Appendix A
In this appendix, we shall present the supplementary material for this chapter.
A.1 Detailed Expressions Regarding First Law
Let us start with evaluating the following expression in GNC coordinates introduced in the main text, which leads to
$$\begin{aligned} \frac{1}{2}\Big (E^{u}_{u}&+E^{r}_{r}\Big )=E^{u}_{u}=E^{r}_{r} \nonumber \\&=-\frac{1}{2}\frac{1}{16\pi }\frac{1}{2^{m}}\delta ^{ra_{1}b_{1}\ldots a_{m}b_{m}}_{rc_{1}d_{1}\ldots c_{m}d_{m}}R^{c_{1}d_{1}}_{a_{1}b_{1}}\ldots R^{c_{m}d_{m}}_{a_{m}b_{m}} \nonumber \\&=-\frac{m}{16\pi }\frac{1}{2^{m-1}}\delta ^{ruPA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{ruQC_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{uQ}_{uP}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&-\frac{m(m-1)}{16\pi }\frac{1}{2^{m-1}}\delta ^{ruPA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{ruQC_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{QC_{1}}_{uP}R^{uD_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&-\frac{1}{16\pi }\frac{1}{2^{m+1}}\delta ^{rA_{1}B_{1}\ldots A_{m}B_{m}}_{rC_{1}D_{1}\ldots C_{m}D_{m}}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m}D_{m}}_{A_{m}B_{m}} \end{aligned}$$
(6.41)
Then we obtain:
$$\begin{aligned} T^{r}_{r}&=2E^{r}_{r}=-\frac{m}{8\pi }\frac{1}{2^{m-1}}\delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{uQ}_{uP}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&-\frac{m(m-1)}{8\pi }\frac{1}{2^{m-1}}\delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{QC_{1}}_{uP}R^{uD_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&-\frac{1}{16\pi }\frac{1}{2^{m}}\delta ^{A_{1}B_{1}\ldots A_{m}B_{m}}_{C_{1}D_{1}\ldots C_{m}D_{m}}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m}D_{m}}_{A_{m}B_{m}} \end{aligned}$$
(6.42)
Now we have the following expression for components of Riemann tensor as:
$$\begin{aligned} R^{uQ}_{uP}&=-\frac{1}{2}q^{QE}\partial _{u}\partial _{r}q_{PE}-\frac{1}{2}q^{QE}\partial _{P}\beta _{E}-\frac{1}{2}\alpha q^{QE}\partial _{r}q_{PE}-\frac{1}{4}\beta ^{Q}\beta _{P} \nonumber \\&+\frac{1}{4}\left( q^{QE}\partial _{r}q_{PF}\right) \left( q^{FL}\partial _{u}q_{EL}\right) +\frac{1}{2}q^{QE}\beta _{A}\hat{\Gamma }^{A}_{EP} \end{aligned}$$
(6.43a)
$$\begin{aligned} R^{AM}_{CD}&=\hat{R}^{AM}_{CD}-\frac{1}{4}q^{AE}q^{MB}\Big \lbrace \partial _{u}q_{CE}\partial _{r}q_{BD}+\partial _{r}q_{CE}\partial _{u}q_{BD}-\left( C\leftrightarrow D\right) \Big \rbrace \end{aligned}$$
(6.43b)
$$\begin{aligned} R^{uN}_{CD}&=-\frac{1}{2}q^{MN}\partial _{r}\partial _{C}q_{MD}-\frac{1}{4}\beta _{C}q^{MN}\partial _{r}q_{MD}-\frac{1}{2}\left( q^{MN}\partial _{r}q_{CE}\right) \hat{\Gamma } ^{E}_{~MD}-\left( C\leftrightarrow D\right) \end{aligned}$$
(6.43c)
$$\begin{aligned} R^{AB}_{uC}&=q^{BD}\Big [\partial _{u}\hat{\Gamma } ^{A}_{~DC}-\frac{1}{2}q^{AE}\partial _{C}\partial _{u}q_{DE}-\frac{1}{2}\partial _{C}q^{AE}\partial _{u}q_{DE}+\frac{1}{4}\beta ^{A}\partial _{u}q_{CD} \nonumber \\&+\frac{1}{2}q^{AF}\partial _{u}q_{EF}\hat{\Gamma } ^{E}_{~CD}-\frac{1}{4}\beta _{D}q^{AE}\partial _{u}q_{EC}-\frac{1}{2}q^{EF}\partial _{u}q_{FD}\hat{\Gamma } ^{A}_{~CE}\Big ] \end{aligned}$$
(6.43d)
