Lovelock theory of gravity

The Lagrangian of the theory is given by a sum of dimensionally extended Euler densities, and it can be written as follows

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\ \sum \limits _{n=0}^{t}\alpha _{n}\ {\mathcal {R}}^{n},\qquad {\mathcal {R}}^{n}={\frac {1}{2^{n}}}\delta _{\alpha _{1}\beta _{1}...\alpha _{n}\beta _{n}}^{\mu _{1}\nu _{1}...\mu _{n}\nu _{n}}\prod \limits _{r=1}^{n}R_{\quad \mu _{r}\nu _{r}}^{\alpha _{r}\beta _{r}}}$

where R_μν^αβ represents the Riemann tensor, and where the generalized Kronecker delta δ is defined as the antisymmetric product

${\displaystyle \delta _{\alpha _{1}\beta _{1}\cdots \alpha _{n}\beta _{n}}^{\mu _{1}\nu _{1}...\mu _{n}\nu _{n}}=(2n)!\delta _{\lbrack \alpha _{1}}^{\mu _{1}}\delta _{\beta _{1}}^{\nu _{1}}\cdots \delta _{\alpha _{n}}^{\mu _{n}}\delta _{\beta _{n}]}^{\nu _{n}}.}$

Each term ${\displaystyle {\mathcal {R}}^{n}}$ in ${\displaystyle {\mathcal {L}}}$ 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 α_n in the Lagrangian ${\displaystyle {\mathcal {L}}}$ have dimensions of [length]^{2n − D}, although it is usual to normalize the Lagrangian density in units of the Planck scale

${\displaystyle \alpha _{1}=(16\pi G)^{-1}=l_{P}^{2-D}\,.}$

Expanding the product in ${\displaystyle {\mathcal {L}}}$, the Lovelock Lagrangian takes the form

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\ (\alpha _{0}+\alpha _{1}R+\alpha _{2}\left(R^{2}+R_{\alpha \beta \mu \nu }R^{\alpha \beta \mu \nu }-4R_{\mu \nu }R^{\mu \nu }\right)+\alpha _{3}{\mathcal {O}}(R^{3})),}$

where one sees that coupling α₀ corresponds to the cosmological constant Λ, while α_n with n ≥ 2 are coupling constants of additional terms that represent ultraviolet corrections to Einstein theory, involving higher order contractions of the Riemann tensor R_μν^αβ. In particular, the second order term

${\displaystyle {\mathcal {R}}^{2}=R^{2}+R_{\alpha \beta \mu \nu }R^{\alpha \beta \mu \nu }-4R_{\mu \nu }R^{\mu \nu }}$

is precisely the quadratic Gauss–Bonnet term, which is the dimensionally extended version of the four-dimensional Euler density.

By noting that

${\displaystyle T={\sqrt {-g}}{\mathcal {R}}^{2}={\sqrt {-g}}\left(R^{2}+R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }-4R_{\mu \nu }R^{\mu \nu }\right)}$

is a topological constant, we can eliminate the Riemann tensor term and thus we can put the Lovelock Lagrangian into the form

${\displaystyle S=-\int d^{D}x{\sqrt {-g}}\left(\alpha R_{\mu \nu }R^{\mu \nu }-\beta R^{2}+\gamma \kappa ^{-2}R\right)}$

which has the equations of motion

${\displaystyle \alpha \left(-{\frac {1}{2}}R_{\rho \sigma }R^{\rho \sigma }g_{\mu \nu }-\nabla _{\nu }\nabla _{\mu }R-2R_{\rho \nu \mu \sigma }R^{\sigma \rho }+{\frac {1}{2}}g_{\mu \nu }\Box R+\Box R_{\mu \nu }\right)+}$${\displaystyle \beta \left({\frac {1}{2}}R^{2}g_{\mu \nu }-2RR_{\mu \nu }-2\nabla _{\nu }\nabla _{\mu }R+2g_{\mu \nu }\Box R\right)+}$${\displaystyle \gamma \left(-{\frac {1}{2}}\kappa ^{-2}Rg_{\mu \nu }+\kappa ^{-2}R_{\mu \nu }\right)=0.}$[2]

Because Lovelock action contains, among others, the quadratic Gauss–Bonnet term (i.e. the four-dimensional Euler characteristic extended to D 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.

• Lovelock's theorem
• f(R) gravity
• Gauss–Bonnet gravity
• Curtright field

• Lovelock, D. (1971). "The Einstein tensor and its generalizations". Journal of Mathematical Physics. 12 (3): 498–502. Bibcode:1971JMP....12..498L. doi:10.1063/1.1665613. Archived from the original on 2013-02-24.
• Lovelock, D. (1969). "The uniqueness of the Einstein field equations in a four-dimensional space". Archive for Rational Mechanics and Analysis. 33 (1): 54–70. Bibcode:1969ArRMA..33...54L. doi:10.1007/BF00248156.
• Lovelock, D. (1972). "The four-dimensionality of space and the Einstein tensor". Journal of Mathematical Physics. 13 (6): 874–876. Bibcode:1972JMP....13..874L. doi:10.1063/1.1666069.
• Lovelock, David; Rund, Hanno (1989), Tensors, Differential Forms, and Variational Principles, Dover, ISBN 978-0-486-65840-7
• Navarro, A.; Navarro, J. (2011). "Lovelock's theorem revisited". Journal of Mathematical Physics. 61 (10): 1950–1956. arXiv:1005.2386. Bibcode:2011JGP....61.1950N. doi:10.1016/j.geomphys.2011.05.004.
• Zwiebach, B. (1985). "Curvature Squared Terms and String Theories". Phys. Lett. B. 156 (5–6): 315. doi:10.1016/0370-2693(85)91616-8..
• Boulware, D.; Deser, S. (1985). "String Generated Gravity Models". Phys. Rev. Lett. 55 (24): 2656. doi:10.1103/PhysRevLett.55.2656. PMID 10032204.