Spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet Gravity

Spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet Gravity

Kang Zhou, Zhan-Ying Yang111Email:zyyang@nwu.edu.cn, De-Cheng Zou and Rui-Hong Yue222 Email:yueruihong@nbu.edu.cn Department of Physics, Northwest University, Xi’an, 710069, China
Faculty of Science, Ningbo University, Ningbo 315211, China
July 19, 2019
Abstract

We explore the gravitational collapse of a spherically symmetric dust cloud in the Einstein-Gauss-Bonnet gravity without a cosmological constant, and obtain three families of LTB-like solutions. It is shown that the Gauss-Bonnet term has a profound influence on the nature of singularities, and the global structure of space-time changes drastically from the analogous general relativistic case. Interestingly, the formation of a naked, massive and uncentral singularity, allowed in 5-dimensional space-time, is forbidden if . Moreover, such singularity is gravitational strong and a serious counter example to CCH.

Einstein-Gauss-Bonnet gravity, gravitational collapse, singularity
pacs:
04.20.Dw, 04.50.Kd, 04.40.Nr

I Introduction

The seminal work of Oppenheimer and Snyder led to the establishment viewpoint that the end state of spherically symmetric gravitational collapse is a black hole which have an event horizon to external observers Oppenheimer:1939ue (). Hawking and Penrose proven that, according to general relativity, there must be a singularity of infinite density of matter and space-time curvature formed by gravitational collapse Penrose:1964wq (); Hawking:1969vz (); Hawking:1973uf (). At the singularity, the laws of science and the ability to predict the future would break down. This led Penrose to propose the (weak and strong) cosmic censorship hypothesis (CCH) Penrose:1969 (), which to date remain unproven. The weak CCH asserts that there can be no singularity visible from future null infinity, light from singularity is completely blocked by the event horizon, while the strong CCH prohibits singularity’s visibility by any observer. Despite almost 40 years of effort we still don’t have a general proof of CCH Joshi:2000fk (). On the contrary, it was shown that the work of Oppenheimer and Snyder Oppenheimer:1939ue () which forbidden the formation of the naked singularity is not a typical model, and the singularities formed in generic collapse are naked and observable Christodoulou:1984mz (); Nolan:2002mz (). Recently, Virbhadra, Ellis, Keeton, etc showed that black holes and naked singularities can be observationally differentiated through their gravitational lensing effects. They classified naked singularities into 3 categories: weakly naked, marginally strongly naked, strongly naked singularities Ellis:2002mz (); Virbhadra:2008mz (), and hypothesize a new cosmic censorship: Generically, marginally and strongly naked singularities do not occur in a realistic gravitational collapse Virbhadra:2009mz (). They conjecture that in case the weak CCH of Penrose turns out to be incorrect, the new cosmic censorship will hold good.

In recent years, superstring/M theory caused a renewed interest about higher order gravity in more than dimensions. The most extensively studied model in the effective low-energy action of superstring theory Lust:1989tj () is so-called Einstein-Gauss-Bonnet gravity (EGB) Lovelock:1971yv (). An interesting question will be, how does Gauss-Bonnet term change the collapse of a massive body? Recently, Maeda considered the spherically symmetric gravitational collapse of a inhomogeneous dust with the -dimensional action including the Gauss-Bonnet term without finding the explicit form of the solution Maeda:2006pm (). Then, Jhingan and Ghosh presented a exact model of the gravitational collapse of a inhomogeneous dust in five dimensional EGB Jhingan:2010zz (); Ghosh:2010jm (). The objective of this paper is to extend their work to a more general case. We investigate the spherically symmetric gravitational collapse of an incoherent dust cloud by considering a Lemaitre-Tolman-Bondi type space-time in EGB without the cosmological constant, and give three families of LTB-like solutions, which are hyperbolic, parabolic and elliptic, respectively. In order to find the final fate of the collapse, we discuss the formation of singularities, and analyze the nature of them. In particular, we consider that whether such singularities would be hidden or be visible to outside observers. We also investigate the gravitational strength of naked singularities, and discuss that whether the naked singularities in EGB theory is a serious threat to CCH.

This paper is organized as follows. In Sec II, we give the field equations in EGB without a cosmological constant, and derive the LTB-like solutions. In Sec III, we investigate the formation of the singularity of the spherically symmetric gravitational collapse, and analyze the global structure of space-time. The strength of the singularity is demonstrated in Sec IV. Sec V is devoted to conclusions and discussions.

Throughout this paper we use units such that .