where \(\hat{A}\) denotes an object A constructed solely from the transverse metric \(q_{AB}\). Note that for \(\partial _{u}g_{AB}=0\), we have:
$$\begin{aligned} R^{uQ}_{uP}&=-\frac{1}{2}q^{QE}\partial _{P}\beta _{E}-\frac{1}{2}\alpha q^{QE}\partial _{r}q_{PE}-\frac{1}{4}\beta ^{Q}\beta _{P}+\frac{1}{2}q^{QE}\beta _{A}\hat{\Gamma }^{A}_{EP} \end{aligned}$$
(6.44a)
$$\begin{aligned} R^{AM}_{CD}&=\hat{R}^{AM}_{CD} \end{aligned}$$
(6.44b)
$$\begin{aligned} R^{uN}_{CD}&=-\frac{1}{2}q^{MN}\partial _{r}\partial _{C}q_{MD}-\frac{1}{4}\beta _{C}q^{MN}\partial _{r}q_{MD}-\frac{1}{2}\left( q^{MN}\partial _{r}q_{CE}\right) \hat{\Gamma } ^{E}_{~MD}-\left( C\leftrightarrow D\right) \end{aligned}$$
(6.44c)
$$\begin{aligned} R^{AB}_{uC}&=0 \end{aligned}$$
(6.44d)
Thus we finally arrive at the following expression:
$$\begin{aligned} T^{r}_{r}&=-\frac{m}{8\pi }\frac{1}{2^{m-1}}\delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}\left( -\frac{1}{2}\alpha q^{QE}\partial _{r}q_{PE}\right) R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&-\frac{m}{8\pi }\frac{1}{2^{m-1}}\delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}\Big [-\frac{1}{2}q^{QE}\partial _{u}\partial _{r}q_{PE}-\frac{1}{2}q^{QE}\partial _{P}\beta _{E}-\frac{1}{4}\beta ^{Q}\beta _{P} \nonumber \\&+\frac{1}{4}\left( q^{QE}\partial _{r}q_{PF}\right) \left( q^{FL}\partial _{u}q_{EL}\right) +\frac{1}{2}q^{QE}\beta _{A}\hat{\Gamma }^{A}_{EP}\Big ]R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&-\frac{m(m-1)}{8\pi }\frac{1}{2^{m-1}}\delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{QC_{1}}_{uP}R^{uD_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&-\frac{1}{16\pi }\frac{1}{2^{m}}\delta ^{A_{1}B_{1}\ldots A_{m}B_{m}}_{C_{1}D_{1}\ldots C_{m}D_{m}}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m}D_{m}}_{A_{m}B_{m}} \end{aligned}$$
(6.45)
which can be simplified and finally leads to the following expression:
$$\begin{aligned} T^{r}_{r}&=2E^{r}_{r}=\left( E^{u}_{u}+E^{r}_{r}\right) \nonumber \\&=\frac{m}{8}\frac{1}{2^{m-1}}\left( \frac{\alpha }{2\pi }\right) \left( \delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}}\right) \left( q^{QE}\partial _{r}q_{PE}\right) \nonumber \\&-\frac{m}{8\pi }\frac{1}{2^{m-1}}\delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}\Big [-\frac{1}{2}q^{QE}\partial _{u}\partial _{r}q_{PE}-\frac{1}{2}q^{QE}\partial _{P}\beta _{E}-\frac{1}{4}\beta ^{Q}\beta _{P} \nonumber \\&+\frac{1}{4}\left( q^{QE}\partial _{r}q_{PF}\right) \left( q^{FL}\partial _{u}q_{EL}\right) +\frac{1}{2}q^{QE}\beta _{A}\hat{\Gamma }^{A}_{EP}\Big ]R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&-\frac{m(m-1)}{8\pi }\frac{1}{2^{m-1}}\delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{QC_{1}}_{uP}R^{uD_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&-\frac{1}{16\pi }\frac{1}{2^{m}}\delta ^{A_{1}B_{1}\ldots A_{m}B_{m}}_{C_{1}D_{1}\ldots C_{m}D_{m}}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m}D_{m}}_{A_{m}B_{m}} \end{aligned}$$
(6.46)
This is the expression used in the text. We also have entropy density to be:
$$\begin{aligned} s&=4\pi m\sqrt{q}\mathcal {L}_{m-1}^{(D-2)} \nonumber \\&=4\pi m \sqrt{q}\left( \frac{1}{16\pi }\frac{1}{2^{m-1}}\delta ^{A_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{C_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \right) \end{aligned}$$
