Does Birkhoff’s theorem really hold?

Abstract

Birkhoff’s theorem states that the external gravitational field of any spherically-symmetric system is static and is described by Schwarzschild metric. In this paper, we demonstrate that it is inconsistent with the direct post-Newtonian solution to Einstein field equations in the weak field limit. Moreover, we identify the flaw in the derivation of Birkhoff’s theorem—the consequences of making the coordinate transformations for solving Einstein field equations are ignored, which is due to the misuse of the covariance in general relativity.

Keywords:

• Birkhoff’s theorem
• general covariance
• post-Newtonian approximation

AMS Subject Classifications:

• 04.20.-q
• 04.20.Cv
• 04.25.Nx
• 95.30.Sf

Public Interest Statement

Before solving Einstein field equations, the energy and momentum distributions of gravitational source have to be specified. This implies that there exists a reference frame and the solution corresponds to this reference frame by default. It is usually believed that based on the general covariance one can make any coordinate transformation in solving Einstein field equations. However, the covariance in general relativity only means that the fundamental equations, e.g. Einstein field equations, take the same form for any space-time coordinates. It does not mean that the solution of Einstein field equations will be kept unchanged after making coordinate transformations, since in relativity the coordinate transformations include changes of reference frame. Therefore, the consequences of making any coordinate transformation should be taken into account when the resultant solution is related to the original problem, otherwise wrong conclusions may be reached. Here we take Birkhoff’s theorem as an example to illustrate this important issue.

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China [grant number 11647314] and the Program for New Century Excellent Talents in University [grant number NCET-10-0702].

Notes on contributors

Wenbin Lin

Wenbin Lin graduated from Xiamen University in 1992, and received two PhD degrees, one in Astrophysics from University of Science & Technology of China in 1999 and the other in Electrical & Computer Engineering from Duke University in 2005. His research interests include general relativity and gravitation, cosmology, electromagnetic field and microwave technology, and scientific computing.

The research reported in this paper demonstrates that Birkhoff’s theorem is not correct, which is resulted from the misuse of the covariance of general relativity. It is pointed out that the coordinate transformations taken in solving Einstein field equations may make the final solution not directly correspond to the original problem. This conclusion is very important, since the coordinate transformations have often been made for solving Einstein field equations in many literatures.