\frac{M}{r^3}}}{\sqrt{1-\frac{3M}{r}}} \Bigg)\]

in Schwarzschild cordinates, which suggests a new coordinate \tilde t := t\sqrt{1-3M/r}. This is null at

    \[\frac{|t|}{M} = \frac{2}{3}\Big(\frac{r}{M}\Big)^2 \frac{1-\frac{3M}{r}}{1-\frac{2M}{r}}\]

which occurs instantly in the limit r\rightarrow 3M, and slowly for r\gg 3M.

In conclusion, proper time hypersurfaces can become timelike:

  • quickly, for high acceleration of the worldlines
  • quickly, for high vorticity of the worldlines
  • slowly, for mild but sustained acceleration or vorticity, a cumulative effect

This investigation was sparked by a lunchtime conversation with Pierre Mourier and Prof. David Wiltshire today, at the University of Canterbury in Christchurch, New Zealand. My forthcoming paper “Time, black holes, and infinity” research paper  will discuss simultaneity in Schwarzschild spacetime.

Update (next day): Mourier clarifies that proper time hypersurfaces have been studied, but often in the zero-vorticity case. Hence they remain orthogonal to the worldlines (in a cosmological context it is taken for granted that the worldlines are geodesics). So the issue of turning timelike does not come up. See perhaps §6.6.1 of Relativistic Cosmology as cited above, or look up “synchronous coordinates”. Mourier has also looked at rotation in Minkowski spacetime, different from the rigid rotation example above, and found similar effects. Wiltshire comments he and collaborators have looked at rotation effects in Lemaître-Tolman-Bondi models and van Stockum dust.