Are the intersections of past and future light cones spacelike?

Question

Given a timelike reference worldline (not necessarily geodesic), we can define light-cone coordinates $\tau^+$ and $\tau^-$ so that the 3-D hypersurfaces of constant $\tau^+$ are past light cones of events on the reference worldline, the value of $\tau^+$ being reference proper time. Similarly $\tau^-$ for future light cones.

• Is there a common name for the 2-D surfaces of constant $\tau^+$ and $\tau^-$ induced by a timelike worldline?
• Is it correct that these are "locally spacelike" in the sense that any curve entirely within a surface is spacelike?
• Under what assumptions are they also "globaly spacelike" in the sense that there are no causal (timelike or null) curves between any two points on a surface?

From this we can also do a simple coordinate change to $r^\star = \frac{\tau^+ - \tau^-}{2c}$ and time $\tau^\star = \frac{\tau^+ + \tau^-}{2}$. In flat spacetime with a geodesic reference worldline, the 3-D hypersurfaces of constant $\tau^\star$ are just the usual orthogonal simultaneous spaces. [EDIT: should have been $r^\star = \frac{\tau^+ - \tau^-}{2}c$ ]

• Is there a common name for the hypersurfaces of constant $\tau^\star$ ?
• Are they also locally spacelike?
• Under what (very strong I expect) assumptions are they globally spacelike?

Addendum

In response to the request to clarify the co-ordinates:

To illustrate, suppose spacetime is flat, and suppose the chosen reference worldline is geodesic. It can then be taken to be stationary at the spatial origin of Minkowski co-ordinates WLOG. In that case, $\tau^+$ is $t + r/c$ and $\tau^-$ is $t - r/c$, since an event at time $t$ and distance $r$ would meet the past and future lightcones originating from the reference worldline at those times, respectively. As a result $\tau^\star = t$ and $r^\star = r$.

Of course this is not the case I am asking about. It is a general definition, not restricted to a geodesic reference worldline, nor to flat spacetime.

Summary: if I am the reference worldline, then $\tau^+$ is what my watch is reading when I see the event. $\tau^-$ is what my watch is reading when the event sees me.

Answer 1

I finally stumbled upon the term radar to describe this use of a reference timelike worldline. A good starting resource is: Perlick, Volker. "On the radar method in general-relativistic spacetimes.", which points to a bunch of other resources.

The $r^\star$ and $\tau^\star$ are in fact the radar distance and radar time resp. I still haven't found any name for the surface, though radar bubble suggests itself. There is a region around the reference worldline where these bubbles are globally spacelike and topologically $S^2$ (a bubble). In flat space this region is the entire space, though a gravitating object will make bubbles self-intersect at certain distance.

A radar bubble can fail to be even locally spacelike: if a null geodesic intersects the worldline twice, then the corresponding bubble would include that geodesic. A black hole can do this. It can act as a gravitational mirror, and bounce light coming in on a certain angle from a distant source back.

Addendum

Instead of the light cones themselves, the above also works using the boundary of the chronological future (past). The chronological future (past) of an event $p$ are the events that can be reached from $p$ (can reach $p$) by a timelike path. They are designated $I^\pm(p)$, and their boundaries $\delta I^\pm(p)$. In a globally hyperbolic spacetime, these are subsets of the past or future light cone, excluding the parts where it self-intersects.

The intersection of a future and past boundary (not to be confused with the boundary of the intersection) $\delta I^+(p) \cap \delta I^-(q)$, is indeed globally spacelike, though not always topologically $S^2$.

Given a timelike path $p(\lambda)$, let's label the bubble $B_p(\lambda,\mu) = \delta I^+(p(\lambda-\mu)) \cap \delta I^-(p(\lambda+\mu))$. Then for a given $\lambda$, the union of bubbles $\bigcup_{\mu \ge 0} B_p(\lambda,\mu)$ is a globally spacelike 3-D hypersurface. This family of hypersurfaces indexed by $\lambda$ foliates the part of the spacetime reachable from the path (unlike the "instantaneous simultaneous spaces" -- even in special relativity, two instantaneous simultaneous spaces at different events of an accelerated worldline will intersect each other).

In particular this is true if the path parameter is just the proper time along the path (i.e. $\tau^\star = \lambda$ and $r^\star = \mu c$).

Answer 2

Lightcones are necessarily null submanifolds of your base manifold. In the base manifold, the null surface will have in intrinsic 3-metric which satisfies the relationship:

$$v_{a}v^{a} \geq 0$$

for all tangent vectors (I'm using the [-,+,+,+] metric convention).

Any intersection of this surface with another null surface will necessarily have a different null tangent vector from your first surface in the region of contact, so the common submanifold will ''project out'' the null direction, and the resultant space will necessarily be spacelike.