Particle Collision near 1+1- Dimensional Horava-Lifshitz Black Hole and Naked Singularity

The unbounded center of mass (CM) energy of oppositely moving colliding particles near horizon emerge also in 1+1- dimensional Horava-Lifshitz gravity. This theory has imprints of renormalizable quantum gravity characteristics in accordance with the method of simple power counting. Surpris- ingly the result obtained is not valid for a 1- dimensional Compton- like process between an outgoing photon and an infalling massless/ massive particle. It is possible to achieve unbounded CM energy due to collision between infalling photons and particles. The source of outgoing particles may be at- tributed to an explosive process just outside the horizon for a black hole and the naturally repulsive character for the case of a naked singularity. It is found that absence of angular momenta in 1+1- dimensions does not yield unbounded energy for collisions in the vicinity of naked singularities.


I. INTRODUCTION
It is known that in spacetime dimensions less than four gravity has no life of its own unless supplemented by external sources. With that addition we can have lower dimensional gravity and we can talk of black holes, wormholes, geodesics, lensing effect etc. in analogy with the higher dimensions. One effect that attracted much interest in recent times is the process of particle colllisions near the horizon of black holes due to Banados, Silk and West [1] which came to be known as the BSW effect. This problem arose as a result of imitating the rather expensive venture of high energy particle collisions in laboratory. From curiosity the natural question arises: is there a natural laboratory ( a particle accelerator) in our cosmos that we may extract information/energy in a cheaper way? This automatically drew attentions to the strong gravity regions such as near horizon of black holes. Rotating black holes host greater energy resorvoir due to their angular momenta and attentions naturally focussed therein first [2,3]. In case the black hole is not spinning there are enough reasons yet to consider the collision process in the near horizon geometry of black holes.
The sama idea can be tested in lowest dimensional black holes as well. One considers the radial geodesics and upon energy-momentum conservation in the centerof-mass (CM) frame the near horizon limit is checked whether the energy is bounded/unbounded. Our aim in this study is to consider black hole solutions in 1+1dimensional Hořava-Lifshitz (HL) gravity [4] and check the BSW effect in such reduced dimensional theory. For a number of reasons HL gravity is promising as candidate for a renormalizable quantum gravity physics has been yearning for a long time [5]. The key idea in HLgravity is the inhomogenous scaling properties of time and space coordinates which violates the Lorentz invariance. Arnowitt-Deser-Misner (ADM) splitting of space and time [6] constitutes its geometrical background. BSW effect in 3+1-dimensions has been worked out by many authors [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Following the similar idea we consider black hole solutions in 1+1-dimensions and derive the same effect in this lower dimension. It should be added that with 1+1-dimensional HL theory the simplest nontrivial solution is the solution describing an accelerated particle in the flat space in Rindler frame. This justifies also the meaning of the vector field (a i ) as the acceleration in the HL-gravity. The role of Rindler acceleration in 3+1-dimensions as a possible source of flat rotation curves and geodesics motion has been discussed recently [22] . It is our belief that the results in lower dimensions are informative for higher dimensions and as a toy model can play the role as precursons in this regard. Even a Compton process can be considered at the toy level between a massless Hawking photon radiated from a black hole and a particle falling in. The diverging CM energy shows itself once more in the case of photon-particle collision in 1+1-dimensions.
Organization of the paper as follows. In Section II, we review in brief the 1+1-D HL theory with a large class of black hole solutions. CM energy of colliding particles near horizon is considered in Section III. Section IV proceeds with applications to particular examples. The case of particle-photon collision is studied separetely in Section V. The paper ends with our conclusion in Section VI.

II. 1+1-D HL BLACK HOLES
HL formalism in 3+1-D makes use of the ADM splitting of time and space components as follows where N (t) and N i are the lapse and shift functions, respectively. The action of this theory is where K ij is the extrinsic curvature tensor with trace K and Planck mass M P l . V (φ) stands for the potensial function of a scalar field φ, and λ is a constant (λ > 1). Reduction from 3+1-D to 1+1-D results in the action [4] where η=constant, α=constant, a 1 =(ln N )′ in which a 'prime' denotes d dx . We note that the first term in S is inherited from the geometric part of the action while the other two terms are from the scalar field source. For simplicity we have set also M P l = 1 .
It has been shown in [4] that by variational principle a general class of solutions is obtained as follows in which C 2 , A, C 1 , B and C are integration constants.
The line element is Note that the associated potential is and the Ricci scalar is calculated as In the case of C 2 = 1/2, B = −2M , η = 1 and A = C = C 1 = 0 , it gives a Schwarzschild-like solution; On the other hand, the choice of the parameters, for C 2 = 1/2,B = −2M , C = 3Q 2 , η = 1 and A = C 1 = 0 gives a Reissner-Nordström-like solution.
The new black hole solution which is derived by Bazeia et. al. [4]is found by taking C 1 = 0, C 2 = 0, B = 0 and This solution develops the following horizons As ∆ = 0 they degenerate, i.e., The Hawking temperature is given in terms of the outer (x + h ) horizon as follows For the special case C 2 = 0, C 1 = −M and B = −2M the horizons are independent of the mass M : The temperature is then given simply by This is a typical relation between the Hawking temperature and the mass of black holes in 1 + 1 dimensions [23].

