An old and seemingly discredited notion of ether has found its new incarnation in a "metric theory of gravity" by I. Schmelzer, who managed to find for his new ether a consistent and instructive interpretation in terms of condensed matter physics. Prof.V.A.Petrov

# General Lorentz Ether Theory (GLET)

is a metric theory of gravity on Newtonian background with condensed matter interpretation and the Lagrangian $L = \frac{1}{8\pi G}(\Xi g^{ii} - \Upsilon g^{00})\sqrt{-g}+ L_{GR}(g_{\mu\nu}) + L_{matter}(g_{\mu\nu},\phi_{m})$ which is derived from simple first principles. It contains, additionally to the standard GR Lagrangian, two additional terms which are not covariant, but depend on preferred coordinates $$\mathfrak{x}^{i}, \mathfrak{t}$$ which we denote here with the old German fraktal letters for $$x^i, t$$.

• The preferred coordinates $$\mathfrak{x}^{i}, \mathfrak{t}$$ describe a Newtonian background of absolute space and time. These preferred coordinates are harmonic.
• The Einstein Equivalence Principle holds exactly: The equations for the matter fields do not depend on the preferred coordinates.
• The Einstein equations of GR in harmonic gauge appear in the natural limit $$\Xi,\Upsilon \to 0$$. This limit defines an ether interpretation of the Einstein equations of GR. As a consequence, most of the observational support for GR does not allow to distinguish GR from GLET, and is, therefore, observational support for GLET as well. But, on the other hand, that means that observational problems of GR will also become observational problems for GLET. In particular, in GLET we also have to assume some dark matter as well as inflation. Problems shared by GR and GLET will be considered here.

## Why should one revive ether theory?

Because this ether theory allows to solve some serious problems of modern physics, namely:

### Quantization of gravity

The principles of General Relativity (GR) and quantum theory appear to be in a deep conceptual conflict. As a consequence, quantization of GR becomes a serious problem.

Ether theory presents the gravitational field as a condensed matter theory on an Euclidean background. But we already know how such theories have to be quantized. All the conceptual problems of GR quantization simply disappear if we have a fixed Newtonian background and absolute time. And even non-renormalizability is no longer problematic: The continuous ether theory anyway has to be replaced, below some unknown critical distance, by a different theory, which describes some sort of "atomic ether".

### The violation of Bell's inequality

Quantum theory is non-local. This is a consequence of Bell's theorem, which shows that Bell's inequality has to hold in any Einstein-causal realistic theory. So, or one has to give up Einstein causality, or realism. The notion of "realism" which one would have to give up is a quite weak one, essentially one needs only the EPR criterion of reality to prove the theorem. Thus, to give up realism would be very strange.

Even more, there is a variant of Bell's theorem which does not even use this weak notion of realism, but relies on causality alone, in particular on Reichenbach's principle of common cause. Thus, one anyway would have to weaken Einstein causality, in such a serious way, that Reichenbach's common cause principle no longer works. That means, we would have correlations which would have to remain unexplainable - nor a direct causal influence, nor some hidden common cause could explain them.

But if one gives up Einstein causality, one essentially has to go back to the Lorentz ether, and, then, to find a way to extend it to gravity. Which is what is done here.

### Singularities of General Relativity

GR has singularities, and these singularities appear in physically important solutions, and, even worse, remain there even for small modifications. A singularity always means that the theory is wrong, that it has to be replaced by a different theory near the singularity. The ether theory presented here gets rid of the most well-known and unavoidable singularities of GR - the big bang and black hole singularities.

### Local energy and momentum conservation laws

GR has a big conceptual problem with energy and momentum conservation. This is not a practical problem - because there is a replacement for energy and momentum conservation laws, known in different variants, but they all have a problem: what is conserved is a so-called pseudotensor, and such a pseudotensor does not allow a physical interpretation in the standard spacetime interpretation of GR, where all physical fields have to be tensor fields.

Instead, the ether theory has standard local energy and momentum densities also for the gravitational field.

## Papers for professionals

I. Schmelzer, A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Foundations of Physics, vol. 39, nr. 1, p. 73 (2009), resp. arxiv:0908.0591. This paper contains, in appendix A, also a short introduction into GLET.

I. Schmelzer, Black Holes or Frozen Stars? A Viable Theory of Gravity without Black Holes, in Bauer, A.J., Eiffel, D.G. (eds.), Black Holes: Evolution, Theory and Thermodynamics, Nova Science Publishers (2012), pp. 117-138, also at arxiv:1003.1446. This paper considers the viability of the theory for the case $$\Upsilon>0$$, where the theory predicts stable frozen stars, slightly greater than their black hole horizon size, instead of black holes.

