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In [[physics]] the '''Einstein-aether theory''', also called '''aetheory''', is the name coined in 2004 for a modification of [[general relativity]] that has a [[preferred frame|preferred reference frame]] and hence violates [[Lorentz invariance]]. These  [[general covariance|generally covariant]] theories describes a [[spacetime]] endowed with both a [[Metric (general relativity)|metric]] and a unit timelike [[vector field]] named the [[Aether theories|aether]]. The ''aether'' in this theory is "a Lorentz-violating vector field"<ref>{{Cite journal |last1=Clifton |first1=Timothy |last2=Ferreira |first2=Pedro G. |last3=Padilla |first3=Antonio |last4=Skordis |first4=Constantinos |date=March 2012 |title=Modified gravity and cosmology |url=https://linkinghub.elsevier.com/retrieve/pii/S0370157312000105 |journal=Physics Reports |language=en |volume=513 |issue=1–3 |pages=1–189 |doi=10.1016/j.physrep.2012.01.001|arxiv=1106.2476 |bibcode=2012PhR...513....1C }}</ref> unrelated to older [[luminiferous aether]] theories; the "Einstein" in the theory's name comes from its use of Einstein's general relativity equation.<ref>{{Cite journal |last1=Eling |first1=Christopher |last2=Jacobson |first2=Ted |date=2004-03-08 |title=Static post-Newtonian equivalence of general relativity and gravity with a dynamical preferred frame |url=https://link.aps.org/doi/10.1103/PhysRevD.69.064005 |journal=Physical Review D |language=en |volume=69 |issue=6 |page=064005 |doi=10.1103/PhysRevD.69.064005 |arxiv=gr-qc/0310044 |bibcode=2004PhRvD..69f4005E |s2cid=15888510 |issn=1550-7998}}</ref>

== Relation to other theories of gravity ==
An Einstein-aether theory is an alternative theory of gravity that adds a vector field to the theory of general relativity. There are also scalar field modifications, including [[Brans–Dicke theory]], all included with [[Horndeski's theory]]. Going the other direction, there are theories that add tensor fields, under the name [[Bimetric gravity]] or both scalar and vector fields can be added, as in [[Tensor–vector–scalar gravity]].<ref name=CliftonFerreiraPadilla/>{{rp|30}}

==History==
The name "Einstein-aether theory" was coined in 2004 by T. Jacobson and D. Mattingly.<ref>{{Cite journal |last1=Jacobson |first1=T. |last2=Mattingly |first2=D. |date=2004-07-19 |title=Einstein-aether waves |url=https://link.aps.org/doi/10.1103/PhysRevD.70.024003 |journal=Physical Review D |language=en |volume=70 |issue=2 |page=024003 |doi=10.1103/PhysRevD.70.024003 |arxiv=gr-qc/0402005 |bibcode=2004PhRvD..70b4003J |s2cid=119355560 |issn=1550-7998 |quote=We refer to the system of the metric coupled to the aether as “Einstein-aether theory”}}</ref> This type of theory originated in the 1970s with the work of C.M.Will and K. Nordtvedt Jr. on gravitationally coupled vector field theories.<ref name=CliftonFerreiraPadilla>{{Cite journal |last1=Clifton |first1=Timothy |last2=Ferreira |first2=Pedro G. |last3=Padilla |first3=Antonio |last4=Skordis |first4=Constantinos |date=March 2012 |title=Modified gravity and cosmology |url=https://linkinghub.elsevier.com/retrieve/pii/S0370157312000105 |journal=Physics Reports |language=en |volume=513 |issue=1–3 |pages=1–189 |doi=10.1016/j.physrep.2012.01.001|arxiv=1106.2476 |bibcode=2012PhR...513....1C }}</ref>{{rp|42}}

In the 1980's Maurizio Gasperini added a [[scalar field]], which intuitively corresponded to a universal notion of [[time]], to the metric of [[general relativity]].<ref>{{cite journal |title=Singularity Prevention and Broken Lorentz Symmetry |journal=Classical and Quantum Gravity |volume=4 |issue=2 |pages=485–494 |doi=10.1088/0264-9381/4/2/026 |year=1987 |last1=Gasperini |first1=M. |bibcode=1987CQGra...4..485G |s2cid=250814796 }}</ref>  Such a theory will have a preferred [[Frame of reference|reference frame]], that in which the universal time is the actual time.

