fbpx
Wikipedia

De Sitter invariant special relativity

January 01, 1970

In mathematical physics, de Sitter invariant special relativity is the speculative idea that the fundamental symmetry group of spacetime is the indefinite orthogonal group SO(4,1), that of de Sitter space. In the standard theory of general relativity, de Sitter space is a highly symmetrical special vacuum solution, which requires a cosmological constant or the stress–energy of a constant scalar field to sustain.

The idea of de Sitter invariant relativity is to require that the laws of physics are not fundamentally invariant under the Poincaré group of special relativity, but under the symmetry group of de Sitter space instead. With this assumption, empty space automatically has de Sitter symmetry, and what would normally be called the cosmological constant in general relativity becomes a fundamental dimensional parameter describing the symmetry structure of spacetime.

First proposed by Luigi Fantappiè in 1954, the theory remained obscure until it was rediscovered in 1968 by Henri Bacry and Jean-Marc Lévy-Leblond. In 1972, Freeman Dyson popularized it as a hypothetical road by which mathematicians could have guessed part of the structure of general relativity before it was discovered.[1] The discovery of the accelerating expansion of the universe has led to a revival of interest in de Sitter invariant theories, in conjunction with other speculative proposals for new physics, like doubly special relativity.

Introduction edit

Main article: de Sitter space

De Sitter suggested that spacetime curvature might not be due solely to gravity[2] but he did not give any mathematical details of how this could be accomplished. In 1968 Henri Bacry and Jean-Marc Lévy-Leblond showed that the de Sitter group was the most general group compatible with isotropy, homogeneity and boost invariance.[3] Later, Freeman Dyson[1] advocated this as an approach to making the mathematical structure of general relativity more self-evident.

Minkowski's unification of space and time within special relativity replaces the Galilean group of Newtonian mechanics with the Lorentz group. This is called a unification of space and time because the Lorentz group is simple, while the Galilean group is a semi-direct product of rotations and Galilean boosts. This means that the Lorentz group mixes up space and time such that they cannot be disentangled, while the Galilean group treats time as a parameter with different units of measurement than space.

An analogous thing can be made to happen with the ordinary rotation group in three dimensions. If you imagine a nearly flat world, one in which pancake-like creatures wander around on a pancake flat world, their conventional unit of height might be the micrometre (μm), since that is how high typical structures are in their world, while their unit of distance could be the metre, because that is their body's horizontal extent. Such creatures would describe the basic symmetry of their world as SO(2), being the known rotations in the horizontal (x–y) plane. Later on, they might discover rotations around the x- and y-axes—and in their everyday experience such rotations might always be by an infinitesimal angle, so that these rotations would effectively commute with each other.

The rotations around the horizontal axes would tilt objects by an infinitesimal amount. The tilt in the x–z plane (the "x-tilt") would be one parameter, and the tilt in the y–z plane (the "y-tilt") another. The symmetry group of this pancake world is then SO(2) semidirect product with R2, meaning that a two-dimensional rotation plus two extra parameters, the x-tilt and the y-tilt. The reason it is a semidirect product is that, when you rotate, the x-tilt and the y-tilt rotate into each other, since they form a vector and not two scalars. In this world, the difference in height between two objects at the same x, y would be a rotationally invariant quantity unrelated to length and width. The z-coordinate is effectively separate from x and y.

Eventually, experiments at large angles would convince the creatures that the symmetry of the world is SO(3). Then they would understand that z is really the same as x and y, since they can be mixed up by rotations. The SO(2) semidirect product R2 limit would be understood as the limit that the free parameter μ, the ratio of the height range μm to the length range m, approaches 0. The Lorentz group is analogous—it is a simple group that turns into the Galilean group when the time range is made long compared to the space range, or where velocities may be regarded as infinitesimal, or equivalently, may be regarded as the limit c → ∞, where relativistic effects become observable "as good as at infinite velocity".

The symmetry group of special relativity is not entirely simple, due to translations. The Lorentz group is the set of the transformations that keep the origin fixed, but translations are not included. The full Poincaré group is the semi-direct product of translations with the Lorentz group. If translations are to be similar to elements of the Lorentz group, then as boosts are non-commutative, translations would also be non-commutative.

In the pancake world, this would manifest if the creatures were living on an enormous sphere rather than on a plane. In this case, when they wander around their sphere, they would eventually come to realize that translations are not entirely separate from rotations, because if they move around on the surface of a sphere, when they come back to where they started, they find that they have been rotated by the holonomy of parallel transport on the sphere. If the universe is the same everywhere (homogeneous) and there are no preferred directions (isotropic), then there are not many options for the symmetry group: they either live on a flat plane, or on a sphere with a constant positive curvature, or on a Lobachevski plane with constant negative curvature. If they are not living on the plane, they can describe positions using dimensionless angles, the same parameters that describe rotations, so that translations and rotations are nominally unified.

In relativity, if translations mix up nontrivially with rotations, but the universe is still homogeneous and isotropic, the only option is that spacetime has a uniform scalar curvature. If the curvature is positive, the analog of the sphere case for the two-dimensional creatures, the spacetime is de Sitter space and its symmetry group is the de Sitter group rather than the Poincaré group.

De Sitter special relativity postulates that the empty space has de Sitter symmetry as a fundamental law of nature. This means that spacetime is slightly curved even in the absence of matter or energy. This residual curvature implies a positive cosmological constant Λ to be determined by observation. Due to the small magnitude of the constant, special relativity with its Poincaré group is indistinguishable from de Sitter space for most practical purposes.

Modern proponents of this idea, such as S. Cacciatori, V. Gorini and A. Kamenshchik,[4] have reinterpreted this theory as physics, not just mathematics. They postulate that the acceleration of the expansion of the universe is not entirely due to vacuum energy, but at least partly due to the kinematics of the de Sitter group, which would replace the Poincaré group.

