Dynamical Henneaux-Teitelboim Gravity

We consider a modified gravity model which we call"dynamical Henneaux-Teitelboim gravity"because of its close relationship with the Henneaux-Teitelboim formulation of unimodular gravity. The latter is a fully diffeomorphism-invariant formulation of unimodular gravity, where full diffeomorphism invariance is achieved by introducing two additional non-dynamical fields: a scalar, which plays the role of a cosmological constant, and a three-form whose exterior derivative is the spacetime volume element. Dynamical Henneaux-Teitelboim gravity is a generalization of this model that includes kinetic terms for both the scalar and the three-form with arbitrary couplings. We study the field equations for the cases of spherically symmetric and homogeneous, isotropic configurations. In the spherically symmetric case, we solve the field equations analytically for small values of the coupling to obtain an approximate black hole solution. In the homogeneous and isotropic case, we perturb around de Sitter space to find an approximate cosmological background for our model.


A. Overview
Cosmic inflation solves a number of fundamental problems in cosmology by positing an early period of rapid expansion driven by a constant part of the stress energy tensor-it provides a framework for explaining why the observable universe is nearly flat and isotropic [1][2][3], and gives a mechanism for large-scale structure to form from density perturbations in the early universe [4].Meanwhile, the present cosmological epoch is dominated by dark energy that is usually attributed to a cosmological constant, albeit a vastly smaller one [5,6].In light of these facts, as well as the recent controversy over the Hubble tension [7,8], and the long-standing cosmological constant problems [9][10][11], simple modifications of general relativity where the cosmological constant is promoted to a dynamical field are worthy of consideration.
One route to accomplishing this is to consider modified gravity models where the Bianchi identities no longer imply conservation of the stress energy tensor, but only the weaker condition that the divergence of the stress tensor is a total derivative.A minimal modification of general relativity in which this can be achieved is unimodular gravity [12].In the standard formulation of unimodular gravity, the spacetime volume element is constrained to be equal to a prescribed background density, with the result that the equations of motion become the trace-free Einstein equations Taking the divergence of (1) and using the contracted Bianchi identity ∇ µ R µν = 1 2 ∇ ν R, leads to If one imposes conservation of the stress tensor, then we learn that Λ ≡ 1 4 (R + κT ) is a constant, which is just the pure-trace part of the Einstein equations, in which case unimodular gravity is classically equivalent to general relativity.However, if we are willing to consider non-conserved stress tensors, then unimodular gravity is physically distinct from general relativity even at the classical level. 1 Some recent proposals for modified gravity models that draw on ideas from unimodular gravity and topological field theory lead to the same type of non-conservation of the stress tensor, and consequent fluctuations in the effective cosmological constant [14][15][16].Here, we promote the cosmological constant to a dynamical field by considering a minimal modification of a fully diffeomorphism invariant, background independent reformulation of unimodular gravity due to Henneaux and Teitelboim [17] and we study the implications for spherically symmetric black holes and homegeneous, isotropic cosmology.
The paper is structured as follows: in section I B, we give a brief review of unimodular gravity; in section II, we introduce dynamical Henneaux-Teitelboim gravity; section III is devoted to the study of spherically symmetric black holes, and in section IV we take a first look at the implications of dynamical Henneaux-Teitelboim gravity for a simple homogeneous, isotropic cosmological model.

B. Unimodular Gravity
The action for unimodular gravity is simply the Einstein-Hilbert action where the spacetime volume element √ −g has been replaced with a fixed background density ϵ: where g µν is a metric satisfying √ −g = ϵ, and ψ represents any matter fields that are present.Varying (3) with respect to g µν leads to the trace-free Einstein equation (1).The introduction of the background density ϵ breaks diffeomorphism invariance down to transverse diffeomorphism invariance-i.e., invariance with respect to diffemorphisms generated by vector fields ξ satisfying In order to preserve full diffeomorphism invariance while retaining the spirit of unimodular gravity, Henneaux and Teitelboim [17] considered the action 2 where φ is a scalar, H is a three-form, and Vol(g) = √ −g d 4 x is the volume form associated with g µν .Varying with respect to g µν gives the full set of Einstein's equations with φ playing the role of the cosmological constant: On the other hand, varying with respect to H leads to dφ = 0, so the gravitational equations of motion are equivalent to the usual Einstein equations.Finally, varying with respect to φ gives Vol(g) = dH, so while H is varied in the action, it can be thought of as a potential for the spacetime volume in much the same way that ϵ fixes the spacetime volume in the standard formulation of unimodular gravity.Since there are no background structures, the action ( 5) is invariant under the full group of diffeomorphisms.There is also a gauge symmetry under H → H + dΩ for any two-form Ω.
