- #1
ProfDawgstein
- 80
- 1
I have a short question about the derivative of a given EM-Tensor.
##\rho## = mass density
##U^\mu## = 4 velocity
##T^{\mu\nu} = \rho U^\mu U^\nu##
Now I do ##\partial_\mu##
Should I get
a) ##\partial_\mu T^{\mu\nu} = (\partial_\mu \rho) (U^\mu U^\nu) + \rho (\partial_\mu U^\mu) U^\nu + \rho U^\mu (\partial_\mu U^\nu)##
b) ##\partial_\mu T^{\mu\nu} = (\partial_\mu \rho) (U^\mu U^\nu) + \rho (\partial_\mu U^\mu) U^\nu##
This should be the product rule, right?
So I should get 3 terms?
##(\partial_\mu \rho) (U^\mu U^\nu) = 0##
because
##(\rho \partial_\mu U^\mu) = 0## (mass conservation)
Solution is
##\partial_\mu T^{\mu\nu} = \rho (U^\mu \partial_\mu) U^\nu##
where ##\partial_\mu (\rho U^\mu) = 0##
What am I missing?
##\rho## = mass density
##U^\mu## = 4 velocity
##T^{\mu\nu} = \rho U^\mu U^\nu##
Now I do ##\partial_\mu##
Should I get
a) ##\partial_\mu T^{\mu\nu} = (\partial_\mu \rho) (U^\mu U^\nu) + \rho (\partial_\mu U^\mu) U^\nu + \rho U^\mu (\partial_\mu U^\nu)##
b) ##\partial_\mu T^{\mu\nu} = (\partial_\mu \rho) (U^\mu U^\nu) + \rho (\partial_\mu U^\mu) U^\nu##
This should be the product rule, right?
So I should get 3 terms?
##(\partial_\mu \rho) (U^\mu U^\nu) = 0##
because
##(\rho \partial_\mu U^\mu) = 0## (mass conservation)
Solution is
##\partial_\mu T^{\mu\nu} = \rho (U^\mu \partial_\mu) U^\nu##
where ##\partial_\mu (\rho U^\mu) = 0##
What am I missing?