Remarks on Null Geodesics of Born-Infeld Black Holes

Abstract

We present interesting properties of null geodesics of static charged black holes in Einstein-Born-Infeld gravity. These null geodesics represents the path for gravitons. In addition, we also study the path of photons for the Born-Infeld black hole which are null geodesics of an effective geometry. We will present how the bending of light is effected by the non-linear parameter 𝛽 of the theory. Some other properties, such as the horizon radius and the temperature are also discussed in the context of the nonlinear parameter 𝛽.

1. Introduction

In Maxwell theory, the field of a point-like charge is singular at the origin. Hence, it has infinite self-energy. To avoid this, Born-Infeld proposed a theory of electrodynamics which is nonlinear in nature which is now known as Born-Infeld electrodynamics [1]. In this theory, the electric filed of a point charge is given as 𝐸_𝑟=𝑄/√𝑟⁴+𝑄²/𝛽² which is regular at the origin. Also, its total energy is finite. Born-Infeld theory has received renewed interest since it turns out to play an important role in string theory. Born-Infeld actions naturally arises in open superstrings and in D-branes [2]. Review articles on the aspects of the Born-Infeld theory in string theory is written by Gibbons [3] and Tseytlin [4].

In this paper, we study null geodesics in the black holes of Einstein-Born-Infeld gravity. The particular black hole in consideration is the nonlinear generalization of the well known Reissner-Nordstrom black hole characterized by charge 𝑄, 𝑀, and 𝛽. Black hole solutions for Born-Infeld gravity was obtained by Garcia et al. [5] in 1984. Two years later, Demianski [6] also presented a solution known as EBIon. There are many papers written in the literature, addressing various aspects of black holes in Einstein-Born-Infeld gravity. Due to the long list, we will mention only a few recent work here.

Kruglov published on generalized Born-Infeld electrodynamics in [7]. Thermodynamics of third-order Lovelock-Born-Infeld black holes were studied by peng et al. [8]. Thin shells in Einstein-Born-Infeld theory were studied by Eiroa and Simeone [9]. Linear alanlogs of the Born-Infeld and other nonlinear theories were presented by Milgrom [10]. Test particle trajectories for the static-charged Born-Infeld black hole were discussed by Breton [11]. Gibbons and Herdeiro [12] derived a Melvin Universe-type solution describing a magnetic field. The current author has studied the gravitational, scalar, and Dirac perturbations of the Born-Infeld black holes in [13–15], respectively. Non-abelian black hole solutions to Born-Infeld gravity were presented by Mazharimousavi et al. [16]. Hairy mass bound in the Einstein-Born Infeld black holes were given by Myung and Moon [17].

2. Static Charged Black Hole in Einstein-Born-Infeld Gravity

The Einstein-Born-Infeld gravity is given by the action 𝑆=𝑑⁴𝑥√−𝑔𝑅16𝜋𝐺+𝐿(𝐹),(2.1) where 𝐿(𝐹) is a function of the field strength 𝐹_𝜇𝜈 given as 𝐿(𝐹)=4𝛽²⎛⎜⎜⎝1−1+𝐹^𝜇𝜈𝐹_𝜇𝜈2𝛽²⎞⎟⎟⎠(2.2) Here, 𝛽 has dimensions 𝑙𝑒𝑛𝑔𝑡ℎ⁻² and 𝐺   𝑙𝑒𝑛𝑔𝑡ℎ². We will assume 16𝜋𝐺=1 in the rest of the paper. Note that when the nonlinear parameter 𝛽→∞, the function 𝐿(𝐹) approaches the one for Maxwell’s electrodynamics given by −𝐹².

