In general relavity, many theorems provides deep understanding of gravity.

Derivation of Field Equation

From postulations

1. General covirance
2. Linear approximation should be compatible with Newton’s thoery/Weak field and slow motion limit is Newton’s thoery of gravity
3. In theory regarding the metric, no higher than second derivative is envolved and the terms of second derivative is linear.

The first point is for the invariance of frames/coordinates. The second point is for the success of Newtonian’s theory on our earth.

Why do we believe the third point? The answer is that we don’t have to. Here we propose it is because the simplicity of such quasilinear equations, i.e., $F(\phi, \partial \phi) \partial^2\phi + G(\phi, \partial\phi) = 0$ We have a bunch of theorems on this system, including its existance of solutions, Couchy problem, wave propagation etc.

We can use both 1&2 and 1&3 to derive Einstein’s equation. That is 2 and 3 are identical when 1 is considered.

From Action

This is an application of stationary principal and Hilbert action or Hilbert action plus a $\Lambda$.

Lovelock’s Theorem

The only possible second-order Euler-Lagrange expression obtainable in a four dimensional space from a scalar density of the form $\scr L = \scr L(g_{\mu\nu})$ is

$$E^{\mu\nu} = \alpha \sqrt{-g} [ R^{\mu\nu} - \frac{1}{2} g^{\mu\nu}R ] +\lambda \sqrt{-g} g^{\mu\nu}$$

Thus modification could be

• Metric tensor not a fundamental tensor
• Higher than second order derivatives of the metric in the field equations
• Not a four dimension space
• Not rank (2,0) tensor field equations, non-symmetry of field equations under exchange of indices, or divergence field equations
• non-locality

Birkhoff’s Theorem

All spherically symmetric solutions of Einstein’s equations in vacuum must be static and asymptotically flat, without $\Lambda$

Actually, this can be extended to a $\Lambda$ space only keeping the static result.

No-hair Theorems

The generic final state of gravitational collapse is a Kerr-Newman black hole, fully specified by its mass, angular momentum and charge

Also, “in the context of General Relativity with a cosmological constant all expanding universe solutions should evolve towards de Sitter space.”[¹] This is only valid in some situation.

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