Lovelock's theorem explained

Lovelock's theorem should not be confused with Lovelock theory of gravity.

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]} The theorem was described by British physicist David Lovelock in 1971.

Statement

In four dimensional spacetime, any tensor

A ^{\mu\nu} whose components are functions of the metric tensor g ^{\mu\nu} and its first and second derivatives (but linear in the second derivatives of g ^{\mu\nu} ), and also symmetric and divergence-free, is necessarily of the formA ^{\mu\nu} = a G ^{\mu\nu} + b g ^{\mu\nu} where a and b are constant numbers and 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

l{L}= l{L}(g _{\mu\nu} ) is^[1] E ^{\mu\nu} = \alpha \sqrt{ - g} \left[R ^{\mu\nu} - (1)/(2) g ^{\mu\nu} R \right] + λ \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

• Lovelock theory of gravity
• Vermeil's theorem

Notes and References

1. Timothy . Clifton . Pedro G. . Ferreira . Antonio . Padilla . Constantinos . Skordis . 1 . Modified Gravity and Cosmology . Physics Reports . 513 . 1–3 . 1–189 . March 2012 . 1106.2476 . 10.1016/j.physrep.2012.01.001 . 2012PhR...513....1C . 119258154 .
2. D. . Lovelock . The Einstein Tensor and Its Generalizations . Journal of Mathematical Physics . 12 . 3 . 498–501 . 1971 . 10.1063/1.1665613 . 1971JMP....12..498L . free .
3. Lovelock . David . 10 January 1972 . The Four-Dimensionality of Space and the Einstein Tensor . Journal of Mathematical Physics . 13 . 6 . 874–876 . 10.1063/1.1666069. 1972JMP....13..874L.