Mixed Gauge-Global Symmetries, Elliptic Modes, and Black Hole Thermodynamics in Hořava-Lifshitz Gravity

One of the most important characteristics of any physical theory is the symmetries imposed on the action. Requiring local gauge invariance is an integral part of the standard model of particle physics, local Lorentz symmetry dictates much of the form of special relativity, and the active diffeomorphism invariance of general relativity is a key component of holography and the AdS/CFT correspondence. As a consequence, symmetries also play a key role in the physics of causal horizons. The first law of black hole mechanics can be viewed as an application of Noether’s theorem, the spacetime metric is affected by the gauge field charges on the black hole, etc. In this paper we discuss the consequences for black hole thermodynamics of a “mixed” symmetry, which is neither a global nor local gauge symmetry, that exists in Hořava-Lifshitz gravity, a putative complete theory of quantum gravity. Hořava-Lifshitz gravity also possesses black holes UH and strong indications of a good thermodynamics of the corresponding universal horizons Mechanics ; toward_thermo ; Stefano ; StefanoVisser ; raytracing for flat and Lifshitz asymptotics. The mixed symmetry has not so far been explicitly examined in universal horizon thermodynamics, however. Additionally, the thermodynamics for asymptotically AdS has proven problematic, in that the first law is not consistent for all asymptotically AdS spacetimes.

The existence of a mixed symmetry and a failure of black hole thermodynamics in AdS may immediately raise concerns about the overall consistency of Hořava-Lifshitz gravity. Standard lore in quantum gravity, especially in the context of AdS/CFT, states that there are no global symmetries in a consistent quantum theory of gravity, but rather only local gauge symmetries Harlow_Main ; Harlow_PRL . Hence one might suspect that the failure of black hole thermodynamics in AdS is related to the existence of this mixed symmetry, which in turn may indicate that Hořava-Lifshitz is not actually a consistent quantum gravity theory but instead belongs to the swampland. We indeed find that the mixed symmetry and the prior failures of black hole thermodynamics in AdS are intimately related. However, instead of being an indication of inconsistency, the mixed symmetry, associated charge, and other seemingly unrelated features of Hořava-Lifshitz gravity all conspire to fix the current approaches to black hole thermodynamics when properly accounted for. While the focus of this paper is on black hole thermodynamics, a side conclusion is therefore that in some quantum gravity theories mixed symmetries may actually play a key role instead of signaling problems.

We now briefly explain the three key features of Hořava-Lifshitz gravity that make this possible to provide a conceptual picture for the reader and return to each in detail in later sections.

Feature: Foliations and reparameterization invariance. Hořava-Lifshitz gravity Horava_Lifshitz ; Horava_Membranes is a theory of gravity where spacetime is endowed with a dynamical spacelike foliation, the leaves of which are labelled by a scalar field $T$, the khronon. The existence of this foliation allows one to write a theory of gravity that is Lifshitz in the ultraviolet and ghost-free Horava_Lifshitz ; Horava_Membranes . The ultraviolet Lifshitz symmetry of the action then permits the theory to be renormalizable Blas , which yields a diffeomorphism invariant and seemingly complete and well-behaved theory of quantum gravity. The cost of the renormalizability of this theory is thus exact Lorentz invariance; there are tight experimental constraints multimessenger ; Kostelecky , but these can be theoretically evaded via the right hierarchies and renormalization group flow Pospelov ; Status . The structural difference between Hořava-Lifshitz gravity and general relativity is the existence of the foliation. Crucially, only the foliation is added to the theory - the actual labels of the foliation, the values of $T$, are irrelevant. This introduces a symmetry in the theory: the theory must be invariant under monotonic reparameterizations of $T$, i.e. $T\rightarrow f(T)$ is a symmetry as long as $f(T)$ is monotonic. This heavily restricts the form of the action. The reparameterization symmetry is not a global symmetry - the corresponding global symmetry would be a shift symmetry $T\rightarrow T+T_{0}$ where $T_{0}$ is a constant. However, it is also not fully local since $f(T)$ depends only on $T$ and not on the position on any one leaf of the foliation; it is, in the parlance of Hořava-Lifshitz gravity, projectable. We will consider in this paper the reparameterization symmetry as a partially gauged global symmetry, local in $T$ but global along each leaf, and the ramifications. We denote such a symmetry as a mixed gauge-global symmetry, or mixed symmetry for short. There is a corresponding charge, but this charge does not source any fully local gauge field.

