A question of Einstein field equation

[chinared]

I got some trouble from this question:
For a given metric: ds² =t^-2(dx²-dt²), derive the energy-momentum tensor which satisfies the Einstein equation: R_αβ- 1/2Rg_αβ=8$\pi$GT_αβ.

I got the Ricci scalar R=2, but Tαβ=0 for all α,β. Does this means a curved spacetime without any source(energy-momentum tensor)? Is this possible? Or this result implies that I have made some mistakes in my calculation?

Thanks for answering this question!

 

[Nabeshin]

A schwarzschild black hole is a curved spacetime with no stress-energy tensor, so yes.

 

[Mentz114]

A correction to the EFE as you've written them

R_αβ- (R/2)g_αβ=8πGT_αβ.

I hope it is a typo. (added later ) I see you fixed it after I posted.

Your result is possible as Nabeshin has said.

[edit]
I did the calculation and your results are correct.

 

[TheEtherWind]

I've got a question about this problem. I'm not completely new to GR, but I'm new to actual calculations because I've focused mostly on concepts and I haven't taken a GR class.

How can one find the value of the Ricci Scalar from a given metric? And what about the Stress Energy tensor?

 

[Mentz114]

From the Christoffel symbols,
$${\Gamma ^{m}}_{ab}=\frac{1}{2}g^{mk}(g_{ak,b}+g_{bk,a}-g_{ab,k})$$
the Riemann tensor follows,
$${R^{r}}_{mqs} = \Gamma ^{r}_{mq,s}-\Gamma ^{r}_{ms,q}+\Gamma ^{r}_{ns}\Gamma ^{n}_{mq}-\Gamma ^{r}_{nq}\Gamma ^{n}_{ms}$$
from which
$$R_{ms}={R^{r}}_{mrs}$$
and so
$$R=g^{ms}R_{ms}$$

Use Maxima or some other CAS to calculate this stuff - it takes days by hand.