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under a simultaneous re-scaling of [[mass]], [[energy]] and [[momentum]]<ref name=dessrintro>R Aldrovandi, J.P. Beltrán Almeida, J.G. Pereira, [http://www.iop.org/EJ/abstract/0264-9381/24/6/002/ "de Sitter Special Relativity"], Class. Quantum Grav. 24 1385-1404 Issue 6 (21 March 2007), [http://arxiv.org/pdf/gr-qc/0606122 arxiv 0606122]</ref>, and is consequently
under a simultaneous re-scaling of [[mass]], [[energy]] and [[momentum]]<ref name=dessrintro>R Aldrovandi, J.P. Beltrán Almeida, J.G. Pereira, [http://www.iop.org/EJ/abstract/0264-9381/24/6/002/ "de Sitter Special Relativity"], Class. Quantum Grav. 24 1385-1404 Issue 6 (21 March 2007), [http://arxiv.org/pdf/gr-qc/0606122 arxiv 0606122]</ref>, 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<ref>
valid at all energy scales. A relationship between doubly special relativity, de Sitter space and general relativity is described by Derek Wise<ref>
[http://arxiv.org/PS_cache/gr-qc/pdf/0611/0611154v1.pdf MacDowell–Mansouri Gravity and Cartan Geometry]</ref>. See also [[MacDowell-Mansouri action]].
[http://arxiv.org/abs/gr-qc/0611154v1 MacDowell–Mansouri Gravity and Cartan Geometry]</ref>. See also [[MacDowell-Mansouri action]].


===Newton-Hooke: de Sitter special relativity in the limit as v<<c===
===Newton-Hooke: de Sitter special relativity in the limit as v<<c===


In the limit as v<<c, the de Sitter group contracts to the Newton-Hooke group<ref>
In the limit as v<<c, the de Sitter group contracts to the Newton-Hooke group<ref>
[http://arxiv.org/PS_cache/gr-qc/pdf/9801/9801100v2.pdf Non–Relativistic Spacetimes with Cosmological Constant]
[http://arxiv.org/abs/gr-qc/9801100v2 Non–Relativistic Spacetimes with Cosmological Constant]
</ref>. This implies 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.
</ref>. This implies 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.


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What this means is that in a spacetime with non-vanishing curvature, gravity is modified from Newtonian gravity<ref>
What this means is that in a spacetime with non-vanishing curvature, gravity is modified from Newtonian gravity<ref>
[http://arxiv.org/PS_cache/hep-th/pdf/0411/0411004v2.pdf Mechanics and Newton-Cartan-Like Gravity on the Newton-Hooke Space-time]</ref>. At distances comparable to the radius of the space, objects feel an additional linear repulsion from the center of coordinates.
[http://arxiv.org/abs/hep-th/0411004v2 Mechanics and Newton-Cartan-Like Gravity on the Newton-Hooke Space-time]</ref>. 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===
===History of de Sitter invariant special relativity===
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*Recent papers by other authors include: dSR and the fine structure constant<ref>
*Recent papers by other authors include: dSR and the fine structure constant<ref>
Shao-Xia Chen, Neng-Chao Xiao, Mu-Lin Yan, [http://mp.ihep.ac.cn/qikan/epaper/zhaiyao.asp?bsid=7371 "Variation of the Fine-Structure Constant from the de Sitter Invariant Special Relativity"], Chinese Physics C 2008 32 (08): 612—616 ISSN: 1674-1137 CN: 11-5641/O4, [http://arxiv.org/PS_cache/astro-ph/pdf/0703/0703110v1.pdf arxiv 0703110]</ref>; dSR and dark energy<ref>
Shao-Xia Chen, Neng-Chao Xiao, Mu-Lin Yan, [http://mp.ihep.ac.cn/qikan/epaper/zhaiyao.asp?bsid=7371 "Variation of the Fine-Structure Constant from the de Sitter Invariant Special Relativity"], Chinese Physics C 2008 32 (08): 612—616 ISSN: 1674-1137 CN: 11-5641/O4, [http://arxiv.org/abs/astro-ph/0703110v1 arxiv 0703110]</ref>; dSR and dark energy<ref>
C G Bohmer, T Harko, [http://www.springerlink.com/content/q2x3004h7018u773/ "Physics of dark energy particles"], Foundations of Physics, Volume 38, Issue 3, pp.216-227, [http://arxiv.org/PS_cache/gr-qc/pdf/0602/0602081v6.pdf arxiv 0602081]</ref>; dSR Hamiltonian Formalism<ref>
C G Bohmer, T Harko, [http://www.springerlink.com/content/q2x3004h7018u773/ "Physics of dark energy particles"], Foundations of Physics, Volume 38, Issue 3, pp.216-227, [http://arxiv.org/abs/gr-qc/0602081v6 arxiv 0602081]</ref>; dSR Hamiltonian Formalism<ref>
[http://arxiv.org/PS_cache/hep-th/pdf/0512/0512319v2.pdf Hamiltonian Formalism of the de-Sitter Invariant Special Relativity]</ref>; and De Sitter Thermodynamics from Diamonds’s Temperature<ref>
[http://arxiv.org/abs/hep-th/0512319v2 Hamiltonian Formalism of the de-Sitter Invariant Special Relativity]</ref>; and De Sitter Thermodynamics from Diamonds’s Temperature<ref>
[http://arxiv.org/PS_cache/gr-qc/pdf/0504/0504040v3.pdf De Sitter Thermodynamics from Diamonds’s Temperature]</ref>
[http://arxiv.org/abs/gr-qc/0504040v3 De Sitter Thermodynamics from Diamonds’s Temperature]</ref>