Ii Equations and solutions

The action of EGB can be written as

(1)

where the coupling constant can be regarded as the inverse of string tension and be assumed in this paper. For future simplicity, we take the coefficient . The second order (Gauss-Bonnet) term of Lovelock Lagrangian is given by

Varying the action Eq. (1), we obtain the field equations as

(2)

Assume that the system consists of a freely falling perfect fluid whose energy-momentum tensor in comoving frame is

(3)

where is the D dimensional velocity vector field. The metric takes the form

(4)

where the coordinate is the comoving radial coordinate, the proper time of freely falling shells. Both and are un-negative function of and . As well known, in general relativity, such metric leads to the Lemaitre-Tolman-Bondi (LTB) solution Lemaitre:1933gd (), which has been extensively used not only in spherical collapse, but also in cosmology as well Maartens:1995dx (); Garfinkle:2006sb (). Plugging the metric Eq. (4) into Eq. (2), we find that

(5)

where an over-dot and prime denote the partial derivative with respect to and , respectively. Eq. (5) leads to two families of solutions

(6)

and

(7)

where is an arbitrary function of . since Eq. (7) doesn’t recover the solution in Einstein gravity in the limit , we shall only consider the case here. The freedom of re-labeling spherical dust shell by enables us to fix the hypersurface such that at , coincide with the area radius

(8)

It is clear that the equations of momentum conservation are automatically satisfied, and the -component reads

Thus we have

and the mass function is

(9)

which is positive and increases with increasing for the non-negative energy density .

According to Eq. (6), other field equations can be written as

(10)
(11)
(12)

With the help of Eq. (9), Eq. (10) reduces to

namely

(13)

It is straightforward to check that the Eq. (13) ensures Eq. (11) and Eq. (12) to be satisfied. Generally, the solution of Eq. (13) can be classified into three types hyperbolic, parabolic and elliptic corresponding , or respectively. The case is a marginally bound case in which the metric Eq. (4) takes the form of Minkowski metric on the hypersurface . In the present discussion we shall assume that , which is equivalent to that . For gravitational collapse, shall decrease respecting to , it gives

(14)

For arbitrary initial data of energy density , Eq. (14) completely specify the dynamical evolution of collapsing dust shells. In the general relativistic limit with marginally bound case, Eq. (14) can be integrated to yield

where is an function of integration and equals to by using Eq. (8). Notice that the expression for function is similar to LTB and LTB-like (in the space-time with more than dimensions) solutions in general relativity Lemaitre:1933gd (); Ghosh:2001fb ().

Now, we return to discuss the above LTB-like solution attached at the boundary of a dust cloud to the outside vacuum region. The boundary is represented by a finite constant comoving radius , and the outside vacuum is

(15)

with

If we define that

(16)

and follow Maeda’s method in Maeda:2006pm (), we find that in present case, the LTB-like solution attached at the hypersurface to the outside vacuum solution smoothly.

Iii The Final fate of collapse

First, we consider the formation of the singularity. There are two kinds of singularities which are shell crossing singularities and shell focusing singularities, defined by and , respectively. The characteristic of a singularity in space-time manifold is the divergence of the Riemann tensor and the energy density Hawking:1973uf (). In our case, the Kretschmann invariant scalar for the metric Eq. (4) is

(17)

and can be shown to be finite on the initial data surface. The energy density of the fluid dust sphere is . It is clear that both Kretschmann scalar and energy density diverge when and . Hence, we have the shell crossing and shell focusing singularities. Shell crossing singularities can be naked, and has been shown to be gravitationally weak in the LTB case, which can not be consider seriously in the context of the CCH Newman:1985gt (). On the other hand, in general relativity, shell focusing singularities can also be naked and gravitationally strong as well. Henceforth, shell focusing singularities are considered to be the genuine singularities in space-time manifold Christodoulou:1984mz (); Deshingkar:1998cb (). Here we only limit in the shell focusing singularities.

It is known that the gravitational collapse in the 5D EGB with the marginally bound case leads to a shell focusing singularity, and positive delays the formation of the singularity Jhingan:2010zz (). In higher dimensional EGB, one can easily find

(18)

That is, the collapsing of the shell from the finite initial data is accelerated. Thus, will vanish in finite proper time for the comoving observer, the shell focusing singularity is inevitably formed in the three cases , and . Furthermore, using Eq. (14) we obtain

(19)

where and . Apparently we have , and then the velocity of collapsing decreases with increasing . This result indicates that positive delays the formation of the shell crossing singularity for , and in higher dimensional space-times.

In the homogeneous collapse, the metric Eq. (4) takes the form as the Robertson-Walker metric Robertson:1935 (); Walker:1936 ()

(20)

where is a constant. In order to satisfies Eq. (8), we can normalize the radial coordinate so that . It is clear that the time when the shell hits shell focusing singularity is completely determined by . Therefore it is independent on , as in the general relativity case. We can check such a consequence by using Eq. (14). In the homogeneous case, is written in the form

(21)

Considering Eq. (20), Eq. (21) and Eq. (14), we get

(22)

This equation does not contain the variable , that is, is a function of . This implies that every shell collapse into the shell focusing singularity at the same time.

