A Linearized equations of the higher-derivative modification of the gravitational equations in higher dimensions

Head-on collision of ultrarelativistic particles in ghost-free theories of gravity


We study linearized equations of a ghost-free gravity in four- and higher-dimensional spacetimes. We consider versions of such a theory where the nonlocal modification of the operator has the form , where or . We first obtain the Newtonian gravitational potential for a point mass for such models and demonstrate that it is finite and regular in any number of spatial dimensions . The second result of the paper is calculation of the gravitational field of an ultrarelativistic particle in such theories. And finally, we study a head-on collision of two ultrarelativistic particles. We formulated conditions of the apparent horizon formation and showed that there exists a mass gap for mini-black-hole production in the ghost-free theory of gravity. In the case when the center-of-mass energy is sufficient for the formation of the apparent horizon, the latter has two branches, the outer and the inner ones. When the energy increases the outer horizon tends to the Schwarzschild-Tangherlini limit, while the inner horizon becomes closer to .

04.70.-s, 04.50.+h, 04.50.Kd

I Introduction

Singularities are inherent properties of general relativity. It is generally believed that the Einstein-Hilbert action should be modified in spacetime domains where the curvature becomes large. Such a modification is required, for example, when one includes in the theory quantum corrections, connected with particle creation and vacuum polarization effects. At a more fundamental level, the modification of the gravity equation might be required if the gravity is described as an emergent phenomenon. In such a case the Einstein equations are nothing but the low energy limit of the corresponding more fundamental background theory. The string theory is a well-known example. It is convenient to introduce two (generally different) energy scale parameters and . The corresponding length scales are and . We assume that when the spacetime curvature is much less than , the corrections to the Einstein equations are small. These corrections become comparable with other terms of the Einstein equations at , and for higher values of the curvature they play an important role. We assume that one can use the classical metric for the description of the gravitational field. For example, one can understand it as a quantum average of some metric operator, . This means that the quantum gravity effects, and in particular fluctuation of the metric, are small. In other words, one can use the effective action approach to study spacetime properties in this domain. The second parameter, , defines the scale when effective action description breaks down and the quantum nature of the gravitational field becomes important.

In studies of the singularity problem in modified gravity it is usually assumed that . In the present paper we also use this assumption and discuss some aspects of the singularity problem in the framework of the classical modified gravity equations.

There exist a wide class of the modified theories of gravity proposed to solve fundamental problems of black holes and cosmology. We consider a special class of such theories, namely theories with higher derivatives. Important features of such theories can be clarified already in a simple approximation when the gravitational field is weak and can be described as the perturbation on the flat spacetime background. Such an analysis was performed by Stelle Stelle (1978). In particular, he demonstrated that the Newtonian gravitational potential of a point mass located at can be made finite at this point, if the higher derivative terms are included in the gravity equations. Detailed analyses of this problem can be found in recent papers Modesto et al. (2015); Asorey et al. (1997).

However, the higher derivative gravity, as well as any theory with higher derivatives, has a fundamental problem. In a general case the propagator of such a theory contains two or more poles, and, as a result, it almost always contains ghostlike excitations (see, e.g., Stelle (1978) and Biswas et al. (2006),Barnaby and Kamran (2008)). Presence of the excitations with negative energy results in an instability of the theory and the possibility of an empty space decay. This is a special case of a very general phenomenon known as Ostrogradsky instability Ostrogradsky (1850) (see discussion in Barnaby and Kamran (2008)).

In the higher-derivative theory a standard box operator , which enters the field equations is changed to the operator , where is a polynomial. The poles of correspond to additional degrees of freedom. However, there exists an interesting option of theories where is an entire function of and hence it does not have poles in the complex plane. Such a modification of the gravitational equations is called ghost-free (GF) gravity (see, e.g.,Tomboulis (1997); Biswas et al. (2012); Modesto (2012a, b); Biswas et al. (2014, 2013); Modesto and Rachwal (2014); Tomboulis (2015a, b) and references therein). GF gravity contains an infinite number of derivatives and, hence, it is nonlocal. Theories of this type were considered a long time ago (see, e.g., Efimov (1967, 1972); Efimov et al. (1977); Efimov (1974, 1968)). They appear naturally also in the context of noncommutative geometry deformation of the Einstein gravity Nicolini et al. (2006); Spallucci et al. (2006) (see a review Nicolini (2005) and references therein). The initial value problem in nonlocal theories was studied in Barnaby and Kamran (2008); Barnaby (2011).The application of the ghost-free theory of gravity to the problem of singularities in cosmology and black holes can be found in Biswas et al. (2010); Modesto et al. (2011); Hossenfelder et al. (2010); Calcagni et al. (2014); Zhang et al. (2015); Conroy et al. (2015a); Li et al. (2015). Static and dynamical solutions of the linearized equations of the ghost-free gravity in four and higher dimensions were studied in Frolov (2015); Frolov et al. (2015). Recently the consequences of the ghost-free modifications of higher-dimensional gravity on the entropy of black holes and on cosmological models have been studied Conroy et al. (2015b).

