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In this paper, we examine the emergence of conserved charges on the horizonof a particular class of extremal non-dilatonic black $p$-branes (which reduceto extremal dilatonic black holes in $D=4$ dimensions upon toroidalcompactification) 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 Gaussiannull coordinate system which allows us to get a non-singular near-horizongeometry description. We find that the near-horizon geometry is$AdS_{p+2}\times S^2$ and that the $AdS_{p+2}$ section has an internalstructure which can be seen as a warped product of $AdS_{2}\times S^{p}$ inGaussian null coordinates. We show that the bulk scalar field satisfying thefield equations is expanded in terms of non-normalizable and normalizablemodes, which for certain suitable quantization conditions are well-behaved atthe boundary of $AdS_{p+2}$ space. Furthermore, we show that picking thenormalizable modes results in the existence of conserved quantities on thehorizon. We discuss the impact of these conserved quantities in the late timeregime.

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