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MisnerThorneWheeler, p.92, Box 4.1, typo? 
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#1
Oct2609, 11:53 AM

P: 6

In MisnerThorneWheeler Gravitation, Chapter 4, Page 92, Box 4.1, at section 4, there is a formula for the contraction of a pform and a pvector. Now, it states that the contraction of a pform basis with a pvector basis gives the antisymmetrizer symbol, [tex]\left\langle {\omega ^{i_1 } \wedge \ldots \wedge \omega ^{i_p } ,e_{j_1 } \wedge \ldots \wedge e_{j_p } } \right\rangle = \delta ^{i_1 \ldots i_p } _{j_1 \ldots j_p } [/tex] and there is a reference to exercises 3.13 and 4.12. I tried this part many many times and I always find the result to be p! times the antisymmetrizer. I also compared it for the case p=2 using the definition of the symbol from exercise 3.13, still the same result, I get an overall 2. Can anybody please explain what am I doing wrong here?



#2
Oct2609, 05:00 PM

P: 136

No typo. The symbol [tex]\delta^{ij}_{kl}\equiv \delta^{[i}_{k}\delta^{j]}_{l}\equiv \frac{1}{2!}\left(\delta^{i}_{k}\delta^{j}_{l}\delta^{j}_{k}\delta^{i}_{l}\right)[/tex] (which generalizes to n indices with a 1/n! factor), and basis 1forms act on basis vectors as [tex]\omega^{i}(e_{j})=\delta^{i}_{j}[/tex].



#3
Oct2609, 05:24 PM

P: 6

Thanks for the reply. However, if you look on page 88, you will see that the definition for [tex] \delta ^{\alpha \beta } _{\mu \nu } = \delta ^\alpha _\mu \delta ^\beta _\nu  \delta ^\alpha _\nu \delta ^\beta _\mu [/tex] according to MTW does not carry the 1/2! factor that you mention. Furthermore, if you expand the wedge products into tensor products within the contraction symbol, you get [tex]\left\langle {\omega ^\alpha \wedge \omega ^\beta ,e_\mu \wedge e_\nu } \right\rangle = \left\langle {\omega ^\alpha \otimes \omega ^\beta  \omega ^\beta \otimes \omega ^\alpha ,e_\mu \otimes e_\nu  e_\nu \otimes e_\mu } \right\rangle[/tex]
[tex] = \left\langle {\omega ^\alpha \otimes \omega ^\beta ,e_\mu \otimes e_\nu } \right\rangle  \left\langle {\omega ^\alpha \otimes \omega ^\beta ,e_\nu \otimes e_\mu } \right\rangle  \left\langle {\omega ^\beta \otimes \omega ^\alpha ,e_\mu \otimes e_\nu } \right\rangle + \left\langle {\omega ^\beta \otimes \omega ^\alpha ,e_\nu \otimes e_\mu } \right\rangle [/tex] [tex] = \delta ^\alpha _\mu \delta ^\beta _\nu  \delta ^\alpha _\nu \delta ^\beta _\mu  \delta ^\beta _\mu \delta ^\alpha _\nu + \delta ^\beta _\nu \delta ^\alpha _\mu = 2\left( {\delta ^\alpha _\mu \delta ^\beta _\nu  \delta ^\alpha _\nu \delta ^\beta _\mu } \right) = 2 \delta ^{\alpha \beta } _{\mu \nu } [/tex] again, using all the conventions of the book up to this point. 


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