Lovelock's theorem
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. The theorem was described by British physicist David Lovelock in 1971.Statement
In four dimensional space, any tensor whose components are function of metric tensor and its first and second derivatives, and also symmetric and divergenceless, then field equation in vacuum, then only possible form of is
where and are just simple constant numbers and is the Einstein tensor.
The only possible second-order Euler–Lagrange expression obtainable in a four-dimensional space from a scalar density of the form isConsequences
Lovelock's theorem means that if we want to modify the Einstein field equations, then we have five options.
- Add other fields rather than the metric tensor
- Use more or less 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.
