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I haven't been able to find a simple self-contained definition of the stress-energy tensor as used in the Einstein field equations.

Suppose I have $N$ classical (not quantum) point particles with position $x_i$, velocity $v_i$ and mass $m_i$, with forces $F_i(x_1, v_1,...,x_N,v_N)$ in some appropriate reference frame.

How would a physicist then compute the components of $T$? Please state it in the simplest possible self-contained way (i.e. no analogies, no references to elsewhere-defined concepts like flux).

Qmechanic
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user56834
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2 Answers2

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The energy-momentum tensor of a set of point massive particles (in special relativity), not including energy and momentum in the field due to their interaction, is, by definition,

$$ T_{matter}^{\mu\nu}(\mathbf x,t) = \sum_a m_a \frac{1}{\gamma_a}\delta\big(\mathbf x - \mathbf r_a(t)\big) u_a{}^\mu u_a{}^\nu $$ where $\gamma_a$ is the Lorentz factor of $a$-th particle $$ \gamma_a = \frac{1}{\sqrt{1-\frac{v_a^2}{c^2}}} $$ and $u_a$ is four-velocity of $a$-th particle: $$ u_a{}^0 = \gamma_a c $$ $$ u_a{}^k = \gamma_a v^k $$ or, abusing the index notation, $$ u_a{}^\mu = (\gamma_a c,\gamma_a \mathbf v_a). $$

Ján Lalinský
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The expression for SET which Ján provides, describes a system of free N particles. And because they are free, they will satisfy the usual conservation rule for energy, momentum and angular momentum. As Ján mentions in the comment, if we now add an interaction force $F_i(\{x_j\}, \{v_j\})$, then the conservation equations need to be modified. For eg, the momentum equation will look like: $$\frac{d}{dt}\sum_i \rho v^a_i = f^a_i$$ where $f_i$ is the interaction force density on i-th particle. The LHS of the above equation can be thought of as $\partial_0T^{a0}_{free}$, where $T^{ab}_{free}$ is the SET of free particles. In general, if you have some interaction terms, then one can generalize the continuity equation for SET as $$\partial_bT^{ab}=F^a$$ As an example, look at divergence of SET for EM in presence of some 4-current. If one can write $F^a = \partial_bV^{ab}$, then the new SET defined as $T^{ab}-V^{ab}$ will satisfy the continuity equation

KP99
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