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So consider a collection of matter in spacetime. We can assign each individual particle a four-momentum, but this would be highly impractical. However, for certain collections of matter, like a cloud of gas for example, we can approximate the entire collection with a single four-momentum vector. But this four-momentum vector isn't enough to entirely describe the collection. So we will use a covariant stress-energy tensor. Given a coordinate system, let ##\{e_0, e_1, e_2, e_3\}## be an orthonormal basis for our space. We can define the components of our tensor, in this coordinate system, by its action on the basis vectors:

1) We define ##T(e_0, e_0) = T_{00}## to be the energy density (mass-energy per unit volume) as measured by an observer whose four-velocity is ##e_0##.

2) Given ##i \neq j##, we define ##-T(e_i, e_j)## to be the momentum density of the matter in the ##e_j## direction, as measured by the observer with a four-velocity of ##e_i##.

3) Given ##i,k \neq j##, we define ##T(e_i, e_k)## to be the ##i##'th component of the momentum in the ##e_k## direction, as measured by the observer with four-velocity ##e_j##.

The action of ##T## on arbitrary vectors is thus determined by the above.

Question 1: Why do we require the tensor to be symmetric?\

Question 2: In 2), there is a negative sign infront of the ##T(e_i, e_j)##. Why is this?

Just in case you're wondering, I used Wald's explanation of the stress-energy tensor on page 61-62, but replaced his arbitrary vectors with basis vectors as I want to really examine the components of the tensor.