The Gravitational Equation in Higher Dimensions

Abstract

Like the Lovelock Lagrangian which is a specific homogeneous polynomial in Riemann curvature, for an alternative derivation of the gravitational equation of motion, it is possible to define a specific homogeneous polynomial analogue of the Riemann curvature, and then the trace of its Bianchi derivative yields the corresponding polynomial analogue of the divergence free Einstein tensor defining the differential operator for the equation of motion. We propose that the general equation of motion is G (n) ab = −Λg ab + κnT ab for d = 2n + 1, 2n + 2 dimensions with the single coupling constant κn, and n = 1 is the usual Einstein equation. It turns out that gravitational behavior is essentially similar in the critical dimensions for all n. All static vacuum solutions asymptotically go over to the Einstein limit, Schwarzschild-dS/AdS. The thermodynamical parameters bear the same relation to horizon radius, for example entropy always goes as r d−2n h and so for the critical dimensions it always goes as r h , r 2 h . In terms of the area, it would go as A 1/n . The generalized analogues of the Nariai and Bertotti-Robinson solutions arising from the product of two constant curvature spaces, also bear the same relations between the curvatures k1 = k2 and k1 = −k2 respectively.

A distinguishing gravitational property for gravitational equation in higher dimensions

“…The newly recognized kinematic property is clearly its distinguishing feature. It is thus the right gravitational equation [6] in higher dimensions, d = 2N + 1, 2N + 2. That is, for each N , the equation is only for the corresponding two odd and even dimensions; for instance for N = 1, the Einstein equation is only for d = 3, 4, for N = 2, the pure GB equation only for d = 5, 6, and so on.…”

“…Note that the pure Lovelock equation [6] has several interesting and desirable features. For instance even though the equation is completely free from the Einstein term yet a static vacuum solution with asymptotically goes over to an Einstein-dS solution in the given dimension [11].…”

“…That is what we shall probe next by appealing to a general principle and some gravitational properties. Next we will argue that the pure Lovelock case is thus a proper gravitational equation [6] in higher dimensions and it is the only one that obeys kinematicity of gravity in all odd, d = 2N +1, dimensions. It is valid only for two, odd, d = 2N +1, and even, d = 2N +2, dimensions.…”

Black Holes in Higher Dimensions (Black Strings and Black Rings)

“…While general Lovelock theories include all terms allowed for a given dimension in the action, thus combining the action of General Relativity with higher order Lovelock terms, in pure Lovelock theory the Einstein-Hilbert action is present only in three and four dimensions, whereas in higher dimensions always only the respective Lovelock term L N , characteristic for the dimension D, where N = ⌊(D − 1)/2⌋, together with the cosmological constant is considered [137][138][139][140] .…”

“…In his view pure Lovelock gravity is the most natural generalization of General Relativity because it uniquely retains the second order field equations, while its action is a homogeneous polynomial built from the Riemann tensor [137][138][139][140] . Indeed, in all odd dimensions D = 2N + 1, in pure Lovelock theory gravity in kinematic 141,142 .…”