Einstein–Æther Gravity

Einstein–æther gravity couples the metric to a dynamical unit timelike vector. The vector selects a local rest frame without sacrificing spacetime diffeomorphism invariance, and its derivative couplings mix metric and æther perturbations into spin-2, spin-1, and spin-0 sectors. The central calculation is therefore not merely to count a vector field: it is to derive the characteristic speeds and kinetic signs of the coupled, constrained system and to identify which gravitational constant a laboratory actually measures.

Required background. Frames, Tetrads, and Spinors supplies orthonormal frames and local Lorentz decompositions used to define the æther rest frame.

Helpful background. Einstein Equations supplies the metric field equation; Gravitational Actions and Boundary Terms supplies the gravitational variational problem; Formulations of General Relativity separates constraints from evolution; and Gravity Beyond General Relativity locates this theory among alternative gravitational actions.

The unit-timelike action

We use the site convention $(-,+,+,+)$, $c=\hbar=k_B=1$, and explicit $G_{\mathrm{ae}}$. The æther satisfies $u^a u_a=-1$, matter couples minimally to $g_{ab}$, and the two-derivative action is

$$S=\frac{1}{16\pi G_{\mathrm{ae}}}\int d^4x\,\sqrt{-g}\left[ R-K^{ab}{}_{mn}\nabla_a u^m\nabla_b u^n +\lambda(u^a u_a+1)\right]+S_m[g,\psi],$$

with

$$K^{ab}{}_{mn}=c_1g^{ab}g_{mn}+c_2\delta^a_m\delta^b_n +c_3\delta^a_n\delta^b_m-c_4u^a u^b g_{mn}.$$

The four $c_i$ are dimensionless. The multiplier equation enforces the unit constraint, while variation of $u^a$ and $g_{ab}$ gives coupled second-order equations. Jacobson writes the review action with signature $(+,-,-,-)$ and $u^2=+1$; the expressions above translate the whole package—Einstein–Hilbert sign, unit constraint, $c_4$ term, and index contractions—rather than changing $u^2$ alone. Compare Jacobson 2008, §1, Eqs. (1.1)–(1.2), PDF.

The combinations that recur are

$$c_{13}=c_1+c_3,\qquad c_{14}=c_1+c_4,\qquad c_{123}=c_1+c_2+c_3.$$

No direct æther–matter coupling is included. That assumption fixes the metric light cone as the matter cone and makes comparisons of mode speeds meaningful; changing the matter metric changes the physical interpretation even if a field redefinition simplifies the vacuum action.

Linear modes and their cones

Linearize about the asymptotically flat æther-rest branch,

$$g_{ab}=\eta_{ab}+h_{ab},\qquad u^a=(1,0,0,0)+v^a,$$

and use Fourier modes $e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)}$. The norm condition gives $-2v^0+h_{00}=0$, hence $v^0=h_{00}/2$; it is not an independent wave amplitude. Diffeomorphism constraints remove gauge data. Five physical polarizations remain: two spin-2, two spin-1, and one spin-0. Solving the coupled principal equations gives

$$\begin{aligned} s_2^2&=\frac{1}{1-c_{13}},\\ s_1^2&=\frac{2c_1-c_1^2+c_3^2}{2c_{14}(1-c_{13})},\\ s_0^2&=\frac{c_{123}(2-c_{14})} {c_{14}(1-c_{13})(2+c_{13}+3c_2)}. \end{aligned}$$

These are speeds relative to the æther rest frame, not relative to an arbitrary coordinate grid. They reproduce Jacobson 2008, §4, Eqs. (4.1)–(4.3), PDF and the direct plane-wave calculation in Jacobson and Mattingly 2004, §§II–III and Table I, PDF. A sector with speed $s_I$ has characteristic covectors satisfying

$$\left[g^{ab}+(1-s_I^{-2})u^a u^b\right]k_a k_b=0.$$

In the rest frame this becomes $-\omega^2/s_I^2+\mathbf{k}^2=0$, an immediate sign check on the effective inverse metric. Each sector therefore has its own cone. A positive $s_I^2$ rules out a short-wavelength exponential instability on this background, but it does not by itself establish positive energy, nonlinear stability, or well-posedness on another branch.

The quadratic energies supply a separate test. In this convention their signs agree with the invariant statements in Jacobson 2008, §4, immediately after Eqs. (4.4)–(4.6), PDF: the spin-2 energy is positive after choosing the usual gravitational sign; the spin-1 sign follows $(2c_1-c_1^2+c_3^2)/(1-c_{13})$; and the spin-0 sign follows $c_{14}(2-c_{14})$. Denominators that vanish do not describe ordinary propagating modes and require a fresh constraint analysis.

The mode count can be seen without trusting a field-component tally. Decompose $h_{ab}$ and $v^a$ into irreducible representations of spatial rotations about the constant æther. In the tensor sector, the transverse-traceless spatial metric supplies two amplitudes. In the vector sector, the transverse æther perturbation mixes with the transverse metric shift; the shift equation is a constraint, leaving two amplitudes. In the scalar sector, the norm condition first fixes $v^0$, while the lapse, longitudinal shift, and nondynamical scalar combinations impose the remaining constraints; one scalar amplitude survives. Thus the physical count is $2+2+1=5$. The polarizations and the constraint relation are displayed explicitly in Jacobson and Mattingly 2004, §III, Eqs. (38)–(45), PDF.