Feature: Universal Horizon thermodynamics Hořava-Lifshitz possesses causally trapped surfaces known as universal horizons with seemingly good thermodynamics Mechanics ; toward_thermo ; Stefano ; StefanoVisser ; raytracing ; Pacilio:2017emh ; Pacilio:2017swi ; DelPorro:2023lbv . The ultraviolet Lifshitz symmetry dictates that ultraviolet field excitations travel infinitely fast. This does not mean that causality is lost; signals still must propagate to future of each leaf. However, relativistic causality is meaningless in Hořava-Lifshitz gravity. As a result, the causal horizons are not the usual black hole horizons of general relativity - for static spacetimes the event horizon is not a Killing horizon. Instead, the dynamics of the leaves of the foliation is such that they can “bend over” in static spacetimes with a central mass. While at infinity the timelike Killing vector $\chi$ in such a spacetime can be orthogonal to the leaves, if the central mass is concentrated enough the bending of the leaves will always be sufficient so that at some finite radius inside the Killing horizon the Killing vector becomes tangential to the leaves. At this radius, since every signal propagates to the future in the foliation, a trapped surface and hence causal boundary is formed: the universal horizon. This universal horizon has a good thermodynamics and shows promise for a full holographic description with boundary Lifshitz field theories Horava_Lifshitz ; Horava_Membranes ; Iranian ; CFTno ; wedges . Key for the compatibility of mixed symmetries, their charges, and black hole thermodynamics is that all leaves outside the foliation asymptote to the universal horizon for static spacetimes and extend to spatial infinity on the other side.

Feature: The presence of an elliptic mode The requirement of a foliation coupled with the fully dynamical nature of all fields leads to an additional equation of motion for the lapse $N$, which is itself a function of $T$. Importantly, this is an elliptic equation of motion. Unlike mathematically similar constraint equations, such as the Gauss constraint, this elliptic equation is not automatically preserved by the Hamiltonian; it is not a constraint. Rather, it must be imposed as an actual equation of motion on every leaf Henneaux . The existence of an elliptic mode in a theory with black holes seems immediately problematic, as elliptic modes couple boundary data on the ends of any leaf, and if the leaf penetrated any causal boundary then one would be able to probe any central singularity by looking at the boundary data on the other side. This would violate assumed properties such as cosmic censorship. However, because all the leaves outside a universal horizon asymptote to it, the boundary data for the leaf couples points at spatial infinity not to central singularity but instead to the universal horizon data itself. In this way the elliptic mode provides a new, unique mechanism by which boundary data at a causal horizon, but not inside, can be transmitted to spatial infinity. It is in this way that the charge associated with this new symmetry can theoretically be emitted from the universal horizon despite there being no local gauge field to generate a chemical potential for different local charged, emitted particles Hook:2014mla . We note that this behavior is only possible with a) the imposition of Lifshitz symmetry (which was originally required for renormalizablility) which makes the universal horizon the causal horizon and b) foliation dynamics, which is required for holography and also generates the elliptic mode needed to pull charge off a causal boundary without a local gauge field. Specific to holographic constructions, the mixed symmetry is possible because the elliptic mode violates the criterion of splittability Harlow_Main ; Harlow_PRL used in current proofs.

