General relativity tensors

  • Thread starter vidi
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  • #1
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This is a question on the nitty-gritty bits of general relativity.

Would anybody mind teaching me how to work these indices?

**Definitions**:

Throughout the following, repeated indices are to be summed over.

Hodge dual of a p-form [itex]X[/itex]:
[tex](*X)_{a_1...a_{n-p}}\equiv \frac{1}{p!}\epsilon_{a_1...a_{n-p}b_1...b_p}X^{b_1...b_p}[/tex]
Exterior derivative of p-form [itex]X[/itex]: [tex](dX)_{a_1...a_{p+1}}\equiv (p+1) \nabla_{[a_1}X_{a_2...a_{p+1}]}[/tex]

Given the relation
[tex]\epsilon^{a_1...a_p c_{p+1}...c_n}\epsilon_{b_1...b_pc_{p+1}...c_n}\equiv p!(n-p)! \delta^{a_1}_{[b_1}...\delta^{a_p}_{b_p]}\,\,\,\,\,\,\,\,\,(\dagger)[/tex]
where [itex]\epsilon_{a_1...a_n}[/itex] is an orientation of the manifold.

Why then is
[tex](*d*X)_{a_1...a_{p-1}}=(-1)^{p(n-p)}\nabla^b X_{a_1...a_{p-1}b}[/tex]?


Firstly, I believe [itex](*d*X)[/itex] means [itex]*(d(*X))[/itex]?
[tex](d*X)_{c_1...c_{n-p+1}}=\frac{n-p+1}{p!}\nabla_{[c_1}\epsilon_{c_2...c_{n-p+1}]b_1...b_p}X^{b_1...b_p}[/tex]
Then [tex]*(d*X)_{d_1...d_{p-1}}=\frac{n-p+1}{(n-p+1)!p!}\epsilon_{d_1...d_{p-1}c_1...c_{n-p+1}}\nabla^{[c_1}\epsilon^{c_2...c_{n-p+1}]b_1...b_p}X_{b_1...b_p}[/tex]
[tex]=\frac{1}{(n-p)!p!}\epsilon_{d_1...d_{p-1}c_1...c_{n-p+1}}\nabla^{[c_1}\epsilon^{c_2...c_{n-p+1}]b_1...b_p}X_{b_1...b_p}[/tex]

Now I know that I should apply [itex](\dagger)[/itex] but I don't know how to given the antisymmetrisation brackets. Would someone mind explaining it to me please? Thank you!
 
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Answers and Replies

  • #2
WannabeNewton
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Ok let's take it slow because you messed up in one of the steps. This is a bit messy (but not much). So we start off with ##d(*\alpha)_{c b_1...b_{n-p}} = (n - p + 1)\nabla_{[c}*\alpha_{b_1...b_{n-p}]}## hence ##*d(*\alpha)_{a_1...a_{p-1}} = \frac{n - p + 1}{(n-p + 1)!}\epsilon_{d_1...d_{n-p + 1}a_1...a_{p-1}}\nabla^{[d_1}*\alpha^{d_2...d_{n-p+1}]}##. Before proceeding, notice that ##\nabla^{[d_1}*\alpha^{d_2...d_{n-p+1}]} = \delta^{[d_1}_{e_1}...\delta^{d_{n-p +1}]}_{e_{{n-p+1}}}\nabla^{e_1}*\alpha^{e_2...e_{n-p+1}} =\\ \frac{1}{(p - 1)!(n - p +1)!}\epsilon^{f_1...f_{p-1}d_1...d_{n-p + 1}}\epsilon_{f_1...f_{p-1}e_{1}...e_{n-p+1}}\nabla^{e_1}*\alpha^{e_2...e_{n-p+1}}##

Now ##\epsilon^{f_1...f_{p-1}d_1...d_{n-p + 1}}\epsilon_{d_1...d_{n-p + 1}a_{1}...a_{p-1}} = (-1)^{(p-1)(n - p + 1)}(p - 1)!(n - p + 1)!\delta^{[f_1}_{a_1}...\delta^{f_{p-1}]}_{a_{p-1}}##
(the ##(-1)^{(p-1)(n - p + 1)}## comes from rearranging the indices on ##\epsilon^{f_1...f_{p-1}d_1...d_{n-p + 1}}## so that it takes the right form)
so ##*d(*\alpha)_{a_1...a_{p-1}} = \frac{(n - p + 1)(-1)^{(p -1)(n - p + 1)}}{(n - p +1)!}\epsilon_{a_1...a_{p-1}e_{1}...e_{n-p+1}}\nabla^{e_1}*\alpha^{e_2...e_{n-p+1}}##. Take it from there.
 
  • #3
WannabeNewton
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Just to be on the safe side, let me know if you need more help and/or if the above isn't clear enough (thankfully there is very little you have to do beyond what I posted above). Also, I didn't mention this before because it isn't really important but for a Lorentzian space-time manifold, the relation you gave involving the orientation of the space-time has an overall negative sign. Finally, keep in mind that ##\nabla_{b}\epsilon_{a_1...a_n} = 0##.
 
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  • #4
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I've been following this thread because I'm interested in the solution, but sadly the OP never came back. Can somebody give it anyway?
 
  • #5
WannabeNewton
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Sure, there's very little to do beyond what was in post #2 anyways. So we left off at ##*d(*\alpha)_{a_1...a_{p-1}} = \frac{(n - p + 1)(-1)^{(p -1)(n - p + 1)}}{(n - p +1)!}\epsilon_{a_1...a_{p-1}e_{1}...e_{n-p+1}}\nabla^{e_1}*\alpha^{e_2...e_{n-p+1}}##.

Now ##(-1)^{(p -1)(n - p + 1)}\epsilon_{a_1...a_{p-1}e_{1}...e_{n-p+1}}\nabla^{e_1}*\alpha^{e_2...e_{n-p+1}} \\ = \frac{(-1)^{p(n - p)}}{p!}\epsilon_{e_2...e_{n-p+1}a_1...a_{p-1}e_{1}}\epsilon^{e_2...e_{n-p + 1}d_1...d_p }\nabla^{e_1}\alpha_{d_1...d_p} \\ =(-1)^{p(n - p)}(n - p)! \nabla^{e_1}\alpha_{[a_1...a_{p-1}e_1]} ##

hence ##*d(*\alpha)_{a_1...a_{p-1}} = \frac{(-1)^{p(n - p)}(n - p + 1)(n - p)!}{(n - p + 1)!} \nabla^{e_1}\alpha_{[a_1...a_{p-1}e_1]} = (-1)^{p(n - p)}\nabla^{e_1}\alpha_{a_1...a_{p-1}e_1} ##
where ##\alpha_{[a_1...a_{p-1}e_1]} = \alpha_{a_1...a_{p-1}e_1}## by definition of a differential form. Thus we have the desired result.
 

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