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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?

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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.

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  • $\begingroup$ Thank you. Conservation identities all make sense now. But may I ask which time are you meaning when you are saying per unit time.Proper time? $\endgroup$
    – Timur9717
    Jan 16, 2020 at 18:15
  • $\begingroup$ I think so, such that you get the proper force for $dp^{\mu}/d\tau$ $\endgroup$
    – Leo L.
    Jan 16, 2020 at 18:20

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