2 Smolin has shown in [18] that certain aspects of the cosmological constant problem can be resolved using path integral methods based on the action (5).
In addition to restoring background independence and full diffeomorphism invariance, the Henneaux-Teitelboim formulation of unimodular gravity automatically has a conserved stress tensor, unlike the standard formulation.However, if one introduces additional terms into the Lagrangian that couple the new fields φ and H to the metric, this may no longer be the case.As we shall see in the following section, introducing kinetic terms for both φ and H is sufficient to violate conservation of the stress tensor just enough to induce fluctuations in the effective cosmological constant φ, although including a kinetic term for just one of them is not.

II. DYNAMICAL HENNEAUX-TEITELBOIM GRAVITY
We now consider "dynamical Henneaux-Teitelboim gravity," defined by the action which enjoys the same symmetries as the Henneaux-Teitelboim formulation of unimodular gravity-it is background independent and fully diffeomorphisminvariant, and it retains the gauge symmetry H → H + dΩ for any two-form Ω.Here, and for the remainder of the article we use geometrized units c = G N = 1, which renders κ dimensionless, and gives the action dimensions of (length) 2 .As a result, α acquires dimensions of (length) −2 , while β ∼ (length) 4 .
Varying (7) with respect to g µν leads to the Einstein field equations with the dynamical field φ playing the role of the cosmological constant, and where the stress tensor gets contributions from the old matter Lagrangian L M (g, ψ), as well as the kinetic terms for H and φ: where with Q = dH, and Q = 1 4! Q abcd ϵ abcd .The trace-free part of the gravitational equation ( 8) is just the trace-free Einstein equation (1), which, as we noted in the introduction, implies that the divergence of the stress tensor is a total derivative ∇ µ T µν = 1 4 ∇ ν (R + κT ).On the other hand, the pure trace part of ( 8) is simply φ = 1 4 (R+κT ).
Putting these together gives So we see that fluctuations in φ are sourced by nonconservation of the stress tensor of precisely the type we have alluded to.As for the scalar and three-form equations of motion, we can vary (7) with respect to φ to obtain while varying (7) with respect to H gives Taking the Hodge star of ( 14) and inserting the result into (15) leads to while taking the Hodge star of ( 15) and plugging into (14) gives where d † ≡ * d * .From equations ( 16) and ( 17), we see that if either α = 0 or β = 0, then we recover dH = Vol(g) and dφ = 0, and the model is equivalent to the Henneaux-Teitelboim formulation of unimodular gravity.However, when αβ ̸ = 0, we obtain non-trivial dynamics for φ.

III. BLACK HOLES
We would like to investigate the implications of this model for black holes.Our approach is similar to [19], in which metrics for black holes and other compact objects were studied for the general case of unimodular gravity with a non-conserved stress tensor, although here we focus on the black holes solutions of dynamical Henneaux-Teitelboim gravity in particular.For this, we consider the static, spherically symmetric ansatz where the ansatz for H is the most general spherically symmetric, static three-form up to gauge transformations, and dΩ 2 2 = dθ 2 + sin 2 θdϕ 2 is the round metric on the unit two-sphere.Additionally, we will only consider contributions to the stress tensor from H and φ, i.e., we will set L M = 0 in (7).
Since the stress tensor for H is pure trace, it will be convenient to divide the Einstein equations into tracefree and pure trace parts.The trace-free equations are (1), while the pure-trace equation is where and for the static, spherically symmetric ansatz ( 18)- (20).
