A generalization of the Lorentz ether to gravity with general-relativistic limit
Ilja Schmelzer\thanks {WIAS Berlin}
We define a class of condensed matter theories in a Newtonian framework with a Lagrange formalism so that a variant of Noether's theorem gives the classical conservation laws: \begin{eqnarray*} \partial_t \rho + \partial_i (\rho v^i) &= &0 \\ \partial_t (\rho v^j) + \partial_i(\rho v^i v^j + p^{ij}) &= &0. \end{eqnarray*} We show that for the metric $g_{\mu\nu}$ defined by \begin{eqnarray*} \hat{g}^{00} = g^{00} \sqrt{-g} &= &\rho \\ \hat{g}^{i0} = g^{i0} \sqrt{-g} &= &\rho v^i \\ \hat{g}^{ij} = g^{ij} \sqrt{-g} &= &\rho v^i v^j + p^{ij} \end{eqnarray*} these theories are equivalent to a metric theory of gravity with Lagrangian \begin{equation*} L = L_{GR} + L_{matter}(g_{\mu\nu},\varphi^m) - (8\pi G)^{-1} (\Upsilon g^{00}-\Xi \delta_{ij}g^{ij})\sqrt{-g}. \end{equation*} with covariant $L_{matter}$, which defines a generalization of the Lorentz ether to gravity. Thus, the Einstein equivalence may be derived from simple condensed matter axioms. The Einstein equations appear in a natural limit $\Xi,\Upsilon\to 0$.
gr-qc0205035.tex
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\begin{document}
\newtheorem{theorem}{Theorem}
\newtheorem{axiom}{Axiom}
\newtheorem{definition}{Definition}
\newtheorem{thesis}{Thesis}
\newtheorem{corollary}{Corollary}
\author{Ilja Schmelzer\thanks
       {WIAS Berlin}}

\title{A generalization of the Lorentz ether to gravity with
general-relativistic limit} \sloppypar

\maketitle
%\tableofcontents
\begin{abstract}

We define a class of condensed matter theories in a Newtonian
framework with a Lagrange formalism so that a variant of Noether's
theorem gives the classical conservation laws:

\begin{eqnarray*}
\partial_t \rho + \partial_i (\rho v^i) &= &0 \\
\partial_t (\rho v^j) + \partial_i(\rho v^i v^j + p^{ij}) &= &0.
\end{eqnarray*}

We show that for the metric $g_{\mu\nu}$ defined by

\begin{eqnarray*}
 \hat{g}^{00} = g^{00} \sqrt{-g} &=  &\rho \\
 \hat{g}^{i0} = g^{i0} \sqrt{-g} &=  &\rho v^i \\
 \hat{g}^{ij} = g^{ij} \sqrt{-g} &=  &\rho v^i v^j + p^{ij}
\end{eqnarray*}

these theories are equivalent to a metric theory of gravity with
Lagrangian

\begin{equation*}
L = L_{GR} + L_{matter}(g_{\mu\nu},\varphi^m) 
  - (8\pi G)^{-1} (\Upsilon g^{00}-\Xi \delta_{ij}g^{ij})\sqrt{-g}.
\end{equation*}

with covariant $L_{matter}$, which defines a generalization of the
Lorentz ether to gravity.  Thus, the Einstein equivalence may be
derived from simple condensed matter axioms.  The Einstein equations
appear in a natural limit $\Xi,\Upsilon\to 0$.

\end{abstract}

\section{Introduction}

The aim of this paper is to propose a generalization of the classical
Lorentz ether to gravity.  The theory we have named ``General Lorentz
Ether Theory'' (GLET) is a metric theory of gravity, with a Lagrangian
which has the GR Lagrangian as a limit $\Xi,\Upsilon\to 0$:

\begin{equation}\label{L}
L = L_{GR} + L_{matter}(g_{\mu\nu},\varphi^m) 
  - (8\pi G)^{-1} (\Upsilon g^{00}-\Xi \delta_{ij}g^{ij})\sqrt{-g}.
\end{equation}

On the other hand, the theory is defined like a classical ether
theory, that means, a classical condensed matter theory in a Newtonian
framework $\mathbb{R}^3\otimes\mathbb{R}$: We assume classical
condensed matter variables (density $\rho$, velocity $v^i$, pressure
tensor $p^{ij}$) which follow classical conservation laws (Euler and
continuity equations):

\begin{eqnarray*}
\partial_t \rho + \partial_i (\rho v^i) &= &0 \\
\partial_t (\rho v^j) + \partial_i(\rho v^i v^j + p^{ij}) &= &0.
\end{eqnarray*}

together with Lagrange formalism and a connection between them as
required by Noether's theorem.  The only ``strange'' assumption is
that the pressure tensor should be negative definite.

The transformation of this theory into a metric theory of gravity is
defined by the following simple definition of the gravitational field
$g_{\mu\nu}$:

\begin{eqnarray*}
\nonumber       \hat{g}^{00} = g^{00} \sqrt{-g} &=  &\rho \\
\label{gdef}    \hat{g}^{i0} = g^{i0} \sqrt{-g} &=  &\rho v^i \\
\nonumber       \hat{g}^{ij} = g^{ij} \sqrt{-g} &=  &\rho v^i v^j + p^{ij}
\end{eqnarray*}

Matter fields $\varphi^m$ are identified with other degrees of freedom
(material properties) of the ether.  The general axioms do not specify
these material properties.  Therefore, they do not fix the matter
degrees of freedom and the matter Lagrangian.  But the general axioms
are already sufficient to derive the general Lagrangian (\ref{L}) and
the Einstein equivalence principle (EEP).

The existence of a viable ether theory is interesting already in
itself, even if only for playing devil's advocate. In any case, we
should reevaluate the arguments against the ether.  At least one
argument against the ether -- non-existence of a viable ether theory
of gravity -- should be thrown away.  Some other arguments against the
Lorentz ether are also no longer valid: Relativistic symmetry is no
longer ad hoc, but derived from simple axioms.  The violation of the
``action equals reaction'' principle by the static, incompressible
Lorentz ether no longer holds in the generalized theory.

But there may be even advantages in different domains in comparison
with relativity:

\begin{itemize}
\item Predictions: The additional terms give interesting predictions.
GLET predicts a flat universe, inflation (in the technical meaning of
$a''(\tau)>0$ for very dense states of the universe), a stop of the
gravitational collapse immediately before horizon formation, and a
homogeneous dark energy term with $p=-(1/3)\varepsilon$.  Especially,
we have no black hole and big bang singularities 
(app. \ref{appPredictions}).

\item Quantization: We know, in principle, how to quantize usual
condensed matter.  We can try to follow this scheme to quantize a
Lorentz ether (app. \ref{appQuantization}).  Famous quantization
problems disappear: There is no ``problem of time'', and also no
``black hole information loss problem''.

\item Classical realism: Bohmian mechanics (BM) prove that realistic
hidden variable theories are possible.  But in the relativistic domain
BM requires a preferred frame.  Thus, to generalize BM into the domain
of relativistic gravity we need something like a generalization of the
Lorentz ether to gravity.  The violation of Bell's inequality proves
that this is a necessary property of all realistic theories
(app. \ref{appRealism}).

\end{itemize}

Moreover, we have well-defined conservation laws with local energy and
momentum densities for the gravitational field.  

While we focus our attention on the presentation of an ether theory of
gravity, the main technical results -- the derivation of a metric
theory with GR limit from condensed-matter-like axioms -- may be,
possibly, applied also in a completely different domain: Search for
classical condensed matter theories which fit into this set of axioms
as analog models of general relativity. This possibility we discuss in
app. \ref{appCondensedMatter}. 

Once the aim of this paper is to define the theory in an axiomatic way
and to derive the Lagrangian of the theory, the paper is organized as
follows.  We start with the description of the explicit formalism
which is used in the formulation of the axioms.  Then we define and
discuss the axioms of our class of condensed matter theories.  After
this, we derive the general Lagrangian and prove the EEP for internal
observers.  We also consider the GR limit.

Then we have added a lot of appendices.  On one hand, they seem
necessary to address common arguments against the ether concept.  On
the other hand, a short consideration of various questions which are
beyond the scope of this paper seems necessary to motivate the
reintroduction of the ether.  Some of these appendices, especially the
parts about analog models, quantization, realism and Bohmian
mechanics, and predictions, have to be considered in separate papers
in more detail.

