Deriving the MTW version of the Einstein-Hilbert density

In summary: N^2K^i_iN^j(\frac{1}{N^2}))_{,j}####∴\sqrt{-g}^{(4)}\nabla_μ (K^i_in^μ) = (\sqrtγK^i_i)_{,0} - (\sqrtγK^i_iN^j)_{,j}####∴\sqrt{-g}^{(4)}\nabla_μ (K^i_in^μ) = (\sqrtγK^i_i)_{,0} - (√γ∂_j(K^i_iN^j) - √γK^i_i
  • #1
TerryW
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Homework Statement



I've been working through a paper by Alexey Golovnev, title 'ADM and massive gravity' arXiv.1302.0687v4 [gr-qc] 26 March 2013. I am hoping to use his result for the Einstein-Hilbert density to achieve my aim of finding a way to derive Equation 21.90 in MTW. I have worked my way through the paper and can see that all the equations given are OK, with one crucial exception:

Homework Equations



The final link in the chain is this:

##-2 \sqrt{-g}\nabla_μ (K^i_in^μ) = -2∂_0(\sqrt{γ}K^i_i) + 2\sqrt{γ}^{(3)}\nabla_j(K^i_iN^j)##

My attempt to prove this identity ends up with an extra term. Can anyone tell me where I've gone wrong?

The Attempt at a Solution



Using the derivations in the paper:

##-2 \sqrt{-g}\nabla_μ (K^i_in^μ) = -2 \sqrt{-g}^{(4)}\nabla_0 (K^i_in^0) - 2 \sqrt{-g}^{(4)}\nabla_j (K^i_in^j)##

##= -2 ^{(4)}\nabla_0 (\sqrt{γ}NK^i_i\frac{1}{N}) - 2 \sqrt{-g}^{(4)}\nabla_j (K^i_in^j)##

##= -2 ^{(4)}\nabla_0 (\sqrt{γ}K^i_i) - 2 \sqrt{-g}[∂_j(K^i_in^j) + ^{(4)}Γ^j_{αj}(K^i_in^α)]##

##= -2 ∂_0 (\sqrt{γ}K^i_i) - 2 \sqrt{-g}∂_j(K^i_in^j) -2\sqrt{-g} ^{(4)}Γ^j_{0j}(K^i_in^0) -2\sqrt{-g} ^{(4)}Γ^j_{kj}(K^i_in^k)##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{-g}∂_j(K^i_i\frac{N^j}{N}) -2\sqrt{-g} ^{(4)}Γ^j_{0j}(K^i_i\frac{1}{N}) +2\sqrt{-g} ^{(4)}Γ^j_{kj}(K^i_i\frac{N^k}{N})##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{-g}∂_j(K^i_iN^j)(\frac{1}{N}) +2 \sqrt{-g}(K^i_iN^j)∂_j(\frac{1}{N})-2\sqrt{γ}N^{(4)}Γ^j_{0j}(K^i_i\frac{1}{N})##
## +2\sqrt{γ} N^{(4)}Γ^j_{kj}(K^i_i\frac{N^k}{N})##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{γ}∂_j(K^i_iN^j) +2\sqrt{γ}^{(4)}Γ^j_{kj}K^i_iN^k+2\sqrt{-g}(K^i_iN^j)(\frac{-1}{N^2})∂_jN -2\sqrt{γ} ^{(4)}Γ^j_{0j}(K^i_i)##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{γ}∂_j(K^i_iN^j) +2\sqrt{γ}(^{(3)}Γ^j_{kj} +\frac{N^j}{N}K_{jk})K^i_iN^k)-2\sqrt{γ}K^i_i(\frac{N^j}{N}∂_jN + ^{(4)}Γ^j_{0j})##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{γ}^{(3)}\nabla_j(K^i_iN^j) +2\sqrt{γ}\frac{N^j}{N}K_{jk}K^i_iN^k -2\sqrt{γ}K^i_i\frac{N^j}{N}∂_jN -2\sqrt{γ}K^i_{i}## ##^{(4)}Γ^j_{0j}##

The first two terms are what I was aiming for. The three remaining terms are 'surplus' to requirements!