Similarly, the trace-free equations reduce to where we have defined In the above, ( 24), ( 25) and ( 26) are the tt, rr and θθ components of the trace-free Einstein equations, while the ϕϕ component is identical to the θθ component, and all the off-diagonal components vanish.Equations ( 24)-( 26) are linearly dependent: by either subtracting (26) from (24) or adding (26) to (25), we obtain which can be integrated to give Inserting the above into (26) and using (27) leads to Doing the same for the trace part of the Einstein equation ( 21), ( 22), (23) and inserting the result into (30) gives which is a first-order linear equation for A(r).The gen-eral solution is (32) where the mass m appears as an integration constant, the lower endpoint of integration ℓ = 3 Λ0 , and Λ 0 is fixed by matching to the (A)dS-Schwarzschild metric in the "decoupling limit," on which we expand below, and we have defined From ( 14) and ( 15), we notice that we can recover the (A)dS-Schwarzschild metric in the limit as β → 0, 1/α → 0 and 1/(αβ) → 0, which we call the decoupling limit.Notice that this limit does not reduce the order of the equation of motions for the scalar φ (16) or for the three form H (17).In this case, equation ( 14) leads to dH = Vol(g) and consequently d * dH = 0 in equation (15).Then equation ( 17) is redundant, while equation ( 16) becomes d□φ = 0 for which φ = constant is a consistent solution.When dH = Vol(g), the stress energy tensor for H, is just a constant times the metric, which can be absorbed into the cosmological constant.At the level of the action, we can integrate out H by inserting the solution to the equation of motion for φ which is dH = Vol(g).Then, the gauge kinetic term just becomes S H = α Vol(g), which contributes as a cosmological constant term.We now turn our attention to the equations of motion ( 16), (17) for H and φ.In order to make progress, through the decoupling limit, we replace g µν in equations ( 16), (17) with the (A)dS-Schwarzschild metric, g (0) µν .Now the equation of motion for H (17) can be integrated once to give where A 0 (r) = 1 − 2m r − Λ 3 r 2 is the metric function for an (A)dS-Schwarzschild black hole.This is a linear, inhomogeneous equation for h(r), so the general solution is of the form where h 0 (r) is the solution of the homogeneous equation and h p (r) is a particular solution.One can check that h p (r) = 1 3 r 3 +C is a satisfactory particular solution.The corresponding homogeneous equation can be solved using the method of Frobenius-i.e., we make a generalized power series ansatz The allowed values of q and coefficients c n are calculated in A. Since the equation is second order, we find two possible values for q corresponding to the two linearly independent solutions of the homogeneous equation corresponding to (34): Now the solution to the homogeneous equation becomes where the coefficients are determined by the following base cases and recursion relation The scalar field equation of motion is given by where A and φ both depend on the radius r.Again, we work around the Schwarzschild-(A)dS metric A(r) = A 0 (r), and apply the method of Frobenius, with our ansatz given by We once again refer the reader to appendix A for the computation of the allowed values of s and the coefficients a n .The roots of the indicial equation are so again, we have two independent solutions, as we would expect for a second order equation.Now the solution becomes where the coefficients are (47) Note that ℜ(s ± ) < 0 for all values of Λ, α and β, so that lim r→∞ φ(r) = Λ, with s ± being real for Λ < 0 or Λ ≥ 1 3αβ .On the interval 0 < Λ < 1 3αβ , s ± become complex, including the physically realistic case when Λ is small and positive.One might worry that such a situation precludes a physically meaningful solution for the most relevant region of parameter space, however this is not the case.So far, we have been assuming that the integration constants a + 0 and a − 0 are real and independent, but they still define a solution of the homogeneous equation even when they are complex, albeit a generically complex-valued one.If we instead take a − 0 = ā+ 0 , then φ is real when s ± are complex conjugates.After some algebra, one finds the explicitly real form where w n = ℜ(a + n ) and z n = ℑ(a + n ).The series expansions above completely characterize the solutions for φ and H in the asymptotic region.However, the series will not converge for all values of r.We will provide expressions for the radii of convergence of all of our series solutions at the end of this section.With this in mind, we now search for a solution to the equations of motion for small r.The equation of motion for h(r) is still given by (34), but now we guess a power series of the form h 0 (r) = ∞ n=0 cn r n+q .The indicial equation is simply q(q − 3) = 0, giving the