Wednesday, March 7, 2007

General Relativity Is Hard

So I've been sick, and tired, and busy. I have nothing to say except that my physics course is kicking my ass. Here is a recent email from my professor:

GR 6333 students

Hint for evaluation the Riemann tensor.

The Riemann curvature tensor Rabcd has a maximum of 20 independent components. When all 4 indices are down it has the symmetries of equations (21.29) of Hartle. You can think of Rabcd (all down) as a 6x6 symmetric matrix Mij =Mji with ab being one index i and cd being the second index j. Because of the anti-symmetry on ab and cd each pair can take on 4x3/2=6 independent values, e.g. i = (t,r),(t,theta),(t,phi),(r,theta),(r,phi),(theta,phi). Because Rabcd=Rcdab, Mij is symmetric and can have at most 6x7/2=21 components. (21,29d) removes 1 additional component making a total of 20.

To work this problem I would suggest using equation (21.20) with your connection symbols for Schwarzschild, to evaluate Rabcd (a up and bcd down).

Look at 6 expressions, e.g., one of which is Rabtr, and see which of the values of (a,b) give a non-zero component. Because the metric is diagonal ab =(t,r) is a possibility but you don't need to compute ab=(r,t) because it will not be independent of ab=(t,r).

There are only 6 independent non-vanishing components of Rabcd the Schwarzschild metric and they can be found on page 554 of Hartle.

RK

2 comments:

blythe said...

my head just blew up. jesus. and i thought chinese was hard.

get better soon. there are gymnastics meets to be attended.

Dan said...

I took a class on G-Rel in college. I went to every class, every office hour of my professor, studied my ass off, and still could rarely do more than label the axes of the graphs on the tests. With the curve, that got you an A in that class. Yeah, hard. I put my tail between my legs and got back to that little puff piece, The Critique of Pure Reason.