In this paper we continue study of the linearized equations of the GF gravity. In Secs. II-IV we study solutions for a static gravitational field in the Newtonian approximation in different models of the GF gravity. Namely, we consider a class of the theories of gravity with . A static solution of the linearized equations for in four-dimensional spacetime was found in Biswas et al. (2012); Modesto et al. (2011) (see also Frolov et al. (2015)). In this paper we generalize this result to the higher-dimensional case and obtain new solutions for theories in the spacetime with an arbitrary number of spatial dimensions. In Sec. V we used these results to obtain a solution of the GF gravity describing a gravitational field of an ultrarelativistic particle. We succeeded to find a generalization of the famous Aichelburg-Sexl solution Aichelburg and Sexl (1971) to the GF gravity in an arbitrary number of dimensions. In Sec. VI we used the obtained solutions to study the apparent horizon formation in head-on collision of two ultrarelativistic particles. This problem for the general theory of relativity in four dimensions was first solved by Penrose Penrose (). Later, this result was generalized for a collision with a nonzero impact parameter in four and higher dimensions Eardley and Giddings (2002); Yoshino and Nambu (2003); Yoshino and Rychkov (2005); Constantinou and Taliotis (2013). In the present paper we show that in the GF gravity a similar process has two important new features: (i) the apparent horizon is not formed if the center-of-mass energy of the particles, , is smaller than some critical value , which depends on the scale parameter , the type of the theory, and the number of spacetime dimensions; (ii) if the energy is larger than the apparent horizon besides the usual outer part always has another inner branch. We discuss the obtained results in the last section.

In the present paper we use units in which and sign conventions adopted in the book Misner et al. (1974).

Ii Newtonian limit of higher-dimensional higher-derivative equations

Let us consider a static gravitational field perturbation on a flat background and write the corresponding metric in the form


Here and later we denote by a number of spatial dimensions. We also have

By substituting these expressions into the gravity equations (73) one gets

Here and is the total number of spacetime dimensions. In the Newtonian approximation and the first of these equations takes the form


For the gravity theory with the equations simplify and one obtains


and the metric (1) takes the form


For a point mass the energy density has the form . Then for the Einstein gravity, where one has


In four dimensions


Iii Static solutions of linearized equations in ghost-free gravity

iii.1 Ghost-free gravity

The Newtonian potential (5) is evidently singular at . One can regularize it and make it finite at by modifying the gravity equations in the ultraviolet (UV) domain. For example, one may assume that and are polynomials of the operator. If these functions obey the condition , the theory correctly reproduces the standard results of general relativity in the infrared regime, that is in the domain where . In a general case such a theory possesses ghosts. These ghosts are new degrees of freedom which are connected with extra poles of the operators and which give contributions to the propagator with a wrong (negative) sign. However, there exists an option to use such functions and that are entire functions of the complex -variable which do not have poles. It happens, for example, when and are of the form , where is a polynomial. A modified gravity which contains such regular formfactors is called ghost-free (GF) gravity. In the present paper we focus on the special class of the theories of GF gravity. Namely, we assume that


We denote such a theory . We restrict ourselves by considering the cases and , which are of the most interest for applications.

The exponent of the operator can be written in the form of a convergent series of the powers of this operator. However, it is not a good idea to “approximate” the exponent by the polynomial which is obtained by keeping a finite number of terms in this series. The inverse operator will have extra poles and the ghost will be present for such truncation. That is why our first goal is to present these nonlocal objects in the form of an integral transform which contains a well-defined kernel.

iii.2 Potential and Green functions in GF theories

Consider the equation for the potential created by a point massive particle placed at a point


where the operator is defined on the -dimensional Euclidean space. It is assumed to be a function of the Laplace operator


The Euclidean Green function of this operator is the solution of the problem


with vanishing boundary conditions at infinity. Formally it can be treated as a matrix element


of the operator


The momentum space calculations of are presented in Appendix B. The result reads (80)


In Sec. V and Sec. VI we will use this Green function to study a gravitational field created by ultrarelativistic particles. For this purpose it is useful to have another representation of the Green function, where the Bessel function is replaced by its integral representation

