Formulations of special relativity
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The theory of special relativity was initially developed in 1905 by Albert Einstein. However, other interpretations of special relativity have been developed, some on the basis of different foundational axioms. While some are mathematically equivalent to Einstein's theory, others aim to revise or extend it.
Einstein's formulation was based on two postulates, as detailed below. Some formulations modify these postulates or attempt to derive the second postulate by deduction. Others differ in their approach to the geometry of
Einstein's two postulates
As formulated by Albert Einstein in 1905, the theory of special relativity was based on two main postulates:
- The inertial frame(a frame of reference that is not accelerating in any direction).
- The speed of light is constant: In all inertial frames, the speed of light c is the same whether the light is emitted from a source at rest or in motion. (Note that this does not apply in non-inertial frames, indeed between accelerating frames the speed of light cannot be constant.[1] Although it can be applied in non-inertial frames if an observer is confined to making local measurements.[2])
Einstein developed the theory of special relativity based on these two postulates. This theory made many predictions which have been experimentally verified, including the relativity of simultaneity, length contraction, time dilation, the relativistic velocity addition formula, the relativistic Doppler effect, relativistic mass, a universal speed limit, mass–energy equivalence, the speed of causality and the Thomas precession.[3][4]
Single-postulate approaches
Several physicists have derived a theory of special relativity from only the first postulate - the principle of relativity - without assuming the second postulate that the speed of light is constant.[1][5][6][7] The term "single-postulate" is misleading because these formulations may rely on unsaid assumptions such as the cosmological principle, that is, the isotropy and homogeneity of space.[8][9] As such, the term does not refer to the exact number of postulates, but is rather used to distinguish such approaches from the "two-postulate" formulation. Single postulate approaches generally deduce, rather than assume, that the speed of light is constant.
Without assuming the second postulate, the Lorentz transformations can be obtained. However, there is a free parameter k, which renders it incapable of making experimental predictions unless further assumptions are made. The case k = 0 is equivalent to Newtonian physics.[10]
Lorentz ether theory
Hendrik Lorentz and Henri Poincaré developed their version of special relativity in a series of papers from about 1900 to 1905. They used Maxwell's equations and the principle of relativity to deduce a theory that is mathematically equivalent to the theory later developed by Einstein.
Taiji relativity
Taiji relativity is a formulation of special relativity developed by Jong-Ping Hsu and Leonardo Hsu.[1][11][12][13] The name of the theory, Taiji, is a Chinese word which refers to ultimate principles which predate the existence of the world. Hsu and Hsu claimed that measuring time in units of distance allowed them to develop a theory of relativity without using the second postulate in their derivation.
It is the principle of relativity, that Hsu & Hsu say, when applied to 4d spacetime, implies the invariance of the 4d-spacetime interval . The difference between this and the spacetime interval in Minkowski space is that is invariant purely by the principle of relativity whereas requires both postulates. The "principle of relativity" in spacetime is taken to mean invariance of laws under 4-dimensional transformations. They show that there are versions of relativity which are consistent with experiment but have a definition of time where the "speed" of light is not constant. They develop one such version called common relativity which is more convenient for performing calculations for "relativistic many body problems" than using special relativity.
Several authors have made the case that Taiji relativity still assumes a further postulate - the cosmological principle that time and space look the same in all directions.[14] Behara (2003) wrote that "the postulation on the speed of light in special relativity is an inevitable consequence of the relativity principle taken in conjunction with the idea of the homogeneity and isotropy of space and the homogeneity of time in all inertial frames".[15]
Test theories of special relativity
Test theories of special relativity are
Geometric formulations
Minkowski spacetime
Minkowski space (or Minkowski spacetime) is a mathematical setting in which special relativity is conveniently formulated. Minkowski space is named for the German mathematician Hermann Minkowski, who around 1907 realized that the theory of special relativity (previously developed by Poincaré and Einstein) could be elegantly described using a four-dimensional spacetime, which combines the dimension of time with the three dimensions of space.
Mathematically, there are a number of ways in which the four-dimensions of Minkowski spacetime are commonly represented: as a four-vector with 4 real coordinates, as a four-vector with 3 real and one complex coordinate, or using tensors.
