Spherically Symmetric Configurations in Unimodular Gravity

Unimodular gravity (UG) is considered, under many aspects, equivalent to General Relativity (GR), even if the theory is invariant under a more restricted diffeomorphic class of transformations. We discuss the conditions for the equivalence between the two formulations by applying the UG to the static and spherically symmetric configurations being the energy-momentum tensor sourced by a scalar field or by the electromagnetic field. We argue that the equivalence between UG and GR may be broken when analyzing the stability of the solutions at perturbative level.


I. INTRODUCTION
General Relativity (GR) is the modern theory of gravitational interaction.The gravitational phenomena is considered as the structure of the space-time itself induced dynamically by matter.GR is considered a very successful theory: all local tests confirms the predictions of GR within high precision.At cosmological scales, it leads to the Standard Cosmological Model (SCM) which also addresses consistently all available observations, from scales of galaxies up to the larger structures of the universe.It accounts also successfully to the different phases of the evolution of the universe, including the primordial phases at least to the primordial nucleosynthesis scales.See from this point of view, the SCM, based on GR, is an almost perfect model to describe the entire evolution of the universe.
However, seen from a different perspective, the SCM is at least problematic.To account for the observations at the different scales, it demands the introduction of two until now undetected components in the matter/energy content of the universe.The dynamics of galaxies and cluster of galaxies, and even the formation of such structures, asks for an additional pressureless component, dubbed dark matter, which manifests only indirectly.Moreover, to explain the present accelerated phase of the universe, the CMB spectrum and to obtain an age of the universe consistent with the age of globular clusters, the SCM asks for another component, with negative pressure, which does not agglomerate, dubbed dark energy.
Dark energy is now frequently associated to the vacuum energy as predicted by quantum field theory.However, its observed value seems not consistent with the theoretical predictions by dozens of order of magnitude [1][2][3].There are many proposals to cope with this problem.One of them is to replace it by a self-interacting scalar field, called quintessence [4].However, it must be explained, in the quintessence program, why the vacuum energy must be exactly or, at least, nearly zero.Therefore, the vacuum energy must, somehow, degravitate [5][6][7].There are many mechanisms to implement such degravitating mechanism, but until now such proposals are, in some sense, in construction.For a general overview of the dark energy problem, see Ref. [8].
One interesting approach to the cosmological constant problem described above is through the unimodular gravity (UG) class of theories [9][10][11][12] where the determinant of the metric, g, is fixed.Originally, g = 1, but other possibilities can be explored, see next section.UG leads to traceless gravitational equations.The energy-momentum tensor does not conserve necessarily anymore, since UG is not invariant by the full diffeomorphism group, but by a more restricted structure called transverse diffeomorphism [13].If the conservation of the energy-momentum tensor is imposed, GR is recovered with a cosmological term that appears as an integration constant.If the conservation of the energymomentum tensor is not imposed, as we will prove below, a class of dynamical vacuum theories is obtained, implying an interaction of the matter sector with the decaying cosmological term.
In previous works we have explored the distinction between GR and UG mainly at perturbative level in the cosmological context, see [14] and references therein (see also [15]).Here we will extend such studies to the static, spherically symmetric configurations.In UG, with the imposition of the conservation of the energy-momentum tensor, the static, spherically symmetric solutions are identical to those of GR, but now containing a cosmological constant.The non-conservation of the energy-momentum tensor, on the other hand, can be mapped in the GR structure with a dynamical cosmological term.Indeed, the non-conservation of the energy-momentum tensor is allowed in this context, leading to a new formulation of the UG theory.It is also worth mentioning that such a non-conservation mechanism appears in many other situations.For a review, see Ref. [16].All these aspects are discussed in the next section.In section III the general equations for a static, spherically symmetric configuration are settled out.Some examples of interacting models, resulting from the non-conservation of the energy-momentum tensor, will be shown in section IV, both in presence of an electromagnetic field as well as of a self-interacting scalar field in section V.For the latter case, we perform, in section VI, a perturbative analysis aiming to show how the usual results of GR change in the unimodular context.In particular, the unimodular condition on the determinant of the metric implies in vanishing perturbations at linear level.The results obtained are discussed in section VII.