\section{Conservation Laws as Euler-Lagrange Equations for Preferred
Coordinates}

Below we consider classical condensed matter theories with Lagrange
formalism and classical conservation laws as known from condensed
matter theory (continuity and Euler equations).  We know from
Noether's theorem that translational invariance of the Lagrange
density leads to conservation laws.  It seems natural to identify the
conservation laws obtained via Noether's theorem from the Lagrange
formalism with the conservation laws as known them from condensed
matter theory.  The problem with this identification is that the
conservation laws are not uniquely defined.  If a tensor
$T^{\alpha\beta}$ fulfills $\partial_\alpha T^{\alpha\beta}=0$, then
the tensor $T^{\alpha\beta}+\partial_\gamma \psi^{\alpha\beta\gamma}$
fulfills the same equation if $\psi$ is an arbitrary tensor which
fulfills the condition
$\psi^{\alpha\beta\gamma}=\psi^{\alpha\gamma\beta}$.

We solve this problem by proposing a variant of Noether's theorem
which seems privileged because it is especially simple, natural and
beautiful.  Then this especially beautiful variant we identify with
the known, also especially simple, classical conservation laws. 

\subsection{Explicit Dependence on Preferred Coordinates}

As the Lagrangian of a classical condensed matter theory in a
classical Newtonian framework $\mathbb{R}^3\otimes\mathbb{R}$, our
Lagrange density is not covariant, does depend on the preferred
coordinates $X^\mu$.  Now, there are various possibilities to express
a given dependence on preferred coordinates.  For example, we have
$u^0 \equiv u^\mu X^0_{,\mu}$.  But on the right hand side $u^\mu
X^0_{,\mu}$ the dependence on the preferred coordinates in given in a
way we name {\em explicit\/}.  This explicit expression is, in some
sense, more informative.  Especially, in $u^\mu X^0_{,\mu}$ we have a
manifest dependence on $X^0$, we can see that there is no dependence
on the other coordinates $X^i$ for $i>0$.  This is not obvious in the
implicit form $u^0$ --- indeed, the similarly looking expression $u_0$
depends on all four coordinates $X^\mu$, not only on $X^0$.

Let's define now explicit dependence on the preferred coordinates
in a formal way:

\begin{definition} An expression
$F(\phi^k,\phi_{,i}^k,\ldots,X^\alpha,X^\alpha_{,\mu},\ldots)$
explicitly depends on the coordinates $X^\alpha$ if the expression
$F(\phi^k,\phi_{,i}^k,\ldots,U^\alpha,U^\alpha_{,\mu},\ldots)$, where
the coordinates $X^\alpha(x)$ have been replaced by four independent
scalar fields $U^\alpha(x)$, is covariant.
\end{definition}

Indeed, if the second expression is covariant, no implicit dependence
on the preferred coordinates $X^\mu$ has been left.  Now, it may be
asked if every dependence on preferred coordinates can be made
explicit.  In this context, we propose the following

\begin{quote}
For every physically interesting non-covariant function
$F(\phi^k,\phi_{,i}^k,\ldots)$ there exists a function
$\hat{F}(\phi^k,\phi_{,i}^k,\ldots,X^\alpha,X^\alpha_{,\mu},\ldots) =
F(\phi^k,\phi_{,i}^k,\ldots)$ which explicitly depends on the
preferred coordinates $X^\alpha$.
\end{quote}

Note that the thesis is not obvious.  The definition of explicit
dependence already requires that we know how to transform the other
fields $\phi^k$.  For a non-covariant theory this is possibly not yet
defined (see app. \ref{appGeneral}).  In this sense, this thesis may
be considered as analogous to Kretschmann's thesis \cite{Kretschmann}
that every physical theory allows a covariant formulation.  In the
following, we do not assume that this thesis holds.  If it holds, the
assumption of axiom \ref{aLagrange} is less restrictive.

\subsection{Euler-Lagrange Equations for Preferred Coordinates and
Noether's Theorem}

Now, if the dependence of the Lagrangian on the preferred coordinates
is explicit, we can vary not only over the fields, but also over the
preferred coordinates.  The point is that, despite the special geometric
nature of coordinates we can apply the Lagrange formalism as usual and
obtain Euler-Lagrange equations for the preferred coordinates
$X^\alpha$:

\begin{theorem}\label{Euler-Lagrange}
If the Lagrange density
$L(\phi^k,\phi_{,i}^k,\ldots,X^\alpha,X^\alpha_{,\mu},\ldots)$
explicitly depends on the preferred coordinates $X^\mu$, then for the
extremum of $S = \int L$ the following Euler-Lagrange equation holds:

\[ \frac{\delta S}{\delta X^\mu} = \frac{\partial L}{\partial X^\mu}
- \partial_\nu \frac{\partial L}{\partial X^\mu_{,\nu}}
+ \partial_\nu\partial_\lambda \frac{\partial L}{\partial X^\mu_{,\nu\lambda}}
-\cdots = 0 
\]
\end{theorem}

For the proof, see appendix \ref{appCoordinates}.  This does not mean
that the special geometric nature of the $X^\alpha$ is not important.
We see in appendix \ref{GRCF} that a theory with ``the same''
Lagrangian but usual scalar fields $U^\alpha(x)$ instead of
coordinates $X^\alpha(x)$ is a completely different physical theory.

We know from Noether's theorem that symmetries of the Lagrangian lead
to conservation laws.  Especially, conservation of energy and momentum
follows from translational symmetry in the preferred coordinates.
Now, in the case of explicit dependence on the preferred coordinates
$X^\alpha$, translational symmetry $X^\alpha\to X^\alpha+c^\alpha$ has
an especially simple form.  We obtain the conservation laws
automatically as Euler-Lagrange equations for the coordinates!
Indeed, translational symmetry immediately gives the Lagrangian does
not depend on the $X^\alpha$ them-self, only on their partial
derivatives $X^\mu_{,\nu},X^\mu_{,\nu\lambda}\ldots$.  Therefore the
Euler-Lagrange equations for the $X^\alpha$ {\em already have the form
of conservation laws\/}.

\begin{theorem}\label{Noether1}

If a Lagrangian with explicit dependence on the preferred coordinates
has translational symmetry $X^\alpha \to X^\alpha + c^\alpha$, then
the Euler-Lagrange equations for the preferred coordinates $X^\alpha$
are conservation laws:

\begin{equation} 
\frac{\delta S}{\delta X^\alpha} = \partial_\mu {T^\mu_\alpha} = 0 
\end{equation}

\end{theorem}

We know that in agreement with Noether's second theorem these
conservation laws disappear if we have general covariance.  Again, in
our formulation we do not have to do much to see this -- if there is
no dependence on the $X^\alpha$, the Euler-Lagrange equation for the
$X^\alpha$, which are the conservation laws, disappear automatically:

\begin{theorem}\label{Noether2}

If L is covariant, then

\begin{equation} \label{eqNoether2}
\frac{\delta S}{\delta X^\alpha} \equiv 0 
\end{equation}

\end{theorem}

In above cases, the proof is an obvious consequence of theorem
\ref{Euler-Lagrange}.  Of course, there is not much to wonder about,
we have simply used a set of variables $X^\alpha$ appropriate for the
symmetries we have considered -- translational symmetry.  

Based on these theorems we identify the Euler-Lagrange equation
$\frac{\delta S}{\delta X^0}$ with the {\em energy conservation law\/}
and $\frac{\delta S}{\delta X^i}$ with the {\em momentum conservation
law\/}. Note also that the most general GR Lagrangian is given by the
conditions (\ref{eqNoether2}).
\footnote{``Most general'' means that higher order derivatives of the
metric and non-minimal interactions with matter fields are allowed.
This is in agreement with the modern concept of effective field
theory, as, for example, expressed by Weinberg \cite{Weinberg}: ``I
don't see any reason why anyone today would take Einstein's general
theory of relativity seriously as the foundation of a quantum theory
of gravitation, if by Einstein's theory is meant the theory with a
Lagrangian density given by just the term $\sqrt{g}R/16\pi G$.  It
seems to me there's no reason in the world to suppose that the
Lagrangian does not contain all the higher terms with more factors of
the curvature and/or more derivatives, all of which are suppressed by
inverse powers of the Planck mass, and of course don't show up at
energy far below the Planck mass, much less in astronomy or particle
physics.  Why would anyone suppose that these higher terms are
absent?''  We follow this point of view.}

It seems worth to note that until now we have only considered a quite
general formalism which is in no way restricted to the application in
condensed matter theory below.