I. Schmelzer, The background as a quantum observable: Einstein's hole argument in a quasiclassical context, arxiv:0909.1408. This paper describes a thought experiment for quantum gravity, which shows that in quantum gravity the background becomes an observable, so that background-free theories like GR cannot be quantized in a reasonable way.

It is worth to note that the equations of GLET are identical to those of Logunov's "Relativistic Theory of Gravity" (RTG). This allows to reuse some of the mathematical results reached for RTG in GLET. Nonetheless, GLET has some important differences. In particular, it allows for other signs of the constants, while RTG is restricted to the case $$\Xi,\Upsilon>0$$ and a negative cosmological constant. It contains also another, more reasonable causality condition. In particular, some solutions of GLET would be forbidden in RTG because of its causality condition.

The FAQ is, instead, written for laymen.

### Condensed matter (ether) interpretation

For the preferred coordinates $$\mathfrak{x}^{i}, \mathfrak{t}$$ old German fraktal letters for x and t will be used. The time coordinate $$\mathfrak{x}^{0}=\mathfrak{t}$$ has to be time-like, the spatial coordinates $$\mathfrak{x}^{i}, 1\le i\le 3$$, have to be spacelike. This leads to a variant of the ADM decomposition: The metric $$g_{\mu\nu}(\mathfrak{x}^i, \mathfrak{t})$$ decomposes into a scalar, positive density $$\rho(\mathfrak{x}^{i}, \mathfrak{t})>0$$ of the ether, a spatial vector field $$v^{i}(\mathfrak{x}^{i}, \mathfrak{t})$$ which defines the velocity of the ether, and a negative-definite metric tensor field $$\sigma^{ij}(\mathfrak{x}^{i}, \mathfrak{t})$$ which defines the stress tensor of the ether: $\begin{eqnarray} g^{00}\sqrt{-g} &=& \rho\\ g^{i0}\sqrt{-g} &=& \rho v^i\\ g^{ij}\sqrt{-g} &=& \rho v^i v^j + \sigma^{ij} \end{eqnarray}$

The harmonic condition, often used in GR as a coordinate condition, is in this theory a physical equation: $\square \mathfrak{x}^\nu = \frac{\partial}{\partial \mathfrak{x}^\mu} \left(g^{\mu\nu} \sqrt{-g}\right) = 0.$

In the ether variables, this gives classical continuity and Euler equations: $\begin{eqnarray} \frac{\partial}{\partial \mathfrak{t}} \rho + \frac{\partial}{\partial \mathfrak{x}^i} (\rho v^i) &=& 0,\\ \frac{\partial}{\partial \mathfrak{t}} (\rho v^j) + \frac{\partial}{\partial \mathfrak{x}^i} (\rho v^i v^j + \sigma^{ij}) &=& 0. \end{eqnarray}$

The GLET Lagrangian can be derived from first principles of such an ether interpretation. The basic principle is that continuity and Euler equations of the ether appear as Noether conservation laws in some special variant of the Noether theorem.

### Predictions

The limit $$\Xi, \Upsilon \to 0$$ gives the Einstein equations of General Relativity (GR) in harmonic coordinates, or, in other words, an ether interpretation of the Einstein equations of GR. This allows to recover all the predictions of GR which have been successfully tested (Solar system predictions, gravitational lenses, pulsar orbits). Nonetheless, even in this limit a few qualitative differences remain:

• There are no solutions with non-trivial topology, in particular no wormholes and no closed universe. The topology is always trivial, and Euclidean space $$\mathbb{R}^3$$ together with absolute time $$\mathbb{R}$$.
• There are no solutions with closed causal loops, like the Gödel solution of a rotating universe in GR.
• The flat universe is preferred for symmetry reasons. Different from GR, only the flat universe is really homogeneous and isotropic.

For non-zero values of the parameters, we have

• For $$\Upsilon>0$$, there is a time-symmetric Big Bounce instead of the Big Bang, without Big Bang singularity. This solves the horizon problem of GR cosmology.
• For $$\Upsilon>0$$, there will be stable gravastars (or frozen stars) instead of black holes, with radius slightly greater than their Schwarzschild radius.
• We obtain some massless scalar dark matter fields.
• The parameter $$\Xi>0$$ leads to a cosmological term which may be used to fit some of the dark energy in cosmology.
• The critical length, where an atomic ether structure becomes important, differs from the Planck length — it seems to increase together with the universe.