Lovelock's theorem

Lovelock's theorem Not to be confused with Lovelock theory of gravity. General relativity {displaystyle G_{mu nu }+Lambda g_{mu nu }={kappa }T_{mu nu }} IntroductionHistory Mathematical formulation Tests show Fundamental concepts show Phenomena show EquationsFormalisms show Solutions show Scientists  Physics portal Category vte Lovelock's theorem of general relativity says that from a local gravitational action which contains only second derivatives of the four-dimensional spacetime metric, then the only possible equations of motion are the Einstein field equations.[1][2][3] The theorem was described by British physicist David Lovelock in 1971.

Contents 1 Statement 2 Consequences 3 See also 4 References Statement In four dimensional space, any tensor {displaystyle A^{mu nu }} whose components are function of metric tensor {displaystyle g^{mu nu }} and its first and second derivatives (but linear in the second derivatives of {displaystyle g^{mu nu }} ), and also symmetric and divergenceless, then field equation in vacuum {displaystyle A^{mu nu }=0} , then only possible form of {displaystyle A^{mu nu }} is {displaystyle A^{mu nu }=aG^{mu nu }+bg^{mu nu }} where {displaystyle a} and {displaystyle b} are just simple constant numbers and {displaystyle G^{mu nu }} is the Einstein tensor.[3] The only possible second-order Euler–Lagrange expression obtainable in a four-dimensional space from a scalar density of the form {displaystyle {mathcal {L}}={mathcal {L}}(g_{mu nu })} is[1] {displaystyle E^{mu nu }=alpha {sqrt {-g}}left[R^{mu nu }-{frac {1}{2}}g^{mu nu }Rright]+lambda {sqrt {-g}}g^{mu nu }} Consequences Lovelock's theorem means that if we want to modify the Einstein field equations, then we have five options.[1] Add other fields rather than the metric tensor; Use more or fewer than four spacetime dimensions; Add more than second order derivatives of the metric; Non-locality, e.g. for example the inverse d'Alembertian; Emergence – the idea that the field equations don't come from the action. See also Physics portal Lovelock theory of gravity Vermeil's theorem References ^ Jump up to: a b c Clifton, Timothy; et al. (March 2012). "Modified Gravity and Cosmology". Physics Reports. 513 (1–3): 1–189. arXiv:1106.2476. Bibcode:2012PhR...513....1C. doi:10.1016/j.physrep.2012.01.001. S2CID 119258154. ^ Lovelock, D. (1971). "The Einstein Tensor and Its Generalizations". Journal of Mathematical Physics. 12 (3): 498–501. Bibcode:1971JMP....12..498L. doi:10.1063/1.1665613. ^ Jump up to: a b Lovelock, David (10 January 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.

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