Analytic solutions in non-linear massive gravity
We study spherically symmetric solutions in a covariant massive gravity model, which is a candidate for a ghost-free non-linear completion of the Fierz-Pauli theory. There is a branch of solutions that exhibits the Vainshtein mechanism, recovering General Relativity below a Vainshtein radius given by , where is the graviton mass and is the Schwarzschild radius of a matter source. Another branch of exact solutions exists, corresponding to Schwarzschild-de Sitter spacetimes where the curvature scale of de Sitter space is proportional to the mass squared of the graviton.
Introduction: It is a fundamental question whether there exists a consistent covariant theory for massive gravity, where the graviton acquires a mass and leads to a large distance modification of General Relativity (GR). The quest for massive gravity dates back to the work by Fierz and Pauli (FP) in 1939 Fierz:1939ix (). They considered a mass term for linear gravitational perturbations, which explicitly breaks the gauge invariance of GR. As a result, there exist five degrees of freedom in the graviton, instead of the two found in GR. There have been intensive studies on what happens going beyond the linearised theory. In 1972, Boulware and Deser (BD) found that, at the non-linear level, there appears a sixth mode in the graviton that becomes a ghost in the FP model Boulware:1973my (). This problem was reexamined using the effective theory approach ArkaniHamed:2002sp (), where additional (Stückelberg) fields were introduced to restore the gauge invariance, and whose scalar part represents the helicity-0 mode of the graviton. In the FP model, the scalar acquires non-linear interaction terms that contain more than two time derivatives, signaling the existence of the ghost.
The Stückelberg approach also sheds light on the other puzzle in the FP gravity: if one linearises the system, the solutions in the FP theory do not reduce to GR solutions in the massless limit. This is known as the van Dam, Veltman, Zakharov (vDVZ) discontinuity vanDam:1970vg (); Zakharov:1970cc (). However, in this massless limit the scalar mode becomes strongly coupled and one cannot linearise the system. Therefore, due to strong coupling, the scalar interaction is shielded and GR can be recovered. This is known as the Vainshtein mechanism Vainshtein:1972sx (). The strong coupling scale in the FP model is identified as where is the Planck scale and is the graviton mass. This scale is tightly connected with the non-linear interactions of the scalar mode that contain more than two time derivatives. In the decoupling limit, where and , while the strong coupling scale is kept fixed, one obtains an effective theory for the scalar mode, where it is possible to study the consistency of the theory in more detail.
Until recently, it was believed that there is no consistent way to extend the FP model Deffayet:2005ys (); Creminelli:2005qk () to get a ghost free model at all orders. A breakthrough came with a 5D braneworld model known as Dvali-Gabadadze-Porrati (DGP) model Dvali:2000hr (). In this model there appears a continuous tower of massive gravitons from a four dimensional perspective. The non-linear interactions of the helicity-0 mode of massive gravitons contain no more than two derivatives, which is crucial to avoid the BD ghost. Due to this fact, the strong coupling scale in this theory is given by instead of , where and is a cross-over scale between 5D and 4D gravity Luty:2003vm (); Nicolis:2004qq (). Further studies have considered more general non-linear interactions which contain no more than two derivatives. In 4D, only a finite number of terms satisfy this condition; these are dubbed Galileon terms because of a symmetry under field transformations of the form Nicolis:2008in (). Ref. deRham:2010ik () constructed the extension of the FP theory that gives the Galileon terms in the decoupling limit, by choosing the correct parameters in the Lagrangian up to quintic order in perturbations. Ref. claudia () proposed a covariant non-linear action that automatically ensures this property to all orders, which we will discuss below.
A remaining crucial question is whether this property, holding in the decoupling limit, is sufficient to ensure the absence of the BD ghost or not. In Ref. claudia (), it was shown that there is no BD ghost in the decoupling limit to all orders in perturbation theory, but only up to and including quartic order away from this limit. However, it is very hard to show the absence of the BD ghost at all orders if one starts from Minkowski and studies non-linear interactions perturbatively. Therefore, it is important to obtain non-perturbative background solutions in this theory, and study fluctuations around them. Moreover, it is interesting to find solutions in this covariant non-linear theory, that can describe features of the observed universe. These are the topics of the present work.
Covariant non-linear massive gravity: We first construct the action for generalised FP model based on Ref. deRham:2010ik (); claudia (). We define the tensor as a covariantization of the metric perturbations:
The Stückelberg fields transform as scalars, while corresponds to a non-dynamical background metric that is needed to define the potential, which is assumed to be the Minkowski metric. The covariant tensor can then be expanded as
and under the coordinate transformation , transforms as
Before proceeding with the massive gravity theory, indices are raised/lowered, from now on, with the dynamical metric ; for example .
