discussion

The Ether Interpretation of the Einstein equations of General Relativity

It is almost unknown that the equations of General Relativity (GR), the Einstein Equations, also allow an ether interpretation.

Short introduction

In this ether interpretation, there exists a set of preferred coordinates. I denote them using the old, fraktal German letters (\(\mathfrak{x}\) (for x) and (\(\mathfrak{t}\) (for t), which have been often used at the time when ether theory was popular, so that we have \(\mathfrak{x}^i, \mathfrak{t}=\mathfrak{x}^0 \). They are defined by the harmonic condition: \[ \square \mathfrak{x}^{\nu} = \left(\frac{\partial}{\partial \mathfrak{x}^\mu} g^{\mu\kappa} \sqrt{-g}\frac{\partial}{\partial \mathfrak{x}^\kappa} \right) \mathfrak{x}^{\nu} = \frac{\partial}{\partial \mathfrak{x}^\mu} \left(g^{\mu\nu} \sqrt{-g}\right) = 0\]

The harmonic coordinates are often used in GR, because, as one would expect from preferred coordinates, they essentially simplify the Einstein equations. The four harmonic coordinates \(\mathfrak{x}^m,\, 0\le m \le 3\) define an absolute Euclidean background space with the Cartesian spatial coordinates \(\mathfrak{x}^i,\, 1\le i\le 3\) and absolute time \(\mathfrak{t}=\mathfrak{x}^0\). The time coordinate has to be time-like and the spatial coordinates space-like.

The ether interpretation of the gravitational field is now defined in the following way:

The consequence of these definitions is that the harmonic condition becomes, in terms of the ether variables, the continuity equation: \[ \partial_{\mathfrak{t}} \rho + \partial_i (\rho v^i) = 0,\] as well as the Euler equations: \[ \partial_{\mathfrak{t}} (\rho v^j) + \partial_i (\rho v^i v^j - \sigma^{ij}) = 0.\]

Thus, the ether in this interpretation fulfills well-known equations of classical condensed matter physics.

While the preferred coordinates \(\mathfrak{x}^i\), \(\mathfrak{t}\) define absolute space and time, these cannot be measured with usual clocks and rulers. What is measured by clocks is defined by the same formula as in GR, namely by GR "proper time": \[ \tau = \int_\gamma \sqrt{g_{mn} \frac{d\gamma^m}{dt}\frac{d\gamma^n}{dt}} dt\] if the clock moves along a trajectory \(\gamma^m(t)\). This formula shows that clocks do not show the true time \(\mathfrak{t}\), but are distorted by the gravitational field \(g_{mn}(\mathfrak{x},\mathfrak{t})\) as well as by the velocity of the clock itself. Similarly, rulers are distorted too by the gravitational field as well as their velocity.

Difference between this ether interpretation and the spacetime interpretation of GR

It should be noted that this ether interpretation is not always possible. The possibility of this ether interpretation requires some, but not many, additional restrictions.

But, first of all, locally, in some small environment of some event, the ether interpretation is always possible. Indeed, to find some local harmonic coordinates is always possible, and to identify among them one time-like and three space-like coordinates is also always possible.

Trivial topology

Globally, this is different. First, the ether interpretation requires that the solution is defined on \(\mathbb{R}^4 \cong \mathbb{R}^3 \times \mathbb{R}\). This excludes all GR solutions with nontrivial topology.

No causal loops

Second, the time coordinate \(\mathfrak{t}\) should be globally a time-like function. That \(\mathfrak{t}\) is time-like is necessary, because this translates into the condition that the ether density \(\rho\) has to be positive. This also excludes some solutions of the Einstein equations, namely those with closed causal loops, like the Gödel solution for a rotating universe.

The ether is complete if defined for all coordinates

There are other differences: First, the ether interpretation leads to a different concept of completeness of a solution. It is complete if it is defined for all values of the preferred coordinates \(-\infty < \mathfrak{x}^m < \infty\). Instead, a GR solution is complete only if the metric is geodetically complete. As the result, what is complete from point of view of the ether interpretation may be incomplete from point of view of the spacetime interpretation.

Frozen stars instead of black holes

This is relevant for the interpretation of the gravitational collapse. The solution itself is the same as in GR. The preferred harmonic time coordinate, which is, initially, simply the Minkowski time coordinate, becomes infinite at horizon formation. But this does not define a singularity - it simply means that the complete solution does not contain the part behind the horizon. This is not in conflict with the interpretation of what happens with clocks - it means, the clocks become near the horizon exponentionally slower, and essentially simply stop. So we have a frozen star.

No singularity at the big bang

A similar reinterpretation happens near the big bang singularity. Here, the preferred harmonic time coordinate also reaches \(-\infty\). Thus, in terms of the ether interpretation, the universe starts from a frozen state where nothing moves.

The background has to be symmetric too

The last difference is that the notion of symmetry of a solution is different. A symmetric solution does not only have to have a symmetry for the metric, but this symmetry has to be a symmetry of the background as well. There can be, of course, symmetries of observables only, which do not have a corresponding symmetry transformation of the background. But such a symmetry would be only some artefact, not a fundamental symmetry of that solution.