\section{Axioms for General Lorentz Ether Theory}

Let's define now our set of axioms for the class of
condensed-matter-like theories we call ``Lorentz ether theories''.  It
looks quite natural, nonetheless, it should be noted that it is in
some essential points an unorthodox formalism for condensed matter
theories.  Especially it remains open if there exist classical
condensed matter theories which fit into this set of axioms.

\begin{axiom}[independent variables] \label{afirst}
The independent variables of the theory are the following fields
defined on a Newtonian framework ${\mathbb R^3\otimes R}$ with
preferred coordinates $X^i,T$: a positive density $\rho(X^i,T)$, a
velocity $v^i(X^i,T)$, a symmetric pressure tensor $p^{ij}(X^i,T)$,
which form the classical energy-momentum tensor $t^{\mu\nu}(X^i,T)$:

\begin{eqnarray*}
 t^{00} &=  &\rho \\
 t^{i0} &=  &\rho v^i \\
 t^{ij} &=  &\rho v^i v^j + p^{ij}
\end{eqnarray*}

Moreover, there is an unspecified number of ``inner degrees of
freedom'' (material properties) $\varphi^m(X^i,T)$ with well-defined
transformation rules for arbitrary coordinate transformations.
\end{axiom}

That we use a pressure tensor $p^{ij}$ instead of a scalar $p$ is a
quite natural and known generalization.  Instead, the use of $p^{ij}$
as an independent variable is non-standard.  Usually pressure is
defined as an algebraic ``material function'' of other independent
variables.

\begin{axiom}[Lagrange formalism] \label{aLagrange}
There exists a Lagrange formalism with explicit dependence of the
Lagrange density on the preferred coordinates:

\begin{equation*}
L = L(t^{\mu\nu},t^{\mu\nu}_{,\kappa},\ldots,
      \varphi^m,\varphi^m_{,\kappa},\ldots,
      X^\alpha,X^\alpha_{,\kappa},\ldots)
\end{equation*}
\end{axiom}

The existence of a Lagrange formalism is certainly a non-trivial
physical restriction (it requires an ``action equals reaction''
symmetry) but is quite common in condensed matter theory.  Instead,
the requirement of explicit dependence on the $X^\mu$ is non-standard.
In appendix \ref{appGeneral} we argue that this additional requirement
is not really a restriction: Every physical theory with Lagrange
formalism can be transformed into this form.  But this remains a
general thesis which cannot be proven in a strong way, only supported
by showing how this can be done in particular cases.

\begin{axiom}[energy conservation law] \label{acontinuity}
The conservation law related by theorem \ref{Noether1} with
translational symmetry in time $X^0\to X^0+c^0$ is proportional to the
continuity equation:

\begin{equation}
\frac{\delta S}{\delta X^0} \sim \partial_t \rho + \partial_i (\rho
 v^i).
\end{equation}

\end{axiom}

\begin{axiom}[momentum conservation law] \label{aEuler} 
The conservation laws related by theorem \ref{Noether1} with
translational symmetry in space $X^i\to X^i+c^i$ is proportional to
the Euler equation:

\begin{equation}
\frac{\delta S}{\delta X^j} \sim \partial_t (\rho v^j) +
 \partial_i(\rho v^i v^j+p^{ij}).
\end{equation}
\end{axiom}

These axioms also look quite natural and are well motivated by our
version of Noether's theorem.  But it should be noted that this is not
the only possibility to connect continuity and Euler equations with
the Lagrange formalism.  The energy-momentum tensor is not uniquely
defined.  The main purpose of the introduction of our ``explicit
dependence'' formalism was to motivate this particular choice as the
most natural one.\footnote{If a tensor $T^{\alpha\beta}$ fulfills
$\partial_\alpha T^{\alpha\beta}=0$, then the tensor
$T^{\alpha\beta}+\partial_\gamma \psi^{\alpha\beta\gamma}$ fulfills
the same equation if $\psi$ is an arbitrary tensor which fulfills the
condition $\psi^{\alpha\beta\gamma}=\psi^{\alpha\gamma\beta}$.}  Thus,
by our choice of axioms \ref{acontinuity}, \ref{aEuler} we fix a
particular, even if especially simple, relation between Lagrange
formalism and continuity and Euler equations.

Despite the non-standard technical features of these axioms, none of
them looks like being made up to derive the results below.  And we
also have not fixed anything about the inner structure of the medium:
The inner degrees of freedom (material properties) and the material
equations are not specified.  From point of view of physics, the
following non-trivial restrictions have been made:

\begin{itemize}

\item The existence of a Lagrange formalism requires a certain
symmetry - the equations are self-adjoint (the principle ``action
equals reaction'').

\item This medium is conserved (continuity equation).

\item There is only one, universal medium.  There are no external
forces and there is no interaction terms with other types of matter
(as follows from the Euler equation).

\end{itemize}

Last not least, there is one additional axiom which sounds quite
strange:

\begin{axiom}[negative pressure] \label{alast}\label{apressure}
The pressure tensor $p^{ij}(X^i,T)$ is negative definite
\end{axiom}

Instead, for most usual matter pressure is positive.  On the other
hand, the Euler equation defines pressure only modulo a constant, and
if we add a large enough negative constant tensor $-C\delta^{ij}$ we
can make the pressure tensor negative definite.  The physical meaning
of this axiom is not clear.

These are all axioms we need.  Of course, these few axioms do not
define the theory completely.  The material properties are not
defined.  Thus, our axioms define a whole class of theories.  Every
medium which meets these axioms we name ``Lorentz ether''.

\section{The Derivation of the General Lagrangian}

The technical key result of the paper is the following derivation:

\begin{theorem}\label{main}
Assume we have a theory which fulfills the axioms
\ref{afirst}-\ref{alast}.

Then there exists an ``effective metric'' $g_{\mu\nu}(x)$ of signature
$(1,3)$, which is defined algebraically by $\left\{\rho, v^i,
p^{ij}\right\}$, and some constants $\Xi,\Upsilon$ so that for every
solution $\left\{\rho(X,T),v^i(X,T),p^{ij}(X,T),\varphi^m(X,T)\right\}$
the related fields $g_{\mu\nu}(x),\varphi^m(x)$ together with the
preferred coordinates $X^\alpha(x)$ are also solutions of the
Euler-Lagrange equations of the Lagrangian {\rm }

\begin{equation}
L = -(8\pi G)^{-1}\gamma_{\alpha\beta}X^\alpha_{,\mu}X^\beta_{,\nu}
        g^{\mu\nu}\sqrt{-g} + L_{GR}(g_{\mu\nu}) +
        L_{matter}(g_{\mu\nu},\varphi^m)
\end{equation}

where $L_{GR}$ denotes the general Lagrangian of general relativity,
$L_{matter}$ some general covariant matter Lagrangian compatible with
general relativity, and the constant matrix $\gamma_{\alpha\beta}$ is
defined by two constants $\Xi,\Upsilon$ as
$\mbox{diag}\left\{-\Upsilon,\Xi,\Xi,\Xi\right\}$.

\end{theorem}

Proof: The metric $g_{\mu\nu}$ defined by equation \ref{gdef} has
signature $(1,3)$. This follows from $\rho>0$ (axiom \ref{afirst}) and
negative definiteness of $p^{ij}$ (axiom \ref{apressure}.  Equation
\ref{gdef} allows to express $\rho,v^i, p^{ij}$ in terms of
$g^{\mu\nu}$, so that the Lagrangian
$L(\rho(x),v^i(x),p^{ij}(x),\varphi^m(x),X^i(x),T(x))$ may be
expressed as a function of the metric variables
$L(g^{\mu\nu}(x),\varphi^m(x),X^i(x),T(x))$.  Now, the key
observation is what happens with our four classical conservation laws
during this change of variables.  They (essentially by construction)
become the harmonic condition for the metric $g_{\mu\nu}$:

\begin{equation}
\partial_\mu (g^{\alpha\mu}\sqrt{-g}) = 
\partial_\mu (g^{\beta\mu}\sqrt{-g}) \partial_\beta X^\alpha 
\equiv \square X^\alpha = 0 
\end{equation}

where $\square$ denotes the harmonic operator of the metric
$g_{\mu\nu}$.  We define now the constants $\Upsilon,\Xi$ as the
proportionality factors between the energy and momentum conservation
laws and the continuity resp. Euler equations (axioms
\ref{acontinuity}, \ref{aEuler}).  For convenience a common factor
$(4\pi G)^{-1}$ is introduced:

\begin{eqnarray}
\frac{\delta S}{\delta T}   &=& - (4\pi G)^{-1} \Upsilon \square T\\
\frac{\delta S}{\delta X^i} &=&   (4\pi G)^{-1} \Xi \square X^i
\end{eqnarray}

Defining the diagonal matrix $\gamma_{\alpha\beta}$ by
$\gamma_{00}=\Upsilon,\gamma_{ii}=-\Xi$ we can write these equations
in closed form as
\footnote{Note that the coefficients $\gamma_{\alpha\beta}$ are only
constants of the Lagrange density, the indices enumerate the scalar
fields $X^\alpha$.  They do not define any fundamental, predefined
object of the theory.  Instead, variables of the Lagrange formalism,
by construction, are only $g_{\mu\nu}$, $\varphi^m$ and $X^\alpha$.}

\begin{equation}
\frac{\delta S}{\delta X^\alpha}
        \equiv-(4\pi G)^{-1}\gamma_{\alpha\beta}\square X^\beta
\end{equation}

Let's find now the general form of a Lagrangian which gives these
equations.  Its easy to find a particular solution $L_0$:

\begin{equation}
L_{0} = -(8\pi G)^{-1}\gamma_{\alpha\beta}X^\alpha_{,\mu}X^\beta_{,\nu}
        g^{\mu\nu}\sqrt{-g}
\end{equation}

Then, let's consider the difference $L-L_{0}$.  Together with $L$ and
$L_0$ it's dependence on the preferred coordinates is explicit.
Moreover we obtain:

\begin{equation} 
\frac{\delta \int(L-L_{0})}{\delta X^\alpha} \equiv 0 
\end{equation}

But that means covariance, and is our condition for the most
general Lagrangian of general relativity.  So we obtain:

\begin{equation}
L = -(8\pi G)^{-1}\gamma_{\alpha\beta}X^\alpha_{,\mu}X^\beta_{,\nu}
        g^{\mu\nu}\sqrt{-g} + L_{GR}(g_{\mu\nu}) +
        L_{matter}(g_{\mu\nu},\varphi^m).
\end{equation}

{\bf qed}.

