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I would like to know on what depends the sign of the stress energy tensor in the following formula :

$T_{\mu\nu}=\pm(\rho c^2+P)u_{\mu}u_{\nu} \pm P g_{\mu\nu}$

In my case the metric is equal to $g_{\mu\nu}=\pmatrix{-c^2 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1}$ and $\rho$ is the mass density.

The problem is that we have:

So why so many different signs, and what are the right ones in my case ?

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I guess it depends on the definition of the relativistic position vector. One definition is $a^\mu = (x,y,z,ct)$, the other is $a^\mu = (x,y,z,ict)$. This can cause inconsistency of all quantities that are derived from them. Hope this helps to track this down... – DaP May 3 '13 at 10:29
If I would guess, it should be the 1st one. Read carefully this section. They use the same metric as you do. [wiki]:… – Hydro Guy May 3 '13 at 13:24
@DaPhil: No, that's incorrect. (1) Position isn't a vector in GR. (2) Nobody uses imaginary components in GR. – Ben Crowell May 3 '13 at 13:27
up vote 5 down vote accepted

First off, please don't use units with $c\ne 1$ in GR. It makes everything horribly messy.

What we normally think of as a ruler or clock measurement is represented in GR by an upper index quantity like $\Delta x^\mu$. Therefore in a Cartesian coordinate system in the fluid's rest frame, we are guaranteed that $u^\mu=(1,0,0,0)$, not $(-1,0,0,0)$. This is independent of the choice of signature or other signs. For this reason, it's better to express everything in the upper-index form, not the lower-index form that that you gave.

Let $T^{\mu\nu}=s_1(\rho+P)u^{\mu}u^{\nu} +s_2 P g^{\mu\nu}$, where $s_1=\pm 1$ and $s_2=\pm 1$.

We want the time-time component of T in the fluid's rest frame to depend only on $\rho$, not $P$. For people who use a metric with signature $(-,+,+,+)$, this requires $s_1=s_2$. For people who use $(+,-,-,-)$, it requires $s_1=-s_2$.

In addition to choices of signature, the GR literature is blessed with several other arbitrary sign conventions that are not consistent from one author to another. MTW has a handy table of these on a page in the back of the book. For example, the Einstein field equations may be written $G=8\pi T$ or $G=-8\pi T$. The Einstein and Riemann tensors can also be defined with either sign. I think this explains the difference between #2 and #3-5 on your list.

The English Wikipedia articles on GR were almost all originally written by one guy, Chris Hillman, so they probably all follow a consistent sign convention. Clearly the French and German wikipedias don't follow the same sign conventions as the English one, and the French wikipedia doesn't seem to be internally consistent.

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