User:JRSpriggs/Conventions for general relativity

I have noticed that there is considerable disagreement about which conventions should be used in special and general relativity for such things as: the signature of the metric, whether certain tensors should be defined as covariant or contravariant; whether they should be tensor densities or ordinary tensors; where the c for light-speed should appear; etc.. Consequently, I feel that there is a need for me to set forth my personal preferences in these matters which I do in this file.

• I feel that it is important to get all the constant factors correct and especially the signs in physical equations. And for practical purposes, we need to be able to convert formulas from the generally invariant form to the form used in non-relativistic classical physics and vice versa. So I adopt the rule that the purely spatial components of a tensor should have the same units (and sign) as the corresponding quantity in non-relativistic physics. In particular, this implies that the spatial part of the metric (in a locally inertial frame of reference) should be the dimensionless numbers

${\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}\,.$

• For the same reason, I want to use the SI units which are the accepted standard in non-relativistic physics. So while the spatial coordinates (in a Cartesian coordinate system) are given in meters, the time coordinate must be given in seconds. Thus the entire metric tensor in a locally inertial frame of reference is

$\eta _{\alpha \beta }={\begin{pmatrix}-c^{2}&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}=\mathrm {diag} [-c^{2},+1,+1,+1]\,.$

Some people object to this on the grounds that they believe all the components of a tensor should have the same units. However, there is no basis in the definition of tensors for such a requirement.

Consequently, components of a tensor involving time would then have the units of the purely spatial components multiplied by (second/meter)^(#contravariant_time_indices − #covariant_time_indices).

If (instead of using a Cartesian coordinate system) you want to use cylindrical or spherical coordinates (e.g. for a Schwarzschild metric), then multiply the above units by (radian/meter)^(#contravariant_angular_indices − #covariant_angular_indices).

In any real world situation, the coordinates themselves will not be arbitrary but must be defined by some physical process. Thus one should expect that the coordinates will have non-trivial units (e.g. meters, seconds, or radians) arising from the process by which they are determined.

Please see my comments at Wikipedia talk:WikiProject Physics/Taskforces/Relativity/Archive 1#Agreeing on a consistent metric for SR. My source is "Post, E.J., Formal Structure of Electromagnetics: General Covariance and Electromagnetics, Dover Publications Inc. Mineola NY, 1962 reprinted 1997.".