The three features above combine into a scenario where there is a mixed symmetry which is global along leaves of the foliation yet gauged in (khronon) time, a corresponding conserved charge which sources no local gauge field, causal boundaries and black hole thermodynamics, and an elliptic mode mechanism that allows horizon data and the charge to be transmitted to infinity without gauge fields. This is a novel extension of black hole thermodynamics with qualitatively new symmetry constructions.¹¹1Of note here is the recent surge of interest in generalized gauge theories Gaiotto ; Harlow_Main ; Heckman:2024oot - for an introduction see Gen_Sym_Review . The shift operator maps leaves into themselves, i.e. it acts intrinsically on extended objects rather than local ones. Whether the corresponding reparameterization symmetry can be embedded into the framework of gauged generalized global symmetries is currently not known. For the purposes of this paper, when we refer to global symmetries we will not be considering generalized global symmetries, but rather the usual “0-form” global symmetries that can act on local objects.

The above picture, while perhaps conceptually compelling, is a large amount of work to quantitatively check and verify all aspects. In the rest of this paper we concentrate on a simple first calculation to establish the relevance of the mixed gauge-global charge for black hole thermodynamics in Hořava-Lifshitz theory. We show that explicitly requiring the mixed gauge-global charge to be conserved fixes the prior problem with asymptotically AdS spacetimes - non-alignment of the normal vector corresponds to an external flux that changes the mixed gauge-global charge. Requiring a first law between equilibrium solutions for all charges therefore removes this possibility and explains why the existing derivations of thermodynamics for universal horizons failed in this case.

The structure of the paper is as follows. In Section 2 we introduce Hořava-Lifshitz gravity, including the basic foliation structure, the relation to Einstein-aether theory, action, and notation. Of the three necessary features, the elliptic mode is introduced in 3, reparameterization invariance as a mixed symmetry and the corresponding charge is discussed in Section 4, while the third, universal horizons and their thermodynamics, is explained in Section 5. We then show how the existing thermodynamics fails in the asymptotically AdS case in Section 6. Resolving this failure using the three features is done in Section 7, and finally we conclude with some remarks and future directions regarding the usual narratives around global/gauge symmetries and splittability in Section 8.

One of the unique features of Hořava-Lifshitz gravity is the so-called “elliptic mode”, which arises since space and time are treated on different footings in a dynamical manner. The equations of motion contain an elliptic equation that allows one to solve for the lapse function, $N$, on the entirety of a given slice $\Sigma_{T}$ when given the appropriate boundary data. This means that elliptic mode not only needs to be taken into account when on-shell reducing various quantities (which will be done in subsequent sections), but also has important ramifications regarding black hole thermodynamics.

The elliptic mode is most clearly understood as being born out of the constraint algebra. While it is not a proper constraint under the Dirac formalism, it is a condition that must be satisfied by the lapse to enforce closure of the algebra. We detail this construction below. In the GR limit, the constraint algebra closes automatically and the need for an elliptic condition on the lapse function vanishes.

For this section, we will chose the notation of the canonical, Hamiltonian formulation of Hořava-Lifshitz gravity rather than that of the covariant Einstein-æther theory. The mixed symmetry of Hořava Lifshitz gravity is most easily understood in the canonical formalism since it arises due to the different treatments of space and time in the action. Additionally, the canonical formulation is the most natural for studying this constraint algebra. How the mixed symmetry and elliptic charge can be understood more broadly in the context of covariant phase space methods of Wald_Phase ; Carlip ; Harlow_Phase ; Tony is an open question beyond the scope of this paper.

The review in this section of the constraint algebra and elliptic mode will initially closely follow Consistency while additionally including the acceleration term to ensure a well behaved theory at low energies Blas . After demonstrating how the elliptic mode emerges in asymptotically flat spaces, we will then generalize to more general asymptotics by adding the bare cosmological constant to the Hamiltonian (a more detailed review of asymptotics in Hořava-Lifshitz gravity will be covered in Section 5.3 below).