\section{Lorentz Ether Theory as a Metric Theory of Gravity}

Now, the explicit formalism has been quite useful to formulate the
axioms of GLET and to derive the effective Lagrangian in theorem
\ref{main}.  But it is only a technical formalism.  Once theorem
\ref{main} has been proven, we can as well return to the usual,
implicit non-covariant formulation of the theory.

\begin{theorem}
Assume we have a theory which fulfills axioms
\ref{afirst}-\ref{alast}.  Then formula \ref{gdef} defines an
isomorphism between solutions of this theory and the solutions of the
(non-covariant) metric theory of gravity defined by the Lagrangian

\begin{equation}
L_{GLET} = L_{GR} + L_{matter}
  - (8\pi G)^{-1}(\Upsilon g^{00}-\Xi \delta_{ij}g^{ij})\sqrt{-g}
\end{equation}

and the additional ``causality condition'' $g^{00}>0$.
\end{theorem}

Indeed, the Lagrangian $L_{GLET}$ is simply the Lagrangian $L$ of
theorem \ref{main} written in the preferred coordinates $X^\alpha$.
It follows from \ref{gdef} and $\rho>0$ that for solutions of GLET the
causality condition $g^{00}>0$ is fulfilled.  For a given solution of
GLET formula \ref{gdef} also allows to define the pre-image solution
in terms of $\rho, v^i, p^{ij}$. {\bf qed}.

Thus, our class of condensed matter theories is equivalent to a metric
theory of gravity with a Newtonian framework as a predefined
background.  As equations of the theory in the preferred coordinates
we obtain the Einstein equations of general relativity with two
additional terms:

\begin{equation}\label{GLETeq}
G^\mu_\nu  = 8\pi G (T_m)^\mu_\nu
   + (\Lambda +\gamma_{\kappa\lambda}g^{\kappa\lambda}) \delta^\mu_\nu
   - 2g^{\mu\kappa}\gamma_{\kappa\nu}
\end{equation}

as well as the harmonic equations -- the conservation laws

\begin{equation} 
\partial_\mu  (g^{\mu\alpha}\sqrt{-g}) =  0. 
\end{equation}

Note that the ``energy-momentum tensor of matter'' $(T_m)^\mu_\nu$
should not be mingled with the classical energy-momentum tensor
$t^\mu_\nu$ of the ether defined in axiom \ref{afirst} which is the
full energy-momentum tensor.  There is also another form of the
conservation laws. The basic equation may be simply considered as a
decomposition of the full energy-momentum tensor
$g^{\mu\kappa}\sqrt{-g}$ into a part $(T_m)^\mu_\nu$ which depends on
matter fields and a part $(T_g)^\mu_\nu$ which depends on the
gravitational field. Thus, for

\begin{equation}
(T_g)^\mu_\nu = (8\pi G)^{-1}\left(\delta^\mu_\nu(\Lambda
              + \gamma_{\kappa\lambda}g^{\kappa\lambda})
              - G^\mu_\nu\right)\sqrt{-g}
\end{equation}

we obtain a second form of the conservation law

\begin{equation}
\partial_\mu  ((T_g)^\mu_\nu +  (T_m)^\mu_\nu\sqrt{-g}) = 0
\end{equation}

Thus, we have even two equivalent forms for the conservation laws, one
with a subdivision into gravitational and matter part, and another one
where the full tensor depends only on the gravitational field.

\section{Internal Observers and the Einstein Equivalence Principle}

The explicit form of the Lagrangian is, if we replace the preferred
coordinates $X^\mu$ with scalar fields $U^\mu$, formally equivalent to
the Lagrangian of ``general relativity with clock fields'' (GRCF)
considered by Kuchar and Torre \cite{Kuchar} in the context of GR
quantization in harmonic gauge:

\begin{equation}
L_{GRCF} = -(8\pi G)^{-1}\gamma_{\alpha\beta}U^\alpha_{,\mu}U^\beta_{,\nu}
        g^{\mu\nu}\sqrt{-g} + L_{GR}(g_{\mu\nu}) +
        L_{matter}(g_{\mu\nu},\varphi^m).
\end{equation}

The only difference is that in GRCF we have scalar fields
$U^\alpha(x)$ instead of preferred coordinates $X^\alpha(x)$.  But
this makes above theories completely different as physical theories.
We show this in detail in app. \ref{GRCF}.  Despite these differences,
internal observers of our condensed matter theories have a very hard
job to distinguish above theories by observation:

\begin{corollary} 
In GLET internal observers cannot falsify GRCF by any local observation.
\end{corollary}

Indeed, internal observers are creatures described by field
configurations of the fields $\left(\rho(X,T), v^i(X,T), p^{ij}(X,T),
\varphi^m(X,T)\right)$.  Whatever they observe may be described by a
solution $\left(\rho(X,T), v^i(X,T), p^{ij}(X,T),
\varphi^m(X,T)\right)$.  Now, following theorem \ref{main}, they can
describe all their experiments and observations also as local
solutions of GRCF for some ``fields'' $g^{\mu\nu}, \varphi^m(x),
U^i(x), U^0(x)$.  {\bf qed.}
\footnote{Note that in the reverse direction the corollary doesn't
hold: There are local solutions of GRCF where the fields $U^\mu(x)$
cannot be used as coordinates even locally.}

Based on this observation we can prove now the Einstein Equivalence
Principle.  For this purpose we have to clarify what is considered to
be a matter field.  This question is theory-dependent, and there are
two possibilities for this: First, the internal observers of GLET may
believe into GRCF.  In this case, they consider the four special
scalar fields $X^\alpha(x)$ as non-special scalar matter fields
together with the other matter fields $\varphi^m(x)$.  Second, they
may believe into GLET.  Then they consider only the fields
$\varphi^m(x)$ as non-gravitational matter fields.  This gives two
different notions of non-gravitational experiments and therefore
different interpretations of the EEP. But in above interpretations of
the EEP it holds exactly:

\begin{corollary} For internal observers of a condensed matter 
theory GLET which fulfills axioms \ref{afirst}-\ref{alast} the
Einstein Equivalence Principle holds in above interpretations.
\end{corollary}

Indeed, GRCF is a variant of general relativity where the EEP holds
exactly.  Because the EEP is a local principle (it holds if it holds
for all local observations) and internal observers cannot falsify GRCF
by local observations, they cannot falsify the EEP by
observation. Thus, in the first interpretation the EEP holds.  In the
second case, only the $\varphi^m$ are interpreted as matter fields.
Nonetheless, we have also a metric theory of gravity coupled with the
matter fields $\varphi^m$ in the usual covariant way, and therefore
the EEP holds.{\bf qed.}

\section{The GR Limit}

In the limit $\Xi,\Upsilon\to 0$ we obtain the Lagrangian of general
relativity.  Thus, we obtain the classical Einstein equations.
Nonetheless, some points are worth to be discussed here:

\subsection{The Remaining Hidden Background}

In this limit the absolute background becomes a hidden variable.