The static-charged black hole solution with spherical symmetry for the above action in (2.1) is given as 𝑑𝑠²=−𝑓(𝑟)𝑑𝑡²+𝑓(𝑟)⁻¹𝑑𝑟²+𝑟²𝑑𝜃²+sin²(𝜃)𝑑𝜑²(2.3)

with 𝑓(𝑟)=1−2𝑀𝑟+2𝛽²𝑟²3⎛⎜⎜⎝1−1+𝑄²𝑟⁴𝛽²⎞⎟⎟⎠+4𝑄²3𝑟²²𝐹₁14,12,45,−𝛽𝑄²𝑟²⁴.(2.4)

Here ₂𝐹₁ is the hypergeometric function.

The electric field is given by 𝐹_𝑡𝑟=𝐸(𝑟)=−𝑄√𝑟⁴+𝑄²/𝛽².(2.5)

In this case, the 𝐿(𝐹) reduces to 𝐿(𝐹)=4𝛽²1−1−𝐸²𝛽².(2.6)

One can observe that there is an upper bound for the electric field as |𝐸|≤𝛽. This is one of the leading characteristics of Born-Infeld electrodynamics which leads to finite self-energy of the electron as compared to Maxwell electrodynamics.

When the non-linear parameter 𝛽→∞, the function 𝑓(𝑟) approaches 𝑓(𝑟)_RN=1−2𝑀𝑟+𝑄𝑟²²,(2.7) which is the metric function 𝑓(𝑟) for the static charged black hole in Einstein-Maxwell gravity which is known as the Reissner-Nordstrom black hole. Reissner-Nordstrom black hole has horizons at 𝑟_±=𝑀±𝑀²−𝑄².(2.8)

For the Born-Infeld black hole, near the origin, the function 𝑓(𝑟) has the behavior as, 𝑓(𝑟)≈1−(2𝑀−𝐴)𝑟−2𝛽𝑄+23𝛽²𝑟²+𝛽³5𝑟⁴.(2.9)

Here, 𝐴=13𝛽𝜋𝑄^3/2Γ14².(2.10)

Hence, for 2𝑀>𝐴, 𝑓(𝑟)→−∞ for small 𝑟 and for 2𝑀<𝐴, 𝑓(𝑟)→+∞ for small 𝑟. When 𝑟→∞, for all values of 𝑀 and 𝐴, 𝑓(𝑟)→1. In fact graphically, we will show that for various values of 𝑀, 𝑄 and 𝛽, that 𝑓(𝑟) could have two roots, one root, or none. Extreme black holes are possible when 𝑓(𝑟)=0 and 𝑓^(𝑟)=0, leading to the horizon radius as 𝑟_ex=√4𝛽²𝑄²−12𝛽.(2.11)

It is clear that extreme black holes exist only if 𝑄𝛽>1/2. In Figure 1, the function 𝑓(𝑟) is plotted for both 𝑄𝛽<1/2 and 𝑄𝛽>1/2.

(a)

(b)

Figure 1 

The function 𝑓(𝑟) for various values of 𝑀 for fixed values of 𝑄 and 𝛽. In the top graph (a), 𝑄=1 and 𝛽=1. In the bottom graph (b), 𝑄=0.3 and 𝛽=1.

In Figure 2, the horizon radius 𝑟₊ is computed for various values of 𝛽. The horizon radius decreases with 𝛽. For the same values of 𝑀 and 𝑄, the horizon radius for the Reissner-Nordstrom black hole is given by 𝑟₊=1.42765.

Figure 2 

The horizon radius 𝑟+ for various values of 𝛽. Here, 𝑄=1 and 𝑀=1.06405.

The Hawking temperature of the Born-Infeld black hole is given by 𝑇=14𝜋⎡⎢⎢⎢⎣𝑟1₊+2𝛽⎛⎜⎜⎜⎝𝑟₊𝛽−𝑄²+𝑟²⁺𝛽²𝑟₊⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦.(2.12)

Here, 𝑟₊ is the event horizon of the black hole such that 𝑓(𝑟)=0. In Figure 3, a graph for temperature versus 𝛽 is plotted. It seems the temperature has a maximum before decreasing for this particular values of 𝑀 and 𝑄.

