Lovelock theory of gravity

In theoretical physics, Lovelock's theory of gravity (often referred to as Lovelock gravity) is a generalization of Einstein's theory of general relativity introduced by David Lovelock in 1971. It is the most general metric theory of gravity yielding conserved second order equations of motion in arbitrary number of spacetime dimensions D. In this sense, Lovelock's theory is the natural generalization of Einstein's General Relativity to higher dimensions. In three and four dimensions (D = 3, 4), Lovelock's theory coincides with Einstein's theory, but in higher dimensions the theories are different. In fact, for D > 4 Einstein gravity can be thought of as a particular case of Lovelock gravity since the Einstein–Hilbert action is one of several terms that constitute the Lovelock action. In theoretical physics, Lovelock's theory of gravity (often referred to as Lovelock gravity) is a generalization of Einstein's theory of general relativity introduced by David Lovelock in 1971. It is the most general metric theory of gravity yielding conserved second order equations of motion in arbitrary number of spacetime dimensions D. In this sense, Lovelock's theory is the natural generalization of Einstein's General Relativity to higher dimensions. In three and four dimensions (D = 3, 4), Lovelock's theory coincides with Einstein's theory, but in higher dimensions the theories are different. In fact, for D > 4 Einstein gravity can be thought of as a particular case of Lovelock gravity since the Einstein–Hilbert action is one of several terms that constitute the Lovelock action. The Lagrangian of the theory is given by a sum of dimensionally extendedEuler densities, and it can be written as follows where Rμναβ represents the Riemann tensor, and where the generalized Kronecker delta δ is defined as the antisymmetric product Each term R n {displaystyle {mathcal {R}}^{n}} in L {displaystyle {mathcal {L}}} corresponds to the dimensional extension of the Euler density in 2n dimensions, so that these only contribute to the equations of motion for n < D/2. Consequently, without lack of generality, t in the equation above can be taken to be D = 2t + 2 for even dimensions and D = 2t + 1 for odd dimensions. The coupling constants αn in the Lagrangian L {displaystyle {mathcal {L}}} have dimensions of 2n − D, although it is usual to normalize the Lagrangian density in units of the Planck scale Expanding the product in L {displaystyle {mathcal {L}}} , the Lovelock Lagrangian takes the form where one sees that coupling α0 corresponds to the cosmological constant Λ, while αn with n ≥ 2 are coupling constants of additional terms that represent ultraviolet corrections to Einstein theory, involving higher order contractions of the Riemann tensor Rμναβ. In particular, the second order term is precisely the quadratic Gauss–Bonnet term, which is the dimensionally extended version of the four-dimensional Euler density.