*In 2006 Ignazio Licata wrote about generalizing the Poincare group to the de Sitter group.<ref>
*In 2006 Ignazio Licata wrote about generalizing the Poincare group to the de Sitter group.<ref>
Line 118: Line 118:


*In 2007 Leonardo Chiatti explored the recent cosmological evidence for the Fantappié-Arcidiacono theory of relativity<ref name=Chiatti_evidences>
*In 2007 Leonardo Chiatti explored the recent cosmological evidence for the Fantappié-Arcidiacono theory of relativity<ref name=Chiatti_evidences>
Leonardo Chiatti, [http://www.ejtp.info/articles/ejtpv4i15p17.pdf "Fantappié-Arcidiacono theory of relativity versus recent cosmological evidences : a preliminary comparison"], EJTP 15 (4), 17-36 (2007), [http://arxiv.org/ftp/physics/papers/0702/0702178.pdf arXiv:physics/0702178]</ref>
Leonardo Chiatti, [http://www.ejtp.info/articles/ejtpv4i15p17.pdf "Fantappié-Arcidiacono theory of relativity versus recent cosmological evidences : a preliminary comparison"], EJTP 15 (4), 17-36 (2007), [http://arxiv.org/abs/physics/0702178 arXiv:physics/0702178]</ref>


*In 2008 Ignazio Licata and Leonardo Chiatti said Fantappié-Arcidiacono theory of relativity was the same thing as the recent work on de Sitter relativity<ref name=Licata_Chiatti2008>
*In 2008 Ignazio Licata and Leonardo Chiatti said Fantappié-Arcidiacono theory of relativity was the same thing as the recent work on de Sitter relativity<ref name=Licata_Chiatti2008>
Ignazio Licata [http://arxiv.org/ftp/arxiv/papers/0808/0808.1339.pdf The archaic universe: Big Bang, cosmological term, and the quantum origin of time in projective cosmology]</ref>
Ignazio Licata [http://arxiv.org/abs/0808.1339 The archaic universe: Big Bang, cosmological term, and the quantum origin of time in projective cosmology]</ref>


*In 2008 S. Cacciatori, V. Gorini and A. Kamenshchik<ref name=c21st/> published a paper about the kinematics of de Sitter relativity.
*In 2008 S. Cacciatori, V. Gorini and A. Kamenshchik<ref name=c21st/> published a paper about the kinematics of de Sitter relativity.

Revision as of 23:08, 6 February 2009

In mathematical physics, de Sitter invariant special relativity is a speculative idea that the fundamental symmetry group of spacetime is 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 deSitter symmetry, and what would normally be called the cosmological constant in General Relativity becomes a fundamental dimensional parameter describing the symmetry structure of space-time.

