Finding the correct units for the energy-momentum tensor? I'm trying to understand the energy-momentum tensor $T^{\mu\nu}$ but I'm confused about the units. My textbook says the components of $T^{\mu\nu}$ are $\mathrm{Jm^{-3}}$. Four-momentum is is given by$$P^{\mu}=\left(E/c,\mathbf{p}\right)=\left(E/c,p_{x},p_{y},p_{z}\right)$$
The $E/c$ component of $P^{\mu}$ has units $\mathrm{Jsm^{-1}}$. The definition of $T^{\mu\nu}$ is “the rate of flow of the $\mu$ component of four-momentum across a surface of constant $\nu$.” Using this definition, how do you get the rate of flow of the $E/c$ component of four-momentum across a surface of constant time (ie the $T^{00}$ component) to have the correct units of $\mathrm{Jm^{-3}}$? Surely you need to multiply $E/c$ by something with $\mathrm{s^{-1}m^{-2}}$ units, but what exactly?
Thank you
 A: You say

The definition of $T^{\mu\nu}$ is “the rate of flow of the $\mu$
  component of four-momentum across a surface [of unit area] of constant $\nu$.”

which means that the dimensions of $T^{\mu\nu}$ ought to be
$$\frac{Momentum}{Area \, \times \, Time.}$$
A: Notation: I will denote SI units between brackets, not to be confused with a multiplication.
Lets start by a well known case, the stress tensor of a perfect fluid:
$$ T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta} $$
from it, we could say that stress energy tensor units are from momentum density ($\rho u^\alpha$) multiplied by velocity $u^\beta$ (momentum flow), or pressure $p$ multiplied by metric $g^{\alpha\beta}$.

*

*When spacetime basis is $(ct,x,y,z)$, all stress energy tensor components (in contravariant and covariant form) have the same units
$$ T^{\alpha\beta} \left[ \left( \frac{kg\,m}{s} \frac{1}{m^3} \right) \frac{m}{s} = \frac{kg}{ms^2} \right] $$ equal to pressure units or energy density [ $ \frac{J}{m^3}=Pa=\frac{kg}{ms^2}$ ]. It corresponds to what is said in the book that is mentioned at the question.


*When spacetime basis is $(t,x,y,z)$ we have (assume $i,j \in \{1,2,3\}$):
$T^{00} \left[kg/m^3\right]$ : mass spatial density.
$T^{0j}, T^{i0} \left[ \frac{kg}{m^3} \frac{m}{s} = \frac{kg}{m^2s} \right]$ : momentum per unit of space volume.
$T^{ij} \left[ \frac{kg}{m^3} \frac{m}{s} \frac{m}{s} = \frac{kg}{ms^2} \right]$ : energy per unit of space volume.
These units corresponds to wikipedia paragraphs about stress energy tensor.
A: According to http://en.wikipedia.org/wiki/Stress-energy_tensor#Identifying_the_components_of_the_tensor $T_{00}$ is the density of relativistic mass not energy, so you have to divide by $c^2$.
For the momentum components:
$$\frac{dp^\alpha}{dV} = -T^\alpha _\beta u^\beta$$
giving you units of density.
