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| In [[physics]] the '''Einstein æther theory''', also called '''æ-theory''', is a [[general covariance|generally covariant]] modification of [[general relativity]] which describes a [[spacetime]] endowed with both a [[Metric (mathematics)|metric]] and a unit timelike [[vector field]] named the [[Aether theories|æther]]. The theory has a [[preferred frame|preferred reference frame]] and hence violates [[Lorentz invariance|Lorentz invariant]].
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| ==History==
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| Einstein-æther theories were popularized by Maurizio Gasperini in a series of papers, such as [http://www.iop.org/EJ/abstract/0264-9381/4/2/026 Singularity Prevention and Broken Lorentz Symmetry] in the 1980s. In addition to the metric of [[general relativity]] these theories also included a [[scalar field]] which intuitively corresponded to a universal notion of [[time]]. Such a theory will have a preferred [[Frame of reference|reference frame]], that in which the universal time is the actual time. The dynamics of the scalar field is identified with that of an [[Aether theories|æther]] which is at rest in the preferred frame. This is the origin of the name of the theory, it contains Einstein's gravity plus an æther.
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| Einstein æther theories returned to prominence at the turn of the century with the paper [http://xxx.lanl.gov/abs/gr-qc/0007031 Gravity and a Preferred Frame] by Ted Jacobson and David Mattingly. Their theory contains less information than that of Gasperini, instead of a scalar field giving a universal time it contains only a unit [[vector field]] which gives the direction of time. Thus observers who follow the æther at different points will not necessarily age at the same rate in the Jacobson–Mattingly theory.
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| The existence of a preferred, dynamical time vector breaks the [[Lorentz symmetry]] of the theory, more precisely it breaks the invariance under [[Lorentz boost|boost]]s. This symmetry breaking may lead to a [[Higgs mechanism]] for the graviton which would alter long distance physics, perhaps yielding an explanation for recent [[supernova]] data which would otherwise be explained by a [[cosmological constant]]. The effect of breaking Lorentz invariance on [[quantum field theory]] has
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| a long history leading back at least to the work of Markus Fierz and [[Wolfgang Pauli]] in 1939. Recently it has regained popularity with, for example, the paper [http://xxx.lanl.gov/abs/hep-th/0210184 Effective Field Theory for Massive Gravitons and Gravity in Theory Space] by [[Nima Arkani-Hamed]], [[Howard Georgi]] and Matthew Schwartz. Einstein-æther theories provide a concrete example of a theory with broken Lorentz invariance and so have proven to be a natural setting for such investigations.
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| ==Consistency==
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| It is still not known whether Einstein æther theories exist as quantum theories. One immediate concern might be that the time vector, which breaks Lorentz invariance, will lead to [[Faddeev–Popov ghost]]s which fail to decouple and ruin the theory. This problem is avoided because the vector is of unit length in a timelike direction, and so its oscillations are spacelike. Therefore it does not contribute extra time derivatives to the denominator of the [[propagator]], which could have led to [[pole (complex analysis)|pole]]s with a wrong-sign [[residue (complex analysis)|residue]] and so could have ruined the [[unitarity]] of the [[S-matrix]].
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| ==The action==
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| The action of the Einstein æther theory is generally taken to consist of the sum of the [[Einstein–Hilbert action]] with a [[Lagrange multiplier]] λ that ensures that the time vector is a unit vector and also with all of the covariant terms involving the time vector ''u'' but having at most two derivatives.
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| In particular it is assumed that the [[action (physics)|action]] may be written as the [[integral]] of a local [[Lagrangian density]]
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| :::<math>S=\frac{1}{16\pi G_N}\int d^4x\sqrt{-g}\mathcal L</math>
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| where ''G''<sub>N</sub> is [[Newton's constant]] and ''g'' is a [[Metric (mathematics)|metric]] with [[Minkowski signature]]. The Lagrangian density is | |
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| ::<math>\mathcal L=-R-K^{ab}_{mn}\nabla_a u^m\nabla_bu^n-\lambda (g_{ab}u^au^b-1).</math>
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| Here ''R'' is the [[Ricci scalar]], <math>\nabla</math> is the [[covariant derivative]] and the tensor ''K'' is defined by
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| ::<math>K^{ab}_{mn}=c_1g^{ab}g_{mn}+
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| c_2\delta^a_m\delta^b_n
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| +c_3\delta^a_n\delta^b_m+c_4u^au^bg_{mn}.
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| </math>
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| Here the ''c''<sub>i</sub> are dimensionless adjustable parameters of the theory.
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| ==Solutions==
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| ===Stars===
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| Several spherically symmetric solutions to æ-theory have been found. Most recently [[Christopher Eling]] and [[Ted Jacobson]] have found solutions resembling [[star]]s in [http://xxx.lanl.gov/abs/gr-qc/0603058 Spherical Solutions to Einstein-Æther Theory: Static Æther and Stars] and solutions resembling [[black hole]]s in [http://xxx.lanl.gov/abs/gr-qc/0604088 Black Holes in Einstein-Æther Theory].
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| In particular they have demonstrated that there are no spherically-symmetric solutions in which stars are constructed entirely from the æther. Solutions without additional matter always have either [[naked singularity|naked singularities]] or else two asymptotic regions of spacetime, resembling a [[wormhole]] but with no [[horizon]]. They have argued that static stars must have ''static æther'' solutions, which means that the æther points in the direction of a timelike [[Killing vector]].
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| ===Black holes and potential problems===
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| However this is difficult to reconcile with static black holes, as at the [[event horizon]] there are no timelike Killing vectors available and so the black hole solutions cannot have static æthers. Thus when a star collapses to form a black hole, somehow the æther must eventually become static even very far away from the collapse.
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| In addition the [[stress tensor]] does not obviously satisfy the [[Raychaudhuri equation]], one needs to make recourse to the equations of motion. This is in contrast with theories with no æther, where this property is independent of the equations of motion.
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| ==Experimental constraints==
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| In [http://arxiv.org/abs/hep-ph/0407034 Universal Dynamics of Spontaneous Lorentz Violation and a New Spin-Dependent Inverse-Square Law Force] [[Nima Arkani-Hamed]], Hsin-Chia Cheng, Markus Luty and Jesse Thaler have examined experimental consequences of the breaking of boost symmetries inherent in æther theories. They have found that the resulting [[Goldstone boson]] leads to, among other things, a new kind of [[Cherenkov radiation]].
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| In addition that have argued that spin sources will interact via a new inverse square law force with a very unusual angular dependence. They suggest that the discovery of such a force would be very strong evidency for an æther theory, although not necessarily that of Jacobson, ''et al.''
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| ==See also==
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| *[[Aether theories]]
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| *[[Modern searches for Lorentz violation]]
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| ==References==
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| {{cite book|author=Christopher Eling, Ted Jacobson and David Mattingly|title=DESERFEST. A Celebration of the Life and Works of Stanley Deser|contribution=Einstein Æther Theory|arxiv=gr-qc/0410001|year=2004|isbn=981-256-082-3|publisher=WorldScientific|location=Singapore}} A review of the status of Einstein æther theory as of 2004.
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| ==External links==
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| *[http://arxiv.org/find/all/1/all:+AND+theory+AND+Einstein+aether/0/1/0/all/0/1 Einstein aether theory on arxiv.org]
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| [[Category:Aether theories]]
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| [[Category:Theories of gravitation]]
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