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dpdt

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I am trying to follow the calculation by samalkhaiat in this thread: https://www.physicsforums.com/threa...n-from-the-stress-energy-tensor.547502/page-2 (post number 36). I am having some difficulty getting the equation above equation (11) (it was an unnumbered equation) from the preceding equations:

In particular, the equation states that:

## \frac{dX^c}{dx^0} \int d^3 x \sqrt{-g} T^{a0} + X^c \int d^3x \partial_b (\sqrt{-g} T^{ab} + X^c \Gamma^a_{bd}\int d^3x \sqrt{-g} T^{bd} ) = \int d^3x \sqrt{-g} T^{ac} ## (*)

I am confused to where the ## \frac{d X^c}{dx^0} ## comes from. I managed to massage equations so that I obtain equation (*), except that instead of

## \frac{dX^c}{dx^0} \int d^3 x \sqrt{-g} T^{a0} ## (**)

I have instead

## \int d^3 x \partial_0 (\sqrt{-g} T^{a0} \delta x^c) ## (***)

Help! Can someone help me see why the two equations above (equations (**) and (***) ) are equal to each other?

Thank you so much for any help, this calculation is frying my brain! I can present my calculation up to this point if it is helpful at all.

Any help will be much appreciated!