Gaussian null coordinates, near-horizon geometry and conserved charges on the horizon of extremal non-dilatonic black $p$-branes

Abstract

In this paper, we examine the emergence of conserved charges on the horizon of a particular class of extremal non-dilatonic black $p$-branes (which reduce to extremal dilatonic black holes in $D=4$ dimensions upon toroidal compactification) in the presence of a probe massless scalar field in the bulk. This result is achieved by writing the black $p$-brane geometry in a Gaussian null coordinate system which allows us to get a non-singular near-horizon geometry description. We find that the near-horizon geometry is $Ad{S}_{p+2}\times S^2$ and that the $Ad{S}_{p+2}$ section has an internal structure which can be seen as a warped product of $Ad{S}_2\times S^p$ in Gaussian null coordinates. We show that the bulk scalar field satisfying the field equations is expanded in terms of non-normalizable and normalizable modes, which for certain suitable quantization conditions are well-behaved at the boundary of $Ad{S}_{p+2}$ space. Furthermore, we show that picking the normalizable modes results in the existence of conserved quantities on the horizon. We discuss the impact of these conserved quantities in the late time regime.

arXiv comment25 pages, 1 figure, Contribution to Selected Papers of the Fifth Amazonian Symposium on Physics

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