Then after the change of the integration variable


the Green function can be written in the form


Note that the last integral contains the expression which is known as the heat kernel of the Laplace operator in a -dimensional flat Euclidean space


The heat kernel obeys the equation


and the condition


It describes the amplitude


In flat space, because of the symmetries of the system in question, both the Green function and the potential are the functions of a distance between the points only


The potential at the point created by the massive particle located at the point is


Iv Gravitational potential in linearized GF gravity theories

iv.1 General properties of GF theories

All GF theories of gravity are assumed to reproduce Einstein gravity in the low energy regime, i.e., at large scales. In particular it means that the functions and approach smoothly to 1 at small :


Then we have the functions and . This property and (13),(21) guarantee that in the limit of large distances one gets a universal asymptotic for the potential for all these GF theories:


Obviously, as it should be, it exactly reproduces the gravitational potential (5) in the higher-dimensional Einstein gravity theory.

The asymptotic of the potential at small distances is theory dependent. Our particular interest is in theories, where


and or an even integer number. The parameter characterizes the scale where the nonlocality becomes important. One can show that for all gravities the potential is finite at small . For these theories the asymptotic at can be computed explicitly. Let us substitute (24) to (12),(13),(21) and change the integration variable . Then we have


One can see that in the limit when only small arguments of the Bessel function contribute to the integral (25). Therefore, one can substitute there an expansion


Then taking the integrals in (25) one obtains


One can see that the leading term is finite and proportional to . Moreover the next term in the expansion is proportional to that guarantees regularity of the metric at .

There are other interesting universal properties of the potentials in generic GF gravities. For example, because the distance in the integral (13) does not enter the function and due to the properties of the derivatives of Bessel functions it is clear that there is a universal relation


For the potentials, considered as functions of the radial distance , this property leads to the relation


provided the mass parameter is the same in and dimensions.

iv.2 Potential in theory

The static potential in the theory satisfies the equation


so that


Substitution of this expression into (13) and change of the integration variable leads to


where is the lower incomplete gamma function Olver et al. (2010). At large distance this expression reproduces the static Green function of the -dimensional Laplace operator


For small distances the Green function is a regular function of and is of the form


The potential is given by


In four-dimensional spacetime () we reproduce the results of Biswas et al. (2012); Modesto et al. (2011); Frolov et al. (2015); Nicolini (2005); Gruppuso (2005)


In the case of five-dimensional spacetime () we obtain even simpler expression


The potentials in an arbitrary number of dimensions qualitatively look alike. They are negative and finite at . At larger distances they become more shallow and at quickly approach the Einstein asymptotic (23).

iv.3 Potential in theory

When the operator corresponds to


and, hence,


Then the potential takes the form




and is the generalized hypergeometric function (see, e.g.,Olver et al. (2010)).

Qualitatively the potentials for different parameters and in different dimensions look similar. Figs. 1-2 show examples of the gravitational potential for and in two cases, and .

Figure 1: This plot shows the function for and .
Figure 2: This plot shows the function for and .

iv.4 Potential in theories

Similar results in terms of the generalized hypergeometric functions can be derived for an arbitrary theory. For all these theories the asymptotic at large distances is governed by the (23) and the asymptotic at small distances is given by (27). Let us present here only one more explicit example of the potential in gravity



and the coefficient

Expressions for the potentials become more complicated for higher and we do not present them here.

V Penrose limit

Let us demonstrate now, that obtained static solutions of the GF gravity can be used to find the gravitational field of an ultrarelativistic object. In the standard 4D Einstein gravity such a limiting metric is known as an Aichelburg-Sexl metric Aichelburg and Sexl (1971). This metric was generalized to the case of higher dimensions and for the spinning objects (called gyratons) in papers Frolov and Fursaev (2005); Frolov et al. (2005); Frolov and Zelnikov (2011). In this section we obtain a metric created by an ultrarelativistic object moving in -dimensional spacetime (nonspinning gyraton metric) in theories of gravity. As we shall see a key role in this derivation is played by the heat kernel representation (15) of the Green function .

Consider the metric in the following form


Let us boost this metric in the -direction


and introduce null coordinates


In the relativistic limit, when the boost velocity is close to the speed of light, i.e., , the boost factor . In this limit and . Then the line element (43) becomes




For a point particle of mass the Penrose limit corresponds to ultrarelativistic limit with the condition that an energy of the particle is kept fixed.

The gravitational potential (see (21),(15)) can be presented in the form


One can see that the boost affects only the last exponent in this integral representation.

Taking into account that after the boost


and using the delta-function representation


we obtain




Comparison of this integral expression with (48) leads to the observation that the function is proportional to the gravitational potential defined in space of one dimension less, i.e., in the space orthogonal to the particle motion:


This property is valid for arbitrary theories of gravity.

