Circular Orbits in Einstein-Gauss-Bonnet Gravity

Circular Orbits in Einstein-Gauss-Bonnet Gravity

Valéria M. Rosa111e-mail: vmrosa@ufv.br Departamento de Matemática, Universidade Federal de Viçosa, 36570-000 Viçosa, M.G., Brazil    Patricio S. Letelier222e-mail: letelier@ime.unicamp.br Departamento de Matemática Aplicada-IMECC, Universidade Estadual de Campinas, 13081-970 Campinas, S.P., Brazil
Abstract

The stability under radial and vertical perturbations of circular orbits associated to particles orbiting a spherically symmetric center of attraction is study in the context of the n-dimensional: Newtonian theory of gravitation, Einstein’s general relativity, and Einstein-Gauss-Bonnet theory of gravitation. The presence of a cosmological constant is also considered. We find that this constant as well as the Gauss-Bonnet coupling constant are crucial to have stability for .

04.50.-h

I Introduction

The physical laws for nonrelativistic phenomena are usually written in a three dimensional space and for relativistic phenomena in a four dimensional spacetime. In both cases these laws admit natural extensions to spaces with arbitrary number of dimensions, these can be taken as the possible existence of an operative principle (or principles) which in additions to these laws lead us to determine the “true dimensions” of the spacetime, the arena where the physical phenomena occur.

Kant observed that the three-dimensionality of the usual space could be related with Newton’s inverse square law; “ The reason for the three-dimensionality of the space is yet unknown. It is likely that the three-dimensionality of space results from the law according to which the forces on the substances act into each other. The three-dimensionality seems to result from the fact that the substances of the existing work act into each other in such a way that the strength of the effect behaves like the inverse of the square of the distances” kant1 ()kant2 ().

Since in a first approximation, the galaxies and planetary systems can be considered as formed by particles moving in circular orbits around an attraction center, a possible principle to determine the dimensionality of the space is the existence of stable circular orbits. The study of the stability of circular orbits moving in potentials solutions of the n-dimensional Laplace’s equations can be found in routh (). These studies on stability culminate with the Ehrenfest work ehrenfest () where he shows that in the Newtonian-Keplerian problem there exists stable limited orbit if and only if the dimension of the space is three (the potential must be null at infinity).

In Einstein’s gravity Tangherlini tangherlini () study the stability under radial perturbation of orbits associated to test particles moving around a -dimensional Schwarzschild black hole. He found stability for only, i.e., in complete concordance with the Newtonian case. The dimension of the constant time slice, , will be denoted by

Generalization of the Newton theory that consists in the addition of a cosmological constant was studied by several authors, see for instance bondi () (cosmological context) and wilkins () (potential theory context). Clearly this generalization is motivated by the success of Einstein’s gravity with cosmological constant. An example of model that makes use of a cosmological constant is the Cold Dark Matter model (CDM) lamcdm () that explains in a simple way the observed acceleration of the Universe as well as the cosmic microwave background.

The Hilbert Lagrangian (Ricci scalar times square root of the metric determinant) is a divergence in two dimensions. We have that the Hilbert action is proportional to the Euler characteristic of the two dimensional manifold (Gauss-Bonnet theorem manfredo ()). In four dimensions we have that the Bach-Lanczos Lagrangian bach ()lanczos () (a particular combination of terms quadratic in the Riemann-Christoffel curvature tensor) is a divergence, so in four dimensions this Lagrangian has no dynamical significance, like the Hilbert Lagrangian in two dimensions. The simplest generalization of the Hilbert Lagrangian for -dimensional spaces () can be obtained adding the Bach-Lanczos Lagrangian to the Hilbert one. This new theory of gravitation, Einstein-Gauss-Bonnet (EGB) theory, has some remarkable mathematical properties, see for instance nathalie ().

Another outstanding feature of the Einstein-Gauss-Bonnet theory is that the Bach-Lanczos action appears as the leading quadratic correction term of the Hilbert action in the expansion of the supersymmetric string theory deser (). To be more precise the Bach-Lanczos action appears exactly in the expansion of the heterotic string model duff ().

The aim of this paper is to study the stability under radial perturbations as well as vertical perturbations of circular orbits related to test bodies orbiting a spherically symmetric center of attraction in the above mentioned theories of gravity in an arbitrary number of dimensions.

