Black holes in the TeVeS theory of gravity and their thermodynamics

Black holes in the TeVeS theory of gravity and their thermodynamics

Eva Sagi    Jacob D. Bekenstein Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
July 14, 2019

TeVeS, a relativistic theory of gravity, was designed to provide a basis for the modified Newtonian dynamics. Since TeVeS differs from general relativity (e.g., it has two metrics, an Einstein metric and a physical metric), black hole solutions of it would be valuable for a number of endeavors ranging from astrophysical modeling to investigations into the interrelation between gravity and thermodynamics. Giannios has recently found a TeVeS analogue of the Schwarzschild black hole solution. We proceed further with the program by analytically solving the TeVeS equations for a static spherically symmetric and asymptotically flat system of electromagnetic and gravity fields. We show that one solution is provided by the Reissner-Nordström metric as physical metric, the TeVeS vector field pointing in the time direction, and a TeVeS scalar field positive everywhere (the last feature protects from superluminal propagation of disturbances in the fields). We work out black hole thermodynamics in TeVeS using the physical metric; black hole entropy, temperature and electric potential turn out to be identical to those in general relativity. We find it inconsistent to base thermodynamics on the Einstein metric. In light of this we reconsider the Dubovsky–Sibiryakov scenario for violating the second law of thermodynamics in theories with Lorentz symmetry violation.

04.70.-s, 97.60.Lf, 04.70.Bw, 04.70.Dy


I Introduction

There are significant discrepancies between the visible masses of galaxies and clusters of galaxies and their masses as inferred from Newtonian dynamics. In particular, the accelerations of stars and gas in the outskirts of galaxies or those of galaxies in clusters are much too large, and the disk rotation curves of spiral galaxies, which are naively expected to drop as away from galaxy centers, tend to remain flat to the last optically or radio measured point. These discrepancies are also manifest in the gravitational lensing by galaxies and clusters. It is commonly assumed that these problems stem from the existence in the said systems of large amounts of “dark matter”. For instance, galaxies are assumed to be enshrouded in roundish halos of dark matter that dominate the gravitational fields far from the galaxy centers.

But the putative dark matter has yet to be identified or detected directly. Furthermore dark matter models of galaxies require much fine tuning of the dark halo parameters to fit the data, and there are some sharp problems outstanding such as the observationally inferred absence of cusps in the dark matter density at galaxy centers, cusps which are predicted by dark matter cosmogony. Thus many have wondered if dark matter is the whole story. An alternative, if less orthodox, approach is formalized in the modified Newtonian dynamics (MOND) paradigm Milgrom (1983), which proposes that Newtonian gravity progressively fails as accelerations drop below a characteristic scale which is typical of galaxy outskirts. MOND assumes that for accelerations of order or well below it, the Newtonian relation is replaced by


where the function smoothly interpolates between at and the Newtonian expectation at . This relation with suitable standard choice of in the intermediate range has proved successful not only in in rationalizing the asymptotical flatness of galaxy rotation curves where acceleration scales are much below , but also in explaining detailed shapes of rotation curves in the inner parts in terms of the directly seen mass, and in giving a precise account of the observed Tully-Fisher law which correlates luminosity of a disk galaxy with its asymptotic rotational velocity Bekenstein (2006).

The pristine MOND paradigm does not fulfill the usual conservation laws, does not make it clear if the departure from Newtonian physics is in the gravity or in the inertia side of the equation , and does not teach us how to handle gravitational lensing or cosmology in the weak acceleration regimes. All these things are done by TeVeS Bekenstein (2004), a covariant field theory of gravity which has MOND as its low velocity, weak accelerations limit, while its nonrelativistic strong acceleration limit is Newtonian and its relativistic limit is general relativity (GR). TeVeS sports two metrics, the “physical” metric on which all matter fields propagate, and the Einstein metric which interacts with the additional fields in the theory: a timelike dynamical vector field, , a scalar field, , and a nondynamical auxiliary scalar field . The theory also involves a free function , a length scale , and two positive dimensionless constants and .