We define a new tensor as
where the square root is formally understood as . This allows us to represent the complete potential for gravitational interactions as
By expanding the potential in , we get an infinite sum of interaction terms for , with the FP term at the lowest order.
In order to study exact solutions associated with the previous Lagrangian, it is convenient to express in terms of matrices, namely
where denotes the identity matrix, and we have used . The potential in four dimensions then reads
In general, the task is to calculate the trace of . Given that is a square matrix, the Schur decomposition theorem ensures that it can be expressed as
for some unitary matrix , and an upper triangular matrix . The diagonal entries of are the eigenvalues of , and we call these eigenvalues . Then, since , one can express the traces in the formulae above, in terms of eigenvalues, as
Plugging these expressions into the potential, Eq. (7), we find the following expression for
Spherically symmetric configurations: We now focus on analyzing properties of spherically symmetric configurations in this set-up. We start by considering static configurations in the unitary gauge, (see Ref. Damour:2002gp () for spherical symmetric solutions in the FP theory). The most general form of the metric respecting spherical symmetry is
where . We choose to write the non-dynamical flat metric as . Notice that in GR one can set by a coordinate transformation, but this is not possible here, since we have already fixed the gauge completely. In order to simplify the analysis, it is convenient to define the combination . We plug the previous metric into the Einstein equations
where the energy momentum tensor from the potential of Eq. (10) is defined as . The Einstein tensor satisfies the identity , which implies the algebraic constraint
The previous condition can be satisfied in two ways, which lead to two different branches of solutions. We can either set , and focus on diagonal metrics, or alternatively, set . The fact that there are two branches of solutions indicate that, unlike in GR where Birkhoff’s theorem holds, there is no uniqueness theorem for spherically symmetric solutions in this theory. We analyze the two branches in turn.
Diagonal-metric branch - asymptotically flat solutions: The case of diagonal metrics, in Eq. (11), leads to equations which in general cannot be solved analytically. Thus, we will analyze them perturbatively, showing that they lead to asymptotically flat solutions (see Ref. us () for a more detailed discussion on this branch of solutions). We expand the functions and as
and truncate the field equations to first order in , and . It is more convenient to introduce a new radial coordinate , so that the linearised metric is expressed as
where and a prime denotes a derivative with respect to . In this new coordinate, the solutions for and are then given by
where we fix the integration constant so that is the mass of a point particle at the origin and . See the right plot of Fig. 1 for the general behaviour of these solutions. Notice that these configurations, as anticipated, are asymptotically flat and exhibit the vDVZ discontinuity, i.e. they do not agree with the GR solutions () in the limit . However, in order to understand what really happens in this limit, one should take into account the non-linear behaviour of . Let us consider scales below the Compton wavelength , and at the same time ignore higher order terms in . Under these approximations, the equations of motion can still be truncated to linear order in and , but since is not necessarily small, we keep all non-linear terms in . Then we obtain the following equations
We should stress that these are exact equations in the limit , , i.e. there are no higher order corrections in . For large radial values, one can linearise the equations in , recovering the solution in Eqs. (16), to first order in . On the other hand, the Vainshtein mechanism applies, and below the so-called Vainshtein radius, , becomes larger than one and the non-linear terms in become important, recovering GR close to a matter source. Actually, for the solution for is simply given by . The latter solution for and Eq. (17) imply
Therefore, corrections to the GR solutions are indeed small for , as shown in the left plot of Fig. 1.
The Vainshtein mechanism becomes also transparent in a non-unitary gauge. Indeed by performing the coordinate transformation , we excite the component of the Stückelberg field (see Eq. (3)), . Thus the strong coupling nature of is encoded in in this coordinate. It is possible to construct an effective theory for this Stückelberg field in the so-called decoupling limit deRham:2010ik (): first we introduce a scalar so that , where . Then the covariantization of metric perturbations is written as
where and . Formally, the decoupling limit is achieved by taking and , but keeping fixed. By substituting Eq. (19) into the action, one can show that the kinetic terms of become total derivatives and a mixing appears between and , which can be diagonalised using the definition
The Lagrangian is then written as
where = , , and is the linearised Einstein-Hilbert action for . The terms containing the scalar field are known as Galileon terms, which give rise to the second order differential equations, as explained in the Introduction. For the spherically symmetric case, the equation of motion for simplifies to Nicolis:2008in (); Wyman:2011mp ()
where the integration constant is again chosen so that is a mass of a particle at the origin. Using the relation between and , , we can show that the solutions for , and given by Eq. (17) agree with the solutions Eq. (20) and Eq. (22).