Maeda proved that in the homogeneous collapse with the marginally bound case , the shell crossing singularity is ingoing null for and is space-like for Maeda:2006pm (). More generally, we can extend such a consequence to cases and . Considering Eq. (22) we have

(23)

Since the energy density have a finite non-negative value, we find that the factor behaves as near the singularity . We take the line element of the FRW solutions with the factor to the conformally flat form as

(24)

where

The range of for is and , for and , respectively, where is a constant. Thus, the shell focusing singularity is ingoing null for , while it is space-like for , as shown in FIG.1.

Next, we consider whether such a shell focusing singularity is naked or covered by an event horizon. An important constructions in general relativity is that of the apparent horizon, which is the outermost marginally trapped surface for the outgoing null geodesics Hayward:1993wb (). The apparent horizon is local in time and can be located at a given space-like hypersurface, while the event horizon coincides in case of static or stationary space-time and is non-local. Moreover, the event horizon is a outer covering surface of apparent horizons, that is, without the presence of apparent horizons there is no event horizon. The condition for the existence of the apparent horizon with outward normals null is

(25)

Using Eq. (6) and Eq. (13), the apparent horizon condition becomes

(26)

then we obtain

(27)

Clearly, the coupling constant produces a change in the location of these horizons. Jhingan and Ghosh found that, in 5D case the positive forbids apparent horizon from reaching the coordinate center thereby making the singularity massive and eternally visible, which is forbidden in the corresponding general relativistic scenario Christodoulou:1984mz (); Cooperstock:1996yv (). We will show, in the space-time, the naked singularity would not be formed except at the coordinate center. It is easy to find the condition that Eq. (27) could not be satisfied before the formation of the singularity is , in the 5D space-time, and the apparent horizon hits the singularity when . In these cases, the shell focusing singularity can be visible from infinity with certain initial data of energy density and leave open even the weak form of CCH Christodoulou:1984mz (). Furthermore, the shell focusing singularity is forever visible from future null infinity after it has been formed if in the 5D case, where is the boundary of the collapsing dust cloud. In other words, there is no black hole can be formed if the mass function takes sufficiently small value.

However, it can be seen obviously that in the space-time the apparent horizon lies in front of the singularity unless , thus the solution forbids the formation of the naked singularity except the singular point at the center, as same as in the general relativity Christodoulou:1984mz ().

For the homogeneous collapse, it is easy to find

(28)

Obviously, . Since is negative, the shell with greater coordinate value reaches earlier, hence the singular shells which hit the singularity at the same time could not be visible from infinity with the presence of apparent horizons. As demonstrated above, our solution can be attached to the outside vacuum solution at the boundary of the dust cloud. Hence, using the conclusions of the global structure of space-time mentioned in Torii:2005xu (), we get that in the outside vacuum region, the singularity is outgoing time-like if it is eternally visible, is outgoing space-like if the apparent horizon lies backward it, and is outgoing null if the apparent horizon hits it. We can give Penrose diagrams for the homogeneous collapse as shown in FIG.1.

(a)
(b)
(c)
(d)
Figure 1: Penrose diagram of the homogeneous collapse of a spherically symmetric dust cloud in EGB. Zigzag lines represent the shell focusing singularities, corresponds to the future (past) null infinity, BEH and CH stand for the black hole event horizon and the Cauchy horizon, respectively.

In addition, it can be noted that Eq. (26) takes the same form for cases , and , therefore the discussion about the apparent horizon in the case which is given by Jhingan and Ghosh can be applied to cases and Jhingan:2010zz ().

Iv Naked singularity and CCH

In this section, we consider whether CCH is violated by naked singularities formed in the gravitational collapse in EGB. First we discuss the new CCH propose by Virbhadra, Ellis, Keeton, etc. This group classified naked singularities into 3 categories by their gravitational lensing effects: weakly naked, marginally strongly naked, and strongly naked singularities. Naked singularities contained within at least one photon sphere are termed weakly naked singularities. Marginally strongly naked and strongly naked singularities are not covered within any photon spheres, and strongly naked singularities can produce images of gravitational lensing with negative time delays Ellis:2002mz (); Virbhadra:2008mz (). The new CCH is that only weakly naked singularities can be formed in the realistic gravitational collapse. The photon sphere equation obtained by them is applied to a general static spherically symmetric metric. Since the naked singularities formed in EGB is visible for only one moment and the metric is not static in this moment, so we only consider observational classification of naked singularities which formed in 5D EGB.

