Null asymptotics of solutions of the Einstein–Maxwell equations in general relativity and gravitational radiation

Abstract

We prove that for spacetimes solving the Einstein-Maxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In [5] Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a permanent displacement of test masses in a laser interferometer gravitational-wave detector. Christodoulou derived a precise formula for this permanent displacement in the Einstein vacuum (EV) case. We prove in Theorem 2.6 that for the EM equations this permanent displacement exhibits a term coming from the electromagnetic field. This term is at the same highest order as the purely gravitational e-print archive: http://lanl.arXiv.org/abs/1011.2267 LYDIA BIERI, PONING CHEN AND SHING-TUNG YAUterm that governs the EV situation. On the other hand, in Section 3, we show that to leading order, the presence of the electromagnetic field does not change the instantaneous displacement of the test masses. Following the method introduced by Christodoulou in [5] and asymptotics derived by Zipser in [8,9], we investigate gravitational radiation at null infinity in spacetimes solving the EM equations. We study the Bondi mass loss formula at null infinity derived in [9]. We show that the mass loss formula from [9] is compatible with the one in Bondi coordinates obtained in [4]. And we observe that the presence of the electromagnetic field increases the total energy radiated to infinity up to leading order. Moreover, we compute the limit of the area radius at null infinity in Theorem 2.7.

Retarded fields of null particles and the memory effect

“…Thorne [2] and Wiseman and Will [3] soon interpreted Christodoulou's nonlinear memory effect as simply corresponding to the linear memory effect, but with the nonlinear effective stress energy of gravitational waves replacing the "particle" sources. Further support for this interpretation can be found in the fact that a similar nonlinear memory effect occurs when a flux of electromagnetic radiation reaches null infinity [5], thereby showing that the nonlinear memory effect is not special to gravitational waves. Very recently, Bieri and Garfinkle [6] have shown that the linear memory effect for null matter can be derived in close parallel to Christodoulou's derivation, thus further confirming that the nonlinear memory effect can be interpreted as being the same as the linear memory effect, with the effective stress energy of gravitational radiation replacing the ordinary stress energy of null matter.…”

“…Consider, in Newtonian gravity, a particle of mass E traveling with velocity c along the z axis. The Newtonian potential produced by such a particle at time t is 5 In particular, the Ricci component R 00 is easily computed by adding together the first two lines of their Eq. (3.12) and does not agree with the (correct) expression they give below Eq.…”

Gravitational wave memory in de Sitter spacetime

“…Gravitational wave memory, a permanent displacement of the gravitational wave detector after the wave has passed, has been known since the work of Zel'dovich and Polnarev [1], extended to the full nonlinear theory of general relativity by Christodoulou [2], and treated by several authors [3][4][5][6][7][8][9][10][11][12][13][14][15]. As is usual in the treatment of isolated systems, all these works consider asymptotically flat spacetimes.…”

Black hole formation and stability: A mathematical investigation

“…In fact, L. Bieri and D. Garfinkle showed [11] that these are two different effects. For memory, see also the following works by Braginsky and Grishchuk [17], Braginsky and Thorne [18], Blanchet and Damour [14], Frauendiener [38], Wiseman and Will [76], and more recently Bieri, Chen, and S.-T. Yau [8], [9], Bieri and Garfinkle [10], [11], Tolish and Wald [73], Flanagan and Nichols [35], [36], and Favata [33]. We refer to these articles for further references.…”