Null Surface Geometry and Associated Thermodynamics

Abstract

The emergent gravity paradigm interprets the gravitational field equations as describing the thermodynamic limit of the underlying statistical mechanics of the microscopic degrees of freedom associated with the spacetime structure. The connection is established by attributing a heat density to the null surfaces. The explicit form of the entropy functional determines the nature of the gravitational theory. We explore the consequences of this paradigm for an arbitrary null surface and highlight the key thermodynamic interpretations of various geometrical quantities. In particular, we show that: three distinct projections of gravitational momentum related to an arbitrary null surface in the spacetime lead to three different equations, all of which have thermodynamic interpretation. The first one reduces to a Navier-Stokes equation for the transverse drift velocity. The second can be written as a thermodynamic identity, while the third one describes the time evolution of the null surface in terms of suitably defined surface and bulk degrees of freedom.

Appendices

Appendix A

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

A.1 General Analysis Regarding Null Surfaces

We will start with a null vector $\ell _{a}=A\nabla _{a}B$, which satisfies the condition $\ell ^{2}=0$ only over a single surface, which is the null surface under our consideration. Then we obtain

$$\begin{aligned} \ell ^{a}\nabla _{a}\ell ^{b}=\kappa \ell ^{b} \end{aligned}$$

(7.48)

where we have the following expression for $\kappa$

$$\begin{aligned} \kappa =\ell ^{a}\partial _{a}\ln A+\tilde{\kappa };\qquad \tilde{\kappa }=-\frac{1}{2}k^{a}\partial _{a}\ell ^{2} \end{aligned}$$

(7.49)

The last relation defining $\tilde{\kappa }$ can also be written as $\nabla _{a}\ell ^{2}=2\tilde{\kappa }\ell _{a}$. The derivation of the result goes as follows, let us expand $\nabla _{b}\ell ^{2}$ in canonical null basis, i.e., $\nabla _{a}\ell ^{2}=C\ell _{a}+Dk_{a}+E_{A}e^{A}_{a}$. Then both $E_{A}=e^{a}_{A}\nabla _{a}\ell ^{2}$ and $D=-\ell ^{a}\nabla _{a}\ell ^{2}$ vanishes, since variation of $\ell ^{2}$ along the null surface vanishes. This shows that the only non-zero component of $\nabla _{a}\ell ^{2}$ is along $\ell _{a}$. Then it turns out that [16]

$$\begin{aligned} \nabla _{i}\ell ^{i}=\Theta +\kappa +\tilde{\kappa } \end{aligned}$$

(7.50)

where, $\Theta =q^{ma}q_{mb}\nabla _{a}\ell ^{b}$. Note that the term $\tilde{\kappa }$ enters the picture as $\ell ^{2}= 0$ only on the null surface. With this setup let us now find out $R_{ab}\ell ^{a}\ell ^{b}$ in detail, which leads to,

$$\begin{aligned} R_{ab}\ell ^{a}\ell ^{b}&=\ell ^{j}\left( \nabla _{i}\nabla _{j}\ell ^{i}-\nabla _{j}\nabla _{i}\ell ^{i}\right) \nonumber \\&=\nabla _{i}\left( \ell ^{j}\nabla _{j}\ell ^{i}\right) -\nabla _{j}\left( \ell ^{j}\nabla _{i}\ell ^{i}\right) -\nabla _{i}\ell ^{j}\nabla _{j}\ell ^{i}+\left( \nabla _{i}\ell ^{i}\right) ^{2} \nonumber \\&=\nabla _{i}\left( \ell ^{j}\nabla _{j} \ell ^{i}-\ell ^{i}\nabla _{j}\ell ^{j}\right) -\left( \nabla _{i}\ell ^{j}\nabla _{j}\ell ^{i}-\left( \nabla _{i}\ell ^{i}\right) ^{2}\right) \end{aligned}$$

(7.51)

However in general, $\ell ^{j}\nabla _{j}\ell ^{i}=\kappa \ell ^{i}$ is not true, it only holds on the null surface (when $\ell ^{2}=0$ everywhere this relation is also true everywhere). Since we were doing the calculation for the most general case, $\ell ^{2}\ne 0$ in the above expression we cannot substitute $\ell ^{j}\nabla _{j}\ell ^{i}=\kappa \ell ^{i}$, since it appears inside the derivative. Thus for the special case when $\ell ^{2}=0$ everywhere, we will arrive at the following result

$$\begin{aligned} R_{ab}\ell ^{a}\ell ^{b}=-\nabla _{i}\left( \Theta \ell ^{i}\right) -\left( \nabla _{i}\ell ^{j}\nabla _{j}\ell ^{i}-\left( \nabla _{i}\ell ^{i}\right) ^{2}\right) \end{aligned}$$

(7.52)

In order to simplify things quiet a bit we will compute the last term $\left( \nabla _{i}\ell ^{j}\nabla _{j}\ell ^{i}-\left( \nabla _{i}\ell ^{i}\right) ^{2}\right)$ which we designate by $\mathcal {S}$. Then we start by calculating the following object on the null surface

$$\begin{aligned} \Theta ^{ab}&=q^{a}_{m}q^{b}_{n}\nabla ^{m}\ell ^{n} \nonumber \\&=\left( \delta ^{a}_{m}+\ell ^{a}k_{m}+k^{a}\ell _{m}\right) \left( \delta ^{b}_{n}+\ell ^{b}k_{n}+k^{b}\ell _{n}\right) \nabla ^{m}\ell ^{n} \nonumber \\&=\nabla ^{a}\ell ^{b}+\ell ^{a}k_{m}\nabla ^{m}\ell ^{b}+\kappa k^{a}\ell ^{b}+\ell ^{b}k_{n}\nabla ^{a}\ell ^{n}+\ell ^{a}\ell ^{b}k_{m}k_{n}\nabla ^{m}\ell ^{n} \nonumber \\&-\kappa k^{a}\ell ^{b}+\tilde{\kappa }\ell ^{a}k^{b}-\tilde{\kappa }\ell ^{a}k^{b}+\kappa k^{a}k^{b}\ell ^{2} \nonumber \\&=\nabla ^{a}\ell ^{b}+\ell ^{a}k_{m}\nabla ^{m}\ell ^{b}+\ell ^{b}k_{n}\nabla ^{a}\ell ^{n}+\ell ^{a}\ell ^{b}k_{m}k_{n}\nabla ^{m}\ell ^{n} \end{aligned}$$

(7.53)

In arriving at the third line we have used the following results: $\ell ^{a}\nabla _{a}\ell ^{i}=\kappa \ell ^{i}$ and $\ell _{a}\nabla _{b}\ell ^{a}=\tilde{\kappa }\ell _{b}$. Then we can reverse the above equation leading to

$$\begin{aligned} \nabla _{a}\ell _{b}=\Theta _{ab}-\ell _{a}k^{m}\nabla _{m}\ell _{b}-\ell _{b}k_{n}\nabla _{a}\ell ^{n} -\ell _{a}\ell _{b}k_{m}k_{n}\nabla ^{m}\ell ^{n} \end{aligned}$$

(7.54)

From the above equation we can derive two very important identity:

$$\begin{aligned} \left( \nabla _{a}\ell _{b}\right) \left( \nabla ^{a}\ell ^{b}\right)&= \Theta _{ab}\Theta ^{ab}+\ell ^{a}\ell _{b}k^{m}\nabla _{m}\ell ^{b}k^{n}\nabla _{a}\ell _{n} +\ell _{a}\ell ^{b}k_{n}\nabla ^{a}\ell ^{n}k^{m}\nabla _{m}\ell _{b} \nonumber \\&=\Theta _{ab}\Theta ^{ab}+2\left( \tilde{\kappa }k^{m}\ell _{m}\right) \left( \kappa \ell _{n}k^{n}\right) \nonumber \\&=\Theta _{ab}\Theta ^{ab}+2\kappa \tilde{\kappa } \end{aligned}$$

(7.55)

In the same spirit we will arrive at

$$\begin{aligned} \left( \nabla _{a}\ell _{b}\right) \left( \nabla ^{b}\ell ^{a}\right)&= \Theta _{ab}\Theta ^{ab}+\ell ^{a}\ell _{b}k_{n}\nabla _{a}\ell ^{n}k^{m}\nabla ^{b}\ell _{m} +\ell _{a}\ell ^{b}k^{m}\nabla _{m}\ell ^{a}k^{n}\nabla _{n}\ell _{b} \nonumber \\&=\Theta _{ab}\Theta ^{ab}+\left( \tilde{\kappa }k^{m}\ell _{m}\right) \left( \tilde{\kappa }k^{n}\ell _{n}\right) +\left( \kappa \ell _{n}k^{n}\right) \left( \kappa \ell _{m}k^{m}\right) \nonumber \\&=\Theta _{ab}\Theta ^{ab}+\kappa ^{2}+\tilde{\kappa }^{2} \end{aligned}$$

(7.56)

The extrinsic curvature for null surfaces, i.e., $\Theta _{ab}$ can be given a very natural interpretation. This essentially follows from [16]. There the expression for $\Theta _{ab}$ in terms of Lie variation of $q_{ab}$ along the null generator $\ell _{a}$ was obtained as,

$$\begin{aligned} \Theta _{ab}=\frac{1}{2}q^{m}_{a}q^{n}_{b}\pounds _{\ell }q_{mn} \end{aligned}$$

(7.57)

Now expanding out the Lie derivative term we obtain,

$$\begin{aligned} \pounds _{\ell }q_{mn}=\ell ^{i}\partial _{i}q_{mn}+q_{ma}\partial _{n}\ell ^{a}+q_{an}\partial _{m}\ell ^{a} \end{aligned}$$

(7.58)

Which on being substituted in Eq. (7.57) immediately leads to,

$$\begin{aligned} \Theta _{ab}=\frac{1}{2}q^{m}_{a}q^{n}_{b}\ell ^{i}\partial _{i}q_{mn}+\frac{1}{2}q_{ai}q^{n}_{b}\partial _{n}\ell ^{i}+\frac{1}{2}q_{bi}q^{m}_{a}\partial _{m}\ell ^{i} \end{aligned}$$

(7.59)

Now on the null surface $q_{ab}=q_{AB}$ as the only non-zero component. Hence the above equation can be written as,

$$\begin{aligned} \Theta _{ab}=\Theta _{AB}=\frac{1}{2}\dfrac{d}{d\lambda }q_{AB}+\frac{1}{2}q_{AC}\partial _{B}\ell ^{C}+\frac{1}{2}q_{BC}\partial _{A}\ell ^{C} \end{aligned}$$

(7.60)

On the null surface $q^{a}_{b}\ell ^{b}=0$, which in this coordinate system leads to $\ell ^{A}=0$ on the null surface. Since $\partial _{A}\ell ^{2}$ represent derivatives on the null surface it also vanishes. If $\ell ^{2}=0$ everywhere, then also $\ell ^{A}$ is identically zero everywhere. Hence we have

$$\begin{aligned} \Theta _{ab}=\frac{1}{2}\dfrac{d}{d\lambda }q_{AB} \end{aligned}$$

(7.61)

There is another way to get this result. If $e^{a}_{A}$ are the basis vectors on the null surface and if $\ell _{a},e^{a}_{A}$ forms coordinate basis vectors, then $q_{AB}=q_{ab}e^{a}_{A}e^{b}_{B}$ is a scalar under 4-dimensional coordinate transformation. This immediately leads to the previous expression. For more discussions along identical lines see [16].

