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.^{[1]} 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.
Lagrangian density
The Lagrangian of the theory is given by a sum of dimensionally extended Euler densities, and it can be written as follows

\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

\delta _{\alpha _{1}\beta _{1} \cdots \alpha _{n}\beta _{n}}^{\mu _{1}\nu _{1}...\mu _{n}\nu _{n}}=\frac{1}{n!}\delta _{\lbrack \alpha _{1}}^{\mu _{1}}\delta _{\beta _{1}}^{\nu _{1}}\cdots \delta _{\alpha _{n}}^{\mu _{n}}\delta _{\beta _{n}]}^{\nu _{n}}.
Each term \mathcal{R}^{n} in \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.
Coupling constants
The coupling constants α_{n} in the Lagrangian \mathcal{L} have dimensions of [length]^{2n − D}, although it is usual to normalize the Lagrangian density in units of the Planck scale

\alpha _{1}=(16\pi G)^{1}=l_{P}^{2D}\,.
Expanding the product in \mathcal{L}, the Lovelock Lagrangian takes the form

\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 α_{0} 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

\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 fourdimensional Euler density.
Other contexts
Because Lovelock action contains, among others, the quadratic Gauss–Bonnet term (i.e. the fourdimensional 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 sixdimensional Calabi–Yau compactifications of Mtheory. 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 ghostfree 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 mid2000s the theory was considered as a testing ground to investigate the effects of introducing highercurvature terms in the context of AdS/CFT correspondence.
See also
Notes

^ D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498.
References






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.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.