Lovelock theory of gravity
In 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 . In this sense, Lovelock's theory is the natural generalization of Einstein's General Relativity to higher dimensions. In three and four dimensions (
), Lovelock's theory coincides with Einstein's theory, but in higher dimensions the theories are different. In fact, for
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 extended Euler densities, and it can be written as follows
where represents the Riemann tensor, and where the generalized Kronecker
-function is defined as the
antisymmetric product
Each term in
corresponds to the dimensional
extension of the Euler density in
dimensions, so that these only
contribute to the equations of motion for
. Consequently, without
lack of generality,
in the equation above can be taken to be
for
even dimensions and
for odd dimensions.
The coupling constants in Lagrangian
have
dimensions of [length]
, although it is usual to normalize the
Lagrangian density in units of the Planck scale
. Expanding the product in
, the Lovelock
Lagrangian takes the form
where one sees that coupling corresponds to the cosmological constant
, while
with
are coupling
constants of additional terms that represent ultraviolet corrections to
Einstein theory, involving higher order contractions of the Riemann tensor
. 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.
Due to the fact that Lovelock action contains, among others, the quadratic Gauss–Bonnet
term (i.e. the four-dimensional Euler characteristic extended to dimensions), it is usually said that Lovelock theory resembles string theory
inspired models of gravity. This is because a quadratic term is present in the
low energy effective action of heterotic string theory, and it also appears
in six-dimensional Calabi–Yau compactifications of M-theory. In the mid
1980s, a decade after Lovelock proposed his generalization of the Einstein
tensor, physicists began to discuss the quadratic Gauss–Bonnet term within the context of string theory, with particular
attention to its property of being ghost-free in Minkowski space.
The theory is known to be free of ghosts about other exact backgrounds as
well, e.g. about one of the branches of the spherically symmetric solution
found by Boulware and Deser in 1985. In general, Lovelock's theory
represents a very interesting scenario to study how the physics of gravity
is corrected at short distance due to the presence of higher order
curvature terms in the action, and in the mid 2000s the theory was
considered as a testing ground to investigate the effects of introducing
higher-curvature terms in the context of AdS/CFT correspondence.
See also
References
- D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498.
- D. Lovelock, The four-dimensionality of space and the Einstein tensor, J. Math. Phys. 13 (1972) 874.
- D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles, Dover Publications 1989.
- A. Navarro and J. Navarro, Lovelock's theorem revisited, J. Geom. Phys. 61 (2011) 1950-1956. (PDF)
- B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. B156 (1985) 315.
- D. Boulware and S. Deser, String Generated Gravity Models, Phys. Rev. Lett. 55 (1985) 2656.
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