A modification of this idea allows   to change with time, so that inflation may come from the cosmological constant being larger near the Big Bang than nowadays. It can also be viewed as a different approach to the problem of quantum gravity.[5]

High energy edit

The Poincaré group contracts to the Galilean group for low-velocity kinematics, meaning that when all velocities are small the Poincaré group "morphs" into the Galilean group. (This can be made precise with İnönü and Wigner's concept of group contraction.[6])

Similarly, the de Sitter group contracts to the Poincaré group for short-distance kinematics, when the magnitudes of all translations considered are very small compared to the de Sitter radius.[5] In quantum mechanics, short distances are probed by high energies, so that for energies above a very small value related to the cosmological constant, the Poincaré group is a good approximation to the de Sitter group.

In de Sitter relativity, the cosmological constant is no longer a free parameter of the same type; it is determined by the de Sitter radius, a fundamental quantity that determines the commutation relation of translation with rotations/boosts. This means that the theory of de Sitter relativity might be able to provide insight on the value of the cosmological constant, perhaps explaining the cosmic coincidence. Unfortunately, the de Sitter radius, which determines the cosmological constant, is an adjustable parameter in de Sitter relativity, so the theory requires a separate condition to determine its value in relation to the measurement scale.

When a cosmological constant is viewed as a kinematic parameter, the definitions of energy and momentum must be changed from those of special relativity. These changes could significantly modify the physics of the early universe if the cosmological constant was greater back then. Some speculate that a high energy experiment could modify the local structure of spacetime from Minkowski space to de Sitter space with a large cosmological constant for a short period of time, and this might eventually be tested in the existing or planned particle collider.[7]

Doubly special relativity edit

Main article: Doubly special relativity

Since the de Sitter group naturally incorporates an invariant length parameter, de Sitter relativity can be interpreted as an example of the so-called doubly special relativity. There is a fundamental difference, though: whereas in all doubly special relativity models the Lorentz symmetry is violated, in de Sitter relativity it remains as a physical symmetry.[8][9] 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, de Sitter relativity is found to be invariant under a simultaneous re-scaling of mass, energy and momentum,[10] and is consequently valid at all energy scales. A relationship between doubly special relativity, de Sitter space and general relativity is described by Derek Wise.[11] See also MacDowell–Mansouri action.

Newton–Hooke: de Sitter special relativity in the limit v ≪ c edit

In the limit as v ≪ c, the de Sitter group contracts to the Newton–Hooke group.[12] This has the effect that in the nonrelativistic limit, objects in de Sitter space have an extra "repulsion" from the origin: objects have a tendency to move away from the center with an outward pointing fictitious force proportional to their distance from the origin.

While it looks as though this might pick out a preferred point in space—the center of repulsion, it is more subtly isotropic. Moving to the uniformly accelerated frame of reference of an observer at another point, all accelerations appear to have a repulsion center at the new point.

What this means is that in a spacetime with non-vanishing curvature, gravity is modified from Newtonian gravity.[13] At distances comparable to the radius of the space, objects feel an additional linear repulsion from the center of coordinates.

History of de Sitter invariant special relativity edit

  • "de Sitter relativity" is the same as the theory of "projective relativity" of Luigi Fantappiè and Giuseppe Arcidiacono first published in 1954 by Fantappiè[14] and the same as another independent discovery in 1976.[15]
  • In 1968 Henri Bacry and Jean-Marc Lévy-Leblond published a paper on possible kinematics[3]
  • In 1972 Freeman Dyson[1] further explored this.
  • In 1973 Eliano Pessa described how Fantappié–Arcidiacono projective relativity relates to earlier conceptions of projective relativity and to Kaluza Klein theory.[16]
  • R. Aldrovandi, J.P. Beltrán Almeida and J.G. Pereira have used the terms "de Sitter special relativity" and "de Sitter relativity" starting from their 2007 paper "de Sitter special relativity".[10][17] This paper was based on previous work on amongst other things: the consequences of a non-vanishing cosmological constant,[18] on doubly special relativity[19] and on the Newton–Hooke group[3][20][21] and early work formulating special relativity with a de Sitter space[22][23][24]
  • In 2008 S. Cacciatori, V. Gorini and A. Kamenshchik[4] published a paper about the kinematics of de Sitter relativity.
  • Papers by other authors include: dSR and the fine structure constant;[25] dSR and dark energy;[26] dSR Hamiltonian Formalism;[27] and De Sitter Thermodynamics from Diamonds's Temperature,[28] Triply special relativity from six dimensions,[29] Deformed General Relativity and Torsion.[30]

Quantum de Sitter special relativity edit

There are quantized or quantum versions of de Sitter special relativity.[31][32]

Early work on formulating a quantum theory in a de Sitter space includes:[33][34][35][36][37][38][39]

