- #1
dman12
- 13
- 0
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∇eRabcd = 2Raedf Rbecf − 2Raecf Rbedf − Rabef Rcdef
I have started from the Bianchi identity:
Rabcd;e + Rabec;d + Rabde;c = 0
To give:
∇e∇e Rabcd = -∇e∇d Rabec - ∇e∇c Rabde
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, Rab = 0 ?
Any help on how to prove this relation would be very much appreciated!
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∇eRabcd = 2Raedf Rbecf − 2Raecf Rbedf − Rabef Rcdef
I have started from the Bianchi identity:
Rabcd;e + Rabec;d + Rabde;c = 0
To give:
∇e∇e Rabcd = -∇e∇d Rabec - ∇e∇c Rabde
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, Rab = 0 ?
Any help on how to prove this relation would be very much appreciated!