# Absence of Black Holes at LHC due to Gravity’s Rainbow

###### Abstract

In this paper, we investigate the effect of Planckian deformation of quantum gravity on the production of black holes at colliders using the framework of gravity’s rainbow. We demonstrate that a black hole remnant exists for Schwarzschild black holes in higher dimensions using gravity’s rainbow. The mass of this remnant is found to be greater than the energy scale at which experiments were performed at the LHC. We propose this as a possible explanation for the absence of black holes at the LHC. Furthermore, we demonstrate that it is possible for black holes in six (and higher) dimensions to be produced at energy scales that will be accessible in the near future.

###### Keywords:

## 1 Introduction

Black holes are one of the most important objects in quantum gravity. However, there is little hope of detecting a four dimensional black hole directly in particle accelerators. This is because in order to produce black holes, an energy of the order of the Planck energy ( GeV) is needed, and this energy is way beyond what can be achieved in the near future. However, if large extra dimensions exist, then there is a hope of observing black holes at colliders, in the near future. This is because the existence of large extra dimensions can lower the effective Planck scale to TeV scales at which experiments can be done Arkani-Hamed et al. (1998). This lowering of Planck scale occurs in Type I and Type II string theories by localizing the standard model particles on a D-brane, while gravity propagates freely in the higher dimensional bulk. Using this model, it was predicted that due to this lowering of effective Planck scale, black holes could be produced at the LHC Banks and Fischler (1999); Giddings and Thomas (2002); Dimopoulos and Landsberg (2001); Emparan et al. (2000); Meade and Randall (2008); Antoniadis et al. (1998); da Rocha and Coimbra-Araujo (2006). Furthermore, the production of such black holes would also serve to prove the existence of extra dimensions, and thus provide a strong indication for string theory to be a correct theory describing the natural world (since string theory is critically based on the existence of higher dimensions).

In the experiments performed at the LHC, no black holes have been detected Chatrchyan et al. (2012a, b). This result has been interpreted to imply the absence of large extra dimensions, at least at the energy scale at which experiments have been performed at the LHC. However, in this paper, we will demonstrate that these results should rather be interpreted as an indication of a suppression of higher dimensional black hole production due to Planckian deformation of quantum gravity. Since large extra dimensions can lower the effective Planck scale to scales at which such experiments are talking place, it becomes very important to consider the Planckian deformation of quantum gravity. We can implement the Planckian deformation of quantum gravity by introducing rainbow functions in the original classical metric using a formalism called gravity’s rainbow.

Gravity’s rainbow is motivated by doubly special relativity (DSR), which in turn is motivated by the fact that almost all approaches to quantum gravity suggest that standard energy-momentum dispersion relation gets deformed near Planck scale. This deformation of the energy-momentum relation has been predicted from spacetime discreteness ’t Hooft (1996), spontaneous symmetry breaking of Lorentz invariance in string field theory Kostelecky and Samuel (1989), spacetime foam models Amelino-Camelia et al. (1998), spin-network in loop quantum gravity (LQG) Gambini and Pullin (1999), non-commutative geometry Carroll et al. (2001), and Horava-Lifshitz gravity Horava (2009a, b). As such a deformation of the dispersion relation is a common prediction of various approaches to quantum gravity, we can expect that this will even hold in any quantum theory of gravity. The modification of the dispersion relation generally takes the form,

(1) |

where is the Planck energy, and the functions and satisfy

(2) |

The modified dispersion relation occurs in DSR because there is a maximum invariant energy scale in addition to the speed of light Amelino-Camelia (2002); Magueijo and Smolin (2002). The most compelling argument for the existence of such a maximum energy scale comes from string theory. This is because it is not possible to probe spacetime below the string length scale. Thus, string theory comes naturally equipped with a minimum length scale, which can be translated into a maximum energy scale Amati et al. (1989); Garay (1995). DSR can naturally incorporate this maximum energy scale corresponding to string length scale Ali et al. (2009, 2011). The gravity’s rainbow is the generalization of DSR to curved spacetime. This is done by incorporating the functions and in general curved spacetime metric. So, in gravity’s rainbow the structure of spacetime depends on the energy used to probe it Magueijo and Smolin (2004).