Figure 3 

The temperature 𝑇 for various values of 𝛽. Here, 𝑄=1 and 𝑀=1.06405.

To compare the temperature of te Born-Infled black hole to the Reissner-Nordstrom black hole, one can compute the temperature for the outer horizon given in (2.8) as, 𝑇_RN=14𝜋−2𝑟𝑄²₊+2𝑀𝑟₊.(2.13)

For the same values of 𝑀 and 𝑄 given in the Figure 3, the temperature for the Reissner-Nordstrom black hole is given as 𝑇_RN=0.356786. Hence, the Reissner-Nordstrom black hole is “hotter” compared to its counterpart in Born-Infeld gravity. The zeroth and the first law of the Born-Infeld black holes are discussed by Rasheed in [18].

3. Null Geodesics of the Born-Infeld Black Hole

The motion of graviton in the background of the Born-Infeld black hole is given by the null geodesics. In general, it is also the path of photons, which is not the case here. This will be discussed in Section 4.

The geodesic equations for the Born-Infeld black hole can be derived from the Lagrangian equation ℒ=−12−𝑓(𝑟)𝑑𝑑𝜏𝑡²+1𝑓(𝑟)𝑑𝑑𝜏𝑟²+𝑟²𝑑𝜃𝑑𝜏²+𝑟²sin²𝜃𝑑𝜙𝑑𝜏².(3.1)

Here, 𝜏 is an affine parameter along the geodesics. The derivation is clearly given in the well known book by Chandrasekhar [19]. Therefore, we will skip some of the details here. Since the Born-Infeld black holes have two Killing vectors 𝜕_𝑡 and 𝜕_𝜙, there are two constants of motion which can be labeled as 𝐸 and 𝐿 given as 𝑟𝑓𝑡̇=𝐸,₂sin²𝜃𝜙̇=𝐿.(3.2)

Here, we will choose 𝜃=𝜋/2 and 𝜃̇=0 as the initial conditions, which leads to, 𝜃̈=0. Hence, 𝜃 will remain at 𝜋/2 and the geodesics will be described in an invariant plane at 𝜃=𝜋/2. From (3.2), 𝑓𝑟(²𝑟𝜙)̇𝑡=̇=𝐿𝐸;.((33..34))

By substituting these values to the Lagrangian in (3.1), one obtains the geodesics as, 𝑟²̇+𝑓(𝑟)𝐿²𝑟²+ℎ=𝐸².(3.5)

Here, 2ℒ=ℎ. ℎ=0 corresponds to null geodesics and ℎ=1 corresponds to time-like geodesics. Note that (3.5) can be written as, 𝑟²̇+𝑉_eff=𝐸², with the effective potential, 𝑉_eff=𝐿²𝑟²+ℎ𝑓(𝑟).(3.6)

From (3.3) and (3.5), one can get a relation between 𝜙 and 𝑟 as follows: 𝑑𝜙𝑑𝑟=𝑟𝐿²1𝐸²−𝑉_eff.(3.7)

3.1. Effective Potential for Null Geodesics

With ℎ=0, V_eff=𝑓(𝑟)𝐿²𝑅(𝑟)².(3.8)

We will only consider the gravitons with nonzero angular momentum here. In Figure 4, the 𝑉_eff is given for various values of 𝛽. The height for the Born-Infled black hole is shorter in comparison with the Reisnner-Nordstrom black hole.

Figure 4 

The graph shows the relation of 𝑉eff with the parameter 𝛽. Here, 𝑀=1.06405, 𝐿=1 and 𝑄=1.

In Figure 5, the effective potential is plotted for three different energy levels, 𝐸₁, 𝐸_𝑐, and 𝐸₂. This gives different scenarios of motion of the particles which are described below.