Nonetheless, the theory remains different from general relativity even
in this limit.  The hidden background leads to some important global
restrictions: Non-trivial topologies are forbidden, closed causal
loops too.  There always exists a global harmonic time-like function.
\footnote{There is a general prejudice against such hidden variables.
It is sometimes argued that a theory without such restrictions should
be preferred as the ``more general'' theory.  But this prejudice is
not based on scientific methodology as described by Popper's criterion
of empirical content.  If this limit of GLET is true, an internal
observer cannot falsify GR in the domain of applicability of this
limit.  On the other hand, if GR is true, GLET is falsifiable by
observation: If we observe non-trivial topology, closed causal loops,
or situations where no global harmonic time exists, we have falsified
GLET.  In this sense, the GR limit of GLET should be preferred by
Popper's criterion of predictive power in comparison with GR.}

\subsection{The Absence of Fine Tuning in the Limit}

Note that we need no fine-tuning to obtain the GR limit
$\Xi,\Upsilon\to 0$.  All non-covariant terms depend only on the
metric $g^{\mu\mu}\sqrt{-g}$, not on its derivatives $\partial_\kappa
g_{\mu\nu}$.  That means, in a region where we have well-defined upper
and lower bounds $0<c^\mu_{min}<|g^{\mu\mu}\sqrt{-g}|<c^\mu_{max}$ (in
the preferred coordinates) all we have to do is to consider low
distance high frequency effects.  In comparison with global cosmology,
solar system size effects obviously fit into this scheme.  Thus, the
non-covariant terms may be reasonably considered as cosmological
terms, similar to Einstein's cosmological constant $\Lambda$.

On the other hand, there may be effects in solar-system-like dimension
where these cosmological terms cannot be ignored: In the preferred
coordinates, the terms $g^{\mu\mu}\sqrt{-g}$ may become large.  An
example is the region near horizon formation for a collapsing star.
As shown in app. \ref{appPredictions}, in the case of $\Upsilon>0$
this term prevents black hole formation, and we obtain stable ``frozen
star'' solutions.

\subsection{Comparison with Other Approaches to GR}

Part of the beauty of GR is that there many different ways to general
relativity.  First, there are remarkable formulations in other
variables (ADM formalism \cite{ADM}, tetrad, triad, and Ashtekar
variables \cite{Ashtekar}) where the Lorentz metric $g_{\mu\nu}$
appears as derived.  Some may be considered as different
interpretations of general relativity (like ``geometrodynamics'').  On
the other hand we have theories where the Lorentz metric is only an
effective metric and the Einstein equations appear in a limit (spin
two field in QFT on a standard Minkowski background \cite{Feynman}
\cite{Weinberg65} \cite{Deser}, string theory \cite{Polchinski}),
Sakharov's approach \cite{Sakharov}.

In this context, the derivation of GLET, combined with the limit
$\Xi,\Upsilon\to 0$, may be considered as ``yet another way to GR''.
In this interpretation, it should be classified as part of the second
group of derivations: The metric is, as in these approaches, not
fundamental.  Moreover, a flat hidden background remains.

\section{The No-Gravity Limit}

Last not least, let's consider shortly the no gravity limit of GLET.
For a metric theory of gravity, it is the Minkowski limit
$g_{\mu\nu}(x)=\eta_{\mu\nu}$.  In this limit, the additional terms of
GLET in comparison with GR no longer vary, therefore, may be omitted.
Thus, the no gravity limit of GLET coinsides with the no gravity limit
of GR, and is, therefore, equivalent to SR.

In this limit, we have 

\begin{eqnarray*}
\rho(x)  &=  &\rho_0 \\
v^i(x)   &=  &0 \\
p^{ij}(x)&=  &p_0 \delta^{ij}
\end{eqnarray*}

with constants $\rho_0>0, p_0<0$.  Thus, the ether is static and
incompressible.  It follows from the properties of the no gravity
limit of the matter Lagrangian, which is identical in GR and GLET,
that clocks and rulers behave as they behave in SR.  

Thus, in the no gravity limit we obtain a static and incompressible
ether in a Newtonian framework, so that clocks and rulers show time
dilation and contraction following the formulas of special relativity.
These properties of the no gravity limit of GLET are sufficiently
close to the properties of the classical Lorentz ether.  This
justifies the interpretation of GLET as a generalization of Lorentz
ether theory to gravity.

Nonetheless, it seems worth to note a minor difference: In the
classical ether theories, the ether has been considered only as a
substance which explains electromagnetic waves.  Usual matter has been
considered to be something different.  In GLET, a key point of the
derivation is that all fields -- including the fields which describe
fermionic particles -- describe degrees of freedom of the ether
itself.  Thus, there is no matter which ``interacts with the ether''.
There is only a single universal ether, and nothing else.

\section{Conclusions}

We have presented a new metric theory of gravity starting with a
remarkable set of axioms for a classical condensed matter theory in a
Newtonian background.  Relativistic symmetry (the Einstein equivalence
principle) is derived here: Internal observers of the medium (called
``general Lorentz ether'') are unable to distinguish all properties of
their basic medium.  All their observations fulfill the Einstein
equivalence principle.

While we have used an unorthodox formalism -- explicit dependence of a
non-covariant Lagrangian on the preferred coordinates -- the basic
principles we have used in the axioms are well-known and accepted:
continuity and Euler equations, Lagrange formalism and Noether's
theorem which relates them with translational symmetry.

The differences between GR and our theory are interesting: The new
theory predicts inflation and a stop of the gravitational collapse,
preventing black hole and big bang singularities.  As a condensed
matter theory in a Newtonian framework, classical canonical
quantization concepts known from quantum condensed matter theory may
be applied to quantize the theory.

The new theory is able to compete with general relativity even in the
domain of beauty: It combines several concepts which are beautiful
already by them-self: Noether's theorem, conservation laws, harmonic
coordinates, ADM decomposition.  Instead of the ugly situation with
local energy and momentum of the gravitational field in GR, we have
nice local conservation laws.  Moreover we have no big bang and black
hole singularities.

To establish this theory as a serious alternative to general
relativity, this paper may be only a first step.  We have to learn
more about the predictions of GLET which differ from GR, experimental
bounds for $\Xi, \Upsilon$ have to be obtained, the quantization
program has to be worked out in detail.  The methodological
considerations related with such fundamental notions like EPR realism
have to be worked out too.  Nonetheless, the results we have obtained
are already interesting enough, and the new theory seems able to
compete with general relativity in all domains we have considered --
cosmology, quantization, explanatory power, and even simplicity and
beauty.

\begin{appendix}

\section{Can every Lagrangian be rewritten in explicit form?}
\label{appGeneral}

In axiom \ref{aLagrange} we assume that the Lagrangian is given in
explicit form.  Now, the question we want to discuss here is if this
is a non-trivial restriction or not.   If it is not, then the
following thesis holds:

\begin{thesis}
For every physically interesting non-covariant function
$F(\phi^k,\phi_{,i}^k,\ldots)$ there exists a function
$\hat{F}(\phi^k,\phi_{,i}^k,\ldots,X^\alpha,X^\alpha_{,\mu},\ldots) =
F(\phi^k,\phi_{,i}^k,\ldots)$ which explicitly depends on the
preferred coordinates $X^\alpha$.
\end{thesis}

In some sense, this can be considered as a corollary to Kretschmann's
\cite{Kretschmann} thesis that every physical theory may be
reformulated in a general covariant way.  We would have to reformulate
a given theory with Lagrangian $F$ into a general covariant form, and
when, as far as possible, try to replace all objects used to describe
the absolute Newtonian framework in this covariant description by a
description which depends explicitly on the preferred coordinates
$X^\alpha(x)$.
\footnote{Remember that there has been some confusion about the role
of covariance in general relativity.  Initially it was thought by
Einstein that general covariance is a special property of general
relativity.  Later, it has been observed that other physical theories
allow a covariant description too.  The classical way to do this for
special relativity (see \cite{Fock}) is to introduce the background
metric $\eta_{\mu\nu}(x)$ as an independent field and to describe it
by the covariant equation $R^\mu_{\nu\kappa\lambda}[\eta]=0$.  For
Newton's theory of gravity a covariant description can be found in
\cite{MTW}, \S 12.4.  The general thesis that every physical theory
may be reformulated in a general covariant way was proposed by
Kretschmann \cite{Kretschmann}.}

In this sense, let's try to describe how the classical geometric
objects related with the Newtonian framework may be described in
an explicit way:

\begin{itemize}

\item The preferred foliation: It is immediately defined by the
coordinate $T(x)$.

\item The Euclidean background space metric: A scalar product
$(u,v)=\delta_{ij}u^iv^j$ may be presented in explicit form as
$(u,v)=\delta_{ij}X^i_{,\mu}X^j_{,\nu}u^\mu v^\nu$.

\item Absolute time metric: Similarly, the (degenerate) metric of
absolute distance in time may be defined by
$(u,v)=T_{,\mu}T_{,\nu}u^\mu v^\nu$.

