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Metric tensor problem

471
9
Homework Statement
Derive the metric tensors for the following spacetimes, need help with (1)
Homework Equations
##ds^2 = g_{\mu \nu} dX^{\mu} dX^{\nu}##
Screenshot 2019-08-26 at 4.17.38 PM.png


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

Science Advisor
Insights Author
1,616
806
[tex]U = y - t, \ \ \ \ V = y + t[/tex]
 

haushofer

Science Advisor
Insights Author
2,227
562
Cross-terms in the interval means off-diagonal terms in the metric :)
 
471
9
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

Science Advisor
Homework Helper
Gold Member
3,540
181
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.
 

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