# Einstein aether theory

In physics the Einstein æther theory, also called æ-theory, is a generally covariant modification of general relativity which describes a spacetime endowed with both a metric and a unit timelike vector field named the æther. The theory has a preferred reference frame and hence violates Lorentz invariance.

## History

Einstein-æther theories were popularized by Maurizio Gasperini in a series of papers, such as Singularity Prevention and Broken Lorentz Symmetry in the 1980s.[1] 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 reference frame, that in which the universal time is the actual time. The dynamics of the scalar field is identified with that of an 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.

Einstein æther theories returned to prominence at the turn of the century with the paper Gravity and a Preferred Frame by Ted Jacobson and David Mattingly.[2] 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.

The existence of a preferred, dynamical time vector breaks the Lorentz symmetry of the theory, more precisely it breaks the invariance under boosts. 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.[3] 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. In 2004, Eling, Jacobson and Mattingly wrote a review of the status Einstein æther theory as of 2004.[4]

## The action

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.

In particular it is assumed that the action may be written as the integral of a local Lagrangian density

${\displaystyle S={\frac {1}{16\pi G_{N}}}\int d^{4}x{\sqrt {-g}}{\mathcal {L}}}$

where GN is Newton's constant and g is a metric with Minkowski signature. The Lagrangian density is

${\displaystyle {\mathcal {L}}=-R-K_{mn}^{ab}\nabla _{a}u^{m}\nabla _{b}u^{n}-\lambda (g_{ab}u^{a}u^{b}-1).}$

Here R is the Ricci scalar, ${\displaystyle \nabla }$ is the covariant derivative and the tensor K is defined by

${\displaystyle K_{mn}^{ab}=c_{1}g^{ab}g_{mn}+c_{2}\delta _{m}^{a}\delta _{n}^{b}+c_{3}\delta _{n}^{a}\delta _{m}^{b}+c_{4}u^{a}u^{b}g_{mn}.}$

Here the ci are dimensionless adjustable parameters of the theory.

## Solutions

### Stars

Several spherically symmetric solutions to æ-theory have been found. Most recently Christopher Eling and Ted Jacobson have found solutions resembling stars[5] and solutions resembling black holes.[6]

In particular, they 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 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.

### 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 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.

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.

## Experimental constraints

In 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.

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 evidence for an æther theory, although not necessarily that of Jacobson, et al.