Relativistic density In the non-relativistic region of the space, with matter of mass $M$, and volume $V$ the density is given by
\begin{equation}
\rho=\frac{M}{V}
\end{equation}
If we encounter an equation, for example,
\begin{equation}
F=F(\rho,...)
\end{equation}
we simply substitute the first equation. However, if we want to consider relativity, then what is the density?
 A: Matter density is the $00$ component of the stress-energy tensor $T^{\mu \nu}$. In order to find out what this component is in a different reference frame, simply perform a Lorentz transformation on the two indices
$$T^{\mu \nu} \rightarrow T'^{\mu \nu}= \Lambda^{\mu}_{\,\,\alpha} \Lambda^{\nu}_{\,\,\beta} T^{\alpha \beta},$$
and pick out the $00$ component in the new (primed) frame.
A: I was going to ask this same question, but let me propose what I suspect the answer to be :
Relativistic M = $mc^2 \gamma$ where $\gamma$ is the familiar $1 \over {\sqrt{1 - \beta}}$
or, fully expanded ${1} \over {\sqrt{ 1 - {{v^2} \over {c^2}}}}$
Relativistic x (of which V = $x^3$) = $1 \over \gamma$
Putting that together, then, I expect relativistic density (M/V) for a non-moving body to be ${mc^2 \gamma^4}\over{V }$
I was planning to post a question here to confirm that perspective. Is there anyone here who can confirm?
A: The proper way to do this is to start with the stress energy tensor in the resting frame and transform it to a moving frame. Then the $T^{tt}$ component of the stress energy tensor in the moving frame is the density (in natural units). The details depend on the specific stress energy tensor used:
For a fluid at rest with density $\rho$ and pressure $p$ we have: $$T^{\mu\nu}=\left(
\begin{array}{cccc}
 \rho  & 0 & 0 & 0 \\
 0 & p & 0 & 0 \\
 0 & 0 & p & 0 \\
 0 & 0 & 0 & p \\
\end{array}
\right) $$ so $$T^{\mu'\nu'}={\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu T^{\mu\nu} = \left(
\begin{array}{cccc}
 \gamma ^2 \left(p v^2+\rho \right) & v \gamma ^2 (p+\rho ) & 0 & 0 \\
 v \gamma ^2 (p+\rho ) & \gamma ^2 \left(\rho  v^2+p\right) & 0 & 0 \\
 0 & 0 & p & 0 \\
 0 & 0 & 0 & p \\
\end{array}
\right) $$
Other stress energy tensors would need to be worked out similarly. Note that the density in the moving frame is not merely a function of the density in the rest frame, but also includes terms due to the other components of the stress energy tensor in the rest frame.
