Aspects of Quasi-local energy for gravity coupled to gauge fields

The notion of a local mass density of pure gravity is non-existent due to the equivalence principle. For an isolated self-gravitating system where the spacetime is asymptotically flat, one defines the notion of mass as a flux integral over a space-like topological 2-sphere located at infinity. For such a system, this so-called ADM mass satisfies the desired positivity property [1, 2, 3]. One also defines a notion of mass at the null-infinity [4, 5] as well. On the other hand, a notion of mass in-between scales is of extreme importance due to the fact that most physical models extend over a finite region. For example, if one were to study the kinematics of a black hole on a curved background, the first entity that one would require is the mass associated with a topological $2-$ sphere that encloses the black hole. There are several problems in classical general relativity that require a notion of quasi-local mass as well. Penrose’s singularity theorem [6] essentially entails the study of the dynamics of space-like $2-$surfaces foliating the future null cone of an arbitrary point in the spacetime. In other words, the formation of black-hole is hinted at by the congruence of the future null geodesics measured by the trace of the null second fundamental form of topological $2-$spheres foliating the outgoing null-hypersurfaces generated by such null geodesics(later [7] proved the dynamical formation of such trapped surfaces from initial data that did not contain the ‘trapped’ characteristics in case of pure vacuum gravity). In such a case, one would want to study the evolution of the gravitational energy that is contained within such topological $2-$spheres and understand if such an evolution exhibits any special characteristics that hint towards a possible singularity formation. In order to even study such an evolution problem, one would need an expression of the energy contained within the surface of interest necessitating a formulation of quasi-local mass/energy. In addition, a formulation (let alone proof) of the hoop conjecture [8, 9] of general relativity requires a notion of quasi-local mass since this conjecture essentially deals with the question of the implosion of an object to a limit where a circular hoop of circumference $2\pi r_{s}$ ($r_{s}$ being the Schwarzschild radius) can be placed around it. Such a limit corresponds to a point of no return or eventual black hole formation similar to the formation of a trapped surface. One requires a notion of mass of the object under study that can be formulated as a quasi-local mass of a topological $2-$sphere bounding the object to address such conjecture. In addition, the study of black hole collision and merging also requires an appropriate notion of quasi-local mass.

Motivated by the aforementioned physically relevant questions, several authors defined different notions of quasi-local mass over the years. A necessary property of such a quasi-local mass would be that it should be positive for a space-like $2-$surface embedded in a curved spacetime and vanishes identically for any such $2-$surface in the Minkowski space. In addition, it should encode the information about pure gravitational energy (Weyl curvature effect) as well as the stress-energy tensor of any source fields present in the spacetime. Based on a Hamilton-Jacobi analysis, Brown-York [10, 11] and Li-Yau [12, 13] defined a quasi-local mass by isometrically embedding the $2-$surface into the reference Euclidean $3-$ space and comparing the extrinsic geometry (the formulation relied on the embedding theorem of Pogorelov [27] i.e., the topological $2-$sphere needed to posses everywhere non-negative sectional curvature). However, [14] discovered surfaces in Minkowski space that do have strictly positive Brown-York and Li-Yau mass. It appeared that these formulations were lacking a prescription of momentum information. This led Wang and Yau [15, 16] to define the most consistent notion (till today) of the quasi-local mass associated with a space-like topological $2-$surface. It involves isometric embedding of the topological $2-$sphere bounding a space-like domain in the physical spacetime satisfying dominant energy condition (energy can not flow into a past light cone of an arbitrary point in spacetime; essentially finite propagation speed) into the Minkowski spacetime instead of Euclidean $3-$space. This formulation relies on a weaker condition on the sectional curvature of the $2-$surface of interest and solvability of Jang’s equation [17] with prescribed Dirichlet boundary data. The Wang-Yau quasi-local mass is then defined as the infimum of the Wang-Yau quasi-local energy among all physical observers that is found by solving an optimal isometric embedding equation. This Wang-Yau quasi-local mass possesses several good properties that are desired on a physical ground. This mass is strictly positive for $2-$surfaces bounding a space-like domain in a curved spacetime that satisfies the dominant energy condition and it identically vanishes for any such $2-$surface in Minkowski spacetime. In addition, it coincides with the ADM mass at space-like infinity [18] and Bondi-mass at null infinity [19], and reproduces the time component of the Bel-Robinson tensor (a pure gravitational entity) together with the matter stress-energy at the small sphere limit, that is when the $2-$sphere of interest is evolved by the flow of its null geodesic generators and the vertex of the associated null cone is approached [20]. In addition, explicit conservation laws were also discovered at the asymptotic infinity [21].

These physically desired features led to the belief that Wang-Yau quasi-local mass may be the one to consider as an appropriate notion of a quasi-local mass modulo few technicalities such as the mean curvature vector of the $2-$surface of interest is restricted to be space-like [15, 16]. Therefore, it is important to apply the Wang-Yau formalism to physically interesting spacetimes, explicitly compute the mass of a space-like domain bounded by a topological $2-$surface, and understand its properties. In this article, we are interested in spacetimes where an additional gauge field is coupled to gravity. The gauge fields are particularly interesting since their evolution is not free due to gauge invariance and therefore constraints must be solved on each Cauchy hypersurface (spatial hypersurface that is met exactly once by every inextendible causal curve) mush like the pure gravity problem itself. Due to these constraints, additional boundary terms appear in the expression of quasi-local energy through Hamilton-Jacobi analysis that is not controlled by the Wang-Yau quasi-local energy expression (unlike sources with non-gauge degrees of freedom where Wang-Yau quasi-local energy is sufficient to control their energy content in addition to pure gravitational energy as well). This additional contribution arises as a consequence of non-vanishing charge associated with the gauge field. Therefore it turns out to be extremely important in the context of charged black-holes (electromagnetic or Yang-Mills) since we expect there must be a charge-mass inequality in order for the interior singularity to be hidden or in technical terms, the black-holes should be of sub-extremal type since exposing the interior singularity to a causal observer located in the domain of outer communication signals a pathological breakdown of the classical general relativity. Since we consider the notion of mass to be Wang-Yau quasi-local mass, this translates to obtaining a suitable inequality relating the charge of the gauge field and the W-Y quasi-local mass of membranes enclosing the black hole.