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 deSitter invariant theories, in conjunction with other speculative proposals for new physics, like doubly special relativity.

Introduction

See also: de Sitter space.

De Sitter himself suggested that space-time 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 so 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 , since that's how high typical structures are in their world, while their x and y axis could be the meter, because that's the size of their body. Such creatures would describe the basic symmetry structure of their world as SO(2), rotations in the x-y plane. Later on, they might discover rotations into the z axis--- and in their every-day experience such rotations would always be by an infinitesimal angle, so that these z-rotations would commute with each other.

The rotations into the z-axis would tilt objects by an infinitesimal amount. The tilt in the x-z plane would be one parameter, and the tilt in the y-z plane another. The symmetry group of this pancake world is 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 completely separate from x and y.

But eventually, experiments at large angles would convince the creatures that the actual 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 , the ratio of the height-unit to the length-unit , approaches 0. The Lorentz group is analogous--- it is a simple group that turns into the Galilean group when the unit of time is made long compared to the unit of space, which is the limit .

But the symmetry group of special relativity is not entirely simple because there are still translations. The Lorentz group are 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. But if you take the unification idea to its logical conclusion then not only are boosts non-commutative but translations should be non-commutative too.

In the pancake world, this would happen if the creatures were living on an enormous sphere, not 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 (homogenous) 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 everywhere 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 options are that space-time has a uniform scalar curvature. If the curvature is positive, the analog of the sphere case for the two-dimensional creatures, the space-time is de Sitter and the symmetry group of spacetime is a 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 is caused by a positive cosmological constant to be determined by observation. Due to the small magnitude of the constant, then special relativity with the Poincaré group is more than accurate enough for all 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 believe that the acceleration of the expansion of the universe is not all due to vacuum energy, but at least partly due to the kinematics of the de Sitter group, which in their view is the correct symmetry group of space time, replacing the Lorentz 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

The Poincaré group generalizes the Galilean group for high–velocity kinematics, meaning that when all velocities are small the Lorentz group 'becomes' the Galilean group. (This can be made precise with Inönü and Wigner's concept of group contraction[6].) Similarly, the de Sitter group generalizes Poincaré for long distance kinematics, meaning that when magnitudes of all translations are small compared to the de Sitter radius, the de Sitter group becomes the Lorentz group[5]. In quantum mechanics, short distances are probed by high energies, so that for energies larger than a very small cosmological scale, 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 is interchangeable with the cosmological constant, is an adjustable parameter in de Sitter relativity, so the theory requires a separate condition to determine its value.

When applied to the propagation of ultra–high energy photons, some claim that the theory explains a controversial time delay possibly observed in extragalactic gamma ray flares. More precisely, there are claims, not yet accepted by the mainstream physics community, that very–high energy extragalactic gamma–ray flares travel slower than lower energy ones[7]. If this is confirmed by future experiments, it will constitute a clear violation of special relativity.

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 modify significantly the physics of the early universe, if the cosmological constant was bigger 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 colliders[8].

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[9][10]. 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[11], 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[12]. See also MacDowell-Mansouri action.

Newton-Hooke: de Sitter special relativity in the limit as v<<c

In the limit as v<<c, the de Sitter group contracts to the Newton-Hooke group[13]. This implies 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 this setup looks like it picks out a preferred point in space--- the center of repulsion--- it is secretly isotropic. If you move to another point, you should transform to the uniformly accelerated frame of reference of an observer at this point, which changes all accelerations to shift the repulsion center to the new origin of coordinates.

What this means is that in a spacetime with non-vanishing curvature, gravity is modified from Newtonian gravity[14]. 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

