The Lovelock gravity in the critical spacetime dimension

Abstract

It is well known that the vacuum in the Einstein gravity, which is linear in the Riemann curvature, is trivial in the critical (2+1=3) dimension because vacuum solution is flat. It turns out that this is true in general for any odd critical $d=2n+1$ dimension where $n$ is the degree of homogeneous polynomial in Riemann defining its higher order analogue whose trace is the nth order Lovelock polynomial. This is the "curvature" for nth order pure Lovelock gravity as the trace of its Bianchi derivative gives the corresponding analogue of the Einstein tensor \cite{bianchi}. Thus the vacuum in the pure Lovelock gravity is always trivial in the odd critical (2n+1) dimension which means it is pure Lovelock flat but it is not Riemann flat unless $n=1$ and then it describes a field of a global monopole. Further by adding Lambda we obtain the Lovelock analogue of the BTZ black hole.Comment: 3 pages, revised version with two minor changes and an author added. Accepted for publication in Physics Letters

Energetics and optical properties of 6-dimensional rotating black hole in pure Gauss–Bonnet gravity

“…That is, in all odd and even D = 2N + 1, 2N + 2 dimensions gravitational dynamics should be similar [29]. This is what has been verified in various situations; for instance the black hole entropy is always proportional to square of the horizon's radius [6,7] and bound orbits around a static black hole exist only for pure Lovelock gravity in all even D = 2N + 2 dimensions [26]. This motivates us to examine this general feature in all possible situations and that is what we wish to do it in this paper.…”

“…To make the problem tractable, it was assumed, for obtaining dimensionally continued black holes [5], that all the couplings were given in terms of the unique ground state λ N . On the other hand, there is a strong case for pure Lovelock gravity [6,7] where there is only one N th order term in the action with λ N . That is, it does not include even the usual Einstein-Hilbert term.…”

“…That is, it does not include even the usual Einstein-Hilbert term. This has the remarkable unique property that gravitational dynamics is distinguished in all odd and even dimensions [7]. In all odd D = 2N + 1 dimensions, gravity is kinematic meaning the N th order Lovelock analog Riemann tensor vanishes whenever the corresponding Ricci tensor vanishes [8].…”

Dimension of quantum channel of radiation in pure Lovelock black holes

“…(28) shows that in an odd D dimensional spacetime, entropy emission from a pure Lovelock black hole is D dimensional. In odd dimension pure Lovelock gravity has a similar behavior as a BTZ black hole [14]. In both of BTZ black hole and pure lovelock black hole in odd dimensions the dimension of quantum channel is obtained to be the same as spacetime dimensions.…”

“…The objective of the present paper is to calculate the dimensionality of entropy transmission from pure Lovelock black holes [13][14][15][16].…”