Now the expression for the quantity $\mathcal {S}$ can be obtained as

$$\begin{aligned} \mathcal {S}&=\nabla _{i}\ell ^{j}\nabla _{j}\ell ^{i}-\left( \nabla _{i}\ell ^{i}\right) ^{2} \nonumber \\&=\Theta _{ab}\Theta ^{ab}+\kappa ^{2}+\tilde{\kappa }^{2}-\left( \Theta +\kappa +\tilde{\kappa }\right) ^{2} \nonumber \\&=\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) -2\Theta \left( \kappa +\tilde{\kappa }\right) -2\kappa \tilde{\kappa } \end{aligned}$$

(7.62)

Using the general expression for $R_{ab}\ell ^{a}\ell ^{b}$ we obtain the following form:

$$\begin{aligned} R_{ab}\ell ^{a}\ell ^{b}&=\nabla _{i}\left( \ell ^{j}\nabla _{j} \ell ^{i}-\ell ^{i}\nabla _{j}\ell ^{j}\right) -\mathcal {S} \nonumber \\&=\nabla _{i}\left( \ell ^{j}\nabla _{j} \ell ^{i}-\ell ^{i}\nabla _{j}\ell ^{j}\right) -\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +2\Theta \left( \kappa +\tilde{\kappa }\right) +2\kappa \tilde{\kappa } \end{aligned}$$

(7.63)

For the situation where, $\ell ^{2}=0$ everywhere we finally arrive at the following simplified expression

$$\begin{aligned} R_{ab}\ell ^{a}\ell ^{b}&=-\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +2\Theta \kappa +\nabla _{i}\left( \kappa \ell ^{i}-\left[ \Theta +\kappa \right] \ell ^{i}\right) \nonumber \\&=-\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +\Theta \kappa -\frac{1}{\sqrt{q}}\dfrac{d}{d\lambda }\left( \sqrt{q}\Theta \right) \end{aligned}$$

(7.64)

Let us now try to derive the Raychaudhuri equation starting from the basic properties of null surfaces. We start with the following result

$$\begin{aligned} \ell ^{a}\nabla _{a}\left( \nabla _{c}\ell _{d}\right)&=\ell ^{a}\nabla _{a}\nabla _{c}\ell _{d} \nonumber \\&=R_{dbac}\ell ^{a}\ell ^{b}+\ell ^{a}\nabla _{c}\nabla _{a}\ell _{d} \nonumber \\&=\nabla _{c}\left( \ell ^{a}\nabla _{a}\ell _{d}\right) -\nabla _{a}\ell _{d}\nabla _{c}\ell ^{a}-R_{bdac}\ell ^{b}\ell ^{a} \end{aligned}$$

(7.65)

Then contraction of the indices c, d leads to the following result

$$\begin{aligned} \ell ^{a}\nabla _{a}\left( \nabla _{c}\ell ^{c}\right)&=\nabla _{c}\left( \ell ^{a}\nabla _{a}\ell ^{c}\right) -\nabla _{a}\ell _{b}\nabla ^{b}\ell ^{a}-R_{ab}\ell ^{a}\ell ^{b} \end{aligned}$$

(7.66)

Otherwise we can rewrite it in a different manner which exactly coincide with Eq. (7.51). On using Eq. (7.56) and the decomposition: $\Theta _{ab}=(1/2)\Theta q_{ab}+\sigma _{ab}+\omega _{ab}$ we arrive at

$$\begin{aligned} \ell ^{a}\nabla _{a}\left( \nabla _{c}\ell ^{c}\right)&-\nabla _{c}\left( \ell ^{a}\nabla _{a}\ell ^{c}\right) =-\Theta _{ab}\Theta ^{ab}-\kappa ^{2}-\tilde{\kappa }^{2}-R_{ab}\ell ^{a}\ell ^{b} \nonumber \\&=-\frac{1}{2}\Theta ^{2}-\sigma ^{ab}\sigma _{ab}+\omega _{ab}\omega ^{ab} -\kappa ^{2}-\tilde{\kappa }^{2}-R_{ab}\ell ^{a}\ell ^{b} \end{aligned}$$

(7.67)

For the situation where, $\ell ^{2}=0$ the left hand side is just: $d\left( \Theta +\kappa \right) /d\lambda$ and the first term on the right hand side is $d\kappa /d\lambda +\kappa (\Theta +\kappa )$ the above equation leads to

$$\begin{aligned} \dfrac{d\Theta }{d\lambda }&=\kappa \Theta +\kappa ^{2}-\nabla _{a}\ell _{b}\nabla ^{b}\ell ^{a}-R_{ab}\ell ^{a}\ell ^{b} \nonumber \\&=\kappa \Theta -\Theta _{ab}\Theta ^{ab}-R_{ab}\ell ^{a}\ell ^{b} \nonumber \\&=\kappa \Theta -\frac{1}{2}\Theta ^{2}-\sigma ^{ab}\sigma _{ab}+\omega _{ab}\omega ^{ab} -R_{ab}\ell ^{a}\ell ^{b} \end{aligned}$$

(7.68)

where to arrive at the last line we have used the following decomposition: $\Theta _{ab}=(1/2)\Theta q_{ab}+\sigma _{ab}+\omega _{ab}$. This is precisely the one obtained in [19] though in a completely different manner.

The next object to consider is the quantity $\ell _{a}J^{a}(\ell )$. This can be obtained by using the identity for Noether current leading to,

$$\begin{aligned} \frac{1}{A}\ell _{a}J^{a}(\ell )&=\nabla _{b}\left( \left\{ \ell ^{a}\ell ^{b} -\ell ^{2}g^{ab}\right\} \frac{\nabla _{a}A}{A^{2}}\right) \nonumber \\&=\nabla _{b}\left[ \frac{1}{A}\ell ^{b}\left( \kappa -\tilde{\kappa }\right) \right] -\nabla _{b}\left( \ell ^{2}\frac{\nabla ^{b}A}{A^{2}}\right) \nonumber \\&=\frac{1}{A}\nabla _{i}\left[ \left( \kappa -\tilde{\kappa }\right) \ell ^{i}\right] -\frac{1}{A}\left( \kappa -\tilde{\kappa }\right) ^{2}-\frac{\nabla ^{b}A}{A^{2}}\nabla _{b}\ell ^{2} \nonumber \\&=\frac{1}{A}\nabla _{i}\left[ \left( \kappa -\tilde{\kappa }\right) \ell ^{i}\right] -\frac{1}{A}\left( \kappa -\tilde{\kappa }\right) ^{2} -\frac{2}{A}\tilde{\kappa }\left( \kappa -\tilde{\kappa }\right) \end{aligned}$$

(7.69)

This can be written in a slightly modified manner as,

$$\begin{aligned} \ell _{a}J^{a}(\ell )&=\nabla _{i}\left[ \left( \kappa -\tilde{\kappa }\right) \ell ^{i}\right] -\left( \kappa ^{2} -\tilde{\kappa }^{2}\right) \nonumber \\&=\ell ^{i}\nabla _{i}\left( \kappa -\tilde{\kappa }\right) +\left( \kappa -\tilde{\kappa }\right) \left( \Theta +\kappa +\tilde{\kappa }\right) -\left( \kappa ^{2} -\tilde{\kappa }^{2}\right) \nonumber \\&=\dfrac{d}{d\lambda }\left( \kappa -\tilde{\kappa }\right) +\Theta \left( \kappa -\tilde{\kappa }\right) \end{aligned}$$

(7.70)

The above expression can be simplified significantly by noting that $\Theta =d(\ln \sqrt{q})/d\lambda$, which leads to

$$\begin{aligned} \ell _{a}J^{a}(\ell )=\frac{1}{\sqrt{q}}\dfrac{d}{d\lambda }\left[ \left( \kappa -\tilde{\kappa }\right) \sqrt{q}\right] \end{aligned}$$

(7.71)

Again, we have

$$\begin{aligned} D_{a}\left[ \left( \kappa -\tilde{\kappa }\right) \ell ^{a}\right]&=\left( g^{ab}+\ell ^{a}k^{b}+\ell ^{b}k^{a}\right) \nabla _{a}\left[ \left( \kappa -\tilde{\kappa }\right) \ell _{b}\right] \nonumber \\&=\nabla _{a}\left[ \left( \kappa -\tilde{\kappa }\right) \ell ^{a}\right] +\left( \ell ^{a}k^{b}+\ell ^{b}k^{a}\right) \left[ \left( \kappa -\tilde{\kappa }\right) \nabla _{a}\ell _{b} +\ell _{b}\nabla _{a} \left( \kappa -\tilde{\kappa }\right) \right] \nonumber \\&=\nabla _{a}\left[ \left( \kappa -\tilde{\kappa }\right) \ell ^{a}\right] -\kappa \left( \kappa -\tilde{\kappa }\right) -\tilde{\kappa }\left( \kappa -\tilde{\kappa }\right) -\dfrac{d}{d\lambda }\left( \kappa -\tilde{\kappa }\right) \end{aligned}$$

(7.72)

$$\begin{aligned}&=\left( \kappa -\tilde{\kappa }\right) \left( \Theta +\kappa +\tilde{\kappa }\right) -\left( \kappa ^{2}-\tilde{\kappa }^{2}\right) \nonumber \\&=\left( \kappa -\tilde{\kappa }\right) \dfrac{d \ln \sqrt{q}}{d\lambda } \end{aligned}$$