two roots q1 = 3 and q2 = 0.In general, the full solution for the case of roots q1 > q2 differing by an integer is given by where the procedure for determining the coefficients dn depends on whether or not k vanishes.If k does vanish, the dn can be obtained using the usual recurrence relation, and otherwise they must be computed using a more complicated approach.In general, the value of k is determined by For this particular differential equation, the recurrence relation for general q is given by c1 (q) = c2 (q) = 0 (52) cn+3 (q) = 1 8αβm(n + 3 + q)(n + q) × 4αβc n+2 (n + 2 + q)(n − 1 + q) (53) from which we conclude that Hence, the general solution becomes where the coefficients are given by: and with n ≥ 0. The equation of motion for φ(r) is given by ( 42), but now we guess a power series of the form giving repeated root s1 = s2 = 0, while the recurrence relation for general s is given by In the special case of a repeated root, the solution takes the general form where K is an integration constant.Here the coefficients a n are given by ã1 = ã2 = 0 (64) and the coefficients b n are related to the a n by bn = with n ≥ 0. The radius of convergence of each power series solution is the distance in the complex plane between the point around which the series is expanded and the nearest other regular singular point of the differential equation.For both (34) and ( 42), the regular singular points are located at r = 0, r = ∞, and the zeros of A 0 (r).Thus, the radii of convergence are determined by the roots of a general depressed cubic equation The roots can be written as where If Λ < 0, the discriminant is always negative and R 1 will be real, while R ± are complex conjugates.The moduli are So the radius of convergence of the r-series is given by 0 < r < |R 1 | while the radius of convergence of the 1 r -series is given by 0 < 1 r < 1 |R±| .If Λ > 0, we have two cases.If Λ > 1 9m 2 then the discriminant is negative and we still have R 1 real with R ± being complex conjugates.Since S + < 0 and S − < 0, |R 1 | > |R ± |.Then the radius of convergence of the r-series is given by 0 < r < |R ± | while the radius of convergence of the 1 r -series is given by 0 < 1 r < 1 |R1| .Finally, if 0 < Λ < 1 9m 2 , then S + and S − become complex conjugates and we have three real roots.This can be seen by writing S ± = (a ± ib) 1/3 = ρ 1/3 e ±iθ/3 , where Hence, the radius of convergence of the r-series is given by 0 < r < |R − | , while the radius of convergence of the 1 r -series is given by 0 < 1 r < 1 |R1| .With the series solutions for φ and H in hand, one can now, take into account the approximate back-reaction on the (A)dS-Schwarzschild background by substituting the solutions for the fields into the general solution for the metric function (32).
Using the series solutions for large r, one can perform a numerical integration to plot the profile function A(r) whose zeros are the black hole and cosmological horizons for different values of a − 0 and ā+ 0 , as shown in figure 1.As expected, for appropriate values of the couplings and integration constants, we find a pair of horizons which tend to the dS-Schwarzschild ones as a ± 0 → 0. Note that the discrepancy from A 0 (r) of the inner horizon is greater than the outer horizon.However, as φ is allowed to fluctuate more strongly, i.e., as a ± 0 becomes large, the solution does not have a horizon.In figure 2 we show the results for different values of Λ 0 .In order for the solution to approach dS-Schwarzschild in the decoupling limit, we fix Λ 0 = Λ + ακ.For Λ 0 < 0, the lower endpoint of integration in (32) becomes imaginary and the solution becomes complex and unphysical.Hence, we see that the model has a dynamical preference for black holes with Λ 0 ≥ 0. This completes our analysis of spherically symmetric black holes in dynamical Henneaux-Teitelboim gravity.Another very natural question to consider is how the dynamics of the effective cosmological constant modify the standard cosmological background of general relativity, which we study in the following section.

IV. COSMOLOGICAL ANALYSIS
We will now consider homogeneous, isotropic configurations sourced only by the fields φ and H with no additional matter content.Our analysis runs along the same lines as other work on cosmology in unimodular gravity with a non-conserved stress tensor, e.g.[20], where the model we consider here serves as a concrete realization of the general proposal.⊙ and chosen the particular solution h(r) = 1 3 r3 .We see that for small values of Λ0 both horizons occur for larger values of r, while for larger values of Λ0 the horizons are pushed inward and closer together until at some critical value, the horizons degenerate and the black hole becomes extremal.For even larger values of Λ0, no horizons exist and the solution has a naked singularity.
