# Doubly special relativity

Doubly special relativity[1][2][3] (DSR) – also called deformed special relativity or, by some, extra-special relativity – is a modified theory of special relativity in which there is not only an observer-independent maximum velocity (the speed of light), but an observer-independent maximum energy scale and minimum length scale (the Planck energy and Planck length).[4]

## History

First attempts to modify special relativity by introducing an observer-independent length were made by Pavlopoulos (1967), who estimated this length at about 10−15 metres.[5][6] In the context of quantum gravity, Giovanni Amelino-Camelia (2000) introduced what now is called doubly special relativity, by proposing a specific realization of preserving invariance of the Planck length 16.162×10−36 m.[7][8] This was reformulated by Kowalski-Glikman (2001) in terms of an observer-independent Planck mass.[9] A different model, inspired by that of Amelino-Camelia, was proposed in 2001 by João Magueijo and Lee Smolin, who also focused on the invariance of Planck energy.[10][11]

It was realized that there are indeed three kinds of deformation of special relativity that allow one to achieve an invariance of the Planck energy, either as a maximum energy, as a maximal momentum, or both. DSR models are possibly related to loop quantum gravity in 2+1 dimensions (two space, one time), and it has been conjectured that a relation also exists in 3+1 dimensions.[12][13]

The motivation to these proposals is mainly theoretical, based on the following observation: The Planck energy is expected to play a fundamental role in a theory of quantum gravity, setting the scale at which quantum gravity effects cannot be neglected and new phenomena might become important. If special relativity is to hold up exactly to this scale, different observers would observe quantum gravity effects at different scales, due to the Lorentz–FitzGerald contraction, in contradiction to the principle that all inertial observers should be able to describe phenomena by the same physical laws. This motivation has been criticized on the grounds that the result of a Lorentz transformation does not itself constitute an observable phenomenon.[14] DSR also suffers from several inconsistencies in formulation that have yet to be resolved.[15][16] Most notably it is difficult to recover the standard transformation behavior for macroscopic bodies, known as the soccer-ball-problem. The other conceptual difficulty is that DSR is a priori formulated in momentum space. There is as yet no consistent formulation of the model in position space.

There are many other Lorentz-violating models in which, contrary to DSR, the principle of relativity and Lorentz invariance is violated by introducing preferred frame effects. Examples are the effective field theory of Sidney Coleman and Sheldon Lee Glashow, and especially the Standard-Model Extension which provides a general framework for Lorentz violations. These models are capable of giving precise predictions in order to assess possible Lorentz violation, and thus are widely used in analyzing experiments concerning the standard model and special relativity (see Modern searches for Lorentz violation).

## Predictions

Experiments to date have not observed contradictions to special relativity (see Modern searches for Lorentz violation).

It was initially speculated that ordinary special relativity and doubly special relativity would make distinct physical predictions in high-energy processes, and in particular the derivation of the GZK limit on energies of cosmic rays from distant sources would not be valid. However, it is now established that standard doubly special relativity does not predict any suppression of the GZK cutoff, contrary to the models where an absolute local rest frame exists, such as effective field theories like the Standard-Model Extension.

Since DSR generically (though not necessarily) implies an energy-dependence of the speed of light, it has further been predicted that, if there are modifications to first order in energy over the Planck mass, this energy-dependence would be observable in high energetic photons reaching Earth from distant gamma ray bursts. Depending on whether the now energy-dependent speed of light increases or decreases with energy (a model-dependent feature) highly energetic photons would be faster or slower than the lower energetic ones.[17] However, the Fermi-LAT experiment in 2009 measured a 31 GeV photon, which nearly simultaneously arrived with other photons from the same burst, which excluded such dispersion effects even above the Planck energy.[18] It has, moreover, been argued that DSR with an energy-dependent speed of light is inconsistent and first order effects are ruled out already because they would lead to non-local particle interactions that would long have been observed in particle physics experiments.[19]

## de Sitter relativity

Since the de Sitter group naturally incorporates an invariant length parameter, de Sitter relativity can be interpreted as an example of doubly special relativity, because de Sitter spacetime incorporates invariant velocity, as well as length parameter. There is a fundamental difference, though: whereas in all doubly special relativity models the Lorentz symmetry is violated, in de Sitter relativity it remains as a physical symmetry. A drawback of the usual doubly special relativity models is that they are valid only at the energy scales where ordinary special relativity is supposed to break down, giving rise to a patchwork relativity. On the other hand, de Sitter relativity is found to be invariant under a simultaneous re-scaling of mass, energy and momentum, and is consequently valid at all energy scales.