(7.73)

Thus we arrive at

$$\begin{aligned} \ell _{a}J^{a}(\ell )&=D_{a}\left[ \left( \kappa -\tilde{\kappa }\right) \ell ^{a}\right] +\dfrac{d}{d\lambda }\left( \kappa -\tilde{\kappa }\right) +\left( \kappa -\tilde{\kappa }\right) \left( \kappa +\tilde{\kappa }\right) -\left( \kappa ^{2} -\tilde{\kappa }^{2}\right) \nonumber \\&=D_{a}\left[ \left( \kappa -\tilde{\kappa }\right) \ell ^{a}\right] +\dfrac{d}{d\lambda }\left( \kappa -\tilde{\kappa }\right) \end{aligned}$$

(7.74)

From the expression of the Noether current we get,

$$\begin{aligned} \ell _{a}J^{a}(\ell )=2R_{ab}\ell ^{a}\ell ^{b}+\ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij} \end{aligned}$$

(7.75)

The above equation can be used to write, $g^{ab}\pounds _{\ell }N^{c}_{ab}$ in terms of $\kappa$ and $R_{ab}$. For that purpose we use Eq. (2.30) and insert Eq. (7.70) leading to

$$\begin{aligned} \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij}&=\ell _{a}J^{a}(\ell )-2R_{ab}\ell ^{a}\ell ^{b} \nonumber \\&=\dfrac{d}{d\lambda }\left( \kappa -\tilde{\kappa }\right) +\Theta \left( \kappa -\tilde{\kappa }\right) -2R_{ab}\ell ^{a}\ell ^{b} \nonumber \\&=\frac{1}{\sqrt{q}}\dfrac{d}{d\lambda }\left[ \sqrt{q}\left( \kappa -\tilde{\kappa }\right) \right] -2R_{ab}\ell ^{a}\ell ^{b} \nonumber \\&=\frac{2}{\sqrt{q}}\dfrac{d}{d\lambda }\left( \sqrt{q}\kappa \right) -2R_{ab}\ell ^{a}\ell ^{b} -\frac{1}{\sqrt{q}}\dfrac{d}{d\lambda }\left[ \sqrt{q}\left( \kappa +\tilde{\kappa }\right) \right] \end{aligned}$$

(7.76)

The above expression when integrated over the null surface with integration measure, $\sqrt{q}d^{2}xd\lambda$ and then being divided by $16\pi$ leads to

$$\begin{aligned} \frac{1}{16\pi }\int d^{2}xd\lambda \sqrt{q}&\left\{ \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij} +\frac{1}{\sqrt{q}}\dfrac{d}{d\lambda }\left[ \sqrt{q}\left( \kappa +\tilde{\kappa }\right) \right] \right\} \nonumber \\&=\frac{1}{8\pi }\int d^{2}x\sqrt{q}\kappa \vert _{1}^{2} -\frac{1}{8\pi }\int d^{2}xd\lambda \sqrt{q}R_{ab}\ell ^{a}\ell ^{b} \end{aligned}$$

(7.77)

Then on using the field equation: $R_{ab}=8\pi \left[ T_{ab}-(1/2)g_{ab}T\right] =8\pi \bar{T}_{ab}$ and then being substituted in Eq. (7.77) we arrive at the following expression

$$\begin{aligned} \frac{1}{4}\int d^{2}x \sqrt{q}\left( \frac{\kappa }{2\pi }\right) \vert _{1}^{2}&-\int d^{2}xd\lambda \sqrt{q}~T_{ab}\ell ^{a}\ell ^{b} \nonumber \\&=\frac{1}{16\pi }\int d^{2}xd\lambda \sqrt{q}\left\{ \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij} +\frac{1}{\sqrt{q}}\dfrac{d}{d\lambda }\left[ \sqrt{q}\left( \kappa +\tilde{\kappa }\right) \right] \right\} \end{aligned}$$

(7.78)

where the last equality follows from the fact that $\ell ^{2}=0$ on the null surface. The above equation can be written in a more abstract form as

$$\begin{aligned} \frac{1}{16\pi }\int d^{2}xd\lambda \sqrt{q} \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij}&= \left[ \frac{1}{4}\int _{\lambda _{2}} d^{2}x \sqrt{q}\left( \frac{\kappa }{2\pi }\right) -\frac{1}{4}\int _{\lambda _{1}} d^{2}x \sqrt{q}\left( \frac{\kappa }{2\pi }\right) \right] \nonumber \\&-\int d^{2}xd\lambda \sqrt{q}~T_{ab}\ell ^{a}\ell ^{b} -\frac{1}{16\pi }\int d^{2}x d\lambda \dfrac{d}{d\lambda }\left[ \sqrt{q}\left( \kappa +\tilde{\kappa }\right) \right] \end{aligned}$$

(7.79)

As an illustration when $\ell ^{2}=0$ everywhere, we have $\tilde{\kappa }=0$, then Eq. (7.79) leads to,

$$\begin{aligned} \frac{1}{16\pi }\int d^{2}xd\lambda \sqrt{q} \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij}&=\frac{1}{2}\left[ \int _{\lambda _{2}} d^{2}x \frac{\sqrt{q}}{4}\left( \frac{\kappa }{2\pi }\right) -\int _{\lambda _{1}} d^{2}x \frac{\sqrt{q}}{4}\left( \frac{\kappa }{2\pi }\right) \right] \nonumber \\&-\int d^{2}xd\lambda \sqrt{q}~T_{ab}\ell ^{a}\ell ^{b} \end{aligned}$$

(7.80)

We will now try to obtain an expression for the quantity $\ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij}$ independently. For that we start with the symmetric and anti-symmetric part of the derivative $\nabla _{a}\ell _{b}$ such that:

$$\begin{aligned} S^{ab}=\nabla ^{a}\ell ^{b}+\nabla ^{b}\ell ^{a},\qquad J^{ab}=\nabla ^{a}\ell ^{b}-\nabla ^{b}\ell ^{a} \end{aligned}$$

(7.81)

Then we have the following result: $\nabla ^{a}\ell ^{b}=(1/2)(S^{ab}+J^{ab})$, which on being substituted in the identity:

$$\begin{aligned} \nabla _{b}\left( \nabla ^{a}\ell ^{b}\right) -\nabla ^{a}\left( \nabla _{b}\ell ^{b}\right) =R^{a}_{b}\ell ^{b} \end{aligned}$$

(7.82)

leads to the following identification:

$$\begin{aligned} g^{ab}\pounds _{\ell }N^{c}_{ab}=-\nabla _{b}\left( S^{bc}-g^{bc}S\right) \end{aligned}$$

(7.83)

Hence we arrive at the following relation:

$$\begin{aligned} \ell _{a}g^{bc}\pounds _{\ell }N^{a}_{bc}&=-\ell _{a}\nabla _{b}\left[ \nabla ^{a}\ell ^{b}+\nabla ^{b}\ell ^{a}-2g^{ab}\nabla _{c}\ell ^{c}\right] \nonumber \\&=-\nabla _{b}\left[ \ell ^{a}\nabla _{a}\ell ^{b}+\ell _{a}\nabla ^{b}\ell ^{a}-2\ell ^{b}\left( \nabla _{c}\ell ^{c}\right) \right] \nonumber \\&+\nabla _{a}\ell _{b}\nabla ^{b}\ell ^{a}+\nabla _{a}\ell _{b}\nabla ^{a}\ell ^{b} -2\left( \nabla _{c}\ell ^{c}\right) ^{2} \nonumber \\&=-\nabla _{b}\left[ \ell ^{a}\nabla _{a}\ell ^{b}+\ell _{a}\nabla ^{b}\ell ^{a}-2\ell ^{b}\left( \nabla _{c}\ell ^{c}\right) \right] \nonumber \\&+2\Theta _{ab}\Theta ^{ab}+\left( \kappa +\tilde{\kappa }\right) ^{2}-2\left( \Theta +\kappa +\tilde{\kappa }\right) ^{2} \nonumber \\&=-\nabla _{b}\left[ \ell ^{a}\nabla _{a}\ell ^{b}+\ell _{a}\nabla ^{b}\ell ^{a}-2\ell ^{b}\left( \nabla _{c}\ell ^{c}\right) \right] \nonumber \\&+2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) -4\Theta \left( \kappa +\tilde{\kappa }\right) -\left( \kappa +\tilde{\kappa }\right) ^{2} \end{aligned}$$

(7.84)

For the case $\ell ^{2}=0$ the term within bracket can be written in a simplified manner such that Lie derivative term gets simplified leading to:

$$\begin{aligned} \ell _{a}g^{bc}\pounds _{\ell }N^{a}_{bc}&=2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) -4\Theta \kappa -\kappa ^{2}+\nabla _{b}\left[ \left( 2\Theta +\kappa \right) \ell ^{b}\right] \nonumber \\&=2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) -4\Theta \kappa -\kappa ^{2} +\left( 2\Theta +\kappa \right) \left( \Theta +\kappa \right) +\dfrac{d}{d\lambda }\left( 2\Theta +\kappa \right) \nonumber \\&=2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) -\Theta \kappa +\dfrac{d}{d\lambda }\kappa +\frac{2}{\sqrt{q}}\dfrac{d}{d\lambda }\left( \sqrt{q}\Theta \right) \end{aligned}$$

(7.85)

If the null generator is affinely parametrized then $\kappa =0$ and Eq. (7.85) reduces to:

$$\begin{aligned} \ell _{a}g^{bc}\pounds _{\ell }N^{a}_{bc}=2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +\frac{2}{\sqrt{q}}\dfrac{d}{d\lambda }\left( \sqrt{q}\Theta \right) \end{aligned}$$

(7.86)

While for the null generator $\ell ^{a}$ in GNC we have (see Appendix A.3 Eq. (7.133)):

$$\begin{aligned} \ell _{a}g^{bc}\pounds _{\ell }N^{a}_{bc}=2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +\frac{2}{\sqrt{q}}\dfrac{d ^{2}\sqrt{q}}{d\lambda ^{2}}+2\dfrac{d\kappa }{d\lambda }-\frac{2}{\sqrt{q}}\dfrac{d}{d\lambda }\left( \sqrt{q}\kappa \right) \end{aligned}$$

(7.87)

Through the above analysis we have obtained expressions for $\ell _{a}J^{a}(\ell )$, $R_{ab}\ell ^{a}\ell ^{b}$ and $\ell _{a}g^{bc}\pounds _{\ell }N^{a}_{bc}$.

