# Finding a Local Inertial Frame

• PeroK
In summary, the conversation discusses finding a local inertial frame for a given metric. The attempt at a solution involves transforming the coordinates and setting the metric to equal η at a particular point. However, the first derivatives with respect to the transformed coordinates do not vanish, even with an arbitrary velocity. The question is raised about whether the primed coordinate curves are geodesics, and it is suggested to look at coordinates centered on the particular point under consideration.

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## Homework Statement

I am trying to find a local inertial frame for the following metric:

$ds^2 = -(1+\Phi(x))dt^2 + (1-\Phi(x))dx^2$

I can get the transformed metric to equate to η at any point, but I can't get the first derivates wrt the transformed coordinates to vanish.

## Homework Equations

Co-ordinate transformations.

## The Attempt at a Solution

With a transformation of the form:

##t = \frac{\gamma}{a}(t' + vx') \ , \ x = \frac{\gamma}{b}(vt' + x')##

And setting ##a= \frac{1}{\sqrt{1+\Phi(x_p)}}## and ##b= \frac{1}{\sqrt{1-\Phi(x_p)}}## for a particular point ##x_p## I get

##g'_{αβ} = η## as required

I've got my one degree of freedom, in terms of the arbitrary velocity ##v## to try to make the first derivative of ##g'## wrt ##x'## vanish. But, for example, I get:

##g'_{00} = \gamma^2(-\frac{(1+\Phi(x))}{a^2} + \frac{v^2(1-\Phi(x))}{b^2})##

Giving:

##\frac{\partial g'_{00}}{\partial x' } = - \frac{\gamma^3}{b} \Phi '(x) (\frac{1}{a^2} + \frac{v^2}{b^2})##

Which is not going away. The general argument is that one can always find ccordinates where all the first derivates vanish, but I don't see it in this case.

This was an example I set myself to try to see how the general process of finding a local IRF worked.

Are your primed coordinate curves geodesics? For example, if ##x_p## is the event labeled by ##\left( x' , t' \right) = \left( 0 , 0 \right)##, are the curves ##t' = 0## and ##x' = 0## geodesics?

George Jones said:
Are your primed coordinate curves geodesics? For example, if ##x_p## is the event labeled by ##\left( x' , t' \right) = \left( 0 , 0 \right)##, are the curves ##t' = 0## and ##x' = 0## geodesics?

The primed co-ordinates have the same origin as the unprimed. Should I look instead at primed co-ordinates centred on ##x_p##?