III. CM ENERGY OF PARTICLE COLLISION NEAR THE HORIZON OF THE 1+1 -D HL BLACK HOLE
Here we will derive the equations of motion of an uncharged massive test particle by using the method of geodesic Lagrangian. Such equations can be derived from the Lagrangian equation, Here, τ is the proper time for time-like geodesics ( or massive particles) The canonical momenta calculated as The 1+1-D HL black hole have only one Killing vector ∂ t . Hence, there is only one conserved quantity along the motion of the particle which can be labeled as E . From eq. (14), E is related to N (x) 2 as, The two-velocity of the particles are given by u µ = dx µ dτ . We have already obtained u t in the above derivation. To find u x =ẋ, the normalization condition for time-like particles, u µ u µ = −1 can be used as, By substituting u t to eq.(18), one can obtain u x as, where V ef f is the effective potential for the motion, given by Now, the two-velocities can be written as, We proceed now to present the CM energy of two particles with two-velocity u µ 1 and u µ 2 . We will assume that both have rest mass m 0 = 1. The CM energy is given by, So, the lowest order term gives the CM energy of two particles as There are two cases for this CM energy, when E 1 E 2 < 0 , the CM energy is reduced to which is unbounded for x −→ x h .
On the other case, when E 1 E 2 > 0 , the CM energy is independent from metric function, hence it gives always the finite energy.
So it should have E 1 E 2 < 0 to obtain an unbounded CM energy near to horizon of the HL black holes when we have the limiting value as x −→ x h .

A. Schwarzchild-like Solution
In the case of C 2 = 1/2, B = −2M , η = 1 and A = C = C 1 = 0 , it gives Schwarzschild-like solution where and For the CM energy on the horizon, we have to compute the limiting value of eq.(24) as x −→ x h = 2M , where is the horizon of the black hole. Setting E 1 E 2 < 0 as is, the CM energy near the event horizon for 1+1 D Schwarzchild BH is From the case of E 1 E 2 > 0 , it is shown that the CM energy is finite. This result for 4-D Schwarzchild Black hole is already calculated by Baushev [21]. When the location of particle 1 which has positive energy approachs the horizon , on the other hand the particle 2 escaping from the horizon with negative energy might give us the BSW effect E 2 cm −→ ∞ so there is BSW effect for 1+1 Schwarzchild-like Solution when the condition E 1 E 2 < 0 is satisfied.

B. Reissner-Nordstrom-like solution
On the other hand, the choice of the parameters, for C 2 = 1/2,B = −2M , C = 3Q 2 , η = 1 and A = C 1 = 0 gives the Reissner-Nordström-like solution. and so the CM energy is calculated by using the limiting value of eqn. 26 so there is a BSW effect.

C. The Non-Black Hole case
The simplest solution in [4] without scalar potential case is given as follows This is not a black hole solution and is transformable to the Rindler metric in 1+1-D.
For the CM energy on the horizon, we have to compute the limiting value of eq.(26) as x −→ x h = 1 2M , where lies the horizon.
After some calculations, we get the limiting value of eq.(26):

D. The Extremal case of the Reissner-Nordstrom like black hole
For the extremal case we have with M = Q, from eq. (31) so that it also gives the same answer from eq.(26) as

E. Specific New Black Hole Case
The new 3-parametric black hole solution given by Bazeia, Brito and Costa [4] is chosen as with the potensial For the special case C 2 = 0, with suitable potensial which is The CM energy of two colliding particles is calculated by taking the limiting values of eq. (26) Hence the BSW effect arises here as well.

F. Near Horizon Coordinates
We have explored the region near the horizon by replacing r by a coordinate ρ. The proper distance from the horizon ρ : The first example is the Schwarzchild-like solution which is The new metric is whereρ ≃ 2 2M (x − 2M ), gives approximately which is once more the Rindler line element.
The CM energy of two colliding particles is given by

HAWKING PHOTON VERSUS AN INFALLING PARTICLE
Hawking radiation is accepted as a reality in the world of black holes. The massless photon of such an emission can naturally scatter an infalling particle or vice versa. This phenomenou is analogous to a Compton scattering taking place in 1+1-dimensions. Null-geodesics for a photon can be described simply by where λ is an affine parameter and E 1 stands for the photon energy. Defining E 1 = ℏω 0 , where ω 0 is the frequency ( with the choice ℏ = 1) we can parametrize energy of the photon by ω 0 alone. The center-of-mass energy of a Hawking photon and the infalling particle can be taken now as in which p µ and k µ refer to the particle and photon, 2momenta, respectively. This amounts to since we have for the particle p µ = m E2 N 2 , E 2 2 − N 2 and for the photon k µ = E1 N 2 , −E 1 . One obtains In the near horizon limit this reduces to Note that for E 2 < 0 we have E 2 cm given by which is finite and therefore is not of interest. On the other hand for E 2 > 0 we obtain an unbounded E 2 cm .

VI. CONCLUSION
Particle collision problem is considered near the horizon of 1+1-dimensional Hořava-Lifshitz (HL) black holes. Our aim is to show that the BSW effect which arises in higher dimensional black holes applies also in the 1+1-D. The theory we adapted is not general relativity but instead the recently popular HL gravity. We employed the class of 5-parametric black hole solutions found recently [4]. The class has particular limits of flat Rindler, Schwarzschild and Reissner-Nordstrom like solutions. For each case we have calculated the centerof-mass (CM) energy of the particles and shown that the energy can grow unbounded. In other words the strong gravity near the event horizon effects the collision process with unlimeted source to turn it into a natural accelerator. The model we use applies also to the case of a photon/particle collision with similar characteristics. Finally, we must admit that absence of rotational effects in 1+1-D confines the problem to the level of a toy model.