Branes in Gravity’s Rainbow
Eur. Phys. J. C
Branes in Gravity's Rainbow
Amani Ashour 2
Mir Faizal 1
Ahmed Farag Ali 0
Fayçal Hammad 3 4
0 Department of Physics, Faculty of Science, Benha University , Benha 13518 , Egypt
1 Department of Physics and Astronomy, University of Lethbridge , Lethbridge, AB T1K3M4 , Canada
2 Mathematics Department, Faculty of Science, Damascus University , Damascus , Syria
3 Physics Department, Champlain College-Lennoxville , Sherbrooke, QC J1M 0C8 , Canada
4 Physics Department & STAR Research Cluster, Bishop's University , 2600 College Street, Sherbrooke, QC J1M 1Z7 , Canada
In this work, we investigate the thermodynamics of black p-branes (BB) in the context of Gravity's Rainbow. We investigate this using rainbow functions that have been motivated from loop quantum gravity and κ -Minkowski non-commutative spacetime. Then for the sake of comparison, we examine a couple of other rainbow functions that have also appeared in the literature. We show that, for consistency, Gravity's Rainbow imposes a constraint on the minimum mass of the BB, a constraint that we interpret here as implying the existence of a black p-brane remnant. This interpretation is supported by the computation of the black p-brane's heat capacity that shows that the latter vanishes when the Schwarzschild radius takes on a value that is bigger than its extremal limit. We found that the same conclusion is reached for the third version of rainbow functions treated here but not with the second one for which only standard black p-brane thermodynamics is recovered.
1 Introduction
One common feature among most of semi-classical
approaches to quantum gravity is a Lorentz invariance
violation due to a departure from the usual relativistic dispersion
relation caused by a redefinition of the physical momentum
and physical energy at the Planck scale. The source of this
departure comes from many approaches, such as loop
quantum gravity [
1, 2
] spacetime discreteness [3], spontaneous
symmetry breaking of Lorentz invariance in string field
theory [
4
], spacetime foam models [
5
] and spin-networks [
6
]. A
more recent approach that also predicts Lorentz invariance
violation is non-commutative geometry [
7
]. All these
findings suggest that Lorentz violation might be a generic and
an essential property when it comes to constructing a
quantum theory of gravity. Mathematically, the departure from
Lorentz invariance is expressed in the form of a modified
dispersion relation (MDR). This modification could be behind
anomalies that might occur in ultra-high-energy cosmic rays
and TeV photons [
5, 8, 9
]. Modern observations are recently
gaining the needed sensitivity to measure such effects, and
they are expected to be further improved in the coming few
years.1 For a recent detailed review of MDR theories and
the possibility of getting physical observations, we refer the
reader to Ref. [
2
].
The theory that naturally produces the MDR is the
socalled doubly special relativity (DSR) [
11, 12
]. DSR is
considered as an extension of the special theory of relativity that
extends the invariant quantities to include the Planck energy
scale besides the speed of light. The simplest realization of
the DSR is based on a nonlinear Lorentz transformation in
momentum space. This nonlinear transformation implies a
deformed Lorentz symmetry such that the usual dispersion
relations of special relativity become modified by corrections
relevant only at the Planck scale. It should be mentioned that
Lorentz invariance violation and Lorentz invariance
deformation are in general conceptually different scenarios. Here
we shall adopt Lorentz invariance deformation by
considering DSR and its extension in models of Gravity’s Rainbow.
In the framework of DSR, the definition of the dual
position space suffers a nonlinearity of the Lorentz
transformation. To resolve this issue, Magueijo and Smolin [
13
]
proposed a doubly general relativity in which one assumes that
1 Threshold anomalies are only predicted by MDR scenarios with a
preferred reference frame in which they imply a full violation of relativistic
symmetries. These anomalies are, however, not predicted by scenarios
in which MDR is due to a deformation of relativistic symmetry with no
preferred reference frame [
10
].
the spacetime background felt by a test particle depends
on the energy of the latter. Therefore, there will not be a
single metric describing the spacetime as seen by test
particles, but a one-parameter family of metrics that depends
on the energy and momentum of these test particles,
forming in a sense a ‘rainbow’ of metrics or geometries. This
idea is usually known as Gravity’s Rainbow (and sometimes
Rainbow Gravity). This is based on new Lorentz
transformations [
1
], which lead to a modified dispersion relation.
It may be noted that a modified equivalence principle has
been proposed in Ref. [
13
], and this has led to the
development of Gravity’s Rainbow. Gravity’s Rainbow depends (...truncated)