See also edit

References edit

  1. ^ a b c Freeman Dyson (1972). "Missed opportunities" (pdf). Bull. Am. Math. Soc. 78 (5): 635–652. doi:10.1090/S0002-9904-1972-12971-9. MR 0522147.
  2. ^ W. de Sitter (1917). "On the curvature of space". Proc. Roy. Acad. Sci. Amsterdam. 20: 229–243.
  3. ^ a b c Henri Bacry; Jean-Marc Lévy-Leblond (1968). "Possible Kinematics". Journal of Mathematical Physics. 9 (10): 1605. Bibcode:1968JMP.....9.1605B. doi:10.1063/1.1664490.
  4. ^ a b S. Cacciatori; V. Gorini; A. Kamenshchik (2008). "Special Relativity in the 21st century". Annalen der Physik. 17 (9–10): 728–768. arXiv:0807.3009. Bibcode:2008AnP...520..728C. doi:10.1002/andp.200810321. S2CID 119191753.
  5. ^ a b R. Aldrovandi; J. G. Pereira (2009). "de Sitter Relativity: a New Road to Quantum Gravity?". Foundations of Physics. 39 (2): 1–19. arXiv:0711.2274. Bibcode:2009FoPh...39....1A. doi:10.1007/s10701-008-9258-5. S2CID 15298756.
  6. ^ E. Inönü; E.P. Wigner (1953). "On the Contraction of Groups and Their Representations". Proc. Natl. Acad. Sci. USA. 39 (6): 510–24. Bibcode:1953PNAS...39..510I. doi:10.1073/pnas.39.6.510. PMC 1063815. PMID 16589298.
  7. ^ Freydoon Mansouri (2002). "Non-Vanishing Cosmological Constant Λ, Phase Transitions, And Λ-Dependence Of High Energy Processes". Phys. Lett. B. 538 (3–4): 239–245. arXiv:hep-th/0203150. Bibcode:2002PhLB..538..239M. doi:10.1016/S0370-2693(02)02022-1. S2CID 13986319.
  8. ^ Aldrovandi, R.; Beltrán Almeida, J. P.; Pereira, J. G. (2007). "Some Implications of the Cosmological Constant to Fundamental Physics". AIP Conference Proceedings. 910: 381–395. arXiv:gr-qc/0702065. Bibcode:2007AIPC..910..381A. doi:10.1063/1.2752487. hdl:11449/69891. S2CID 16631274.
  9. ^ R. Aldrovandi; J.P. Beltran Almeida; C.S.O. Mayor; J.G. Pereira (2007). "Lorentz Transformations in de Sitter Relativity". arXiv:0709.3947 [gr-qc].
  10. ^ a b R. Aldrovandi; J.P. Beltrán Almeida; J.G. Pereira (2007). "de Sitter Special Relativity". Class. Quantum Grav. 24 (6): 1385–1404. arXiv:gr-qc/0606122. Bibcode:2007CQGra..24.1385A. doi:10.1088/0264-9381/24/6/002. S2CID 11703342.
  11. ^ Wise (2010). "MacDowell–Mansouri Gravity and Cartan Geometry". Classical and Quantum Gravity. 27 (15): 155010. arXiv:gr-qc/0611154. Bibcode:2010CQGra..27o5010W. doi:10.1088/0264-9381/27/15/155010. S2CID 16706599.
  12. ^ Aldrovandi; Barbosa; Crispino; Pereira (1999). "Non–Relativistic Spacetimes with Cosmological Constant". Classical and Quantum Gravity. 16 (2): 495–506. arXiv:gr-qc/9801100. Bibcode:1999CQGra..16..495A. CiteSeerX 10.1.1.339.919. doi:10.1088/0264-9381/16/2/013. S2CID 16691405.
  13. ^ Yu Tian; Han-Ying Guo; Chao-Guang Huang; Zhan Xu; Bin Zhou (2004). "Mechanics and Newton–Cartan-Like Gravity on the Newton–Hooke Space–time". Physical Review D. 71 (4): 44030. arXiv:hep-th/0411004. Bibcode:2005PhRvD..71d4030T. doi:10.1103/PhysRevD.71.044030. S2CID 119378100.
  14. ^ Licata, Ignazio; Leonardo Chiatti (2009). "The archaic universe: Big Bang, cosmological term, and the quantum origin of time in projective cosmology". International Journal of Theoretical Physics. 48 (4): 1003–1018. arXiv:0808.1339. Bibcode:2009IJTP...48.1003L. doi:10.1007/s10773-008-9874-z. S2CID 119262177.
  15. ^ Dey, Anind K. (2001). "An extension of the concept of inertial frame and of Lorentz transformation". Proc. Natl. Acad. Sci. USA. 73 (5): 1418–21. Bibcode:1976PNAS...73.1418K. doi:10.1073/pnas.73.5.1418. PMC 430307. PMID 16592318.
  16. ^ The De Sitter Universe and general relativity
  17. ^ R. Aldrovandi; J. G. Pereira (2009). "De Sitter Special Relativity: Effects on Cosmology". Gravitation and Cosmology. 15 (4): 287–294. arXiv:0812.3438. Bibcode:2009GrCo...15..287A. doi:10.1134/S020228930904001X. S2CID 18473868.
  18. ^ R. Aldrovandi; J.P. Beltran Almeida; J.G. Pereira (2004). "Cosmological Term and Fundamental Physics". Int. J. Mod. Phys. D. 13 (10): 2241–2248. arXiv:gr-qc/0405104. Bibcode:2004IJMPD..13.2241A. doi:10.1142/S0218271804006279. S2CID 118889785.
  19. ^ Giovanni Amelino-Camelia (2001). "Testable scenario for Relativity with minimum-length". Phys. Lett. B. 510 (1–4): 255–263. arXiv:hep-th/0012238. Bibcode:2001PhLB..510..255A. doi:10.1016/S0370-2693(01)00506-8. S2CID 119447462.
  20. ^ G.W. Gibbons; C.E. Patricot (2003). "Newton–Hooke spacetimes, Hpp-waves and the cosmological constant". Class. Quantum Grav. 20 (23): 5225. arXiv:hep-th/0308200. Bibcode:2003CQGra..20.5225G. doi:10.1088/0264-9381/20/23/016. S2CID 26557629.
