View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Dynamic wormholes Sean A. Hayward Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan [email protected] (Revised 14th December 1998) ative energy is required generically. A new framework is proposed for general dynamic worm- The fundamental problem is that there is no agreed holes, unifying them with black holes. Both are generically idea of what wormholes actually are in general. This let- defined locally by outer trapping horizons, temporal for worm- ter proposes such a definition and discusses widespread holes and spatial or null for black and white holes. Thus consequences. The concepts involved are mostly familiar wormhole horizons are two-way traversible, while black-hole in relativity, e.g. Hawking & Ellis [4], and otherwise de- and white-hole horizons are only one-way traversible. It fol- lows from the Einstein equation that the null energy condition fined in the author’s previous papers on black holes [5,6], is violated everywhere on a generic wormhole horizon. It is so will not be explained in detail here. Similarly, proofs suggested that quantum inequalities constraining negative en- will be avoided since they are simple extensions or con- ergy break down at such horizons. Wormhole dynamics can verses of previous results for black holes and will anyway be developed as for black-hole dynamics, including a reversed be spelled out in a longer article [7]. second law and a first law involving a definition of worm- A static wormhole may be defined simply by a mini- hole surface gravity. Since the causal nature of a horizon mal surface in a static spatial (spacelike) hypersurface, so can change, being spatial under positive energy and temporal that it is locally the narrowest section of a tunnel between under sufficient negative energy, black holes and wormholes two larger regions. Morris & Thorne gave further worm- are interconvertible. In particular, if a wormhole's negative- hole criteria which will not be considered here, except for energy source fails, it may collapse into a black hole. Con- the basic context of Einstein gravity. This effectively in- versely, irradiating a black-hole horizon with negative energy cludes other gravitational theories like Brans-Dicke which could convert it into a wormhole horizon. This also suggests involve what can be treated as an Einstein equation with a possible final state of black-hole evaporation: a stationary other fields, as emphasised by Hochberg & Visser. wormhole. The new framework allows a fully dynamical de- To define dynamic wormholes, an analogous idea (re- scription of the operation of a wormhole for practical trans- fined below) is that the spatial surface is minimal in some port, including the back-reaction of the transported matter spatial hypersurface. This implies that the surface is on the wormhole. As an example of a matter model, a Klein- generically a (future or past) trapped surface, usually as- Gordon field with negative gravitational coupling is a source sociated with black or white holes. Here genericity means for a static wormhole of Morris & Thorne. stability in the sense that any sufficiently small variation 04.70.-s, 04.20.-q of a trapped surface is also a trapped surface, and there- fore also minimal in some spatial hypersurface.1 Thus a wormhole must generically be a region of space-time. Space-time wormholes, short cuts between otherwise This differs qualitatively from a static wormhole, which distant or even unconnected regions of the universe, have is the temporal (timelike) hypersurface generated by the become a common feature in science fiction. That they minimal surfaces. For a wormhole region to be two- might be science fact is suggested by the simplest station- way traversible, it should have two boundaries, referred ary black-hole solutions to the Einstein equations, which to here as wormhole horizons and elsewhere as worm- do indeed have the global structure of wormholes, albeit hole throats, which are in causal contact. Local two-way just barely failing to be traversible. Morris & Thorne [1] traversibility suggests that these horizons should be tem- revived interest in the possibility of wormholes for prac- poral hypersurfaces. Moreover, one expects the horizons tical travel, showing that it is theoretically possible in to be foliated by marginal surfaces (marginally trapped Einstein gravity, provided sufficient negative energy can surfaces), as occurs for black-hole apparent horizons ac- be maintained. Although mundane matter has positive cording to Hawking & Ellis. Proving this is non-trivial energy, quantum theory predicts negative energy during particle creation, such as in the Casimir effect. Morris & Thorne considered the simplest case of static, spheri- cally symmetric wormholes, and subsequently there has 1The non-generic case is a surface which is doubly marginal been some controversy about whether negative energy is somewhere: arbitrary variations of such a surface are usually necessary in general, or if so, whether it can be avoided not minimal in any spatial hypersurface. This class includes by travellers navigating the wormhole. This is discussed static minimal surfaces and suggests a minor generalisation in detail by Hochberg & Visser [2,3], who independently from static to stationary wormholes: a stationary outer trap- reach the conclusion (confirmed in this article) that neg- ping horizon. 