A new asymptotical flat and spherically symmetric solution in the generalized Einstein-Cartan-Kibble-Sciama gravity and gravitational lensing

A new asymptotical flat and spherically symmetric solution in the generalized Einstein-Cartan-Kibble-Sciama gravity and gravitational lensing

Songbai Chen111Corresponding author: csb3752@hunnu.edu.cn, Lu Zhang, Jiliang Jing 222jljing@hunnu.edu.cn Institute of Physics and Department of Physics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
Key Laboratory of Low Dimensional Quantum Structures
and Quantum Control of Ministry of Education, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China
Abstract

Abstract

In this paper we firstly present a new asymptotical flat and spherically symmetric solution in the generalized Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity and then investigate the propagation of photon in this background. This solution possesses three independent parameters which affect sharply photon sphere, deflection angle of light ray and gravitational lensing. Since the condition of existence of horizons is not inconsistent with that of photon sphere, there exists a special case where there is horizon but no photon sphere in this spacetime. Especially, we find that in this special case, the deflection angle of a light ray near the event horizon tends to a finite value rather than diverges, which is not explored in other spacetimes. We also study the strong gravitational lensing in this spacetime with the photon sphere and then probe how the spacetime parameters affects the coefficients in the strong field limit.

pacs:
04.70.Dy, 95.30.Sf, 97.60.Lf

I Introduction

General relativity is the most beautiful theory of gravity at present and it is a fundamental theoretical setting for the modern astrophysics and cosmology. However, the observed accelerating expansion of the current Universe t5 (); t501 (); t51 (); t6 (); t61 () implies that some important ingredients could be missing in this theory. One of ingredients vanished Einstein theory is torsion, which is the antisymmetric part of the general affine connection. In the gravity theories with torsion, the gravitational field is described by both of spacetime metric and the torsion field, which means that the emergence of torsion will modify the feature of the gravitational interaction.

One of natural extensions of Einstein’s theory of gravity is the so-called ECKS theory of gravity Cartan (); Kibble (). In this theory with torsion, the curvature and the torsion, respectively, are assumed to couple with the energy and momentum and the intrinsic angular momentum of matter. The gravitational repulsion effect arising from such a spinor-torsion coupling can avoid the formation of spacetime singularities in the region with extremely large densities, for example, in the interior of black holes and the very early stage of Universe avoid (); avoid1 (); avoid2 (); avoid3 (). In the low densities region, the ECSK theory and Einstein’s general relativity give indistinguishable predictions since the contribution from torsion to the Einstein equations is negligibly small. However, the torsion field is not dynamical in the ECSK theory since the torsion equation is an algebraic constraint rather than a partial differential equation, which means that the torsion field outside of matter distribution vanishes because it can not propagate as a wave in the spacetime.

In order to construct a dynamical torsion field, one can generalize ECSK theory by introducing higher order corrections in Lagrangian gK1 (); gK2 (). These coupling terms between the spacetime torsion and curvature yield that both equations of motion for spacetime torsion and curvature are dynamical equations, and then ensure that the spacetime torsion can propagate in the spacetime even in the absence of spin of matter. Recently, some non-trivial static asymptotical flat vacuum solutions sbh1 () are obtained in this generalized ECSK theory by solving Einstein field equations and torsion field equation. These non-trivial solutions describe the spacetimes with special structures, which is useful for detecting the effects originating from spacetime torsion within the gravitational interactions.

A natural question is that there are other asymptotical flat and spherically symmetric solutions in this generalize ECSK theory. In this paper, we will present a new asymptotical flat and spherically symmetric solution. Our new solution can recover to Schwarzschild solution and Reissner-Nordström one, which is different from those obtained in Ref. sbh1 (). Especially, as the parameters take certain special values, this solution can also reduce to the black hole solutions in the brane-world BHbrane (). Gravitational lensing is a phenomenon of the deflection of light rays in the curve spacetime. It is well known that gravitational lensing can provide us a lot of important signatures about compact objects, which could help to identify black hole and verify alternative theories of gravity in their strong field regime Ein1 (); Darwin (); KS1 (); KS2 (); KS4 (); VB1 (); VB2 (); VB202 (); Gyulchev (); Gyulchev1 (); Fritt (); Bozza1 (); Eirc1 (); whisk (); Bhad1 (); Song1 (); Song2 (); TSa1 (); AnAv (); agl1 (); agl101 (); gr1 (); gr2 (); gr201 (); gr3 (); gr4 (); gr401 (). Therefore, in this paper, we also further study the gravitational lensing in the spacetime described by our new solution.

