Alternative Geometrical Variables in Lanczos-Lovelock Gravity

Abstract

It has been shown in the context of general relativity that there exists a pair of canonically conjugate variables, which also act as thermodynamically conjugate variables on any horizon. We generalize these results to Lanczos-Lovelock gravity in this chapter by identifying two such conjugate variables. Our results do reduce to that of general relativity in the appropriate limit and provides a complete geometrical understanding of Lanczos-Lovelock gravity.

Appendices

Appendix

In this appendix, we shall present the supplementary material for this chapter.

A.1 Identities Regarding Lie Variation of $P^{abcd}$

In this section we shall derive some identities related to Lie variation of the entropy tensor, $P^{abcd}$. For that purpose we first consider Lie variation of the Lagrangian treated as a scalar function of the metric $g_{ab}$ and $R_{abcd}$ leading to

$$\begin{aligned} \pounds _{\xi }L\left( g_{ab},R_{ijkl}\right)&=\xi ^{m}\nabla _{m}L\left( g_{ab},R_{ijkl}\right) \nonumber \\&=\frac{\partial L}{\partial g_{ab}}\xi ^{m}\nabla _{m}g_{ab}+\frac{\partial L}{\partial R_{ijkl}} \xi ^{m}\nabla _{m}R_{ijkl} \nonumber \\&=P^{ijkl}\xi ^{m}\nabla _{m}R_{ijkl} \end{aligned}$$

(3.29)

where we have used the fact that covariant derivative of metric tensor vanishes. Then for the Lagrangian which is homogeneous function of degree m we get

$$\begin{aligned} \pounds _{\xi }L=\xi ^{m}\nabla _{m}\left( \frac{1}{m}P^{ijkl}R_{ijkl}\right) \end{aligned}$$

(3.30)

Then using Eq. (3.29) we readily obtain

$$\begin{aligned} R_{abcd}\xi ^{m}\nabla _{m}P^{abcd}=(m-1)P^{abcd}\xi ^{m}\nabla _{m}R_{abcd} \end{aligned}$$

(3.31)

We also have the following relation:

$$\begin{aligned} P^{ijkl}\pounds _{\xi }R_{ijkl}&=P^{ijkl}\Bigg (\xi ^{m}\nabla _{m} R_{ijkl}+R_{ajkl}\nabla _{i}\xi ^{a}+R_{iakl}\nabla _{j}\xi ^{a} \nonumber \\&+R_{ijal}\nabla _{k}\xi ^{a}+R_{ijka}\nabla _{l}\xi ^{a}\Bigg ) \nonumber \\&=P^{ijkl}\xi ^{m}\nabla _{m} R_{ijkl}+4\nabla _{i}\xi _{m}R^{i}_{~jkl}P^{mjkl} \nonumber \\&=P^{ijkl}\xi ^{m}\nabla _{m} R_{ijkl}+4\nabla _{i}\xi _{m}\mathcal {R}^{im} \end{aligned}$$

(3.32)

Again we can also write, $m\pounds _{\xi }L=P^{ijkl}\pounds _{\xi }R_{ijkl}+R_{ijkl}\pounds _{\xi }P^{ijkl}$. Then we obtain:

$$\begin{aligned} R_{ijkl}\pounds _{\xi }P^{ijkl}= & {} m\pounds _{\xi }L-P^{ijkl}\pounds _{\xi }R_{ijkl} \nonumber \\= & {} m\pounds _{\xi }L-P^{ijkl}\xi ^{m}\nabla _{m} R_{ijkl}-4\nabla _{i}\xi _{m}\mathcal {R}^{im} \nonumber \\= & {} (m-1)P^{abcd}\xi ^{m}\nabla _{m}R_{abcd}-4\nabla _{i}\xi _{m}\mathcal {R}^{im} \end{aligned}$$

(3.33)

This equation can also be casted in a different form as

$$\begin{aligned} R_{abcd}\left( \pounds _{\xi }P^{abcd}-\xi ^{m}\nabla _{m}P^{abcd}\right) =-4\nabla _{i}\xi _{m}\mathcal {R}^{im} \end{aligned}$$

(3.34)

Now we can rewrite the metric as a function of $g_{ab}$ and $R^{a}_{~bcd}$, in which case the Lie variation leads to

$$\begin{aligned} \pounds _{\xi }L\left( g_{ij},R^{a}_{~bcd}\right) =P_{i}^{~jkl}\xi ^{m}\nabla _{m}R^{i}_{~jkl} \end{aligned}$$