1 [8], so it seems more practical to assume it as part of the ping horizons are spatial or null, reflecting the intuitive definition, as follows. idea that one may enter but not leave a black hole, and A hypersurface foliated by marginal surfaces will be leave but not enter a white hole. The above negative- called a trapping horizon, following Ref. [5]. The energy theorem is a converse of this result. Thus the marginal surfaces are a dynamic generalisation of the framework actually unifies black holes and wormholes, re- static minimal surfaces, except for the so-called flare-out vealing that they are similar objects, geometrically and condition selecting minimal surfaces from extremal ones. physically. Indeed, black holes and wormholes are lo- For the generic flare-out condition, this letter proposes cally identical except for the causal nature of the hori- an outer trapping horizon as defined in Ref. [5]. This zons. If the horizon is temporal and therefore two-way is a strict inequality, excluding cases of equality which traversible, we can call it a wormhole horizon, while if it describe degenerate wormholes, for reasons explained be- is spatial or null and therefore only one-way traversible, low. It reduces in the static case to the strict static flare- we can call it a black-hole or white-hole horizon, respec- out condition. In short, a generic wormhole horizon is tively if the trapped region is to the future or past. One defined as a temporal outer trapping horizon. may refer to all collectively simply as space-time holes. The definition implies that the marginal surfaces are This connection between black holes and wormholes minimal in the null direction of marginality. Thus each seems to have gone unnoticed elsewhere. Indeed, even surface is locally the narrowest section of a null hyper- the most seminal authors [1,2] have instead stressed that surface, naturally generalising the static definition. This wormholes are quite different from black holes, sternly sort of idea was first suggested by Page, according to warning against comparisons. This seems to be due to Morris & Thorne. A precise formulation, similar to that misconceptions engendered by the old global paradigm proposed here, is suggested independently by Hochberg for black holes in terms of event horizons. The connection & Visser, and indeed their strict flare-out condition is arises instead from the new local, dynamical paradigm for implied by the above definition. However, the converse black holes in terms of trapping horizons [5,6]. is not so, since Hochberg & Visser specifically allow spa- This theory of black-hole dynamics induces a corre- tial horizons, in which case their flare-out condition se- sponding theory of wormhole dynamics, as will be de- lects maximal rather than minimal surfaces in a time- tailed in Ref. [7]. Some basic laws of wormhole dynamics symmetric hypersurface. can be obtained simply by adapting those for black and An immediate consequence of the definition and the white holes. In particular, the second law of black-hole Einstein equation is that the null energy condition is vio- dynamics of Ref. [5] becomes reversed under negative en- lated everywhere on a generic wormhole horizon [7]. This ergy, yielding a second law of wormhole dynamics: fu- negative-energy theorem offers a clean resolution to the ture, past or stationary wormhole horizons respectively above controversy. This simple result gives more infor- have decreasing, increasing or constant area. mation than the topological censorship theorem of Fried- In spherical symmetry, the definition of surface gravity man et al. [9], which showed that the null energy con- proposed in Ref. [6] for dynamic black or white holes may dition excludes wormholes as defined globally, in asymp- also be applied to wormholes, providing a physical mea- totically flat space-times. Such global assumptions have sure of the strength of the wormhole. This seems to have been avoided here in favour of a local, dynamical defini- been lacking even for static wormholes, for which one tion of wormhole horizon. Similarly, the negative-energy finds that the surface gravity over circumference equals property is purely local, applying to each point of the tension minus energy density. Positivity of the surface wormhole horizon. Thus the definition could be relaxed gravity is equivalent to the strict flare-out condition, giv- so that the horizon need be temporal only somewhere, in ing the latter a physical meaning.