Using the property (29), which is also valid for a generic gravity, we derive a relation


This relation will be useful for the study of gravitational effects in collisions of ultrarelativistic particles (gyratons Frolov et al. (2005); Yoshino et al. (2007); Frolov and Zelnikov (2011)) in the next sections.

Vi Apparent horizon formation for head-on collision of the ultrarelativistic particles

Our next goal is to use the obtained results to study head-on collision of the ultrarelativistic particles in the GF theories of gravity. We use an approach developed by Penrose Penrose () and D’Eath and Payne D’Eath and Payne (1992a, b, c) and approximate the colliding particles by gyratons. A schematic picture of such a process is shown in Fig. 3. It shows two-particle motion in the center-of-mass frame. Each of the particles moves with the velocity of light. Particle 1 moves from the left to the right along the -direction, while particle 2 moves in the opposite direction. The null lines, representing their trajectories, belong to and null planes, correspondingly. The gravitational field of these particles is localized on the plane (for particle 1) and (for particle 2). The intersection of two null planes is the -dimensional transverse plane. In the regions , , and , outside the and null planes the metric is flat and null rays in these domains are nothing but null straight lines. However, when such a ray passes either through or planes, it is scattered by the gravitational field of the corresponding particle.

Figure 3: Head-on collision of two ultrarelativistic particles.

Our purpose is to study formation of the apparent horizon in such a process. Let us remember that a trapped surface is a compact spacelike -dimensional surface which has the property that both of the null congruences orthogonal to it, are not expanding. We focus on the outgoing congruence. One calls a trapped surface a marginally trapped surface if the outer normals to it have zero convergence Hawking and Ellis (2011). In a spherically symmetric spacetime one may consider spherical slices and define an apparent horizon as -dimensional surface which on each of the slices coincides with the marginally trapped surface.

The problem of ultrarelativistic particle collision in general relativity was discussed recently in connection with possible mini-black-hole creation in colliders Eardley and Giddings (2002); Yoshino and Nambu (2003); Yoshino and Rychkov (2005); Yoshino et al. (2007). Eardley and Giddings Eardley and Giddings (2002) demonstrated that a problem of existence of the apparent horizon can be reduced to a special boundary-value problem for an elliptic (Poisson) equation in a flat spacetime. Generalizations of these results to the collision of shock waves on AdS background were also considered in Gubser et al. (2008); Kiritsis and Taliotis (2012). The problem is greatly simplified for the case of the head-on collision and can be solved analytically in any number of spacetime dimensions. In the present paper we follow their approach. Let us write the metric (46) in the form


It is possible to show that geodesics and their tangent vectors are not continuous in these coordinates (see e.g. Eardley and Giddings (2002)). One can change the coordinates so that both geodesics and their tangent vectors will be continuous in the new coordinates. The new coordinates in the domain are defined as follows


A similar transformation (with a change ) should be made in the domain .

The metric (55) in the new coordinates takes the form


We consider a special marginally trapped surface which consists of two parts . In coordinates a position of and on two incoming null planes is described by equations


respectively. These two -dimensional surfaces intersect at -dimensional boundary , located at . The function is positive inside the boundary and vanishes at . The internal (induced) geometry of and are the geometry of a half of a -dimensional round sphere, their intersection being a round -dimensional sphere. For the head-on collision the function , which enters both equations in (59), is the same. In Eardley and Giddings (2002) it was shown that the outer null normals have zero convergence in and if


A condition that both normals (in and ) coincide at their boundary implies


Denote and by the radius at the boundary. Then


Hence one can put inside so that


The condition (61) takes the form


Using (54) one gets


In terms of a dimensionless coordinate , dimensionless energy , and a dimensionless profile function


this condition reads

Figure 4: The plot shows function for N=1,2.
Figure 5: The plot shows the radius of outer and inner apparent horizons as a function of the energy for and N=1,2.

All functions look similar (see Figs. 4 and 6). They vanish at and then grow, reach maximum, and then decrease to a universal asymptotic, that does not depend on the parameter , though depends on . The plots Figs. 5 and 7 show solutions of Eq.(67). The apparent horizon exists for the energy obeying the condition . In this energy domain it has at least two branches, inner and outer. At they meet and the apparent horizon disappears 1. This behavior resembles qualitatively that of the colliding relativistic extended sources Taliotis (2013). This resemblance is not accidental. One can rearrange Laplace opeators in (8) and move to the right-hand side of the equation. Then it can be identically rewritten as


When acting on the localized source, the operator delocalizes it and makes to become effectively an extended current for the traditional Laplace equation (68). In this sense the analogy of effects in the ghost-free gravities and for the colliding extended sources Taliotis (2013) becomes evident.