II. FIELD EQUATIONS
The Einstein-Hilbert action, in presence of the cosmological term and the matter Lagrangian, implies in the following field equations: The application of the Bianchi identities leads to the energy-momentum tensor T µν conservation: The conservation laws related to the energy-momentum tensor can be alternatively deduced from the invariance of the Einstein-Hilbert Lagrangian by diffeomorphic transformations [17].
In order to obtain the UG equations, we introduce a constraint in the action via a Lagrange Multiplier χ and an external field ξ [15]: The presence of the external field allows one to use a suitable coordinate system according to the problem under analysis, for example, the usual spherical coordinates or the quasi-global coordinates employed in spherical symmetric space-time.The final field equations for this case are The above equation ( 6) is obtained by using the Bianchi identities in (5).
As highlighted in Ref. [17], it is important to note that in UG, the conservation of the energy-momentum tensor cannot be derived through the conventional diffeomorphism invariance because the theory only exhibits invariance with respect to a limited set of diffeomorphisms, referred to as transverse diffeomorphisms.The latter implies that the energy-momentum divergence tensor is equal to the gradient of a (undetermined) scalar function: On one hand, it is entirely permissible to set the gradient of Θ to zero.If this is done, we recover (2), with Λ appearing as an integration constant.On the other hand, one can also choose, From now on, we will identify Θ ≡ −Λ.With this identification, Λ becomes a dynamical term.If Λ is constant, as already stressed, we return to the GR equations in presence of a cosmological constant.But, if Λ is a function of the space-time coordinates, we end up with the following set of equations, This is equivalent (up to the restriction in the diffeomorphic class of transformation) to the RG equations in presence of a dynamical cosmological term: provided that Λ is identified with Θ as given by ( 8).Hence, the non-conservation of the energy-momentum tensor allows to map the UG theory into GR equipped with a dynamical cosmological term, implying in an interacting like model in the GR context.It is also convenient, for reasons that will become clear later on in the work, to write down the UG equations in a more compact form such as with the definitions, We will call E µν the unimodular gravitational tensor and τ µν the unimodular energy-momentum tensor.

III. EQUATIONS FOR A SYMMETRIC AND STATIC CONFIGURATION
In this section, we will write down the general expressions for a symmetric and static configuration.In the appendix A the corresponding expressions with a time dependence will be derived, which are necessary to perform the perturbative analysis to be described later.
Let us consider the metric, The non-vanishing Christoffel symbols are the following.
Also, the non-vanishing components of the Ricci tensor and the Ricci scalar are the following.
Consequently, the non-vanishing components of the unimodular gravitational tensor defined in ( 14) are the following: The left-hand side of the UG field equations for the symmetric and static configuration has been set up.The next step is to characterize the source field.In the next couple of sections, the electromagnetic field and a scalar field will be considered as sources of the gravitational field.

IV. THE ELECTROMAGNETIC FIELD
For the case of a electromagnetic field as the source of the energy-momentum tensor one has, It is worth mentioning that it has zero trace: Equations ( 5) and ( 6) become, Remark that, contrarily to GR, the traceless character of the energy-momentum tensor does not imply R = 0, unless the Maxwell equations are obeyed.Imposing the spherical symmetry, the only non-vanishing component is F 01 = E ≡ E(r).Then, the equations are: with R given by (24).Until now, no coordinate condition has been imposed.Adding (33) and (34), we obtain, The use of the quasi-global coordinates, with α = −γ, leads to, As in the usual Reissner-Nordström (RN) solution in GR, there is a center at r = 0. Equation ( 8), with T = 0 and with the identification of Θ with −Λ implies in R = −4Λ.The equations of motion reduce to, In order to proceed further, we must impose a condition.This is a crucial step in working with UG as already stressed in [14].One possibility it to fix R = −4Λ ≡ constant.This leads to the Reisnner-Nordström-de Sitter (RNdS) solution.In fact, this implies to recover the conservation law F µν ;µ = 0.If Λ = 0, we re-obtain the RN solution.If Λ > 0, the RNdS solution is obtained, and if Λ < 0, the Reisnner-Nordström-(Anti) de Sitter (RNAdS) solution is recovered, as it will be seen below.On the other hand, there also also other possibilities that to be explored since Λ can be non constant, covering the possibility of a dynamical cosmological term.Three cases will be considered, namely a constant and two dynamical cosmological terms, corresponding to either the usual or the modified conservation laws.

A. Constant cosmological term
If Λ = constant, after identifying an integration constant with the total charge Q.The Coulomb law is recovered, as in the RN solution.
Using the quasi-global coordinate condition and the solution for the electric field E, equation (33) becomes: Defining A = e 2γ , the equation takes the form, This is a second order, linear, non-homogeneous differential equation whose solution is C 1,2 being integration constants.Inserting this solution into the condition R = −Λ, we obtain that it is satisfied provided C 2 = −Λ, while C 1 remains arbitrary, being fixed by using the newtonian limit.
The final solution is given by, This is the RNdS solution.It coincides with the static and spherically symmetric solution in GR with an electromagnetic field and a cosmological constant.This could be expected from the beginning since UG (satisfying the usual conservation laws) leads to the same field equations as RG with a cosmological term, with the only (but important, as we will see later) difference that UG is restricted to transverse diffeormophism instead of the full diffeomorphism.

