A version of relativity where one very high energy looks the same to everyone.
Magueijo-Smolin Doubly Special Relativity
The Planck energy becomes an invariant, built in by adding a rescaling to every Lorentz boost.
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
A modified special relativity in which the Planck energy joins the speed of light as an invariant, while all inertial frames stay equivalent and Einstein's theory returns at low energies.
§2 · Why it might be true
Magueijo and Smolin proposed in 2002 a second route to the same goal as Amelino-Camelia. Here the extra invariant is an energy, which may be the Planck energy. All inertial frames remain fully equivalent, and the theory agrees with Einstein's at everyday energies.
The construction changes how a boost acts on energy and momentum. Each boost is combined with a rescaling, a dilatation, tuned so that the Planck energy never changes. The symmetry is still the Lorentz group with its usual algebra, but it now acts non-linearly on momentum space.
Picture a camera that refocuses each time it moves, so one chosen object always appears the same size. Each boost carries its own small refocusing, chosen so that the Planck energy stays fixed for every observer.
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 algebra keeps unmodified structure constants, so the symmetry remains the Lorentz group and only its action on energy and momentum changes.
- Freidel, Kowalski-Glikman and Smolin showed in 2004 that gravity in 2+1 dimensions coupled to point particles is a nontrivial example of DSR.
- The 2003 follow-up gave a general formalism for non-linear Lorentz transformations on momentum space, with generalized energy and momentum conservation.
- Judes and Visser derived exact energy-momentum rules for composite systems in both this model and Amelino-Camelia's in 2003.
§3 · What you'd need to test it
- Every inertial observer measures the same Planck-scale energy, alongside the same speed of light
- Physics agrees with special relativity at energies far below the invariant scale
- In some versions of the 2003 generalization, elementary particles have a maximum momentum or energy and the speed of light varies with energy
- Field theory and the equivalence principle are modified, with consequences for how the theory might embed in general relativity
§4 · Where it breaks
- The 2003 generalization admits many examples with different maximum energies and photon speeds, so the framework alone does not fix one prediction.
- Versions with a speed of light that changes at first order in energy over the Planck energy face the Fermi and LHAASO gamma-ray burst limits.
- The 2002 paper only suggests a modified equivalence principle, so a full gravitational version of the model was left for later work.
Go deeper
The 2002 model modifies the action of the Lorentz group on momentum space by adding a dilatation to each boost, so that the Planck energy E_p is invariant. The structure constants of the algebra are unmodified. The authors point out the similarity with a transformation proposed earlier by Fock. Judes and Visser call this model DSR2 and Amelino-Camelia's DSR1, and in both the Lorentz group acts non-linearly on energy and momentum, so these stop being additive.
The 2003 Phys. Rev. D paper generalizes to any non-linear action of Lorentz transformations on momentum space, and proposes rules for composite systems. It also proposes identifying spatial coordinates with the generators of translations in Minkowski spacetime. In some examples this yields a commutative geometry with an energy-dependent Planck constant.
Magueijo also proposed, with Albrecht in 1999, varying-speed-of-light cosmology (Ch. 1, Alternatives to Inflation). That theory lets c change over cosmic time to solve the horizon problem. The Magueijo-Smolin model is a separate theory: c stays invariant at low energies, and an invariant energy scale is added.
▸§5 · Who built it, and when(7 sources, 7 established)
- EstablishedMagueijo & Smolin (2002). 'Lorentz invariance with an invariant energy scale.' Phys. Rev. Lett. 88, 190403
- EstablishedMagueijo & Smolin (2003). 'Generalized Lorentz invariance with an invariant energy scale.' Phys. Rev. D 67, 044017
- EstablishedJudes & Visser (2003). 'Conservation laws in Doubly special relativity.' Phys. Rev. D 68, 045001
- EstablishedFreidel, Kowalski-Glikman & Smolin (2004). '2+1 gravity and doubly special relativity.' Phys. Rev. D 69, 044001
- EstablishedHossenfelder (2014). 'The Soccer-Ball Problem.' SIGMA 10, 074
- EstablishedVasileiou et al. (2013). 'Constraints on Lorentz Invariance Violation from Fermi-Large Area Telescope Observations of Gamma-Ray Bursts.' Phys. Rev. D 87, 122001
- EstablishedLHAASO Collaboration (2024). 'Stringent Tests of Lorentz Invariance Violation from LHAASO Observations of GRB 221009A.' Phys. Rev. Lett. 133, 071501
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