In Section II we study the stability of circular orbits in a -dimensional Newtonian context with cosmological constant. In Section III we consider the stability of circular orbits in the EGB theory. We divide the study of stability in two. First, in Sect. IV, we consider the case without cosmological constant. In the next section (Sect. V) we analyze the case where this constant is not null. We end summarizing our results in Sect. V.

Ii d-dimensional Newtonian center of attraction with cosmological constant

The Laplace equation with cosmological constant is

 ∇2Φ+Λ=0, (1)

where is the Laplacian operator in -dimensions, and are the Newtonian potential and the cosmological constant, respectively. For spherical symmetry Eq. (1) reduces to

 1rn−2∂∂r(rn−2∂∂rΦ)+Λ=0 (2)

that has as solution for ,

 Φ=−Cn−31rn−3−Λ2(n−1)r2, (3)

where is an integration constant, that for () it is proportional to the mass of the center of attraction (, with the Newton’s constant of gravitation).

The Newton equations for an axially symmetric potential can be written as

 ¨R−R˙φ2=−∂Φ∂R, (4) ddt(R2˙φ)=0, (5) ¨zi=−∂Φ∂zi, (6)

where the overdots denote derivation with respect to , are -dimensional cylindrical coordinates, and . Defining the effective potential , with (=constant), we find,

 ¨R=−∂Φeff∂R,¨zi=−∂Φeff∂zi. (7)

Now let us study the stability of the circular orbit . We shall assume that the potential has reflection symmetry on the planes , i.e., . Expanding the effective potential around this circular orbit we find,

 Φeff=Φeff(R0,0,...,0)+12∂2Φeff(R0,0,...,0)∂R2(R−R0)2+12d−2∑i=1∂2Φeff(R0,0,...,0)∂(zi)2(zi)2+⋯. (8)

From Newton’s equations we find the motion for the perturbations and ,

 ¨ρ+κ2ρ=0,¨ζi+ν2iζi=0, (9)

where

 k2=∂2Φeff(R0,0,...,0)∂R2=∂2Φ(R0,0,...,0)∂R2+3R∂Φ(R0,0,...,0)∂R, (10) ν2i=∂2Φeff(R0,0,...,0)∂(zi)2=∂2Φ(R0,0,...,0)∂(zi)2. (11)

Thus when we have that the radial perturbation remains bounded, i.e., we have linear stability. This perturbations are usually named as epicyclic perturbations binney (). In a similar way when we have that the “vertical” perturbation also remains bounded. In this case we say that we have linear stability under perturbations along the direction . For the spherically symmetric potential (3) we find

 κ2=(5−n)CR1−n0−4Λ/(n−1), (12) ν2i=ν2=CR1−n0−Λ/(n−1). (13)

To derive the previous equations we used . Note that because of the spherical symmetry of the potential we have that all the frequencies for all the vertical perturbations are equal.

The relations (12) and (13) tell us that when we have stability only for . For and we have stability whenever . These last relation puts an upper limit to the size of the stable galactic structures. For and we have and the circular orbits are not stable under epicyclic perturbations. Now for we have stable circular orbits for and for we have stability for limited by the lower bound . In other words for certain values of the presence of a cosmological constant can stabilize the circular orbits, sometimes with limitations in the size of the orbit.

Iii Einstein-Gauss-Bonnet theory of gravitation

The Einstein-Gauss-Bonnet action is,

 S=∫dnx√−g[1κ2n(R−2Λ+αLGB)]+Smatter, (14)

where is the -dimensional Ricci scalar (), is defined as , where is the -dimensional gravitational constant. We use units such that the speed of light is one (). The Gauss-Bonnet term is given by

 LGB=R2−4RμνRμν+RμνρδRμνρδ, (15)

and is a coupling constant. is the part of the action that describes the matter. We use the conventions, , and . The action (14) is the simplest one for built with topological terms. For the general case see lovelock ().