  21. ^ Yu Tian; Han-Ying Guo; Chao-Guang Huang; Zhan Xu; Bin Zhou (2005). "Mechanics and Newton–Cartan-Like Gravity on the Newton–Hooke Space–time". Phys. Rev. D. 71 (4): 044030. arXiv:hep-th/0411004. Bibcode:2005PhRvD..71d4030T. doi:10.1103/PhysRevD.71.044030. S2CID 119378100.
  22. ^ F. G. Gursey, "Introduction to the de Sitter group", Group Theoretical Concepts and Methods in Elementary Particle Physics edited by F. G. Gursey (Gordon and Breach, New York, 1965)
  23. ^ L. F. Abbott; S. Deser (1982). "Stability of gravity with a cosmological constant". Nucl. Phys. B (Submitted manuscript). 195 (1): 76–96. Bibcode:1982NuPhB.195...76A. doi:10.1016/0550-3213(82)90049-9.
  24. ^ J. Kowalski-Glikman; S. Nowak (2003). "Doubly special relativity and de Sitter space". Class. Quantum Grav. 20 (22): 4799–4816. arXiv:hep-th/0304101. Bibcode:2003CQGra..20.4799K. doi:10.1088/0264-9381/20/22/006. S2CID 16875852.
  25. ^ Shao-Xia Chen; Neng-Chao Xiao; Mu-Lin Yan (2008). . Chinese Physics C. 32 (8): 612–616. arXiv:astro-ph/0703110. Bibcode:2008ChPhC..32..612C. doi:10.1177/0022343307082058. S2CID 143773103. Archived from the original on 2011-07-07.
  26. ^ C G Bohmer; T Harko (2008). "Physics of dark energy particles". Foundations of Physics. 38 (3): 216–227. arXiv:gr-qc/0602081. Bibcode:2008FoPh...38..216B. doi:10.1007/s10701-007-9199-4. S2CID 16361512.
  27. ^ Mu-Lin Yan; Neng-Chao Xiao; Wei Huang; Si Li (2007). "Hamiltonian Formalism of the de-Sitter Invariant Special Relativity". Communications in Theoretical Physics. 48 (1): 27–36. arXiv:hep-th/0512319. Bibcode:2007CoTPh..48...27Y. doi:10.1088/0253-6102/48/1/007. S2CID 250880550.
  28. ^ Yu Tian (2005). "De Sitter Thermodynamics from Diamonds's Temperature". Journal of High Energy Physics. 2005 (6): 045. arXiv:gr-qc/0504040v3. Bibcode:2005JHEP...06..045T. doi:10.1088/1126-6708/2005/06/045. S2CID 119399508.
  29. ^ S. Mignemi (2008). "Triply special relativity from six dimensions". arXiv:0807.2186 [gr-qc].
  30. ^ Gibbons, Gary W.; Gielen, Steffen (2009). "Deformed General Relativity and Torsion". Classical and Quantum Gravity. 26 (13): 135005. arXiv:0902.2001. Bibcode:2009CQGra..26m5005G. doi:10.1088/0264-9381/26/13/135005. S2CID 119296100.
  31. ^ Ashok Das; Otto C. W. Kong (2006). "Physics of Quantum Relativity through a Linear Realization". Phys. Rev. D. 73 (12): 124029. arXiv:gr-qc/0603114. Bibcode:2006PhRvD..73l4029D. doi:10.1103/PhysRevD.73.124029. S2CID 30161988.
  32. ^ Han-Ying Guo; Chao-Guang Huang; Yu Tian; Zhan Xu; Bin Zhou (2007). "Snyder's Quantized Space–time and De Sitter Special Relativity". Front. Phys. China. 2 (3): 358–363. arXiv:hep-th/0607016. Bibcode:2007FrPhC...2..358G. doi:10.1007/s11467-007-0045-0. S2CID 119368124.
  33. ^ N. D. Birrell; P. C. W. Davies (1982). Quantum fields in curved space. Cambridge University Press. ISBN 978-0521233859.
  34. ^ J. Bros; U. Moschella (1996). "Two-point functions and quantum fields in de Sitter universe". Rev. Math. Phys. 8 (3): 327–392. arXiv:gr-qc/9511019. Bibcode:1996RvMaP...8..327B. doi:10.1142/S0129055X96000123. S2CID 17974712.
  35. ^ J. Bros; H. Epstein; U. Moschella (1998). "Analyticity properties and thermal effects for general quantum field theory on de Sitter space–time". Commun. Math. Phys. 196 (3): 535–570. arXiv:gr-qc/9801099. Bibcode:1998CMaPh.196..535B. doi:10.1007/s002200050435. S2CID 2027732.
  36. ^ J. Bros; H. Epstein; U. Moschella (2008). "Lifetime of a massive particle in a de Sitter universe". Transactions of the American Fisheries Society. 137 (6): 1879. arXiv:hep-th/0612184. Bibcode:2008JCAP...02..003B. doi:10.1577/T07-141.1.
  37. ^ U. Moschella (2006), "The de Sitter and anti-de Sitter sightseeing tour", in Einstein, 1905–2005 (T. Damour, O. Darrigol, B. Duplantier, and V. Rivesseau, eds.), Progress in Mathematical Physics, Vol. 47, Basel: Birkhauser, 2006.
  38. ^ Moschella U (2007). "Particles and fields on the de Sitter universe". AIP Conference Proceedings. 910: 396–411. Bibcode:2007AIPC..910..396M. doi:10.1063/1.2752488.
  39. ^ E. Benedetto (2009). "Fantappiè–Arcidiacono Spacetime and Its Consequences in Quantum Cosmology". Int J Theor Phys. 48 (6): 1603–1621. Bibcode:2009IJTP...48.1603B. doi:10.1007/s10773-009-9933-0. S2CID 121015516.