(6.47)
Then under variation along the radial coordinate i.e. along \(k^{a}\) parametrized by \(\lambda \) we have:
$$\begin{aligned} \delta _{\lambda }s&=4\pi m \left( \frac{1}{2}q^{AB}\delta _{\lambda }q_{AB}\right) \sqrt{q}\mathcal {L}_{m-1}^{(D-2)} \nonumber \\&-4\pi m \sqrt{q}\left( \frac{m-1}{16\pi }\frac{1}{2^{m-1}}\delta ^{A_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{C_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{C_{1}A}_{A_{1}B_{1}}q^{D_{1}B}\delta _{\lambda }q_{AB}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \right) \nonumber \\&=-4\pi m \sqrt{q}\delta _{\lambda }q_{AB}\Big (-\frac{1}{2}q^{AB}\mathcal {L}_{m-1}^{(D-2)} \nonumber \\&+\frac{m-1}{16\pi }\frac{1}{2^{m-1}}q^{BD_{1}}\delta ^{A_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{D_{1}C_{1}\ldots C_{m-1}D_{m-1}}R^{AC_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}}\Big ) \nonumber \\&=-4\pi m\sqrt{q}E^{AB}\delta _{\lambda }q_{AB} \end{aligned}$$
(6.48)
where we have:
$$\begin{aligned} E^{A}_{B}&=-\frac{1}{2}\delta ^{A}_{B}\mathcal {L}_{m-1}^{(D-2)}+\frac{m-1}{16\pi }\frac{1}{2^{(m-1)}}\delta ^{A_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{BC_{1}\ldots C_{m-1}D_{m-1}}R^{AC_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&=-\frac{1}{2}\frac{1}{16\pi }\frac{1}{2^{m-1}}\delta ^{AA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{BC_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \end{aligned}$$
(6.49)
Hence we obtain:
$$\begin{aligned} \delta _{\lambda }s&=-4\pi m\sqrt{q}\delta _{\lambda }q_{AB}\Big (-\frac{1}{2}\frac{1}{16\pi }\frac{1}{2^{m-1}}\delta ^{AA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{BC_{1}D_{1}\ldots C_{m-1}D_{m-1}} \nonumber \\&\times R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}}\Big ) \nonumber \\&=\frac{m}{8 2^{m-1}}\sqrt{q}\left( \delta ^{AA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{BC_{1}D_{1}\ldots C_{m-1}D_{m-1}} R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}}\right) q^{BC}\delta _{\lambda }q_{AC} \end{aligned}$$
(6.50)
Finally using Eq. (6.50) in Eq. (6.46) we obtain the most general expression for energy as
$$\begin{aligned} \delta _{\lambda }E&=\delta \lambda \int d\Sigma \Big \lbrace \frac{m}{8\pi }\frac{1}{2^{m-1}}\delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}\Big [-\frac{1}{2}q^{QE}\partial _{u}\partial _{r}q_{PE}-\frac{1}{2}q^{QE}\partial _{P}\beta _{E}-\frac{1}{4}\beta ^{Q}\beta _{P} \nonumber \\&+\frac{1}{4}\left( q^{QE}\partial _{r}q_{PF}\right) \left( q^{FL}\partial _{u}q_{EL}\right) +\frac{1}{2}q^{QE}\beta _{A}\hat{\Gamma }^{A}_{EP}\Big ]R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&+\frac{m(m-1)}{8\pi }\frac{1}{2^{m-1}}\delta ^{PA_{1}B_{1}\ldots A_{m-1}B_{m-1}}_{QC_{1}D_{1}\ldots C_{m-1}D_{m-1}}R^{QC_{1}}_{uP}R^{uD_{1}}_{A_{1}B_{1}}\ldots R^{C_{m-1}D_{m-1}}_{A_{m-1}B_{m-1}} \nonumber \\&+\frac{1}{16\pi }\frac{1}{2^{m}}\delta ^{A_{1}B_{1}\ldots A_{m}B_{m}}_{C_{1}D_{1}\ldots C_{m}D_{m}}R^{C_{1}D_{1}}_{A_{1}B_{1}}\ldots R^{C_{m}D_{m}}_{A_{m}B_{m}}\Big \rbrace \end{aligned}$$
(6.51)
where \(d\Sigma =d^{D-2}x\sqrt{q}\) is the integration measure on the null surface.