The paper is organized as follows. In Sec. II, we will firstly present a new asymptotical flat and spherically symmetric solution in the generalized Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity. In Sec. III, we will investigate investigate the propagation of photon in this background and probe the effects of the spacetime parameters on the photon sphere, the deflection angles for light ray. We also analyze the coefficients in the strong field limit in the cases with photon sphere. Finally, we present a summary.

Ii A black hole solution in the generalized Einstein-Cartan-Kibble-Sciama gravity

In this section, we will present a new black hole solution with torsion in the generalized ECKS theory of gravity. Let us start with the action sbh1 ()

(1)

where is a coupling constant, is ECKS Lagrangian gK1 (); gK2 (); sbh1 ()

(2)

and is

(3)

Here is Ricci Scalar. The tensor describes the spacetime torsion, which is defined by

(4)

It is the antisymmetric part of the general affine connection . The affine connection is related to the Levi-Civita Christoffel connection by

(5)

where is the contorsion tensor with a form

(6)

As in Ref.sbh1 (), we focus on a static spherically symmetric vacuum solution with the metric

(7)

and suppose that the non-vanishing components of the torsion tensor for the solution (7) have the forms

(8)

Here , , , and are only functions of the polar coordinate . The choice (8) for the torsion tensor satisfies the constraint , which has been applied in cosmological setting tensorcos1 (). Inserting Eqs.(7) and (8) into the action (1), we can find that the total Lagrangian has a form

(9)

Making use of Euler-Lagrange equation, we can obtain four second order differential equations

(10)
(11)
(12)
(13)

Setting and as in Ref.sbh1 (), one can find that the equation (II) is satisfied naturally and other three equations can be reduced to a single equation

(14)

which is consistent with that in Ref.sbh1 (). This means that once is selected as a proper form, we can obtain the function by solving the second order differential equation (14). Letting and , we can obtain the solution

(15)

where is an integral constant. If taking and , we can get the solution

(16)

Interestingly, these two solutions (15) and (16), respectively, have the same forms as the black hole solutions and in the brane-world BHbrane (). This implies that there exist certain a unknown connection between the generalized ECKS theory of gravity (1) and the braneworld gravity BHbrane (). Here, we set the function and the coupling parameter , and then obtain a new black hole solution in the generalized ECKS theory of gravity (1)

(17)

where , , and are constants. This solution is asymptotically flat since both of functions and tend to as approaches to spatial infinity. As , we find that the solution (II) reduces to the Reissner-Nordström type. As , it becomes the black hole solution (16). As in the case of the black hole solutions and in the brane-world BHbrane (), the solution (II) will be expressed in terms of the ADM mass and the parameterized post-newtonian (PPN) parameter . The parameter is similar to a “tidal-like charge”. The position of outer event horizon lies at

(18)

which is defined by equation . Especially, the surface gravity constant of event horizon for the black hole (II) is

(19)

Thus, the presence of parameter will bring some particular spacetime properties differed from those in the Einstein’s general relativity, which could make a great deal influence on the propagation of photon in the spacetime with metric functions (II).