Further reading edit

  • R. Aldrovandi; J. G. Pereira (2009). "Is Physics Asking for a New Kinematics?". International Journal of Modern Physics D. 17 (13 & 14): 2485–2493. arXiv:0805.2584. Bibcode:2008IJMPD..17.2485A. doi:10.1142/S0218271808013972. S2CID 14403086.
  • S Cacciatori; V Gorini; A Kamenshchik; U Moschella (2008). "Conservation laws and scattering for de Sitter classical particles". Class. Quantum Grav. 25 (7): 075008. arXiv:0710.0315. Bibcode:2008CQGra..25g5008C. doi:10.1088/0264-9381/25/7/075008. S2CID 118544579.
  • S Cacciatori (2009). "Conserved quantities for the Sitter particles". arXiv:0909.1074 [gr-qc].
  • Aldrovandi; Beltran Almeida; Mayor; Pereira; Adenier, Guillaume; Khrennikov, Andrei Yu.; Lahti, Pekka; Man'Ko, Vladimir I.; Nieuwenhuizen, Theo M. (2007). "de Sitter Relativity and Quantum Physics". AIP Conference Proceedings. 962: 175–184. arXiv:0710.0610. Bibcode:2007AIPC..962..175A. doi:10.1063/1.2827302. hdl:11449/70009. S2CID 1178656.
  • Claus Lämmerzahl; Jürgen Ehlers (2005). Special Relativity: Will it Survive the Next 101 Years?. Springer. ISBN 978-3540345220.
  • Giuseppe Arcidiacono (1986). Projective Relativity, Cosmology, and Gravitation. Hadronic Press. ISBN 978-0911767391.