  • "de Sitter relativity" is the same as the theory of "projective relativity" of Luigi Fantappiè and Giuseppe Arcidiacono first published in 1954 by Fantappiè[15] and the same as another independent discovery in 1976[16].
  • In 1973 Eliano Pessa described how Fantappié-Arcidiacono projective relativity relates to earlier conceptions of projective relativity and to Kaluza Klein theory.[17]
  • 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"[11]. This paper was based on previous work on amongst other things: the consequences of a non-vanishing cosmological constant[31], on doubly special relativity[32] and on the Newton-Hooke group[3][33][34] and early work formulating special relativity with a de Sitter space[35][36][37]
  • Recent papers by other authors include: dSR and the fine structure constant[38]; dSR and dark energy[39]; dSR Hamiltonian Formalism[40]; and De Sitter Thermodynamics from Diamonds’s Temperature[41]
  • In 2006 Ignazio Licata wrote about generalizing the Poincare group to the de Sitter group.[42]
  • In 2007 Leonardo Chiatti explored the recent cosmological evidence for the Fantappié-Arcidiacono theory of relativity[43]
  • In 2008 Ignazio Licata and Leonardo Chiatti said Fantappié-Arcidiacono theory of relativity was the same thing as the recent work on de Sitter relativity[15]
  • In 2008 S. Cacciatori, V. Gorini and A. Kamenshchik[4] published a paper about the kinematics of de Sitter relativity.

Quantum de Sitter special relativity

There are quantized or quantum versions of de Sitter special relativity[44][45][46].

Early work on formulating a quantum theory in a de Sitter space includes: [47][48][49][50][51][52]. [53]

