Einstein-Æther Theories  

In General > s.a. hořava-lifshitz gravity; modified general relativity.
* Idea: General relativity coupled to a dynamical, unit timelike vector field u; A generally covariant theory with a preferred vector field, or a preferred frame.
* Dynamics: The general form of the action is a sum of an Einstein-Hilbert term for the metric and terms involving u; The leading ones are

S[g, u] = (16πG)−1 ∫ d4x (−g)1/2 [R + c1 (∇a ub) (∇a ub) + c2 (∇a ua)2 + c3 (∇a ub) (∇b ua) + c4 (uaa ub)2] ,

but one can add terms with higher derivatives of u and/or higher-order terms in the curvature; Stability and energy positivity imply some inequalities for the coefficients ci.
@ General References: Jacobson & Mattingly PRD(01)gq/00, PRD(01)gq/00, gq/01-conf [in general relativity], PRD(04)gq; Eling et al gq/04-proc; Foster PRD(05); Foster & Jacobson PRD(06) [PN parameters]; Eling & Jacobson PRD(06)gq [2D theory]; Jacobson proc(08)-a0711, PoS-a0801 [rev]; Withers CQG(09)-a0905 [quantum effective field theory]; Donnelly & Jacobson PRD(10)-a1008 [stability]; Garfinkle et al PRE(12)-a1207 [constraint equations and the weak gravitational field limit]; Haghani et al a1404 [with ua = ∇aφ]; Sarbach et al CQG(19)-a1902 [Cauchy formulation]; > s.a. Ether.
@ Constraints: Bonvin et al PRD(08) [solar system]; Yagi et al PRD(14)-a1311 [binary pulsars].
@ Variations: Pujolàs & Sibiryakov JHEP(12)-a1109 [supersymmetric extension, and emergent Lorentz symmetry]; Jacobson & Speranza a1405 [on the scalar version]; Balakin & Lemos AP(14)-a1407 [with a Maxwell field]; Balakin PRD(16)-a1606 [axionic extension, and cosmology].

Phenomenology > s.a. black-hole thermodynamics; horizons [universal horizons]; positive-energy theorem.
* Newtonian limit: The slow-motion, weak-field approximation can be made to agree with that of general relativity, and in the limit one gets

GN = G/[1−(c1+c4)/2] .

* Propagating modes: There are 5, two spin-2 gravity modes {traveling at speed v2 = 1/[1 − (c1 + c3)], which leads to very strong bounds on the coefficients from the absence of vacuum Cerenkov radiation}, two spin-1 modes and one spin-0 mode {each of which has to travel at or above the speed of light, or possibly just a little bit below, without this leading to causality violations}; Gravitational waves can have a longitudinal component.
@ Cosmology in general: Tartaglia & Radicella PRD(07); Li et al PRD(08)-a0709 [primordial perturbations]; Zuntz et al PRL(08)-a0808; Carruthers & Jacobson PRD(11)-a1011 [deviations from isotropy and alignment]; Meng & Du PLB(12) [as an alternative to dark energy]; Barrow PRD(12)-a1201 [accelerating-universe solutions]; Saga et al PRD(13)-a1302 [generation of magnetic fields]; Trinh et al PRD(19)-a1811 [and the cmb]; > s.a. cmb polarization.
@ Cosmological solutions: Latta et al JCAP(16)-a1606 [perfect-fluid Kantowski-Sachs]; Campista et al CJP-a1807; van den Hoogen et al JCAP(18)-a1809 [scalar field Kantowski-Sachs]; Roumeliotis et al EPJC(20)-a1911 [Bianchi V]; Casalino et al PRD(20)-a2003 [non-singular FLRW].
@ Spherical solutions: Eling & Jacobson CQG(06); Coley et al JCAP(15)-a1508 [with a tilted perfect fluid source]; Coley & Leon GRG(19)-a1905; Chan et al a2003 [vacuum static, general solution].
@ Other solutions: Bertolami & Páramos PRD(05)ht [vacuum solutions]; Garfinkle et al PRD(07)gq [gravitational collapse]; Eling et al PRD(07)-a0705 [neutron stars]; Foster PRD(07)-a0706 [binaries, strong field effects]; Oost et al GRG(18)-a1804 [plane-fronted gravitational waves]; Paliathanasis & Leon EPJC(20)-a2004 [Einstein-æther-scalar]; > s.a. black holes; gödel solutions; White Hole [collapse].
@ And observations: Oost et al PRD(18)-a1802 [constraints, after GW170817]; Khodadi & Saridakis a2012 [EHT observations of M87*].

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