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I know that the energy-momentum tensor for a perfect fluid in General Relativity is given by $$T_{\alpha \beta} = (\rho +p)u_{\alpha}u_{\beta}-p\,g_{\alpha \beta}, $$ where $\rho$ is the density, $p$ is the pressure and $u_{\alpha}$ is the four-velocity of the fluid. I also know that if my space is filled by a scalar field, I can relate the above expression to the one of the energy-momentum tensor for the wave equation: $$T_{\alpha \beta} = \partial_{\alpha }\varphi \partial_{\beta} \varphi - \frac{1}{2} g_{\alpha \beta} \partial^{\gamma} \varphi \partial_{\gamma} \varphi$$ if I choose $u^{\alpha} = \frac{\partial^{\alpha} \varphi}{\sqrt{\partial^{\gamma} \varphi \partial_{\gamma} \varphi}}$ and $\rho=p=\frac{1}{2} \partial^{\gamma} \varphi \partial_{\gamma} \varphi$. Here, I am assuming that $\nabla_{\alpha} \partial^{\alpha} \varphi=0$ and $\partial_{\alpha} = \frac{\partial}{\partial x^{\alpha}}$.

My questions are:

  1. This relation and formulas hold for $(+, -, -, -)$ signature. How about $(-, +, +, +)$? Of course it should work, but now I need that $u_{\alpha} u^{\alpha}=-1$ and I am not sure how I can make it work (without using complex numbers).

  2. I would expect $\rho = T_{00}$, since the $00-$component of the energy-momentum tensor is referred to as "the energy density". However, this does not seem the case from the previous formulas. Is there something wrong with the above expressions, or is my analogy wrong?

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  • $\begingroup$ First subquestion is a duplicate of physics.stackexchange.com/q/228185/2451 Consider to only ask 1 question per post. $\endgroup$
    – Qmechanic ♦
    Commented May 15, 2021 at 9:36
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    $\begingroup$ The second question can be answered by a comment - $\rho$ is the energy density w.r.t. to the time coordinate of the observer moving along $u^\alpha$. Usually, one chooses $u^\alpha=(1,0,0,0)$ so in this case the energy density is simply $T_{00}$. More generally the energy density is $u^\alpha u^\beta T_{\alpha\beta}$ which is always $\rho$. $\endgroup$
    – Prahar
    Commented May 15, 2021 at 9:39
  • $\begingroup$ @Qmechanic as regards the first question: even if I change sign to the energy-momentum tensor, how can I write $u^{\alpha}$ such that $u^{\alpha}u_{\alpha}=-1$? I could introduce a minus in the square root, but how can I know if what is inside the square root is positive or negative? $\endgroup$
    – moonknight
    Commented May 17, 2021 at 9:45
  • $\begingroup$ Does this help? physics.stackexchange.com/q/619657/226902 $\endgroup$
    – Quillo
    Commented Feb 14 at 18:52

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