January 01, 1970
sitter, invariant, special, relativity, mathematical, physics, sitter, invariant, special, relativity, speculative, idea, that, fundamental, symmetry, group, spacetime, indefinite, orthogonal, group, that, sitter, space, standard, theory, general, relativity, . In mathematical physics de Sitter invariant special relativity is the speculative idea that the fundamental symmetry group of spacetime is the indefinite orthogonal group SO 4 1 that of de Sitter space In the standard theory of general relativity de Sitter space is a highly symmetrical special vacuum solution which requires a cosmological constant or the stress energy of a constant scalar field to sustain The idea of de Sitter invariant relativity is to require that the laws of physics are not fundamentally invariant under the Poincare group of special relativity but under the symmetry group of de Sitter space instead With this assumption empty space automatically has de Sitter symmetry and what would normally be called the cosmological constant in general relativity becomes a fundamental dimensional parameter describing the symmetry structure of spacetime First proposed by Luigi Fantappie in 1954 the theory remained obscure until it was rediscovered in 1968 by Henri Bacry and Jean Marc Levy Leblond In 1972 Freeman Dyson popularized it as a hypothetical road by which mathematicians could have guessed part of the structure of general relativity before it was discovered 1 The discovery of the accelerating expansion of the universe has led to a revival of interest in de Sitter invariant theories in conjunction with other speculative proposals for new physics like doubly special relativity Contents 1 Introduction 1 1 High energy 1 2 Doubly special relativity 1 3 Newton Hooke de Sitter special relativity in the limit v c 1 4 History of de Sitter invariant special relativity 1 5 Quantum de Sitter special relativity 2 See also 3 References 4 Further readingIntroduction editMain article de Sitter space De Sitter suggested that spacetime curvature might not be due solely to gravity 2 but he did not give any mathematical details of how this could be accomplished In 1968 Henri Bacry and Jean Marc Levy Leblond showed that the de Sitter group was the most general group compatible with isotropy homogeneity and boost invariance 3 Later Freeman Dyson 1 advocated this as an approach to making the mathematical structure of general relativity more self evident Minkowski s unification of space and time within special relativity replaces the Galilean group of Newtonian mechanics with the Lorentz group This is called a unification of space and time because the Lorentz group is simple while the Galilean group is a semi direct product of rotations and Galilean boosts This means that the Lorentz group mixes up space and time such that they cannot be disentangled while the Galilean group treats time as a parameter with different units of measurement than space An analogous thing can be made to happen with the ordinary rotation group in three dimensions If you imagine a nearly flat world one in which pancake like creatures wander around on a pancake flat world their conventional unit of height might be the micrometre mm since that is how high typical structures are in their world while their unit of distance could be the metre because that is their body s horizontal extent Such creatures would describe the basic symmetry of their world as SO 2 being the known rotations in the horizontal x y plane Later on they might discover rotations around the x and y axes and in their everyday experience such rotations might always be by an infinitesimal angle so that these rotations would effectively commute with each other The rotations around the horizontal axes would tilt objects by an infinitesimal amount The tilt in the x z plane the x tilt would be one parameter and the tilt in the y z plane the y tilt another The symmetry group of this pancake world is then SO 2 semidirect product with R2 meaning that a two dimensional rotation plus two extra parameters the x tilt and the y tilt The reason it is a semidirect product is that when you rotate the x tilt and the y tilt rotate into each other since they form a vector and not two scalars In this world the difference in height between two objects at the same x y would be a rotationally invariant quantity unrelated to length and width The z coordinate is effectively separate from x and y Eventually experiments at large angles would convince the creatures that the symmetry of the world is SO 3 Then they would understand that z is really the same as x and y since they can be mixed up by rotations The SO 2 semidirect product R2 limit would be understood as the limit that the free parameter m the ratio of the height range mm to the length range m approaches 0 The Lorentz group is analogous it is a simple group that turns into the Galilean group when the time range is made long compared to the space range or where velocities may be regarded as infinitesimal or equivalently may be regarded as the limit c where relativistic effects become observable as good as at infinite velocity The symmetry group of special relativity is not entirely simple due to translations The Lorentz group is the set of the transformations that keep the origin fixed but translations are not included The full Poincare group is the semi direct product of translations with the Lorentz group If translations are to be similar to elements of the Lorentz group then as boosts are non commutative translations would also be non commutative In the pancake world this would manifest if the creatures were living on an enormous sphere rather than on a plane In this case when they wander around their sphere they would eventually come to realize that translations are not entirely separate from rotations because if they move around on the surface of a sphere when they come back to where they started they find that they have been rotated by the holonomy of parallel transport on the sphere If the universe is the same everywhere homogeneous and there are no preferred directions isotropic then there are not many options for the symmetry group they either live on a flat plane or on a sphere with a constant positive curvature or on a Lobachevski plane with constant negative curvature If they are not living on the plane they can describe positions using dimensionless angles the same parameters that describe rotations so that translations and rotations are nominally unified In relativity if translations mix up nontrivially with rotations but the universe is still homogeneous and isotropic the only option is that spacetime has a uniform scalar curvature If the curvature is positive the analog of the sphere case for the two dimensional creatures the spacetime is de Sitter space and its symmetry group is the de Sitter group rather than the Poincare group De Sitter special relativity postulates that the empty space has de Sitter symmetry as a fundamental law of nature This means that spacetime is slightly curved even in the absence of matter or energy This residual curvature implies a positive cosmological constant L to be determined by observation Due to the small magnitude of the constant special relativity with its Poincare group is indistinguishable from de Sitter space for most practical purposes Modern proponents of this idea such as S Cacciatori V Gorini and A Kamenshchik 