Rindler Wedge: Timelike or Spacelike? Intuition & Light Cone

[knowwhatyoudontknow]

TL;DR

Rindler wedge - timelike or spacelike

Intuitively, the Rindler wedge is timelike in Minkowski coordinates and an object crossing the horizon enters a spacelike region. This seems
at odds with my understanding of the light cone where the 2 regions are reversed. I think this may be related to the signature of the metric but I'm not sure. What am I missing?

 

[Dale]

Summary:: Rindler wedge - timelike or spacelike

a spacelike region

I am not sure what you mean by “a spacelike region”. Do you mean a spacelike surface? Or a foliation?

Usually I would use the word “region” for a 4D open set in the spacetime. But then timelike and spacelike wouldn’t make sense. So I am not sure what you are asking.

 

[PeterDonis]

Intuitively, the Rindler wedge is timelike in Minkowski coordinates and an object crossing the horizon enters a spacelike region.

If this is your intuition then your intuition needs to be retrained. It makes no sense to say a region of spacetime is "timelike" or "spacelike"; those terms only make sense for worldlines or vectors.

You might be thinking of integral curves of the "boost" Killing vector field in Minkowski spacetime, which are timelike hyperbolas in the Rindler wedge but are spacelike hyperbolas above the future horizon or below the past horizon. However, those are curves, not regions.

This seems
at odds with my understanding of the light cone where the 2 regions are reversed.

I think what you are trying to say here is that, if we look at curves passing through the origin of Minkowski coordinates, timelike curves lie inside the light cone, null curves lie on the light cone, and spacelike curves lie outside the light cone. That is correct.

However, this has nothing whatever to do with worldlines in the Rindler wedge, since none of those worldlines pass through the origin. There are timelike worldlines in the Rindler wedge (i.e., the region of spacetime with $x > 0$ and $|t| < x$) that stay in that wedge forever, and there are other timelike worldlines in that wedge that enter the wedge from below the past Rindler horizon (the line $t = -x$) and/or exit the wedge to above the future Rindler horizon (the line $t = x$). But those worldlines stay timelike everywhere regardless of their behavior relative to the wedge and its boundaries.

 

[Ibix]

Summary:: Rindler wedge - timelike or spacelike

Intuitively, the Rindler wedge is timelike in Minkowski coordinates and an object crossing the horizon enters a spacelike region. This seems
at odds with my understanding of the light cone where the 2 regions are reversed. I think this may be related to the signature of the metric but I'm not sure. What am I missing?

Are you aware that "the" light cone is wrong? There's a light cone associated with every event, and all events are outside some light cones, inside others, and on yet others. On a Minkowski diagram the Rindler wedge appears to correspond to one side of the region spacelike separated from the origin, sure, but that just tells you that you can't get into that particular wedge if you pass through the origin.

Note that "the" Rindler wedge is also a slight misnomer. Rindler coordinates have the center of their hyerbolae at the origin of Minkowski coordinates - but the origin of Minkowski coordinates is arbitrary. There is a different Rindler wedge associated with different families of Rindler observers.

Finally, it should be noted that the Rindler wedge doesn't match the exterior of a light cone except in 1d. It's the exterior of the causal future of the plane $x=0,\ t=0$, while a light cone is normally the causal future of a single event. $x=0,\ t=0$ is only an event in 1d - it's a line or plane in 2d or 3d.

 

[Ibix]

Just to illustrate my last point, here's a Minkowski diagram with two spatial dimensions. The past and future light cones of some arbitrary event are shown in yellow and the boundary of a Rindler wedge whose center happens to pass through that event is shown in red. A few of the relevant Rindler observers are shown in blue.

If you sliced this diagram perpendicular to y (i.e., extracted a normal x/t Minkowski diagram) through the point of the cone you would recover the usual diagram of Rindler coordinates, because in that plane the slope of the light cone and the Rindler wedges are the same (and you could place the light cone center anywhere on the seam of the wedge - there's nothing special about where I've chosen to draw it). But shown in (2+1)d the light cone is a completely different shape from the Rindler wedge.