In 2000, Ted Jacobson and David Mattingly developed a model that allows the consequences of preferred frames to be studied.<ref>{{Cite journal |last1=Jacobson |first1=Ted |last2=Mattingly |first2=David |date=2001-06-26 |title=Gravity with a dynamical preferred frame |url=https://link.aps.org/doi/10.1103/PhysRevD.64.024028 |journal=Physical Review D |language=en |volume=64 |issue=2 |page=024028 |arxiv=gr-qc/0007031 |doi=10.1103/PhysRevD.64.024028 |bibcode=2001PhRvD..64b4028J |issn=0556-2821 |s2cid=119372246}}</ref>  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 aether at different points will not necessarily age at the same rate in the Jacobson–Mattingly theory.  In 2008 Ted Jacobson presented a status report on Einstein-aether theory.<ref>{{Cite arXiv |last=Jacobson |first=Ted |title=Einstein-aether gravity: A status report |date=2008-01-10 |class=gr-qc |eprint=0801.1547v2 }}</ref>

== Breaking Lorentz symmetry ==
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 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 ''Effective Field Theory for Massive Gravitons and Gravity in Theory Space'' by [[Nima Arkani-Hamed]], [[Howard Georgi]] and Matthew Schwartz.<ref>{{cite journal |arxiv=hep-th/0210184 |title=Effective Field Theory for Massive Gravitons and Gravity in Theory Space |journal=Annals of Physics |volume=305 |issue=2 |pages=96–118 |doi=10.1016/S0003-4916(03)00068-X |year=2003 |last1=Arkani-Hamed |first1=Nima |last2=Georgi |first2=Howard |last3=Schwartz |first3=Matthew D. |bibcode=2003AnPhy.305...96A |s2cid=1367086 }}</ref>  Einstein-aether theories provide a concrete example of a theory with broken Lorentz invariance and so have proven to be a natural setting for such investigations.

==Action==

The action of the Einstein-aether 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.

In particular it is assumed that the [[action (physics)|action]] may be written as the [[integral]] of a local [[Lagrangian density]]

:::<math>S=\frac{1}{16\pi G_N}\int d^4x\sqrt{-g}\mathcal L</math>

where ''G''<sub>N</sub> is [[Newton's constant]] and ''g'' is a [[Metric (general relativity)|metric]] with [[Minkowski signature]].  The Lagrangian density is

::<math>\mathcal L=-R-K^{ab}_{mn}\nabla_a u^m\nabla_bu^n-\lambda (g_{ab}u^au^b-1).</math>

Here ''R'' is the [[Ricci scalar]], <math>\nabla</math> is the [[covariant derivative]] and the tensor ''K'' is defined by

::<math>K^{ab}_{mn}=c_1g^{ab}g_{mn}+
c_2\delta^a_m\delta^b_n
+c_3\delta^a_n\delta^b_m+c_4u^au^bg_{mn}.
</math>

Here the ''c''<sub>i</sub> are dimensionless adjustable parameters of the theory.

==Solutions==

===Stars===

Several spherically symmetric solutions to ae-theory have been found.  Most recently Christopher Eling and [[Ted Jacobson]] have found solutions resembling [[star]]s<ref>{{cite journal |arxiv=gr-qc/0603058 |title=Spherical Solutions to Einstein-aether Theory: Static Aether and Stars |doi=10.1088/0264-9381/23/18/008 |last1=Jacobson |first1=Ted |last2=Mattingly |first2=David |journal=Classical and Quantum Gravity |year=2006 |volume=23 |issue=18 |pages=5625–5642 |bibcode=2006CQGra..23.5625E |s2cid=120259601 }}</ref> and solutions resembling [[black hole]]s.<ref>{{cite journal |arxiv=gr-qc/0604088 |title=Black Holes in Einstein-aether Theory |journal=Classical and Quantum Gravity |volume=23 |issue=18 |pages=5643–5660 |doi=10.1088/0264-9381/23/18/009 |year=2006 |last1=Eling |first1=Christopher |last2=Jacobson |first2=Ted |bibcode=2006CQGra..23.5643E |s2cid=119488152 }}</ref>

In particular, they demonstrated that there are no spherically symmetric solutions in which stars are constructed entirely from the aether. 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 aether'' solutions, which means that the aether points in the direction of a timelike [[killing vector]].

===Black holes and potential problems===

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 aethers.  Thus when a star collapses to form a black hole, somehow the aether must eventually become static even very far away from the collapse.

In addition the [[Stress–energy tensor|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 aether, where this property is independent of the equations of motion.

==Experimental constraints==

In a 2005 paper,<ref>{{cite journal |arxiv=hep-ph/0407034 |doi=10.1088/1126-6708/2005/07/029 |title=Universal dynamics of spontaneous Lorentz violation and a new spin-dependent inverse-square law force |date=2005 |last1=Arkani-Hamed |first1=Nima |last2=Cheng |first2=Hsin-Chia |last3=Luty |first3=Markus |last4=Thaler |first4=Jesse |journal=Journal of High Energy Physics |volume=2005 |issue=7 |page=029 |bibcode=2005JHEP...07..029A }}</ref>  [[Nima Arkani-Hamed]], Hsin-Chia Cheng, Markus Luty and [[Jesse Thaler]] have examined experimental consequences of the breaking of boost symmetries inherent in aether theories.  They have found that the resulting [[Goldstone boson]] leads to, among other things, a new kind of [[Cherenkov radiation]].

In addition they 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 evidence for an aether theory, although not necessarily that of Jacobson, ''et al.''

==See also==
*[[Aether theories]]
*[[Modern searches for Lorentz violation]]

==References==
{{reflist}}


[[Category:Aether theories]]
[[Category:Theories of gravity]]