Harmonic coordinates

In General Relativity (GR), coordinates play only a secondary role, and according to the spacetime interpretation there are no preferred coordinates at all, all coordinates are on equal foot. But what if one thinks that there nonetheless exist some preferred coordinates \(\mathfrak{x}^{\nu}\), preferred for reasons not yet known, in particular preferred by some theory of quantum gravity, or simply some more fundamental theory? What would be the candidates for reasonable coordinate conditions?

The surprising answer is that there is, essentially, only one candidate for preferred coordinates, namely the harmonic coordinates: \[\square \mathfrak{x}^{\nu} = \partial_{\mu} (g^{\mu\nu} \sqrt{-g}) = 0.\]

The harmonic coordinates are quite old, they have been proposed independently by de Donder [1] and Lanczos [2]. It essentially simplifies the Einstein equations, and, moreover, in a qualitatively important way. The Einstein tensor obtains the form \[G^{mn} = g^{ab}\partial_a\partial_b g^{mn} + F^{mn}(g^{pq}, \partial_k g^{pq})\] or \[G^{mn}\sqrt{-g} = \square g^{mn} + \tilde{F}^{mn}(g^{pq}, \partial_k g^{pq}).\] This simplification has allowed Bruhat [3] to prove, in harmonic coordinates, the first local existence and uniqueness theorems for GR. So, one of the most important properties of preferred coordinates, namely the simplification of the equations, is clearly fulfilled.

Fock has in [4] provided a lot of arguments in favor of interpreting harmonic coordinates as preferred. One argument was a proof that for insular configurations (all masses concentrated in some local region) with natural boundary conditions at infinity the harmonic coordinates are unique.

There is one property, which would be required if the preferred coordinates define some homogeneous flat background, like a Newtonian absolute space or so: Given some set of preferred coordinates, one can use rotations and translations of these coordinates to obtain other coordinates, and these other coordinates would have to be preferred coordinates too. This seems harmless, but the result is that the equation for the preferred coordinates has to be a linear equation for them. Once this linear equation defines the preferred coordinates in terms of other, general coordinates, it should be itself a covariant equation. With the need for a covariant linear equation for the preferred coordinates, the harmonic condition \(\square \mathfrak{x}^\nu=0\) becomes the obvious choice. Note that to add a mass-like term \(\square \mathfrak{x}^\nu - c \mathfrak{x}^\nu =0\) would destroy translational symmetry, the equation should not depend on \(\mathfrak{x}^\nu\), only on derivatives, to have translational symmetry.

Harmonic coordinates for our universe

The large scale universe is usually described in GR with an FLRW ansatz. This ansatz allows for a global curvature parameter, but observation has not given any indication for a nontrivial curvature parameter, so we can ignore nontrivial curvature and consider only the case of a flat universe. In this case, the ansatz is \[ ds^2 = d\tau^2 - a^2(\tau) (dx^2+dy^2+dz^2).\] Here, the spatial coordinates are already harmonic coordinates. All one has to do to get all coordinates harmonic is to introduce harmonic time. But this appears easy, and the ansatz in harmonic coordinates will be \[ds^2=a^6(\mathfrak{t}) d\mathfrak{t}^2 - a^2(\mathfrak{t})(d\mathfrak{x}^2+d\mathfrak{y}^2+d\mathfrak{z}^2).\] So, for the spatial coordinates it appears that the most natural choice of coordinates - Euclidean coordinates comoving with the matter, in rest relative to the background radiation - are also harmonic coordinates.

The harmonic time coordinate has the surprising nice property that the Big Bang singularity disappears: For \(a(\tau)=\tau^\alpha\) we obtain \(d\tau=a^3(\tau) d\mathfrak{t}= \tau^{3\alpha} d\mathfrak{t}\), \(\frac{d\mathfrak{t}}{d\tau} = \tau^{-3\alpha}\), \(\mathfrak{t}= \frac{1}{1-3\alpha} \tau^{1-3\alpha}\), thus, as for a matter-dominated \(\alpha=\frac{2}{3}\), as for a radiation-dominated \(\alpha=\frac12\) for \(\tau\to 0\) we obtain \(\mathfrak{t}\to -\infty\).


  1. T. de Donder, La gravifique einsteinienne, Paris, 1921
  2. K. Lanczos, Ein vereinfachendes Koordinatensystem fuer die Einsteinschen Gravitationsgleichungen Phys. Z., 23, 537 (1922)
  3. Y. Bruhat, The Cauchy Problem, in L. Witten (ed.) Gravitation: an Introduction to Current Research, Wiley, New York (1962)
  4. V. Fock, The Theory of Space Time and Gravitation, Pergamon Press, Oxford 1964,
    В.А. Фок, Теория пространства времени и тяготения, Москва, 1955