TeVeS is an attempt to recast MOND into a full physical theory in which the latter’s novel behavior is due to the gravitational field. Some checks of its consistency and comparisons with hard facts have been made. Thus Bekenstein showed that TeVeS’s weak acceleration limit reproduces MOND, and that it also has a Newtonian limit, and calculated its parametrized post-Newtonian coefficients and , which agree with the results of solar system tests Bekenstein (2004, 2005); Giannios (2005). Skordis et al. Skordis et al. (2006) and Dodelson and Liguori Dodelson and Liguori (2006) studied the evolution of homogenous and isotropic model universes in TeVeS, and showed that it may reproduce key features of the power spectra of the cosmic microwave background and the galaxy distribution. TeVeS has also been tested against a multitude of data on gravitational lensing (for some references see Ref. Bekenstein, 2006). All the above refer principally to situations where the gravitational potential is small on scale . Since neutron stars and black holes exist in nature one must also understand strong gravity systems in the TeVeS framework.

A beginning in the investigation of the strong gravity regime of TeVeS has been made by Giannios Giannios (2005). For vacuum spherically symmetric and static situations he showed that under a simplifying limit (which we shall detail below), the Schwarzschild metric qua physical metric and a particular scalar field distribution together constitute a black hole solution of TeVeSs. This motivates us to look in this paper at more complicated cases, such as that of the charged nonrotating black hole in TeVeS. We find that the Reissner-Nordström (RN) metric as physical metric and the usual electric field together with a special configuration of TeVeS’s scalar field constitute a black hole solution in TeVeS. Using this solution we investigate the thermodynamics of spherical black holes in TeVeS.

In Sec. II we recapitulate the fundamentals of TeVeS, while in in Sec. III we describe Giannios’ results for the nonrotating vacuum black hole. In Sec. IV we go on to solve the TeVeS equations for the case of a charged nonrotating black hole, obtaining a physical metric which coincides with the RN metric of GR. Sec. V presents a resolution of the problem pointed out by Giannios: the uncharged black hole solution he found seems to permit superluminal propagation near the black hole horizon. Next in Sec. VI we examine how the familiar concepts of black hole thermodynamics apply to our black hole solutions, and check their consistency using several prescriptions. We calculate the relevant thermodynamic quantities using the physical metric, and show that the Einstein metric is inappropriate for discussing thermodynamics. In this light we discuss anew the potential thermodynamic inconsistency described by Dubovsky and Sibiryakov for theories with broken Lorentz symmetry Dubovsky and Sibiryakov (2006).

Ii The TeVeS equations

The acronym TeVeS refers to the Tensor-Vector-Scalar content of the theory. The tensor part pertains to the two metrics, , dubbed the Einstein metric, on which the vector and the scalar fields propagate, and the physical metric , on which matter, electromagnetic fields, etc. propagate. The physical metric is obtained from the Einstein metric through the following relation:


Thus one passes from the space of to that of by stretching spacetime along the vector by a factor , and shrinking it by the same factor orthogonally to that vector. This prescription retains MOND phenomenology, while augmenting the gravitational lensing by clusters and galaxies to fit observations.

The dynamics of the metrics and the fields are derivable from an action principle. The action in TeVeS is the sum of four terms. The first two are the familiar Hilbert-Einstein action and the matter action for field variables collectively denoted :


Next comes the vector field’s action ( is a dimensionless positive coupling constant)


which includes a constraint that forces the vector field to be timelike (and unit normalized); is the corresponding Lagrange multiplier. The presence of a nonzero establishes a preferred Lorentz frame, thus breaking Lorentz symmetry. Finally, we have the scalar’s action ( is a dimensionless positive parameter while is a constant with the dimensions of length, and a dimensionless free function)


Above with . The scalar action is here written differently than in Ref. Bekenstein, 2004; we have eliminated the nondynamical field and redefined the function . The new form makes it easier to understand the strong acceleration limit of the theory, which is especially relevant to the present work.