(3.35)

With the Lagrangian as homogeneous function of curvature tensor to mth order leads to

$$\begin{aligned} R^{a}_{~bcd}\pounds _{\xi }P_{a}^{~bcd}=(m-1)P_{i}^{~jkl}\xi ^{m}\nabla _{m}R^{i}_{jkl} \end{aligned}$$

(3.36)

Then we arrive at the following identity

$$\begin{aligned} P^{abcd}\pounds _{\xi }\left( g_{am}R^{m}_{~bcd}\right) =P_{a}^{~bcd}\xi ^{m}\nabla _{m}R^{a}_{~bcd} +4\nabla _{i}\xi _{m}\mathcal {R}^{im} \end{aligned}$$

(3.37)

or

$$\begin{aligned} P_{a}^{~bcd}\pounds _{\xi }R^{a}_{~bcd}= & {} P_{a}^{~bcd}\xi ^{m}\nabla _{m}R^{a}_{~bcd} +4\nabla _{i}\xi _{m}\mathcal {R}^{im}-\mathcal {R}^{am}\mathcal {L}_{\xi }g_{am} \nonumber \\= & {} P_{a}^{~bcd}\xi ^{m}\nabla _{m}R^{a}_{~bcd} +2\nabla _{i}\xi _{m}\mathcal {R}^{im} \end{aligned}$$

(3.38)

This leads to the following relation:

$$\begin{aligned} R^{a}_{~bcd}\left( \pounds _{\xi }P_{a}^{~bcd}-\xi ^{m}\nabla _{m}P_{a}^{~bcd}\right) =-2\nabla _{i}\xi _{m}\mathcal {R}^{im} \end{aligned}$$

(3.39)

If we proceed along the same lines we readily obtain another such relation given as:

$$\begin{aligned} R^{ij}_{kl}\left( \pounds _{\xi }P^{kl}_{ij}-\xi ^{m}\nabla _{m}P^{kl}_{ij}\right) =0 \end{aligned}$$

(3.40)

These relations illustrate the Lie variation of $P^{abcd}$ when contracted with the curvature tensor.

A.2 Derivation of Various Identities Used in Text

We will consider the $p\partial q$ and $q\partial p$ structure arising from the identification of $\tilde{f}^{ab}$ as coordinate and $\tilde{N}^{c}_{ab}$ as momentum in Lanczos-Lovelock gravity. For the calculation, the following identity will be used here and there:

$$\begin{aligned} 0=\nabla _{c}Q_{ab}^{cd}= & {} \partial _{c}Q_{ab}^{cd}+\Gamma ^{c}_{ck}Q_{ab}^{kd} \nonumber \\- & {} \Gamma ^{k}_{ca}Q_{kb}^{cd}-\Gamma ^{k}_{cb}Q_{ak}^{cd}+\Gamma ^{d}_{ck}Q_{ab}^{ck} \end{aligned}$$

(3.41)

However $Q_{ab}^{cd}$ being antisymmetric in (c,d) while $\Gamma ^{c}_{ab}$ being symmetric in (a,b) the last term in the above expansion vanishes. Thus ordinary derivative of the quantity $Q_{ab}^{cd}$ has the following expression

$$\begin{aligned} \partial _{c}Q_{ab}^{cd}=-\Gamma ^{c}_{ck}Q_{ab}^{kd}+\Gamma ^{k}_{ca}Q_{kb}^{cd} +\Gamma ^{k}_{cb}Q_{ak}^{cd} \end{aligned}$$

(3.42)

Note that we can include $\sqrt{-g}$ in the above expression leading to:

$$\begin{aligned} \partial _{c}\left( \sqrt{-g}Q_{a}^{~bcd}\right) =\left( \sqrt{-g}Q_{p}^{~bcd}\right) \Gamma ^{p}_{ac}- \left( \sqrt{-g}Q_{a}^{pcd}\right) \Gamma ^{b}_{cp} \end{aligned}$$

(3.43)