I can reduce these three terms down to just one as follows

##2\sqrt{γ}\frac{N^j}{N}K_{jk}K^i_iN^k -2\sqrt{γ}K^i_i\frac{N^j}{N}∂_jN -2\sqrt{γ}K^i_{i} ## ##^{(4)}Γ^j_{0j}##

##= 2\sqrt{γ}K^i_i[\frac{N^j}{N}K_{jk}N^k -\frac{N^j}{N}∂_jN - ^{(4)}Γ^j_{0j}]##

##= 2\sqrt{γ}K^i_i[\frac{N^j}{N}K_{jk}N^k -\frac{N^j}{N}∂_jN - [-\frac{N^j}{N}∂_jN -N(γ^{jk} - \frac{N^jN^k}{N^2})K_{jk} + ^{(3)}\nabla_j(N^j)]##

##= 2\sqrt{γ}K^i_i[Nγ^{jk}K_{jk} - ^{(3)}\nabla_j(N^j)]##

##= 2\sqrt{γ}K^i_i[Nγ^{jk}K_{jk} - γ^{jk}## ##^{(3)}\nabla_j(N_k)]##

##= -2\sqrt{γ}K^i_iγ^{jk}\,^{(4)}Γ_{k0j}##

I'd really appreciate it if someone can help me get rid of this last term!
 

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  • #2
TerryW said:

Homework Statement



I've been working through a paper by Alexey Golovnev, title 'ADM and massive gravity' arXiv.1302.0687v4 [gr-qc] 26 March 2013. I am hoping to use his result for the Einstein-Hilbert density to achieve my aim of finding a way to derive Equation 21.90 in MTW. I have worked my way through the paper and can see that all the equations given are OK, with one crucial exception:

Homework Equations



The final link in the chain is this:

##-2 \sqrt{-g}\nabla_μ (K^i_in^μ) = -2∂_0(\sqrt{γ}K^i_i) + 2\sqrt{γ}^{(3)}\nabla_j(K^i_iN^j)##

My attempt to prove this identity ends up with an extra term. Can anyone tell me where I've gone wrong?

The Attempt at a Solution



Using the derivations in the paper:

##-2 \sqrt{-g}\nabla_μ (K^i_in^μ) = -2 \sqrt{-g}^{(4)}\nabla_0 (K^i_in^0) - 2 \sqrt{-g}^{(4)}\nabla_j (K^i_in^j)##

##= -2 ^{(4)}\nabla_0 (\sqrt{γ}NK^i_i\frac{1}{N}) - 2 \sqrt{-g}^{(4)}\nabla_j (K^i_in^j)##

##= -2 ^{(4)}\nabla_0 (\sqrt{γ}K^i_i) - 2 \sqrt{-g}[∂_j(K^i_in^j) + ^{(4)}Γ^j_{αj}(K^i_in^α)]##

##= -2 ∂_0 (\sqrt{γ}K^i_i) - 2 \sqrt{-g}∂_j(K^i_in^j) -2\sqrt{-g} ^{(4)}Γ^j_{0j}(K^i_in^0) -2\sqrt{-g} ^{(4)}Γ^j_{kj}(K^i_in^k)##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{-g}∂_j(K^i_i\frac{N^j}{N}) -2\sqrt{-g} ^{(4)}Γ^j_{0j}(K^i_i\frac{1}{N}) +2\sqrt{-g} ^{(4)}Γ^j_{kj}(K^i_i\frac{N^k}{N})##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{-g}∂_j(K^i_iN^j)(\frac{1}{N}) +2 \sqrt{-g}(K^i_iN^j)∂_j(\frac{1}{N})-2\sqrt{γ}N^{(4)}Γ^j_{0j}(K^i_i\frac{1}{N})##
## +2\sqrt{γ} N^{(4)}Γ^j_{kj}(K^i_i\frac{N^k}{N})##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{γ}∂_j(K^i_iN^j) +2\sqrt{γ}^{(4)}Γ^j_{kj}K^i_iN^k+2\sqrt{-g}(K^i_iN^j)(\frac{-1}{N^2})∂_jN -2\sqrt{γ} ^{(4)}Γ^j_{0j}(K^i_i)##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{γ}∂_j(K^i_iN^j) +2\sqrt{γ}(^{(3)}Γ^j_{kj} +\frac{N^j}{N}K_{jk})K^i_iN^k)-2\sqrt{γ}K^i_i(\frac{N^j}{N}∂_jN + ^{(4)}Γ^j_{0j})##

##= -2 ∂_0 (\sqrt{γ}K^i_i) + 2 \sqrt{γ}^{(3)}\nabla_j(K^i_iN^j) +2\sqrt{γ}\frac{N^j}{N}K_{jk}K^i_iN^k -2\sqrt{γ}K^i_i\frac{N^j}{N}∂_jN -2\sqrt{γ}K^i_{i}## ##^{(4)}Γ^j_{0j}##

The first two terms are what I was aiming for. The three remaining terms are 'surplus' to requirements!