We consider the FRW metric along with the homogeneous and isotropic fields (73) Again, we are only considering contributions to the stress tensor from H and φ, so we set L M = 0 in (7).
We decompose the Einstein equations into trace-free and pure trace parts, recalling that the stress tensor for H is pure trace.The trace-free equation is given by ( 1), and plugging in our symmetry-reduced expressions for the metric, three-form, and scalar, we find a single independent equation: The trace equation (21) gives which can be combined with (74) to give The equations of motion for the scalar and three-form imply ( 16) and ( 17 where the coefficients are If we impose initial conditions such that A u = A v = 0, which imply that A H = 0, then φ relaxes to a constant at late times, allowing for a de Sitter-like phase of accelerated expansion in the late universe.However, in general both the exponentially growing and decaying modes will be present, so we expect that the solution that is compatible with observations of our current cosmological epoch will be unstable to small perturbations around this background.Still, having a solution of the linearized equations of motion which approaches de Sitter at late times may be useful for more sophisticated analyses that include matter other than H and φ.

V. DISCUSSION
In this work we have introduced dynamical Henneaux-Teitelboim gravity, a minimal modification of general relativity based closely on the Henneaux-Teitelboim formulation of unimodular gravity that realizes fluctuations in the cosmological constant sourced by non-conservation of the energy-momentum tensor.Furthermore, we have considered the implications of this model for spherically symmetric black holes and homogeneous, isotropic configurations.In order to make progress analytically, we have made a number of simplifying assumptions that should ultimately be lifted in order to probe more realistic physical scenarios.For the analysis of black holes, we have only presented an approximate solution of all the field equations by perturbing around the Schwarzschildde Sitter metric and using series methods.In order to establish whether such an approximate solution is physically viable we would need, at a minimum, to study the geodesic motion of a massive particle orbiting such a black hole around the equator and compare with astrophysical observations.The assumption of spherical symmetry dramatically simplifies the equations of motion, reducing them to ordinary differential equations, but of course, in order to model rotating black holes, this restriction must be lifted.Studies of rotating black holes in similar modified gravity models such as dynamical Chern-Simons gravity [22][23][24][25][26] suggest that going to the slowly-rotating limit and making use of numerical methods are likely to be necessary.Finally, the series solutions we have presented for small and large r have finite radii of convergence, leaving the possibility of intermediate regions, possibly containing the event horizon, which are not covered by either series solution.Here again, numerical methods for interpolating between the solutions for small and large r might be useful.We leave such numerical studies for the future, but note that for the solutions we have studied in the decoupling limit that are close to Schwarzschild-de Sitter, the radius of convergence is well below the scale of the event horizon.On the cosmology side, we have only considered the contribution to the stress tensor from the fields H and φ for simplicity.In order to make realistic predictions, we would need, at a minimum, to include contributions from general baryonic matter and radiation as well, consider the effects of cosmological perturbations, and, if they are stable, study the matter-power spectrum for deviations from ΛCDM.We have also seen that while a solution of the linearized equations of motion that approaches de Sitter at late times exists, this solution is apparently unstable.It would be interesting to see whether this instability persists in a more realistic analysis including additional matter and radiation, or whether, perhaps, the exponentially growing modes might be dynamically suppressed by the inclusion of additional matter degrees of freedom.The model we have considered in this article is a natural extension of the Henneaux-Teitelboim formulation of unimodular gravity because the kinetic terms for the scalar and threeform fields are invariant under the local symmetries of the latter-spacetime diffeomorphisms and two-form-valued gauge transformations.Consequently, the kinetic terms are forced upon us by "Gell-Mann's totalitarian principle," which dictates that "everything that is not forbidden (by local symmetries) is compulsory."To put it plainly, renormalization group flow will generate every term that is compatible with the local symmetries of the action (5), and that includes kinetic terms for the scalar and three-form fields.In this sense, the model we have studied in this article follows straightforwardly from (5) and general principles of effective field theory.