For a static spherically symmetric space-time

the photon sphere equation take the form of Ellis:2002mz (); Virbhadra:2008mz ()

(29)

Since our solution can be attached to the outside vacuum solution, after all shells collapse into the singularity, the space-time can be described by Eq. (15), and the photon equation Eq. (29) becomes

(30)

The positive real solution of Eq. (30) is

which is equivalent to that . Under such condition, however, the shell focusing singularity is not naked in 5D EGB as shown in the Penrose diagram. Hence, the photon equation have no real solution while the naked singularity has been formed in 5D EGB. Consequently, naked, massive and uncentral singularities formed in 5D EGB are marginally strongly or strongly naked singularities, which is a counter example of the new CCH suggested by Virbhadra Virbhadra:2009mz (). Physically, the gravitation collapse is a dynamical procedure. One should apply the dynamical photon sphere equation given in Claudel:2000yi () to analyze the collapse. The equation takes the form Claudel:2000yi ()

with and . Since our concern is the final fate of the collapse rather than the dynamic process, we are regretful to abandon the discussion of this equation.

Next, we consider the strength of the singularity Tipler:1977zza (). The importance of the strength of the singularity lies in the fact that even if a naked singularity occurs, if it is gravitationally weak in some suitable sense, it may not have any physical implications and it may perhaps be removable by extending the space-time Clarke:1994cw (). A sufficient condition for a strong curvature singularity is that in the limit of approach to the singularity, we must have along at least one causal geodesic Deshingkar:1998cb (); Clarke:1987 (),

(31)

where is the tangent vector to the geodesic. The idea captured here is that in the limit of approach to such a singularity, the physical objects get crushed to a zero size, and so the idea of extension of space-time through it would not make sense, characterizing this to be a genuine space-time singularity.

Jhingan and Ghosh pointed out that in the 5D space-time the central singularity which can be naked may be gravitational weak Jhingan:2010zz (). We will prove that although the Gauss-Bonnet term weakens the strength of singularity, the central singularity in the case which can also be visible is gravitational strong. We consider radial time-like causal geodesics with the marginally bound case , here the affine parameter is the proper time along particle trajectories. According to such definition, satisfies

that is,

(32)

Using the geodesic equation, we have

(33)

Substituting Eq. (32) into Eq. (33), we obtain that radial time-like geodesics must satisfy

(34)

The simplest solution is the worldline of a freely falling particle, which is . In terms of proper time we can describe it as

(35)

We consider the expansion of near

(36)

it specifies that behaves as near the coordinate center, where is a function of and vanishes at . Thus, we get

(37)

With the help of Eq. (18), we obtain that

(38)

where is defined as therefore has finite value. In the limit we have

thus,

(39)

Consequently,

(40)

the strong curvature condition is satisfied in the case and is not satisfied in the case .

Hence, the central shell focusing singularity is gravitational strong in the space-time while it may be gravitational weak in the 5D space-time. The difference between two cases is that the singularity can be eternally visible in the 5D case while it is disallowed in the case. Therefore, the singularity is gravitational strong when the solution represents the black hole formation for arbitrary initial data. If a naked singularity is gravitational weak, it may not have any significant physical consequences and so may not be a serious threat to the weak CCH of Penrose and the new CCH of Ellis’s group. Thus, the naked singularity in the case violates the CCH, as in the general relativity Christodoulou:1984mz (); Deshingkar:1998cb (), while the naked singularity may not be regarded as an essential counter example to the CCH seriously in the 5D case.

Furthermore, such a central singularity in the 5D case is gravitational strong in the general relativistic limit. In the limit , in space-time, Eq. (38) takes the form

and Eq. (39) becomes

Thus,

(41)

the strong curvature condition is satisfied in the 5D space-time. Comparing Eq. (41) to Eq. (40), we find that high order Lovelock terms weaken the strength of the singularity.

V conclusions and discussions

In this paper, we have investigated the gravitational collapse of a spherically symmetric dust cloud in the EGB without a cosmological constant, and extended works of Maeda, Jhingan and Ghosh to a more general case. As we demonstrated, the final fate of the collapsing dust cloud is a shell focusing singularity which can be either hidden or naked. We analyzed the global structure of the space-time and gave the Penrose diagram for the homogeneous case. It turns out that, the contribution of Gauss-Bonnet curvature corrections alters the course of collapse and the time of formation of singularities, modifies apparent horizon formation and the location of apparent horizons, and changes the strength of singularities. Unlike the 5D case, there is a serious threat to the CCH caused by the naked, gravitational strong singularity in the case.

In addition, it is interesting to study the spherically symmetric gravitational collapse of a dust cloud in higher order Lovelock gravity. We will discuss them elsewhere.

Acknowledgment This work has been supported by the Natural Science Foundation of China under grant No. 10875060, 10975180 and 11047025.

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