\item The preferred coordinates also define tetrad and cotetrad
fields: $dX^\alpha = X^\alpha_{,\mu} dx^\mu$, for the vector fields
$\partial/\partial X^\alpha =
(J^{-1})^\mu_\alpha(X^\beta_{,\nu})\partial/\partial x^\mu$ we need
the inverse Jacobi matrix $J^{-1}$, which is a rational function of
the $X^\beta_{,\nu}$.

\item Arbitrary upper tensor components $t^{\alpha_1,..\alpha_n}$ may
be transformed into $X^{\alpha_1}_{,\mu_1}\cdots
X^{\alpha_n}_{,{\mu_n}}t^{\mu_1,..\mu_n}$.  For lower indices we have
to use again the inverse Jacobi matrix $J^{-1}$.

\item Together with the background metrics we can define also the
related covariant derivatives.

\end{itemize}

Now, these examples seem sufficient to justify our hypothesis.  Of
course, if we want to transform a given non-covariant function into
explicit form, we have no general algorithm.  For example, a scalar in
space in a non-relativistic theory may be a spacetime scalar, the time
component of a four-vector, and so on for tensors.  But uniqueness is
not the problem we have to solve here.  For our hypothesis it is
sufficient that there exists at least one way.

Let's note again that our hypothesis is not essential for our
derivation.  Simply, if the hypothesis is false, our assumption that
the Lagrangian is given in explicit form (which is part of axiom
(\ref{aLagrange})) is a really non-trivial restriction.  This would be
some loss of generality and therefore of beauty of the derivation, but
in no way fatal for the main results of the paper.  The considerations
given here already show that the class of theories which allows an
explicit formulation is quite large.

\section{Justification of the Euler-Lagrange Equations for the 
Preferred Coordinates}
\label{appCoordinates}

In the Euler-Lagrange formalism for a Lagrangian with explicit
dependence on the preferred coordinates we propose to handle the
preferred coordinates like usual fields.  Of course, every valid set
of coordinates $X^\alpha(x)$ defines a valid field configuration.  But
the reverse is not true.  To define a valid set of coordinates, the
functions $X^\alpha(x)$ have to fulfill special local and global
restrictions: the Jacobi matrix should be non-degenerated everywhere,
and the functions should fulfill special boundary conditions.  In
app. \ref{GRCF} we find that this makes the two theories different as
physical theories.

Here we want to prove theorem \ref{Euler-Lagrange} that nonetheless
the Lagrange formalism works as usual.  The description of the
Lagrange function in explicit form is, in comparison with an implicit
description, only another way to describe the same minimum problem.
To solve this minimum problem we can try to apply the standard
variational calculus.  Especially we can also vary the preferred
coordinates $X^\alpha$, which are, together with the other fields,
functions of the manifold.  But there is a subtle point which has to
be addressed.  The point is that not all variations $\delta
X^\alpha(x)$ are allowed -- only the subset with the property that
$X^\alpha(x)+\delta X^\alpha(x)$ defines valid global coordinates.
Therefore, to justify the application of the standard Euler-Lagrange
formalism we have to check if this subset of allowed variations is
large enough to give the classical Euler-Lagrange equations.

Fortunately this is the case.  To obtain the Euler-Lagrange equations
we need only variations with compact support, so we don't have a
problem with the different boundary conditions.  Moreover, for
sufficiently smooth variations ($\delta X^\alpha \in C^1({\mathbb
R}^4)$ is sufficient) there is an $\varepsilon$ so that
$X^\alpha(x)+\varepsilon\delta X^\alpha(x)$ remains to be a system of
coordinates: indeed, once we have an upper bound for the derivatives
of $\delta X^\alpha$ (because of compact support), we can make the
derivatives of $\varepsilon\delta X^\alpha$ arbitrary small,
especially small enough to leave the Jacobi matrix of
$X^\alpha+\varepsilon\delta X^\alpha$ non-degenerated.  Such
sufficiently smooth variations are sufficient to obtain the
Euler-Lagrange equations. 

Now, to obtain the Euler-Lagrange equations we need only small
variations.  Thus, the geometrical restrictions on global coordinates
do not influence the derivation of the Euler-Lagrange equations for
the $X^\alpha(x)$.  For the preferred coordinates we obtain usual
Euler-Lagrange equation, as if they were usual fields, despite their
special geometric nature.

\section{Discussion of the Derivation}
\label{appDerivation}

The extremely simple derivation of exact general-relativistic symmetry
in the context of a classical condensed matter theory looks very
surprising.  It may be suspected that too much is hidden behind the
innocently looking relation between Lagrange formalism and the
condensed matter equations, or somewhere in our unorthodox ``explicit
formalism''.  Therefore, let's try to understand on a more informal,
physical level what has happened, and where we have made the
physically non-trivial assumptions which lead to the very non-trivial
result -- the EEP.

A non-trivial physical assumption is, obviously, the classical Euler
equation.  This equation contains non-trivial information -- that
there is only one medium, which has no interaction, especially no
momentum exchange, with other media.  This is, indeed, a natural but
very strong physical assumption.  And from this point of view the
derivation looks quite natural: What we assume is a single, universal
medium.  What we obtain with the EEP is a single, universal type of
clocks.  An universality assumption leads to an universality result.

From point of view of parameter counting, all seems to be nice too.
We have explicitly fixed four equations, and obtain an independence
from four coordinates.  All the effective matter fields are by
construction fields of a very special type -- inner degrees of freedom,
material properties, of the original medium.  Therefore it is also not
strange to see them closely tied to the basic degrees of freedom
(density, velocity, pressure) which define the effective gravitational
field.

The non-trivial character of the existence of a Lagrange formalism
seems also worth to be mentioned here.  A Lagrange formalism leads to
a symmetry property of the equation -- they should be self-adjoint.
This is the well-known symmetry of the ``action equals reaction''
principle.  It can be seen where this symmetry has been applied if we
consider the second functional derivatives.  The EEP means that the
equations for effective matter fields $\varphi^m$ do not depend on the
preferred coordinates $X^\alpha$.  This may be formally proven in this
way:

\begin{equation} 
   \frac{\delta}{\delta X^\mu}     \frac{\delta S}{\delta \varphi^m}
 = \frac{\delta}{\delta \varphi^m} \frac{\delta S}{\delta X^\mu}
 = \frac{\delta}{\delta \varphi^m} \mbox{[cons. laws]} = 0 
\end{equation}

Thus, we have applied here the ``action equals reaction'' symmetry of
the Lagrange formalism and the property that the fundamental classical
conservation laws (connected with translational symmetry by Noether's
theorem) do not depend on the material properties $\varphi^m$.

Our considerations seem to indicate that the relativistic symmetry
(EEP) we obtain is explained in a reasonable way by the physical
assumptions which have been made, especially the universality of the
medium and the special character of the ``effective matter'' degrees of
freedom -- assumptions which are implicit parts of the Euler equation
-- and the ``action equals reaction'' symmetry of the Lagrange
formalism.

It seems also necessary to clarify that the states which cannot be
distinguished by local observation of matter fields are not only
different states, they can be distinguished by gravitational
experiments.  The EEP holds (we have a metric theory of gravity) but
not the Strong Equivalence Principle (SEP).  The SEP holds only in the
GR limit.  For the definitions of EEP and SEP used here see, for
example, \cite{Will}.

\section{What Happens During Evolution with the Causality Condition?}
\label{appFailure}

We have proven that solutions of $GRCF$ which are images of solutions
of GLET fulfill some restrictions (time-like $T(x)$, $X^\alpha(x)$ are
coordinates).  Now, it is reasonable to ask if these restrictions are
compatible with the evolution equations of GRCF.  Assume that we have
a solution of GRCF which fulfills all necessary restrictions for some
time $T(x)< t_0$.  This solution now evolves following the evolution
equations of GRCF.  Is there any warranty that these global
restrictions remain to be fulfilled also for $T(x)\ge t_0$?  The
answer is no.  There is not such general warranty.  But, as we will
see, there is also no necessity for such a warranty.

For a given Lorentz ether theory (with fixed material Lagrangian) such
a warranty would follow from a global existence and uniqueness
theorem.  Indeed, having such a theorem for the condensed matter
theory, it would be sufficient to consider the image of the global
solution.
\footnote{In this context it seems not uninteresting to note that the
famous local existence and uniqueness theorems for GR given by
Choquet-Bruhat \cite{Choquet-Bruhat} are based on the use of harmonic
gauge.  That means, they may be interpreted as local existence and
uniqueness theorems for the limit $\Xi,\Upsilon\to 0$ of $GLET$,
combined with considerations that local existence and uniqueness for
this limit (in this gauge) is the same as local existence and
uniqueness for general relativity.}

But in such a general class of theories as defined by our axioms
\ref{afirst}-\ref{alast} where is no hope for general global existence
theorems and, as a consequence, for a warranty that the restrictions
remain valid for $T(x)>t_0$.  Moreover, theories without global
existence theorems are quite reasonable as condensed matter theories.
It happens in reality that some material tears.  In this case, the
continuous condensed matter theory which is appropriate for this
material should have a limited domain of application, and there should
be solutions which reach $\rho=0$ somewhere.  But according to
equation \ref{gdef} $\rho=0$ corresponds to a violation of the
condition that $T(x)$ is time-like.  Thus, if such a material is
described by a theory which fulfills our axioms, and a solution
describes a state which at some time starts to tear, the restriction
that $T(x)$ is time-like no longer holds.

