Lovelock's theorem of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
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
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
.
The theorem was described by British physicist
David Lovelock in 1971.
Statement
In four dimensional spacetime, any tensor
whose components are functions of the metric tensor
and its first and second derivatives (but linear in the second derivatives of
), and also symmetric and divergence-free,
is necessarily of the form
:
where
and
are constant numbers and
is the
Einstein tensor
In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field e ...
.
The only possible second-order Euler–Lagrange expression obtainable in a four-dimensional space from a scalar density of the form
is
[
]
Consequences
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 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
References
General relativity
Theorems in general relativity
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