Variation of the total action with respect to yields the TeVeS Einstein equations for ;


The sources here are the usual matter energy-momentum tensor , the variational derivative of with respect to , as well as the energy-momentum tensors for the scalar and vector fields:




Each choice of defines a separate TeVeS theory, and is similar in nature to the function in MOND. In particular, corresponds to high acceleration, i.e., to the Newtonian limit.

The equations of motion for the vector and scalar fields are, respectively,


Additionally, there is the normalization condition on the vector field:


The lagrange multiplier can be calculated from the vector equation.

Iii Neutral spherical Black Holes

In his work on black holes, Giannios Giannios (2005) worked in the limit which also entails . Since we shall later work in the same limit, we shall here justify it in more detail than he did. Near the horizon of a black hole of mass , the Newtonian acceleration amounts to . Thus even for the most massive black holes suspected () the accelerations are strong on scale out to at least a million times the gravitational radius, i.e. well into the asymptotically flat region which determines the metric properties. This means MOND effects are suppressed while the full complexity of the TeVeS equations is still evident. In the said limit, and under the assumption that the vector field points in the time direction (which has support in the more general context of static solutions Bekenstein (2004)), Giannios obtained an exact spherically symmetric analytical solution to the TeVeS equations, for metric, scalar and vector fields.

The Einstein metric is taken in isotropic coordinates, and ,


where henceforth . Since , are functions of only, and the vector field points in the time direction, its dependence is fully determined by the normalization condition Eq. (13) and the requirement that be future pointing:


Then the relation between the physical and fields metrics reduces to


Giannios first solved the TeVeS equations assuming that , thus decoupling the vector field from the theory, and then performed a transformation involving , which recovered the more general solution. For the Einstein’s and equations are, respectively,


and the scalar equation takes the form


Since there are only three unknown functions, and , one of the four equations is obviously superfluous.

Combining the and Einstein equations gives the simple differential equation


This has the solution


where the additive integration constant has been set to zero in order to have an asymptotically flat spacetime, namely, when .

The second integration constant, , can be evaluated by expanding above in and comparing with the expansions (with ) of the metric coefficients of the exterior solution for a spherical mass Bekenstein (2004); Giannios (2005),


Here is a mass scale Bekenstein (2004) defined by the expansion


for the solution of Eq. (21). For a ball of nonrelativistic fluid, is very close to the Newtonian mass, and is a scale of length that can be linked to the object’s mass Bekenstein (2004). However, the relation between and depends on the system under consideration, and is different for stars and black holes. At any rate, for , is found to be


Making the educated guess that


Giannios determines to be


The correctness of Eqs. (28) and (29) can be checked by substituting them into the sum of Eq. (18) and Eq. (19), or the difference of  Eq. (18) and Eq. (20); both combinations are independent of the equation pair already used. The determination of the Einstein metric for is completed by the trivial integration of Eq. (28). Finally, the scalar field is found now by integrating Eq. (21) and fixing the two integration constants just as in Eq. (26),


Going on to the more general case , Giannios finds that just by replacing Eq. (27) by


in the above solutions for and will produce an exact solution of the TeVeS equations for [equations which are the case of Eqs. (49)-(52) below].

The physical metric now follows from Eqs. (16)-(17):


with . In order for this result to represent a black hole, the candidate event horizon must have bounded surface area, and must not be a singular surface. The surface area is proportional to , which has a factor ; for this to be bounded requires . The Ricci scalar of the above metric is


We notice that will blow up as unless or . Thus, only the value describes a black hole. The definition of then gives another relation between and ,


and the physical metric takes the final form


which we recognize as the Schwarzschild metric in isotropic coordinates.

