Quantum tunneling of fermions from Grumiller black hole
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ORIGINAL PAPER
Quantum tunneling of fermions from Grumiller blach hole F S Mirekhtiary1* and I Sakalli2 1
Faculty of Engineering, Near East University, Nicosia, North Cyprus, Mersin-10, Turkey
2
Department of Physics, Arts and Sciences Faculty, Eastern Mediterranean University, G. Magusa, North Cyprus, Mersin-10, Turkey Received: 27 October 2018 / Accepted: 09 August 2019
Abstract: In this paper, we examine the Hawking radiation of the Grumiller black hole via the quantum tunneling of fermions. To this end, we consider the Dirac equation in that black hole geometry. It is shown that the surface temperatures of the event horizon in either coordinate systems [singular and non-singular (Painle´ve–Gullstrand)] correspond to the standard Hawking temperature of the Grumiller black hole. Keywords: Hawking radiation; Grumiller black hole; Fermion; Temperature PACS Nos.: 04.70.Dy; 04.40.Nr
1. Introduction In 2010, Grumiller [1] constructed a model for the gravity of a central object at large distances. The effective potential in Grumiller’s modified gravity includes the Newtonian potential and a Rindler term [2]. This modified gravity is also a good theoretical model to explain the flat rotation curves of galaxies [3]. First, we briefly review the Grumiller’s spacetime, which possesses the spherical symmetry in four dimensions and can be split in the following way [4, 5]: ds2 ¼ gij dxi dx j þ U2 ðdh2 þ sin2 hd/2 Þ;
ð1Þ
wherein gij is a two-dimensional metric and U stands for the surface radius [6]. Both gij and U depend on xi ¼ ft; rg. (Throughout the paper, we use Latin letters to indicate twodimensional indices and Greek letters for four-dimensional metrics.) It is possible to define the dynamics of the fields gij and U in two dimensions, because the metric g and scalar field U are both solely two-dimensional objects. Each solution of the equations of motion of this two-dimensional theory leads to the above four-dimensional line element. The test particles are expected to move in the background of the four-dimensional line element. The procedure of ‘‘spherical
*Corresponding author, E-mail: [email protected]
reduction’’ [7] abbreviates the four-dimensional Einstein– Hilbert action to specified two-dimensional dilaton gravity model. The two-dimensional theory was proposed by Grumiller [1] with the following action, which has been very recently discussed in [8]: Z i 1 pffiffiffiffiffiffiffih S ¼ 2 d2 x g f ðUÞR þ 2ðoUÞ2 2VðUÞ : ð2Þ j The free functions f and V are analytic in the limit of a large dilaton field U, like in spherically reduction. The coupling function f that multiplies the Ricci scalar R must have the following structure: f ¼ U2 , so that one has the Newton potential Mr 1 . If one used f ¼ cU2 instead, then 1 the potential would change to Mr c . There is an experimental bound on c as jc 1j\1010 [9]. On the other hand, relying on the data obtained through the observations, Grumiller [1] made a conservative assumption: f ¼ U2 should not be renormalized in the infrared region (IR). In the seq
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