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whatisreality
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Homework Statement
Using conservation of energy, show that the 4-velocity ##u^{\mu}## of dust satisfies:
##u^{\mu}\nabla_{\mu}u^{\nu} = f u^{\nu}##
Explicitly identify ##f##, which is some function of ##u^{\mu}##, ##\rho## and their derivatives. And show that this equation becomes
##\dot{\rho} + 3 \left(\frac{\dot{a}}{a}\right)\rho = 0##
For comoving observers in a homogeneous Friedmann-Robertson-Walker Universe.
Homework Equations
The Attempt at a Solution
I don't think I've done the first bit right, I'm not really sure how to treat the ##\rho##. For dust, ##T^{\mu\nu} = \rho u^{\mu}u^{\nu}## and energy conservation means:
##\nabla_{\mu}T^{\mu\nu} = 0##
So
##\nabla_{\mu} (\rho u^{\mu}u^{\nu})##
Using the product rule,
##\rho u^{\mu} (\nabla_{\mu}u^{\nu}) + \rho u^{\nu} (\nabla_{\mu}u^{\mu}) + u^{\mu}u^{\nu}\nabla_{\mu}\rho = 0##
I think ##\nabla_{\mu} \rho =0##, but if it is then I could just divide through by ##\rho## and I could rearrange to get the LHS
I want:
##u^{\mu}\nabla_{\mu}u^{\nu}##
But the RHS wouldn't be a function of ##\rho##. So if ##\nabla_{\mu} \rho## isn't zero, then I get:
##u^{\mu}\nabla_{\mu}u^{\nu} = -u^{\nu}\nabla_{\mu}u^{\mu} - \frac{u^{\nu}u^{\mu}}{\rho} \nabla_{\mu}\rho##
##= \left(-\nabla_{\mu}u^{\mu}-\frac{u^{\mu}}{\rho}\nabla_{\mu}\rho \right)u^{\nu}##
Is that right? Is ##\nabla_{\mu}\rho## not zero?