See also

References

  1. ^ a b c F. J. Dyson, Missed opportunities, Bull. Am. Math. Soc. 78, 635–652 (1972).
  2. ^ W. de Sitter, On the curvature of space, Proc. Roy. Acad. Sci. Amsterdam 20 (1917), pp. 229–243.
  3. ^ a b c Henri Bacry, Jean-Marc Lévy-Leblond,Possible Kinematics Journal of Mathematical Physics, October 1968, Volume 9, Issue 10, page 1605
  4. ^ a b , S. Cacciatori, V. Gorini, A. Kamenshchik, "Special Relativity in the 21st century", Annalen der Physik, Volume 17 Issue 9-10, Pages 728 - 768, Published Online: 15 Aug 2008 arxiv 08073009
  5. ^ a b R. Aldrovandi, J. G. Pereira "de Sitter Relativity: a New Road to Quantum Gravity?", Foundations of Physics, Springer Netherlands, Published online: 15 November 2008 arxiv 0711.2274
  6. ^ E. Inönü, E.P. Wigner: "On the Contraction of Groups and their Representations", Proc. Nat. Acad. Sc., 39 (1953), pp. 510-24
  7. ^ J. Albert et al (for the MAGIC Collaboration), J. Ellis, N. E. Mavromatos, D. V. Nanopoulos, A. S. Sakharov and E. K. G. Sarkisyan, "Probing quantum gravity using photons from a flare of the active galactic nucleus Markarian 501 observed by the MAGIC telescope", Phys. Lett. B 668, 253 (2008), arxiv astro-ph/0708.2889v2
  8. ^ Freydoon Mansouri, "Non-Vanishing Cosmological Constant , Phase Transitions, And -Dependence Of High Energy Processes", Phys.Lett. B538 (2002) 239-245, arxiv hep-th/0203150
  9. ^ R. Aldrovandi, J. P. Beltran Almeida, J. G. Pereira "Some Implications of the Cosmological Constant to Fundamental Physics", COSMOLOGY AND GRAVITATION: XIIth Brazilian School of Cosmology and Gravitation. AIP Conference Proceedings, Volume 910, pp. 381-395 (2007), arxiv 0702065
  10. ^ R. Aldrovandi, J.P. Beltran Almeida, C.S.O. Mayor and J.G. Pereira, Lorentz Transformations in de Sitter Relativity, [gr-qc/0709.3947].
  11. ^ a b R Aldrovandi, J.P. Beltrán Almeida, J.G. Pereira, "de Sitter Special Relativity", Class. Quantum Grav. 24 1385-1404 Issue 6 (21 March 2007), arxiv 0606122
  12. ^ MacDowell–Mansouri Gravity and Cartan Geometry
  13. ^ Non–Relativistic Spacetimes with Cosmological Constant
  14. ^ Mechanics and Newton-Cartan-Like Gravity on the Newton-Hooke Space-time
  15. ^ a b Ignazio Licata The archaic universe: Big Bang, cosmological term, and the quantum origin of time in projective cosmology
  16. ^ An extension of the concept of inertial frame and of Lorentz transformation
  17. ^ The De Sitter Universe and general relativity
  18. ^ Han-Ying Guo, Chao-Guang Huang, Zhan Xu, Bin Zhou, "On Special Relativity with Cosmological Constant", Phys.Lett. A331 (2004) 1-7, arxiv hep-th/0403171
  19. ^ Han-Ying Guo, Chao-Guang Huang, Yu Tian, Zhan Xu, Bin Zhou, "On de Sitter Invariant Special Relativity and Cosmological Constant as Origin of Inertia", arxiv hep-th/0405137
  20. ^ Han-Ying Guo, Chao-Guang Huang, Hong-Tu Wu, "Yang's Model as Triply Special Relativity and the Snyder's Model--de Sitter Special Relativity Duality", Physics Letters B 663 (2008) 270-274 arxiv 0801.1146
  21. ^ Han-Ying Guo, Chao-Guang Huang, Yu Tian, Hong-Tu Wu, Zhan Xu, Bin Zhou, "Snyder's Model -- de Sitter Special Relativity Duality and de Sitter Gravity", Class. Quantum Grav. 24 4009-4035, Published 31 July 2007, arxiv 0703078
  22. ^ Wu Hong-Tu, Huang Chao-Guang, Guo Han-Ying, "From the Complete Yang Model to Snyder's Model, de Sitter Special Relativity and Their Duality", Chinese Phys. Lett. 25 2751-2753, Issue 8 (August 2008), arxiv 0809.3560
  23. ^ Han-Ying Guo, "On Principle of Inertia in Closed Universe", Phys. Lett. B653 (2007) 88-94, arxiv hep-th/0611341
  24. ^ Han-Ying Guo, "Special Relativity and Theory of Gravity via Maximum Symmetry and Localization", Science in China A, Vol.51 [4] (2008) 568-603, arxiv 0707.3855
  25. ^ "Our Universe Prefers The De Sitter Special Relativity And Its Localization"
  26. ^ H.-Y. Guo, C.-G. Huang, Z. Xu, and B. Zhou, “On beltrami model of de Sitter spacetime,” Mod. Phys. Lett., vol. A19, pp. 1701–1710, 2004
  27. ^ H.-Y. Guo, B. Zhou, Y. Tian, and Z. Xu, "The triality of conformal extensions of three kinds of special relativity", Phys. Rev., vol. D75, p. 026006, 2007. arxiv hep-th/0611047
  28. ^ Chang Zhe, Chen Shao-Xia, and Huang Chao-Guang, Absence of GZK Cutoff and Test of de Sitter Invariant Special Relativity, 2005 Chinese Phys. Lett. 22 791-794
  29. ^ H.-Y. Guo, C.-G. Huang and B. Zhou, Temperature at horizon in de Sitter spacetime, Europhys. Lett. 72 1045-1051, arxiv hep-th/0404010