From (14) we find

 Gμν−αHμν−Λgμν=κ2nTμν, (16)

where

 Gμν=Rμν−12gμνR, Hμν=2[RRμν−2RμαRαν−2RαβRμανβ+RαβγμRναβγ]−12gμνLGB. (17)

For vacuum () spherically symmetric solutions to the EGB equations there is not lost of generality in choosing the metric,

 ds2=f(r)dt2−f(r)−1dr2−r2dΩ2n−2, (18)

where is the metric of a -sphere. The case in which this sphere is changed by a space of negative or null curvature has also been considered gleiser ().

From the fact that the the Einstein tensor, , as well as the Gauss Bonnet tensor, , are divergence free lovelock () we have that equation (16) tells us that is also divergence free. From this last fact we conclude that test particles move along geodesics fock (). We shall analyze the stability of the circular geodesics whose parametric equation is,

 t=t(s),r=r0,θj=π2,j=2,…,n−1,θn=ϕ∈[0,2π]. (19)

The evolution of a small perturbation, , of is given by the geodesic deviation equation,

 D2ξμds2+Rμναβuνξαuβ=0, (20)

where , and . Equation (20) can be cast in a simpler, but not manifestly covariant way, as

 ¨ξμ+2Γμαβuα˙ξβ+Γμαβ,νuαuβξν=0, (21)

where the comma denotes ordinary differentiation. For the curve (19) the system (21) reduces to,

 ¨ξ0+f′(r0)f(r0)˙t˙ξ1=0, (22) ¨ξ1+κ2ξ1=0, (23) ¨ξj+ν2ξj=0, (24) ¨ξn+2r0˙ϕ˙ξ1=0, (25)

where

 κ2=f(r0)(r0f′′(r0)+3f′(r0))−2r0f′(r0)2r0(2f(r0)−r0f′(r0)), ν2=f′(r0)r0(2f(r0)−r0f′(r0)). (26)

We shall do a study of cases depending on the values of and . The analysis starts with the stability of circular orbits for the pure Einsteinian case (no Gauss-Bonnet term). Then we study the stability of these orbits when a coupling term and/or the cosmological constant are considered.

For and , the model of spacetime reduces to the -dimensional Schwarzschild solution. The equation for the metric function is

 rf′−(n−3)(1−f)=0, (27)

which has the general solution . is an integration constant that will taken as positive in order to have the Schwarzschild solution when (). In this case the constants and can be cast as,

 ν2=C(n−3)rn−30rn−10(2rn−30−C(n−1)), κ2=C(n−3)((5−n)rn−30−C(n−1))rn−10(2rn−30−C(n−1)).

Therefore, we have if and only if , this imply that if and only if , and it only occurs when and . Hence, we have stable circular orbits only in a 4-dimensional spacetime as it was expected.

Iv Einstein-Gauss-Bonnet Theory with Λ=0.

Taking in (16) with , the equation of metric function is

 rf′−(n−3)(1−f)[−1+α(n−4)r2(2rf′−(n−5)(1−f))]=0, (28)

which has the general solution

 f±(r)=1+r22~α[1±√1+4~αCrn−1], (29)

where and is an integration constant. The sign of is chosen analyzing the behavior of for . Let be the function where the sign of square root is positive and let be the other case. When is small, the asymptotic behavior of is given by

 f−≈1−Crn−3. (30)

So, considering the Gauss-Bonnet term as a perturbation of the -dimensional Schwarzschild solution, must be a positive constant.

When is small, the asymptotic behavior of is given by

 f+≈1+r2~α+Crn−3. (31)

For , it corresponds to Schwarzschild-anti-de Sitter spacetime with negative gravitational mass, with the standard energy definition in background anti-de Sitter abott ().

Then, there are two families of solutions corresponding to the sign in front of the square root that appears in (29). Following the definition given in nozawa (), the family with minus (plus) sign will be called general relativity (GR) branch (non-GR branch) solution. The spacetime structure of these solutions of the EGB theory is studies in torii ().

iv.1 The GR branch solution.

By chosing the solution of (28) from (26) we find,

 ν2=rn0(1−√β)+Cr0~α(5−n)~α(r30C(n−1)−2rn0√β), κ2=A1+A2+A3−2rn0β√β~α(−2rn0√β+r30C(n−1)), (32)

where and

 A1=8r2n0(1−√β)+rn+30C(n2−1)(1−√β), A2=2rn+10[32−√β(n2−8n+39)]+2r40C2~α(n−1)[n+7−√β(n+3)], A3=4r20C2~α2[32−2√β~αC(n−5)(n−9)]−8r5−n0C~α(n−5)(n−1). (33)

The behavior of the functions and are not simple, but it is possible to find , , and such that and are both positive, i.e., there are stable circular orbits. For example, we have stable circular orbits when , , and ; , , and ; , , and ; , , and ; and , , and . In all these examples we have in the specified region. Hence, the Gauss-Bonnet term can stabilize circular orbits in some higher dimensions spacetimes.

