Stationary spacetimes—sounds fairly simple, unchanging. Static—even more boring. But are they?
Consider an experiment of emitting a photon along a closed path—closed either due to a constraining light-tube or due to a topological closure in the spacetime—and finding the time till the photon returns to the starting point. (Our naive expectation is for the time to be the same as the length of the path, if our clocks and measuring rods are in geometric units, set to show speed of light is unity.) Now turn around and emit a photon along the same path but in the reverse direction—does it take the same time to traverse the path backwards? Not necessarily! Not even if the spacetime is static, we are a static observer, we use our static clocks to measure time, and the track length is measured in our static rest-space.
For a static spacetime, at least the discrepancy between forward and backward travel times is independent of a small change in the path; for stationary spacetimes, we don’t even have that consolation: the discrepancy is a continuous function of the path.
Examining these phenomena in detail—classifying behaviors and looking at examples simple and complex—is the topic of my paper, published in Classical and Quantum Gravity.
Simple example, a Minkowski cylinder with a temporal shift: Take a vertical strip of Minkowski 2-space, , such as all with , and close it up by identifying with for some constant ; choosing is just a standard static spacetime built on a circle, but we still get a globally hyperbolic spacetime so long as . Say we’re the static observer sitting at . At time , we emit a photon to the right; the photon’s world-line, beginning at , includes the event , which is also : we note the photon’s return at time , and that is also the elapsed trip-time. We turn around at time and emit a photon to the left; this second photon has its world-line begin at , which is the same as , and it also contains the event : we note departure at time and return at time , for a trip-time in this direction of . And the actual length of the path, in the static rest-frame, is .
This example quite captures the flavor of what can happen in static spacetimes: it all comes down to making topological identifications on a simpler, more well-behaved spacetime—specifically, a standard static spacetime, which is basically a product spacetime (with Killing field the obvious choice, ).
There’re no surprises for a standard static spacetime. Causal behavior is just what one expects, determined by the distance function in : , and is globally hyperbolic if is complete using that distance function (only complication is that if the Killing field varies in length, depending on the point in , then one uses the conformally related metric, dividing out by ). But making topological identifications in the larger picture can change things up, as the Minkowski cylinder illustrates.
For a general stationary spacetime , we use Geroch’s notion of looking at the space of stationary orbits (i.e. the world-lines of stationary observers, the integral curves of the timelike Killing field ), which automatically comes equipped with a Riemannian metric , reflecting the unchanging nature of the spacelike perp-spaces of the orbits. Then is a line-bundle over , which means that although is topologically the product of and the line, the geometry is more complex than that; and while there is a natural projection (taking a point to the stationary orbit through ), there is lots of gauge freedom in selecting the other coördinate ; we always make sure , so accurately measures stationary time.
The basic tool to understanding the “causal curiosities” in a stationary or static spacetime is to simply look at the following: For any closed loop in the orbit-space —say, beginning and ending at some orbit —let be the future-null curve starting at some point in the orbit and returning to at some point ; then is some units further advanced along from , measuring by our stationary clock. Let be the length of the path in the orbit-space , measured in units conformally adjusted for . Then is the crucial ingredient, measuring the discrepancy from naivité of the elapsed travel-time for a photon moving along the path .
The amounts to an operator on loops of . In fact, we can express the set of directed but unparametrized loops (all with the same basepoint ) as the “cycles” of , an abelian group with concatenation as the operation; and then we’ve defined a group homomorphism, , the “fundamental cocyle” of , with . For any gauge choice of coördinate , there is an accompanying “drift form” on (allowing us to write the spacetime metric as ), and is then realizable as . is static if is closed, which means depends only on homotopy class.
What is particularly important is a sort of norm for this fundamental cocyle, called its “weight” , defined as , with being length measured in the conformal metric; this controls the larger causal properties of the spacetime, as is shown in the main results of this paper: If then is chronologically vicious (all events timelike related to all others); if , then is strongly causal and, in fact, globally hyperbolic if is complete in the conformal metric. There are several subcases for the critical value of : If there is some loop that realizes this weight, then is chronological but not causal; if weight = 1 is realized only by a sequence of loops with , then is causal but not strongly causal; if there is a weight-realizing sequence of loops with bounded away from , then is strongly causal; and if weight-realizing sequences all obey and is complete in the conformal metric, then is globally hyperbolic.
For instance, the Kerr spacetime (with angular momentum and mass obeying ) is stationary outside the outer horizon; in this portion it has . The Minkowski cylinder above has . If we start with a cosmic string in Minkowski space of finite diameter and angular defect , cut it apart, and re-stitch it together with a temporal shift of , then we end up with . The paper explores similarly re-stitched cosmic strings in Rindler, Schwarzschild, and Kerr backgrounds (always needing to delete regions around and to keep the weight finite), as applications of general theorems about how global causality differs between a stationary spacetime and its quotient by a group-action. A primary tool for analysis is a sort of asymmetric quasi-distance on , ; for then we have in , .
Why look at this topic? My original motivation was to understand the global behavior of non-standard static spacetimes, in order to determine the causal boundary (José Flores and I having previously explored in detail the causal boundary of standard static spacetimes). While standard static spacetimes have causal properties solely determined by their rest-spaces, for the non-standard case, it’s the fundamental cocycle—in particular, its weight—that supplies the additional information needed to characterize global causality behavior. Preliminary examination suggests that it is the critical value of 1 for this weight that can result in interesting variations for the causal boundary.
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