Thus we get the following expression from Eq. (3.42):

$$\begin{aligned} \partial _{c}\tilde{N}^{c}_{ab}= & {} \partial _{c} \left[ Q_{bp}^{cq}\Gamma ^{p}_{aq}+Q_{ap}^{cq}\Gamma ^{p}_{bq} \right] \nonumber \\= & {} \left( \partial _{c}Q_{bp}^{cq} \right) \Gamma ^{p}_{aq}+Q_{bp}^{cq}\partial _{c}\Gamma ^{p}_{aq} +\left( \partial _{c}Q_{ap}^{cq}\right) \Gamma ^{p}_{bq}+Q_{ap}^{cq}\partial _{c}\Gamma ^{p}_{bq} \nonumber \\= & {} Q_{kp}^{cq}\Gamma ^{k}_{cb}\Gamma ^{p}_{aq}+Q_{bk}^{cq}\Gamma ^{k}_{cp}\Gamma ^{p}_{aq} -Q_{bp}^{kq}\Gamma ^{p}_{aq}\Gamma ^{c}_{ck}+Q_{kp}^{cq}\Gamma ^{k}_{ca}\Gamma ^{p}_{bq} \nonumber \\+ & {} Q_{ak}^{cq}\Gamma ^{k}_{cp}\Gamma ^{p}_{bq}-Q_{ap}^{kq}\Gamma ^{c}_{ck}\Gamma ^{p}_{bq} +Q_{bp}^{cq}\partial _{c}\Gamma ^{p}_{aq}+Q_{ap}^{cq}\partial _{c}\Gamma ^{p}_{bq} \end{aligned}$$

(3.44)

where in order to arrive at the last equality Eq. (3.42) has been used. Now contracting the above expression with $\tilde{f}^{ab}$ we readily obtain

$$\begin{aligned} \tilde{f}^{ab}\partial _{c}\tilde{N}^{c}_{ab}= & {} \sqrt{-g}g^{ab} \left[ 2Q_{ap}^{cq}\partial _{c}\Gamma ^{p}_{bq}+2Q_{kp}^{cq}\Gamma ^{k}_{ca}\Gamma ^{p}_{bq} +2Q_{ak}^{cq}\Gamma ^{k}_{cp}\Gamma ^{p}_{bq}-2Q_{ap}^{kq}\Gamma ^{c}_{ck}\Gamma ^{p}_{bq} \right] \nonumber \\= & {} -2\sqrt{-g}Q_{p}^{~bcq}\left( \partial _{c}\Gamma ^{p}_{bq}+\Gamma ^{p}_{ck}\Gamma ^{k}_{bq} \right) \nonumber \\+ & {} 2\sqrt{-g}Q_{p}^{~bkq}\Gamma ^{c}_{ck}\Gamma ^{p}_{bq} +2\sqrt{-g}g^{ab}Q_{kp}^{cq}\Gamma ^{k}_{ca}\Gamma ^{p}_{bq} \nonumber \\= & {} -\sqrt{-g}Q_{p}^{~bqc}R^{p}_{~bqc}+2\sqrt{-g}Q_{p}^{~bkq}\Gamma ^{c}_{ck}\Gamma ^{p}_{bq} +2\sqrt{-g}g^{ab}Q_{kp}^{cq}\Gamma ^{k}_{ca}\Gamma ^{p}_{bq}. \end{aligned}$$

(3.45)

Note that in the Einstein-Hilbert limit the last two terms adds up to yield $-\sqrt{-g}L_{quad}$. Then consider the other combination which can be expressed as

$$\begin{aligned} \tilde{N}^{c}_{ab}\partial _{c}\tilde{f}^{ab}= & {} \left( Q_{ap}^{cq}\Gamma ^{p}_{qb} +Q_{bp}^{cq}\Gamma ^{p}_{qa}\right) \partial _{c}\left( \sqrt{-g}g^{ab}\right) \nonumber \\= & {} \sqrt{-g}\left( Q_{ap}^{cq}\Gamma ^{p}_{qb}+Q_{bp}^{cq}\Gamma ^{p}_{qa}\right) \left( \partial _{c}g^{ab}+g^{ab}\Gamma ^{p}_{cp} \right) \nonumber \\= & {} 2\sqrt{-g}Q_{ap}^{cq}\Gamma ^{p}_{qb}\partial _{c}g^{ab} +2\sqrt{-g}Q_{p}^{~bqc}\Gamma ^{p}_{qb}\Gamma ^{m}_{cm} \nonumber \\= & {} 2\sqrt{-g}Q_{p}^{~bcq}\Gamma ^{l}_{bc}\Gamma ^{p}_{ql} +2\sqrt{-g}Q_{p}^{~bqc}\Gamma ^{p}_{qb}\Gamma ^{m}_{cm} -2\sqrt{-g}g^{bm}Q_{ap}^{cq}\Gamma ^{p}_{qb}\Gamma ^{a}_{cm} \nonumber \\= & {} \sqrt{-g}L_{quad}+2\sqrt{-g}Q_{p}^{~bqc}\Gamma ^{p}_{qb}\Gamma ^{m}_{cm} -2\sqrt{-g}g^{bm}Q_{ap}^{cq}\Gamma ^{p}_{qb}\Gamma ^{a}_{cm}. \end{aligned}$$

