Einstein-aether theory

In physics the Einstein-æther theory is a controversial generally covariant generalization of general relativity which describes a spacetime endowed with both a metric and a unit timelike vector field named the æther. In particular such theories have a prefered reference frame.

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. 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. 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. 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.

Consistency

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 ghosts 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 poles with a wrong-sign residue and so could have ruined the unitarity of the S-matrix.

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

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

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

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

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

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

Here the c_i are dimensionless adjustable parameters of the theory.