Physics Asked by Adika on January 22, 2021
I am trying to understand the calculations of the latest Charles Dalang’s paper "Scalar and Tensor Gravitational Waves", arXiv:2009.11827.
Since I just learned basic general relativity, I found it hard to prove equation (13) in that paper.
Here is my calculation for the variation of scalar field $phi$ for ${Largevarepsilon}_{phi}^{(2)}$:
$$delta S_2 = int d^4x delta(sqrt{-g}G_2(phi, X)) = int d^4x sqrt{-g} delta(G_2(phi, X)) $$ where $X = -Largefrac{1}{2}$$g^{munu}partial_{mu}phipartial_{nu}phi$. I think this can be solved by using Euler-Lagrange:
$$frac{partial G_2}{partial phi}=G_{2,phi}$$
$$nabla_{mu} ({frac{partial G_2}{partial(nabla_{mu} phi)}})= nabla_{mu}(frac{partial G_2}{partial X}frac{partial X}{partial(nabla_{mu}phi)})= nabla_{mu}(G_{2,X}frac{partial X}{partial(nabla_{mu}phi)})= ??? $$ I can’t solve this part.
Here the solution for ${Largevarepsilon}_{phi}^{(2)}$ given in that paper (Eq. 13):
$${Largevarepsilon}_{phi}^{(2)} = G_{2,phi} + G_{2,X}squarephi – 2XG_{2,Xphi} + G_{2,XX}phi^{,mu}X_{,mu}.$$
Actually I’m interested in finding $Large{varepsilon}$$_{phi}^{(2)}$ and $Large{varepsilon}$$_{munu}^{(4)}$. I’d appreciate it so much if you give an explicit answer and details in the calculation to prove the equation.
$$nabla_{mu} left({frac{partial G_2}{partial(nabla_{mu} phi)}}right)= nabla_{mu}left(frac{partial G_2}{partial X}frac{partial X}{partial(nabla_{mu}phi)}right)= nabla_{mu}left(G_{2,X}frac{partial X}{partial(nabla_{mu}phi)}right)= ? $$
Since in the bracket is not a tensor, we may change $nabla_mu$ with $partial_mu$, so then we have
$$nabla_{mu}left(G_{2,X}frac{partial X}{partial(nabla_{mu}phi)}right)longrightarrow partial_{mu}left( G_{2,X}frac{partial X}{partial(partial_{mu}phi)} right)$$ Now we can easily calculate $partial_mu$ term $$ partial_{mu}left( G_{2,X}frac{partial X}{partial(partial_{mu}phi)} right) = (partial_{mu}G_{2,X})frac{partial X}{partial(partial_{mu}phi)} + G_{2,X}partial_{mu}left(frac{partial X}{partial(partial_{mu}phi)} right)$$
$$ = left(frac{partial G_{2,X}}{partial phi} frac{partial phi}{partial x^{mu}} +frac{partial G_{2,X}}{partial X} frac{partial X}{partial x^{mu}} right)(-g^{munu}partial_nu phi) + G_{2,X}partial_mu(-g^{munu}partial_nu phi)$$
where $Largefrac{partial X}{partial(partial_{mu}phi)}$ $= -g^{munu}partial_nu phi$
$$left(frac{partial G_{2,X}}{partial phi} frac{partial phi}{partial x^{mu}} +frac{partial G_{2,X}}{partial X} frac{partial X}{partial x^{mu}} right)(-g^{munu}partial_nu phi) + G_{2,X}partial_mu(-g^{munu}partial_nu phi) = G_{2,Xphi} partial_mu phi (-g^{munu}partial_nu phi) + G_{2,XX}X_{,mu}(-g^{munu}partial_nu phi) - G_{2,X}square phi$$ $$ = 2XG_{2,Xphi} - G_{2,XX} phi^{,mu}X_{,mu}-G_{2,X}square phi$$
Correct answer by Adika on January 22, 2021
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