In Fig. 1 we show the epicyclic and vertical frequencies for several values of and . For the first two figures and for the third . In the first graphic take the values (top curve), and (bottom curve). In the second graphic, counting along the line , takes the values (top curve), and (bottom curve). There is no stable circular orbit when . And for the third graphic, counting along the same line, (top line), (bottom line). In this case here is no stable circular orbit when , but for we have stability.

The vertical frequency, , is always positive no matter the value of but the range of where is positive decreases when the dimension increases.

For and and also for and , we calculate the functions and into a grid of points with (step ), and (step into , step into , step into ) and . In the case we work with , with the same steps for the corresponding intervals. In all these numerical examples we obtain and, therefore, no circular orbit is stable.

iv.2 The non-GR branch solution.

For completeness we study the case when is the solution of (28). In this case the constants and are given as in (32) and (33) replacing by .

Also in this case, the stability depends on the sign of . When , in the expression for , we have a numerator, , non negative and a denominator, , always negative. So, is not positive, hence there are no stable circular orbits.

The parameters and are positive. When the denominators of and are always positive (). The same happens with the functions , , and with the coefficients of in and (Note that is the function after replacing by , ). These positive coefficients assure that, for sufficiently large, the numerator of , , and the numerator of , , are positive. Hence, we conclude that it is possible to find stable circular orbits for and .

Fig. 2 shows the functions and , for , , and (top line), (bottom line). We have stability for , where depends on the dimension .

V Einstein-Gauss-Bonnet Theory with Λ≠0.

Taking and in (16) with , the metric function obeys the equation,

 rf′−(n−3)(1−f)+2Λr2n−2=0, (34)

which has the general solution,

 f(r)=1−Crn−3−2Λr2(n−1)(n−2), (35)

the integration constant will be considered positive as before. The frequencies (26) in this case are and , where

 N1=Cr20(n−1)(n−2)(n−3)2−4Λrn+10;D1=r40(n−1)(n−2)(2rn−30−C(n−1)), N2=Cr20(n−2)(n−3)2((5−n)rn−30−C(n−1))−2Λrn+10(8rn−30−C(n2−1)), D2=rn+10(2rn−30−C(n−1))(n−1)(n−2). (36)

When , the numerator of is positive for all , and its denominator is positive if and only if . Note that change of sign with . The function is positive when , and this condition is satisfied when . So, it is necessary to find the values of for which the numerator of is positive. For better visualization, we write in

 N2=16λr2n−20−2Cλ(n2−1)rn+10−C(n−2)(n−3)(n−5)rn−10−C2(n−2)(n−3)(n−1)r20.

Analyzing the variation of the signs of the polynomial , we can see that there is at least one positive root for . Adding the fact that for large , is positive, we can conclude that it is possible to find a value such that and for . Therefore, when and for there exist stable circular orbits.

When , in order to have it is necessary that, at least,

 2rn−3−C(n−1)>0. (37)

Under this condition the denominators and are positive. Now it is necessary that the numerators and be also positive. The polynomial is positive when

 Cr20(n−1)(n−2)(n−3)2−4Λrn+10>0. (38)

In four dimensions we have and when we take, for example, , and . But when the numerators, and , can not be positive at same time. We can see this by doing an analysis of the polynomial . Using (37) and (38) we obtain

 N2<−2Λrn+10(4C(n−1)−C(n2−1))−Cr20(n−2)(n−3)2((5−n)C(n−1)/2−C(n−1))= =C(n−1)(n−3)[4Λrn+10−Cr20(n−1)(n−2)(n−3)2]/2<0. (39)

So, for and there is no stable circular orbit. It is interesting to note that the addition of a negative cosmological constant stabilize, former not stable, circular orbits for any .