A.2 Various Identities Used in the Text Regarding Lanczos-Lovelock Gravity
In this subsection we will collect derivation of important identities used in the text while describing the generalization to Lanczos-Lovelock gravity. We will order the derivations as in text.
1.1 A.2.1 Gravitational Momentum and related derivations for Einstein-Hilbert Action
In this section we provide derivation to various identities used in Sect. 6.4.1. We start by giving the result for variation of Lanczos-Lovelock Lagrangian:
$$\begin{aligned} \delta \left( \sqrt{-g}L\right) =\sqrt{-g}E_{ab}\delta g^{ab} -\partial _{c}\left( 2\sqrt{-g}P_{p}^{~qrc}\delta \Gamma ^{p}_{qr}\right) \end{aligned}$$
(6.52)
The Noether current can be written as:
$$\begin{aligned} J^{a}(q)&=2\mathcal {R}^{a}_{b}q^{b}+2P_{p}^{~qra}\pounds _{q}\Gamma ^{p}_{qr} \nonumber \\&=2E^{a}_{b}q^{b}+2P_{p}^{~qra}\pounds _{q}\Gamma ^{p}_{qr}+Lq^{a} \nonumber \\&=2E^{a}_{b}q^{b}-P^{a}(q) \end{aligned}$$
(6.53)
Then multiplying both sides by the four velocity \(u_{a}\) and taking \(q_{a}=\xi _{a}\) we readily obtain:
$$\begin{aligned} -u_{a}P^{a}(\xi )&=u_{a}J^{a}(\xi )-2E^{ab}\xi _{a}u_{b} \nonumber \\&=D_{\alpha }\left( 2N\chi ^{\alpha }\right) -2NE^{ab}u_{a}u_{b} \end{aligned}$$
(6.54)
Let us now consider the variation of the gravitational momentum corresponding to \(q^{a}\). This has the following expression:
$$\begin{aligned} -\delta \left( \sqrt{-g}P^{a}\right)&=q^{a}\delta \left( \sqrt{-g}L\right) +\delta \left( 2\sqrt{-g}P_{p}^{~qra}\pounds _{q}\Gamma ^{p}_{qr}\right) \nonumber \\&=q^{a}\left( \sqrt{-g}E_{pq}\delta g^{pq}\right) -q^{a}\partial _{c}\left( 2\sqrt{-g}P_{p}^{~qrc}\delta \Gamma ^{p}_{qr} \right) \nonumber \\&+\pounds _{q}\left( 2\sqrt{-g}P_{p}^{~qra}\delta \Gamma ^{p}_{qr} \right) \nonumber \\&+\delta \left( 2\sqrt{-g}P_{p}^{~qra}\right) \pounds _{q}\Gamma ^{p}_{qr} -\pounds _{q}\left( 2\sqrt{-g}P_{p}^{qra}\right) \delta \Gamma ^{p}_{qr} \nonumber \\&=\sqrt{-g}E_{pq}\delta g^{pq}q^{a}+\sqrt{-g}\omega ^{a} +\partial _{c}\left( 2\sqrt{-g}P_{p}^{~qr[a}q^{c]}\delta \Gamma ^{p}_{qr}\right) \end{aligned}$$
(6.55)
where in arriving at the last line we have used the following relation:
$$\begin{aligned} \pounds _{q}Q^{a}-q^{a}\partial _{c}Q^{c}=\partial _{c}\left( q^{[c}Q^{a]}\right) \end{aligned}$$
(6.56)
for the tensor density \(Q^{c}=2\sqrt{-g}P_{p}^{~qrc}\delta \Gamma ^{p}_{qr}\). Also since \(q^{a}\) is a constant vector \(\delta \) and \(\pounds _{q}\) are assumed to commute. Also the object \(\omega ^{a}\) is defined as,