In addition, one of the other main motivation of the current study is the Bekenstein’s inequality relating total energy of a relativistic object bounded by a membrane, its charge, and the angular momentum. Based on physical arguments, Bekenstein proposed an upper bound of the entropy of the object which takes the following form in the natural unit $\hbar=c=\kappa=1$ [22]

$\displaystyle\mathcal{S}\leq\sqrt{\mathcal{E}^{2}R^{2}-\mathcal{J}^{2}}-\frac{Q^{2}}{2},$

(1)

where $\mathcal{S},\mathcal{E},\mathcal{J},Q$, and $R$ are the objects entropy, total energy, angular momentum, charge, and size (the radius of smallest sphere containing it), respectively. This inequality proves to be difficult to establish in a rigorous way. Nevertheless, if one simply assumes the non-negativity of the entropy, then a weaker version of the above inequality reads

$\displaystyle\mathcal{E}^{2}\geq\frac{\mathcal{J}^{2}}{R^{2}}+\frac{Q^{4}}{4R^{2}}$

(2)

and for an object with non-negative angular momentum

$\displaystyle\mathcal{E}\geq\frac{Q^{2}}{2R}.$

(3)

As a first step of the proof of this inequality, one requires a physically reasonable definition of the energy contained within a region of non-zero charge bounded by a membrane in the fully general relativistic setting. A natural choice would be to consider the Wang-Yau quasi-local energy and try to establish the previous inequality. We note that recently [23] studied this inequality for several definitions of the quasi-local energy.

The structure of the article is as follows. Starting with the classical Hilbert action of the Einstein-Yang-Mills system (including a $U(1)$ Yang-Mills as well), we derive the boundary contribution to the quasi-local energy that arises solely due to the gauge fields through an ADM Hamilton-Jacobi analysis. Later, we specialize in the Kerr-Newman spacetime and explicitly compute the boundary contribution. Simultaneously, we compute the Wang-Yau energy functional and establish the necessary condition for it to be well behaved. We later show that the trivial data solves the optimal isometric embedding equation and the associated mass is positive by construction if the angular momentum of the black hole is not too large. This yields an inequality between the charge of the black hole and the quasi-local energy of membranes enclosing it. This total quasi-local energy is shown to exhibit monotonic decay in the radially outward direction from the outer horizon under the assumption of a small angular momentum. A few physical conclusions are drawn on the basis of our results and a few additional problems are discussed that are to be handled in the potential future.

In this section, we derive the contribution of the gauge field to the expression of the quasi-local mass. We consider a ‘1+3’ dimensional $C^{\infty}$ globally hyperbolic spacetime $\mathbb{R}\times M$, where $M$ is diffeomorphic to a Cauchy hypersurface, equipped with a Lorentzian metric $\hat{g}$. Since this globally hyperbolic spacetime is foliated by the space-like submanifolds $M_{t}$ ($t$ is a time function which is well defined for a globally hyperbolic spacetime), we may use a lapse function $N$ and a $M-$parallel shift vector field $Y$ to represent $\hat{g}$ in the following ADM form [24]

$\displaystyle\hat{g}:=-N^{2}dt\otimes dt+g_{ij}(dx^{i}+Y^{i}dt)\otimes(dx^{j}+Y^{j}dt),$

(4)

where $g$ is the Riemannian metric induced on $M$ by the embedding $i:M\hookrightarrow\mathbb{R}\times M$. The second fundamental form $K$ measuring the extrinsic geometry of $M$ in $\mathbb{R}\times M$ is defined by $K(X,Z):=-\hat{g}(\nabla[\hat{g}]_{X}\mathbf{n},Z),$ for all $X,Z$ being the sections of $TM$. Here $\mathbf{n}$ is the $t=$constant hypersurface orthogonal unit time-like vector field defined as $\mathbf{n}:\frac{1}{N}(\partial_{t}-Y)$.

To formulate the Yang-Mills theory over the spacetime $\mathbb{R}\times M$, we first choose a compact semisimple Lie group $G$. If a section of the principle $G-$bundle defined over $\mathbb{R}\times M$ is chosen and the connection is pulled back to the base manifold, then it yields a $1-$form field on the base which takes values in the Lie algebra $\mathfrak{g}$ of $G$. Let us consider the dimension of the group $G$ to be $dim_{G}$ and since $\mathfrak{g}:=T_{e}G$, it has a natural vector space structure. Assume that the vector space $\mathfrak{g}$ has a basis $\{\chi_{A}\}_{A=1}^{dim_{G}}$ given by a set of $k\times k$ real-valued matrices ($k$ being the dimension of the representation $V$ of the Lie algebra $\mathfrak{g}$). The connection $1-$form field is then defined to be

$\displaystyle\hat{A}:=\hat{A}^{A}_{\mu}\chi_{A}dx^{\mu}=\hat{A}^{A}_{\mu}(\chi_{A})^{a}_{b}dx^{\mu}=\hat{A}^{a}~{}_{b\mu}dx^{\mu},~{}a,b=1,2,3,...,k.$

(5)

From now on by the connection 1-form field $\hat{A}_{\mu}$, we will always mean $\hat{A}^{a}~{}_{b\mu}$. In the current setting $\hat{A}\in\Omega^{1}(\mathbb{R}\times M;End(V))$, where $End(V)$ denotes the space of endomorphisms of the vector space $V$. The curvature of this connection is defined to be the Yang-Mills field $F\in\Omega^{2}(\mathbb{R}\times M;End(V))$

$\displaystyle\hat{F}^{a}~{}_{b\mu\nu}:=\partial_{\mu}\hat{A}^{a}~{}_{b\nu}-\partial_{\nu}\hat{A}^{a}~{}_{b\mu}+[\hat{A},\hat{A}]^{a}~{}_{b\mu\nu},$

(6)

where the bracket is defined on the Lie algebra $\mathfrak{g}$ and given by commutator of matrices under multiplication. The Yang-Mills coupling constant is set to unity. Since $G$ is compact, it admits a positive definite adjoint invariant metric on $\mathfrak{g}$. We choose a basis of $\mathfrak{g}$ such that this adjoint invariant metric takes the Cartesian form $\delta_{AB}$ and work with representations for which the bases satisfy

$\displaystyle-\tr(\chi_{A}\chi_{B})=(\chi_{A})^{a}_{b}(\chi_{A})^{b}_{a}=\delta_{AB}.$

(7)

One may for convenience decompose the $1+3$ $\mathfrak{g}$-valued connection $1-$form field $A^{a}~{}_{b\mu}$ into its component parallel and perpendicular to $M$ as follows

$\displaystyle\hat{A}^{a}~{}_{b}=A^{a}~{}_{b}-\hat{g}(A^{a}~{}_{b},\mathbf{n})\mathbf{n}$