Iii The deflection angle for light ray and strong gravitational lensing in the spacetime with torsion

Let us to study the deflection angle for light ray and the corresponding gravitational lensing in the background of a black hole spacetime with torsion (II). Due to the nature of the spherically symmetric spacetime, we here consider only the case where both the source and the observer lie in the equatorial plane so that the orbit of the photon is limited on the same plane in the background spacetime. With the condition , the metric (II) can be expressed as

(20)

with

(21)

The geodesics for the photon in the spacetime (20) obey

(22)
(23)
(24)

where is the angular momentum of the photon and is an affine parameter along the null geodesics. Here, the energy of photon is set to . For a black hole spacetime (II), the impact parameter for a photon can be expressed as

(25)

where is the closest distance between the photon and black hole. It is well known that the photon sphere plays an important role in the propagation of photon in a curve spacetime. In the background of a black hole spacetime (II), the radius of the photon sphere is the largest real root of the equation

(26)
Figure 1: Change of the photon sphere radius with parameters and in the background of a black hole spacetime with torsion (II). Here, we set .

As expected, the radius of the photon sphere depends on both the parameters and of the black hole. However, the appearance of yields that the form of the photon sphere radius become very complicated in this case. In Fig.1, we present the variety of the photon sphere with the parameter and of by solving eq. (26) numerically. It is show that the photon sphere radius increases with the parameter and decreases with . The change of with is similar to that in the Reissner-Nordström spacetime.

Figure 2: The boundary of the existence of horizon (red dashed line) and of the photon sphere (blue line) in the background of a black hole spacetime with torsion (II). Here, we set .

Moreover, we find that the photon sphere exists only in the regime . The value of the upper limit depends on the parameter , and its form can be expressed as

(27)

where

(28)

Comparing Eq. (18) with Eq. (27), it is obvious that the condition of existence of horizons is not inconsistent with that of the photon sphere. This means that in the parameter panel , the whole region is split into four regions - by those critical curves. It is shown in Fig.(2) where the red dashed line and the blue solid line, respectively, denote the boundary of the existence of horizon and of the photon sphere. When the parameters lie in the region , there exist both horizon and photon sphere radius , which is similar to that of in the static black hole spacetime in the Einstein’s general relativity. When lie in the region , there is no horizon and no photon sphere, which corresponds to case of strong naked singularity where the singularity is completely naked KS4 (); Gyulchev1 (). When is located in region , there exists only photon sphere but no horizon, which corresponds to case of weak naked singularity where the singularity is covered by the photon sphere KS4 (); Gyulchev1 (). When lies in the region , there is horizon but no photon sphere. These four situations are similar to those in the black hole spacetime with a torsion zchen (). Thus, the presence of torsion changes the spacetime structure which will affect the propagation of photon in the background spacetime.

Let us now to discuss the behavior of the deflection angle of light ray in the spacetime described by a metric with torsion (II). For the photon coming from infinite, the deflection angle in a curve spacetime can be expressed as

(29)

where is the closest approach distance and is Ein1 ()

(30)
Figure 3: Deflection angle as a function of the closest distance of approach for the cases with photon sphere. The panels in the upper and bottom rows correspond to the cases in which the parameters () are located in the regions and in Fig.(2), respectively. Here, we set .
Figure 4: Deflection angle as a function of the closest distance of approach for the case without photon sphere and horizon in which the parameters () lie in the region in Fig.(2). Here, we set .
Figure 5: Deflection angle as a function of the closest distance of approach for the case with horizon and no photon sphere in which the parameters () lie in the region in Fig.(2). Here, we set .

In Figs.(3)-(5), we plot the change curves of the deflection angle with the distance of approach for different parameters and in the spacetime with torsion (II). For the cases with photon sphere, i.e., the parameters () lie in the region or in Fig.(2), the deflection angle for different and strictly increases with the decreases of the closest distance of approach and finally becomes infinite as tends to the respective photon sphere radius , i.e., , which is shown in Fig.(3). In Fig.(4), we present the deflection angle in the case in which the parameters () lie in the region in Fig.(2), there is no horizon and no photon sphere and singularity is naked completely. It is shown that the deflection angle of the light ray closing to the singularity tends to a finite value for different and , which means that the photon could not be captured by the compact object so that the photon goes back along the original direction in this situation. This behavior can be regarded as a common feature of gravitational lensing by strong naked singularity. As the parameters () lie in the region in Fig.(2), there exists horizon but no photon sphere, we find that the deflection angle of the light finally becomes a finite value as tends to the respective event horizon radius , i.e., . This behavior differs from those in the black hole with a torsion considered in Ref.zchen () in which the deflection angle of the light finally becomes unlimited large as tends to the event horizon radius. It could be understand by a fact that due to nonexistence of photon sphere the photon is captured directly by black hole before it make infinite complete loops around the central object in this case. Moreover, we find that the deflection angle increases with the parameter and decreases with . In the far-field limit, the deflection angle can be approximated as