The choice of the rainbow functions and is important for making predictions. This choice should be phenomenologically motivated. Different aspects of Gravity’s Rainbow with various choices of rainbow functions have been studied in Galan and Mena Marugan (2004); Hackett (2006); Garattini and Mandanici (2012); Garattini and Majumder (2014); Garattini (2013); Garattini and Majumder (2014); Leiva et al. (2009); Li et al. (2009); Ali et al. (2014); Awad et al. (2013); Barrow and Magueijo (2013); Liu and Zhu (2008); Ali and Khalil (2014); Gim and Kim (2014). Among these choices, the rainbow functions proposed by Amelino-Camelia, et al. Amelino-Camelia et al. (1997, 1998), are both phenomenologically important and theoretically interesting,

(3) |

where is an integer , and is a constant of order unity, because naturalness says that the parameter is set to be one, unless the observations or measurements prove differently. Besides, in gravity’s rainbow, the Planck energy is an invariant scale, and if eta were much greater than one, this would be analogous to reducing the energy scale below the Planck energy.

These rainbow functions lead to the most common form of MDR in the literature. This MDR is compatible with some results from non-critical string theory, loop quantum gravity and -Minkowski non-commutative spacetime Amelino-Camelia (2013). Furthermore, this MDR was first used to study the possible dispersion of electromagnetic waves from gamma ray bursters Amelino-Camelia et al. (1998), and it resolved the ultra high energy gamma rays paradox Amelino-Camelia and Piran (2001); Kifune (1999). In fact, it was used for providing an explanation for the 20 TeV gamma rays from the galaxy Markarian 501 Amelino-Camelia and Piran (2001); Protheroe and Meyer (2000). Apart from that, it also provides stringent constraints on deformations of special relativity and Lorentz violations Aloisio et al. (2000); Myers and Pospelov (2003). A detailed analysis of the phenomenological aspects of these functions has been done in Amelino-Camelia (2013).

An outline of the paper is as follows. In section 2, we review the thermodynamics of higher dimensional Schwarzschild black holes, and in section 3, we study their modified thermodynamics using gravity’s rainbow with the rainbow functions Eq. (3). This is the higher dimensional study of rainbow Schwarzschild black hole which was studied by one of the authors in Ali (2014), and reached the conclusion that black holes end in a remnant. In section 4, we discuss this result and compare it with the energy scale of the LHC. Finally, in section 5, we set bounds on the parameter from LHC experiments. In this paper, we use natural units, in which , , and .

## 2 Schwarzschild Black Holes in Higher Dimensions

In this section, we will review the Schwarzschild black holes in higher dimensions. This will be used to motivate a similar analysis based on gravity’s rainbow, in the next section. The metric of Schwarzschild black holes in dimensions takes the form Emparan and Reall (2008); Aman and Pidokrajt (2006)

(4) |

where the mass parameter is given by

(5) |

where is Newton’s constant in dimensions, which is related to the Planck mass via Dimopoulos and Landsberg (2001)

(6) |

and is the volume of the unit sphere

(7) |

The horizon radius is evaluated by solving leading to

(8) |

The Hawking temperature can be calculated via the relation Angheben et al. (2005)

(9) |

This relation applies to any spherically symmetric black hole with a metric of the form

(10) |

From the Schwarzschild metric in Eq. (4), . Thus, we get the temperature

(11) |

and when we substitute the value of from Eq. (8) we get Cavaglia et al. (2003)

(12) |

Since , the temperature goes to infinity as . Figure 2 is a plot of this equation for and , with the generic values , , and ; different values lead to the same qualitative behavior.

