Status of special relativity

See also: Special relativity

Scope of special relativity

Special relativity is only accurate when gravitational effects are negligible or very weak, otherwise it must be replaced by general relativity. At very small scales, such as at the Planck length and below, it is also possible that special relativity breaks down, due to the effects of quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is now universally accepted by the physics community and experimental results which appear to contradict it are widely believed to be due to unreproducible experimental error.

Dependence on definition of units

Because of the freedom one has to select how one defines units of length and time in physics, it is possible to make one of the two postulates of relativity a tautological consequence of the definitions, but one cannot do this for both postulates simultaneously, as when combined they have consequences which are independent of one's choice of definition of length and time. For instance, if one defines units of length and time in terms of a physical object (e.g. by defining units of time in terms of transitions of a caesium atom, or length in terms of wavelengths of a krypton atom) then it becomes tautological that the law determining that unit of length or time will be the same in all reference frames, but then the invariance of c is non-trivial. Conversely, if one defines units of length and time in such a way that c is defined to be constant, then the second postulate becomes tautological, but the first one does not; for instance, if the length unit is defined in terms of the time unit and a predetermined fixed value of c, then there is no a priori reason why the number of wavelengths of krypton in a unit of length will be the same in all reference frames (or even in all orientations).

Experimental evidence

A number of experiments support special relativity when compared against other theories. These include:

• The Michelson-Morley experiment disproved the possibility of ether drift, and tested the directional invariance of the speed of light
• The Hamar experiment - obstruction of ether flow
• The Trouton-Noble experiment - torque on a capacitor
• The Kennedy-Thorndike experiment - time contraction
• Experiments to test emitter theory demonstrated that the speed of light is independent of the speed of the emitter.

The Sagnac effect, which is a phenomenon which is taken in to account in GPS synchronisation procedures, is an effect which applies equally in both special relativity and Galilean relativity and is thus neutral evidence as far as special relativity is concerned.

Mathematical consistency

At a purely mathematical level, special relativity is internally consistent, being nothing more than the geometry of Minkowski space, together with a requirement that all laws of nature be Lorentz-invariant. Thus it is not possible to create a thought-experiment within special relativity that creates a genuine logical paradox, unless one assumes the existence of objects which do not seem to exist in special relativity (see below). However it is certainly possible to create thought-experiments which have unintuitive consequences, and which contradict some other theories.

Theories incompatible with relativity

Special relativity is not compatible with the physical existence of the following objects, forces, or laws (except in the nonrelativistic limit in which all speeds are much less than c):

1. Infinitely rigid rods, or any other object which transmits forces at infinite speeds. Note that this would require the existence of a new force which is not currently explained by any of the laws of physics.
2. Tachyon particles, unless these particles cannot transmit any information at superluminal speeds, or are somehow not subject to the laws of cause and effect.
3. Rulers which are immune to Lorentz contraction. Again, this would require a new force not currently explained by the laws of physics.
4. Devices which can record absolute position . Note that the existence of such devices would also contradict Galilean relativity.
5. Clocks that are immune to time dilation. Again, this would require a new force not currently explained by the laws of physics.
6. Clocks which can record absolute time. Indeed, the concept of absolute time is philosophically inconsistent with Einstein's interpretation of special relativity.
7. Forces which can act instantaneously at a distance; this includes Newton's law of gravity and Coulomb's law of electrostatics. Note however that these two laws can be modified (to general relativity and Maxwell's equations respectively) in a manner consistent with or generalizing the theory of special relativity. There are also some laws of physics which act non-locally but do not transmit information at superluminal speeds, and which are thus technically (if not philosophically) consistent with special relativity; the primary example here is the collapse of the wave function.
8. Laws of nature which are Galilean invariant instead of Lorentz invariant, or which are not invariant under either of these two transformations.
9. The Newtonian velocity addition law v = v₁ + v₂; this law is replaced by the relativistic addition law.
10. The Newtonian linear relationship p = mv between momentum and velocity, and the Newtonian quadratic relationship between kinetic energy and velocity. These should be replaced by the equations p = Ev / c² and E² = m²c⁴ + p²c². Similarly, Newton's second law in the form F = ma is no longer valid, but must be replaced by F = dp / dt (which is in fact closer to Newton's original formulation of this law).
11. The Schrodinger equation, which is the quantization of non-relativistic equation E = p² / 2m + V from Newtonian mechanics. This can be replaced by the Dirac equation, Klein-Gordon equation, or quantum field theory.
12. Nonrelativistic fluid equations such as the Euler equations and Navier-Stokes equations; these must be replaced by relativistic fluid equations such as the Relativistic Euler equations.
13. Additivity of mass; the total mass of a system (as determined by solving the equation E² = m²c⁴ + p²c², where E is the total energy and p is the total momentum) is not necessarily equal to the sum of the masses of its components, just as the length of a sum of vectors is not necessarily equal to the sum of the lengths of the individual vectors. Indeed there is a triangle inequality which says that the total mass is always greater than or equal to the sum of the individual masses. However, the total mass of a system remains conserved (this is a consequence of conservation of energy and momentum).
14. Conservation of particle number is compatible with relativity, but once quantum mechanics is also added, it is possible that this conservation law breaks down, leading to spontaneous particle creation and annihilation. This phenomenon is usually studied within the framework of quantum field theory.
15. Wormholes or other objects which affect the topology of spacetime. However, these objects can be compatible with general relativity.