As in general relativity, we take the 3-metric of a leaf $p_{ij}$ to be the generalized position coordinate. In the case of Hořava gravity, we take this 3-metric to be that of a leaf in the preferred foliation. It can be shown that the momentum conjugate to $p_{ij}$ is given as Consistency :

$$\pi^{ij}=\sqrt{p}\,G^{ijkl}K_{kl}$$

(22)

Making use of (5) and performing a Legendre transformation, one can find that the Hamiltonian density for the theory is given by

$$\mathcal{H}_{tot}=N\mathcal{H}+N_{i}\mathcal{H}^{i}$$

(23)

where

	$\displaystyle\mathcal{H}=$	$\displaystyle\frac{\pi^{ij}\pi^{kl}}{\sqrt{p}}\mathcal{G}_{ijkl}-\sqrt{p}\;(\xi\prescript{(3)}{}{R}+\eta a_{i}a^{i})$		(24)
	$\displaystyle\mathcal{H}^{i}=$	$\displaystyle-2N_{i}\nabla_{j}\pi^{ij}=\pi^{ij}_{\;\;;\,j}.$

The total Hamiltonian is unsurprisingly obtained by integrating (23) over a given leaf of the foliation, namely:

$$H=\int_{\Sigma}d^{3}x\,\mathcal{H}_{tot}$$

(25)

As in general relativity, the Einstein field equations imply the vanishing of $\mathcal{H}$ and $\mathcal{H}^{i}$, and thus also $H$. The constraint $\mathcal{H}=0$ is referred to as the Hamiltonian constraint, also in keeping with the convention of general relativity.

Recall that to enforce preservation in time of the primary constraint $\mathcal{H}$, it is sufficient to compute its Poisson bracket with itself. Preservation in time of a constraint $\phi$ amounts to enforcing that $\{\phi,H\}=0$; however, since $\mathcal{H}^{i}$ generates spatial diffeomorphisms, Poisson brackets between $\mathcal{H}^{i}$ and $H$ correspond to infinitesimal coordinate transformations and lead to no new constraints Consistency ; DeWitt . Thus the only remaining term in the bracket $\{\mathcal{H},H\}$ is $\{\mathcal{H},\mathcal{H}\}$.

Proceeding along these lines, we find that the Poisson bracket $\{\langle A\mathcal{H}\rangle,\langle B\mathcal{H}\rangle\}$, where $A,B$ are in general distributions and the angled brackets denote integration over $\Sigma$, is given by

$$\{\langle A\mathcal{H}\rangle,\langle B\mathcal{H}\rangle\}=2\int d^{3}x\,d^{3}y\,A\;\pi^{ij}\bigg{[}\nabla_{i}\nabla_{j}B-\bigg{(}\frac{\lambda-1}{3\lambda-1}\bigg{)}g_{ij}\nabla^{2}B\bigg{]}-(A\leftrightarrow B)$$

(26)

Now we can take $A=\delta(x-y)$ and $B=N$ the lapse and integrate by parts to obtain

$$\{\mathcal{H},\langle N\mathcal{H}\rangle\}=2\bigg{(}\frac{\lambda-1}{3\lambda-1}\bigg{)}\big{(}N\nabla^{2}\pi+2\partial_{i}N\nabla^{i}\pi\big{)}+2\mathcal{H}^{i}\partial_{i}N+N\nabla_{i}\mathcal{H}^{i}$$

(27)

Recalling that $\mathcal{H}^{i}=0$, we find that enforcing $\{\mathcal{H},\langle N\mathcal{H}\rangle\}=0$ is equivalent to enforcing that

$$N\nabla^{2}\pi+2\partial_{i}N\nabla^{i}\pi=0$$

(28)

Since $N\neq 0$, it is clear that this constraint is equivalent to requiring

$$\nabla_{i}(N^{2}\nabla^{i}\pi)=0$$

(29)

Multiplying this quantity by $\pi/\sqrt{p}$ and integrating by parts over $\Sigma$, we obtain

$$\int_{\Sigma}\mathrm{Vol}_{\Sigma}\,N^{2}(\nabla_{i}\pi)^{2}=0$$

(30)

Again, since $N^{2}>0$ on the entirety of $\Sigma$, this is equivalent to

$$\nabla_{i}\pi=0$$

(31)

