Einstein's relativity, rebuilt so every observer agrees on one tiny length.
Amelino-Camelia's Doubly Special Relativity
The Planck length joins the speed of light as a quantity that every inertial observer measures the same.
Placeholder for a 3D visualisation of Deformed Special Relativity. The interactive scene will land in Phase 3. Special relativity rests on one quantity that every observer agrees on, the speed of light. Deformed special relativity, also called doubly special relativity, adds a second one at the Planck scale, a length or an energy. Under ordinary Lorentz transformations, moving observers would measure such a scale differently, which seemed to demand a preferred frame. DSR avoids that by changing how boosts act on energy and momentum at very high energies. Every inertial observer stays equal, and Einstein's theory returns at everyday energies. The later relative-locality programme goes further. It treats momentum space as curved and makes the spacetime each observer reconstructs depend on the energy used to probe it.
§1 · The claim, in one sentence
The rules of special relativity can be rewritten so that all inertial observers agree on a Planck-scale length, without picking out any preferred frame.
§2 · Why it might be true
Special relativity has one speed that every observer measures the same, the speed of light. Various theoretical arguments suggest that quantum gravity makes the Planck length physically special as well. Under ordinary Lorentz transformations, observers moving at different speeds would measure that length differently. A fixed Planck length therefore seemed to require a preferred class of observers.
Amelino-Camelia showed in 2002 that this expectation is wrong. He reformulated the relativity postulates so that the Planck length is observer-independent, like the speed of light. The price is that boosts act on energy and momentum in a modified, non-linear way. Every inertial observer then agrees on a deformed relation between a particle's energy, momentum and mass.
One concrete realization uses the κ-Poincaré algebra. There, boosts can never push a particle's momentum past the inverse of the Planck length. The limit works like the speed of light in Einstein's theory, because no change of observer can cross it.
The family stance
Spacetime keeps a relativity principle down to the Planck scale, but its rules are deformed there. All inertial observers agree on the speed of light and on a Planck-scale length or energy, and none of them is preferred. What that Planck scale means mathematically varies between formulations, and none is established by experiment. In a later geometric version, relative locality, the arena every observer shares is phase space. Each observer builds a spacetime from it, and distant observers can disagree about whether an interaction happened at one point.
§2.5 · Evidence
- The 2002 paper showed that the relativity postulates stay consistent with an observer-independent length scale, removing the main argument for a preferred frame.
- Bruno, Amelino-Camelia and Kowalski-Glikman derived explicit finite boosts from the κ-Poincaré algebra in 2001, giving the idea a concrete mathematical form.
- Amelino-Camelia, Ellis, Mavromatos, Nanopoulos and Sarkar showed in 1998 that gamma-ray burst timing could reach sensitivity near the Planck energy, so such effects are testable.
§3 · What you'd need to test it
- All inertial observers agree on a Planck-scale length as well as on the speed of light, and no preferred frame exists
- Energy, momentum and mass obey a deformed relation with Planck-length corrections, and every observer finds the same relation
- In particular κ-Poincaré realizations, boosts saturate at a maximum momentum equal to the inverse Planck length
- If the deformation makes photon speed depend on energy, high-energy gamma rays from distant bursts arrive at different times than low-energy ones
§4 · Where it breaks
- The 2002 paper proves consistency for certain types of deformation only, so the general case is still open.
- Fermi's 2009 limit from GRB 090510 disfavours versions in which photon speed changes linearly with energy at the Planck scale.
- The framework allows several deformations. Amelino-Camelia's own 2010 review describes DSR scenarios with and without a modified dispersion relation, so one null result rules out only some versions.
Go deeper
The 2002 paper keeps the relativity principle and adds the Planck length L_p as a second observer-independent scale. All inertial observers then agree on a dispersion relation E^2 - c^2 p^2 - c^4 m^2 + f(E, p, m; L_p) = 0, at least for certain functions f. Judes and Visser later labelled this construction DSR1.
Bruno, Amelino-Camelia and Kowalski-Glikman built finite boosts from the Lorentz sector of the κ-Poincaré Hopf algebra in the bicrossproduct basis. With the deformation parameter set to the Planck length, the boosts saturate at momentum 1/L_p. Kowalski-Glikman's 2005 lectures link this structure to non-commutative κ-Minkowski spacetime (compare Ch. 4, Non-Commutative Geometry).
The gamma-ray burst test is older than DSR. The 1998 Nature paper proposed that photon arrival times could reveal velocity differences of order E/E_QG. In DSR the modified dispersion relation holds for every observer, while in preferred-frame models (compare Hořava-Lifshitz gravity in this chapter) it holds in one frame. Rosati and collaborators showed in 2015 that the standard Jacob-Piran ansatz for travel times in an expanding universe assumes translations are unaffected by the Planck scale. In DSR models that assumption can fail, with significant quantitative effects.
▸§5 · Who built it, and when(7 sources, 7 established)
- EstablishedAmelino-Camelia (2002). 'Relativity in space-times with short distance structure governed by an observer independent (Planckian) length scale.' Int. J. Mod. Phys. D 11, 35
- EstablishedBruno, Amelino-Camelia & Kowalski-Glikman (2001). 'Deformed boost transformations that saturate at the Planck scale.' Phys. Lett. B 522, 133
- EstablishedAmelino-Camelia, Ellis, Mavromatos, Nanopoulos & Sarkar (1998). 'Potential sensitivity of gamma-ray burster observations to wave dispersion in vacuo.' Nature 393, 763
- EstablishedFermi GBM/LAT Collaborations (2009). 'A limit on the variation of the speed of light arising from quantum gravity effects.' Nature 462, 331
- EstablishedAmelino-Camelia (2010). 'Doubly-Special Relativity: Facts, Myths and Some Key Open Issues.' Symmetry 2, 230
- EstablishedKowalski-Glikman (2005). 'Introduction to doubly special relativity.' Lect. Notes Phys. 669, 131
- EstablishedRosati, Amelino-Camelia, Marciano & Matassa (2015). 'Planck-scale-modified dispersion relations in FRW spacetime.' Phys. Rev. D 92, 124042
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