(8)

where $A^{a}~{}_{b}$ is a $\mathfrak{g}$ valued $1-$form field parallel to the spatial manifold $M$ i.e., $A\in\Omega^{1}(M;End(V))$. Importantly note that $\hat{A}^{a}~{}_{bi}=A^{a}~{}_{bi}$ but $\hat{A}^{a}~{}_{b}^{i}\neq A^{a}~{}_{b}^{i}$ unless the shift vector field $Y$ vanishes. Similarly, we decompose the Yang-Mills field strength $\hat{F}$ as follows

$\displaystyle\hat{F}^{a}~{}_{b}=\frac{1}{\mu_{g}}(\mathcal{E}^{a}~{}_{b}\otimes\mathbf{n}-\mathbf{n}\otimes\mathcal{E}^{a}~{}_{b})+F^{a}~{}_{b},$

(9)

where $\mathcal{E}\in\Omega^{1}(M;End(V))$ is the electric field, $F\in\Omega^{2}(M;End(V))$ is related to the magnetic field, and $\mu_{g}:=\sqrt{\det(g_{ij})}$. Once again note $\hat{F}^{a}~{}_{bij}=F^{a}~{}_{bij}$ but $\hat{F}^{a}~{}_{b}~{}^{ij}\neq F^{a}~{}_{b}~{}^{ij}$. The Einstein-Hilbert action for a gauge-gravity system in the natural unit $G=1=\wp$ ($\wp$ is the Yang-Mills coupling constant) may be written as follows [25]

	$\displaystyle 8\pi\mathcal{S}$	$\displaystyle:=$	$\displaystyle\frac{1}{2}\int_{I\times M}\mathcal{R}(\hat{g})\mu_{\hat{g}}-\frac{1}{4}\int_{I\times M}\hat{F}^{a}~{}_{b\mu\nu}F^{b}~{}_{a\alpha\beta}\hat{g}^{\alpha\mu}\hat{g}^{\beta\nu}\mu_{\hat{g}}$
			$\displaystyle+\int_{M_{t}-M_{t_{0}}}\tr_{g}K\mu_{\hat{g}}|_{M},$

where $I\subset\mathbb{R}$ denotes a closed interval on the real line i.e., $I:=[t_{0},t]$, $\mu_{\hat{g}}:=\sqrt{|\det(\hat{g}_{\mu\nu})|}$, $K\tr_{g}$ is the mean extrinsic curvature of the constant time hypersurface $M$, and $\mu_{\hat{g}}|_{M}$ is the volume form induced on $M$. $M_{t}-M_{t_{0}}$ denotes the difference of the integrals over hypersurfaces $t=t$ and $t=t_{0}$. Through the ADM decomposition (4), one may introduce the canonical pairs ($g_{ij},\pi^{ij}:=-\mu_{g}(K^{ij}-\tr_{g}Kg^{ij})$) for the gravity and $(A^{a}~{}_{bi},\mathcal{E}^{a}~{}_{b}~{}^{i}:=\frac{\mu_{g}}{N}g^{ik}(\hat{F}^{a}~{}_{b0k}-\hat{F}^{a}~{}_{bjk}Y^{j}))$ for the Yang-Mills theory. Now, using the same ADM formalism, we may obtain an expression for the quasi-local energy associated with a $2-$surface bounding a domain in $M$ for this gauge-gravity coupled system through the Hamilton-Jacobi analysis. The following lemma describes the result.

Lemma 1: Let $\Sigma$ be a $2-$surface bounding a domain $\Omega$ in $M$ i.e., $\partial\Omega=\Sigma$ and $\nu$ be its space-like outward unit normal vector. Also assume that the Einstein-Hilbert action (2) for a gauge-gravity coupled system is defined on $[t_{0},t]\times\Omega$. Then the quasi-local Hamiltonian defined by $\mathcal{H}_{ql}:=-\partial_{t}\mathcal{S}$ for this Einstein-Yang-Mills system with a compact semi-simple gauge group verifies the following expression

$\displaystyle\mathcal{H}_{ql}:=-\frac{\partial\mathcal{S}}{\partial t}=-\frac{1}{8\pi}\int_{\Sigma_{t}}(kN-\frac{\pi^{ij}}{\mu_{\Sigma}}\nu_{i}Y_{j}-\frac{\mathcal{E}^{a}~{}_{b}~{}^{i}}{\mu_{\Sigma}}A^{b}~{}_{a0}\nu_{i})\mu_{\Sigma}d^{2}x,$

(11)

where $k=\tr_{g}K$, and $\mu_{\Sigma}$ is the induced volume form on $\Sigma$.

Proof: Let us start with the Einstein-Hilbert action $\mathcal{S}:=\frac{1}{2}\int_{[t_{0},t]\times\Omega}\mathcal{R}(\hat{g})\mu_{\hat{g}}-\frac{1}{4}\int_{[t_{0},t]\times\Omega}\hat{F}^{a}~{}_{b\mu\nu}F^{b}~{}_{a\alpha\beta}\hat{g}^{\alpha\mu}\hat{g}^{\beta\nu}\mu_{\hat{g}}+\int_{\Omega_{t}-\Omega_{t_{0}}}\tr_{g}K\mu_{\hat{g}}|_{\Omega}$ and reduce it utilizing the ADM splitting (4). Straightforward calculations yield

$\displaystyle\mathcal{R}(\hat{g})=R(g)+K_{ij}K^{ij}-(\tr_{g}K)^{2}-2(\nabla_{\mu}(n^{\nu}\nabla_{\nu}n^{\mu})-\nabla_{\nu}(n^{\nu}\nabla_{\mu}n^{\mu}))$

(12)

and

	$\displaystyle\frac{1}{4}\hat{F}^{a}~{}_{b\mu\nu}F^{b}~{}_{a\alpha\beta}$	$\displaystyle=$	$\displaystyle-\frac{1}{2N^{2}}g^{ik}(\hat{F}^{a}~{}_{b0i}-F^{a}~{}_{bji}X^{j})(\hat{F}^{b}~{}_{a0k}-F^{b}~{}_{ajk}X^{j})$
			$\displaystyle+\frac{1}{4}g^{ik}g^{jm}F^{a}~{}_{bij}F^{b}~{}_{akm},$

where we have used the fact that $\hat{F}^{a}~{}_{bij}=F^{a}~{}_{bij}$. Now recall the definitions of the gravitational and Yang-Mills momenta

$\displaystyle\pi^{ij}:=-\mu_{g}(K^{ij}-\tr_{g}Kg^{ij}),~{}\mathcal{E}^{a}_{b}~{}^{i}:=\frac{\mu_{g}}{N}g^{ik}(\hat{F}^{a}~{}_{b0k}-F^{a}~{}_{bjk}Y^{j}),$

