Proving Non-linear Wave Equation for Riemann Tensor

[dman12]

Hello,

I am working through Hughston and Tod "An introduction to General Relativity" and have gotten stuck on their exercise [7.7] which asks to prove the following non- linear wave equation for the Riemann tensor in an empty space:

∇^e∇_eR_abcd = 2R_aedf R_b^e_c^f − 2R_aecf R_b^e_d^f − R_abef R_cd^ef

I have started from the Bianchi identity:

R_abcd;e + R_abec;d + R_abde;c = 0

To give:

∇^e∇_e R_abcd = -∇^e∇_d R_abec - ∇^e∇_c R_abde

But I don't know what to do to get the RHS into the correct form. Do I use the fact that we are considering empty space such that the Ricci tensor vanishes, R_ab = 0 ?

Any help on how to prove this relation would be very much appreciated!

 

[bcrowell]

Interesting. If you're told to prove it in empty space, then clearly you're going to need the fact that the Ricci tensor vanishes.

It seems sensible to imagine that you might need the Bianchi identity at some point, since you're differentiating the Riemann tensor, but this looks to me more like something you'd do for simplification at the very end.

None of the possible ingredients you've suggested so far will result in a curvature polynomial of the form R...R...

The only method of attack that I can think of would be to write out the Riemann tensor in terms of the Christoffel symbols. I'm sure this would work, but I imagine it would be really, really ugly.

You could go to Riemann normal coordinates, which might simplify things somewhat.

 

[ForensicPhysic]

I am with bcrowell on this one, you do not have enough information to efficiently create your proof. Look back and see what you can find that may help, otherwise we can not help you much more than that.

 

[Ben Niehoff]

I have started from the Bianchi identity:

R_abcd;e + R_abec;d + R_abde;c = 0

To give:

∇^e∇_e R_abcd = -∇^e∇_d R_abec - ∇^e∇_c R_abde

But I don't know what to do to get the RHS into the correct form. Do I use the fact that we are considering empty space such that the Ricci tensor vanishes, R_ab = 0 ?

Any help on how to prove this relation would be very much appreciated!

This is the right place to start. Now manipulate the RHS of that to give expressions with commutators of covariant derivatives, $[\nabla_a, \nabla_b]$. Finally, use the fact that the commutator of covariant derivatives gives you a Riemann tensor:

$$[\nabla_a, \nabla_b] V^c = R^c{}_{dab} V^d,$$
etc.