It turns out from the above analysis that $\Theta _{ab}\Theta ^{ab}-\Theta ^{2}$ can be given a more physical meaning by considering Lie variation of gravitational momentum. This can be obtained by considering variation of the gravitational momentum first:

$$\begin{aligned} q_{ab}\delta \Pi ^{ab}&=q_{ab}\delta \left[ \sqrt{q}\left( \Theta ^{ab}-\Theta q^{ab}\right) \right] \nonumber \\&=q_{ab}\sqrt{q}\delta \Theta ^{ab}-2\sqrt{q}\delta \Theta -\sqrt{q}\Theta q_{ab}\delta q^{ab}-\Theta \delta \sqrt{q} \nonumber \\&=\sqrt{q}q_{ab}\delta \Theta ^{ab}-2\sqrt{q}\delta \Theta +\Theta \delta \sqrt{q} \end{aligned}$$

(7.88)

Now specializing to Lie variation we arrive at:

$$\begin{aligned} -q_{ab}\pounds _{\ell }\Pi ^{ab}&=-\Theta \pounds _{\ell }\sqrt{q}-\sqrt{q}q_{ab}\pounds _{\ell }\Theta ^{ab}+2\sqrt{q}\pounds _{\ell }\Theta \nonumber \\&=-\sqrt{q}\pounds _{\ell }\Theta +\sqrt{q}\Theta ^{ab}\pounds _{\ell }q_{ab}-\Theta \pounds _{\ell }\sqrt{q}+2\sqrt{q}\pounds _{\ell }\Theta \nonumber \\&=2\sqrt{q}\left( \Theta ^{ab}\Theta _{ab}-\Theta ^{2}\right) +\pounds _{\ell }\left( \sqrt{q}\Theta \right) \nonumber \\&=\sqrt{q}\ell _{a}g^{bc}\pounds _{\ell }N^{a}_{bc}-\dfrac{d ^{2}\sqrt{q}}{d\lambda ^{2}} \end{aligned}$$

(7.89)

where in the last line we have used Eq. (7.86). Here the quantities $\Theta _{ab}$ and $\Theta$ can be defined as: $\Theta _{ab}=(1/2)\pounds _{\ell }q_{ab}$ and $\Theta =\pounds _{\ell }\ln \sqrt{q}$. In GNC parametrization: $(1/2)\pounds _{\ell }q_{ab}=(1/2)\partial _{u}q_{ab}$ and $\pounds _{\ell }\ln \sqrt{q}=\partial _{u}\ln \sqrt{q}$. Thus the Lie variation of gravitational momentum for affine parametrization is directly related to $\mathcal {D}$, i.e. to $(\Theta _{ab}\Theta ^{ab}-\Theta ^{2})$.

For non-affine parametrization the gravitational momentum associated with null surfaces can be taken as $\Pi ^{ab}=\sqrt{q}(\Theta ^{ab}-(\Theta +\kappa )q^{ab})$. Then we readily arrive at the Lie variation expression:

$$\begin{aligned} -q_{ab}\pounds _{\ell }\Pi ^{ab}&=2\sqrt{q}\left( \Theta ^{ab}\Theta _{ab}-\Theta ^{2}\right) +\pounds _{\ell }\left( \sqrt{q}\Theta \right) +2\sqrt{q}\pounds _{\ell }\kappa \end{aligned}$$

(7.90)

Since $\kappa$ is a scalar Lie variation term can be written as: $\pounds _{\ell }\kappa =d\kappa /d\lambda$, where $\lambda$ is the parameter along $\ell ^{a}$. If we consider the null generator $\ell ^{a}$ from GNC then we arrive at (see Eq. (7.87)):

$$\begin{aligned} -q_{ab}\pounds _{\ell }\Pi ^{ab}=\sqrt{q}\ell _{a}g^{bc}\pounds _{\ell }N^{a}_{bc}-\dfrac{d ^{2}\sqrt{q}}{d\lambda ^{2}}+\frac{2}{\sqrt{q}}\dfrac{d}{d\lambda }\left( \sqrt{q}\kappa \right) \end{aligned}$$

(7.91)

while if we have defined the conjugate momenta $\Pi ^{ab}$ as $\Pi ^{ab}=\sqrt{q}(\Theta ^{ab}-\Theta q^{ab})$, then the above relation could have been written as,

$$\begin{aligned} -q_{ab}\pounds _{\ell }\Pi ^{ab}=\sqrt{q}\ell _{a}g^{bc}\pounds _{\ell }N^{a}_{bc}-\dfrac{d ^{2}\sqrt{q}}{d\lambda ^{2}}+2\kappa \dfrac{d}{d\lambda }\left( \sqrt{q}\right) \end{aligned}$$

(7.92)

A.2 Some Useful Results Associated with Null Foliation

There are a few more geometrical quantities associated with the null vectors associated with GNC, which we will introduce for ready reference later on. The first one corresponds to the induced metric $q_{ab}$ on the null surface, defined as,

$$\begin{aligned} q_{ab}\equiv g_{ab}+\ell _{a}k_{b}+\ell _{b}k_{a};\qquad q^{a}_{b}\equiv \delta ^{a}_{b}+\ell ^{a}k_{b}+\ell _{b}k^{a} \end{aligned}$$

(7.93)

Note that both $\ell _{a}q^{a}_{b}$ and $k_{a}q^{a}_{b}$ identically vanishes on the null surface, thanks to the relation $\ell _{a}k^{a}=-1$; we can think of $q^{a}_{b}$ as a projector on to the $r=0$ surface, which is two-dimensional. Using this projector, we can define extrinsic curvature on a null surface:

$$\begin{aligned} \Theta _{ab}\equiv q^{m}_{a}q^{n}_{b}\nabla _{m}\ell _{n}=\frac{1}{2}q^{m}_{a}q^{n}_{b}\pounds _{\ell }q_{mn} \end{aligned}$$

(7.94)

If $\lambda$ is the parameter along the null generator $\ell _{a}$ on the null surface, the only nonzero components of $\Theta _{ab}$ are (see Eq. (7.61) of Appendix A.1),

$$\begin{aligned} \Theta _{AB}=\frac{1}{2}\dfrac{d}{d\lambda }q_{AB} \end{aligned}$$

(7.95)

The trace of $\Theta _{ab}$, is $\Theta =q_{ab}\Theta ^{ab}$ and it is useful to define the trace free shear tensor $\sigma _{ab}$ as,

$$\begin{aligned} \sigma _{ab}\equiv \Theta _{ab}-\frac{1}{2}q_{ab}\Theta \end{aligned}$$

(7.96)

Then, as described in Ref. [3] we can introduce shear viscosity coefficient, $\eta =(1/16\pi )$ and the bulk viscosity coefficient $\zeta =-(1/16\pi )$ as well as the dissipation term by:

$$\begin{aligned} \mathcal {D}\equiv 8\pi \left( 2\eta \sigma _{ab}\sigma ^{ab}+\zeta \Theta ^{2}\right) =\Theta _{ab}\Theta ^{ab}-\Theta ^{2} \end{aligned}$$

(7.97)

The importance of $\sigma _{ab}$ and $\mathcal {D}$—which will occur repeatedly in our discussion—arises from the following fact: It turns out that Einstein’s equations, when projected on to any null surface in any spacetime, takes the form of a Navier-Stokes equations [2] with $\sigma _{ab}$ acting as a viscous tensor and $\eta ,\zeta$ acting as bulk and shear viscosity coefficients, In that case, the apparent viscous dissipation is given by $\mathcal {D}$. (The conceptual issues related to this ‘dissipation without dissipation’ since there can be no real, irreversible, drain of energy are clarified in [2]). In the GNC, we have on the null surface,

$$\begin{aligned} \mathcal {D}=\frac{1}{4}q^{ac}q^{bd}\partial _{u}q_{ab}\partial _{u}q_{cd}-\left( \partial _{u}\ln \sqrt{q}\right) ^{2} \end{aligned}$$

(7.98)

which vanishes when $\partial _{u}q_{ab}=0$ on the null surface. These results have been used in the main text.

A.3 Derivation of Various Expressions Used in Text

This appendix will contain derivations of most of the results that we have used in the main text. The derivations will be arranged in the same order as that in the main text. First we will present derivations related to the Navier-Stokes equation and then we will present the requisite derivations of subsequent sections.

1.1 A.3.1 Derivation Regarding Navier-Stokes Equation

The first thing to compute is the Lie derivative of the object $N^{c}_{ab}$. This can be obtained starting from the first principle, i.e., using expression for $N^{c}_{ab}$ in terms of $\Gamma ^{a}_{bc}$ and then using Lie variation of the connection. This immediately leads to:

$$\begin{aligned} \pounds _{v}N^{a}_{bc}&=Q^{ad}_{be}\pounds _{v}\Gamma ^{e}_{cd}+Q^{ad}_{ce}\pounds _{v}\Gamma ^{e}_{bd} \nonumber \\&=Q^{ad}_{be}\left( \nabla _{c}\nabla _{d}v^{e}+R^{e}_{~dmc}v^{m}\right) +Q^{ad}_{ce}\left( \nabla _{b}\nabla _{d}v^{e}+R^{e}_{~dmb}v^{m}\right) \nonumber \\&=\frac{1}{2}\left( \delta ^{a}_{b}\nabla _{c}\nabla _{d}v^{d}+\delta ^{a}_{c}\nabla _{b}\nabla _{d}v^{d}\right) -\frac{1}{2}\left( \nabla _{b}\nabla _{c}v^{a}+\nabla _{c}\nabla _{b}v^{a}\right) \nonumber \\&-\frac{1}{2}\left( R^{a}_{~bmc}+R^{a}_{~cmb}\right) v^{m} \end{aligned}$$

(7.99)