4 have reinterpreted this theory as physics not just mathematics They postulate that the acceleration of the expansion of the universe is not entirely due to vacuum energy but at least partly due to the kinematics of the de Sitter group which would replace the Poincare group A modification of this idea allows L displaystyle Lambda nbsp to change with time so that inflation may come from the cosmological constant being larger near the Big Bang than nowadays It can also be viewed as a different approach to the problem of quantum gravity 5 High energy edit The Poincare group contracts to the Galilean group for low velocity kinematics meaning that when all velocities are small the Poincare group morphs into the Galilean group This can be made precise with Inonu and Wigner s concept of group contraction 6 Similarly the de Sitter group contracts to the Poincare group for short distance kinematics when the magnitudes of all translations considered are very small compared to the de Sitter radius 5 In quantum mechanics short distances are probed by high energies so that for energies above a very small value related to the cosmological constant the Poincare group is a good approximation to the de Sitter group In de Sitter relativity the cosmological constant is no longer a free parameter of the same type it is determined by the de Sitter radius a fundamental quantity that determines the commutation relation of translation with rotations boosts This means that the theory of de Sitter relativity might be able to provide insight on the value of the cosmological constant perhaps explaining the cosmic coincidence Unfortunately the de Sitter radius which determines the cosmological constant is an adjustable parameter in de Sitter relativity so the theory requires a separate condition to determine its value in relation to the measurement scale When a cosmological constant is viewed as a kinematic parameter the definitions of energy and momentum must be changed from those of special relativity These changes could significantly modify the physics of the early universe if the cosmological constant was greater back then Some speculate that a high energy experiment could modify the local structure of spacetime from Minkowski space to de Sitter space with a large cosmological constant for a short period of time and this might eventually be tested in the existing or planned particle collider 7 Doubly special relativity edit Main article Doubly special relativity Since the de Sitter group naturally incorporates an invariant length parameter de Sitter relativity can be interpreted as an example of the so called doubly special relativity There is a fundamental difference though whereas in all doubly special relativity models the Lorentz symmetry is violated in de Sitter relativity it remains as a physical symmetry 8 9 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 de Sitter relativity is found to be invariant under a simultaneous re scaling of mass energy and momentum 10 and is consequently valid at all energy scales A relationship between doubly special relativity de Sitter space and general relativity is described by Derek Wise 11 See also MacDowell Mansouri action Newton Hooke de Sitter special relativity in the limit v c edit In the limit as v c the de Sitter group contracts to the Newton Hooke group 12 This has the effect that in the nonrelativistic limit objects in de Sitter space have an extra repulsion from the origin objects have a tendency to move away from the center with an outward pointing fictitious force proportional to their distance from the origin While it looks as though this might pick out a preferred point in space the center of repulsion it is more subtly isotropic Moving to the uniformly accelerated frame of reference of an observer at another point all accelerations appear to have a repulsion center at the new point What this means is that in a spacetime with non vanishing curvature gravity is modified from Newtonian gravity 13 At distances comparable to the radius of the space objects feel an additional linear repulsion from the center of coordinates History of de Sitter invariant special relativity edit de Sitter relativity is the same as the theory of projective relativity of Luigi Fantappie and Giuseppe Arcidiacono first published in 1954 by Fantappie 14 and the same as another independent discovery in 1976 15 In 1968 Henri Bacry and Jean Marc Levy Leblond published a paper on possible kinematics 3 In 1972 Freeman Dyson 1 further explored this In 1973 Eliano Pessa described how Fantappie Arcidiacono projective relativity relates to earlier conceptions of projective relativity and to Kaluza Klein theory 16 R Aldrovandi J P Beltran Almeida and J G Pereira have used the terms de Sitter special relativity and de Sitter relativity starting from their 2007 paper de Sitter special relativity 10 17 This paper was based on previous work on amongst other things the consequences of a non vanishing cosmological constant 18 on doubly special relativity 19 and on the Newton Hooke group 3 20 21 and early work formulating special relativity with a de Sitter space 22 23 24 In 2008 S Cacciatori V Gorini and A Kamenshchik 4 published a paper about the kinematics of de Sitter relativity Papers by other authors include dSR and the fine structure constant 25 dSR and dark energy 26 dSR Hamiltonian Formalism 27 and De Sitter Thermodynamics from Diamonds s Temperature 28 Triply special relativity from six dimensions 29 Deformed General Relativity and Torsion 30 Quantum de Sitter special relativity edit There are quantized or quantum versions of de Sitter special relativity 31 32 Early work on formulating a quantum theory in a de Sitter space includes 33 34 35 36 37 38 39 See also edit nbsp Physics portal Noncommutative geometry Quantum field theory in curved spacetimeReferences edit a b c Freeman Dyson 1972 Missed opportunities pdf Bull Am Math Soc 78 5 635 652 doi 10 1090 S0002 9904 1972 12971 9 MR 0522147 W de Sitter 1917 On the curvature of space Proc Roy Acad Sci Amsterdam 20 229 243 a b c Henri Bacry Jean Marc Levy Leblond 1968 Possible Kinematics Journal of Mathematical Physics 9 10 1605 Bibcode 1968JMP 9 1605B doi 10 1063 1 1664490 a b S Cacciatori V Gorini A Kamenshchik 2008 Special Relativity in the 21st century Annalen der Physik 17 9 10 728 768 arXiv 0807 3009 Bibcode 2008AnP 520 728C doi 10 1002 andp 200810321 S2CID 119191753 a b R Aldrovandi J G Pereira 2009 de Sitter Relativity a New Road to Quantum Gravity Foundations of Physics 39 2 1 19 arXiv 0711 2274 Bibcode 2009FoPh 39 1A doi 10 1007 s10701 008 9258 5 S2CID 15298756 E Inonu E P Wigner 1953 On the Contraction of Groups and Their Representations Proc Natl Acad Sci USA 39 6 510 24 Bibcode 1953PNAS 39 510I doi 10 1073 pnas 39 6 510 PMC 1063815 PMID 16589298 Freydoon Mansouri 2002 Non Vanishing Cosmological Constant L Phase Transitions And L Dependence Of High Energy Processes Phys Lett B 538 3 4 239 245 arXiv hep th 0203150 Bibcode 2002PhLB 538 239M doi 10 1016 S0370 2693 02 02022 1 S2CID 13986319 Aldrovandi R Beltran Almeida J P Pereira J G 2007 Some Implications of the Cosmological Constant to Fundamental Physics AIP Conference Proceedings 910 381 395 arXiv gr qc 0702065 Bibcode 2007AIPC 910 381A doi 10 1063 1 2752487 hdl 11449 69891 S2CID 16631274 R Aldrovandi J P Beltran Almeida C S O Mayor J G Pereira 2007 Lorentz Transformations in de Sitter