# What is the exact meaning of flux in the context of this stress-energy tensor definition?

I have trouble understanding flux in definition of stress-energy tensor.

$$T^{uv}_{}$$ is the flux of four-momentum $$p^{u}$$ across a surface of constant $$x^{v}_{}$$ .

Do we take a surface integral that all students are familiar form Calc 2(multiply vector $$p^{u}$$ by normal of the surface and integrate) or is something else meant by word flux here? I am trying to apply concepts I learned in Calc 2. It seems to me like normal of surface of constant $$x^{0}_{}$$ and $$p^{1}$$ should be orthogonal. It should be orthogonal for all similar cases where indices are different. Shouldn't this fact make all off-diagonal elements of stress energy tensor 0? I know that I miss something very fundamental here. Can you please help me with this?

No, the interpretation of flux for stress-energy tensor is different. Firstly, flux here is a vector quantity, not a scalar quantity as the flux in Calc 2. Flux here means amount of four-momentum $$p^\mu$$ passing through a surface per unit area and per unit time.
To give an example, the flux of $$p^2$$, i.e. momentum in y-direction, across a surface of constant $$x^1$$, i.e. the yz plane, is the amount of momentum in y-direction that passes through the yz plane in unit time and unit area.
• I think so, such that you get the proper force for $dp^{\mu}/d\tau$ Jan 16, 2020 at 18:20