Solving Metric Tensor Problems: My Attempt at g_μν for (2)

[WWCY]

Homework Statement

Derive the metric tensors for the following spacetimes, need help with (1)

Relevant Equations

$ds^2 = g_{\mu \nu} dX^{\mu} dX^{\nu}$

My attempt at $g_{\mu \nu}$ for (2) was
\begin{pmatrix}
-(1-r^2) & 0 & 0 & 0 \\ 0 &\frac{1}{1-r^2} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2(\theta)
\end{pmatrix}

and the inverse is the reciprocal of the diagonal elements.

For (1) however, I can't even think of how to write the vector $X^{\mu}$; what exactly are $U,V$?

Also, what does the question mean by "one of them could describe Minkowski spacetime"? At first glance, the metric tensor for (1) is non-diagonal, which I think rules it out. The metric for (2) is diagonal, and appears to approach the Minkowski metric in the small $r$ limit, which I'm guessing is the answer.

Thanks in advance!

 

[samalkhaiat]

$$U = y - t, \ \ \ \ V = y + t$$

 

[haushofer]

Cross-terms in the interval means off-diagonal terms in the metric :)

 

[WWCY]

Thanks for the responses!

So by setting $X^{\mu} = (U = y-t , x, V = y+t, z)$ and expanding according to the line element expansion given above, I find that this form of spacetime reduces to Minkowski spacetime if $g_{20} + g_{02} = 1$, is this right?

 

[nrqed]

Thanks for the responses!

So by setting $X^{\mu} = (U = y-t , x, V = y+t, z)$ and expanding according to the line element expansion given above, I find that this form of spacetime reduces to Minkowski spacetime if $g_{20} + g_{02} = 1$, is this right?

First write the 4x4 matrix in the variables x, z, U,V. (Hint: dU dV = 1/2 dU dV + 1/2 dV dU).
After that only, make the change of variables and then write the matrix in the variables x,y,z,t.