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Embeddings and Time Evolution of the Schwarzschild Wormhole

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Embeddings and Time Evolution of the Schwarzschild Wormhole Embeddings and time evolution of the Schwarzschild wormhole Peter Collasa) Department of Physics and Astronomy, California State University, Northridge, Northridge, California 91330-8268 David Kleinb) Department of Mathematics and Interdisciplinary Research Institute for the Sciences, California State University, Northridge, Northridge, California 91330-8313 (Received 26 July 2011; accepted 7 December 2011) We show how to embed spacelike slices of the Schwarzschild wormhole (or Einstein-Rosen bridge) in R3. Graphical images of embeddings are given, including depictions of the dynamics of this nontraversable wormhole at constant Kruskal times up to and beyond the “pinching off” at Kruskal times 61. VC 2012 American Association of Physics Teachers. [DOI: 10.1119/1.3672848] I. INTRODUCTION because the flow of time is toward r 0. Therefore, to under- stand the dynamics of the Einstein-Rosen¼ bridge or Schwarzs- 1 Schwarzschild’s 1916 solution of the Einstein field equa- child wormhole, we require a spacetime that includes not tions is perhaps the most well known of the exact solutions. only the two asymptotically flat regions associated with In polar coordinates, the line element for a mass m is Eq. (2) but also the region near r 0. For this purpose, the maximal¼ extension of Schwarzschild 1 2m 2m À spacetime by Kruskal7 and Szekeres,8 along with their global ds2 1 dt2 1 dr2 ¼À À r þ À r coordinate system,9 plays an important role. The maximal extension includes not only the interior and exterior of the r2 dh2 sin2 hd/2 ; (1) þ ð þ Þ Schwarzschild black hole [covered by the coordinates of Eq. (1)] but also a second copy of the exterior, as well as a region where we adopt units for which G 1 c. The realization that Eq. (1) describes what is now called¼ ¼ a black hole was far surrounding a white hole, from which particles may emerge but not enter (see Fig. 2). from immediate. It was not realized until the work of Oppen- 10 heimer and Snyder2 that such objects might exist and could Fuller and Wheeler employed Kruskal coordinates to result from the collapse of sufficiently massive stars. describe the geometry of the Schwarzschild wormhole and In contrast, investigations leading to the study of worm- showed that it is nontraversable, even by a photon. Their holes began almost immediately. Within one year of Ein- paper includes sketches of a sequence of wormhole profiles stein’s final formulation of the field equations, Flamm for particular spacelike slices, illustrating the formation, col- lapse, and subsequent “pinching-off” of the wormhole, but recognized that the Schwarzschild solution could represent 3 what we now describe as a wormhole, and in the 1920s, calculations for the embeddings in R were not provided. Weyl speculated about related possibilities.3 The nontraversability of the Schwarzschild wormhole makes In 1935 Einstein and Rosen4 introduced what would be the embeddings of its stages tricky, and this difficulty might called the Einstein-Rosen bridge as a possible geometrical account for the dearth of explanations in the literature. model of particles which avoided the singularities of points Although several widely used general relativity textbooks with infinite mass or charge densities. In the uncharged case, include qualitative descriptions and sketches of the dynamics 2 of the Schwarzschild wormhole (or its profiles) similar to the bridge arises from the coordinate change y r 2m, which transforms Eq. (1) to ¼ À those in Fig. 1, we are not aware of any published explana- tion for the calculations of embeddings.11 y2 In this article, we show how to embed spacelike slices of ds2 dt2 4 y2 2m dy2 the Schwarzschild wormhole in R3. The embeddings may be ¼Ày2 2m þ ð þ Þ þ thought of as snapshots of the wormhole in its various stages y2 2m 2 dh2 sin2 hd/2 : (2) as measured by a party of explorers (represented by test par- þð þ Þ ð þ Þ ticles) at fixed times, along particular spacelike slices of For < y < these coordinates omit the region inside spacetime. The most natural spacelike slices from the point the eventÀ1 horizon,1 0 < r < 2m, and twice cover the asymptoti- of view of Kruskal coordinates are slices of constant Kruskal cally flat region r 2m. The region near y 0 is the bridge time v, and we consider these first. Other slices are also pos- connecting the two asymptotically flat regions¼ close to sible and give rise to interesting geometries, as we illustrate y and y . Although the Einstein-Rosen bridge