(14)

substitution of which yields the following expression for the action $\mathcal{S}(t)$

	$\displaystyle 8\pi\mathcal{S}(t)$	$\displaystyle=$	$\displaystyle\frac{1}{2}\int_{[t_{0},t]}\int_{\Omega}\left(\pi^{ij}\partial_{t}g_{ij}-N\mu_{g}(|K|^{2}_{g}-(\tr_{g}K)^{2}-R(g))-2\pi^{ij}\nabla_{i}Y_{j}\right)d^{3}xdt$
			$\displaystyle+\int_{I}\int_{\Omega}\left(\mathcal{E}^{a}~{}_{b}~{}^{i}\partial_{t}A^{b}~{}_{ai}-\mathcal{E}^{a}~{}_{bi}F^{b}~{}_{aji}Y^{j}+\mathcal{E}^{a}~{}_{b}~{}^{i}[A_{0},A_{i}]^{b}~{}_{a}-\frac{N}{2\mu_{g}}\mathcal{E}^{a}~{}_{b}~{}^{i}\mathcal{E}^{b}~{}_{ai}\right.$
			$\displaystyle\left.-\frac{1}{4}F^{a}~{}_{bij}F^{b}~{}_{a}~{}^{ij}N\mu_{g}-\mathcal{E}^{a}~{}_{b}~{}^{i}\partial_{i}A^{b}~{}_{a0}\right)d^{3}xdt$
			$\displaystyle+\int_{[t_{0},t]}\int_{\Omega}kN\mu_{\Sigma}d^{2}xdt.$

Here we have used the definition of the second fundamental form $K_{ij}=\hat{g}(\nabla[\hat{g}]_{\partial_{i}}n,\partial_{j})=-\frac{1}{2N}(\partial_{t}g_{ij}-L_{Y}g_{ij})$ and integration by parts of $(\nabla_{\mu}(n^{\nu}\nabla_{\nu}n^{\mu})-\nabla_{\nu}(n^{\nu}\nabla_{\mu}n^{\mu}))N\mu_{g}$ to obtain the last boundary term since the first term cancels out by $\int_{\Omega_{t}-\Omega_{t_{0}}}\tr_{g}K\mu_{\hat{g}}|_{\Omega}$. Now notice the following calculations

$\displaystyle\int_{\Omega}\pi^{ij}\nabla_{i}Y_{j}d^{3}x=\int_{\Omega}\nabla_{i}(\pi^{ij}Y_{j})d^{3}x-\int_{\Omega}Y_{j}\nabla_{i}\pi^{ij}d^{3}x,$

(15)

where we may integrate the first term since $\pi$ is a density and therefore $\nabla_{i}(\pi^{ij}Y_{j})=\partial_{i}(\pi^{ij}Y_{j})$ yielding

$\displaystyle\int_{\Omega}\pi^{ij}\nabla_{i}Y_{j}d^{3}x=\int_{\Sigma}\pi^{ij}Y_{j}\nu_{i}d^{2}x-\int_{\Omega}Y_{j}\nabla_{i}\pi^{ij}d^{3}x.$

(16)

In an exact similar way we may reduce the similar term of the Yang-Mills sector

	$\displaystyle\int_{\Omega}\mathcal{E}^{a}~{}_{b}^{i}\partial_{i}A^{b}~{}_{a0}d^{3}x$	$\displaystyle=$	$\displaystyle\int_{\Omega}\partial_{i}(\mathcal{E}^{a}~{}_{b}^{i}A^{b}~{}_{a0})d^{3}x-\int_{\Omega}\partial_{i}\mathcal{E}^{a}~{}_{b}^{i}A^{b}~{}_{a0}d^{3}x$
		$\displaystyle=$	$\displaystyle\int_{\Omega}\mathcal{E}^{a}~{}_{b}~{}^{i}A^{b}~{}_{a0}\mathcal{\nu}_{i}d^{2}x-\int_{\Omega}\partial_{i}\mathcal{E}^{a}~{}_{b}^{i}A^{b}~{}_{a0}d^{3}x.$

Notice that the terms $\int_{\Omega}Y_{j}\nabla_{i}\pi^{ij}d^{3}x$ and $\int_{\Omega}\partial_{i}\mathcal{E}^{a}~{}_{b}^{i}A^{b}~{}_{a0}d^{3}x$ give rise to the momentum constraint and the Gauss law constraint, respectively. Assembling all the terms together, we obtain the final expression for the action functional

	$\displaystyle 8\pi\mathcal{S}(t)=\frac{1}{2}\int_{[t_{0},t]}\int_{\Omega}\left(\pi^{ij}\partial_{t}g_{ij}-N\mu_{g}(|K|^{2}_{g}-k^{2}-\mathcal{R}(g)+\frac{1}{\mu^{2}_{g}}\mathcal{E}^{a}~{}_{b}~{}^{i}\mathcal{E}^{b}~{}_{ai}\right.$
	$\displaystyle\left.+\frac{1}{2}F^{a}~{}_{bij}F^{b}~{}_{a}~{}^{ij})+(2\nabla_{i}\pi^{i}_{j}-\mathcal{E}^{a}~{}_{b}^{i}F^{b}~{}_{aji})Y^{j}\right)d^{3}xdt$
	$\displaystyle+\int_{[t_{0},t]}\underbrace{\int_{\Sigma}(kN-\frac{\pi^{ij}}{\mu_{\Sigma}}\nu_{i}Y_{j})\mu_{\Sigma}d^{2}x}_{gravitational~{}energy~{}modulo~{}a~{}reference~{}term}dt$
	$\displaystyle+\int_{[t_{0},t]}\int_{\Omega}\left(\mathcal{E}^{a}~{}_{b}~{}^{i}(\partial_{t}A^{b}~{}_{ai}+[A_{0},A_{i}]^{b}~{}_{a})+\partial_{i}\mathcal{E}^{a}~{}_{b}~{}^{i}A^{b}~{}_{a0}\right)d^{3}xdt$
	$\displaystyle-\int_{[t_{0},t]}\underbrace{\int_{\Sigma}\mathcal{E}^{a}~{}_{b}~{}^{i}A^{b}~{}_{a0}\mathcal{\nu}_{i}d^{2}x}_{surface~{}energy~{}from~{}gauge~{}field}dt.$

Therefore using the Hamilton-Jacobi equation $\partial_{t}\mathcal{S}+\mathcal{H}_{q.l}=0$, we obtain

