- #1

WWCY

- 479

- 12

- 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!