Unlike GR’s Schwarzschild black hole, the TeVeS neutral spherical black hole is “dressed” with a scalar field , a solution of Eq. (30). This field does not induce a singularity at the horizon because of the particular structure of the TeVeS equations. However, the logarithmic divergence of at the horizon was a cause of concern to Giannios. It was earlier shown Bekenstein (2004) that absence of superluminal propagation of the various TeVeS fields is guaranteed only when . But here diverges logarithmically at , and becomes already negative sufficiently close to even if . We will show in the next section how this apparent problem is solved.

Iv Charged spherical black holes

The next natural step is to look for an electrovacuum static spherically symmetric solution to the TeVeS equations, the analog of the RN solution of GR. We again take . Again we assume that the vector field points in the time direction, and that both the physical and the Einstein metrics are spherically symmetric. These are essential simplifying assumptions which enable us to find a specific solution to the TeVeS field equations. Other solutions may exist for which the vector field is endowed with a radial component. However, to judge from the neutral case, as analyzed by Giannios Giannios (2005), the PPN parameter of such a solution with very low charge would be in contradiction with recent observations Will (2003) in the solar system. It would be odd if the PPN structure of a black hole’s far field were that different from the sun’s. By contrast, still in the neutral case, a TeVes solution with the vector field pointing in the cosmological time direction yields PPN parameters identical to those of GR Giannios (2005).

We continue to work in isotropic coordinates, for which the transition between physical and Einstein metrics is simplest: as seen earlier, in view of the the normalization condition (13), the transformation (2) is equivalent to


The Einstein metric again takes the form (14), and the physical metric will have similar form, namely


with the following relation among , , and :


Now, the cosmological value of should be nonzero in our evolving universe: . Thus the requirement that the Einstein metric be asymptotically Minkowski (both and vanish as ), needed to maintain consistency with previous work Bekenstein (2004), introduces a factor in the physical metric coefficients,


This is equivalent to a rescaling of the coordinates which depends on cosmological epoch, and will have to be taken into account when considering physical quantities in the framework of TeVeS.

The energy-momentum tensor no longer vanishes; it is given by


with , the electromagnetic field tensor (not its dual), obtained by solving Maxwell’s equations in vacuum written wholly with the metric , namely,


In the isotropic metric Eq. (39), and with the assumption of spherical symmetry and absence of magnetic monopoles, the only nonvanishing component of the electromagnetic field tensor is


The constant of integration will be shown in Sec. IV to coincide with the physical electric charge of the black hole.

Since we assumed the vector field to point in the time direction, then as in the vacuum case, the normalization condition (13) determines its functional dependence:


It follows that the spatial components of the vector equation (12) are identically satisfied, while its temporal component serves to determine the Lagrange multiplier to be substituted in the Einstein equations:


We now turn to the Einstein equations (7), and the scalar equation (II). Upon substitution of the Lagrange multiplier (48) and the electromagnetic field tensor (46), the and equations become, respectively,


The scalar equation is


These are four equations for three unknowns , and , so one of the equations is actually redundant. We shall use two combinations of the three Einstein equations plus the scalar equation.

By adding the and equations, we again obtain as in the vacuum case Eq. (22). This time we write the solution


Here one integration constant has been set so as to have an asymptotically flat spacetime, namely, when . The other constant, , will be set by the boundary conditions on the horizon.

The remaining equations for , and are not immediately solvable. To make progress we shall assume that the physical metric is of RN form, solve for the scalar field in this framework, and check that all TeVeS equations are satisfied. This will give us a pair of charged black hole solution of TeVeS; existence of other solutions is yet to be excluded.

