How wild can a static spacetime get?

Stationary spacetimes—sounds fairly simple, unchanging. Static—even more boring. But are they?

Steve (also known as Stacey) Harris is a mathematical relativist in the Department of Mathematics at Saint Louis University, mostly working on global structures such as causal boundary, and a notorious campus gadfly and rebel, being active in AAUP and fanatical about shared governance. Off campus there’s hiking and playing flute in concert band.

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 $1 + 1$ example, a Minkowski cylinder with a temporal shift: Take a vertical strip of Minkowski 2-space, $\Bbb L^2$ , such as all $(x,t)$ with $0 \le x \le 1$ , and close it up by identifying $(0,t)$ with $(1, t + \lambda)$ for some constant $\lambda$ ; choosing $\lambda = 0$ is just a standard static spacetime built on a circle, but we still get a globally hyperbolic spacetime so long as $|\lambda| < 1$ . Say we’re the static observer sitting at $x = 0$ . At time $t = 0$ , we emit a photon to the right; the photon’s world-line, beginning at $(0,0)$ , includes the event $(1,1)$ , which is also $(0, 1 - \lambda)$ : we note the photon’s return at time $1 - \lambda$ , and that is also the elapsed trip-time. We turn around at time $t = 2$ and emit a photon to the left; this second photon has its world-line begin at $(0,2)$ , which is the same as $(1,2 + \lambda)$ , and it also contains the event $(0,3 + \lambda)$ : we note departure at time $2$ and return at time $3 + \lambda$ , for a trip-time in this direction of $1 + \lambda$ . And the actual length of the path, in the static rest-frame, is $1$ .

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 $N \times \Bbb L^1$ (with Killing field the obvious choice, $\partial/\partial t$ ).

There’re no surprises for a standard static spacetime. Causal behavior is just what one expects, determined by the distance function $d$ in $N$ : $(x,t) \ll (x', t') \iff d(x,x') < t' - t$ , and $M$ is globally hyperbolic if $N$ is complete using that distance function (only complication is that if the Killing field $K$ varies in length, depending on the point $x$ in $N$ , then one uses the conformally related metric, dividing out by $|K|^2$ ). But making topological identifications in the larger picture can change things up, as the Minkowski cylinder illustrates.

For a general stationary spacetime $M$ , we use Geroch’s notion of looking at the space $Q$ of stationary orbits (i.e. the world-lines of stationary observers, the integral curves of the timelike Killing field $K$ ), which automatically comes equipped with a Riemannian metric $h_Q$ , reflecting the unchanging nature of the spacelike perp-spaces of the orbits. Then $M$ is a line-bundle over $Q$ , which means that although $M$ is topologically the product of $Q$ and the line, the geometry is more complex than that; and while there is a natural projection $\pi : M \to Q$ (taking a point $p$ to the stationary orbit through $p$ ), there is lots of gauge freedom in selecting the other coördinate $\tau: M \to \Bbb R^1$ ; we always make sure $(d\tau)K = 1$ , so $\tau$ accurately measures stationary time.

The basic tool to understanding the “causal curiosities” in a stationary or static spacetime $M$ is to simply look at the following: For any closed loop $c$ in the orbit-space $Q$ —say, beginning and ending at some orbit $q$ —let $\bar c$ be the future-null curve starting at some point $x$ in the orbit $q$ and returning to $q$ at some point $x'$ ; then $x'$ is some $T$ units further advanced along $q$ from $x$ , measuring by our stationary clock. Let $L$ be the length of the path $c$ in the orbit-space $Q$ , measured in units conformally adjusted for $|K|^2$ . Then $T - L$ is the crucial ingredient, measuring the discrepancy from naivité of the elapsed travel-time for a photon moving along the path $c$ .

The amounts to an operator on loops of $Q$ . In fact, we can express the set of directed but unparametrized loops (all with the same basepoint $q_0$ ) as the “cycles” $Z(Q)$ of $Q$ , an abelian group with concatenation as the operation; and then we’ve defined a group homomorphism, $\beta_M: Z(Q) \to \Bbb R$ , the “fundamental cocyle” $\beta_M$ of $M$ , with $\beta_M(c) = T- L$ . For any gauge choice of coördinate $\tau: M \to \Bbb R$ , there is an accompanying “drift form” $\omega$ on $Q$ (allowing us to write the spacetime metric as $ds^2 = -|K|^2(d\tau + \omega)^2 + h_Q$ ), and $\beta_M$ is then realizable as $\beta_M(c) = \int_c \omega$ . $M$ is static if $\omega$ is closed, which means $\beta_M$ depends only on homotopy class.

What is particularly important is a sort of norm for this fundamental cocyle, called its “weight” $\text{wt}(\beta_M)$ , defined as $\sup \beta_M(c)/L(c)$ , with $L$ 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 $\text{wt}(\beta_M) > 1$ then $M$ is chronologically vicious (all events timelike related to all others); if $\text{wt}(\beta) < 1$ , then $M$ is strongly causal and, in fact, globally hyperbolic if $Q$ is complete in the conformal metric. There are several subcases for the critical value of $\text{wt}(\beta_M) = 1$ : If there is some loop that realizes this weight, then $M$ is chronological but not causal; if weight = 1 is realized only by a sequence of loops $\{c_n\}$ with $\{L(c_n) - \beta_M(c_n)\} \to 0$ , then $M$ is causal but not strongly causal; if there is a weight-realizing sequence of loops with $\{L(c_n) - \beta_M(c_n)\}$ bounded away from $0$ , then $M$ is strongly causal; and if weight-realizing sequences all obey $\{L(c_n) - \beta_M(c_n)\} \to \infty$ and $Q$ is complete in the conformal metric, then $M$ is globally hyperbolic.

For instance, the Kerr spacetime (with angular momentum $a$ and mass $m$ obeying $a < m$ ) is stationary outside the outer horizon; in this portion it has $\text{wt}(\beta_M) = 1$ . The Minkowski cylinder above has $\text{wt}(\beta_M) = |\lambda|$ . If we start with a cosmic string in Minkowski space of finite diameter $r_0$ and angular defect $\delta$ , cut it apart, and re-stitch it together with a temporal shift of $\lambda$ , then we end up with $\text{wt}(\beta_M) = |\lambda|/(r_0(2\pi - \delta))$ . The paper explores similarly re-stitched cosmic strings in Rindler, Schwarzschild, and Kerr backgrounds (always needing to delete regions around $\theta = 0$ and $\theta = \pi$ 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 $Q$ , $d_\omega(q,q') = \inf\{L(c) - \int_c\omega\,|\, c \text{ goes from } q \text{ to } q' \}$ ; for then we have in $M$ , $p \ll p' \iff d_\omega(\pi(p),\pi(p')) < \tau(p') - \tau(p)$ .

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.

Read the full article in Classical and Quantum Gravity or on arXiv:
Static- and stationary- complete spacetime: algebraic and causal structures
Steven G Harris 2015 Class. Quantum Grav. 32 135026

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