0
$\begingroup$

The stress-energy tensor has 16 components, but this question is only about the 9 components $T^{ij}$ with $i,j=1,2,3$. According to Wikipedia, these components are defined as follows:

The components $T^{kl}$ represent flux of kth component of linear momentum across the $x^l$ surface.

I find it hard to see how this uniquely defines $T^{kl}$. For example, we could add a constant value to any of the $T^{kl}$ and this would give the same time evolution of linear momentum after we apply the continuity equation.

To be more precise, it seems that one can choose 6 of the 9 components $T^{kl}$ arbitrarily; for example, all components with $k\neq l$. To make sure that the continuity equation works, we need that $0=\partial_{\nu}T^{\mu\nu}$ for all $\mu$. This can be done with smart choices of $T^{\mu\mu}$ for $\mu=1,2,3$.

So how can the $T^{kl}$ be uniquely defined?

Improve this question
$\endgroup$
4
  • $\begingroup$ I found out that there might be a simple formula in some cases: the flow of $x_i$ momentum in the $y_j$ direction is $\frac{1}{m} \rho p_i p_j$, where $m$ is the particle mass, $\rho$ is the particle density, and $\vec{p}$ is the average momentum of the particles at that specific location. $\endgroup$
    – Riemann
    Commented Mar 19 at 13:17
  • $\begingroup$ However, I have no source for this $\endgroup$
    – Riemann
    Commented Mar 19 at 13:18
  • 2
    $\begingroup$ Why do you assume that statement is meant to define $T$ uniquely? It's a statement of constraints $T$ satisfies. We need more concrete definition of the physical system first and then we can devise unique definition of $T$. For example, for system of point particles, we have the definition $T_{ik}(\mathbf x)=\sum_a m_a\delta(\mathbf x - \mathbf r_a) v_{a,i} v_{a,k}$. $\endgroup$
    – Ján Lalinský
    Commented Mar 19 at 15:40
  • $\begingroup$ Thank you, that helps a lot $\endgroup$
    – Riemann
    Commented Mar 19 at 18:49

0

Reset to default