  30. ^ GUO Han-Ying, HUANG Chao-Guang, XU Zhan and ZHOU Bin, Three Kinds of Special Relativity via Inverse Wick Rotation,Chinese Phys. Lett. 22 2477-2480 , arxiv hep-th/0508094v1
  31. ^ R. Aldrovandi, J.P. Beltran Almeida, and J.G. Pereira, "Cosmological Term and Fundamental Physics", Int.J.Mod.Phys. D13 (2004) 2241-2248, arxiv gr-qc/0405104
  32. ^ Giovanni Amelino-Camelia, "Testable scenario for Relativity with minimum-length", Phys.Lett. B510 (2001) 255-263, arxiv hep-th/0012238
  33. ^ G.W. Gibbons, C.E. Patricot, "Newton-Hooke spacetimes, Hpp-waves and the cosmological constant", Class.Quant.Grav. 20 (2003) 5225, arxiv hep-th/0308200
  34. ^ Yu Tian, Han-Ying Guo, Chao-Guang Huang, Zhan Xu, Bin Zhou, "Mechanics and Newton-Cartan-Like Gravity on the Newton-Hooke Space-time", Phys.Rev. D71 (2005) 044030, arxiv hep-th/0411004
  35. ^ 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)
  36. ^ L. F. Abbott and S. Deser, “Stability of gravity with a cosmological constant,” Nucl. Phys.,vol. B195, pp. 76–96, 1982.
  37. ^ J. Kowalski-Glikman and S. Nowak, “Doubly special relativity and de Sitter space,” Class. Quant. Grav., vol. 20, pp. 4799–4816, 2003.
  38. ^ Shao-Xia Chen, Neng-Chao Xiao, Mu-Lin Yan, "Variation of the Fine-Structure Constant from the de Sitter Invariant Special Relativity", Chinese Physics C 2008 32 (08): 612—616 ISSN: 1674-1137 CN: 11-5641/O4, arxiv 0703110
  39. ^ C G Bohmer, T Harko, "Physics of dark energy particles", Foundations of Physics, Volume 38, Issue 3, pp.216-227, arxiv 0602081
  40. ^ Hamiltonian Formalism of the de-Sitter Invariant Special Relativity
  41. ^ De Sitter Thermodynamics from Diamonds’s Temperature
  42. ^ Universe Without Singularities. A Group Approach to De Sitter Cosmology
  43. ^ Leonardo Chiatti, "Fantappié-Arcidiacono theory of relativity versus recent cosmological evidences : a preliminary comparison", EJTP 15 (4), 17-36 (2007), arXiv:physics/0702178
  44. ^ Ashok Das (Rochester U), Otto C. W. Kong (Nat'l Central U, Taiwan), "Physics of Quantum Relativity through a Linear Realization", Phys.Rev. D73 (2006) 124029, arxiv gr-qc/0603114
  45. ^ Han-Ying Guo, Chao-Guang Huang, Yu Tian, Zhan Xu, Bin Zhou, "Snyder's Quantized Space-time and De Sitter Special Relativity", Front. Phys. China 2 (2007) 358-363, arxiv hep-th/0607016v2
  46. ^ Han-Ying Guo, "The Beltrami Model of De Sitter Space: From Snyder's quantized space-time to de Sitter invariant relativity", 15 pages. Invited talk given at `International workshop on noncommutative geometry and physics', Beijing, Nov. 7-10, 2005. To appear in the proceedings, arxiv hep-th/0607017
  47. ^ N. D. Birrell and P. C. W. Davies, Quantum fields in curved space. Cambridge (UK): Cambridge University Press, 1982.
  48. ^ J. Bros and U. Moschella, “Two-point functions and quantum fields in de Sitter universe,” Rev.Math. Phys., vol. 8, pp. 327–392, 1996.
  49. ^ J. Bros, H. Epstein, and U. Moschella, “Analyticity properties and thermal effects for general quantum field theory on de Sitter space-time,” Commun. Math. Phys., vol. 196, pp. 535–570,1998.
  50. ^ J. Bros, H. Epstein, and U. Moschella, “Lifetime of a massive particle in a de Sitter universe,” JCAP, 2008.
  51. ^ U. Moschella, “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.
  52. ^ U. Moschella, “Particles and fields on the de Sitter universe,” AIP Conference Proceedings, vol. 910,pp. 396–411, 2007.
  53. ^ E.Benedetto, " Fantappiè-Arcidiacono Spacetime and Its Consequences in Quantum Cosmology" Int J Theor Phys DOI 10.1007/s10773-009-9933-0

Further reading

  • R. Aldrovandi, J. G. Pereira Is Physics Asking for a New Kinematics?
  • S Cacciatori, V Gorini, A Kamenshchik, U Moschella "Conservation laws and scattering for de Sitter classical particles" - CLASSICAL AND QUANTUM GRAVITY, Issue 7 (7 April 2008) - iop.org, arxiv 0710.0315
  • R. Aldrovandi, J. P. Beltran Almeida, C. S. O. Mayor, J. G. Pereira, "de Sitter Relativity and Quantum Physics", Quantum Theory: Reconsideration of Foundations 4, 11-16 June 2007, Vaxjo University, Sweden (AIP Conference Proceedings, ed. by G. Adenier, A. Khrennikov and T. Nieuwenhuizen), arxiv 0710.0610
  • "Special Relativity: Will it Survive the Next 101 Years?", By Claus Lämmerzahl, Jürgen Ehlers, Published by Springer, 2005, ISBN 3540345221, 9783540345220.
  • "Projective Relativity, Cosmology, and Gravitation", By Giuseppe Arcidiacono, Published by Hadronic Press, 1986, ISBN 0911767398, 9780911767391
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