- Art Gallery -

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 an 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 Rμναβ 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 \( \mathcal{R}^{n} \) 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 αn 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

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

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 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

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

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

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]

Other contexts

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.


See also

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

Notes

Lovelock, David (1971). "The Einstein Tensor and Its Generalizations". Journal of Mathematical Physics. AIP Publishing. 12 (3): 498–501. doi:10.1063/1.1665613. ISSN 0022-2488.

"Higher Derivative Theories of Gravity" (PDF). pp. 10, 15.

References

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". Physics Letters B. 156 (5–6): 315. doi:10.1016/0370-2693(85)91616-8..
Boulware, D.; Deser, S. (1985). "String Generated Gravity Models". Physical Review Letters 55 (24): 2656. doi:10.1103/PhysRevLett.55.2656. PMID 10032204.

vte

Theories of gravitation
Standard
Newtonian gravity (NG)

Newton's law of universal gravitation Gauss's law for gravity Poisson's equation for gravity History of gravitational theory

General relativity (GR)

Introduction History Mathematics Exact solutions Resources Tests Post-Newtonian formalism Linearized gravity ADM formalism Gibbons–Hawking–York boundary term

Alternatives to
general relativity
Paradigms

Classical theories of gravitation Quantum gravity Theory of everything

Classical

Einstein–Cartan Bimetric theories Gauge theory gravity Teleparallelism Composite gravity f(R) gravity Infinite derivative gravity Massive gravity Modified Newtonian dynamics, MOND
AQUAL Tensor–vector–scalar Nonsymmetric gravitation Scalar–tensor theories
Brans–Dicke Scalar–tensor–vector Conformal gravity Scalar theories
Nordström Whitehead Geometrodynamics Induced gravity Chameleon Pressuron Degenerate Higher-Order Scalar-Tensor theories

Quantum-mechanical

Unified-field-theoric

Kaluza–Klein theory
Dilaton Supergravity

Unified-field-theoric and
quantum-mechanical

Noncommutative geometry Semiclassical gravity Superfluid vacuum theory
Logarithmic BEC vacuum String theory
M-theory F-theory Heterotic string theory Type I string theory Type 0 string theory Bosonic string theory Type II string theory Little string theory Twistor theory
Twistor string theory

Generalisations /
extensions of GR

Liouville gravity Lovelock theory (2+1)-dimensional topological gravity Gauss–Bonnet gravity Jackiw–Teitelboim gravity

Pre-Newtonian
theories and
toy models

Aristotelian physics CGHS model RST model Mechanical explanations
Fatio–Le Sage Entropic gravity Gravitational interaction of antimatter Physics in the medieval Islamic world Theory of impetus

Related topics

Graviton

Physics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License