Figure 5 

The graph shows the relation of 𝑉eff with the energy 𝐸. Here, 𝑀=1.06405, 𝛽=1, 𝑄=1 and 𝐿=1.

Case 1 (𝐸=𝐸_𝑐). Here, 𝐸²−𝑉_eff=0 leads to circular orbits. From the nature of the potential at 𝑟=𝑟_𝑐, one can conclude that these are unstable circular orbits.

Case 2 (𝐸=𝐸₂). Here, the motion is possible only in the regions where, 𝐸²−𝑉_eff≥0.

Case 3 (𝐸=𝐸₁). Since 𝐸²−𝑉_eff≥0 and 𝑟̇>0 for all 𝑟 values, motion is possible for all 𝑟 values.

For Case 1, one can compute the radius of the circular orbits. The conditions for the circular orbits are 𝑟̇=0𝑑⟹𝑉_eff𝑉_eff=𝐸^2𝑐,(3.9)𝑑𝑟=0.(3.10)

From (3.10), 𝑟_𝑐 can be computed numerically. It is given in Figure 6. 𝑟_𝑐 decrease as 𝛽 increases.

Figure 6 

The graph shows the critical radius 𝑟𝑐 for the Born-Infeld black hole as a function of the nonlinear parameter 𝛽. Here, 𝑀=1.06405 and 𝑄=1.

4. Null Geodesics of the Effective Geometry

In general, the motion of photons are represented by the null geodesics of the space-time. However, in nonlinear electrodynamics, the path of the photons are not given by the null geodesics of the background metric. The path is given by null geodesics of an effective geometry generated by the self-interaction of the electromagnetic field. This effective geometry depends on the particular nonlinear theory considered, and in Einstein-Born-Infeld gravity, the effective geometry is given by 𝑑𝑠²=−ℎ(𝑟)𝑑𝑡²+𝑔(𝑟)⁻¹𝑑𝑟²+𝑅(𝑟)²𝑑𝜃²+sin²(𝜃)𝑑𝜙²,(4.1) where 𝜔(𝑟)=1+𝑄²𝑏²𝑟⁴𝑔(𝑟)=𝑓(𝑟)𝜔(𝑟),_−1/2ℎ(𝑟)=𝑓(𝑟)𝜔(𝑟)_1/2,𝑅(𝑟)₂,=𝑟²𝜔(𝑟)^−1/2.(4.2)

Hence, gravitational lensing, which is related to the bending of light around the black hole can be computed from the knowledge gained from null geodesics of the effective geometry. The derivation leading to the null geodesics are similar to the one given in Section 3. Since the symmetries are the same, there will be two conserved quantities as,𝑅𝑓(𝑟(𝑟)²)𝜙𝑡̇̇==𝐸𝐿.;((44..34))

The equations can be given as 𝑟²̇+𝑉_eff=0, where the effective potential, which depends on 𝐸, and 𝐿 is given as follows: 𝑉_eff=𝐿²𝑔(𝑟)𝑅(𝑟)²−𝐸²𝑔(𝑟)ℎ(𝑟).(4.5)

4.1. Circular Orbits

The conditions for the circular orbits are 𝑟̇=0𝑑𝑉⟹_eff𝑉_eff=0,𝑑𝑟=0.(4.6)

These two conditions lead to the equation ℎ(𝑟)𝑅(𝑟)^2′−ℎ(𝑟)^𝑅(𝑟)²=0.(4.7)

One can obtain a solution for the circular orbits at 𝑟=𝑟_𝑐 by solving (4.7) numerically.

From Figure 7, it is clear that 𝑟_𝑐 decreases for increasing 𝛽 values. The circular orbit at 𝑟=𝑟_𝑐 are unstable. The radius of the circular orbit is related to 𝐸 and 𝐿 as 𝐸^2𝑐𝐿^2𝑐=ℎ𝑟^𝑐𝑅𝑟_𝑐².(4.8)

Figure 7 

The graph shows the critical radius 𝑟𝑐 for the Born-Infeld black hole as a function of the nonlinear parameter 𝛽. Here, 𝑀=1.06405 and 𝑄=1.

