**Doubly-special relativity** — 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 minimum length (the Plank length).

This was first proposed in a paper by Giovanni Amelino-Camelia, though it is at least implicit in a paper by Paul Merriam. An alternate approach to doubly-special relativity theory, inspired by that of Amelino-Camelia, was proposed later by João Magueijo and Lee Smolin. There exist proposals that these theories may be related to loop quantum gravity.

One of the motivations for this work is the observation of high-energy cosmic rays that appear to violate the Greisen-Zatsepin-Kuzmin limit: the so-called Oh-My-God particles.

The theory is highly speculative as of first publishing in 2002. The theory is built using a well-established approach in theoretical physics named invariance under transformation, which is colloquially (even in science) called relativistic. Nevertheless the theory is not considered a promising approach by a majority of members of the high-energy physics community.

DSR is based upon a generalization of symmetry to quantum groups. The Poincaré symmetry of ordinary special relativity is deformed into some noncommutative symmetry and Minkowski space is deformed into some noncommutative space. This theory is not a violation of Poincaré symmetry as much as a deformation of it and this symmetry is exact. This deformation is scale dependent in the sense that the deformation is huge at the Planck scale but negligible at much larger length scales. Many models which are significantly Lorentz violating at the Planck scale are also significantly Lorentz violating in the infrared limit because of nasty radiative corrections. Without any exact Lorentz symmetry to protect them, such Lorentz violating terms will be generated with abandon by quantum corrections. However, DSR models do not succumb to this difficulty since the deformed symmetry is exact and will protect the theory from unwanted radiative corrections — assuming the absence of quantum anomalies.

Jafari and Shariati have constructed canonical transformations that relate both the doubly-special relativity theories of Amelino-Camelia and of Magueijo and Smolin to ordinary special relativity. They claim that doubly-special relativity is therefore only a complicated set of coordinates for an old and simple theory. However, all theories are related to free theories by canonical transformations. Therefore supporters of doubly-special relativity may claim that while it is equivalent to ordinary relativity, the momentum and energy coordinates of doubly-special relativity are those that appear in the usual form of the standard model interactions. This implies that ordinary special relativity and doubly-special relativity make distinct physical predictions in high energy processes, and in particular the derivation of the Greisen-Zatsepin-Kuzmin limit is not valid if one asserts that quantum electrodynamics takes its usual Maxwell form only in the coordinate systems of doubly-special relativity.

## Literature

- Fabio Cardone[1], Roberto Mignani ,
*Energy and Geometry: An Introduction to Deformed Special Relativity*, World Scientific 2004, ISBN 981-238-728-5

## External references

- Giovanni Amelino-Camelia, “Doubly-Special Relativity: First Results and Key Open Problems.”
*Int. J. Mod. Phys.*D11 (2002) 1643. Online copy at http://arxiv.org/abs/gr-qc/0210063 - Giovanni Amelino-Camelia,
*Doubly-Special Relativity.’*http://lanl.arxiv.org/abs/gr-qc/0207049, a non-technical review - Nosratollah Jafari, Ahmad Shariati,
*Doubly Special Relativity: A New Relativity or Not?*http://arxiv.org/abs/gr-qc/0602075