The only regular solution of (31) that satisfies zero momentum at the asymptotic boundary is the second-class constraint

$$\pi=0$$

(32)

One can then obtain the desired elliptic mode by enforcing the preservation in time of the constraint in (32) which once again amounts to computing the Poisson bracket of the constraint with $\mathcal{H}$, as per the above discussion. One then finds

$$\{\pi,\langle N\mathcal{H}\rangle\}=-2\sqrt{p}\xi(\vec{\nabla}^{2}-\prescript{(3)}{}{R})N+2\eta N\sqrt{p}\,a_{i}a^{i}+\frac{3}{2}N\mathcal{H}$$

(33)

Noting again that $\mathcal{H}=0$, we find that enforcing that $\pi=0$ be preserved in time amounts to requiring that the lapse $N$ satisfy the following elliptic equation:

$$\left(\vec{\nabla}^{2}-\prescript{(3)}{}{R}-c_{14}\,a_{i}a^{i}\right)N=0.$$

(34)

Note that (34) is exactly as found in Consistency with the addition of the acceleration term. Here, we have made use of (19) in simplifying (34).

The next natural step, then, is to generalize (34) to more general asymptotics as will be necessary to our study of asymptotically AdS universal horizons. One can do this rather simply by adding a cosmological constant term to the Hamiltonian

	$\displaystyle\tilde{\mathcal{H}}=$	$\displaystyle\mathcal{H}+\mathcal{H}_{\Lambda}$		(35)
	$\displaystyle\mathcal{H}_{\Lambda}=$	$\displaystyle\sqrt{p}\xi\frac{6c_{cc}}{l^{2}}.$

Since $\mathcal{H}_{\Lambda}$ contains no derivatives, it is clear that

$$\{\langle A\tilde{\mathcal{H}}\rangle,\langle B\tilde{\mathcal{H}}\rangle\}=\{\langle A\mathcal{H}\rangle,\langle B\mathcal{H}\rangle\}.$$

(36)

One might note that the constraint $\pi=0$ must be modified since it was obtained through the assumed boundary condition for $\pi_{ij}=0$ at infinity which only holds in asymptotically Minkowski spacetimes. However, since $K$ must be constant at infinity Universal , and since the $\pi=0$ constraint is reached by solving $\nabla_{i}\pi=0$ on a given hypersurface and applying the asymptotically flat boundary condition $\pi_{ij}=0$, if we enforce the boundary condition $\pi=\pi^{\prime}$ some constant at infinity, then we simply enforce in time the constraint $\tilde{\pi}=\pi-\pi^{\prime}=0$. Since $\pi^{\prime}$ is a constant, it is clear that the bracket will be unaffected as it vanishes upon differentiation. The only bracket we must therefore compute is

$$\{\tilde{\pi},\langle N\mathcal{H}_{\Lambda}\rangle\}=\{\pi,\langle N\mathcal{H}_{\Lambda}\rangle\}=-\xi\int_{\Sigma}\mathrm{Vol}_{\Sigma}\frac{9c_{cc}}{l^{2}}.$$

(37)

Since the bracket is linear, we have

$$\{\pi,\langle N\tilde{\mathcal{H}}\rangle\}=-\int_{\Sigma}\mathrm{Vol}_{\Sigma}2\sqrt{p}\left[\xi\left(\vec{\nabla}^{2}-\prescript{(3)}{}{R}+\frac{9c_{cc}}{l^{2}}\right)-\eta a^{2}\right]N-\frac{3}{2}N\tilde{H}.$$

(38)

Thus, by enforcing the equations of motion, which now amounts to saying that $\tilde{\mathcal{H}}=0$, we can see that the elliptic equation for $N$ with general asymptotics is given by

$$\left[\xi\left(\vec{\nabla}^{2}-\prescript{(3)}{}{R}+\frac{9c_{cc}}{l^{2}}\right)-\eta a^{2}\right]N=0.$$

(39)