In the above expression the second term in the last line can be written as:

$$\begin{aligned} \left( \nabla _{b}\nabla _{c}v^{a}+\nabla _{c}\nabla _{b}v^{a}\right)&=2\partial _{b}\partial _{c}v^{a}+2\Gamma ^{a}_{bd}\partial _{c}v^{d}+2\Gamma ^{a}_{cd}\partial _{b}v^{d}-2\Gamma ^{d}_{bc}\partial _{d}v^{a} \nonumber \\&+v^{d}\left( \partial _{b}\Gamma ^{a}_{cd}+\partial _{c}\Gamma ^{a}_{bd}\right) -2\Gamma ^{d}_{bc}\Gamma ^{a}_{de}v^{e} +\Gamma ^{a}_{bd}\Gamma ^{d}_{ce}v^{e}+\Gamma ^{a}_{cd}\Gamma ^{d}_{be}v^{e} \end{aligned}$$

(7.100)

In order to compute the Lie variation of $N^{c}_{ab}$ along the transverse direction we need the two objects $\pounds _{\ell }N^{A}_{ur}$ and $\pounds _{\ell }N^{A}_{BC}$. For the evaluation of $\pounds _{\ell }N^{A}_{ur}$ the following identities will be useful:

$$\begin{aligned} \left( \nabla _{b}\nabla _{c}\ell ^{a}+\nabla _{c}\nabla _{b}\ell ^{a}\right) ^{A}_{ur}&=2\partial _{u}\partial _{r}\ell ^{A}+2\Gamma ^{A}_{ud}\partial _{r}\ell ^{d}+2\Gamma ^{A}_{rd}\partial _{u}\ell ^{d}-2\Gamma ^{d}_{ur}\partial _{d}\ell ^{A} \nonumber \\&+v^{d}\left( \partial _{u}\Gamma ^{A}_{rd}+\partial _{r}\Gamma ^{A}_{ud}\right) -2\Gamma ^{d}_{ur}\Gamma ^{A}_{de}\ell ^{e} +\Gamma ^{A}_{ud}\Gamma ^{d}_{re}v^{e}+\Gamma ^{A}_{rd}\Gamma ^{d}_{ue}\ell ^{e} \nonumber \\&=2\partial _{u}\beta ^{A}+4\alpha \Gamma ^{A}_{ur}+2\beta ^{B}\Gamma ^{A}_{uB}-2\beta ^{A}\Gamma ^{r}_{ur}+\partial _{u}\Gamma ^{A}_{ru}+\partial _{r}\Gamma ^{A}_{uu} \nonumber \\&-2\Gamma ^{d}_{ur}\Gamma ^{A}_{du}+\Gamma ^{A}_{ud}\Gamma ^{d}_{ur}+\Gamma ^{A}_{rd}\Gamma ^{d}_{uu} \end{aligned}$$

(7.101)

and

$$\begin{aligned} \left( \nabla _{b}\nabla _{c}\ell ^{a}+\nabla _{c}\nabla _{b}\ell ^{a}\right) ^{A}_{ur}&+\left[ \left( R^{a}_{~bmc}+R^{a}_{~cmb}\right) \ell ^{m}\right] ^{A}_{ur} \nonumber \\&=2\partial _{u}\beta _{A}+4\alpha \Gamma ^{A}_{ur}+2\beta ^{B}\Gamma ^{A}_{uB}-2\beta ^{A}\Gamma ^{r}_{ur} \nonumber \\&+\partial _{u}\Gamma ^{A}_{ru}+\partial _{r}\Gamma ^{A}_{uu}-2\Gamma ^{d}_{ur}\Gamma ^{A}_{du}+\Gamma ^{A}_{ud}\Gamma ^{d}_{ur}+\Gamma ^{A}_{rd}\Gamma ^{d}_{uu}+R^{A}_{~uur} \nonumber \\&=2\partial _{u}\beta ^{A}+2\beta ^{B}\Gamma ^{A}_{uB}+2\partial _{u}\Gamma ^{A}_{ur} \nonumber \\&=\partial _{u}\beta ^{A}+\beta ^{B}q^{AC}\partial _{u}q_{BC} \end{aligned}$$

(7.102)

While for $\pounds _{\ell }N^{A}_{BC}$ we have:

$$\begin{aligned} \left( \nabla _{b}\nabla _{c}\ell ^{a}+\nabla _{c}\nabla _{b}\ell ^{a}\right) ^{A}_{BC}&+\left[ \left( R^{a}_{~bmc}+R^{a}_{~cmb}\right) \ell ^{m}\right] ^{A}_{BC} \nonumber \\&=-2\beta ^{A}\Gamma ^{r}_{BC}+\partial _{B}\Gamma ^{A}_{uC}+\partial _{C}\Gamma ^{A}_{uB}-2\Gamma ^{d}_{BC}\Gamma ^{A}_{ud} \nonumber \\&+\Gamma ^{A}_{Bd}\Gamma ^{d}_{uC}+\Gamma ^{A}_{Cd}\Gamma ^{d}_{uB}+\partial _{u}\hat{\Gamma }^{A}_{BC}-\partial _{C}\Gamma ^{A}_{Bu}+\Gamma ^{A}_{ud}\Gamma ^{d}_{BC} \nonumber \\&-\Gamma ^{A}_{Cd}\Gamma ^{d}_{uB}+\partial _{u}\hat{\Gamma }^{A}_{BC}-\partial _{B}\Gamma ^{A}_{Cu}+\Gamma ^{A}_{ud}\Gamma ^{d}_{BC}-\Gamma ^{A}_{Bd}\Gamma ^{d}_{uC} \nonumber \\&=-2\beta ^{A}\Gamma ^{r}_{BC}+2\partial _{u}\hat{\Gamma }^{A}_{BC} \end{aligned}$$

(7.103)

These are the expressions used to get expressions in Sect. 7.4.1. From the vector $\Omega _{a}$ given in Eq. (7.23) we can calculate the Lie variation along $\ell ^{a}$ leading to,

$$\begin{aligned} \pounds _{\ell }\Omega _{n}&=\ell ^{m}\partial _{m}\Omega _{n}+\Omega _{m}\partial _{n}\ell ^{m} \nonumber \\&=\left( 0,\frac{1}{2}\beta _{A}\beta ^{A},\frac{1}{2}\partial _{u}\beta _{A}\right) \end{aligned}$$

(7.104)

and equivalently,

$$\begin{aligned} \pounds _{\ell }\Omega ^{n}&=\ell ^{m}\partial _{m}\Omega ^{n}-\Omega ^{m}\partial _{m}\ell ^{n} \nonumber \\&=\left( 0,0,\frac{1}{2}\partial _{u}\beta ^{A}\right) \end{aligned}$$

(7.105)

Also,

$$\begin{aligned} D_{m}\Theta ^{m}_{a}&=\partial _{B}\Theta ^{B}_{A}+\partial _{C}\ln \sqrt{q}\Theta ^{C}_{A}-\hat{\Gamma } ^{C}_{AB}\Theta ^{B}_{C} \nonumber \\&=\frac{1}{2}\partial _{B}\left( q^{BC}\partial _{u}q_{AC}\right) +\frac{1}{2}q^{CD}\partial _{C}\ln \sqrt{q}\partial _{u}q_{AD}-\frac{1}{2}q^{BD}\partial _{u}q_{CD}\hat{\Gamma }^{C}_{AB} \end{aligned}$$

(7.106)

Using these results we finally obtain,

$$\begin{aligned} q^{n}_{a}\pounds _{\ell }\Omega _{n}&+D_{m}\Theta ^{m}_{a}+\Theta \Omega _{a}-D_{a}\left( \Theta +\alpha \right) \nonumber \\&=\frac{1}{2}\partial _{u}\beta _{A}+\partial _{B}\left( \frac{1}{2}q^{BC}\partial _{u}q_{AC}\right) +\frac{1}{2}q^{CD}\partial _{u}q_{AD}\partial _{C}\ln \sqrt{q} \nonumber \\&-\frac{1}{2}q^{BD}\partial _{u}q_{CD}\hat{\Gamma }^{C}_{AB} +\partial _{u}\ln \sqrt{q}\frac{1}{2}\beta _{A}-\partial _{A}\partial _{u}\ln \sqrt{q}-\partial _{A}\alpha \end{aligned}$$

(7.107)

In raising the free index of the above equation the following identities can be useful

$$\begin{aligned} -q^{CD}q^{AB}\partial _{u}q_{BC}\partial _{D}\ln \sqrt{q}&=\partial _{u}\left( q^{AD}\partial _{D}\ln \sqrt{q}\right) -q^{AD}\partial _{u}\partial _{D}\ln \sqrt{q} \end{aligned}$$

(7.108a)

$$\begin{aligned} \partial _{u}\left( q^{AD}\partial _{D}\ln \sqrt{q}\right)&-q^{AB}\partial _{u}q^{CF}\partial _{C}q_{BF}-q^{AB}\partial _{D}\left( q^{CD}\partial _{u}q_{BC}\right) \nonumber \\&=-\partial _{u}\left( q^{BC}\hat{\Gamma }^{A}_{BC}\right) +q^{CF}\partial _{u}q^{AB}\partial _{C}q_{BF}-q^{AB}\partial _{D}q^{CD}\partial _{u}q_{BC} \nonumber \\&=-\partial _{u}\left( q^{BC}\hat{\Gamma }^{A}_{BC}\right) \end{aligned}$$

(7.108b)

$$\begin{aligned} q^{AB}q^{CD}\partial _{u}q_{ED}\hat{\Gamma }^{E}_{BC}&=\hat{\Gamma }^{A}_{FC}\partial _{u}q^{CF}-q^{AB}\partial _{C}q_{BF}\partial _{u}q^{CF} \end{aligned}$$

(7.108c)

Moreover we also have:

$$\begin{aligned} R_{ab}\ell ^{a}q^{b}_{c}&=R_{uA}=G_{uA} \nonumber \\&=\frac{1}{2}\partial _{u}\beta _{A}-\partial _{A}\alpha +\frac{1}{2}q^{BC}\partial _{u}\partial _{B}q_{CA}+\frac{1}{2}\partial _{B}q^{BC}\partial _{u}q_{AC}-\partial _{u}\partial _{A}\ln \sqrt{q} \nonumber \\&+\frac{1}{2}\beta _{A}\partial _{u}\ln \sqrt{q}+\frac{1}{2}q^{BC}\partial _{u}q_{AC}\partial _{B}\ln \sqrt{q}-\frac{1}{2}q^{BD}\partial _{u}q_{CD}\hat{\Gamma }^{C}_{~AB} \end{aligned}$$

(7.109)

It can be checked that this expression coincides exactly with Eq. (7.32) in Sect. 7.4.1 as it should.