The black hole entropy can be calculated from the first law of black hole thermodynamics leading to

(13) |

which goes to zero as .

The specific heat capacity is calculated from the relation

(14) |

By differentiating the temperature from Eq. (12) with respect to we get

(15) |

The emission rate (the energy radiated per unit time) can be calculated from the temperature using the Stefan-Boltzmann law assuming the energy loss is dominated by photons. In -dimensional brane the emission rate of a black body with temperature and surface area is given by Emparan et al. (2000)

(16) |

where is the Stefan-Boltzmann constant in dimensions. Since black holes are radiating mainly on the brane Emparan et al. (2000), so using as in Cavaglia et al. (2003), and since and from Eq. (12) we get that

(17) |

The exact form can be found in Emparan et al. (2000); Cavaglia et al. (2003).

From the relations Eq. (12), (13), (15), and (17), we see that when the black hole evaporates and its mass goes to zero, the temperature and emission rate go to infinity, while the entropy and heat capacity vanish. This means that the black hole reaches a stage of catastrophic evaporation as the black hole mass approaches zero, and this definitely needs a resolution. This problem has been tackled in Adler et al. (2001), and it has been resolved by considering the generalized uncertainty principle Amati et al. (1989) instead of the standard uncertainty principle, and in this picture, black holes end at a remnant that does not exchange hawking radiation with the surroundings. Similar conclusion was obtained by one of the authors in Ali (2014), in which it was studied the thermodynamics of Schwarzschild black holes in the context of gravity’s rainbow, and it was found that the rainbow black hole ends at a remnant at which the specific heat vanishes and hence the catastrophic behavior is again resolved but this time in the context of gravity’s rainbow. In the next section, we shall extend this study into extra dimensions to investigate the phenomenological implications on the productions of black holes at TeV scales.

## 3 Schwarzschild Black Holes in Gravity’s Rainbow

In this section, we will analyze the Schwarzschild black hole in higher dimensions using gravity’s rainbow. The four dimensional Schwarzschild black hole has been analyzed in gravity’s rainbow Ali (2014), and it was found that a remnant forms. In this section, we extend this analysis into higher dimensional Schwarzschild black holes. In gravity’s rainbow, the geometry of spacetime depends on the energy of the particle used to probe it, and so, the rainbow modified metric can be written as Magueijo and Smolin (2004)

(18) |

The energy dependence of the frame fields can be written as

(19) |

where the tilde quantities refer to the energy independent frame fields. So, we can write the modified Schwarzschild metric as Magueijo and Smolin (2004); Liu (2014)

(20) |

where and are the rainbow functions used in the MDR given in Eq. (1).

Thus, the modified temperature can be calculated from Eq. (9) with the change and leading to

(21) |

where we used the rainbow functions from Eq. (3). According to Adler et al. (2001); Cavaglia et al. (2003); Medved and Vagenas (2004); Amelino-Camelia et al. (2004), the uncertainty principle can be translated to a lower bound on the energy of a particle emitted in Hawking radiation, and the value of the uncertainty in position can be taken to be the event horizon radius. Hence,

(22) |

The temperature becomes

(23) |

where we used in natural units.

From Eq. (3), it is clear that the temperature goes to zero at , and below this value the temperature has no physical meaning. This minimum horizon radius corresponds to the minimum mass

(24) |

This implies that the black hole ends in a remnant. Figure 2 is a plot of Eq. (3) for and .

The entropy can be calculated from the first law of black hole thermodynamics using the modified temperature from Eq. (3)

(25) |

This integral cannot be evaluated exactly for general and , but taking as an example and we get

(26) |

which is the same as the expression derived in Ali (2014). Taking as another example and we get

(27) |

The heat capacity can be calculated from Eq. (14) with the modified temperature in Eq. (3), and we get

(28) |

Figures 4 and 4 are plots of the heat capacity for and respectively. We see that the modified heat capacity diverges at a value where the temperature is maximum, then goes to zero at the minimum mass given by Eq. (24). The zero value of the heat capacity means the black hole cannot exchange heat with the surrounding space, and hence predicting the existence of a remnant.

