Cylindrical gravitational waves in expanding universes: Models for waves from compact sources

Abstract

New boundary conditions are imposed on the familiar cylindrical gravitational wave vacuum spacetimes. The new spacetime family represents cylindrical waves in a flat expanding (Kasner) universe. Space sections are flat and nonconical where the waves have not reached and wave amplitudes fall off more rapidly than they do in Einstein-Rosen solutions, permitting a more regular null inifinity.This paper shows how to construct exact solutions for compact cylindrical gravitational wave pulses spreading in an expanding universe that is spatially flat at large cylindrical radius, where the waves have not reached. The universal expansion is restricted to the translation-direction (or z-axis) of cylindrical symmetry. The remaining directions show gravitational waves spreading in a three-dimensional spacetime that can be exactly Minkowskian before the wave pulse. The effective three dimensional spacetime has a completely regular conformal compactification so that a radiation zone at future null infinity is well defined without any rescaling of fields. This simple behavior makes these solutions ideal for testing the numerical simulation codes that are used to predict gravitational waves from compact sources. Because they have the simplicity of Einstein-Rosen solutions but are compatible with the boundary conditions of the real universe, these new solutions may model localized inhomogeneities in Big Bang cosmologies and may also provide an interesting restricted framework for testing formal ideas such as quantum gravity.The original Einstein-Rosen cylindrical wave solutions of Einstein's vacuum field equations[1, 2] inhabit a universe that is conical rather than flat at infinite cylindrical radius [3] and which, apart from the oscillations of the waves, does not evolve in time. Thus, they are not good models for inhomogeneities in our own universe, which is thought to be spatially flat and expanding and do not make convenient test cases for numerical simulations of gravitational waves from compact sources. However, the boundary conditions can be changed. Cylindrically symmetric (G 2 ) exact solutions with closed space sections of S 3 , S 1 × S 2 and S 1 × S 1 × S 1 (three-torus) topologies are known and have proven useful as toy models of inhomogeneous cosmologies.[4, 5, 6] The three-torus solutions, in particular, can be regarded as standing-gravitational-wave inhomogeneous generalizations of spatially flat Kasner universes [7,8,9,10] and have provided useful test cases for validating numerical simulations of Einstein's field equations. [11,12] Here, we will change the boundary conditions again.We will use the notation of earlier papers [4,5,6] on these spacetimes, but with some changes in coordinate names to fit the new circumstances. The isometry group coordinates are z and ϕ (instead of σ and δ used in the earlier papers), with z ranging from −∞ to +∞ and ϕ ranging from 0 to 2π. The remaining coordinates t, r each range from 0 to ∞. The isometry group consists of translations of z and of ϕ modulo 2π. Thus, each group orb...

Visualizing gravitational Bessel waves

“…There is a well-known exact solution of Einstein equations -the already mentioned Einstein-Rosen waves, which were utilized in discussions concerning the reality of gravitational radiation [52]. Results about the underlying topology of a specific class of cylindrical gravitational waves are available as well [53], e.g., spacelike hypersurfaces of these gravitational waves can be homeomorphic for example to a tree-torus. This means that these waves admit compact sources.…”

5D gravitational waves from complexified black rings

“…Here we have complexified the ψ coordinate to have a standard signature. In fact, because of the above flat Kasner form of the 5-D metric, what one obtains here is the first example of the so-called boost-symmetric spacetime in higher dimensions [15], see also [16], and may represent radiation generated by accelerated sources in extra dimensions. It is interesting that if one generates cylindrical waves in a similar manner from a four dimensional black hole, no boost-symmetry is apparently present, it appears in the 5th-dimension due to the coupling of the radiation to the extra compact dimension-the term t 2 dψ 2 in the line element in this limit.…”

“…G µ G µ is associated to the expansion scalar and defines the trapped 3-surfaces for fixed (R, T ). In case G µ G µ > 0, the solution can be interpreted as cylindrical gravitational waves; when G µ G µ varies from point to point then (16) represents gravitational waves propagating along the R-direction in an expanding universe.…”

“…The next step would be to perform the complex trick: ξ → iT (this changes the sign of dξ 2 ) together with t → iz and K → iK; by doing so we arrive to the generalized Einstein-Rosen form (16).…”

Summary of session A1: exact solutions and their interpretation

“…Actually, these metrics were extensively studied in [4], where they were dubbed "Boost-rotation-symmetric spacetimes". The spacetimes are claimed to describe gravitational waves travelling in an expanding background given by flat spacetime in Kasner form ‡ [17], and to contain "rogue waves". Despite some technical difficulties concerning the addition of single pulses at infinity, some wellbehaved waves can be constructed.…”