Sign conventions in general relativity

There are a total of three sign conventions in general relativity . The three signs are in principle freely selectable and are chosen differently by different authors. Usually, authors of books or articles on general relativity indicate in the introduction which signs they use.

Sign of the metric

The sign of the Minkowski metric can be chosen in different ways. One possible convention is

${\displaystyle \eta _{\mu \nu }={\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}}\equiv \operatorname {diag} (1,-1,-1,-1)}$.

The other convention is

${\displaystyle \eta _{\mu \nu }={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\equiv \operatorname {diag} (-1,1,1,1)}$.

In the general theory of relativity, only the signs of the eigenvalues ​​of the metric can be determined. The first possibility mentioned above corresponds to one positive and three negative eigenvalues. One advantage is that time-like vectors, i.e. physically meaningful motion vectors, have a positive length. The second possibility mentioned above corresponds to three positive and one negative eigenvalue. This has the advantage that you have more positive eigenvalues, which often simplifies the calculation by reducing the number of minus signs. It is also the natural generalization of the metric in Euclidean space, since time is added as a special coordinate. As a result, the sign of the determinant of the metric does not change if the theory is expanded to more dimensions.

Sign of the curvature tensor

The Riemann curvature tensor can be defined as

${\displaystyle R(u,v)w=\pm \left([\nabla _{u},\nabla _{v}]w-\nabla _{[u,v]}w\right)}$.

In terms of coordinates, these two possibilities correspond

${\displaystyle {R^{\rho }}_{\lambda \mu \nu }=\pm \left(\partial _{\mu }\Gamma _{\nu \lambda }^{\rho }+\Gamma _{\mu \tau }^{\rho }\Gamma _{\lambda \nu }^{\tau }-\partial _{\nu }\Gamma _{\mu \lambda }^{\rho }-\Gamma _{\nu \tau }^{\rho }\Gamma _{\lambda \mu }^{\tau }\right)}$.

Einstein's sum convention was used in the formula .

Sign in the field equations

In Einstein's field equations ,

${\displaystyle R_{\mu \nu }-{\frac {R}{2}}g_{\mu \nu }=\kappa T_{\mu \nu }=\pm {\frac {8\pi G}{c^{4}}}T_{\mu \nu }}$

write.

Sign of the Ricci tensor

While the three preceding signs are all independent of one another, the sign of the Ricci tensor can be understood as the product of the signs of field equations and curvature tensor. The Ricci tensor can be defined as

${\displaystyle R_{\mu \nu }=\pm {R^{\lambda }}_{\mu \lambda \nu }}$

The upper sign is obtained if the upper or lower sign has been chosen as the sign of field equations and curvature tensor. The lower sign is obtained when one has chosen the upper one for field equations and the curvature tensor and the lower one for the other. So one can select two of the signs of field equations, Ricci tensor and curvature tensor, which are determined by convention. The third is then automatically locked.

literature

• Charles Misner ; Kip S. Thorne , John. A. Wheeler : Gravitation . WH Freeman, San Francisco 1973, ISBN 0-7167-0344-0