Plane gravitational waves

5.11 Plane gravitational waves

A plane gravitational wave is a spacetime with metric

where $f(u)2 + g(u)2$ is not identically zero. For any choice of $f (u)$ and $g(u)$ , the metric (156

) has vanishing Ricci tensor, i.e., Einstein’s vacuum field equation is satisfied. The lightlike vector field $∂v$ is covariantly constant. Non-flat spacetimes with a covariantly constant lightlike vector field are called plane-fronted waves with parallel rays or pp-waves for short. They made their first appearance in a purely mathematical study by Brinkmann [43 ].

In spite of their high idealization, plane gravitational waves are interesting mathematical models for studying the lensing effect of gravitational waves. In particular, the focusing effect of plane gravitational waves on light rays can be studied quite explicitly, without any weak-field or small-angle approximations. This focusing effect is reflected by an interesting light cone structure.

The basic features with relevance to lensing can be summarized in the following way. If the profile functions $f$ and $g$ are differentiable, and the coordinates $(x,y, u,v)$ range over $ℝ4$ , the spacetime with the metric (156 ) is geodesically complete [92 ]. With the exception of the integral curves of $∂ v$ , all inextendible lightlike geodesics contain a pair of conjugate points. Let $q$ be the first conjugate point along a past-oriented lightlike geodesic from an observation event $pO$ . Then the first caustic of the past light cone of $pO$ is a parabola through $q$ . (It depends on the profile functions $f$ and $g$ whether or not there are more caustics, i.e., second, third, etc. conjugate points.) This parabola is completely contained in a hyperplane $u = constant$ . All light rays through $pO$ , with the exception of the integral curve of $∂v$ , pass through this parabola. In other words, the entire sky of $pO$ , with the exception of one point, is focused into a curve (see Figure 29 ). This astigmatic focusing effect of plane gravitational waves was discovered by Penrose [259 ] who worked out the details for “sufficiently weak sandwich waves”. (The name “sandwich wave” refers to the case that $f (u )$ and $g(u)$ are different from zero only in a finite interval $u1 < u < u2$ .) Full proofs of the above statements, for arbitrary profile functions $f$ and $g$ , were given by Ehrlich and Emch [94 , 95 ] (cf. [25 ], Chapter 13). The latter authors also demonstrate that plane gravitational wave spacetimes are causally continuous but not causally simple. This strengthens Penrose’s observation [259 ] that they are not globally hyperbolic. (For the hierarchy of causality notions see [25 ].) The generators of the light cone leave the boundary of the chronological past $I− (pO)$ when they reach the caustic. Thus, the above-mentioned parabola is also the cut locus of the past light cone. By the general results of Section 2.8, the occurrence of a cut locus implies that there is multiple imaging in the plane-wave spacetime. The number of images depends on the profile functions. We may choose the profile functions such that there is no second caustic. (The “sufficiently weak sandwich waves” considered by Penrose [259 ] are of this kind.) Then Figure 29 demonstrates that an appropriately placed worldline (close to the caustic) intersects the past light cone exactly twice, so there is double-imaging. Thus, the plane waves demonstrate that the number of images need not be odd, even in the case of a geodesically complete spacetime with trivial topology.

Figure 29: Past light cone of an event $pO$ in the spacetime (156

) of a plane gravitational wave. The picture was produced with profile functions $f (u ) > 0$ and $g(u) = 0$ . Then there is focusing in the $x$ -direction and defocusing in the $y$ -direction. In the (2 + 1)-dimensional picture, with the $y$ -coordinate not shown, the past light cone is completely refocused into a single point $q$ , with the exception of one generator $λ$ . It depends on the profile functions whether there is a second, third, and so on, caustic. In any case, the generators leave the boundary of the chronological past $− I (pO)$ when they pass through the first caustic. Taking the $y$ -coordinate into account, the first caustic is not a point but a parabola (“astigmatic focusing”) (see Figure 30

). An electromagnetic plane wave (vanishing Weyl tensor rather than vanishing Ricci tensor) can refocus a light cone, with the exception of one generator, even into a point in 3 + 1 dimensions (“anastigmatic focusing”) (cf. Penrose [259 ] where a hand-drawing similar to the picture above can be found).

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Figure 30: “Small wave fronts” of the light cone in the spacetime (156

) of a plane gravitational wave. The picture shows the intersection of the light cone of Figure 29

with the lightlike hyperplane $u = constant$ for three different values of the constant: (a) exactly at the caustic (parabola), (b) at a larger value of $u$ (hyperbolic paraboloid), and (c) at a smaller value of $u$ (elliptic paraboloid). In each case, the hyperplane $u = constant$ does not intersect the one generator $λ$ tangent to $∂v$ ; all other generators are intersected transversely and exactly once.

The geodesic and causal structure of plane gravitational waves and, more generally, of pp-waves is also studied in [162 , 51 ].

One often considers profile functions $f$ and $g$ with Dirac-delta-like singularities (“impulsive gravitational waves”). Then a mathematically rigorous treatment of the geodesic equation, and of the geodesic deviation equation, is delicate because it involves operations on distributions which are not obviously well-defined. For a detailed mathematical study of this situation see [309 , 192 ].

Garfinkle [131 ] discovered an interesting example for a pp-wave which is singular on a 2-dimensional worldsheet. This exact solution of Einstein’s vacuum field equation can be interpreted as a wave that travels along a cosmic string. Lensing in this spacetime was numerically discussed by Vollick and Unruh [339 ].

The vast majority of work on lensing by gravitational waves is done in the weak-field approximation. For the exact treatment and in the weak-field approximation one may use Kovner’s version of Fermat’s principle (see Section 2.9), which has the advantage that it allows for time-dependent situations. Applications of this principle to gravitational waves have been worked out in the original article by Kovner [186 ] and by Faraoni [109 , 110 ].