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Deriving MTW's Equation 21.90 from Equation 21.83

  1. Dec 30, 2018 #1

    TerryW

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    Gold Member

    1. The problem statement, all variables and given/known data

    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!
     
    Last edited: Jan 10, 2019
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