## References

1. ^ Amelino-Camelia, Giovanni (1 November 2009). "Doubly-Special Relativity: Facts, Myths and Some Key Open Issues". Recent Developments in Theoretical Physics. Statistical Science and Interdisciplinary Research. 9. pp. 123–170. arXiv:1003.3942. doi:10.1142/9789814287333_0006. ISBN 978-981-4287-32-6.
2. ^ Amelino-Camelia, Giovanni (1 July 2002). "Doubly Special Relativity". Nature. 418 (6893): 34–35. arXiv:gr-qc/0207049. Bibcode:2002Natur.418...34A. doi:10.1038/418034a. PMID 12097897.
3. ^ Nyambuya, Golden G. (30 August 2012). "On a New Position Space Doubly Special Relativity Theory". Prespacetime Journal – via ResearchGate.
4. ^ Amelino-Camelia, G. (2010). "Doubly-Special Relativity: Facts, Myths and Some Key Open Issues". Symmetry. 2 (4): 230–271. arXiv:1003.3942. Bibcode:2010rdtp.book..123A. doi:10.3390/sym2010230.
5. ^ Pavlopoulos, T. G. (1967). "Breakdown of Lorentz Invariance". Physical Review. 159 (5): 1106–1110. Bibcode:1967PhRv..159.1106P. doi:10.1103/PhysRev.159.1106.
6. ^ Pavlopoulos, T. G. (2005). "Are we observing Lorentz violation in gamma ray bursts?". Physics Letters B. 625 (1–2): 13–18. arXiv:astro-ph/0508294. Bibcode:2005PhLB..625...13P. doi:10.1016/j.physletb.2005.08.064.
7. ^ Amelino-Camelia, G. (2001). "Testable scenario for relativity with minimum length". Physics Letters B. 510 (1–4): 255–263. arXiv:hep-th/0012238. Bibcode:2001PhLB..510..255A. doi:10.1016/S0370-2693(01)00506-8.
8. ^ Amelino-Camelia, G. (2002). "Relativity in space–times with short-distance structure governed by an observer-independent (Planckian) length scale". International Journal of Modern Physics D. 11 (1): 35–59. arXiv:gr-qc/0012051. Bibcode:2002IJMPD..11...35A. doi:10.1142/S0218271802001330.
9. ^ Kowalski-Glikman, J. (2001). "Observer-independent quantum of mass". Physics Letters A. 286 (6): 391–394. arXiv:hep-th/0102098. Bibcode:2001PhLA..286..391K. doi:10.1016/S0375-9601(01)00465-0.
10. ^ Magueijo, J.; Smolin, L (2001). "Lorentz invariance with an invariant energy scale". Physical Review Letters. 88 (19): 190403. arXiv:hep-th/0112090. Bibcode:2002PhRvL..88s0403M. doi:10.1103/PhysRevLett.88.190403. PMID 12005620.
11. ^ Magueijo, J.; Smolin, L (2003). "Generalized Lorentz invariance with an invariant energy scale". Physical Review D. 67 (4): 044017. arXiv:gr-qc/0207085. Bibcode:2003PhRvD..67d4017M. doi:10.1103/PhysRevD.67.044017.
12. ^ Amelino-Camelia, Giovanni; Smolin, Lee; Starodubtsev, Artem (2004). "Quantum symmetry, the cosmological constant and Planck-scale phenomenology". Classical and Quantum Gravity. 21 (13): 3095–3110. arXiv:hep-th/0306134. Bibcode:2004CQGra..21.3095A. doi:10.1088/0264-9381/21/13/002.
13. ^ Freidel, Laurent; Kowalski-Glikman, Jerzy; Smolin, Lee (2004). "2+1 gravity and doubly special relativity". Physical Review D. 69 (4): 044001. arXiv:hep-th/0307085. Bibcode:2004PhRvD..69d4001F. doi:10.1103/PhysRevD.69.044001.
14. ^ Hossenfelder, S. (2006). "Interpretation of Quantum Field Theories with a Minimal Length Scale". Physical Review D. 73 (10): 105013. arXiv:hep-th/0603032. Bibcode:2006PhRvD..73j5013H. doi:10.1103/PhysRevD.73.105013.
15. ^ Aloisio, R.; Galante, A.; Grillo, A.F.; Luzio, E.; Mendez, F. (2004). "Approaching Space Time Through Velocity in Doubly Special Relativity". Physical Review D. 70 (12): 125012. arXiv:gr-qc/0410020. Bibcode:2004PhRvD..70l5012A. doi:10.1103/PhysRevD.70.125012.
16. ^ Aloisio, R.; Galante, A.; Grillo, A.F.; Luzio, E.; Mendez, F. (2005). "A note on DSR-like approach to space–time". Physics Letters B. 610: 101–106. arXiv:gr-qc/0501079. Bibcode:2005PhLB..610..101A. doi:10.1016/j.physletb.2005.01.090.
17. ^ Amelino-Camelia, G.; Smolin, L. (2009). "Prospects for constraining quantum gravity dispersion with near term observations". Physical Review D. 80 (8): 084017. arXiv:0906.3731. Bibcode:2009PhRvD..80h4017A. doi:10.1103/PhysRevD.80.084017.
18. ^ Fermi LAT Collaboration (2009). "A limit on the variation of the speed of light arising from quantum gravity effects". Nature. 462 (7271): 331–334. arXiv:0908.1832. Bibcode:2009Natur.462..331A. doi:10.1038/nature08574. PMID 19865083.
19. ^ Hossenfelder, S. (2009). "The Box-Problem in Deformed Special Relativity". arXiv:0912.0090 [gr-qc].