The emission rate is proportional to , which means that from the modified temperature in Eq. (3), the modified emission rate is

(29) |

which also goes to zero at .

From the calculations in this section, we conclude that in gravity’s rainbow black holes reach a remnant near the Planck scale. In the next section, we investigate whether black hole remnants can be detected in the LHC.

## 4 Black Hole Production at Colliders

In the last section, we found that in gravity’s rainbow, black holes end up in a remnant with the mass in Eq. (24), which we reproduce here for convenience,

(30) |

From this minimum mass, we can calculate the minimum energy needed to form black holes in a collider, such as the LHC. In the ADD model Arkani-Hamed et al. (1998), the reduced Planck constant in extra dimensions is related to the 4D Planck mass GeV via

(31) |

where is the size of the compactified extra dimensions. Fixing at around the electroweak scale TeV, and using Eq. (31), we obtain Beringer et al. (2012). Thus, is clearly ruled out, but not .

When we use the latest experimental limits on from Ref. Chatrchyan et al. (2012a), and assume that the rainbow parameter , we obtain the results given in Table 1. We see that in , black holes can form only at energies not less than TeV, and in the minimal mass is TeV. This energy scale is larger than the energy scale of the current runs of the LHC, which explains why they were not detected in the LHC. Previous work based on theories with large extra dimensions predicted the possibility of forming black holes at energy scales of a few TeVs Giddings and Thomas (2002); Dimopoulos and Landsberg (2001); Emparan et al. (2000); Cavaglia et al. (2003), which has not been experimentally observed at the Compact Muon Solenoid (CMS) detector in LHC where experiments are excluding semiclassical and quantum black holes with masses below to TeV Chatrchyan et al. (2012a, b). We also note that our results may ameliorate the ranges of masses of black holes that has been predicted in the earlier work in Fig. (2) in Dimopoulos and Landsberg (2001) that gave a wide range between around 1.5 TeV and 10 TeV.

By considering our proposed approach of studying black holes in the context of gravity’s rainbow, we may justify why higher energy scales are needed to form black holes. Furthermore, this energy scale will be accessible in the near future. If black holes were produced in future colliders, it will need a collision center-of-mass energy greater than the minimal mass. The emitted radiation from the evaporation will be smaller than the standard case (Eq. (29)), and the emission will stop when the black hole reaches the remnant mass. This will lead to the detection of a missing energy of the order of the remnant mass.

The total cross section of a collision that produces a black hole can be estimated by Dimopoulos and Landsberg (2001)

(32) |

and the differential cross section

(33) |

The maximum number of expected events per second is given by

(34) |

For the LHC, the luminosity , and the total center of mass energy is currently 7 TeV, but can be increased up to 14 TeV in future runs.

[TeV] | [TeV] | [pb] | [pb/100 GeV] | [events/s] | |
---|---|---|---|---|---|

6 | 4.54 | 9.5 | 59.4 | 0.42 | 0.59 |

7 | 3.51 | 10.8 | 99.4 | 0.46 | 0.99 |

8 | 2.98 | 11.8 | 137.8 | 0.47 | 1.38 |

9 | 2.71 | 12.3 | 166.7 | 0.45 | 1.67 |

10 | 2.51 | 11.9 | 194.3 | 0.47 | 1.94 |

Table 1 includes the estimated cross section, differential cross section, and the maximum number of expected events per second. For comparison, the cross section of the Higgs boson is approximately fb, and the number of events per second is . This means that for a collision with energies higher than the remnant mass of the black holes, the production of black holes could be more than that of the Higgs.