To bring out the physics associated with Noether current and its various projections we compute the Noether potential and hence the Noether current completely in GNC for the vector $\xi ^{a}$. To start with we provide all the components of the tensor $\nabla _{a}\xi _{b}$, which are:

$$\begin{aligned} \left( \nabla _{a}\xi _{b}\right) _{uu}&=-r\partial _{u}\alpha ;~~ \left( \nabla _{a}\xi _{b}\right) _{ur}=\alpha +r\partial _{r}\alpha ;\qquad \left( \nabla _{a}\xi _{b}\right) _{ru}=-\alpha -r\partial _{r}\alpha \end{aligned}$$

(7.110)

$$\begin{aligned} \left( \nabla _{a}\xi _{b}\right) _{uA}&=r\partial _{A}\alpha -r\partial _{u}\beta _{A};~~ \left( \nabla _{a}\xi _{b}\right) _{Au}=-r\partial _{A}\alpha ; \end{aligned}$$

(7.111)

$$\begin{aligned} \left( \nabla _{a}\xi _{b}\right) _{rA}&=-\frac{1}{2}\beta _{A}-\frac{1}{2}r\partial _{r}\beta _{A};~~ \left( \nabla _{a}\xi _{b}\right) _{rr}=0 ;~ \left( \nabla _{a}\xi _{b}\right) _{Ar}=\frac{1}{2}\beta _{A}+\frac{1}{2}r\partial _{r}\beta _{A}; \nonumber \\&\left( \nabla _{a}\xi _{b}\right) _{AB}=\frac{1}{2}\partial _{u}q_{AB}+\frac{1}{2}r\left( \partial _{A}\beta _{B}-\partial _{B}\beta _{A}\right) ; \nonumber \\&\left( \nabla _{a}\xi _{b}\right) _{BA}=\frac{1}{2}\partial _{u}q_{AB}-\frac{1}{2}r\left( \partial _{A}\beta _{B}-\partial _{B}\beta _{A}\right) \end{aligned}$$

(7.112)

Then components of Noether potential $J_{ab}=\nabla _{a}\xi _{b}-\nabla _{b}\xi _{a}$ have the following expression:

$$\begin{aligned} J_{uu}&=0;\qquad J_{ur}=2\alpha +2r\partial _{r}\alpha ;\qquad J_{uA}=2r\partial _{A}\alpha -r\partial _{u}\beta _{A} \end{aligned}$$

(7.113)

$$\begin{aligned} J_{rA}&=-\beta _{A}-r\partial _{r}\beta _{A};\qquad J_{AB}=r\left( \partial _{A}\beta _{B}-\partial _{B}\beta _{A}\right) \end{aligned}$$

(7.114)

The upper components of Noether potential can be obtained as

$$\begin{aligned} J^{uu}&=0;\qquad J^{ur}=-2\alpha -2r\partial _{r}\alpha -r\beta _{A}\beta ^{A}-r^{2}\beta ^{A}\partial _{r}\beta _{A} \end{aligned}$$

(7.115)

$$\begin{aligned} J^{uA}&=-\beta ^{A}-rq^{AB}\partial _{r}\beta _{B};\qquad J^{rr}=r^{3}\beta ^{A}\beta ^{B}\left( \partial _{A}\beta _{B}-\partial _{B}\beta _{A}\right) \end{aligned}$$

(7.116)

$$\begin{aligned} J^{rA}&=2rq^{AB}\partial _{B}\alpha -rq^{AB}\partial _{u}\beta _{B}-2r^{2}\alpha q^{AB}\partial _{r}\beta _{B}-r^{3}\beta ^{2}q^{AB}\partial _{r}\beta _{B} \nonumber \\&-r^{2}q^{AB}\beta ^{C}\left( \partial _{B}\beta _{C}-\partial _{C}\beta _{B}\right) +2r^{2}\beta ^{A}\partial _{r}\alpha -r^{3}\beta ^{A}\beta ^{B}\partial _{r}\beta _{B}\end{aligned}$$

(7.117)

$$\begin{aligned} J^{AB}&=-r\beta ^{A}q^{BC}\left( \beta _{C}+r\partial _{r}\beta _{C}\right) \nonumber \\&+r\beta ^{B}q^{AC}\left( \beta _{C}+r\partial _{r}\beta _{C}\right) +rq^{AC}q^{BD}\left( \partial _{C}\beta _{D}-\partial _{D}\beta _{C}\right) \end{aligned}$$

(7.118)

Using the above components of Noether potential the components of Noether current can be obtained as

$$\begin{aligned} J^{u}(\xi )&=-4\partial _{r}\alpha -\beta ^{2}-2\alpha \partial _{r}\ln \sqrt{q}-\frac{1}{\sqrt{q}}\partial _{A}\left( \sqrt{q}\beta ^{A}\right) \end{aligned}$$

(7.119)

$$\begin{aligned} J^{r}(\xi )&=2\alpha \partial _{u}\ln \sqrt{q}+2\partial _{u}\alpha \end{aligned}$$

(7.120)

$$\begin{aligned} J^{A}(\xi )&=\frac{1}{\sqrt{q}}\partial _{u}\left( \sqrt{q}\beta ^{A}\right) +q^{AB}\partial _{u}\beta _{B}-2q^{AB}\partial _{B}\alpha \end{aligned}$$

(7.121)

Note that $k_{a}J^{a}(\xi )=-J^{u}(\xi )$, $q^{a}_{b}J^{b}(\xi )=J^{A}(\xi )$ and finally $\ell _{a}J^{a}(\xi )=J^{r}(\xi )$. As we will see all of them matches with our desired expressions. Also in the stationary limit we have $\partial _{u}\alpha =\partial _{u}\beta _{A}=\partial _{u}q_{AB}=0$, which in particular tells us that $J^{r}=0$. Hence in the static limit Noether current is on the null surface since its component along $k^{a}$ (which is $-\ell _{a}J^{a}(\xi )$) vanishes. Also in this case we have:

$$\begin{aligned} \left( \nabla _{b}\nabla _{c}\xi ^{a}+\nabla _{c}\nabla _{b}\xi ^{a}\right) ^{A}_{ur}&+\left[ \left( R^{a}_{~bmc}+R^{a}_{~cmb}\right) \xi ^{m}\right] ^{A}_{ur} \nonumber \\&=\partial _{u}\Gamma ^{A}_{ru}+\partial _{r}\Gamma ^{A}_{uu}-\Gamma ^{d}_{ur}\Gamma ^{A}_{du}+\Gamma ^{A}_{rd}\Gamma ^{d}_{uu}+R^{A}_{~uur} \nonumber \\&=2\partial _{u}\Gamma ^{A}_{ur}=-\partial _{u}\beta ^{A} \end{aligned}$$

(7.122)

as well as,

$$\begin{aligned} \left( \nabla _{b}\nabla _{c}\xi ^{a}+\nabla _{c}\nabla _{b}\xi ^{a}\right) ^{A}_{BC}&+\left[ \left( R^{a}_{~bmc}+R^{a}_{~cmb}\right) \xi ^{m}\right] ^{A}_{BC} \nonumber \\&=\partial _{B}\Gamma ^{A}_{uC}+\partial _{C}\Gamma ^{A}_{uB}-2\Gamma ^{d}_{BC}\Gamma ^{A}_{ud} \nonumber \\&+\Gamma ^{A}_{Bd}\Gamma ^{d}_{uC}+\Gamma ^{A}_{Cd}\Gamma ^{d}_{uB}+\partial _{u}\hat{\Gamma }^{A}_{BC}-\partial _{C}\Gamma ^{A}_{Bu}+\Gamma ^{A}_{ud}\Gamma ^{d}_{BC} \nonumber \\&-\Gamma ^{A}_{Cd}\Gamma ^{d}_{uB}+\partial _{u}\hat{\Gamma }^{A}_{BC}-\partial _{B}\Gamma ^{A}_{Cu}+\Gamma ^{A}_{ud}\Gamma ^{d}_{BC}-\Gamma ^{A}_{Bd}\Gamma ^{d}_{uC} \nonumber \\&=2\partial _{u}\hat{\Gamma }^{A}_{BC} \end{aligned}$$

(7.123)

Using these two results we arrive at:

$$\begin{aligned} \pounds _{\xi }N^{A}_{ur}&=\frac{1}{2}\partial _{u}\beta ^{A} \end{aligned}$$

(7.124)

$$\begin{aligned} \pounds _{\xi }N^{A}_{BC}&=\frac{1}{2}\delta ^{A}_{B}\partial _{C}\Theta +\frac{1}{2}\delta ^{A}_{C}\partial _{B}\Theta -\partial _{u}\hat{\Gamma }^{A}_{BC} \end{aligned}$$

(7.125)

These are the expressions used in Sect. 7.4.1.

