 #1
TerryW
Gold Member
 168
 6
 Homework Statement:

I'm having another attempt to get a bit of help working out how MTW (21.115) (attached) can be derived from:
##I = \frac{1}{16π}\int[ \dot π^{ij} γ_{ij}  N\mathcal{H} N_i\mathcal{H^i}]d^4x + \int\mathfrak{L}_{Field}d^4x \quad## MTW (21.95)
 Relevant Equations:

In the hope of making progress, I'm going to present my working so far bit by bit to see if anyone can spot where I am going wrong. I'll start with a bit that seems to work OK:
I am first going to vary ##  N\mathcal{H} ## wrt ##g_{ij}##
##\delta( N\mathcal{H}) = \delta(N[\gamma^{\frac{1}{2}}(Trπ ^2  \frac{1}{2}(Trπ )^2 \gamma^{\frac{1}{2}}R])##
and I am going to concentrate on the first part and find ##\delta(N[\gamma^{\frac{1}{2}}(Trπ ^2  \frac{1}{2}(Trπ )^2]) ## wrt ##g_{ij}##.
##\delta(N[\gamma^{\frac{1}{2}}(Trπ ^2  \frac{1}{2}(Tr π )^2]) = \delta(N^2[(g)^{\frac{1}{2}}]^{1}(Trπ ^2  \frac{1}{2}(Tr π )^2) ##
## = N^2(1)[(g)^{\frac{1}{2}}]^{2} \frac{1}{2}(g)^{\frac{1}{2}}g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2)\delta g_{ij} N\gamma^{\frac{1}{2}}\delta(Trπ ^2  \frac{1}{2}(Tr π )^2)##
## = \frac{1}{2}N^2(g)^{\frac{1}{2}} g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2)\delta g_{ij} N\gamma^{\frac{1}{2}}\delta(Trπ ^2  \frac{1}{2}(Tr π )^2)##
## = \frac{1}{2}N\gamma^{\frac{1}{2}} g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2)\delta g_{ij} N\gamma^{\frac{1}{2}}(\delta(g_{js}π^{sm}g_{mi}π^{ij})  \frac{1}{2} 2(Trπ)π^{ij}\delta g_{ij}##
## = \frac{1}{2}N\gamma^{\frac{1}{2}} g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2)\delta g_{ij} N\gamma^{\frac{1}{2}}(π^{im}g_{ms}π^{sj} +g_{ms}π^{sj} π^{im} (Trπ)π^{ij})\delta g_{ij}##
## = [\frac{1}{2}N\gamma^{\frac{1}{2}} g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2) 2N\gamma^{\frac{1}{2}}(π^{im}π_m{}^j +g_{ms}π^{sj} π^{im} \frac{1}{2}(Trπ)π^{ij})]\delta g_{ij}##
This has produced two of the terms in MTW's (21.115) which makes me feel that my process for finding the variation wrt ##g_{ij}## is probably sound.
Can anyone suggest where I might be missing something?
Regards
Terry W
and I am going to concentrate on the first part and find ##\delta(N[\gamma^{\frac{1}{2}}(Trπ ^2  \frac{1}{2}(Trπ )^2]) ## wrt ##g_{ij}##.
##\delta(N[\gamma^{\frac{1}{2}}(Trπ ^2  \frac{1}{2}(Tr π )^2]) = \delta(N^2[(g)^{\frac{1}{2}}]^{1}(Trπ ^2  \frac{1}{2}(Tr π )^2) ##
## = N^2(1)[(g)^{\frac{1}{2}}]^{2} \frac{1}{2}(g)^{\frac{1}{2}}g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2)\delta g_{ij} N\gamma^{\frac{1}{2}}\delta(Trπ ^2  \frac{1}{2}(Tr π )^2)##
## = \frac{1}{2}N^2(g)^{\frac{1}{2}} g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2)\delta g_{ij} N\gamma^{\frac{1}{2}}\delta(Trπ ^2  \frac{1}{2}(Tr π )^2)##
## = \frac{1}{2}N\gamma^{\frac{1}{2}} g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2)\delta g_{ij} N\gamma^{\frac{1}{2}}(\delta(g_{js}π^{sm}g_{mi}π^{ij})  \frac{1}{2} 2(Trπ)π^{ij}\delta g_{ij}##
## = \frac{1}{2}N\gamma^{\frac{1}{2}} g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2)\delta g_{ij} N\gamma^{\frac{1}{2}}(π^{im}g_{ms}π^{sj} +g_{ms}π^{sj} π^{im} (Trπ)π^{ij})\delta g_{ij}##
## = [\frac{1}{2}N\gamma^{\frac{1}{2}} g^{ij} (Trπ ^2  \frac{1}{2}(Tr π )^2) 2N\gamma^{\frac{1}{2}}(π^{im}π_m{}^j +g_{ms}π^{sj} π^{im} \frac{1}{2}(Trπ)π^{ij})]\delta g_{ij}##
This has produced two of the terms in MTW's (21.115) which makes me feel that my process for finding the variation wrt ##g_{ij}## is probably sound.
Can anyone suggest where I might be missing something?
Regards
Terry W