When 𝛽→∞ and 𝑄→0, 𝑟_𝑐→3𝑀 which is the radius of the unstable circular orbit of the Schwarzschild black hole [19].

4.2. Bending of Light

To compute the angle of bending of light, first, let us compute the closest approach 𝑟_𝑜. It is defined by the solutions to the equation 𝑑𝑟/𝑑𝜙=0. From (4.3) and (4.5) 𝑅(1𝑟)²𝑑𝑟𝑑𝜙²=𝑔(𝑟)𝑅(𝑟)²−𝐸²𝐿²𝑔(𝑟)ℎ(𝑟).(4.9)

Since 𝐸²/𝐿²=1/𝐷², where 𝐷 is the impact parameter, the above equation 𝑑𝑟/𝑑𝜙=0 simplifies to, 𝐷²ℎ(𝑟)−𝑅(𝑟)²=0.(4.10)

For various values of 𝐷 and 𝛽, (4.10) can be solved numerically to obtain 𝑟_𝑜. In Figure 8 and Figure 9, the graph 𝑟_𝑜 versus 𝐷 and 𝛽 is given. For large 𝐷, 𝑟_𝑜 becomes larger as expected. For large 𝛽, 𝑟_𝑜 decreases.

Figure 8 

The graph shows the closest approach 𝑟𝑜 for the effective geometry of the Born-Infeld black hole as a function of the impact parameter 𝐷. Here, 𝑀=1.06405, 𝑄=1 and 𝐿=1.

Figure 9 

The graph shows the closest approach 𝑟𝑜 for the effective geometry of the Born-Infeld black hole as a function of the nonlinear parameter 𝛽. Here, 𝐷 is kept fixed at 4.56. Also, 𝑀=1.06405, 𝑄=1.

Gravitational lensing of the photons in the Born-Infeld black hole was studied by Eiroa [20]. In an interesting paper by Amore [21], analytical expression for the bending angle was derived. Here, we will use that expression which is given as 𝛼=4𝑀𝑟_𝑜+24𝑀²𝜋𝑟^2𝑜−3𝜋𝑄²4𝑟^2𝑜+16𝜋0²𝑀𝑟^32𝑜−9𝑀𝑄²𝑟^3𝑜+𝑂𝑟1^4𝑜.(4.11)

One can compute the bending angle as a function of 𝐷 and 𝛽 which is presented in Figures 10 and 11. The bending is grater for large 𝐷 as expected. The angle 𝛼 increases as 𝛽 increases.

Figure 10 

The graph shows the bending angle 𝛼 for the Born-Infeld black hole as a function of the impact parameter 𝐷. Here, 𝑀=1.06405, 𝑄=1 and 𝐿=1.

Figure 11 

The graph shows the bending angle 𝛼 for the Born-Infeld black hole as a function of the 𝛽. Here, the impact parameter is kept fixed at 𝐷=4.56. Also, 𝑀=1.06405 and 𝑄=1.

5. Conclusions

In this paper, we have done a detailed study of null geodesics of the Born-Infeld black hole for the gravitons and the photons. Unlike in other cases, null geodesics of the black hole is not the path of the photons. Path for photons are given by an effective geometry. We have studied the bending of light and showed that the bending angle increases with the nonlinear parameter 𝛽. On the other hand, the bending angle decreases with the impact parameter 𝐷.

We have also discussed the thermodynamics of the black hole and showed how the temperature vary with 𝛽. For the same mass and the charge, the corresponding Reissner-Nordstrom black hole is “hotter”.

The horizon radius also seems to decrease with the increasing 𝛽 for the particular mass and charge considered.

In extending this work, it would be interesting to study massive test particles around the Born-Infeld black hole.