Metric for non-inertial coordinate system

[Dixanadu]

Homework Statement

Hey guys.

So here's the problem:

Consider an ordinary 2D flat spacetime in Cartesian coordinates with the line element
$ds^{2}=-dt^{2}+dx^{2}$

Now consider a non-inertial coordinate system $(t',x')$, given by

$t'=t, x'=x-vt-\frac{1}{2}at^{2}$

(1) What is the metric in these coordinates?

There are some more questions apart from this but I think I can do those if I know how to do this part.

Homework Equations

None

The Attempt at a Solution

Okay so here's why I'm confused. How do I get the line element in these coordinates? Here are the two options in my mind...which one is correct?

OPTION 1
The line element they are looking for is $ds^{2}=-dt^{2}+dx'^{2}$ where $dx'=dx-(v+at)dt$

OPTION 2
The line element they are looking for is $ds^{2}=-dt^{2}+dx^{2}$, where $dx=dx'+(v+at)dt$

Both of these options give different metrics...so which one (if any) is the way to go?

Thanks guys!

 

[WannabeNewton]

All you have to do is calculate $dx$ and $dt$ in terms of the $(t',x')$ coordinates and plug them into $ds^2 = -dt^2 + dx^2$ to get the metric in $(t',x')$ coordinates.