(31)

which means that for all values of parameters and , which is a common feature in all asymptotical flat spacetimes.

Finally, we will study the strong gravitational lensing by a compact object (II) with the photon sphere and then probe how the parameters and affects the coefficients in the strong field limit. Using of the method developed by Bozza VB1 (), one can define a variable

(32)

and then rewrote the integral (30) as

(33)

with

(34)
(35)

The function is regular for all values of and , but diverges as tends to zero. Thus, one can split the integral (33) into the divergent part and the regular part ,

(36)

Expanding the argument of the square root in to the second order in , one can obtain

(37)

with

(38)
Figure 6: Change of the strong deflection limit coefficients and with the parameters and in the spacetime with torsion (II).

It is obvious that the coefficient vanishes as tends to the radius of photon sphere , and then the leading term of the divergence in is . This means that near the photon sphere the deflection angle of light ray can be expressed as VB1 ()

(39)

with

(40)

which indicates clearly that the deflection angle diverges logarithmically where the light is close to the photon sphere. The quantity is the distance between gravitational lens object and observer, and are the strong field limit coefficients which depend on the spacetime functions at the photon sphere. The changes of the coefficients ( and ) with the parameters and is shown in Fig.(6). For the fixed , the coefficient increases monotonously with . With increase of , the coefficient first increases and then decreases for the smaller , but first decreases and then increases for the larger . With increase of the parameter , increases monotonously as and decreases as , first decreases and then increases as . For the case with , increases with as the quantity is small. With the further increase of , the change of shows gradually a tendency of first decrease and then increase.

Iv summary

In this paper we firstly present a new black hole solution in the generalized ECKS gravity with three independent parameters , and and then investigate the propagation of photon in this background. We find that these spacetime parameters affect sharply photon sphere, deflection angle of light ray and strong gravitational lensing. The photon sphere exists only in the regime and the value of depends on the parameter . In the regime where photon sphere exists, the radius of photon sphere increases with the parameter , but decreases with . Moreover, the condition of existence of horizons is not inconsistent with that of photon sphere, which yields that the whole region in the panel () can be split into four regions by the boundaries of the existence of horizon and of the photon sphere. In the cases with photon sphere, the deflection angle of the light ray near the photon sphere diverges logarithmically, which is similar to those in the usual spacetime of a black hole or a weak naked singularity in the strong-field limit. In the case without photon sphere and horizon, the deflection angle of the light ray closing very to the singularity approaches a finite value , which does not depend on spacetime parameters and . It should be a common feature of the deflection angle of light ray near the static strong naked singularity. Furthermore, we also find that there exists a special case in which there is horizon but no photon sphere for the spacetime (II) as in the spacetime with a torsion zchen (). However, we find that the deflection angle of the light ray near the event horizon tends to a finite value in this case, which differs from those in the black hole with a torsion considered in Ref.zchen () where the deflection angle of the light finally becomes diverges logarithmically. It could be attribute to that the photon is captured directly by black hole before it make infinite complete loops around the central object in this case. Finally, we study the strong gravitational lensing by a compact object (II) with the photon sphere and then probe how the parameters and affects the coefficients in the strong field limit.

V Acknowledgments

This work was partially supported by the Scientific Research Fund of Hunan Provincial Education Department Grant No. 17A124. J. Jing’s work was partially supported by the National Natural Science Foundation of China under Grant No. 11475061.

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