Deriving MTW's Equation 21.90 from Equation 21.83

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In summary, the conversation discusses the development of MTW's equation 21.90 from the simple Lagrangian, with the use of equations 21.83, 21.88, and 21.89. The transformation is achieved by establishing the relationship between Trπ, Trπ^2, and TrK, and using Golovnev's identity for -2(-g)^1/2∇μ(Kii nμ). The final result is the equation 16πLgeom= LgeomADM= -gij∂πij/∂t - NH - NiHi - 2[πijNj - 1/2 NiTrπ + N^i(g)^1/2]
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TerryW
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Homework Statement



This isn't a request for assistance, I am just posting this to help anyone else in the future who wants to see how MTW's equation 21.90 can be developed from the simple Lagrangian.

MTW's Equation 21.83 is simply ##16π\mathfrak{L}_{geom} = (-^{(4)}g)^{(4)}R##

One page later, equation 21.90 appears -

##16π\mathfrak{L}_{geom} = \mathfrak{L}_{geomADM} = -g_{ij}∂π^{ij}/∂t - N\mathcal{H} - N^i\mathcal{H}^i -2\big{[}π^{ij}N_j - \frac{1}{2} N^iTr{π}+ N^{|i}(g)^½\big{]}_{,i}##

The intervening paragraphs include some pointers as to how this transformation is achieved but I was unable to work out how equations 21.88 and 21.89 could be used to complete the job. I didn't really want to move on without completing a proof that 21.90 does indeed come from 21.83, so I bought the reproduction of ADM's original paper, but that didn't help either as 21.90 is simply introduced as "an equation which can be derived" from some basic quantities (Unless I am missing something).

I then found a paper by Alex Golovnev on ArXiv, which I have been able to work through to the point where I was able to follow his derivation to establish:

##(-^{(4)}g)^{(4)}R = γ^½N(^{(3)}R +K^{ij}K_{ij} - K^i_iK^j_j) - 2 (-^{(4)}g)^½ ∇_μ (K^i _i n^μ) - 2γ^½{ }^{(3)}ΔN##

where ##γ =\ ^{(3)}g##

All that needs to be done now is to show that Golovnev's equation can be transformed into MTW's 21.90, which can be achieved as follows:

First I established how Trπ, Trπ^2 and TrK are related:

(i) ##Trπ = g_{ij}π^{ij} = γ^½[g_{ij}g^{ij}TrK - g_{ij}K^{ij}] = γ^½(3TrK - TrK) = γ^½(2TrK)##
(ii) ##Trπ^2 = π^{ij}π_{ij} = γ(g^{ij}g_{ij}(TrK)^2 - g^{ij}K_{ij}TrK - g_{ij}K^{ij}TrK +K^{ij}K_{ij})##
∴ ##Trπ^2 = γ(3(TrK)^2 - (TrK)^2-(TrK)^2 + Tr(K^2)) = γ((TrK)^2 + Tr(K^2))##
(edited to correct last term in line above)
So
##γ^½N(^{(3)}R +K^{ij}K_{ij} - K^i_iK^j_j) - 2 (-^{(4)}g)^½ ∇_μ (K^i _i n^μ) - 2γ^½{ }^{(3)}ΔN##
##= γ^½N(^{(3)}R +TrK^2 - (TrK)^2) - 2 (-^{(4)}g)^½ ∇_μ (K^i _i n^μ) - 2γ^½{ }^{(3)}ΔN##
Then using Golovnev's identity for ##- 2 (-^{(4)}g)^½ ∇_μ (K^i _i n^μ)##
##= γ^½N(^{(3)}R +TrK^2 - (TrK)^2) - 2∂_0(γ^½ TrK) +2γ^½{ }^{(3)}∇_j(K^i_iN^j) - 2γ^½{ }^{(3)}ΔN##
Then using (i) above and the general expression for the divergence of a vector (MTW 21.85 p 579):
##= γ^½N(^{(3)}R +TrK^2 - (TrK)^2) - ∂_0(Trπ) +2∂_j(γ^½TrKN^j) - 2γ^½{ }^{(3)}ΔN##
##= -γ^½N( (TrK)^2- TrK^2 -^{(3)}R) - ∂_0(π^{ij}γ_{ij}) +2∂_i(γ^½TrKN^i) - 2γ^½(N^{|i}{}_{|i})##
##= -γ^½N( (TrK)^2- TrK^2 -^{(3)}R) - π^{ij}\dot γ_{ij}- \dot π^{ij}γ_{ij}+(N^iTrπ)_{,i}- 2(γ^½N^{|i})_{,i}##
##= -2γ^½N((TrK)^2- TrK^2)+ γ^½N((TrK)^2- TrK^2)+γ^½N^{(3)}R - π^{ij}\dot γ_{ij}- \dot π^{ij}γ_{ij} +\quad(N^iTrπ- 2γ^½N^{|i})_{,i}##
##= -[2γ^½N(g^{ij}TrK- K^{ij})K_{ij} + π^{ij}\dot γ_{ij}] - γ^½N(TrK^2- (TrK)^2-^{(3)}R) - \dot π^{ij} γ_{ij} \quad+(N^iTrπ- 2γ^½N^{|i})_{,i}##
##= -2π^{ij}[NK_{ij} + ½\dot γ_{ij}] - γ^½N(TrK^2- (TrK)^2-^{(3)}R) - \dot π^{ij} γ_{ij} +(N^iTrπ- 2γ^½N^{|i})_{,i}##
Using Golovnev's Equation (3)...
##= - \dot π^{ij} γ_{ij} - γ^½N(TrK^2- (TrK)^2-^{(3)}R) -2π^{ij}N_{i|j}+(N^iTrπ- 2γ^½N^{|i})_{,i}##
##= - \dot π^{ij} γ_{ij} - N[γ^{-½}[γ(TrK^2+(TrK)^2)-½(4γ(TrK)^2)]-γ^{½{}(3)}R] -2π^{ij}N_{i|j}+(N^iTrπ- 2γ^½N^{|i})_{,i}##
##= - \dot π^{ij} γ_{ij} - N[γ^{-½}[Trπ^2-½(Trπ)^2-γ^{½{}(3)}R] -2π^{ij}N_{i|j}+(N^iTrπ- 2γ^½N^{|i})_{,i}##
##= - \dot π^{ij} γ_{ij} - N\mathcal{H} -2π^{ij}N_{j,i} + 2π^{ij{}(3)}Γ^k_{ji}N_k+(N^iTrπ- 2γ^½N^{|i})_{,i}##
##= - \dot π^{ij} γ_{ij} - N\mathcal{H} -2π^{ij}N_{j,i} -2π^{ij}{}_{,i}N_j +2π^{ij}{}_{,i}N_j+ 2π^{ij{}(3)}Γ^k_{ji}N_k+(N^iTrπ- 2γ^½N^{|i})_{,i}##
##= - \dot π^{ij} γ_{ij} - N\mathcal{H} -(2π^{ij}N_j)_{,i} +2π^{ij}{}_{,i}N_j+ 2π^{ij{}(3)}Γ^k_{ji}N_k+(N^iTrπ- 2γ^½N^{|i})_{,i}##
##= - \dot π^{ij} γ_{ij} - N\mathcal{H} -(2π^{ij}N_j)_{,i} +2π^{ik}{}_{,i}N_k+ 2π^{ij{}(3)}Γ^k_{ji}N_k +2π^{ij{}(3)}Γ^k_{jk}N_i-2π^{ij{}(3)}Γ^k_{jk}N_i \quad+(N^iTrπ- 2γ^½N^{|i})_{,i}##
##= - \dot π^{ij} γ_{ij} - N\mathcal{H} +2π^{ik}{}_{,i}N_k+ 2π^{ij{}(3)}Γ^k_{ji}N_k +2π^{ij{}(3)}Γ^k_{jk}N_i-2π^{ij{}(3)}Γ^k_{jk}N_i \quad-(2π^{ij}N_j-N^iTrπ+2γ^½N^{|i})_{,i}##
Then a little bit of index swapping gives
##= - \dot π^{ij} γ_{ij} - N\mathcal{H} +2π^{ki}{}_{,k}N_i+ 2π^{kj{}(3)}Γ^i_{jk}N_i +2π^{ij{}(3)}Γ^k_{jk}N_i-2π^{ik{}(3)}Γ^j_{kj}N_i \quad-(2π^{ij}N_j-N^iTrπ+2γ^½N^{|i})_{,i}##
Then remembering that ##π^{ki}## is a tensor density
##= - \dot π^{ij} γ_{ij} - N\mathcal{H} -N_i(-2π^{ki}{}_{|k})-(2π^{ij}N_j-N^iTrπ+2γ^½N^{|i})_{,i}##
##= - \dot π^{ij} γ_{ij} - N\mathcal{H} -N_i\mathcal{H^i}-2(π^{ij}N_j-½N^iTrπ+γ^½N^{|i})_{,i}##

MTW's Equation 21.90 at last!
 
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##16π\mathfrak{L}_{geom} = \mathfrak{L}_{geomADM} = -g_{ij}∂π^{ij}/∂t - N\mathcal{H} - N^i\mathcal{H}^i -2\big{[}π^{ij}N_j - \frac{1}{2} N^iTr{π}+ N^{|i}(g)^½\big{]}_{,i}##Homework EquationsSee aboveThe Attempt at a SolutionThis is not a question, so I have provided the solution in the statement
 

FAQ: Deriving MTW's Equation 21.90 from Equation 21.83

1. How do you derive MTW's Equation 21.90 from Equation 21.83?

To derive MTW's Equation 21.90 from Equation 21.83, you must first substitute the values of the variables in Equation 21.83 into Equation 21.90. Then, you must perform the necessary mathematical operations, following the rules of calculus and special relativity, to arrive at the final form of Equation 21.90.

2. What is the significance of MTW's Equation 21.90?

MTW's Equation 21.90 is significant because it relates the curvature of space-time to the mass and energy distribution within a given region. This equation is an important component of Einstein's general theory of relativity and has been used to make predictions and calculations in various fields of physics.

3. Can Equation 21.90 be derived using other methods?

Yes, Equation 21.90 can be derived using other methods, such as the Einstein field equations or the geodesic equation. These methods use different mathematical approaches and may provide additional insights into the nature of space-time curvature.

4. Are there any limitations to the use of Equation 21.90?

Like any equation, there are limitations to the use of Equation 21.90. It is derived within the framework of general relativity, which may not accurately describe extreme conditions such as those near black holes or in the early universe. Additionally, it may not take into account other factors that could affect space-time curvature, such as quantum effects.

5. How is Equation 21.90 related to the concept of gravity?

Equation 21.90 is closely related to the concept of gravity. In general relativity, gravity is understood as the curvature of space-time caused by the presence of mass and energy. Equation 21.90 mathematically describes this relationship, showing how the curvature is determined by the distribution of mass and energy in a given region.

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