In æther notation, this reads

$$\left[\left(\vec{\nabla}^{2}-\prescript{(3)}{}{R}+\frac{9c_{cc}}{l^{2}}\right)-c_{14}a^{2}\right]N=0.$$

(40)

Hořava-Lifshitz gravity is invariant under reparameterization symmetry, $T\rightarrow f(T)$ as long as $f(T)$ is monotonic. As we saw, this constrained the form of the action. The reparameterization invariance can be rewritten as $T\rightarrow T+\tilde{f}(T)$ without loss of generality as long as monotonicity is preserved. In coordinates adapted to the foliation, $T$ simply defines the coordinate time $t$ and so this transformation becomes a spacetime time translation $t\rightarrow t+\tilde{f}(t)$ where the translation parameter is now local in time.

Theories with such a time (and space) dependent translations can be constructed as localizations of theories with global translation symmetries. For example, typical non-relativistic free theories possess global Galilean invariance, where the theory is invariant under $x^{a}\rightarrow x^{a}+\xi^{a}$ where $\xi^{a}$ is a constant. There is a well defined localization procedure for such non-relativistic theories diffR where $\xi^{a}$ becomes coordinate dependent, leading to a gauge theory with a three dimensional diffeomorphism invariance and time reparameterization invariance. We take the same approach here, simply in the foliation labels $T$, which are equivalent to a time coordinate in the foliation frame. However, unlike a full localization in both space and time of a globally Galilean invariant theory only the time localization is needed to match the reparameterization invariance. The reparameterization invariance is then a mixed, or partially localized/gauged invariance that corresponds to the global symmetry $T\rightarrow T+T_{0}$, i.e. a global foliation translation/shift symmetry. Clearly this is a global transformation, preserves the foliation structure, and is a symmetry of the action.

As an aside on nomenclature, functions that are functions of only $T$ are called “projectable” in Hořava-Lifshitz gravity Horava_Membranes ; Horava_Lifshitz . Indeed, originally projectability was a key aspect of the lapse in the initial construction, although this was later shown to be too restrictive Blas . Hence one could also think of mixed symmetries as “projectable” symmetries. However, at the same time the reparameterization transformation is an example of an “unfree” gauge transformation unfree . Unfree gauge transformations are those where the gauge parameter is not a local function of $t,x$ but instead satisfies some differential equation. In the case of Hořava-Lifshitz gravity, the differential equation would simply be $\vec{\nabla}\tilde{f}(x,T)=0$, where $\vec{\nabla}$ is the spatial gradient along a leaf. This, up to a meaningless global shift, restricts $\tilde{f}(x,T)$ to simply $\tilde{f}(T)$, as we have been discussing. Finally, as mentioned previously, whether this symmetry can be embedded into the framework of generalized global symmetries is unknown. We choose the term “mixed” in this paper to emphasize the dual global/gauge nature of the transformation without unnecessarily pigeonholing it into any of the existing conceptual frameworks it is related to.

Given the existence of the above mixed and corresponding global symmetry in Hořava theory, Noether’s theorem dictates a conserved quantity. This is easily determined in the usual fashion by requiring that the action (4) be stationary under variations in $T$ and collecting boundary terms, which we now do.

Using the identities found in Section 2.3, one can easily determine the following variations

	$\displaystyle\delta_{T}u_{a}=$	$\displaystyle-Np_{ab}(\nabla^{b}\delta T)$		(41)
	$\displaystyle\delta_{T}N=$	$\displaystyle-u_{a}N^{2}(\nabla^{a}\delta T)$		(42)
	$\displaystyle\delta_{T}a_{a}=$	$\displaystyle-N\,K_{ab}(\nabla^{b}\delta T)-u_{c}\nabla^{c}(N\,p_{ab})(\nabla^{b}\delta T)$		(43)
		$\displaystyle-u_{c}Np_{ab}(\nabla^{c}\nabla^{b}\delta T)$

where $\delta_{T}$ denotes variation with respect to $T$. In the usual fashion, one obtains

