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?


1 Answer 1


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.