ADM field Lagrangian for a source-free electromagnetic field

[TerryW]

Homework Statement

I am trying to reproduce MTW's ADM version of the field Lagrangian for a source free electromagnetic field:

$4π\mathcal {L} = -\mathcal {E}^i∂A_i/∂t - ∅\mathcal {E}^i{}_{,i} - \frac{1}{2}Nγ^{-\frac{1}{2}}g_{ij}(\mathcal {E}^i\mathcal {E}^i + \mathcal {B}^i\mathcal {B}^i) + N^i[ijk]\mathcal {E}^j\mathcal {B}^k$ ...(21.100)

(I'm using γ instead of $^{(3)}g$ so $(-^{4}g)^{\frac{1}{2}} = Nγ^{\frac{1}{2}}$)

Homework Equations

I have used as my start point "by what in flat spacetime would be"

$\quad\quad \frac{1}{4π}\big{[}A_{μ,ν}F^{μν} + \frac{1}{4} F_{μν}{}^{μν}\big{]} ...(21.99)$

The Attempt at a Solution

To begin, I recast (21.99) as:

$4π\mathcal {L} =\big{[}A_{μ,ν}g^{αμ}g^{βν}F^{αβ} + \frac{1}{4} F_{μν}g^{αμ}g^{βν}F_{αβ}\big{]}$

I then worked on this to produce:

$4π\mathcal {L} =\frac{1}{N}\big{[}(-(γ^{\frac{1}{2}}γ^{ij}F_{i0}A_{j,0}) - A_0\frac{∂}{∂x^j}(γ^{\frac{1}{2}}γ^{ij}F_{i0})\big{]}\hspace{23mm}(A)$

$\quad\quad\quad\quad\quad+γ^{\frac{1}{2}}(A_{j,0} - A_{0,j})(\frac{γ^{ji}}{N})N^k(A_{k,i} - A_{i,k}) \hspace{21mm}(B)$

$\quad\quad\quad\quad\quad-\frac{1}{2}γ^{\frac{1}{2}}\big{[}(A_{i,0} - A_{0,i})(\frac{γ^{ij}}{N})(A_{j,0} - A_{0,j})\hspace{23mm}(C)$

$\quad\quad\quad\quad\quad-\frac{1}{4}γ^{\frac{1}{2}}(A_{j,i} - A_{i,j})Nγ^{ik}γ^{jl}(A_{l,k}-A_{k,l})\hspace{20mm}(D)$

$\quad\quad\quad\quad\quad+\frac{1}{2}γ^{\frac{1}{2}}(A_{j,i} - A_{i,j})γ^{jl}\frac{N^kN^i}{N}(A_{l,k}-A_{k,l})\hspace{20mm}(E)$

From here on, I am working on assumptions which may not be entirely correct:

If $F_{i0} = E_i, γ^{\frac{1}{2}}γ^{ij}F_{i0} = \mathcal{E}^j$

(A) becomes

$\frac{1}{N}(-\mathcal{E}^j\frac{∂A_j}{∂t} +φ\mathcal{E}^i{}_i)$

If $(A_{k,i}- A_{i,k}) = \frac{1}{2}[jki](A_{k,i}- A_{i,k})$

(B) becomes

$\frac{1}{N}(\mathcal{E}^iN^k\mathcal{B}^j)[ijk]$
Where $[ijk]$ is needed because the i in $\mathcal{E}^i$ and the k in $N^k$ are tied to the i,k in $A_{i,k}$

(C ) becomes

$-\frac{1}{2}(\frac{1}{N})γ^{-\frac{1}{2}}\mathcal{E}^i\mathcal{E}^jγ_{ij}$

(D) becomes

$-\frac{N}{4}γ^{\frac{1}{2}}\mathcal{B}^m\mathcal{B}_n\frac{γ_{mn}γ^{mn}}{3}[mij][nlk]γ^{ik}γ^{jl}$

which then becomes

$-\frac{N}{2}γ^{-\frac{1}{2}}\mathcal{B}^m\mathcal{B}_nγ_{mn}$

(E) is a big problem because it is surplus to requirements and I can't see any way of making it disappear.

So basically, I have produced a set of elements (A) to (D) which are almost the same as the elements in MTW (21.100) except for some annoying factors of 'N'.

As I noted at the start, MTW make the point that

$\quad\quad \frac{1}{4π}\big{[}A_{μ,ν}F^{μν} + \frac{1}{4} F_{μν}{}^{μν}\big{]} ...(21.99)$

is "what would in flat space-time be" and we are not in a flat spacetime, but I can't see a way of making a transformation which would be in any way useful. It would be really nice if $E^i$ in flat spacetime could become $NE^i$ as this would solve all the issues with (A) to (D), but that still leaves me with (E).Any ideas anyone??RegardsTerryW

 

[TerryW]

Hi anyone who is following my ramblings.

I've sorted this now! I should have looked further down the page and spotted that MTW give a definition for
$\mathcal{E}^i$ in 21.103.

Using this in my recast of 21.99 introduces a whole lot of extra terms but they all cancel out, including my problematical (E) term and leaving me with just the terms I need for 21.100 with no bits left over and no problems with unwanted 'N's.



TerryW