Relativity arXiv 0709 3947 gr qc a b R Aldrovandi J P Beltran Almeida J G Pereira 2007 de Sitter Special Relativity Class Quantum Grav 24 6 1385 1404 arXiv gr qc 0606122 Bibcode 2007CQGra 24 1385A doi 10 1088 0264 9381 24 6 002 S2CID 11703342 Wise 2010 MacDowell Mansouri Gravity and Cartan Geometry Classical and Quantum Gravity 27 15 155010 arXiv gr qc 0611154 Bibcode 2010CQGra 27o5010W doi 10 1088 0264 9381 27 15 155010 S2CID 16706599 Aldrovandi Barbosa Crispino Pereira 1999 Non Relativistic Spacetimes with Cosmological Constant Classical and Quantum Gravity 16 2 495 506 arXiv gr qc 9801100 Bibcode 1999CQGra 16 495A CiteSeerX 10 1 1 339 919 doi 10 1088 0264 9381 16 2 013 S2CID 16691405 Yu Tian Han Ying Guo Chao Guang Huang Zhan Xu Bin Zhou 2004 Mechanics and Newton Cartan Like Gravity on the Newton Hooke Space time Physical Review D 71 4 44030 arXiv hep th 0411004 Bibcode 2005PhRvD 71d4030T doi 10 1103 PhysRevD 71 044030 S2CID 119378100 Licata Ignazio Leonardo Chiatti 2009 The archaic universe Big Bang cosmological term and the quantum origin of time in projective cosmology International Journal of Theoretical Physics 48 4 1003 1018 arXiv 0808 1339 Bibcode 2009IJTP 48 1003L doi 10 1007 s10773 008 9874 z S2CID 119262177 Dey Anind K 2001 An extension of the concept of inertial frame and of Lorentz transformation Proc Natl Acad Sci USA 73 5 1418 21 Bibcode 1976PNAS 73 1418K doi 10 1073 pnas 73 5 1418 PMC 430307 PMID 16592318 The De Sitter Universe and general relativity R Aldrovandi J G Pereira 2009 De Sitter Special Relativity Effects on Cosmology Gravitation and Cosmology 15 4 287 294 arXiv 0812 3438 Bibcode 2009GrCo 15 287A doi 10 1134 S020228930904001X S2CID 18473868 R Aldrovandi J P Beltran Almeida J G Pereira 2004 Cosmological Term and Fundamental Physics Int J Mod Phys D 13 10 2241 2248 arXiv gr qc 0405104 Bibcode 2004IJMPD 13 2241A doi 10 1142 S0218271804006279 S2CID 118889785 Giovanni Amelino Camelia 2001 Testable scenario for Relativity with minimum length Phys Lett B 510 1 4 255 263 arXiv hep th 0012238 Bibcode 2001PhLB 510 255A doi 10 1016 S0370 2693 01 00506 8 S2CID 119447462 G W Gibbons C E Patricot 2003 Newton Hooke spacetimes Hpp waves and the cosmological constant Class Quantum Grav 20 23 5225 arXiv hep th 0308200 Bibcode 2003CQGra 20 5225G doi 10 1088 0264 9381 20 23 016 S2CID 26557629 Yu Tian Han Ying Guo Chao Guang Huang Zhan Xu Bin Zhou 2005 Mechanics and Newton Cartan Like Gravity on the Newton Hooke Space time Phys Rev D 71 4 044030 arXiv hep th 0411004 Bibcode 2005PhRvD 71d4030T doi 10 1103 PhysRevD 71 044030 S2CID 119378100 F G Gursey Introduction to the de Sitter group Group Theoretical Concepts and Methods in Elementary Particle Physics edited by F G Gursey Gordon and Breach New York 1965 L F Abbott S Deser 1982 Stability of gravity with a cosmological constant Nucl Phys B Submitted manuscript 195 1 76 96 Bibcode 1982NuPhB 195 76A doi 10 1016 0550 3213 82 90049 9 J Kowalski Glikman S Nowak 2003 Doubly special relativity and de Sitter space Class Quantum Grav 20 22 4799 4816 arXiv hep th 0304101 Bibcode 2003CQGra 20 4799K doi 10 1088 0264 9381 20 22 006 S2CID 16875852 Shao Xia Chen Neng Chao Xiao Mu Lin Yan 2008 Variation of the Fine Structure Constant from the de Sitter Invariant Special Relativity Chinese Physics C 32 8 612 616 arXiv astro ph 0703110 Bibcode 2008ChPhC 32 612C doi 10 1177 0022343307082058 S2CID 143773103 Archived from the original on 2011 07 07 C G Bohmer T Harko 2008 Physics of dark energy particles Foundations of Physics 38 3 216 227 arXiv gr qc 0602081 Bibcode 2008FoPh 38 216B doi 10 1007 s10701 007 9199 4 S2CID 16361512 Mu Lin Yan Neng Chao Xiao Wei Huang Si Li 2007 Hamiltonian Formalism of the de Sitter Invariant Special Relativity Communications in Theoretical Physics 48 1 27 36 arXiv hep th 0512319 Bibcode 2007CoTPh 48 27Y doi 10 1088 0253 6102 48 1 007 S2CID 250880550 Yu Tian 2005 De Sitter Thermodynamics from Diamonds s Temperature Journal of High Energy Physics 2005 6 045 arXiv gr qc 0504040v3 Bibcode 2005JHEP 06 045T doi 10 1088 1126 6708 2005 06 045 S2CID 119399508 S Mignemi 2008 Triply special relativity from six dimensions arXiv 0807 2186 gr qc Gibbons Gary W Gielen Steffen 2009 Deformed General Relativity and Torsion Classical and Quantum Gravity 26 13 135005 arXiv 0902 2001 Bibcode 2009CQGra 26m5005G doi 10 1088 0264 9381 26 13 135005 S2CID 119296100 Ashok Das Otto C W Kong 2006 Physics of Quantum Relativity through a Linear Realization Phys Rev D 73 12 124029 arXiv gr qc 0603114 Bibcode 2006PhRvD 73l4029D doi 10 1103 PhysRevD 73 124029 S2CID 30161988 Han Ying Guo Chao Guang Huang Yu Tian Zhan Xu Bin Zhou 2007 Snyder s Quantized Space time and De Sitter Special Relativity Front Phys China 2 3 358 363 arXiv hep th 0607016 Bibcode 2007FrPhC 2 358G doi 10 1007 s11467 007 0045 0 S2CID 119368124 N D Birrell P C W Davies 1982 Quantum fields in curved space Cambridge University Press ISBN 978 0521233859 J Bros U Moschella 1996 Two point functions and quantum fields in de Sitter universe Rev Math Phys 8 3 327 392 arXiv gr qc 9511019 Bibcode 1996RvMaP 8 327B doi 10 1142 S0129055X96000123 S2CID 17974712 J Bros H Epstein U Moschella 1998 Analyticity properties and thermal effects for general quantum field theory on de Sitter space time Commun Math Phys 196 3 535 570 arXiv gr qc 9801099 Bibcode 1998CMaPh 196 535B doi 10 1007 s002200050435 S2CID 2027732 J Bros H Epstein U Moschella 2008 Lifetime of a massive particle in a de Sitter universe Transactions of the American Fisheries Society 137 6 1879 arXiv hep th 0612184 Bibcode 2008JCAP 02 003B doi 10 1577 T07 141 1 U Moschella 2006 The de Sitter and anti de Sitter sightseeing tour in Einstein 1905 2005 T Damour O Darrigol B Duplantier and V Rivesseau eds Progress in Mathematical Physics Vol 47 Basel Birkhauser 2006 Moschella U 2007 Particles and fields on the de Sitter universe AIP Conference Proceedings 910 396 411 Bibcode 2007AIPC 910 396M doi 10 1063 1 2752488 E Benedetto 2009 Fantappie Arcidiacono Spacetime and Its Consequences in Quantum Cosmology Int J Theor Phys 48 6 1603 1621 Bibcode 2009IJTP 48 1603B doi 10 1007 s10773 009 9933 0 S2CID 121015516 Further reading editR Aldrovandi J G Pereira 2009 Is Physics Asking for a New Kinematics International Journal of Modern Physics D 17 13 amp 14 2485 2493 arXiv 0805 2584 Bibcode 2008IJMPD 17 2485A doi 10 1142 S0218271808013972 S2CID 14403086 S Cacciatori V Gorini A Kamenshchik U Moschella 2008 Conservation laws and scattering for de Sitter classical particles Class Quantum Grav 25 7 075008 arXiv 0710 0315 Bibcode 2008CQGra 25g5008C doi 10 1088 0264 9381 25 7 075008 S2CID 118544579 S Cacciatori 2009 Conserved quantities for the Sitter particles arXiv 0909 1074 gr qc Aldrovandi Beltran Almeida Mayor Pereira Adenier Guillaume Khrennikov Andrei Yu Lahti Pekka Man Ko Vladimir I Nieuwenhuizen Theo M 2007 de Sitter Relativity and Quantum Physics AIP Conference Proceedings 962 175 184 arXiv 0710 0610 Bibcode 2007AIPC 962 175A doi 10 1063 1 2827302 hdl 11449 70009 S2CID 1178656 Claus Lammerzahl Jurgen Ehlers 2005 Special Relativity Will it Survive the Next 101 Years Springer ISBN 978 3540345220 Giuseppe Arcidiacono 1986 Projective Relativity Cosmology and Gravitation Hadronic Press ISBN 978 0911767391 Retrieved from https en wikipedia org w index php title De Sitter invariant special relativity amp oldid 1182304001, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.