Register to reply 
Question involving curvature tensor 
Share this thread: 
#1
Apr2906, 02:18 PM

P: 307

Greetings,
I'm working out some of the mathematical relations between YangMill theory and GR. I'm having difficulty working out a somewhat trivial thing, I was wondering if anyone here could help me. To keep things concrete, I'll stick to the case D=4, but I'd like to be able to generalise to higher dimensional cases afterwards. Consider the Riemann curvature tensor. It has 20 independent components. The vacuüm Einstein equations amount to setting the Ricci tensor to zero, which imposes 10 constraints on the Riemann tensor. The additional 10 components are then described the the Weyltensor. Now imaging that we do not impose the Einstein equations. How many constraints would the equation [tex]D^{\mu} R_{\mu\nu\sigma\tau}=0[/tex] (1) impose, together with the second bianchi identity [tex]D_{\left[\rho\right.}R_{\left.\mu\nu\right]\sigma\tau}=&0[/tex] (B2) ? The reason I'm asking is because I've proven that (1)+(B2) implies Einstein equations. I'm wondering if the reverse if true, and if not, what additional constraints must be applied. Thanks in advance. 


Register to reply 
Related Discussions  
How can we tell if a given tensor is a curvature tensor?  Special & General Relativity  6  
Dual of Maxwell tensor as gauge curvature  Quantum Physics  7  
Relation between Ricci and Riemann curvature tensor  Differential Geometry  0  
Energymom tensor does not determine curvature tensor uniquely ?  Special & General Relativity  10  
Riemann curvature tensor derivation  Special & General Relativity  4 