However, the values of the cross section in Table 1 will decease if one takes into account that only a fraction of the energy in a collision is achieved in a parton-parton scattering Dimopoulos and Landsberg (2001). In addition, the minimal mass is sensitive to the value of the parameter . For example, for and , TeV. Also, for and , TeV. Thus, to determine the expected number of produced black holes accurately, we need better constraints on the parameter from other experiments Ali and Khalil (2014), and simulate the production and decay of black hole remnants as was done in Bellagamba et al. (2012); Alberghi et al. (2013).

## 5 Bounds on

In the previous section, we used the value to calculate the expected mass of the remnant. We could do the reverse and constrain the value of the parameter from the measurements of no black holes at LHC up to 5.3 TeV Aad et al. (2014). From Eq. (24), ,

(35) |

which constrains by

(36) |

6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|

0.68 | 0.70 | 0.73 | 0.76 | 0.79 |

Table 2 shows the bounds on in different dimensions, and fig 5 is a plot for the minimal mass vs . To our knowledge, the best upper bound on in the context of gravity’s rainbow is , but can be reduced by 4 orders in the next few years from tests of the weak equivalence principle Ali and Khalil (2014). Combining these two bounds supports the assumption that .

## 6 Conclusions

In this paper, we have analyzed higher dimensional Schwarzschild black holes in gravity’s rainbow. It was expected that black holes will be detected at LHC if large extra dimensions existed. This was because the existence of extra dimensions would lower the effective Planck mass to TeV scale (i.e LHC energy scale). The absence of any black hole at LHC could thus be interpreted as the absence of large extra dimensions, at least at the energy scale of the LHC. However, we argued that black holes were not detected due to Planckian deformation of quantum gravity, which was not taken into account. As the effective Planck scale was reduced due to the existence of large extra dimensions, it is important that these effects are taken into account. When we did that using gravity’s rainbow, we found that the energy needed to form black holes is larger than the energy scale of the LHC, but is within reach of the next particle colliders.

It may be noted that such a suppression was predicted in the framework of generalized uncertainty principle in Cavaglia et al. (2003); Ali (2012); Hossenfelder (2004). The fact that the generalized uncertainty principle can lead to a deformed dispersion relation suggests that this might be a general feature of theories with modified dispersion relation. It would be interesting to analyze this relation in more details. It is worth mentioning, suppression on black hole masses at Tetra scale was studied in non-commutative geometry Nicolini and Winstanley (2011); Mureika et al. (2012). Useful reviews on the remnant of black holes in the framework of noncommutative geometry can be found in Nicolini (2009); Bleicher and Nicolini (2014).

Apart from this phenomenological result, it was demonstrated that a black hole remnant will form for higher dimensional Schwarzschild black holes. Such a remnant forms for a four dimensional Schwarzschild black hole Ali (2014). In fact, recently it was demonstrated that a remnant also forms for black rings Ali et al. (2014). These are strong indications that a remnant might form for all black objects, in gravity’s rainbow. It will be appropriate to extend the investigation into dark matter, cosmological constant, etc in the context of gravity’s rainbow. We hope to report on these in the future.

### Acknowledgments

The research of AFA is supported by Benha University (www.bu.edu.eg) and CFP in Zewail City.