In Schwarzschild coordinates and , the RN metric may be written as


where and are the coordinates of the outer and inner horizons, respectively. We may transform the metric to isotropic form by going over to a new radial coordinate defined implicitly by


This gives

We recall that in GR and satisfy the relations and , where we write the gravitational constant as to distinguish it from plain , the coupling constant in TeVeS. We shall here assume that the physical metric of TeVeS has the above form while leaving the parameters and to be determined later. However, the proposed metric is asymptotically Minkowskian, while as previously mentioned, we require rather that the Einstein metric be asymptotically Minkowskian. This means that in the generic physical metric Eq. (39) we must set


To simplify these note that by Eqs. (40)-(41) we have , whereupon in view of Eq. (53),


We may thus relate and to the integration constant appearing in (53):


Since , is the outer black hole horizon in isotropic coordinates. In terms of and the physical metric coefficients are


It is useful at this point to trade the charge for a dimensionless positive parameter defined by


This replaces the GR relation . The value of will be determined by the Einstein equations (49)-(50).

The only indeterminate function remaining now is the scalar field. The scalar equation (52) can be rewritten in terms of the new parameters , and as


Its general solution is


with and integration constants, the first already familiar. Since we guessed the form of the metric, we need to verify that the Einstein equations are satisfied. From the requirement that Eq. (50) be satisfied, we obtain values for and :


Eq. (49) is then satisfied identically. Since we have already used the sum of Eq. (50) and Eq. (51) to get the solution (53), we see that all TeVeS equations are satisfied. Thus the RN metric from GR with a suitable choice of parameters is the physical metric of TeVeS spherical charged black holes.

We shall soon see that a physically acceptable solution can be had only for . For such solutions the sign of the quantity under the square root in Eq. (66) is positive. The two TeVeS solutions (corresponding to the two signs of ) are most clearly presented in terms of the coefficients , or


In view of Eq. (47) we finally obtain the solutions


V Resolving the superluminal paradox

We have found two black hole solutions for each value of and . The requirement that superluminal propagation be excluded selects one of them as physically viable. As mentioned in Sec. III, in a region where superluminal propagation of the TeVeS fields is not ruled out. This acausal behavior would be unacceptable. Now, since in Eq. (69) is arbitrarily large and negative near enough to the horizon, its coefficient must be negative in order that have a chance to be nonnegative everywhere. It is easy to see that for , is always positive. Thus the solution Eqs. (68)-(70) with lower signs is immediately excluded on grounds that it permits superluminal propagation. But is the second solution viable in this sense?

Focusing on the solution with upper signs, we must now exclude the parameter range ; the equality here corresponds to unbounded and , while the inequality leads to and superluminal propagation. The range , although palatable in this sense, gives . We shall show in Sec. VI that this is unphysical. For we have while . Thus a physically viable black hole solution of TeVeS can exists only for (we continue to assume that ). It is the solution with the lower indices in Eqs. (68)-(70).

Close enough to the horizon, of this solution is necessarily positive because of the which is arbitrarily large. Additionally, the asymptotic value of is , the cosmological value of the scalar, which may be assumed to be positive Bekenstein (2004). Hence the question of whether is positive in the intermediate region hinges on whether it has a negative minimum outside the horizon, or not.

To find out we look at its derivative,


The numerator here is quadratic in and thus has two roots. Now in the case we have , but . Then because [see Eq. (59) and the following discussion], both roots are real. Furthermore, if


both roots are at , so for the field has no minimum and must be everywhere positive.

Focus now on the case


Now does have a minimum outside the horizon. In Fig. 1 we plot for several values of . We see that unless is very large, the dip below the axis (which grows roughly as ) is modest compared to unity. Hence, a modest positive (which is expected from cosmological models Bekenstein (2004)) will be enough to keep positive throughout the black hole exterior, except for black holes with exponentially large values of for which a region of negative will occur near the horizon.

In fact, for , which by Eq. (59) means that , i.e., that the physical metric is extremal RN, the two solutions for are identical:


Thus for and , the variable part of is negative and can be very large for . This is unacceptable as it permits superluminal propagation. We may conclude that provided , and somewhat above zero, the superluminality issue raised by Giannios does not arise for the TeVeS charged black hole solution with the lower signs in Eqs. (68)-(70). The above conclusion does not apply to black holes near the extremally charged case.

