Is the equivalence for the response of static scalar sources in the Schwarzschild and Rindler spacetimes valid only in four dimensions?

Abstract

It was shown recently that in four dimensions scalar sources with fixed proper acceleration minimally coupled to a massless Klein-Gordon field lead to the same responses when they are (i) uniformly accelerated in Minkowski spacetime (in the inertial vacuum) and (ii) static in the Schwarzschild spacetime (in the Unruh vacuum). Here we show that this equivalence is broken if the spacetime dimension is more than four.PACS numbers: 04.70. Dy, 04.62.+v Let us consider a pointlike scalar source with fixed proper acceleration, a 0 = const, minimally coupled to a massless Klein-Gordon field Φ through a small coupling constant q. It was shown recently that the source's response R S (r 0 , M ) to the Hawking radiation (associated with the Unruh vacuum) obtained when it lies at rest with (Schwarzschild) radial coordinate r 0 = const > 2M , outside a chargeless static black hole with mass M , is exactly the same as the response R M (a 0 ) of the source when it is uniformly accelerated (with the same proper acceleration as before) in the inertial vacuum of the Minkowski spacetime, or, equivalently, when it is static in the Fulling-Davies-Unruh thermal bath of the Rindler wedge [1].The fact that this result is surprising can be seen as follows. First let us recall that in Schwarzschild spacetime we can express the source's radial coordinate r 0 in terms of its proper acceleration a 0 and the black hole mass M : r 0 = r 0 (a 0 , M ). Thus, it would be natural to expect that the response would depend on M as well as on a 0 , i.e., R S = R S (a 0 , M ), rather thanWe note that structureless static scalar sources can only interact with zero-energy field modes. Such modes probe the global geometry of spacetime and are accordingly quite different in Schwarzschild and Rindler spacetimes. Indeed the equivalence (1) nor when the massless Klein-Gordon field is replaced by electromagnetic [2] or massive Klein-Gordon [3] one. Moreover, the equivalence was shown to be broken also when the background spacetime is endowed with a cosmological constant [4] or when the black hole is given some electric charge [5]. It is hitherto unclear whether or not the equivalence found in Ref.[1] hides something deeper behind it. Even in the less interesting case where the equivalence turns out to be a "coincidence", it will still be interesting to determine whether or not this is precisely restricted to the number of (macroscopic) dimensions of physical spacetimes, as we will do in this paper. Here we adopt natural units (c = G = = k B = 1) and spacetime signature (+ − · · · −).The line element of a Schwarzschild spacetime with N ≡ p + 2 dimensions (p ≥ 2) is [6]where f (r) = 1−(r H /r) p−1 with r H = (2M )

The Unruh effect and its applications

“…This equivalence was expected when the source is close to the horizon (Grishchuk et al, 1987) but not everywhere. Indeed, by considering other vacua (Higuchi et al, 1998) as in Hartle and Hawking (1976), other spacetimes (Castiñeiras and Matsas, 2000), fields (Castiñeiras et al, 2003;Crispino et al, 1998) or spacetime dimensionalities (Crispino et al, 2004) the equivalence is broken.…”

Scalar radiation from a radially infalling source into a Schwarzschild black hole in the framework of quantum field theory

“…The formalism of QFT applied to the problem of radiation emission has been used for the determination of the radiation emission by sources and charges rotating around a BH, known as synchrotron radiation [18][19][20][21][22][23][24][25]. Furthermore, QFT has been used to investigate radiation emission from an uniformly accelerated source in flat spacetimes [26,27] and also to investigate the interaction of sources with Hawking radiation [28][29][30][31][32][33][34]. In this paper, using QFT at tree level, we investigate in detail the properties of the radiation emission due to the radial infall of a particle source of a massless scalar field into a Schwarzschild BH.…”