This derivation assumes nonzero denominators and a flat, constant-æther background. Its exceptional hypersurfaces are physically informative. At $c_{14}=0$, generic æther speeds diverge because the relevant æther time derivatives disappear from the linear equations. At $c_{123}=0$, the scalar loses its ordinary spatial propagation. At $c_{13}=1$, the spin-2 kinetic coefficient vanishes. These are changes in constraint or principal-symbol rank, not harmless endpoints obtained by substituting into a formula derived away from them. Jacobson and Mattingly 2004, end of §III, PDF discusses the $c_{14}=0$ and other special cases.

The boundary and approximation regime also matter. The calculation uses vacuum plane waves on all of Minkowski space with physical wave numbers below the cutoff of the two-derivative effective action. A curved or time-dependent æther background makes the kinetic matrices position dependent, and higher-spatial-derivative operators can dominate at sufficiently large wave number. The three constants above are therefore infrared characteristic speeds of this branch, not universal ultraviolet front velocities.

Deriving the tensor speed directly

The spin-2 result can be checked without solving the scalar and vector constraints. Keep only $h_{ij}^{\mathrm{TT}}$, with $\partial_i h_{ij}^{\mathrm{TT}}=h_{ii}^{\mathrm{TT}}=0$, and set $h_{0\mu}=v^\mu=0$. Then $\nabla_i u_j=\dot h_{ij}^{\mathrm{TT}}/2$ at first order. The $c_1$ and $c_3$ terms therefore change the time-derivative coefficient, while the expansion and acceleration terms vanish. Up to boundary terms,

$$S_2^{(2)}=\frac{1}{64\pi G_{\mathrm{ae}}} \int d^4x\left[ (1-c_{13})\dot h_{ij}^{\mathrm{TT}}\dot h_{ij}^{\mathrm{TT}} -\partial_kh_{ij}^{\mathrm{TT}}\partial_kh_{ij}^{\mathrm{TT}} \right].$$

Its equation is $(1-c_{13})\ddot h_{ij}^{\mathrm{TT}}-\nabla^2h_{ij}^{\mathrm{TT}}=0$, giving the quoted $s_2^2$. For $G_{\mathrm{ae}}\gt0$, $1-c_{13}\gt0$ also makes the tensor kinetic term positive. This illustrates why a wave speed is a ratio of gradient and kinetic coefficients, not a coupling read directly from the action.

The Newton constant and two limiting checks

The constant multiplying the action is not the Cavendish constant. On the weak-field, static, asymptotically flat branch, the modified Poisson equation may be written

$$\left(1-\frac{c_{14}}{2}\right)\nabla^2\Phi=4\pi G_{\mathrm{ae}}\rho,$$

so

$$G_N=\frac{G_{\mathrm{ae}}}{1-c_{14}/2}.$$

This is Jacobson 2008, §2, Eq. (2.1), PDF. Dimensions check: $G_N$ and $G_{\mathrm{ae}}$ both have mass dimension $-2$, while the correction is dimensionless. With $G_{\mathrm{ae}}\gt0$, attractive finite Newtonian gravity on this branch requires $c_{14}\lt2$.

Two limits diagnose the calculation. First, setting the couplings to zero gives $G_N\to G_{\mathrm{ae}}$ and $s_2\to1$, recovering the GR tensor sector. The vector and scalar formulas become path-dependent ratios such as $c_{123}/c_{14}$: the extra modes do not generically approach free, canonically normalized particles when every kinetic coefficient vanishes. Thus “$c_i\to0$” is a controlled GR limit for metric observables only after sources, normalization, and strong-coupling scales are specified.

Second, the field redefinition $g'_{ab}=g_{ab}+(1-\zeta)u_a u_b$ rescales all vacuum mode cones together. It cannot erase a speed difference relative to matter if matter remains minimally coupled to the original $g_{ab}$. This distinguishes a change of variables from a change of physical matter coupling.

Common pitfalls

Reading $G_{\mathrm{ae}}$ as the measured Newton constant. The æther contributes already in the static constraint. Use $G_N=G_{\mathrm{ae}}/(1-c_{14}/2)$ before comparing with a gravitational observable.

Treating real speeds as complete viability. Real $s_I$ test only a specified linearized branch. Energy signs, constraint degeneracies, Cherenkov kinematics, strong coupling, and nonlinear evolution remain independent questions.

Changing only the norm sign. Translating a $(+,-,-,-)$ source by writing $u^2=-1$ while retaining its remaining action signs changes the theory. Translate the complete sign package and rederive one rest-frame dispersion relation.

Exercises

For $c_1=0.10$, $c_2=0.04$, $c_3=-0.02$, and $c_4=0.03$, compute the three squared speeds and $G_N/G_{\mathrm{ae}}$. What can and cannot be concluded from the result?

Solution

The useful combinations are $c_{13}=0.08$, $c_{14}=0.13$, and $c_{123}=0.12$. Hence

$$s_2^2=\frac1{0.92}\simeq1.087,$$ $$s_1^2=\frac{0.20-0.010+0.0004}{2(0.13)(0.92)} \simeq0.796,$$

and

$$s_0^2=\frac{(0.12)(1.87)}{(0.13)(0.92)(2.20)} \simeq0.853.$$

Also $G_N/G_{\mathrm{ae}}=1/(1-0.065)\simeq1.070$. All three squared speeds are positive, so the frozen-coefficient modes have real frequencies on this Minkowski branch. The spin-1 and spin-0 modes are subluminal relative to minimally coupled matter, which activates the Cherenkov question studied later. These numbers alone do not prove positive energy, nonlinear stability, or acceptable empirical bounds.

References

• Jacobson, T. “Einstein-æther gravity: a status report.” Proceedings of Science QG-PH (2007) 020. DOI. Open PDF.
• Jacobson, T., and D. Mattingly. “Einstein-Aether Waves.” Physical Review D 70 (2004) 024003. DOI. Open PDF.