Physical meaning of the space-space components $T^{ij}$ of the stress-energy tensor $T^{\mu\nu}$

Question

The time-time component $T^{00}$ and the time-space components $T^{0i}$ of the energy-momentum tensor $T^{\mu\nu}$ are respectively called the Hamiltonian (energy) density and the momentum density. Integrated over all space, $\int \mathrm d^3x\,T^{00}\equiv E$ and $\int \mathrm d^3x\,T^{0i}\equiv P^i$ respectively represent the energy and components of the physical momentum carried by a field. In field theories, the quantities of interest are mainly $T^{00}$ and $T^{0i}$ components of the $T^{\mu\nu}$ tensor. This is because they naturally arise as the conserved charges associated with time translation and space translation symmetries respectively. Therefore their meaning is kind of obvious.

Is there a similar interpretation of the space-space components $T^{ij}$? In cosmology, the diagonal space-space components, $T^{ii}$, are called pressures $p$. Since $T^{\mu\nu}$ is a tensor under Lorentz transformation, $T^{ij}$ must transform like a tensor under rotation $T^{ij}=O^i_{~m}O^j_{~n}T^{mn}.$ That much is clear. The next target is to understand that the stress tensor $\sigma^{ij}$ introduced for a fluid or for elastic deformation is somehow related to $T^{ij}$ of a field.

This terminology is used even even during inflation which is described as a classical theory which suggest that this 'pressure' terminology make sense for fields (e.g., the inflaton field)?

Answer 1

All this can be understood by the means of the conservation law

$$ \partial_{\mu}T^{\mu\nu}=0 $$

for every $\nu$. Splitting the $\mu$ and $\nu$ indices into space-like and time-like directions result to the following equations:

$$ \partial_{0}T^{00}+\partial_{i}T^{i0}=0,\qquad \partial_{0}T^{0i}+\partial_{j}T^{ji} = 0 $$

Since $T^{00}$ is the energy density, it follows from the equation above that $T^{i0}$ is the flow of the energy density. In order to see that more clearly just integrate the equation in a space-like volume at fixed $x^{0}$:

$$ \int_{\Sigma}\left(\partial_{0}T^{00}+\partial_{i}T^{i0}\right)=\partial_{0}\int_{\Sigma}T^{00}+\int_{\partial\Sigma}T^{i0}n_{i}(\partial\Sigma)=\partial_{0}E(\Sigma)+\int_{\partial\Sigma}T^{i0}n_{i}(\partial\Sigma)=0 $$

which is saying that the variation of the energy inside the space-like volume $\Sigma$ is equal to the flux of the vector field $T^{i0}$ in the boundary.

Now, looking at the second equation we have something very similar. The equation in integral form is given by:

$$ \partial_{0}P^{i}(\Sigma)+\int_{\partial\Sigma}T^{ji}n_{j}(\partial\Sigma)=0 $$

where $T^{0i}$ is the density for $P^{i}$. The variation of the quantity $P^{i}(\Sigma)$ is equal to the flux of the vector field $T^{ij}$ (i is fixed here) in the boundary of $\Sigma$.

It is important to realize that $E(\Sigma)$ and $P^{i}(\Sigma)$ rotate into each other under boost transformation, which implies that $P^{i}(\Sigma)$ is the momentum of the region $\Sigma$ and $T^{0i}$ is the momentum density. By the definition of pressure (force per area) we conclude that $T^{ji}$ are pressures. In an area element with normal vector $x_{j}$, $T^{ij}$ is the the pressure at the $x_{i}$ direction. This can be translated to the following:

An area element $dA$ with normal vector $n_{j}$ will feel a force $f^{i}$ equal to $f^{i}=n_{j}T^{ji}dA$