$$\delta_{T}S=\int_{\mathcal{M}}\left(d\star J+E_{T}\cdot\delta T\right)$$

(44)

where $E_{T}$ denotes the equation of motion corresponding to $T$, and $J$ the current corresponding to variations in $T$. After a few lines of work and use of the identities in (41)-(43), one finds that the components of the current $J$ are given by

$$J_{b}=\delta T\big{(}f_{b}-\nabla^{a}(\tilde{f}_{ab}+\tilde{f}_{ba})\big{)}$$

(45)

where

	$\displaystyle f_{b}=$	$\displaystyle-u_{b}N\mathcal{L}-c_{13}\big{[}-(-u_{a}a_{b}+K_{ab})(\nabla^{a}N)-N(u_{b}K_{ac}K^{ac})$		(46)
		$\displaystyle-Na^{c}K_{bc}-u_{a}a_{b}\nabla^{a}N-Nu_{b}a^{2}\big{]}$
		$\displaystyle-2c_{2}K\big{[}-NKu_{b}-Na_{b}-\nabla_{b}N\big{]}+2c_{14}\big{[}-Na^{c}K_{bc}-u_{a}a_{b}\nabla^{a}N-Nu_{b}a^{2}\big{]}$
	$\displaystyle\nabla^{a}f_{ab}=$	$\displaystyle 2c_{13}\big{[}K_{ab}\nabla^{a}N+N\nabla^{a}K_{ab}\big{]}+2c_{2}K\big{[}\vec{\nabla}_{b}N+Nu_{b}K+Na_{b}\big{]}$
		$\displaystyle-2c_{14}\big{[}u_{a}a_{b}\nabla^{a}N+u_{a}N\nabla^{a}a_{b}+Na_{b}K\big{]}$
	$\displaystyle\nabla^{a}f_{ba}=$	$\displaystyle 2c_{13}\big{[}K_{ab}\nabla^{a}N+N\nabla^{a}K_{ab}\big{]}+2c_{2}K\big{[}\vec{\nabla}_{b}N+Nu_{b}K+Na_{b}\big{]}$
		$\displaystyle-2c_{14}\big{[}u_{b}a_{a}\nabla^{a}N+u_{b}N\nabla^{a}a_{a}+Na_{a}\nabla^{a}u_{b}\big{]}$

Using this current, one can thus obtain the corresponding charge by integrating the current $\star J$ over a slice $\Sigma$:

$$Q_{T}=\int_{\Sigma}\star J_{T}.$$

(47)

Using (46), one finds that the charge reads

$$Q_{T}=\int_{\Sigma}d^{3}x\sqrt{p}\,N\left[\mathcal{L}+2c_{13}K_{ab}K^{ab}+2c_{2}K^{2}-2c_{14}a^{2}-2c_{13}a^{2}-c_{14}\nabla_{a}a^{a}\right]$$

(48)

where $\mathcal{L}$ denotes the Lagrangian density. One notes that in the GR limit where $c_{i}\to 0$ for all $i$, $\mathcal{L}$ reduces to the Einstein-Hilbert Lagrangian, and the remaining terms of (48) vanish. After an on-shell reduction using $H=0$ (we have dropped the tilde from here on out for simplicity, but it is assumed that $H$ here is the Hamiltonian for general asymptotics), and additionally the elliptic mode equation from section 3, one finds that the charge, which we will call the elliptic charge, is given by

$$Q_{T}=\int_{\Sigma}d^{3}x\,\sqrt{p}\left[2\left(\frac{1}{1-c_{13}}\right)\vec{\nabla}_{i}(Na^{i})+2N\frac{c_{123}}{1-c_{13}}K^{2}-2c_{13}Na^{2}-N\frac{15c_{cc}}{(1-c_{13})\,l^{2}}\right].$$

(49)