$\displaystyle\mathcal{H}_{ql}=-\frac{1}{8\pi}\int_{\Sigma}(kN-\frac{\pi^{ij}}{\mu_{\Sigma}}\nu_{i}Y_{j}-\frac{\mathcal{E}^{a}~{}_{b}~{}^{i}}{\mu_{\Sigma}}A^{b}~{}_{a0}\nu_{i})\mu_{\Sigma}d^{2}x.$

(18)

This concludes the proof of the lemma.                                       $\Box$

Wang and Yau [15, 16] dealt with the pure gravitational and non-gauge part of the quasi-local energy by subtracting a well-defined reference contribution (by isometrically embedding $\Sigma$ into Minkowski space). Therefore, we will use their expression to handle the first term $-\int_{\Sigma}(kN-\frac{\pi^{ij}}{\mu_{\Sigma}}\nu_{i}Y_{j})\mu_{\Sigma}d^{2}x$ while computing the quasi-local energy for a given spacetime. They have shown with a suitable choice of the lapse function $N$ and the shift vector field $Y$ that one can define a notion of quasi-local mass which satisfies several properties that are desired on the physical ground. Let us now describe the Wang-Yau quasi-local mass which we wish to evaluate. Let us assume that the mean curvature vector $\mathbf{H}$ of $\Sigma$ is space-like. Let $\mathbf{J}$ be the reflection of $\mathbf{H}$ through the future outgoing light cone in the normal bundle of $\Sigma$. The data that Wang and Yau use to define the quasi-local energy is the triple ($\sigma,|\mathbf{H}|_{\hat{g}},\alpha_{\mathbf{H}}$) on $\Sigma$, where $\sigma$ is the induced metric on $\Sigma$ by the Lorentzian metric $\hat{g}$ or $\mathbb{R}\times M$, $|\mathbf{H}|_{\hat{g}}$ is the Lorentzian norm of $\mathbf{H}$, and $\alpha_{\mathbf{H}}$ is the connection $1-$form of the normal bundle with respect to the mean curvature vector $\mathbf{H}$ and is defined as follows

$\displaystyle\alpha_{\mathbf{H}}(X):=\hat{g}(\nabla[\hat{g}]_{X}\frac{\mathbf{J}}{|\mathbf{H}|},\frac{\mathbf{H}}{|\mathbf{H}|}).$

(19)

Choose a basis pair ($e_{3},e_{4}$) for the normal bundle of $\Sigma$ in the spacetime that satisfy $\hat{g}(e_{3},e_{3})=1,\hat{g}(e_{4},e_{4})=-1$, and $\hat{g}(e_{3},e_{4})=0$. Now embed the $2-$surface $\Sigma$ isometrically into the Minkowski space with its usual metric $\eta$ i.e., the embedding map $X:x^{a}\mapsto X^{\mu}(x^{a})$ satisfies $\sigma(\frac{\partial}{\partial x^{a}},\frac{\partial}{\partial x^{b}})=\langle\frac{\partial X}{\partial x^{a}},\frac{\partial X}{\partial x^{b}}\rangle_{\eta}$, where $\{x^{a}\}_{a=1}^{2}$ are the coordinates on $\Sigma$. Now identify a basis pair ($e_{30},e_{40}$) in the normal bundle of $X(\Sigma)$ in the Minkowski space that satisfy the exact similar property as $(e_{3},e_{4})$. In addition the time-like unit vector $e_{4}$ is chosen to be future directed i.e., $\hat{g}(e_{4},\partial_{t})<0$. Let $\tau:=-\langle X,\partial_{t}\rangle_{\eta}$, a function on $\Sigma$ be the time function of the embedding $X$. The Wang-Yau quasi-local energy is defined as follows

	$\displaystyle\mathcal{QLE}_{gravity}=\frac{1}{8\pi}\underbrace{\int_{\Sigma_{t}}\left(-\sqrt{1+|\nabla\tau|^{2}_{\sigma}}\langle\mathbf{H}_{0},e_{30}\rangle-\langle\nabla[\eta]_{\nabla\tau}e_{30},e_{40}\rangle\right)\mu_{\Sigma}}_{I:=Contribution~{}from~{}the~{}Minkowski~{}space}$		(20)
	$\displaystyle-\frac{1}{8\pi}\underbrace{\int_{\Sigma_{t}}\left(-\sqrt{1+|\nabla\tau|^{2}_{\sigma}}\langle\mathbf{H},e_{3}\rangle-\langle\nabla[\hat{g}]_{\nabla\tau}e_{3},e_{4}\rangle\right)\mu_{\Sigma}}_{II:=contribution~{}from~{}the~{}physical~{}spacetime}.$

However, there is still a gauge redundancy due to the boost transformations in the normal bundle of $\Sigma$. In other words, one is left with the freedom of choosing $e_{3}$ and $e_{4}$ since one may apply a hyperbolic rotation (boost) to yield another pair $(\hat{e}_{3},\hat{e}_{4})$. Wang and Yau [15, 16] considers the following minimization procedure to get rid of this extra gauge freedom. Choose a fixed basis $(\hat{e}_{3},\hat{e}_{4})$ of the fibres of the normal bundle of $\Sigma$ such that the space-like mean curvature vector $\mathbf{H}$ is expressible as

$\displaystyle\mathbf{H}=-|\mathbf{H}|_{\hat{g}}\hat{e}_{3},$

(21)

where $\langle\hat{e}_{3},\hat{e}_{3}\rangle_{\hat{g}}=1$ and $\langle\hat{e}_{4},\hat{e}_{4}\rangle_{\hat{g}}=-1$. Write any general basis $(e_{3},e_{4})$ through a boost transformation in the normal bundle as follows

	$\displaystyle e_{3}=\cosh\psi\hat{e}_{3}-\sinh\psi\hat{e}_{4},$		(22)
	$\displaystyle e_{4}=-\sinh\psi\hat{e}_{3}+\cosh\psi\hat{e}_{4}$		(23)

and substitute these expressions of $e_{3}$ and $e_{4}$ in the expression of $II$, and use the fact that $\hat{g}(\nabla[\hat{g}]_{e_{3}}\hat{e}_{4},e_{3})+\hat{g}(\nabla[\hat{g}]_{e_{4}}\hat{e}_{4},e_{4})=0$ (by construction) to yield

$\displaystyle II=\int_{\Sigma_{t}}\left(\sqrt{1+|\nabla\tau|^{2}_{\sigma}}|\mathbf{H}|_{\hat{g}}\cosh\psi-\alpha_{\hat{e}_{3}}(\nabla\tau)+\psi\Delta\tau\right)\mu_{\Sigma}.$

(24)