$$\begin{aligned} \sqrt{-g}\omega ^{a}\left( \delta , \pounds _{q}\right) =\delta \left( 2\sqrt{-g}P_{p}^{~qra}\right) \pounds _{q}\Gamma ^{p}_{qr} -\pounds _{q}\left( 2\sqrt{-g}P_{p}^{~qra}\right) \delta \Gamma ^{p}_{qr} \end{aligned}$$
(6.57)
This is the result used in Eq. (6.25). Then we can write the above relation in terms of the Noether current as:
$$\begin{aligned} \delta \left( \sqrt{-g}J^{a}-2\sqrt{-g}E^{a}_{b}q_{b}\right)&=\sqrt{-g}E_{pq}\delta g^{pq}q^{a}+\sqrt{-g}\omega ^{a} \nonumber \\&+\partial _{c}\left( 2\sqrt{-g}P_{p}^{~qr[a}q^{c]}\delta \Gamma ^{p}_{qr}\right) \end{aligned}$$
(6.58)
The above relation can also be written using the Noether potential as:
$$\begin{aligned} \partial _{b}\Bigg \lbrace \delta \left( \sqrt{-g}J^{ab}\right)&-2\sqrt{-g}P_{p}^{~qr[a}q^{b]}\delta \Gamma ^{p}_{qr}\Bigg \rbrace =\sqrt{-g}E_{pq}\delta g^{pq}q^{a}+\sqrt{-g}\omega ^{a} \nonumber \\&+2\delta \left( \sqrt{-g}E^{ab}q_{b}\right) \end{aligned}$$
(6.59)
While for on-shell (i.e., when \(E_{ab}=0\)) we have the following relations:
$$\begin{aligned} -\delta \left( \sqrt{-g}P^{a}\right)&=\sqrt{-g}\omega ^{a} +\partial _{c}\left( 2\sqrt{-g}P_{p}^{~qr[a}q^{c]}\delta \Gamma ^{p}_{qr}\right) \end{aligned}$$
(6.60)
$$\begin{aligned} \partial _{b}\left\{ \delta \left( \sqrt{-g}J^{ab}\right) -2\sqrt{-g}P_{p}^{~qr[a}q^{b]}\delta \Gamma ^{p}_{qr}\right\}&=\sqrt{-g}\omega ^{a} +2\delta \left( \sqrt{-g}E^{ab}q_{b}\right) \end{aligned}$$
(6.61)
Then integrating the second equation over volume \(\mathcal {R}\) with \(d^{D-1}x\sqrt{h}\) as integration measure and \(q^{a}=\zeta ^{a}=Nu^{a}+N^{a}\) we arrive at:
$$\begin{aligned} \delta&\int _{\mathcal {R}}d^{D-1}x\sqrt{h}\left( 2u_{a}E^{ab}\zeta _{b}\right) \nonumber \\&=\int _{\mathcal {R}}d^{D-1}x\partial _{b}\Big \lbrace \delta \left[ \sqrt{h}u_{a}J^{ab}(\zeta )\right] -2\sqrt{h}u_{a}\left( NP_{p}^{~qr[a}u^{b]}+P_{p}^{~qr[a}N^{b]}\right) \delta \Gamma ^{p}_{qr}\Big \rbrace \nonumber \\&-\int _{\mathcal {R}}d^{D-1}x\sqrt{h}u_{a}\omega ^{a}\left( \delta ,\pounds _{q}\right) \nonumber \\&=\int _{\mathcal {R}}d^{D-1}x\partial _{b}\Big \lbrace \delta \left[ \sqrt{h}u_{a}J^{ab}(\zeta )\right] -2\sqrt{h}\left( Nh^{b}_{a}P_{p}^{~qra}+P_{p}^{~qr[a}N^{b]}\right) \delta \Gamma ^{p}_{qr}\Big \rbrace \nonumber \\&-\int _{\mathcal {R}}d^{D-1}x\sqrt{h}u_{a}\omega ^{a}\left( \delta ,\pounds _{q}\right) \end{aligned}$$
(6.62)
Then we want the variation of the Hamiltonian obtained by contracting the momentum along the four velocity \(u_{a}\), such that:
