Theorems Regarding Gravity
In general relavity, many theorems provides deep understanding of gravity.
Derivation of Field Equation
From postulations
 General covirance
 Linear approximation should be compatible with Newton’s thoery/Weak field and slow motion limit is Newton’s thoery of gravity
 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., 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 secondorder EulerLagrange expression obtainable in a four dimensional space from a scalar density of the form $\scr L = \scr L(g_{\mu\nu})$ is
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, nonsymmetry of field equations under exchange of indices, or divergence field equations
 nonlocality
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.
Nohair Theorems
The generic final state of gravitational collapse is a KerrNewman 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.
Footnote

R. M. Wald. Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant. Phys. Rev. D, 28(8):2118–2120, Oct 1983.↩