This is a convex functional of $\psi$ and therefore is minimized for the following boost

$\displaystyle\psi=\sinh^{-1}(-\frac{\Delta\tau}{|\mathbf{H}|\sqrt{1+|\nabla\tau|^{2}_{\sigma}}}).$

(25)

One repeats the same procedure for the surface with metric $\sigma$ embedded in the Minkowski space and write the Wang-Yau quasi-local energy as follows

	$\displaystyle\mathcal{QLE}_{gravity}=\frac{1}{8\pi}\int_{\Sigma}\left(\sqrt{1+|\nabla\tau|^{2}_{\sigma}}|\mathbf{H}_{0}|_{\eta}\cosh\psi_{0}-\alpha_{\hat{e}_{30}}(\nabla\tau)+\psi_{0}\Delta\tau\right)\mu_{\Sigma}$
	$\displaystyle-\frac{1}{8\pi}\int_{\Sigma}\left(\sqrt{1+|\nabla\tau|^{2}_{\sigma}}|\mathbf{H}|_{\hat{g}}\cosh\psi-\alpha_{\hat{e}_{3}}(\nabla\tau)+\psi\Delta\tau\right)\mu_{\Sigma}.$		(26)

The quasi-local mass is defined to be the minimum of $\mathcal{QLE}_{gravity}$ in the space of the residual embedding function $\tau$. This is so because the isometric embedding of a $2-$surface into a $4$-dimensional manifold provides three constraints out of a total of four degrees of freedom. Therefore, the additional leftover degrees of freedom (in this case $\tau$) needs to be obtained by other physical means. In the current context, therefore, the mass is defined through a minimization procedure in the space of $\tau$ motivated by the definition of the rest mass which is the minimum in the space of observers. Through a variational argument, [15] obtained the following fourth-order elliptic equation for $\tau$

	$\displaystyle-(\hat{H}\hat{\sigma}^{ab}-\hat{\sigma}^{ac}\hat{\sigma}^{bd}\hat{h}_{cd})\frac{\nabla[\sigma]_{a}\nabla_{b}\tau}{\sqrt{1+|\nabla\tau|^{2}_{\sigma}}}+\nabla[\sigma]^{a}(\frac{\nabla_{a}\tau\cosh\psi}{\sqrt{1+|\nabla\tau|^{2}_{\sigma}}}|\mathbf{H}|_{\hat{g}}$		(27)
	$\displaystyle-\nabla_{a}\psi-(\alpha_{\hat{e}_{3}})_{a})=0,$

where $\hat{h}$ is the second fundamental form of $\hat{\Sigma}$ while viewed as a surface in $\mathbb{R}^{3}$. From now on we will use the term mass and energy interchangeably if there is no confusion. The compatible condition that is required to guarantee the isometric embedding is the positivity of the Gauss curvature of $\hat{\Sigma}$ i.e., $K_{\Sigma}+(1+|\nabla\tau|^{2}_{\sigma})^{-1}\det(\nabla[\sigma]_{a}\nabla_{b}\tau)>0$, where $K_{\Sigma}$ is the Gauss curvature of $\Sigma$.

Including the contribution from the pure gauge (Yang-Mills) part, we define the total quasi-local energy as follows

	$\displaystyle\mathcal{QLE}:=\mathcal{QLE}_{gravity}+\mathcal{QLE}_{gauge}$		(28)
	$\displaystyle=\frac{1}{8\pi}\int_{\Sigma}\left(\sqrt{1+|\nabla\tau|^{2}_{\sigma}}|\mathbf{H}_{0}|_{\eta}\cosh\psi_{0}-\alpha_{\hat{e}_{30}}(\nabla\tau)+\psi_{0}\Delta\tau\right)\mu_{\Sigma}$
	$\displaystyle-\frac{1}{8\pi}\int_{\Sigma}\left(\sqrt{1+|\nabla\tau|^{2}_{\sigma}}|\mathbf{H}|_{\hat{g}}\cosh\psi-\alpha_{\hat{e}_{3}}(\nabla\tau)+\psi\Delta\tau\right)\mu_{\Sigma}$
	$\displaystyle-\frac{1}{8\pi}\int_{\Sigma}\hat{F}^{a}~{}_{b}(\hat{e}_{3},\hat{e}_{4})A^{b}~{}_{a0}\mu_{\Sigma}.$

Luckily, due to the anti-symmetry as a $2-$form on the spacetime, $\hat{F}^{a}~{}_{b}(e_{3},e_{4})=\hat{F}(\cosh\psi\hat{e}_{3}-\sinh\psi\hat{e}_{4},-\sinh\psi\hat{e}_{3}+\cosh\psi\hat{e}_{4})=\hat{F}^{a}~{}_{b}(\hat{e}_{3},\hat{e}_{4})$ and therefore is boost invariant. However, notice that the term $\int_{\Sigma}\hat{F}^{a}~{}_{b}(\hat{e}_{3},\hat{e}_{4})A^{b}~{}_{a0}\mu_{\Sigma}$ heavily depends on the Yang-Mills gauge choice. Nevertheless, for a stationary space-time, this term is fixed. In addition, this term can be made to vanish if the spacetime does not contain a horizon. This is due to the fact that in the absence of a horizon, $A^{a}~{}_{b0}$ can always be gauged to zero (such a choice is temporal gauge). However, this term does not vanish for a black-hole spacetime that is electrically charged since a gauge transformation that sets $A^{a}~{}_{b0}$ to zero would be singular at the horizon. The reference contribution to the gauge part vanishes since $A^{a}~{}_{b0}$ may be set to zero on a reference spacetime (Minkowski spacetime in this case) which does not contain any horizon.