$$\begin{aligned} -\sqrt{h}u_{a}P^{a}(\xi )=t_{a}\sqrt{-g}P^{a}(\xi ) \end{aligned}$$
(6.63)
where the vector \(t_{a}=-u_{a}/N\) in the coordinate system under consideration. Then varying the above expression (noting that variation of \(t_{a}\) vanishes) we obtain
$$\begin{aligned} -\delta \left[ \sqrt{h}u_{a}P^{a}(\xi )\right]&=t_{a}\delta \left[ \sqrt{-g}P^{a}(\xi )\right] \nonumber \\&=-t_{a}\left[ \sqrt{-g}E_{pq}\delta g^{pq}\xi ^{a}+\sqrt{-g}\omega ^{a} +\partial _{c}\left( 2\sqrt{-g}P_{p}^{~qr[a}\xi ^{c]}\delta \Gamma ^{p}_{qr}\right) \right] \nonumber \\&=\sqrt{h}u_{a}\omega ^{a}-\sqrt{-g}E_{pq}\delta g^{pq} +\partial _{c}\left[ 2\sqrt{-g}u_{a}P_{p}^{~qr[a}u^{c]}\delta \Gamma ^{p}_{qr}\right] \nonumber \\&=\sqrt{h}u_{a}\omega ^{a}-\sqrt{-g}E_{pq}\delta g^{pq} +\partial _{c}\left[ 2\sqrt{-g}h^{c}_{a}P_{p}^{~qra}\delta \Gamma ^{p}_{qr}\right] \end{aligned}$$
(6.64)
where in the second line we have used the standard trick in order to get \(u_{a}\) and in the last line we have used the relation:
$$\begin{aligned} u_{a}P_{p}^{~qr[a}u^{c]}=h^{c}_{a}P_{p}^{~qra} \end{aligned}$$
(6.65)
Defining the gravitational Hamiltonian as:
$$\begin{aligned} \mathcal {H}_{grav}=-\int d^{D-1}x \sqrt{h}u_{a}P^{a}(\xi ) \end{aligned}$$
(6.66)
its variation can be obtained readily from Eq. (6.64) as:
$$\begin{aligned} \delta \mathcal {H}_{grav}=\int d^{D-1}x\sqrt{h}u_{a}\omega ^{a}-\int d^{D-1}x \sqrt{-g}E_{pq}\delta g^{pq}+\int d^{D-2}x~2r_{c}\sqrt{q}P_{p}^{~qrc}\delta \Gamma ^{p}_{qr} \end{aligned}$$
(6.67)
where in order to obtain the last term we have used the result that \(r_{c}h^{c}_{a}=r_{a}\) and \(\sqrt{-g}=N\sqrt{h}\). The above results are true for arbitrary variations. Applying it to Lie variation along \(\xi ^{a}\) we arrive at the following form for Eq. (6.64) as:
$$\begin{aligned} -\pounds _{\xi }\left[ \sqrt{h}u_{a}P^{a}(\xi )\right] =2\sqrt{-g}E_{pq}\nabla ^{p}\xi ^{q}+\partial _{c}\left[ 2\sqrt{-g}h^{c}_{a}P_{p}^{~qra}\pounds _{\xi } \Gamma ^{p}_{qr}\right] \end{aligned}$$
(6.68)
In arriving at the above result we have used the relations: \(\pounds _{\xi }g^{ab}=-(\nabla ^{a}\xi ^{b}+\nabla ^{b}\xi ^{a})\) and \(\omega ^{a}(\pounds _{\xi },\pounds _{\xi })=0\). Now using Bianchi identity \(\nabla _{a}E^{ab}=0\) we arrive at:
$$\begin{aligned} -\pounds _{\xi }\left[ \sqrt{h}u_{a}P^{a}(\xi )\right] =\partial _{c}\left[ 2\sqrt{-g}\left( E^{cd}\xi _{d}+h^{c}_{a}P_{p}^{~qra}\pounds _{\xi } \Gamma ^{p}_{qr}\right) \right] \end{aligned}$$
(6.69)
This is the relation used to arrive at the results in the main text.