1.2 A.3.2 Derivation Regarding Spacetime Evolution

We need to consider the object $\ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij}$ in GNC. This in turn requires us to obtain expressions for $\pounds _{\ell }N^{r}_{ur}$ and $\pounds _{\ell }N^{r}_{AB}$. Then using the identity for Lie variation of $N^{c}_{ab}$ we can obtain both the Lie variations. For that purpose we have:

$$\begin{aligned} \frac{1}{2}\Big (\delta ^{a}_{b}\nabla _{c}\nabla _{d}\ell ^{d}&+\delta ^{a}_{c}\nabla _{b}\nabla _{d}\ell ^{d}\Big )^{r}_{ur}=\frac{1}{2}\partial _{u}\Theta +\partial _{u}\alpha \end{aligned}$$

(7.126)

$$\begin{aligned} \Big (\nabla _{b}\nabla _{c}\ell ^{a}&+\nabla _{c}\nabla _{b}\ell ^{a}\Big )^{r}_{ur}=2\partial _{u}\alpha \end{aligned}$$

(7.127)

$$\begin{aligned} \Big [-\frac{1}{2}\Big (R^{a}_{~bmc}&+R^{a}_{~cmb}\Big )\ell ^{m}\Big ]^{r}_{ur}=0 \end{aligned}$$

(7.128)

$$\begin{aligned} \Big (\nabla _{b}\nabla _{c}\ell ^{a}&+\nabla _{c}\nabla _{b}\ell ^{a}\Big )^{r}_{AB}=\alpha \partial _{u}q_{AB}-\frac{1}{2}q^{CD}\partial _{u}q_{AC}\partial _{u}q_{BD} \end{aligned}$$

(7.129)

$$\begin{aligned} \Big [-\frac{1}{2}\Big (R^{a}_{~bmc}&+R^{a}_{~cmb}\Big )\ell ^{m}\Big ]^{r}_{AB}=-\frac{1}{2}\alpha \partial _{u}q_{AB}+\frac{1}{2}\partial _{u}^{2}q_{AB}-\frac{1}{4}q^{CD}\partial _{u}q_{AC}\partial _{u}q_{BD} \end{aligned}$$

(7.130)

This immediately leads to

$$\begin{aligned} \pounds _{\ell }N^{r}_{ur}&=\frac{1}{2}\partial _{u}^{2}\ln \sqrt{q} \end{aligned}$$

(7.131)

$$\begin{aligned} \pounds _{\ell }N^{r}_{AB}&=-\alpha \partial _{u}q_{AB}+\frac{1}{2}\partial _{u}^{2}q_{AB} \end{aligned}$$

(7.132)

Combining all the pieces and using the results $\Theta =\partial _{u}\ln \sqrt{q}$ and $\Theta _{AB}=(1/2)\partial _{u}q_{AB}$, which is the only non-zero component of $\Theta _{ab}$, [16] we finally obtain

$$\begin{aligned} \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij}&=2\pounds _{\ell }N^{r}_{ur}+q^{AB}\pounds _{\ell }N^{r}_{AB} \nonumber \\&=-2\alpha \partial _{u}\ln \sqrt{q}+2\partial _{u}^{2}\ln \sqrt{q}-\frac{1}{2}\partial _{u}q_{AB}\partial _{u}q^{AB} \nonumber \\&=2\partial _{u}\alpha +2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +\frac{2}{\sqrt{q}}\dfrac{d ^{2}\sqrt{q}}{du^{2}} -\frac{2}{\sqrt{q}}\dfrac{d}{du}\left( \sqrt{q}\alpha \right) \end{aligned}$$

(7.133)

which can also be obtained from a completely different viewpoint. For sake of completeness we will illustrate the alternative methods as well. For the null vector $\ell ^{a}$ in the adapted GNC system we have:

$$\begin{aligned} \left( \ell ^{c}\nabla _{c}\ell ^{a}\right) ^{u}&=\alpha +r\beta ^{2}+\mathcal {O}(r^{2});\qquad \left( \ell ^{c}\nabla _{c}\ell ^{a}\right) ^{r}=r\partial _{u}\alpha +2r\alpha ^{2}+\mathcal {O}(r^{2}) \nonumber \\ \left( \ell ^{c}\nabla _{c}\ell ^{a}\right) ^{A}&=r\alpha \beta ^{A}+rq^{CA}\partial _{C}\alpha +\mathcal {O}(r^{2}) \end{aligned}$$

(7.134)

Hence on $r=0$ surface, we have $\kappa =\alpha$, as well as, $\tilde{\kappa }=-(1/2)k^{a}\nabla _{a}\ell ^{2}=\alpha$. Now we will use the Raychaudhuri equation to get $R_{ab}\ell ^{a}\ell ^{b}$ and hence the Lie variation term. In this case we have, $du=d\lambda$, thus Raychaudhuri equation reduces to the following form (see Eq. (7.66))

$$\begin{aligned} \ell ^{a}\nabla _{a}\left( \Theta +2\alpha \right)&=\nabla _{c}\left( \ell ^{a}\nabla _{a}\ell ^{c}\right) -\nabla _{a}\ell _{b}\nabla ^{b}\ell ^{a}-R_{ab}\ell ^{a}\ell ^{b} \end{aligned}$$

(7.135)

where, the $\Theta +2\alpha$ term comes from $\nabla _{i}\ell ^{i}$. Then we have,

$$\begin{aligned} \nabla _{c}\left( \ell ^{a}\nabla _{a}\ell ^{c}\right)&=\partial _{c}\left( \ell ^{a}\nabla _{a}\ell ^{c}\right) +\ell ^{a}\nabla _{a}\ell ^{c}\partial _{c}\ln \sqrt{q} \nonumber \\&=2\alpha ^{2}+\alpha \partial _{u}\ln \sqrt{q}+2\partial _{u}\alpha \end{aligned}$$

(7.136)

Thus non zero components of $B_{ab}=\nabla _{a}\ell _{b}$ are as follows:

$$\begin{aligned} B_{ur}=\alpha ;\qquad B_{rA}=\frac{1}{2}\beta _{A};\qquad B_{AC}=\frac{1}{2}\partial _{u}q_{AC} \end{aligned}$$

(7.137)

From which it can be easily derived that, $B_{ab}B^{ba}=2\alpha ^{2}-(1/4)\partial _{u}q_{AB}\partial _{u}q^{AB}$. Thus we obtain

$$\begin{aligned} R_{ab}\ell ^{a}\ell ^{b}&=-\partial _{u}\Theta +2\alpha ^{2}+\Theta \alpha -B_{ab}B^{ba} \nonumber \\&=\alpha \Theta -\frac{1}{2}q^{AB}\partial _{u}^{2}q_{AB} -\frac{1}{4}\partial _{u}q_{AB}\partial _{u}q^{AB} \nonumber \\&=\alpha \Theta -\frac{1}{\sqrt{q}}\partial _{u}^{2}\sqrt{q}+\left( \partial _{u}\ln \sqrt{q}\right) ^{2}-\Theta _{ab}\Theta ^{ab} \end{aligned}$$

(7.138)

where $\Theta _{ab}$ has the only non-zero component to be, $\Theta _{AB}=(1/2)\partial _{u}q_{AB}$. For the GNC null normal $\ell _{a}$, the Noether current vanishes, such that Lie variation of $N^{a}_{bc}$ turns out to have the following expression

$$\begin{aligned} \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij}&=-2R_{ab}\ell ^{a}\ell ^{b} \nonumber \\&=2\partial _{u}\alpha +2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +\frac{2}{\sqrt{q}}\dfrac{d ^{2}\sqrt{q}}{du^{2}} -\frac{2}{\sqrt{q}}\dfrac{d}{du}\left( \sqrt{q}\alpha \right) \end{aligned}$$

(7.139)

The components of $S_{ab}=\nabla _{a}\ell _{b}+\nabla _{b}\ell _{a}$ In GNC are as follows:

$$\begin{aligned} S_{uu}&=2r\partial _{u}\alpha -4r\alpha ^{2}+\mathcal {O}(r^{2});\qquad S_{ur}=2\alpha +2r\partial _{r}\alpha +r\beta ^{2}+\mathcal {O}(r^{2}) \nonumber \\ S_{uA}&=-r\beta ^{B}\partial _{u}q_{AB}+2r\partial _{A}\alpha -2r\alpha \beta _{A}+\mathcal {O}(r^{2});\qquad S_{rr}=0 \nonumber \\ S_{rA}&=\beta _{A}+r\partial _{r}\beta _{A}-r\beta ^{C}\partial _{r}q_{CA}+\mathcal {O}(r^{2}) \nonumber \\ S_{AB}&=\partial _{u}q_{AB}+2r\alpha \partial _{r}q_{AB}+r\left( D_{A}\beta _{B}+D_{B}\beta _{A}\right) +\mathcal {O}(r^{2}) \end{aligned}$$

(7.140)

Thus the trace at $r=0$ leads to: $S=4\alpha +2\partial _{u}\ln \sqrt{q}$. Thus we arrive at the following expression (see Eq. (7.83) of Appendix A.1)

$$\begin{aligned} \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij}&=2\partial _{u}\left( \Theta +2\alpha \right) -\partial _{b}S^{rb}-\Gamma ^{r}_{bc}S^{bc}-S^{rc}\partial _{c}\ln \sqrt{q} \end{aligned}$$

(7.141)

Then the upper components of $S_{ab}$ necessary for the above computation are the followings:

$$\begin{aligned} S^{ur}&=S_{ur}+r\beta ^{A}S_{rA}=2\alpha +2r\partial _{r}\alpha +2r\beta ^{2}+\mathcal {O}(r^{2}) \nonumber \\ S^{rr}&=2r\partial _{u}\alpha +4r\alpha ^{2}+\mathcal {O}(r^{2}) \nonumber \\ S^{rA}&=4\alpha r\beta ^{A}+2rq^{AB}\partial _{A}\alpha -2r\alpha \beta ^{A}+\mathcal {O}(r^{2}) \end{aligned}$$

(7.142)

The mixed components leads to nothing new so we have not presented them. From Eq. (7.141) the expression for Lie derivative can be obtained as:

$$\begin{aligned} \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij}&=2\partial _{u}\left( \Theta +2\alpha \right) -\partial _{u}S^{ru}-\partial _{r}S^{rr}-\partial _{A}S^{rA}+4\alpha ^{2}+2\Theta _{ab}\Theta ^{ab} -2\alpha \Theta \nonumber \\&=2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +\frac{2}{\sqrt{q}}\dfrac{d}{d\lambda }\left( \sqrt{q}\Theta \right) -2\alpha \Theta +4\partial _{u}\alpha +4\alpha ^{2} -4\partial _{u}\alpha -4\alpha ^{2} \nonumber \\&=2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +\frac{2}{\sqrt{q}}\dfrac{d}{d\lambda }\left( \sqrt{q}\Theta \right) -2\alpha \Theta \end{aligned}$$

(7.143)

which exactly matches with Eq. (7.139). The same can be ascertained for Eq. (7.138) by computing $R_{ab}\ell ^{a}\ell ^{b}$ on the null surface i.e. in the $r\rightarrow 0$ limit, directly leading to:

$$\begin{aligned} R_{ab}\ell ^{a}\ell ^{b}&=R_{uu}=\partial _{a}\Gamma ^{a}_{uu}-\partial _{u}\Gamma ^{a}_{ua} +\Gamma ^{a}_{uu}\Gamma ^{b}_{ab}-\Gamma ^{a}_{ub}\Gamma ^{b}_{ua} \nonumber \\&=\partial _{u}\Gamma ^{u}_{uu}+\partial _{r}\Gamma ^{r}_{uu}+\partial _{A}\Gamma ^{A}_{uu} -\partial _{u}^{2}\ln \sqrt{q}+\Gamma ^{u}_{uu}\partial _{u}\ln \sqrt{q}-\Gamma ^{a}_{ub}\Gamma ^{b}_{ua} \nonumber \\&=2\alpha ^{2}-\partial _{u}^{2}\ln \sqrt{q}+\alpha \partial _{u}\ln \sqrt{q} -\Gamma ^{u}_{ub}\Gamma ^{b}_{uu}-\Gamma ^{r}_{ub}\Gamma ^{b}_{ur}-\Gamma ^{A}_{ub}\Gamma ^{b}_{uA} \nonumber \\&=-\partial _{u}^{2}\ln \sqrt{q}+\alpha \partial _{u}\ln \sqrt{q}-\Theta _{ab}\Theta ^{ab} \end{aligned}$$

