Cosmological model, geodesics question

[binbagsss]

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.

 

[Orodruin]

The spatial equations yes. You replaced them by the correspondig first integral. You still need to solve the temporal one.

 

[binbagsss]

The spatial equations yes. You replaced them by the correspondig first integral. You still need to solve the temporal one.

okay many thanks.
I am unsure what you mean by 'integral ' however.
But I see clearly I have missed out the time -component equation of $\ddot{x^u}+\Gamma^u_{ab}\dot{x^a}\dot{x^b}=0$

I have found that the Christoffel Symbols $\Gamma^x_{xt} , \Gamma^t_{xt} \neq 0$ , and $\Gamma^t_{tt}=0$
And so for there to be a chance of a geodesic, from my above workings, I assume the quesion means to hold all $x^i$ constant and not just one?

This leaves me to look at whether $\ddot{t}=0$?

To this I check whether there is an affine parameter $s$ such that this is possible. The definition of affine parameter is that $dL/ds=0$

and so if I write

$L=-\dot{t^2} + K_{x_1} + K_{x_1} + K_{x_1}$; where $K_{x_1}$ is the KvF associated with the $x_i$ coordinate.
and then differentiate wrt s to get
$0=\ddot{t}$

is this ok?
have I took a long-winded approach?

 

[Orodruin]

I am unsure what you mean by 'integral ' however

Not ”integral”, ”first integral”.