In the following we analyze the stability of the circular orbits when both and are not null. In this case the equation of metric function is written as

 rf′−(n−3)(1−f)+2Λr2n−2+α(n−4)(n−3)(1−f)r2(2rf′−(n−5)(1−f))=0, (40)

which has the general solution

 f(r)±=1+r22~α[1±√1+4~α(n−1)(n−2)(2Λ+Crn−1)], (41)

where and is a integration constant. As before, let be the function when the sign of square root is positive and let be the other case. In the limit the function reduce to

 f−≈1−2Λr2(n−1)(n−2)−C(n−1)(n−2)rn−3. (42)

By considering the Gauss-Bonnet term as a perturbation of the Schwarzschild-dS (Schwarzschild-adS) spacetime with (), we conclude that must be positive.

When is small, the asymptotic behavior of is given by

 f+≈1+r2~α+2Λr2(n−1)(n−2)+C(n−1)(n−2)rn−3. (43)

As we can see, here again, there are two families of solutions corresponding to the sign in front of the square root in (41). We define two branches following the same criterion, the GR branch and the non-GR branch.

v.1 The GR branch solution (Λ≠0)

The functions and in this case can be written as (26)

 ν2=rn0[(n−1)(n−2)(1−√β∗)+8Λ~α−~αCr1−n0(n−5)]~α(n−1)(−2rn0√β∗(n−2)+r30C), (44) κ2=B4rn+30+B3rn+10+B2rn0+B1r40+B0r202~α(n−1)(−2rn0√β∗(n−2)+r30C)rn0[(n−1)(n−2)+4~α(2Λ+Cr1−n0)] (45)

where

 β∗=1+4~α(n−1)(n−2)(2Λ+Crn−1), (46)

and

 B4=C(8Λ~α+(n−1)(n−2))(n2−1)[−√β∗+1], B3=C[−32~α√β∗(n−1)(n−2)+2~α(n2−8n+39)(8~αΛ+(n−1)(n−2))], B2=8[[(−√β∗+1)(n−1)(n−2)+8~αΛ][(n−1)(n−2)+8~αΛ]], B1=2C2~α(n−1)[−√β∗(5−n)+n+3], B0=4~α2C2[(n−5)(n−9)]. (47)

First, when it is possible to find stable circular orbits. For example, if , , then is stable when (). Note that if and only if

 p(r)=2(−Λ)rn+4+(n−1)(n−2)rn+2+(n−1)(n−2)~αrn−Cr5=0. (48)

Analyzing the variation of signs in the coefficients of , we can see that, if then the number of variations is one and there is only one positive value such that . And for we have . If , the existence of a root of depends on the relation between , and C. In the cases , , , the polynomial has only one positive root. Hence, we have a horizon (at ) with the zone of stability outside this horizon.

Figure 3 shows the behavior of the functions and for , , and , and (top curve), (bottom curve). We see a small region of instability for large . Similar behavior is found for the same parameters, but .

The cases where was numerically analyzed. For example, we have stable circular orbits when , and ; , and ; , , and ; , , and , and , , , and . In all these examples we have into the fixed intervals.

The other cases with ( and or and ) we calculate the functions and into a grid of points and at each point of the grid these function are not positive at same time. The grid is built with (step ), (step ), and (step into , step into , step into ) and . In the case we work with , with the same steps for the corresponding intervals.

In these cases (), each value of has a different set of parameters where both and are positive, and this differences turns quit impossible to repeat the graphical analysis as in the cases.

v.2 The non-GR branch solution (Λ≠0)

Again, for completeness we describe the case where is the solution of (40). In this case, the functions and (26) are given as (45) replacing by .

The condition to existence of () gives us . For the numerator of , , is always positive but the denominator, , is negative, so , and in this case no circular orbit is stable.

For we first analyze the case . In this case all parameters are positive and the denominators of and are always positive. The same happens with , , , , and the expression (Note that is the coefficient after replacing by , ). These positive coefficients assure that, for sufficiently large, the numerator of and are positive. Hence, we conclude that there are stable circular orbits for , and .

If and , it is necessary to set conditions to assure . It happens when , what is true when . In this case, from the expressions for and , we can see that the same argument used in the case is valid. Hence, we conclude that there are stable circular orbits for , and .

Figure (4) shows the functions and for ,