## References

- Arkani-Hamed et al. (1998) N. Arkani-Hamed, S. Dimopoulos, G. Dvali, The Hierarchy problem and new dimensions at a millimeter, Phys.Lett. B429 (1998) 263–272.
- Banks and Fischler (1999) T. Banks, W. Fischler, A Model for high-energy scattering in quantum gravity (1999).
- Giddings and Thomas (2002) S. B. Giddings, S. D. Thomas, High-energy colliders as black hole factories: The End of short distance physics, Phys.Rev. D65 (2002) 056010.
- Dimopoulos and Landsberg (2001) S. Dimopoulos, G. L. Landsberg, Black holes at the LHC, Phys.Rev.Lett. 87 (2001) 161602.
- Emparan et al. (2000) R. Emparan, G. T. Horowitz, R. C. Myers, Black holes radiate mainly on the brane, Phys.Rev.Lett. 85 (2000) 499–502.
- Meade and Randall (2008) P. Meade, L. Randall, Black Holes and Quantum Gravity at the LHC, JHEP 0805 (2008) 003.
- Antoniadis et al. (1998) I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, New dimensions at a millimeter to a Fermi and superstrings at a TeV, Phys.Lett. B436 (1998) 257–263.
- da Rocha and Coimbra-Araujo (2006) R. da Rocha, C. H. Coimbra-Araujo, Extra dimensions in LHC via mini-black holes: Effective Kerr-Newman brane-world effects, Phys.Rev. D74 (2006) 055006.
- Chatrchyan et al. (2012a) S. Chatrchyan, et al. (CMS Collaboration), Search for dark matter and large extra dimensions in monojet events in collisions at TeV, JHEP 1209 (2012a) 094.
- Chatrchyan et al. (2012b) S. Chatrchyan, et al. (CMS Collaboration), Search for microscopic black holes in collisions at TeV, JHEP 1204 (2012b) 061.
- ’t Hooft (1996) G. ’t Hooft, Quantization of point particles in (2+1)-dimensional gravity and space-time discreteness, Class.Quant.Grav. 13 (1996) 1023–1040.
- Kostelecky and Samuel (1989) V. A. Kostelecky, S. Samuel, Spontaneous Breaking of Lorentz Symmetry in String Theory, Phys.Rev. D39 (1989) 683.
- Amelino-Camelia et al. (1998) G. Amelino-Camelia, J. R. Ellis, N. Mavromatos, D. V. Nanopoulos, S. Sarkar, Tests of quantum gravity from observations of gamma-ray bursts, Nature 393 (1998) 763–765.
- Gambini and Pullin (1999) R. Gambini, J. Pullin, Nonstandard optics from quantum space-time, Phys.Rev. D59 (1999) 124021.
- Carroll et al. (2001) S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane, T. Okamoto, Noncommutative field theory and Lorentz violation, Phys.Rev.Lett. 87 (2001) 141601.
- Horava (2009a) P. Horava, Quantum Gravity at a Lifshitz Point, Phys.Rev. D79 (2009a) 084008.
- Horava (2009b) P. Horava, Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point, Phys.Rev.Lett. 102 (2009b) 161301.
- Amelino-Camelia (2002) G. Amelino-Camelia, Relativity in space-times with short distance structure governed by an observer independent (Planckian) length scale, Int.J.Mod.Phys. D11 (2002) 35–60.
- Magueijo and Smolin (2002) J. Magueijo, L. Smolin, Lorentz invariance with an invariant energy scale, Phys.Rev.Lett. 88 (2002) 190403.
- Amati et al. (1989) D. Amati, M. Ciafaloni, G. Veneziano, Can Space-Time Be Probed Below the String Size?, Phys.Lett. B216 (1989) 41.
- Garay (1995) L. J. Garay, Quantum gravity and minimum length, Int.J.Mod.Phys. A10 (1995) 145–166.