B. Varying cosmological term
For a varying cosmological term, it is necessary to impose an ansatz on the behaviour of the function Λ.This is also true in GR when the cosmological term is dynamical.Since a static and spherically symmetric configuration is considered, the cosmological term must be a function on the coordinate r only: Λ ≡ Λ(r).
Let us restrict ourselves again to the condition R = −4Λ.Using the previous results and also identifying β = ln r, α = −γ and A = e 2γ , then: The solution for the homogenous equation is, To obtain the inhomogeneous solution, we write, obtaining, with a solution which depends on r: We will consider two different configurations for the function Λ(r), corresponding to two distinct behavior both asymptotically as well as at the center (r = 0).

Case A
First, it is imposed a power law behavior for Λ(r), with Λ 0,1 constants.The final solution is given by the following expressions.
• p = −4: • p = −4: The case p = −4 is clearly pathological since the electric field becomes imaginary near r = 0 when Λ 1 > 0 or for large r if Λ 1 < 0. For p = −4 a change of sign of E 2 can be avoided by choosing Λ 1 > 0 for −4 < p < 0, or Λ 1 < 0 for p < −4 or p > 0. The values p = 0, −3 correspond to the cases already included in the constants C 1 and C 2 of the homogenous solution.
The solution p = 0, with the required conditions to avoid an imaginary electric field, contains either multiple horizon black holes, with a singularity at r = 0, or naked singularities similarly to the dSRN solution in RG but the metric functions, in the UG case, may present a different shape mainly near the singularity.These solutions are asymptotically non-flat except if Λ 0 = 0 and p > −2.The corresponding equations in GR equipped with a cosmological term with the same functional dependence, using the same symmetries, lead to the same solution as it can be explicitly verified.

Case B
We will exploit now the functional form, If Λ 0 = 0, this functional form represents an asymptotically constant cosmological term near the origin, which becomes zero at infinity.Following the same steps of the previous case, the final form of the metric function is: Again, the same solution is obtained in the GR with a varying cosmological term giving by (56).There are multiple horizons and naked singularities, as in the previous case.No change of sign in the E 2 term can be assured by imposing Λ 1 > 0.
In all the cases discussed above, the presence of the a cosmological term, constant or not, introduces new features in the solutions with respect to the usual RN solution but does not remove the singularity at r = 0.

V. SCALAR FIELD
The energy-momentum tensor for a self-interacting scalar field is, The ordinary scalar field is denoted by ǫ = +1 and the phantom scalar field by ǫ = −1.In GR, in four dimensions, black holes exist only for the phantom case [19,20].
Inserting the expression for the energy-momentum tensor (59) in the UG equations one obtains, One distinguishing feature of the above equations is the absence of the potential (or, as before, the cosmological term, which is the particular case of a constant potential): it naturally disappears due to the traceless structure of the UG equations.Equation (61) can be written as, equations ( 60) and (61) take the following form, In this way, we recover the GR equations equipped with a self-interacting scalar field.
In a static and spherically symmetric configuration, the UG field equations (60) read, Combining these equations, we have the following relations: Remark that in the UG equations, there is no potential, even if it appears in the energy-momentum tensor.Moreover, there are three metric functions (which can be reduced to two functions by gauging the radial coordinate) and the scalar field to be determined, and just two independent equations, (69) and (70).Hence an ansatz must be introduced.From the conservation law, we have the relation, where V (φ) is a function to be determined.We have slightly changed the notation (V instead of Λ) to identify the unknown function with the potential.With this identification, the UG equations become identical to the GR equations with a potential.In GR, the potential must be chosen.In UG a functional form for the scalar field (or for one of the metric function) must be chosen.There is a correspondence between the choice of the functional form of the scalar field and the choice of the potential in GR.
Two possible examples are the following.
1.If the scalar field is chosen such that, we find, Using (71) we find V = 0.This solution represents a black hole only if ǫ = −1.This solution has been determined in the GR context in Ref. [18].
2. The regular black hole determined in Ref. [20], is also solution in the UG case, without a potential.Imposing that the scalar field is given by, the metric is then given, in the quasi-global coordinates, by, In these expressions b, c and ρ 0 are constants.Using relation (71) the potential in GR context is given by, the same used in the GR context.