What about Giannios’ case for which he found conditions conducive to superluminal propagation (see end of Sec. III )? The TeVeS equations (49)-(52) are smooth with , so we may take the limit of their solutions, Eqs. (68)-(70). In this limit according to Eq. (62), while by Eq. (59), . Thus our metric (68) reduces exactly to Giannios’ Eqs. (36)-(37) with the obvious identification ; that is we recover the fact that the physical metric is Schwarzschild. In the same limit our scalar field solutions (69) reduce to the pair


whereas Giannios obtained only one scalar solution. We notice that the solution with upper sign has for all provided that we stick to the parameter ranges , for which . The solution with the lower sign has sufficiently near ; this is Giannios’ solution, and it is indeed excluded because it allows superluminal propagation.

Figure 1: Our solution for as a function of for several values of , the higher , the lower the curve. Both axes are in arbitrary units.

To sum up, in our study of spherical static black holes in TeVeS, we have found a viable charged black hole solution for the parameter range , . The limiting case of this is a viable neutral black hole solution. Since black holes are seen in nature with virtual certainty, the above results tell us that only the range , of TeVeS need be considered as physical. This range includes the values that have been explored in the confrontation of TeVeS with observations Bekenstein (2004, 2006); Skordis et al. (2006); Dodelson and Liguori (2006).

Vi Black Hole Thermodynamics

It has been clear for long that black holes are really thermodynamic systems characterized by temperature and entropy. Thus a discussion of black hole solutions in TeVeS would be incomplete without a survey of their thermodynamic properties. However, before we can talk about thermodynamics for the charged black hole in TeVeS, we must first identify the physical values of attributes of the black hole solution. By physical values we mean the quantities than an asymptotically Minkowski observer would measure using instruments made of matter, measurements which are thus referred to the physical metric. These values need not be identical with those of quantities naively associated with the attributes. For example, we do not know a priori that the masslike quantity and the chargelike quantity appearing in our solution are indeed the physical mass and charge of the black hole. In fact, we shall see that is related to physical mass in a nontrivial fashion.

We first note that the appearing in the TeVeS equations is not Newton’s constant, but, as shown elsewhere Bekenstein and Sagi (), is related to it through


It will be useful to also write the above relation in terms of the constant defined by Eq. (65):


Experimentally ; it also seems natural that the fundamental coupling constant be positive; hence we must require . This explains why in Sec. IV we ruled out the parameter range .

Next, recall that if we use the same coordinates for the Einstein and physical metric, the transformation (2) implies that our physical metric is not asymptotically Minkowski. Thus, asymptotically, the relation between physical distance and the corresponding spatial length-like coordinate (denoted ) must be


Focus now on . According to Eq. (61), we may write the asymptotical expansion for as


Thus here is not physical distance , but is related to it through Eq. (78). Rewriting in terms of the latter gives


From the customary linear approximation we see that and physical mass are related by


Likewise, the physical charge can be easily identified by integrating the flux of the electromagnetic field tensor through a spherical shell at spatial infinity:


Use of in forming the area element guarantees that we are calculating a physical flux: according to Eq. (40) the factor required by Eq. (78) is supplied by the . Substituting from Eq. (46) gives


Thus our charged black hole is characterized by mass and charge as measured by physical asymptotic observers for which the metric is .

In investigating the black hole entropy we start with the assumption that it is given in terms of the physical surface area of the outer horizon by the usual formula


It is true that more complicated forms are known, but they usually appear in gravity theories with higher derivatives; TeVeS is free of these. The proof that our choice is correct ultimately rests on the consistency checks we present later in this section.



From (62) we have for the outer horizon




We now express in terms of physical mass, charge and Newton’s constant using the relations (81), (77):


so that


This is identical to the familiar expression for the entropy of a RN black hole. To it corresponds the thermodynamic temperature , or