- Ali et al. (2009) A. F. Ali, S. Das, E. C. Vagenas, Discreteness of Space from the Generalized Uncertainty Principle, Phys.Lett. B678 (2009) 497–499.
- Ali et al. (2011) A. F. Ali, S. Das, E. C. Vagenas, A proposal for testing Quantum Gravity in the lab, Phys.Rev. D84 (2011) 044013.
- Magueijo and Smolin (2004) J. Magueijo, L. Smolin, Gravity’s rainbow, Class.Quant.Grav. 21 (2004) 1725–1736.
- Galan and Mena Marugan (2004) P. Galan, G. A. Mena Marugan, Quantum time uncertainty in a gravity’s rainbow formalism, Phys.Rev. D70 (2004) 124003.
- Hackett (2006) J. Hackett, Asymptotic flatness in rainbow gravity, Class.Quant.Grav. 23 (2006) 3833–3842.
- Garattini and Mandanici (2012) R. Garattini, G. Mandanici, Particle propagation and effective space-time in Gravity’s Rainbow, Phys.Rev. D85 (2012) 023507.
- Garattini and Majumder (2014) R. Garattini, B. Majumder, Naked Singularities are not Singular in Distorted Gravity, Nucl.Phys. B884 (2014) 125–141.
- Garattini (2013) R. Garattini, Distorting General Relativity: Gravity’s Rainbow and f(R) theories at work, JCAP 1306 (2013) 017.
- Garattini and Majumder (2014) R. Garattini, B. Majumder, Electric Charges and Magnetic Monopoles in Gravity’s Rainbow, Nucl.Phys. B883 (2014) 598–614.
- Leiva et al. (2009) C. Leiva, J. Saavedra, J. Villanueva, The Geodesic Structure of the Schwarzschild Black Holes in Gravity’s Rainbow, Mod.Phys.Lett. A24 (2009) 1443–1451.
- Li et al. (2009) H. Li, Y. Ling, X. Han, Modified (A)dS Schwarzschild black holes in Rainbow spacetime, Class.Quant.Grav. 26 (2009) 065004.
- Ali et al. (2014) A. F. Ali, M. Faizal, B. Majumder, Absence of an Effective Horizon for Black Holes in Gravity’s Rainbow (2014).
- Awad et al. (2013) A. Awad, A. F. Ali, B. Majumder, Nonsingular Rainbow Universes, JCAP 1310 (2013) 052.
- Barrow and Magueijo (2013) J. D. Barrow, J. Magueijo, Intermediate inflation from rainbow gravity, Phys.Rev. D88 (2013) 103525.
- Liu and Zhu (2008) C.-Z. Liu, J.-Y. Zhu, Hawking radiation and black hole entropy in a gravity ’ rainbow, Gen.Rel.Grav. 40 (2008) 1899–1911.
- Ali and Khalil (2014) A. F. Ali, M. M. Khalil, A Proposal for Testing Gravity’s Rainbow (2014).
- Gim and Kim (2014) Y. Gim, W. Kim, Thermodynamic phase transition in the rainbow Schwarzschild black hole, JCAP 10 (2014) 003.
- Amelino-Camelia et al. (1997) G. Amelino-Camelia, J. R. Ellis, N. Mavromatos, D. V. Nanopoulos, Distance measurement and wave dispersion in a Liouville string approach to quantum gravity, Int.J.Mod.Phys. A12 (1997) 607–624.
- Amelino-Camelia et al. (1998) G. Amelino-Camelia, J. R. Ellis, N. Mavromatos, D. V. Nanopoulos, S. Sarkar, Tests of quantum gravity from observations of gamma-ray bursts, Nature 393 (1998) 763–765.
- Amelino-Camelia (2013) G. Amelino-Camelia, Quantum-Spacetime Phenomenology, Living Rev.Rel. 16 (2013) 5.
- Amelino-Camelia and Piran (2001) G. Amelino-Camelia, T. Piran, Planck scale deformation of Lorentz symmetry as a solution to the UHECR and the TeV gamma paradoxes, Phys.Rev. D64 (2001) 036005.
- Kifune (1999) T. Kifune, Invariance violation extends the cosmic ray horizon?, Astrophys.J. 518 (1999) L21–L24.
- Protheroe and Meyer (2000) R. Protheroe, H. Meyer, An Infrared background TeV gamma-ray crisis?, Phys.Lett. B493 (2000) 1–6.
