Sign crazyness on the stress energy tensor? 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:


*

*Wikipedia Stress-energy tensor: $T^{\mu\nu}=(\rho c^2+P)u^{\mu}u^{\nu} + P g^{\mu\nu}$

*Tenseur énergie-impulsion: $T^{\mu\nu}=-(\rho c^2+P)u^{\mu}u^{\nu} + P g^{\mu\nu}$

*Energie-Impuls-Tensor: $T^{\mu\nu}=(\rho c^2+P)u^{\mu}u^{\nu} - P g^{\mu\nu}$

*Fluide parfait: $T^{\mu\nu}=(\rho c^2+P)u^{\mu}u^{\nu} - P g^{\mu\nu}$

*Dérivation des équations de Friedmann: $T_{\mu\nu}=(\rho c^2+P)u_{\mu}u_{\nu} - P g_{\mu\nu}$
So why so many different signs, and what are the right ones in my case ?
 A: 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.
