Variation of Four-Velocity Vector w/ Respect to Metric Tensor

[Hubble_92]

Hi everyone! I'm having some difficulty showing that the variation of the four-velocity,

U^μ=dx^μ/dτ

with respect the metric tensor g^αβ is

δU^μ=1/2 U^μδg^αβU_αU_β

Does anyone have any suggestion?

Cheers,
Rafael.

PD: Thanks in advances for your answers; this is my first post! I think ill be active sharing and discussing in other Physics/ Astrophysics topics ;)

 

[dextercioby]

Hi everyone! [...] PD: Thanks in advances for your answers; this is my first post! I think ill be active sharing and discussing in other Physics/ Astrophysics topics ;)

Just a suggestion for the future. Please, use the LaTex code. This website has MathJax implemented, so that the equations are made to look great. Just search here for a tutorial on how to write with the simple code.

 

[samalkhaiat]

First prove that
$$\delta g^{\alpha\beta}U_{\alpha}U_{\beta} = 2 U_{\mu}\delta U^{\mu}. \ \ \ \ \ \ \ (1)$$ Then, using $U_{\mu}U^{\mu} = 1$, you can write the left-hand-side of (1) as
$$\delta g^{\alpha \beta}U_{\alpha}U_{\beta} \equiv 2 U_{\mu} \left( \frac{1}{2}U^{\mu} \delta g^{\alpha \beta}U_{\alpha}U_{\beta}\right) . \ \ \ \ \ (2)$$
The result follows by comparing (1) with (2).