- #1

binbagsss

- 1,259

- 11

## Homework Statement

I am unsure of Q3 but have posted my solutions to other parts

## Homework Equations

## The Attempt at a Solution

3)

ok so it is clear that because the metric components are independent of ##x^i## each ##x^i## has an associated conserved quantity ##d/ds (\dot{x^i})=0##. (1)

The geodesic equation can be written as ##\frac{d^2x^i}{ds^2}+\Gamma^i_{ab}\dot{x}^a\dot{x}^b=0 ##

I look at the Euler-Lagrange equations and I can quickly show that all Christoffel symbols with an upper index ##^i## are zero and so the above is zero (via comparing the the form above of a geodesic equation and identifying the Christoffel symbols from the e-l equations this way). so the first term is zero i have shown by the KVF and the second term zero. so the geodesic equation is obeyed.

Now here is my probably very stupid question, how is this constant ##x^i ## geodesics that the above geodesic equation describes. . the above geodesic equation would hold for ##\dot{x^i}## constant which implies that ##x^i## is constant ofc, but here I see my thoughts are way of track and i have clearly misunderstood something as just using this line of reasoning the geodesic equation is trivially satisfied...

thanks.