1.2 A.2.2 Characterizing Null Surfaces
Let us now try to generalize the result for null surfaces to Lovelock gravity with the null vector \(\ell _{a}\) such that \(\ell ^{2}=0\) everywhere. For that purpose, we start with the combination:
$$\begin{aligned} \mathcal {R}_{ab}\ell ^{a}\ell ^{b}&=R_{apqr}P_{b}^{~pqr}\ell ^{a}\ell ^{b} \nonumber \\&=P^{bpqr}\ell _{b}\left( R_{apqr}\ell ^{a}\right) \nonumber \\&=-P^{bpqr}\ell _{b}\left( \nabla _{q}\nabla _{r}\ell _{p}-\nabla _{r}\nabla _{q}\ell _{p}\right) \nonumber \\&=-2P^{bpqr}\ell _{b}\nabla _{q}\nabla _{r}\ell _{p} \nonumber \\&=\nabla _{q}\left( -2P^{bpqr}\ell _{b}\nabla _{r}\ell _{p}\right) -\mathcal {S} \end{aligned}$$
(6.70)
where we have defined, the entropy density as:
$$\begin{aligned} \mathcal {S}=-2P^{bpqr}\nabla _{q}\ell _{b}\nabla _{r}\ell _{p} \end{aligned}$$
(6.71)
The Einstein-Hilbert limit can be obtained easily leading to:
$$\begin{aligned} 16\pi \mathcal {S}&=-\left( g^{bq}g^{pr}-g^{br}g^{pq}\right) \nabla _{q}\ell _{b}\nabla _{r}\ell _{p} \nonumber \\&=\nabla _{a}\ell ^{b}\nabla _{b}\ell ^{a}-\left( \nabla _{i}\ell ^{i}\right) ^{2} \end{aligned}$$
(6.72)
as well as,
$$\begin{aligned} 32\pi P^{bpqr}\ell _{b}\nabla _{r}\ell _{p}&=\left( g^{bq}g^{pr}-g^{br}g^{pq}\right) \ell _{b}\nabla _{r}\ell _{p} \nonumber \\&=\ell ^{q}\nabla _{r}\ell ^{r}-\ell ^{r}\nabla _{r}\ell ^{q} \nonumber \\&=\Theta \ell ^{q} \end{aligned}$$
(6.73)
However in Lanczos-Lovelock gravity, the combination, \(2P^{bpqr}\ell _{b}\nabla _{r}\ell _{p}\) cannot be written as, \(\phi \ell ^{q}\), for arbitrary \(\phi \), since \(2P^{bpqr}\ell _{q}\ell _{b}\nabla _{r}\ell _{p}\ne 0\). Next let us consider the object, \(\ell _{a}J^{a}\left( \ell \right) \), for which we define, \(\ell _{a}=A\nabla _{a}B\). Then in Lanczos-Lovelock gravity, we arrive at:
$$\begin{aligned} \frac{1}{A}\ell _{a}J^{a}\left( \ell \right)&=\nabla _{b}\left[ 2P^{abcd}\ell _{a}\ell _{d}\nabla _{c}A\frac{1}{A^{2}} \right] \nonumber \\&=\frac{1}{A}\nabla _{b}\left[ 2P^{abcd}\ell _{a}\ell _{d}\nabla _{c}\ln A \right] -\frac{1}{A}\left[ 2P^{abcd}\ell _{a}\ell _{d}\nabla _{c}\ln A \nabla _{b}\ln A \right] \end{aligned}$$
(6.74)
Let us now expand \(\nabla _{c}\ln A\) in canonical null basis, such that we obtain,
$$\begin{aligned} \nabla _{c}\ln A=-\kappa k_{c}+A\ell _{c}+B_{A}e^{A}_{c} \end{aligned}$$
(6.75)