(3.46)

In the Einstein-Hilbert limit the above term leads to $2\sqrt{-g}L_{quad}$. Next we will derive similar relations which actually behaves as conjugate variables, with the identification, $p\equiv 2\sqrt{-g}Q_{a}^{~bcd}$ and $q\equiv \Gamma ^{a}_{bc}$. Then the respective $p\partial q$ and $q\partial p$ expressions are given in the following results:

$$\begin{aligned} 2\sqrt{-g}Q_{e}^{~bdc}\partial _{c}\Gamma ^{e}_{bd}= & {} \sqrt{-g}Q_{e}^{~bdc} \left( \partial _{c}\Gamma ^{e}_{bd}-\partial _{d}\Gamma ^{e}_{bc} \right) \nonumber \\= & {} \sqrt{-g}Q_{e}^{~bdc}R^{e}_{~bcd}-2\sqrt{-g}Q_{e}^{~bdc}\Gamma ^{e}_{mc}\Gamma ^{m}_{bd} \nonumber \\= & {} -\sqrt{-g}Q_{e}^{~abc}R^{e}_{~abc}-\sqrt{-g}L_{quad} \end{aligned}$$

(3.47)

and

$$\begin{aligned} \Gamma ^{d}_{be}\partial _{c}\left( 2\sqrt{-g}Q_{d}^{~bec} \right)&= 2\sqrt{-g}\Gamma ^{d}_{be}\partial _{c}Q_{d}^{~bec}+2\Gamma ^{d}_{be}Q_{d}^{~bec}\partial _{c}\sqrt{-g} \nonumber \\&= 2\sqrt{-g}\Gamma ^{d}_{be}\left( \Gamma ^{a}_{cd}Q_{a}^{~bec}-\Gamma ^{b}_{ca}Q_{d}^{~aec} -\Gamma ^{c}_{ca}Q_{d}^{~bea} \right) \nonumber \\&+2\Gamma ^{d}_{be}Q_{d}^{~bec}\partial _{c}\sqrt{-g} \nonumber \\&=2\sqrt{-g}L_{quad} \end{aligned}$$

(3.48)

Finally we will explicitly demonstrate the result of one derivative used in the text for Lanczos-Lovelock Lagrangian:

$$\begin{aligned} \frac{\partial \left( \sqrt{-g}L\right) }{\partial \left( \partial _{l}\Gamma ^{u}_{vw}\right) }= & {} m\sqrt{-g}\delta ^{aba_{2}b_{2}\ldots a_{m}b_{m}}_{cdc_{2}d_{2}\ldots c_{m}d_{m}} \frac{\partial R^{cd}_{ab}}{\partial \left( \partial _{l}\Gamma ^{u}_{vw}\right) } R^{c_{2}d_{2}}_{a_{2}b_{2}}\ldots R_{a_{m}b_{m}}^{c_{m}d_{m}} \nonumber \\= & {} m\sqrt{-g}\delta ^{aba_{2}b_{2}\ldots a_{m}b_{m}}_{cdc_{2}d_{2}\ldots c_{m}d_{m}} \left[ g^{dp}\delta ^{c}_{u}\delta ^{v}_{p}\left( \delta ^{l}_{a}\delta ^{w}_{b}- \delta ^{l}_{b}\delta ^{w}_{a}\right) \right] R^{c_{2}d_{2}}_{a_{2}b_{2}}\ldots R_{a_{m}b_{m}}^{c_{m}d_{m}} \nonumber \\= & {} 2m\sqrt{-g}g^{dv}Q^{lw}_{ud}=mU_{u}^{~vlw} \end{aligned}$$

(3.49)