VI. REMARKS ON THE BIRKHOFF THEOREM AND THE STABILITY OF THE SOLUTIONS
Initially, we present a straightforward argument to demonstrate that, in the cases of both electromagnetism and scalar fields, the Birkhoff theorem holds the same significance as it does in GR.For electric charged static solutions in GR, the Birkhoff theorem is valid.The same occur for the corresponding solution in the UG.This can be seen by supposing radial time dependent configurations.The argumentation follows the same reasoning used in GR, see for example [21].From the expression presented in the appendix, and considering only a radial electric field, the 0 − 1 component of the field equations, using the Schwarzschild coordinate system, with β = ln r, implies that α must be time independent, since the right hand side of the equation is zero for a pure radial electric field.Combining equations 0 − 0 and 1 − 1, it comes out that α = −γ.Hence, all metric functions are time independent.
For the scalar field case, the Birkhoff theorem is not valid because the right hand side contains a term of the time φφ ′ which forbids to consider the metric function α as time independent, as it happens in the GR case.The Birkhoff theorem is verified only if the scalar field is static [22].
In the two examples discussed in the previous section, having the scalar field as the source of the geometry, and considering the GR context, the solutions are unstable, except for the regular solution in the very special case the minimum of the areal function coincides with the horizon [23,24].However, this result can change in the UG context since the unimodular condition implies in new relations for the perturbed functions that are absent in GR.
We will illustrate the special features of the perturbative analysis considering the case of black holes with a scalar field.Only radial perturbations will be considered.In the GR context, this is enough to conclude about the instability of the solution [23].We will show that in the UG, if if we try to follow the same procedure as in GR, the perturbations at first order are strictly zero due to the unimodular condition.
The unimodular condition implies, Since the function ξ is fixed, the unimodular condition leads, at linear perturbative order, There is still the freedom to impose a coordinate condition due to the diffeomorphic (even if transverse) invariance.
The choice δβ = 0 is related to the gauge invariant variables [23].Hence, we end up with the conditions, We write down the perturbations in a generic way as The perturbed equations, under the conditions above, are the following: It is clear that the equations are consistent only in the trivial case: δφ = δγ = 0. Hence, it is not possible to obtain informations on the stability of the solution, at least at linear level and following a procedure close to that used in GR.This is a distinguishing feature of unimodular gravity in comparison with GR.

VII. CONCLUSION
Unimodular gravity (UG) is one of the first alternatives to General Relativity (GR).It is a geometric theory which is invariant with respect to a restricted diffeomorphic class of transformations, the transverse diffeomorphism, due to the imposition of a constraint on the determinant of the metric.In UG the usual conservation of the energy-momentum tensor is not assured: The conservation of the energy-momentum tensor is a choice.If it is imposed, UG becomes in principle equivalent to GR with a cosmological term.However, the restriction on the determinant of the metric may lead to some important new features at perturbative level.We have shown here that if the conservation of the energy-moment tensor is relaxed, UG becomes equivalent to GR with a dynamical cosmological term, with still the same important difference due to the UG constraint which can manifested at perturbative level.
We have discussed, in this context, the static and spherically symmetric solutions in UG.For the vacuum configuration, Schwarzschild solution is also verified in UG.The same occurs with the Reissner-Nordström solution, but only if the energy-momentum tensor is conserved.If not, the dynamical cosmological term induces new features, but it does not prevent the appearance of the singularity at r = 0. Similar features appear in the case when a scalar field appears as the main source.In this case, the potential term, representing the self-interaction of the scalar field, disappears in the UG context and an ansatz must be imposed in order to close the set of equations.This mounts, in the GR context, to choose a given potential for the scalar field.For a discussion of the UG in static, spherical configurations but focusing compact objects, see Ref. [25].
We have shown that the Birkhoff theorem follows the same features as in GR, being satisfied for a charged solution, being possibly violated for a dynamical scalar field.The linear radial perturbations have been analyzed when a scalar field is present.Once more, GR black hole solutions are generically unstable in the latter case.In UG, using the gauge invariant approach employed in GR and restricting to radial perturbations, the condition on the determinant of the metric leads to vanishing perturbations at linear order, and possibly also for higher order.As already discussed in the cosmological context, this result seems to point out for a breaking of the equivalence of UG and GR at perturbative level.There are other viewpoints on the implementation of the UG constraints in performing a perturbative analysis, see for example Ref. [26].However, the results reported here indicates that a direct application of the procedures used in GR combined with the unimodular constraint may lead to conclusions different from those obtained in GR.
are the following: For the static case, the above expressions reduce to,