Theories compatible with relativity

Special relativity is compatible with

1. Translation invariance , rotation invariance, time reversal symmetry, and reflection symmetry of the laws of physics. Indeed, special relativity generalizes and unifies these symmetries via the principle of Lorentz invariance.
2. The non-relativistic Doppler shift law, which works fine if time dilation is accounted for; the combination equals "relativistic Doppler".
3. Maxwell's equations of electromagnetism. Conversely, Maxwell's equations combined with the first postulate of special relativity can be used to deduce the second postulate. Actually electromagnetism is greatly simplified by relativity, as magnetism is simply the relativistic effect obtained when the simple law of electrostatics is put into a relativistic Universe.
4. The Lorentz force law in electromagnetism, subject to the caveats concerning Newton's second law mentioned earlier. It is also worth noting that Maxwell's equations, combined with the Lorentz force law, can also be used to mathematically demonstrate several consequences of special relativity such as Lorentz contraction and time dilation, at least for rulers and clocks which operate via electromagnetic forces.
5. Newton's first law and Newton's third law are still compatible with special relativity, though as mentioned earlier all forces must now act locally instead of at a distance (and are most likely mediated via fields with finite speed of propagation).
6. Classical Yang-Mills theory, which generalizes Maxwell's equations and which govern the classical theory of the weak and strong nuclear forces. Indeed, as with Maxwell's equations, one could use the Yang-Mills equations to deduce the second postulate of special relativity from the first, and can also demonstrate relativistic effects such as Lorentz contraction and time dilation for rulers and clocks that operate via nuclear forces (e.g. atomic clocks). Quantum Yang-Mills theory is of course a special case of quantum field theory.
7. In addition to Maxwell's equations and Yang-Mills equations, related equations such as the wave equation, Dirac equation, Klein-Gordon equation, and Yang-Mills-Higgs equation are also compatible with special relativity. See Relativistic wave equations for further discussion.
8. Quantum mechanics, though as mentioned above Schrodinger's equation must now be replaced by another equation. One can view quantum field theory as the natural unification of special relativity with quantum mechanics. However, if one assumes both special relativity and quantum mechanics then one is forced to abandon local hidden variable theories, unless one is willing to adopt interpretations of quantum mechanics such as the many-worlds interpretation; see Bell's theorem for more discussion. For similar reasons the concept of the collapse of a wave function becomes problematic in relativity, though the difficulties are more aesthetic than fundamental in nature. The unification of general relativity and quantum mechanics is a notoriously difficult problem which has not yet been resolved satisfactorily; see quantum gravity.
9. General relativity collapses to special relativity in the limit when the strength of the gravitational field tends to zero.
10. Hamiltonian mechanics, though the Hamiltonian system often has to incorporate not only point particles, but also the fields which mediate the forces between these particles.
11. Conservation laws, such as conservation of mass, energy, momentum, angular momentum and charge. (See however the earlier note about failure of additivity of mass). This can be viewed as a consequence of Noether's theorem from Hamiltonian mechanics. Conservation of particle number is not covered by Noether's theorem and can break down in relativity.
12. Lagrangian mechanics (the principle of least action), although as with Hamiltonian mechanics, the Lagrangian system often needs to incorporate fields as well as particles. Also, the Lagrangian of a point particle often needs to be written using proper time instead of absolute time or time in a co-ordinate frame.
13. Relativistic quantum chemistry

As these examples show, special relativity affects all aspects of physics, and is not purely concerned with light. Indeed, in a very literal sense, light is merely the most visible phenomenon in physics which involves the constant c.

Alternatives to special relativity

1. emitter theory: largely abandoned nowadays
2. ether drag theories: also largely abandoned, due to null results such as the Michelson-Morley experiment
3. Lorentz ether theory : This theory gives the same predictions in any experimental setting as special relativity, however the two theories are ontologically and philosophically very different, and they suggest different ways to extend or modify the theory, for instance to understand gravity.