\section{Reconsideration of Old Arguments Against The Ether}
\label{appOldArguments}

The generalization of Lorentz ether theory we have presented here
removes some old, classical arguments against the Lorentz ether --
arguments which have justified the rejection of the Lorentz ether in
favor of relativity:

\begin{itemize}

\item There was no viable theory of gravity --- we have found now a
theory of gravity with GR limit which seems viable;

\item The assumptions about the Lorentz ether have been ad hoc.  There
was no explanation of relativistic symmetry, the relativistic terms
have had ad hoc character --- the assumptions we make for our medium
seem quite natural, and we {\em derive\/} the EEP instead of
postulating it.

\item There was no explanation of the general character of
relativistic symmetry, the theory was only electro-magnetic --- in the
new concept the ``ether'' is universal, all matter fields describe
properties of the ether, which explains the universality of the
EEP and the gravitational field;

\item There was a violation of the ``action equals reaction''
principle: there was influence of the ether on matter, but no reverse
influence of matter on the stationary and incompressible ether --- we
have now a Lagrange formalism which guarantees the ``action equals
reaction'' principle and have a compressible, instationary ether;

\item There were no observable differences in the predictions of the
Lorentz ether and special relativity --- there are important and
interesting differences between our theory and general relativity (see
appendix \ref{appPredictions}).

\end{itemize}

\section{Non-Equivalence of Lorentz Ether Theory and 
General Relativity with Clock Fields}
\label{GRCF}

In theorem \ref{main} we have proven that solutions of Lorentz ether
theory also fulfill the equations of general relativity with four
additional scalar fields $X^\alpha(x)$ which do not interact in any
way with the other matter.  In the context of GR quantization in
harmonic gauge, this Lagrangian has been considered by Kuchar and
Torre \cite{Kuchar}.  Following them, we name it ``general relativity
with clock fields'' (GRCF).

Now, a very important point is that, despite theorem \ref{main}, GLET
is not equivalent to GRCF as physical theories.  Indeed, what we have
established in theorem \ref{main} is only a very special relationship:
for solutions of GLET we obtain ``image solutions'' which fulfill the
equations of GRCF.  But there are a lot of important differences which
make above theories different from physical point of view:

\begin{itemize}

\item Different notions of completeness: A complete, global solution
of GLET is defined for all $-\infty<X^i,T<\infty$.  It is in no way
required that the effective metric $g_{\mu\nu}$ of this solution is
complete.  Therefore, the image of this solution in GRCF is not
necessarily a complete, global solution of GRCF.  

\item Global restrictions of topology: GRCF has solutions with
non-trivial topology. Every image of a solution of GLET has
trivial topology.

\item Global hyperbolicity: GRCF has solutions which are not globally
hyperbolic, especially solutions with closed causal loops.  Every
image of a solution of GLET is globally hyperbolic.  Moreover,
there exists a global harmonic time-like function: $T(x)$ on images.

\item Special character of $X^i,T$: GRCF contains a lot of
solutions so that the fields $X^i(x),T(x)$ do not define a global
system of coordinates. For every image of a solution of GLET the
fields $X^i(x),T(x)$ have this property.

\item Unreasonable boundary conditions: Physically reasonable
solutions of GRCF are only solutions with reasonable boundary
conditions in infinity.  For example, it seems reasonable to require
that physical fields have some upper bounds.  This leads to
$|X^i(x)|,|T(x)|<C$ for some constant $C$.  In this case, no image of
a solution of GLET defines a global physically reasonable
solution of GRCF.

\end{itemize}

Therefore, the two theories are conceptually very different theories.
They only look similar if we ignore the very special geometric nature
of the preferred coordinates $X^i,T$ of GLET.

\section{Comparison with RTG}\label{RTG}

There is also another theory with the same Lagrangian -- the
``relativistic theory of gravity'' (RTG) proposed by Logunov et
al. \cite{Logunov}.  In this theory, we have a Minkowski background
metric $\eta_{\mu\nu}$.  The Lagrangian of RTG is

\[L = L_{Rosen} + L_{matter}(g_{\mu\nu},\psi^m)
    - \frac{m_g^2}{16\pi}(\frac{1}{2}\eta_{\mu\nu}g^{\mu\nu}\sqrt{-g}
            - \sqrt{-g} - \sqrt{-\eta})
\]

If we identify the Minkowski coordinates in RTG with the preferred
coordinates in GLET, the Lagrangians are equivalent as functions of
$g_{\mu\nu}$ for the following choice of constants:
        $\Lambda=-\frac{m_g^2}{ 2}<0$,
        $\Xi=-\eta^{11}\frac{m_g^2}{ 2}>0$,
        $\Upsilon=\eta^{00}\frac{m_g^2}{ 2}>0$.
In this case, the equations for $g^{\mu\nu}$ coincide.  The harmonic
equation for the Minkowski coordinates hold in RTG \cite{Logunov}.  As
a consequence, the equations of the theories coincide.

These two theories are much closer to each other -- above theories
are metric theories of gravity with a preferred background. 
Nonetheless, there remain interesting differences even in this case.

First, in above theories we have additional restrictions related with
the notion of causality -- causality conditions.  In GLET, causality is
related with the Newtonian background -- the preferred time $T(x)$
should be a time-like function.  This is equivalent to the condition
$\rho>0$.  In RTG, causality is defined by the Minkowski background.
The light cone of the physical metric $g_{\mu\nu}$ should be inside
the light cone of the background metric $\eta_{\mu\nu}$.

Second, the derivation of GLET does not lead to the restrictions for
the sign of $\Xi,\Upsilon,\Lambda$.  Especially the restriction for
$\Lambda$ may be important: The cosmological constant $\Lambda$ in RTG
leads to deceleration, which (in combination with the big bounce
caused by $\Upsilon>0$) leads to an oscillating universe.  Instead,
observation suggests an acceleration of the expansion, thus, the
``wrong'' sign of $\Lambda$.  

But the more important differences are the differences which are
metaphysical at the classical level.  RTG, with its completely
different metaphysics, leads to a completely different quantization
program.  

\section{Analog Models of General Relativity}
\label{appCondensedMatter}

In this paper, we propose an ether theory of gravity, starting from
condensed-matter-like axioms and deriving a metric theory of gravity
with GR limit.  But this connection between condensed matter axioms
and a metric theory with GR limit may be, possibly, applied also in
another direction: Search for classical condensed matter theories
which fit into this set of axioms as analog models of general
relativity.  In this appendix we consider the possibility of such
applications.

\subsection{Comparison With Analog Gravity Research}

The observation that curved Lorentz metrics appear in condensed matter
theories is currently attracting considerable attention.  Though it is
not a priori expected that all features of Einstein gravity can
successfully be carried over to the condensed matter realm, interest
has turned to investigating the possibility of simulating aspects of
general relativity in analog models.  Domains of research in ``over a
hundred articles devoted to one or another aspect of analog gravity
and effective metric techniques'' have been dielectric media,
acoustics in flowing fluids, phase perturbations in Bose-Einstein
condensates, slow light, quasi-particles in superfluids, nonlinear
electrodynamics, linear electrodynamics, Scharnhorst effect, thermal
vacuum, ``solid state'' black holes and astrophysical fluid flows
\cite{Barcelo}.  Asking whether something more fundamental is going
on, Barcelo et al \cite{Barcelo} have found that linearization of a
classical field theory gives, for a single scalar field, a unique
effective Lorentz metric.  As observed by Sakharov \cite{Sakharov},
quantization of the linearized excitations around this background
gives a term proportional to the Einstein-Hilbert action in the
one-loop effective action.  But this approach has serious limitations:
It is not clear how to obtain the Einstein equivalence principle for
different types of excitations which, in general, ``see'' different
effective metrics.  The Einstein-Hilbert action appears only in the
quantum domain.  The most serious question is how to suppress the
non-covariant terms.

