Special relativity

In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 treatment, the theory is presented as being based on just two postulates:^{[p 1]}^[1]^[2]

The first postulate was first formulated by Galileo Galilei (see Galilean invariance).

The – the laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.^{[p 1]}

principle of relativity

The principle of invariant light speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" (from the preface). That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source.

[p 1]

Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light in vacuum and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:^{[p 1]}

The constancy of the speed of light was motivated by Maxwell's theory of electromagnetism^[13] and the lack of evidence for the luminiferous ether.^[14] There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment.^[15]^[16] In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance.

The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.^{[p 6]}

Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations.^[17] But the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the principle of relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is:

Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations. Einstein later derived these transformations from his axioms.

Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.^{[p 7]}

The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′.

Frame S′ moves, for simplicity, in a single direction: the x-direction of frame S with a constant velocity v as measured in frame S.

The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.

Eq. 1: $\Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .$

Eq. 2: $\Delta x=x_{2}-x_{1}\ ,\ \ \Delta t=t_{2}-t_{1}\ .$

Δs² > 0: In this case, the two events are separated by more time than space, and they are hence said to be timelike separated. This implies that $|\Delta x/\Delta t|<c,$ and given the Lorentz transformation $\Delta x'=\gamma \ (\Delta x-v\,\Delta t),$ it is evident that there exists a $v$ less than $c$ for which $\Delta x'=0$ (in particular, $v=\Delta x/\Delta t$ ). In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place. In this frame, the separation in time, $\Delta s/c,$ is called the proper time.

Δs² < 0: In this case, the two events are separated by more space than time, and they are hence said to be spacelike separated. This implies that $|\Delta x/\Delta t|>c,$ and given the Lorentz transformation $\Delta t'=\gamma \ (\Delta t-v\Delta x/c^{2}),$ there exists a $v$ less than $c$ for which $\Delta t'=0$ (in particular, $v=c^{2}\Delta t/\Delta x$ ). In other words, given two events that are spacelike separated, it is possible to find a frame in which the two events happen at the same time. In this frame, the separation in space, ${\textstyle {\sqrt {-\Delta s^{2}}},}$ is called the proper distance, or proper length. For values of $v$ greater than and less than $c^{2}\Delta t/\Delta x,$ the sign of $\Delta t'$ changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. But the temporal order of timelike-separated events is absolute, since the only way that $v$ could be greater than $c^{2}\Delta t/\Delta x$ would be if $v>c.$

Δs² = 0: In this case, the two events are said to be lightlike separated. This implies that $|\Delta x/\Delta t|=c,$ and this relationship is frame independent due to the invariance of $s^{2}.$ From this, we observe that the speed of light is $c$ in every inertial frame. In other words, starting from the assumption of universal Lorentz covariance, the constant speed of light is a derived result, rather than a postulate as in the two-postulates formulation of the special theory.

$\beta =v/c$ and

$\gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}$ is the .

Lorentz factor

Theories of relativity and quantum mechanics[edit]

Special relativity can be combined with quantum mechanics to form relativistic quantum mechanics and quantum electrodynamics. How general relativity and quantum mechanics can be unified is one of the unsolved problems in physics; quantum gravity and a "theory of everything", which require a unification including general relativity too, are active and ongoing areas in theoretical research.

The early Bohr–Sommerfeld atomic model explained the fine structure of alkali metal atoms using both special relativity and the preliminary knowledge on quantum mechanics of the time.^[81]

In 1928, Paul Dirac constructed an influential relativistic wave equation, now known as the Dirac equation in his honour,^{[p 25]} that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation not only described the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron),^{[p 25]}^{[p 26]} and fine structure could only be fully explained with special relativity. It was the first foundation of relativistic quantum mechanics.

On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions. Instead, a theory of particles interpreted as quantized fields, called quantum field theory, becomes necessary; in which particles can be created and destroyed throughout space and time.

The (1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities.

Fizeau experiment

The famous Michelson–Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer were travelling together at the same velocity at all times.

The (1903) showed that the torque on a capacitor is independent of position and inertial reference frame.

Trouton–Noble experiment

The (1902, 1904) showed that length contraction does not lead to birefringence for a co-moving observer, in accordance with the relativity principle.

Experiments of Rayleigh and Brace

Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c² in the region of interest.^[82] In a strong gravitational field, one must use general relativity. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10⁻²⁰)^[83] and thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.

Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields).

Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) – thus Newtonian mechanics can be considered as a special relativity of slow moving bodies. See classical mechanics for a more detailed discussion.

Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905,^[84] and historians have concluded that Einstein was at least aware of the Michelson–Morley experiment as early as 1899 despite claims he made in his later years that it played no role in his development of the theory.^[16]

Particle accelerators accelerate and measure the properties of particles moving at near the speed of light, where their behavior is consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. These machines would simply not work if they were not engineered according to relativistic principles. In addition, a considerable number of modern experiments have been conducted to test special relativity. Some examples:

Max Planck

Hermann Minkowski

Max von Laue

Arnold Sommerfeld

Max Born

Mileva Marić

Einstein, Albert (1920). .

Relativity: The Special and General Theory

Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. 1-56731-136-9

ISBN

Logunov, Anatoly A. (2005). (transl. from Russian by G. Pontocorvo and V. O. Soloviev, edited by V. A. Petrov). Nauka, Moscow.

Henri Poincaré and the Relativity Theory

Einstein's original work in German, Annalen der Physik, Bern 1905

Zur Elektrodynamik bewegter Körper

English Translation as published in the 1923 book The Principle of Relativity.