We have shown that the weak field solutions for the metric Eq. (11) with have three phases (see Fig. 1). On the largest scales, , beyond the Compton wavelength, the gravitational interactions decay exponentially due to the mass of graviton: see Eq. (16) and region 3 in Fig. 1. In the intermediate region , we obtain the gravitational potential but Newton’s constant is rescaled . Moreover, the post-Newtonian parameter is , which reduces to in the limit, instead of of GR, showing the vDVZ discontinuity (see region 2 in Fig. 1). Finally, below the Vainshtein radius , the GR solution is recovered due to the strong coupling of the mode (see Eq. (18) and region 1 in Fig. 1). This background solution provides us with a testing ground for the BD ghost. Instead of expanding the action in around the Minkowski spacetime perturbatively, one can study linear perturbations around this non-perturbative solution using the complete potential Eq. (7). In order to obtain the fully non-linear solution, a numerical approach is necessary. In the next section, we consider the second branch of solutions for this theory, which can instead be obtained analytically.
Non-diagonal-metric branch - Schwarzschild de Sitter solutions: Next, we analyze the second branch of vacuum solutions that solve Eq. (13), where . Interestingly, this branch leads to asymptotically de Sitter configurations. There is another identity , which leads to the condition
The remaining Einstein equations provide the following unique solution (see Ref. us () for detailed derivations)
with arbitrary and . This solution is similar to that in Salam:1976as (), up to numerical factors. Notice that this configuration depends on two integration constants. A sufficient condition to ensure that is real, is to choose and . The form of metric coefficients as in Eq.(24) do not allow a manifest comparison with de Sitter spacetime, since we have already chosen the unitary gauge and cannot do a further coordinate transformation without exciting components of . However, if we allow for a vector of the form , the metric can be rewritten in a diagonal form as
Then we can write down the action in terms of , and , considering them as fields. It is possible to show that the following configuration solves the corresponding equations of motion
while and are the same as in Eq. (24). The resulting metric has then a manifestly de Sitter-Schwarzschild form by making a time rescaling . However, we should note that this time rescaling cannot be done without introducing an additional time dependent contribution to . As expected, the metric in Eq. (26) can be obtained by making the following transformation of the time coordinate to the metric (11); this produces a non-zero time component of , that does not vanish even in the limit for any allowed value of . There are two integration constants, and , in this solution. In GR, corresponds to the mass of a source at the origin but a careful analysis including a matter source is necessary to fully understand the role of these integration constants. Note that there is an apparent singularity at the horizon , both for the metric and .
We can further make a coordinate transformation at the expense of exciting further components of . For example, by setting and making the following coordinate transformations with
the metric becomes that of flat slicing of de Sitter,
where the Hubble parameter is given by . The Stückelberg fields are now given by This is an interesting solution in which the acceleration of the universe is determined by the graviton mass and the Hubble parameter is given by . For , this solution reduces to the “self-accelerating” solution obtained in the decoupling limit in Ref. deRham:2010tw ().
Conclusions: The solutions obtained in the non-linear covariant massive gravity are remarkably similar to those in the DGP braneworld model including the existence of the “self-accelerating” de Sitter solution without cosmological constant Deffayet:2000uy () although there are differences in detail. There are a number of important issues. Firstly, we should confirm that there is no BD ghost in this theory by studying perturbations around the non-perturbative solution obtained in this letter. In the DGP model, the self-accelerating solution suffers from a ghost instability Luty:2003vm (); Nicolis:2004qq (); Koyama:2005tx (), which is related to the ghost in the FP theory on a de Sitter background. It is crucial to study the stability of the de Sitter solution in this model. In fact Ref. deRham:2010tw () showed that there exists a ghost in this self-accelerating background in the decoupling limit for a particular value of the second integration constant . They argue that this ghost can be cured by adding higher order corrections in to the potential. Our formalism is ready to be applied to this extended model. However, we believe a more complete analysis of perturbations about our exact solution is needed. Once these issues are clarified, the massive gravity model presented here provides an interesting playground to study large distance modifications of general relativity.
Acknowledgements.We would like to thank Misao Sasaki for useful comments, and Timothy Clemson for reading the manuscript. KK and GN are supported by the ERC. KK is supported by STFC grant ST/H002774/1 and GT is supported by STFC advanced fellowship. KK acknowledges support from the Leverhulme trust and RCUK.
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