Figure 6: The plot shows function for N=1,2.
Figure 7: The plot shows the radius of outer and inner apparent horizons as a function of the energy for and N=1,2.

Vii Summary and discussion

In this paper we discussed an application of the linearized equations of the ghost-free theory of gravity to three connected problems. First, we calculated the gravitational potential of a point mass in the Newtonian limit and showed that GF modification of gravity works as a regularizer. Namely, this potential is regular at the origin. This property is valid for and theories in any number of spatial dimensions . This is a generalization of the earlier obtained result for for Biswas et al. (2012); Modesto et al. (2011) and for Frolov et al. (2015). The second main result of the paper is calculation of the gravitational field of an ultrarelativistic particle in the theories. The obtained metrics are generalizations of the famous four-dimensional Aichelburg-Sexl metric Aichelburg and Sexl (1971) of general relativity. Again, the obtained metrics are solutions of the equations of the gravity equations ( and ) in a spacetime with an arbitrary number of dimensions . And finally, we used these results to study an apparent horizon formation in the head-on collision of two ultrarelativistic particles. Our main conclusion is that in such a process there exists a mass gap for the mini-black-hole formation. If is the characteristic mass scale of the corresponding ghost-free theory, then in order for a mini-black hole to be formed in the collision, the center-of-mass energy should be of the order of or larger than . Another important feature of the process is that when the apparent horizon is formed, it has two branches: outer and inner marginally trapped surfaces. Both of them have the geometry of the sphere. When the center-of-mass energy increases, the inner part becomes closer to the point until it reaches the scale , where the model we used breaks down.

This result is again valid for any theory ( and ) in any number of dimensions. It can be considered as some indication that for such theories the inner singularity of a black hole might be absent and there exists a closed apparent horizon. Such a model was proposed in Frolov and Vilkovisky (1981) and discussed later in many publications. It should be emphasized that most of the results, related to the study of the models with closed apparent horizons, beyond a linear approximation, were obtained without using concrete dynamical equations. In this sense they are phenomenological. It is a real challenge to obtain solutions for a dynamical collapse in the modifications of the Einstein theory which are UV complete. In particular, in order to arrive at a definite conclusion concerning the structure of a black hole interior in the GF gravity one needs to perform analysis in the complete version of such a theory, which includes nonlinear effects.


The authors thank the Natural Sciences and Engineering Research Council of Canada and the Killam Trust for their financial support.

Appendix A Linearized equations of the higher-derivative modification of the gravitational equations in higher dimensions

In order to obtain linearized equations of a theory of gravity with higher derivatives in higher dimensions one can follow a similar derivation in four dimensions presented in the papers Biswas et al. (2012, 2013). In this appendix we collected the corresponding formulas for further reference.

The main steps of this derivation are the following. One considers first a covariant action which besides the Einstein term contains also a part which is quadratic in curvature. The latter may contain an arbitrary number of covariant derivatives acting on each of the curvature tensors. One can always move the derivatives acting on the first Riemann tensor to the position, when it acts on the other one. This can be achieved by using integration by parts. The number of derivatives may even be infinite, so that a theory is nonlocal. Since each of the Riemann curvature tensors has fourindices, the maximal total number of derivatives with “free” indices is eight. All other derivatives can be combined in functions of the covariant box operator. In order to achieve this it might be required to commute the derivatives. But this operation produces terms which are of the third order in the curvature so that they should be neglected in the adopted approximation. Using symmetry properties of the curvature tensor, Bianchi identities and commutativity of the covariant derivatives in the adopted approximation one finally obtains the following expression for Biswas et al. (2012, 2013)

Here and is the gravitational coupling constant in -dimensional spacetime. In four dimensions the value of this constant is fixed by the requirement that the Poisson equation for the gravitational potential in the Newtonian limit has a standard form. There is an ambiguity in the normalization of in higher dimensions. We fix it by requiring the Einstein-Hilbert action to have the same form in all dimensions.

This general form of the quadratic in curvature action can be further simplified using the following observation Barvinsky and Vilkovisky (1990); Modesto et al. (2015): the ”Gauss-Bonnet structures” of the form ()


in arbitrary dimensions are all of the third and higher order in curvature plus total divergence terms. As a result, the general higher derivative action can be written in the form which contains only two arbitrary functions of the box operator Biswas et al. (2013).

To obtain the linearized equation we write the action in the form

We use the following expressions for the variations of the objects that enter the above action and keep only the terms that are quadratic in perturbations

Let us write the total linearized action in the form


Then we have