- Aloisio et al. (2000) R. Aloisio, P. Blasi, P. L. Ghia, A. F. Grillo, Probing the structure of space-time with cosmic rays, Phys.Rev. D62 (2000) 053010.
- Myers and Pospelov (2003) R. C. Myers, M. Pospelov, Ultraviolet modifications of dispersion relations in effective field theory, Phys.Rev.Lett. 90 (2003) 211601.
- Ali (2014) A. F. Ali, Black Hole Remnant from Gravity’s Rainbow, Phys.Rev. D89 (2014) 104040.
- Emparan and Reall (2008) R. Emparan, H. S. Reall, Black Holes in Higher Dimensions, Living Rev.Rel. 11 (2008) 6.
- Aman and Pidokrajt (2006) J. E. Aman, N. Pidokrajt, Geometry of higher-dimensional black hole thermodynamics, Phys.Rev. D73 (2006) 024017.
- Angheben et al. (2005) M. Angheben, M. Nadalini, L. Vanzo, S. Zerbini, Hawking radiation as tunneling for extremal and rotating black holes, JHEP 0505 (2005) 014.
- Cavaglia et al. (2003) M. Cavaglia, S. Das, R. Maartens, Will we observe black holes at LHC?, Class.Quant.Grav. 20 (2003) L205–L212.
- Adler et al. (2001) R. J. Adler, P. Chen, D. I. Santiago, The Generalized uncertainty principle and black hole remnants, Gen.Rel.Grav. 33 (2001) 2101–2108.
- Liu (2014) C.-Z. Liu, Charged Particle‘s Tunneling in a Modified Reissner-Nordstrom Black Hole, Int.J.Theor.Phys. 53 (2014) 60–71.
- Medved and Vagenas (2004) A. Medved, E. C. Vagenas, When conceptual worlds collide: The GUP and the BH entropy, Phys.Rev. D70 (2004) 124021.
- Amelino-Camelia et al. (2004) G. Amelino-Camelia, M. Arzano, A. Procaccini, Severe constraints on loop-quantum-gravity energy-momentum dispersion relation from black-hole area-entropy law, Phys.Rev. D70 (2004) 107501.
- Beringer et al. (2012) J. Beringer, et al. (Particle Data Group), Review of Particle Physics (RPP), Phys.Rev. D86 (2012) 010001.
- Bellagamba et al. (2012) L. Bellagamba, R. Casadio, R. Di Sipio, V. Viventi, Black Hole Remnants at the LHC, Eur.Phys.J. C72 (2012) 1957.
- Alberghi et al. (2013) G. Alberghi, L. Bellagamba, X. Calmet, R. Casadio, O. Micu, Charged Black Hole Remnants at the LHC, Eur.Phys.J. C73 (2013) 2448.
- Aad et al. (2014) G. Aad, et al. (ATLAS Collaboration), Search for Quantum Black Hole Production in High-Invariant-Mass LeptonJet Final States Using Collisions at 8ââTeV and the ATLAS Detector, Phys.Rev.Lett. 112 (2014) 091804.
- Ali (2012) A. F. Ali, No Existence of Black Holes at LHC Due to Minimal Length in Quantum Gravity, JHEP 1209 (2012) 067.
- Hossenfelder (2004) S. Hossenfelder, Suppressed black hole production from minimal length, Phys.Lett. B598 (2004) 92–98.
- Nicolini and Winstanley (2011) P. Nicolini, E. Winstanley, Hawking emission from quantum gravity black holes, JHEP 1111 (2011) 075.
- Mureika et al. (2012) J. Mureika, P. Nicolini, E. Spallucci, Could any black holes be produced at the LHC?, Phys.Rev. D85 (2012) 106007.
- Nicolini (2009) P. Nicolini, Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review, Int.J.Mod.Phys. A24 (2009) 1229–1308.
- Bleicher and Nicolini (2014) M. Bleicher, P. Nicolini, Mini-review on mini-black holes from the mini-Big Bang, Astron. Nachr. 335 (2014) 605–611.
- Ali et al. (2014) A. F. Ali, M. Faizal, M. M. Khalil, Remnants of Black Rings from Gravity’s Rainbow (2014).