was in Secs. V and VI. The calculations and examples, apart not¼1 successful¼À as1 a model of particles, it emerged as the pro- from supplying the missing material to interested readers, totype wormhole in gravitational physics and led to the study are also suitable for general relativity and differential geome- of traversable wormholes (see, for example, Refs. 3 and 5, try courses. and for visual appearances Ref. 6). Our paper is organized as follows. In Sec. II, we review It can be seen from Eq. (1) that if r < 2m, then r is a time- Kruskal coordinates for the maximal extension of Schwarzs- like coordinate, t is spacelike, and within the event horizon, child spacetime, display the metric in this coordinate system, Schwarzschild spacetime is not static. A test particle within and give a qualitative description of the dynamics of the the event horizon moves inexorably to smaller r values Schwarzschild wormhole. In Sec. III, we develop a general 203 Am. J. Phys. 80 (3), March 2012 http://aapt.org/ajp VC 2012 American Association of Physics Teachers 203 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.166.243.16 On: Sat, 06 Feb 2016 05:00:52 Fig. 1. Schematic for the dynamical evolution of the Schwarzschild wormhole. From left to right: Kruskal time v < 1 prior to the formation of the bridge; formation at v 1; maximum extent at v 0 (middle); separation at v 1; Post separation at Kruskal time v > 1 (far right).À ¼À ¼ ¼ method for embedding space slices as surfaces of revolution in R3. Section IV applies this method to embeddings of con- u2 v2 r u; v 2m 1 W À : (9) stant Kruskal times between 1 and 1 to reveal the full dy- ð Þ¼ þ e namics of the wormhole fromÀ formation to collapse. At Kruskal times v > 1, the two universes, connected by the Figure 2 shows the maximally extended Schwarzschild wormhole at earlier times v < 1, have separated into two spacetime in terms of Kruskal coordinates, with the angular j j connected components. In Sec. V, we show how to treat a coordinates suppressed so that each point in the diagram rep- technical issue in order to embed space slices of nearly con- resents a 2-sphere. stant Kruskal time v > 1 (or v < 1). In Sec. VI, we show by The original Schwarzschild coordinates cover only regions À example how more general embeddings can be done, includ- I and II in Fig. 2. Region II is the interior of the black hole. ing an embedding consisting of three separated components. Region III is a copy of region I, and region IV is the interior Concluding remarks are given in Sec. VII. of a white hole, which lies in the past of any event. The sin- gularity at r 0 is given by the “singularity hyperbola” ¼ 2 II. KRUSKAL COORDINATES vs u 6p1 u . Lines making 45 angles with the coor- dinateð Þ¼ axes areþ null paths (that is, paths of light rays). Addi- Kruskal coordinates u,v and Schwarzschild coordinates tional detailsffiffiffiffiffiffiffiffiffiffiffiffiffi may be found, for example, in Refs. 15 and 16. are related by The dynamics of the wormhole can be described qualita- tively using Figs. 1 and 2. Figure 2 shows two distinct, 2 2 r 2 u v 1 er= m r 0 ; (3) asymptotically flat Schwarzschild manifolds or universes, À ¼ 2m À ð Þ one consisting of regions I and II, and the other of regions III and and IV. Regions I and III are asymptotically flat or Minkow- skian far from the singularities. They may be identified as 2uv t tanh ; (4) the top and bottom horizontal planes, respectively, in each u2 v2 ¼ 2m þ diagram of Fig. 1. If v < 1, the two universes are disconnected, each con- where t and r are the time and radial Schwarzschild coordi- taining anÀ infinite curvature singularity at 0. At time nates, respectively, of Eq. (1) and the angular coordinates are r v 1, the singularities join to form a nonsingular¼ bridge. unchanged. The coordinate v is timelike and it follows from As¼À the universes evolve, the bridge increases until v 0. At Eq. (4) that constant values of the ratio v/u correspond to this instant, the wormhole has maximum width, with¼ the constant Schwarzschild t. In Kruskal coordinates, the observer at 0 exactly on the event horizon 2 . As Schwarzschild metric (1) becomes u r m v increases the¼ observer at u 0 enters the region¼ r < 2m, 3 and the bridge narrows. By the¼ time v 1 the bridge pinches 2 32m r=2m 2 2 2 2 2 2 ds eÀ dv du r dh sin hd/ ; ¼ ¼ r ðÀ þ Þþ ð þ Þ (5) where r r(u,v) is determined implicitly by Eq. (3) and may be expressed¼ in terms of the Lambert W function22 as fol- lows. We divide both sides of Eq. (3) by e and write W f f W f e ð Þ; (6) ¼ ð Þ where in the present case u2 v2 f À : (7) ¼ e The Lambert W function, W(f), satisfies Eq.
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