In this section, we explicitly compute the Wang-Yau quasi-local energy functional as well as the energy contribution arising from the U(1) gauge sector of the Kerr-Newman spacetime. In Boyer-Lindquist coordinates, the Kerr-Newman metric reads

	$\displaystyle\hat{g}$		(29)
	$\displaystyle=-\frac{r^{2}-2rM+a^{2}\cos^{2}\theta+Q^{2}}{r^{2}+a^{2}\cos^{2}\theta}dt\otimes dt+\frac{r^{2}+a^{2}\cos^{2}\theta}{r^{2}-2rM+a^{2}+Q^{2}}dr\otimes dr$
	$\displaystyle+(r^{2}+a^{2}\cos^{2}\theta)d\theta\otimes d\theta+\frac{a\sin^{2}\theta(Q^{2}-2rM)}{r^{2}+a^{2}\cos^{2}\theta}(dt\otimes d\varphi+d\varphi\otimes dt)$
	$\displaystyle+\left(\frac{(r^{2}+a^{2})(r^{2}+a^{2}\cos^{2}\theta)+2rMa^{2}\sin^{2}\theta-a^{2}Q^{2}\sin^{2}\theta}{r^{2}+a^{2}\cos^{2}\theta}\right)\sin^{2}\theta d\varphi\otimes d\varphi,$

where $M$, $a$, and $Q$ are the mass, angular momentum, and electric charge, respectively. They are defined in such a way as to have the same dimensions. Let us recognize the important surfaces associated with this spacetime. The horizons are obtained by solving the equation $\frac{1}{\hat{g}_{rr}}=0$ i.e., $r^{2}-2rM+a^{2}+Q^{2}=0$ which yields

$\displaystyle R^{+}=M+\sqrt{M^{2}-a^{2}-Q^{2}},~{}R^{-}=M-\sqrt{M^{2}-a^{2}-Q^{2}}.$

(30)

The equation of the ergosphere $\hat{g}_{tt}=0$ yields

$\displaystyle R^{+}_{e}(\theta)=M+\sqrt{M^{2}-a^{2}\cos^{2}\theta-Q^{2}},~{}R^{-}_{e}(\theta)=M-\sqrt{M^{2}-a^{2}\cos^{2}\theta-Q^{2}}$

and the ergo region is the region between $R^{+}$ and $R^{+}_{e}$. The outer horizon is the surface $r=R^{+}$. Throughout this article, we will be interested in the region outside of the outer horizon.

We want to understand the quasi-local energy associated with the constant radius surface $\Sigma$ given by $t=$cosntant, $r=R$. The induced metric $\sigma$ on $\Sigma$ is explicitly written as follows

	$\displaystyle\sigma=(R^{2}+a^{2}\cos^{2}\theta)d\theta\otimes d\theta$		(31)
	$\displaystyle+\left(\frac{(R^{2}+a^{2})(R^{2}+a^{2}\cos^{2}\theta)+2rMa^{2}\sin^{2}\theta-a^{2}Q^{2}\sin^{2}\theta}{R^{2}+a^{2}\cos^{2}\theta}\right)\sin^{2}\theta d\varphi\otimes d\varphi.$

Fix a basis $(\partial_{\theta},\partial_{\varphi})$ of the tangent space of $\Sigma$ at each point. We will denote the elements of this basis set by $\partial_{a}$ ($a=\theta,\varphi$). The corresponding solution of the Maxwell equations yields the connection

$\displaystyle A=\frac{Qr}{r^{2}+a^{2}\cos^{2}\theta}dt-\frac{aQr\sin^{2}\theta}{r^{2}+a^{2}\cos^{2}\theta}d\varphi.$

(32)

Notice that on this fixed stationary spacetimes, $A_{0}$ is non-vanishing, and therefore the quasi-local energy contribution from the $U(1)$ gauge sector is non-vanishing as well. The curvature $\hat{F}$ of the connection $A$ is found to be

	$\displaystyle\hat{F}=-\frac{Q(a^{2}\cos^{2}\theta-r^{2})}{(r^{2}+a^{2}\cos^{2}\theta)^{2}}dt\wedge dr-\frac{Qra^{2}\sin 2\theta}{(r^{2}+a^{2})^{2}}dt\wedge d\theta$		(33)
	$\displaystyle-\frac{Qa\sin^{2}\theta(a^{2}\cos^{2}\theta-r^{2})}{(r^{2}+a^{2}\cos^{2}\theta)^{2}}dr\wedge d\varphi-\frac{Qar(r^{2}+a^{2})\sin 2\theta}{(r^{2}+a^{2}\cos^{2}\theta)^{2}}d\theta\wedge d\varphi.$

In order to compute the quasi-local energy of a surface $\Sigma$ defined by $t=$constant, $r=R$ that arises from the $U(1)$ sector, we need to fix a pair $(\hat{e}_{3},\hat{e}_{4})$ of normal vectors to $\Sigma$ satisfying $\hat{g}(\hat{e}_{3},\hat{e}_{3})=1,\hat{g}(\hat{e}_{4},\hat{e}_{4})=-1,\hat{g}(\hat{e}_{3},\hat{e}_{4})=0$

	$\displaystyle\hat{e}_{3}=\frac{1}{\sqrt{g_{rr}}}\partial_{r},$		(34)
	$\displaystyle\hat{e}_{4}=\frac{1}{\sqrt{\hat{g}_{\varphi\varphi}(\hat{g}^{2}_{t\varphi}-\hat{g}_{\varphi\varphi}\hat{g}_{tt})}}\left(\hat{g}_{\varphi\varphi}\partial_{t}-\hat{g}_{t\varphi}\partial_{\varphi}\right).$		(35)

Clearly they satisfy $\langle\hat{e}_{3},\partial_{a}\rangle_{\hat{g}}=\langle\hat{e}_{4},\partial_{a}\rangle_{\hat{g}}=0$. An explicit computation yields the following expression for $\hat{F}(\hat{e}_{4},\hat{e}_{3})$ on $\Sigma$

	$\displaystyle\hat{F}(\hat{e}_{4},\hat{e}_{3})|_{\Sigma}=\frac{1}{\sqrt{\hat{g}_{rr}}\sqrt{\hat{g}_{\varphi\varphi}(\hat{g}^{2}_{t\varphi}-\hat{g}_{tt}\hat{g}_{\varphi\varphi})}}\left(\hat{g}_{\varphi\varphi}\hat{F}(\partial_{t},\partial_{r})-\hat{g}_{t\varphi}\hat{F}(\partial_{\varphi},\partial_{r})\right)$
	$\displaystyle=\frac{1}{\sqrt{\hat{g}_{rr}}\sqrt{\hat{g}_{\varphi\varphi}(\hat{g}^{2}_{t\varphi}-\hat{g}_{tt}\hat{g}_{\varphi\varphi})}}\left(\frac{\hat{g}_{\varphi\varphi}Q(R^{2}-a^{2}\cos^{2}\theta)}{(R^{2}+a^{2}\cos^{2}\theta)^{2}}-\frac{\hat{g}_{t\varphi}Qa\sin^{2}\theta(a^{2}\cos^{2}\theta-R^{2})}{(R^{2}+a^{2}\cos^{2}\theta)^{2}}\right)$

which together with $A_{0}|_{\Sigma}=\frac{QR}{R^{2}+a^{2}\cos^{2}\theta}$ leads to the following expression for the energy density arising from $U(1)$ sector