(7.144)

which under some manipulations will match exactly with Eq. (7.138). Then in GNC we obtain in identical fashion, the following expression for heat density,

$$\begin{aligned} \mathcal {S}&=\nabla _{i}\ell _{j}\nabla ^{j}\ell ^{i}-\left( \nabla _{i}\ell ^{i}\right) ^{2} \nonumber \\&=\left( 2\alpha ^{2}-(1/4)\partial _{u}q _{AB}\partial _{u}q ^{AB}\right) -\left( \Theta +2\alpha \right) ^{2} \nonumber \\&=-2\alpha ^{2}-4\alpha \Theta -\Theta ^{2}+\Theta _{ab}\Theta ^{ab} \end{aligned}$$

(7.145)

This on integration over the null surface leads to,

$$\begin{aligned} \frac{1}{8\pi }\int du d^{2}x\sqrt{q}\mathcal {S}&=\frac{1}{8\pi }\int dud^{2}x\sqrt{q}\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) \nonumber \\&-\frac{1}{4\pi }\int du d^{2}x\sqrt{q}\alpha ^{2}-4\int d^{2}xTds \end{aligned}$$

(7.146)

Let us now write the integral form of $R_{ab}\ell ^{a}\ell ^{b}$, for that we note the integration measure to be $dud^{2}x\sqrt{q}$. Thus on integration with proper measure and $(1/8\pi )$ factor leads to

$$\begin{aligned} \frac{1}{8\pi }\int dud^{2}x\sqrt{q}R_{ab}\ell ^{a}\ell ^{b}&=-\frac{1}{8\pi }\int du d^{2}x\sqrt{q}\mathcal {D} -\frac{1}{8\pi }\dfrac{d\mathcal {A}_{\perp }}{d\lambda }\Big \vert _{1}^{2} \nonumber \\&+\int d^{2}x Ts\vert _{1}^{2}-\int d^{2}x sdT \end{aligned}$$

(7.147)

which can be written in a slightly modified manner as:

$$\begin{aligned} \frac{1}{8\pi }\int dud^{2}x\sqrt{q}R_{ab}\ell ^{a}\ell ^{b}&=-\frac{1}{8\pi }\int du d^{2}x\sqrt{q}\mathcal {D} -\frac{1}{8\pi }\dfrac{d\mathcal {A}_{\perp }}{d\lambda }\Big \vert _{1}^{2} +\int d^{2}x Tds \end{aligned}$$

(7.148)

Also the Lie variation term (with all the surface contributions kept) on being integrated over the null surface we obtain

$$\begin{aligned} \frac{1}{16\pi }\int dud^{2}x\sqrt{q}&\times \ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij} \nonumber \\&=\frac{1}{8\pi }\int du d^{2}x\sqrt{q}\mathcal {D} +\frac{1}{8\pi }\dfrac{d\mathcal {A}_{\perp }}{d\lambda }\Big \vert _{1}^{2} -\int d^{2}x~\left( \frac{\alpha }{2\pi }\right) d\left( \frac{\sqrt{q}}{4}\right) \nonumber \\&=-\int d^{2}x~Tds+\frac{1}{8\pi }\int du d^{2}x\sqrt{q}\mathcal {D} +\frac{1}{8\pi }\dfrac{d\mathcal {A}_{\perp }}{d\lambda }\Big \vert _{1}^{2} \end{aligned}$$

(7.149)

To calculate Lie variation for $\xi ^{a}$ we need to calculate $\nabla _{a}\xi _{b}+\nabla _{b}\xi _{a}=S_{ab}$. This tensor has the following components:

$$\begin{aligned} S_{uu}&=-2r\partial _{u}\alpha ,\qquad S_{ur}=0 \nonumber \\ S_{uA}&=-r\partial _{u}\beta _{A},\qquad S_{rr}=0 \nonumber \\ S_{rA}&=0.\qquad S_{AB}=\partial _{u}q_{AB} \end{aligned}$$

(7.150)

Thus in the null limit obtained from the relation: $r\rightarrow 0$ we arrive at the result that all the components of $S_{ab}$ vanishes except for the $S_{AB}$ components. If we want to satisfy the Killing condition for $\xi ^{a}$ on the null surface we would require $\partial _{u}q_{AB}=0$. From the above relations it is clear that $\nabla _{a}\xi ^{a}=\Theta$. Moreover we also have,

$$\begin{aligned} \kappa&=-k_{b}\xi ^{a}\nabla _{a}\xi ^{b}=-\Gamma ^{b}_{ac}k_{b}\xi ^{a}\xi ^{c} =\Gamma ^{u}_{uu}=\alpha \end{aligned}$$

(7.151a)

$$\begin{aligned} \tilde{\kappa }&=-\frac{1}{2}k_{b}\nabla ^{b}\xi ^{2}=\frac{1}{2}\partial _{r}\left( -2r\alpha \right) =-\alpha \end{aligned}$$

(7.151b)

which shows that for $\xi ^{a}$, $\kappa =\tilde{\kappa }$. Thus even without the condition $\partial _{u}q_{AB}=0$, we arrive at the relation $\kappa =-\tilde{\kappa }=\alpha$. Moreover Lie variation of $N^{a}_{bc}$ along $\xi ^{a}$ can be obtained by computing the following objects:

$$\begin{aligned} \frac{1}{2}\Big (\delta ^{a}_{b}\nabla _{c}\nabla _{d}\xi ^{d}&+\delta ^{a}_{c}\nabla _{b}\nabla _{d}\xi ^{d}\Big )^{r}_{ur}=\frac{1}{2}\partial _{u}\Theta \end{aligned}$$

(7.152)

$$\begin{aligned} \Big (\nabla _{b}\nabla _{c}\xi ^{a}&+\nabla _{c}\nabla _{b}\xi ^{a}\Big )^{r}_{ur}=-2\partial _{u}\alpha \end{aligned}$$

(7.153)

$$\begin{aligned} \Big [-\frac{1}{2}\Big (R^{a}_{~bmc}&+R^{a}_{~cmb}\Big )\xi ^{m}\Big ]^{r}_{ur}=0 \end{aligned}$$

(7.154)

$$\begin{aligned} \Big (\nabla _{b}\nabla _{c}\xi ^{a}&+\nabla _{c}\nabla _{b}\xi ^{a}\Big )^{r}_{AB}=-\alpha \partial _{u}q_{AB}-\frac{1}{2}q^{CD}\partial _{u}q_{AC}\partial _{u}q_{BD} \end{aligned}$$

(7.155)

$$\begin{aligned} \Big [-\frac{1}{2}\Big (R^{a}_{~bmc}&+R^{a}_{~cmb}\Big )\xi ^{m}\Big ]^{r}_{AB}=-\frac{1}{2}\alpha \partial _{u}q_{AB}+\frac{1}{2}\partial _{u}^{2}q_{AB}-\frac{1}{4}q^{CD}\partial _{u}q_{AC}\partial _{u}q_{BD} \end{aligned}$$

(7.156)

which can be used to obtain the Lie variation term associated with $\xi ^{a}$ as,

$$\begin{aligned} \ell _{a}g^{ij}\pounds _{\xi }N^{a}_{ij}&=2\partial _{u}\alpha +2\left( \Theta _{ab}\Theta ^{ab}-\Theta ^{2}\right) +\frac{2}{\sqrt{q}}\partial _{u}^{2}\sqrt{q} \nonumber \\&=\frac{2}{\sqrt{q}}\partial _{u}\left( \alpha \sqrt{q}\right) +\ell _{a}g^{ij}\pounds _{\ell }N^{a}_{ij} \end{aligned}$$

(7.157)

Then using the momentum $\Pi ^{ab}=\sqrt{q}[\Theta ^{ab}-q^{ab}(\Theta +\kappa )]$ conjugate to the induced metric $q_{ab}$ from Eq. (7.90) we immediately arrive at,

$$\begin{aligned} -q_{ab}\pounds _{\xi }\Pi ^{ab}=\sqrt{q}\ell _{a}g^{ij}\pounds _{\xi }N^{a}_{ij}-\dfrac{d^{2}\sqrt{q}}{d\lambda ^{2}} \end{aligned}$$

(7.158)

These expressions are used to obtain Eq. (7.19). Also the variational principles in this context are:

$$\begin{aligned} Q_{1}&=\int d\lambda d^{2}x\sqrt{q}\left( -\frac{1}{8\pi }R_{ab}\ell ^{a}\ell ^{b} +T_{ab}\ell ^{a}\ell ^{b}\right) \nonumber \\&=\int d\lambda d^{2}x\sqrt{q}\Big [\frac{1}{8\pi }\mathcal {D} +T_{ab}\ell ^{a}\ell ^{b}\Big ] -\int d^{2}x~Tds+\frac{1}{8\pi }\dfrac{d\mathcal {A}_{\perp }}{d\lambda }\Big \vert _{1}^{2} \end{aligned}$$

(7.159a)

$$\begin{aligned} Q_{2}&=\int d\lambda d^{2}x \sqrt{q}\left[ \frac{1}{16\pi }\ell _{a}g^{ij}\pounds _{\xi }N^{a}_{ij} +T_{ab}\ell ^{a}\ell ^{b}\right] \nonumber \\&=\int d\lambda d^{2}x \sqrt{q}\Big [\frac{1}{8\pi }\mathcal {D} +T_{ab}\ell ^{a}\ell ^{b}\Big ]+\int d^{2}x~sdT+\frac{1}{8\pi }\dfrac{d\mathcal {A}_{\perp }}{d\lambda }\Big \vert _{1}^{2} \end{aligned}$$

(7.159b)

These are the expressions used in Sect. 7.4.3.