Standing waves in general relativity

Abstract

We propose a covariant definition of standing gravitational waves in general relativity. 1 The same year the problem of gravitational standing waves was investigated by Sir Hermann Bondi in his last paper on general relativity [2]. Standing waves appear naturally in some quasistationary approximations in binary black holes inspirals -an approach introduced by Detweiler [3] and pursued by Price [4] and others [5][6][7][8][9][10].

An example of cross-polarized standing gravitational waves

“…The studies of this type has been initiated by Bondi [2] and Stephani [7]. More recently, the problem has been investigated in the articles [8], [9].…”

Freely falling bodies in a standing-wave spacetime

“…Stephani tackled this problem in the article [12] where he presented an example of an exact standing wave spacetime. An alternative definition was proposed in the article [15]. The intuitive notion of a standing wave implies that there is an alternating energy flow through nodes which averages to zero in time.…”

“…Unfortunately, the energy of gravitational waves cannot be easily localized and the covariant averaging procedure is not known which makes a covariant definition of standing gravitational waves problematic. It has been suggested in the article [15] to evade this problem by taking the high frequency limit which captures the dominant contribution to the average energy flow. According to the definition presented in the paper [15] this limit should correspond to an effective spacetime with the Ricci tensor of the Serge type [(11)1, 1].…”

“…It has been suggested in the article [15] to evade this problem by taking the high frequency limit which captures the dominant contribution to the average energy flow. According to the definition presented in the paper [15] this limit should correspond to an effective spacetime with the Ricci tensor of the Serge type [(11)1, 1].…”

An hyperbolic system of P.D.E. relevant in general relativity

“…This case for its simplicity has been, and still is, the object, in various contexts, of many papers we quote, among others [26], [27], [28], [5] and [17]. In this paper we consider the case ω = 0.…”