Note that the first term of (49), which does not vanish in the general relativistic limit, can be understood as a consequence of how the foliation approaches asymptotic infinity. Essentially, it is similar to a non-orthogonal corner term Hawking_Corner ; Odak:2021axr where there is also a mismatch between the asymptotic Killing vector and the normal to the foliation. To see the foliation geometry more clearly, one can rewrite this first term as

$$I_{1}=\int_{\Sigma}d^{3}x\sqrt{p}\,\vec{\nabla}_{a}(Na^{a})=\int_{\partial\Sigma}d^{2}x\sqrt{\hat{g}}\,(Ns_{a}a^{a})=\int_{\partial\Sigma}d^{2}x\sqrt{\hat{g}}Ns_{a}u_{b}\nabla^{b}u^{a}$$

(50)

Integrating by parts, one finds

$$I_{1}=-\int_{\partial\Sigma}d^{2}x\sqrt{\hat{g}}\,Nu^{a}u^{b}\nabla_{a}s_{b}=-\int_{\partial\Sigma}d^{2}x\sqrt{\hat{g}}\,N(p^{ab}-g^{ab})\nabla_{a}s_{b}.$$

(51)

From the second equality in (51), it is clear that this term measures the difference in extrinsic curvature of the surface defined by $s_{a}$ at infinity when measured with respect to the Hořava-Lifshitz foliation versus the spacetime as a whole. Namely, if the three and four divergences of $s_{a}$ are equal, $I_{1}$ is clearly vanishing. In general relativity, this term does not play as important of a role since one could always chose a foliation where this term would vanish. However, in Hořava Lifshitz gravity, one must work with the preferred foliation, which in general will introduce non-orthogonality of leaves $\Sigma_{T}$ with the timelike boundary $\partial\mathcal{M}$, and in turn introducing non-orthogonal corner-type terms such as $I_{1}$. Further understanding of how such corner-type terms could affect this charge and the thermodynamics of universal horizons in general using the methods of covariant phase space explored in Wald_Phase ; Carlip ; Harlow_Phase ; Tony ; Odak:2021axr will be left for future work.

Additionally, by examining (49), one can see that changing the required boundary data to solve the elliptic mode, namely $\nabla_{i}N$ and $N$, will make a corresponding change to the charge $Q_{T}$, hence the the name “elliptic charge”. Since Hořava-Lifshitz gravity admits black hole solutions, the existence of such a charge raises questions about the form of the first law governing said black holes. First, in a truly static solution that corresponds to a time independent equilibrium thermodynamical state, this charge should be conserved. Second, if this charge gravitates, as in the case of the electromagnetic charge of a Reissner-Nordstrom black hole for instance, the charge will appear in the black hole’s first law for the appropriate ensemble. In the following section, we cover the formulation of black holes in Hořava-Lifshitz gravity, and the difficulties of formulating first laws for universal horizons in asymptotically AdS. We will then see that use of the elliptic charge (49) resolves some of the known issues.

At first glance, Hořava-Lifshitz gravity would seem to admit no causal horizons. The ultraviolet mode behavior is Lifshitz, which implies that the group velocity of wave modes diverges at high energy. This precludes any finite limiting speed for all modes, which would seem to forbid the existence of a causal horizon. Contrary to this expectation, Hořava-Lifshitz gravity has been shown to exhibit causal horizons known as universal horizons. In essence, in a static situation with a timelike Killing vector $\chi^{a}$ and associated Eddington-Finkelstein Killing time coordinate $v$, the bulk dynamics of the foliation leaves themselves allow for solutions where the leaves all asymptote to a leaf not connected to spacelike infinity, denoted as the universal horizon, for infinite negative $v$. Since all modes, even those with infinite group velocity, must propagate to the future in khronon time, the universal horizon acts as a trapped surface and a causal horizon, as we detail below.

A rigorous discussion of these black holes can be found in Causality . In addition, maximally symmetric universal horizon solutions have been studied in Universal , which will largely be reviewed in this section. An important result of Universal is that the static, spherically symmetric black hole solution spaces of Einstein-æther theory and Hořava-Lifshitz are equivalent. Therefore, we will largely use coupling constants and parameters from Einstein-æther theory for simplicity.