	$\displaystyle F(\hat{e}_{4},\hat{e}_{3})A_{0}=-\frac{Q^{2}R}{\sqrt{\hat{g}_{rr}}\sqrt{\hat{g}_{\varphi\varphi}(\hat{g}^{2}_{t\varphi}-\hat{g}_{tt}\hat{g}_{\varphi\varphi})}}$		(36)
	$\displaystyle\left(-\frac{(R^{2}-a^{2}\cos^{2}\theta)\{(R^{2}+a^{2})(R^{2}+a^{2}\cos^{2}\theta)+(2rM-Q^{2})a^{2}\sin^{2}\theta\}}{(R^{2}+a^{2}\cos^{2}\theta)^{4}}\right.$		(37)
	$\displaystyle\left.+\frac{a^{2}\sin^{2}\theta(a^{2}\cos^{2}\theta-R^{2})(Q^{2}-2rM)}{(R^{2}+a^{2}\cos^{2}\theta)^{4}}\right)\sin^{2}\theta$
	$\displaystyle=\frac{Q^{2}R\sin^{2}\theta}{\sqrt{\hat{g}_{rr}}\sqrt{\hat{g}_{\varphi\varphi}(\hat{g}^{2}_{t\varphi}-\hat{g}_{tt}\hat{g}_{\varphi\varphi})}}\frac{(R^{2}+a^{2})(R^{2}-a^{2}\cos^{2}\theta)}{(R^{2}+a^{2}\cos^{2}\theta)^{3}}.$

We could simply use $\hat{e}_{3}$ and $\hat{e}_{4}$ because $\hat{F}(\hat{e}_{3},\hat{e}_{4})$ is boost invariant as mentioned previously and therefore any orthonormal basis pair $(e_{3},e_{4})$ of the normal bundle would suffice. However, while computing the Wang-Yau quasi-local energy functional that arises from the gravity sector, one needs to make sure that $(\hat{e}_{3},\hat{e}_{4})$ satisfy the desired property (21) otherwise we have to choose a different basis that does. Collecting all the terms together, we obtain

	$\displaystyle 8\pi\mathcal{QLE}_{gauge}=Q^{2}\int_{\Sigma}\frac{R(R^{2}+a^{2})(R^{2}-a^{2}\cos^{2}\theta)\sin\theta}{(R^{2}+a^{2}\cos^{2}\theta)^{3}}d\theta d\varphi$		(38)
	$\displaystyle=2\pi Q^{2}R(R^{2}+a^{2})\int_{0}^{\pi}\frac{(R^{2}-a^{2}\cos^{2}\theta)\sin\theta}{(R^{2}+a^{2}\cos^{2}\theta)^{3}}d\theta$

$\mathcal{QLE}_{gauge}$ is manifestly positive and behaves like $\sim\frac{Q^{2}}{2R}$ as $R\to\infty$. Therefore, one may retrieve the electrical charge as follows

$\displaystyle Q:=\lim_{R\to\infty}\sqrt{2R\mathcal{QLE}_{gauge}}.$

(39)

If we set $a=0$ (solution reduces to Reissner Nordstrom solution), then the expression for $\mathcal{QLE}_{gauge}$ simplifies tremendously yielding

$\displaystyle\mathcal{QLE}_{gauge}=\frac{Q^{2}}{2R}~{}\forall R\geq R^{+}.$

(40)

We can, in fact, evaluate the integral exactly for any $0\neq a<M$. Direct integration yields

$\displaystyle\mathcal{QLE}_{gauge}=\frac{1}{8}Q^{2}R(R^{2}+a^{2})\left(\frac{a^{2}+3R^{2}}{R^{2}(a^{2}+R^{2})^{2}}+\frac{1}{aR^{3}}\tan^{-1}(\frac{a}{R})\right).$

(41)

Later we evaluate this term on the outer horizon $R=R^{+}$ assuming a small angular momentum $a$.
Now that we have explicitly computed the contribution of the $U(1)$ sector, we move on to the computation of the quasi-local energy expression for the gravity part. Recall the expression for the quasi-local energy (28)

	$\displaystyle\mathcal{QLE}_{gravity}=\frac{1}{8\pi}\underbrace{\int_{\Sigma}\left(\sqrt{1+|\nabla\tau|^{2}_{\sigma}}|\mathbf{H}_{0}|_{\eta}\cosh\psi_{0}-\alpha_{\hat{e}_{30}}(\nabla\tau)+\psi_{0}\Delta\tau\right)\mu_{\Sigma}}_{MS}$
	$\displaystyle-\frac{1}{8\pi}\underbrace{\int_{\Sigma}\left(\sqrt{1+|\nabla\tau|^{2}_{\sigma}}|\mathbf{H}|_{\hat{g}}\cosh\psi-\alpha_{\hat{e}_{3}}(\nabla\tau)+\psi\Delta\tau\right)\mu_{\Sigma}}_{KN}.$		(42)

We will explicitly compute each term for the physical spacetime (Kerr-Newman spacetime in the current context) contribution $KN$. For the contribution $MS$ that arises from the Minkowski space, we will use a more direct approach since [15] derived the following simpler expression for $MS$

	$\displaystyle MS=\frac{1}{8\pi}\int_{\Sigma}\left(\sqrt{1+|\nabla\tau|^{2}_{\sigma}}|\mathbf{H}_{0}|_{\eta}\cosh\psi_{0}-\alpha_{\hat{e}_{30}}(\nabla\tau)+\psi_{0}\Delta\tau\right)\mu_{\Sigma}$
	$\displaystyle=\frac{1}{8\pi}\int_{\hat{\Sigma}}\hat{H}\mu_{\hat{\Sigma}},$		(43)

where $\hat{\Sigma}$ is the convex shadow of $\Sigma$ onto the complement of $\partial_{t}$ i.e., on a $\tau=$constant Euclidean slice $\mathbb{R}^{3}$, $\hat{H}$ is the mean curvature of $\hat{\Sigma}$ while realized as a $2-$surface in $\mathbb{R}^{3}$, and $\mu_{\hat{\Sigma}}=\sqrt{1+|\nabla\tau|^{2}_{\sigma}}\mu_{\Sigma}$ is the volume form induced on $\hat{\Sigma}$. Now we proceed to compute the physical space contribution to the Wang-Yau quasi-local energy functional.
Lemma 2: For the Kerr-Newman spacetimes, the basis pair $(\hat{e}_{3},\hat{e}_{4})$ defined in equations (34-35) is the canonical pair for which $\mathbf{H}=-|\mathbf{H}|_{\hat{g}}\hat{e}_{3}$ and $\langle\mathbf{H},\hat{e}_{4}\rangle_{\hat{g}}=0$