This changes the interpretation of the homogeneous isotropic FLRW solutions: The FLRW solution with non-zero spatial curvature are no longer homogeneous solutions. Their symmetry is not compatible with the symmetry of the Euclidean background. Only the flat universe is homogeneous in the ether interpretation. So, if we assume, as the cosmological principle, that the universe has to be, in the large distance on average, homogeneous and isotropic, then the consequence is that the spatial curvature has to be zero. So, the consequence of the ether interpretation is the prediction that the universe has to be flat. This is only a weak prediction - it depends on the cosmological principle as an independent hypothesis. But it nicely corresponds to observation.

What Popper's criterion of empirical content tells us

Let's note that the additional restrictions - trivial topology, no causal loops - are in no way weaknesses, but, instead, a strength of the ether interpretation. This is because they define, in comparison with the spacetime interpretation, additional possibilities for falsification of the theory by observation: If we would observe solutions with nontrivial topology, say, some wormholes, the ether interpretation would be falsified, the spacetime interpretation, instead, would survive this. The same holds for the restriction that there cannot be any closed causal loops: Observing such a causal loop would falsify the ether interpretation but not the spacetime interpretation. Thus, in agreement with Popper's criterion of empirical content, the ether interpretation is preferable, because it has a higher empirical content.

Once, in principle, the two interpretations have a different empirical content, they could, as well, be considered as different physical theories. But, given that these differences are quite minimal, and the physical equations of above theories - the Einstein equations in harmonic coordinates - are exactly the same, I prefer to name this theory the ether interpretation of the Einstein equations of GR. This emphasizes that there is a difference between the two theories - else, it would be more natural to name it the ether interpretation of GR itself.

Weaknesses of the ether interpretation

The ether interpretation has, nonetheless, also a weak point. This weak point is that the harmonic condition - which is, in the ether interpretation, a physical equation - is not an Euler-Lagrange equation, is not derived from a Lagrange formalism. In this particular aspect, GR in its spacetime interpretation is preferable, it has a Lagrange formalism, defined by the Hilbert-Einstein Langrangian.

The non-existence of a Lagrange formalism makes some other things problematic too. In particular, we cannot use the Noether theorem to derive energy and momentum conservation laws. This issue is very problematic in the spacetime interpretation too - there is no local energy and momentum density for the gravitational field, all what is available is some pseudo-tensor, which does not allow for a physical interpretation in the spacetime interpretation. And it also makes quantization more difficult - quantization would be much simpler if there would be a local energy density to define the Hamilton operator.

The solution: A different ether theory of gravity

Fortunately, this problem can be solved by a minor modification of the theory. One has to add a term to the GR Lagrangian which gives the harmonic condition as an additional equation. This can be done with the following Lagrangian: \[ L = -\frac{1}{8\pi G} \left(\Upsilon g^{00} - \Xi g^{ii}\right)\sqrt{-g} + L_{GR} + L_{matter}\] This Lagrangian could be rewritten as one which depends explicitly on the preferred coordinates \(\mathfrak{t}, \mathfrak{x}^i\) as function depending on arbitrary other coordinates x: \[ L = -\frac{1}{8\pi G} \left(\Upsilon g^{mn}\partial_m\mathfrak{t}(x)\partial_n\mathfrak{t}(x) - \Xi\delta_{ij} g^{mn}\partial_m\mathfrak{x}^i(x)\partial_n\mathfrak{x}^j(x)\right)\sqrt{-g} + L_{GR} + L_{matter}.\]

This reformulation makes it quite obvious that the Euler-Lagrange equations for the functions \(\mathfrak{t}(x), \mathfrak{x}^i(x)\) will be simply the harmonic equations. But the additional term adds also another term to the Einstein equations. So, this theory becomes a different theory of gravity, with a different equation, and is no longer an ether interpretation of the Einstein equations.

This theory is named General Lorentz Ether Theory and will be in more detail considered here. It has been published in peer-reviewed journal:

I. Schmelzer, A generalization of the Lorentz ether to gravity with general-relativistic limit, Advances in Applied Clifford Algebras 22, 1 (2012), p. 203-242.

The ether interpretation discussed here is, then, simply the limit \(\Xi,\Upsilon \to 0\) of this ether theory, which already has a Lagrange formalism, and, as a consequence, also local energy and momentum conservation laws. Even if in this limit the Lagrange formalism does no longer give the harmonic equation, once it is the same for all \(\Xi,\Upsilon\), thus, does not change during the limit process \(\Xi,\Upsilon \to 0\), it can be considered as a physical equation in this limit too. Moreover, the Noether theorem gives the tensor \[ \mathfrak{g}^{mn} = g^{mn}\sqrt{-g} \] as the energy-momentum tensor of the ether, and it remains conserved, thus, we can interpret this tensor also as the energy-momentum tensor of the ether.