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*] = (16π*G*)^{−1}
∫ d^{4}*x* (−*g*)^{1/2}
[*R* + *c*_{1} (∇_{a}*u _{b}*)
(∇

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 *c _{i}*.

@

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@

**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

*G*_{N}
= *G*/[1−(*c*_{1}+*c*_{4})/2] .

* __Propagating modes__:
There are 5, two spin-2 gravity modes{traveling at speed *v*_{2}
= 1/[1−(*c*_{1}+*c*_{3})],
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];
> s.a. cmb polarization.

@ __Cosmological solutions__: Latta et al JCAP(16)-a1606 [perfect-fluid Kantowski-Sachs].

@ __Spherical solutions__: Eling & Jacobson CQG(06);
Coley et al JCAP(15)-a1508 [with a tilted perfect fluid source].

@ __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 a1804 [plane-fronted gravitational waves];
> s.a. black holes;
gödel solutions; White Hole [collapse].

@ __Related topics__:
Oost et al a1802 [constraints, after GW170817].

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