I can reduce these three terms down to just one as follows

##2\sqrt{γ}\frac{N^j}{N}K_{jk}K^i_iN^k -2\sqrt{γ}K^i_i\frac{N^j}{N}∂_jN -2\sqrt{γ}K^i_{i} ## ##^{(4)}Γ^j_{0j}##

##= 2\sqrt{γ}K^i_i[\frac{N^j}{N}K_{jk}N^k -\frac{N^j}{N}∂_jN - ^{(4)}Γ^j_{0j}]##

##= 2\sqrt{γ}K^i_i[\frac{N^j}{N}K_{jk}N^k -\frac{N^j}{N}∂_jN - [-\frac{N^j}{N}∂_jN -N(γ^{jk} - \frac{N^jN^k}{N^2})K_{jk} + ^{(3)}\nabla_j(N^j)]##

##= 2\sqrt{γ}K^i_i[Nγ^{jk}K_{jk} - ^{(3)}\nabla_j(N^j)]## (***)

##= 2\sqrt{γ}K^i_i[Nγ^{jk}K_{jk} - γ^{jk}## ##^{(3)}\nabla_j(N_k)]##

##= -2\sqrt{γ}K^i_iγ^{jk}\,^{(4)}Γ_{k0j}##

I'd really appreciate it if someone can help me get rid of this last term!

Resolution

If I use the general expression for the divergence of a vector (MTW 21.85 p 579):

##(A^α)_{;α} = \sqrt{-g}^{-½}(\sqrt{-g}A^α)_{,α}##

and then work on ##K_{ij} = -^{(4)}\nabla_in_j## (Golovnev (2)), I find that

##g^{ij}K_{ij} = K^i_i = -g^{ij}## ##^{(4)}\nabla_i n_j## ## = -^{(4)}\nabla_i n^i##

So ##\sqrt{-g}K^i_i = \sqrt{-g}^{(4)}\nabla_i\frac{N^i}{N} = (\sqrt{-g}\frac{N^i}{N})_{,i} = (\sqrtγN^i)_{,i}##

Now use the general expression for the divergence of a vector in the 3-D space

##^{(3)}\nabla_iA^i = \sqrtγ^{-½}(\sqrtγA^i)_{,i}## ...(1)

So ##\sqrtγNK^i_i = (\sqrtγN^i)_{,i} = \sqrtγ^{(3)}\nabla_iN^i##

So ##NK^i_i = ^{(3)}\nabla_iN^i##

which means that (***) above = 0, which achieves my aim of getting rid of the 'surplus to requirements' terms.This resolution was prompted by TSny, who sent me a much more elegant solution, again based on MTW 21.85 as follows;

## \sqrt{-g}^{(4)}\nabla_μ (K^i_in^μ) = (\sqrt{-g}(K^i_in^0)_{,0} - \sqrt{-g}K^i_in^j)_{j}##

##∴\sqrt{-g}^{(4)}\nabla_μ (K^i_in^μ) = (\frac {\sqrt{-g}}{N}K^i_i)_{,0} - (\frac {\sqrt{-g}}{N}K^i_iN^j)_{,j}##

Then using (1) and ##\sqrt{-g} = \sqrtγN##

##\sqrt{-g}^{(4)}\nabla_μ (K^i_in^μ) = (\sqrtγK^i_i)_{,0} - \sqrtγ^{(3)}\nabla_j(K^i_iN^j)##

QED!
 
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FAQ: Deriving the MTW version of the Einstein-Hilbert density

1. What is the Einstein-Hilbert density?

The Einstein-Hilbert density is a mathematical expression used in general relativity to describe the curvature of spacetime. It is derived from the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy.

2. What is the significance of the MTW version of the Einstein-Hilbert density?

The MTW version of the Einstein-Hilbert density, named after the scientists Misner, Thorne, and Wheeler, is a modified version of the original density with additional terms that account for the presence of matter and energy in the universe. It is a more accurate and comprehensive expression for describing the curvature of spacetime.

3. How is the MTW version of the Einstein-Hilbert density derived?

The MTW version of the density is derived by adding additional terms to the original Einstein-Hilbert density, which account for the presence of matter and energy in the universe. These terms are based on the energy-momentum tensor, which describes the distribution of matter and energy in spacetime.

4. What is the role of the MTW version of the Einstein-Hilbert density in general relativity?

The MTW version of the density plays a crucial role in general relativity as it is used to calculate the curvature of spacetime, which in turn determines the motion of objects and the behavior of gravity. It is an essential component in the mathematical formulation of general relativity and its predictions.

5. Are there any other modified versions of the Einstein-Hilbert density?

Yes, there are other modified versions of the density, such as the Lovelock density, which includes higher-order curvature terms, and the Palatini density, which uses a different mathematical approach to derive the field equations. Each version has its own advantages and is used in different contexts within the field of general relativity.

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