Spacetime algebra
Spacetime algebra is a type of geometric algebra that is closely related to Minkowski space, and is equivalent to other formalisms of special relativity. It uses mathematical objects such as bivectors to replace tensors in traditional formalisms of Minkowski spacetime, leading to much simpler equations than in matrix mechanics or vector calculus.
de Sitter relativity
According to the works of Cacciatori, Gorini, Kamenshchik,
Euclidean relativity
Euclidean relativity[18][19][20]
[21]
Very special relativity
Ignoring
have demonstrated that a small subgroup of the Lorentz group is sufficient to explain all the current bounds.The minimal
Doubly special relativity
Doubly special relativity (DSR) is a modified theory of
The motivation to these proposals is mainly theoretical, based on the following observation: the
A drawback of the usual doubly special relativity models is that they are valid only at the energy scales where ordinary special relativity is supposed to break down, giving rise to a patchwork relativity. On the other hand,
See also
- Alternative derivations of special relativity
- Derivations of the Lorentz transformations
- History of special relativity
Notes
- ^ The Minkowski metric describes four-dimensional space-time: the coordinates are time and three spatial dimensions. The Euclidean metric describes four-dimensional Euclidean space: it has four spatial coordinates.
References
- ^ a b c
Hsu, J.-P.; Hsu, L. (2006). A Broader View of Relativity. ISBN 981-256-651-1.
- ISBN 978-0-321-85656-2.
- ISBN 0-471-30932-X.
- ^ von Ignatowsky, W. (1911). "Das Relativitätsprinzip". Archiv der Mathematik und Physik (in German). 17: 1.
- ^
Feigenbaum, M. J. (2008). "The Theory of Relativity - Galileo's Child". arXiv:0806.1234 [physics.class-ph].
- ^ a b
Cacciatori, S.; Gorini, V.; Kamenshchik, A. (2008). "Special relativity in the 21st century". S2CID 119191753.
- S2CID 4064705.
- ^ Einstein, A. (1921). Morgan document.[full citation needed]
- ISSN 1355-2198.
- ^
Hsu, J.-P.; Hsu, L. (1994). "A physical theory based solely on the first postulate of relativity". .
- Erratum Hsu, Jong-Ping; Hsu, Leonardo (1996). "A physical theory based solely on the first postulate of relativity (Physics Letters a 196 (1994)1)".
- ^
Hsu, J.-P.; Hsu, L. (2008). "Experimental tests of a new Lorentz-invariant dynamics based solely on the first postulate of relativity". S2CID 120483040.
- ^
Hsu, J.-P.; Hsu, L. (2008). "Four-dimensional symmetry of taiji relativity and coordinate transformations based on a weaker postulate for the speed of light". S2CID 119831503.
- ^
Ai, Xiao-Bai (1996). "On the Basis of Taiji Relativity". S2CID 250777204.
- ^
Behera, H. (2003). "A comment on the Invariance of the Speed of Light". Bibcode:2003physics...4087B.
- ^
Zhang, Y.-Z. (1997). Special Relativity and Its Experimental Foundations. ISBN 978-981-02-2749-4.
- ^ Bacry, H.; Lévy-Leblond, J.-M. (1968). "Possible Kinematics". .
- ^ Yamashita, Takuya (May 2023). "Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space". Preprints. .
- ^
Hans, H. (2001). "Proper time formulation of relativistic dynamics". S2CID 117357649.
- ^
Gersten, A. (2003). "Euclidean special relativity". S2CID 15496801.
- ^ van Linden, R. F. J. (2006). "Minkowski versus Euclidean 4-vectors" (PDF).
- ^
Crabbe, A. (2004). "Alternative conventions and geometry for Special Relativity" (PDF). Annales de la Fondation Louis de Broglie. 29 (4): 589–608.
- ^
Almeida, J. (2001). "An alternative to Minkowski space-time". arXiv:gr-qc/0104029.
- ^ "Euclidean relativity portal". 28 September 2012. Retrieved 23 July 2014.
- ^
Fontana, G. (2005). "The Four Space‐times Model of Reality". S2CID 118189976.
- ^
Cohen, Andrew G.; Glashow, Sheldon L. (2006). "Very special relativity". S2CID 11056484.