Note that we obtain, \(\ell ^{c}\nabla _{c}\ln A=\kappa \). In a similar fashion, we can expand, \(2P^{abcd}\ell _{a}\ell _{d}\nabla _{c}\ln A\) in the following manner,
$$\begin{aligned} 2P^{abcd}\ell _{a}\ell _{d}\nabla _{c}\ln A=P\ell ^{b}+Qk^{b}+R^{A}e^{b}_{A} \end{aligned}$$
(6.76)
It is evident that \(Q=-2P^{abcd}\ell _{a}\ell _{d}\ell _{b}\nabla _{c}\ln A=0\), due to antisymmetry of \(P^{abcd}\) in the first two indices. Then it turns out that,
$$\begin{aligned} P=-2P^{abcd}\ell _{a}k_{b}\ell _{d}\nabla _{c}\ln A\equiv \mathcal {K} \end{aligned}$$
(6.77)
It is obvious from the Einstein-Hilbert limit, that
$$\begin{aligned} 16\pi \mathcal {K}&=-\left( g^{ac}g^{bd}-g^{ad}g^{bc}\right) \ell _{a}k_{b}\ell _{d}\nabla _{c}\ln A \nonumber \\&=-\left( g^{ac}g^{bd}-g^{ad}g^{bc}\right) \ell _{a}k_{b}\ell _{d}\left( -\kappa k_{c}+B_{A}e^{A}_{c}\right) =\kappa \end{aligned}$$
(6.78)
Also in the expansion for, \(\nabla _{c}\ln A\), we obtain, \(B_{A}\) is completely arbitrary. Then we can use \(B_{A}\) such that the following relation: \(B_{Q}P^{abcd}\ell _{a}\ell _{d}e^{A}_{b}e^{Q}_{c}=\kappa P^{abcd}\ell _{a}k_{c}\ell _{d}e^{A}_{b}\) is satisfied, such that, \(R^{A}\) vanishes. Thus we obtain,
$$\begin{aligned} 2P^{abcd}\ell _{a}\ell _{d}\nabla _{c}\ln A=\mathcal {K}\ell ^{b} \end{aligned}$$
(6.79)
Again, we get,
$$\begin{aligned} 2P^{abcd}\ell _{a}\ell _{d}\nabla _{c}\ln A \nabla _{b}\ln A=\mathcal {K}\ell ^{b}\nabla _{b}\ln A =\kappa \mathcal {K} \end{aligned}$$
(6.80)
Thus, finally we arrive at the following result:
$$\begin{aligned} \ell _{a}J^{a}\left( \ell \right) =\nabla _{a}\left( \mathcal {K}\ell ^{a}\right) -\kappa \mathcal {K} \end{aligned}$$
(6.81)
Again by considering derivative on the null surface, we obtain,
$$\begin{aligned} D_{a}\left( \mathcal {K}\ell ^{a}\right)&=\left( g^{ab}+\ell ^{a}k^{b}+\ell ^{b}k^{a}\right) \nabla _{a}\left( \mathcal {K}\ell _{b}\right) \nonumber \\&=\nabla _{a}\left( \mathcal {K}\ell ^{a}\right) +k^{b}\ell ^{a}\nabla _{a}\left( \mathcal {K}\ell _{b}\right) \nonumber \\&=\nabla _{a}\left( \mathcal {K}\ell ^{a}\right) -\kappa \mathcal {K}-\ell ^{a}\nabla _{a}\mathcal {K} \end{aligned}$$
(6.82)
Thus the Noether current contraction can also be written as:
$$\begin{aligned} \ell _{a}J^{a}\left( \ell \right) =D_{a}\left( \mathcal {K}\ell ^{a}\right) +\dfrac{d\mathcal {K}}{d\lambda } \end{aligned}$$
(6.83)