Now, the approach presented here may be used to solve these problems.
If we are able to find some real condensed matter which fits into our
set of axioms, this matter defines an ideal analog model of general
relativity.  Thus, we have reduced the search for analog models for
general relativity to a completely different question -- the search
for Lagrange formalisms for condensed matter theories which fit into
our set of axioms.  While we have not been able to find such Lagrange
formalisms yet, there are some points which seem worth to be
mentioned:

\begin{itemize}
\item  Our metric has been defined independent of any particular
linearization of waves.  It's definition is closely related to the
energy-momentum tensor:

\begin{equation}
T^\mu_\alpha = -(4\pi G)^{-1}\gamma_{\alpha\beta} g^{\beta\mu}\sqrt{-g}
\end{equation}

It is in no way claimed that some or all excitations which may be
obtained by various linearizations follow this metric.  Such a claim
would be false even in the world of standard general relativity.
Indeed, we know particular excitations which have different speeds --
like light on the background of a medium.  The EEP allows to
distinguish ``fundamental'' excitations (which use its basic metric)
and ``less fundamental'' excitations (like light in a medium).  But
starting with arbitrary waves in a medium described phenomenologically
or obtained by some linearization of approximate equations, without
knowing the EEP-related metric, we have no base to make this
distinction.

Once we do not define the metric as an effective metric of some
excitations, it is not a problem for our approach if different
excitations follow different effective metrics.  The metric is
uniquely defined by the energy-momentum tensor, which is closely
related to translational symmetry.  Therefore it is not strange that
this metric is the metric related with the EEP.

\item The general-relativistic terms appear already in the classical
domain.  Their origin is not quantization, but the general
restrictions for non-covariant terms which follow from our axioms.
Whatever additional terms we add to the Lagrangian -- if the modified
theory does not violate the axioms and does not renormalize
$\gamma_{\alpha\beta}$, the difference between old and new Lagrangian
should be covariant.

\item The non-covariant terms in our approach are uniquely defined and
easy to control: they do not depend on matter fields, they do not
depend on derivatives of $g_{\mu\nu}$.  

\end{itemize}

It remains to find interesting condensed matter theories which fits
into our axioms.

\subsection{How to Find Analog Models}

Let's shortly consider how difficult it is to find such matter.  The
continuity and Euler equations are common.  The negative definiteness
of the pressure tensor $p^{ij}$ seems uncommon.  But pressure is
defined by the Euler equation only modulo a constant.  This constant
should be fixed by the Lagrange formalism.  Thus, the real point is
the Lagrange formalism.  Unfortunately, Lagrange formalisms are not
uniquely defined, and there is a similar uncertainty in the definition
of the energy-momentum tensor.  Moreover, usually the pressure
$p^{ij}$ is not used as an independent variable.  Because of such
problems, it is not straightforward to obtain a Lagrange formalism
which fits into our axioms starting with known condensed matter
Lagrangians (as described, for example, in \cite{Wagner}).

One idea to find Lagrange formalisms which fit into our scheme may be
reverse engineering: we start with a typical GLET Lagrangian, see how
it looks in the condensed matter variables, and try to interpret them.
The background-dependent part of the GLET Lagrangian obtains a quite
simple form:

\begin{equation}
L = L_{GR} + L_{matter}
  - (8\pi G)^{-1}(\Upsilon \rho-\Xi (\rho v^2 + p^{ii}))
\end{equation}

\subsection{Condensed Matter Interpretation of Curvature}

The physical meaning of the general-relativistic Lagrangian can be
understood as related with inner stress.  If at a given moment of time
all spatial components of the curvature tensor vanish, then there
exists a spatial diffeomorphism which transforms the metric into the
Euclidean metric.  Such diffeomorphisms are known as transformations
into stress-free reference states in elasticity theory.  If no such
transformation exists, we have a material with inner stress.  This
understanding is in agreement with a proposal by Malyshev
\cite{Malyshev} to use of the 3D Euclidean Hilbert-Einstein Lagrangian
for the description of stress in the presence of dislocations.

The physical meaning of the components which involve time direction is
less clear.  It seems clear that they describe modifications of inner
stresses, but this has to be better understood.  Nonetheless, our
approach suggests a generalization of Malyshev's proposal -- to use
the full Hilbert-Einstein Lagrangian instead of its 3D part only.

\subsection{Condensed Matter Interpretation of Gauge Fields and Fermions}
\label{appGauge}

Of course, search for analog models requires a better understanding of
the matter part of the theory.  It would be helpful to have a similar
understanding of gauge and fermion fields.  Here the example of
superfluid $^3He-A$ seems to be of special interest: ``the low-energy
degrees of freedom in $^3He-A$ do really consist of chiral fermions,
gauge fields and gravity'' \cite{Volovik}. 

The incorporation of gauge fields and fermions into the theory is not
straightforward.  For gauge fields, the analogy between gauge symmetry
and general relativity suggests that there should be an analog of our
approach for gauge theory too.  How could it look like?  It seems
clear that the analog of the harmonic condition, the Lorenz gauge
condition

\begin{equation}\label{Lorenz}
\partial_\mu A^\mu = 0
\end{equation}

has to be interpreted as a conservation law.  Then, gauge symmetry
will be no longer a fundamental symmetry.  Instead, it should be
derived in a similar way as the EEP, describing the inability of
internal observers to distinguish states which are different in
reality.

Fermions also require also additional considerations.  Some first
proposals have been made in \cite{SchmelzerHypocrystalline}.  It shows
that the kinematics of deformations of a quite simple ether model
gives a particle content which is identical with the fermionic part of
the standard model (in its non-minimal variant with a neutrino as a
usual Dirac spinor).

\section{Quantization of Gravity Based on Ether Theory}
\label{appQuantization}

The most important open problem of fundamental physics is quantization
of gravity.  Here, an ether theory suggests an alternative
quantization concept which at least solves some of the most serious
problems of GR quantization.  Indeed, we already know one simple way
to quantize classical condensed matter theories in a Newtonian
framework: It would be sufficient to use classical Schr\"odinger
quantization of some discrete (atomic) model of these field theories.
Once we have a classical condensed matter theory with Newtonian
framework as the fundamental theory of everything, the analogy with
classical condensed matter theory suggests a quantization program
along the following lines:

\begin{itemize}

\item As a first step we have to switch from Euler (local) coordinates
to Lagrange (material) coordinates.  It is known that a Lagrange
formalism in Euler coordinates gives a Lagrange formalism in Lagrange
coordinates and that this transformation gives a canonical
transformation of the related Hamilton formalisms
\cite{Broer}.

This essentially changes the situations with the constraints. The
condensed matter is no longer described by $\rho, v^i$ but by
$x^i(x^i_0,t)$.  The continuity equation disappears, the Euler equation
becomes a second order equation.

\item Next, we have to find atoms, that means, to discretize the
problem.  The density $\rho$ should be identified with the number of
nodes inside a region, and for functions defined on the nodes we
obtain

\begin{equation} 
\int_A f(x)\rho(x) dx \to \sum_{x_k\in A} f_k. 
\end{equation}

The density $\rho$ does not appear as a discrete variable at all.  Of
course, instead of discretizing the equations them-self, we discretize
the Lagrangian and define the discrete equations as exact
Euler-Lagrange equations for the discretized Lagrangian.

It would be helpful if it would be possible to discretize the
Lagrangian so that the discrete Lagrangian $L_d$ fulfills our axioms
too. Then it would be sufficient to discretize only the lowest order
parts $L_0$ of $L$, that means, not the Einstein-Hilbert Lagrangian
itself.  Following our derivation, we obtain that $L_0-L_d$ should be
covariant but non-zero.  Then the Einstein-Hilbert term appears as the
lowest order term of the discretization error automatically.  On the
other hand, the discrete Lagrangian itself would be only a simple
discretization of the lowest order terms.

\item Note that in this approach we are not forced to quantize in a
way which preserves local symmetry groups like the group of
diffeomorphisms or gauge groups.  Instead, we believe that these
symmetry groups appear only as large distance approximations, moreover
not as fundamental symmetry groups but as groups which describe the
restricted possibilities of internal observers to distinguish
different states -- but such symmetry groups are irrelevant for
quantization.  Of course, we have to develop yet a similar
understanding of gauge symmetry (see app. \ref{appGauge}).

\item The last step is canonical quantization.  We have to derive the
Hamilton formalism from the Lagrange formalism. Then we can apply some
canonical ordering (Weyl quantization or normal ordering) to obtain a
well-defined quantum theory.

\end{itemize}

Note that this quantization approach is closely related to the
consideration of analog models (app. \ref{appCondensedMatter}).  If we
have an analog model, we can use this model as a guide for
quantization.  Nonetheless, the general scheme described here may work
even if we do not find any real analog models.

\subsection{Comparison with Canonical Quantization}

It is worth to note that this quantization program is completely
different from the program of canonical quantization of general
relativity: Space and time remain a classical background and are not
quantized.  We have a different number of degrees of freedom: States
which are different but indistinguishable for internal observers are
nonetheless handled as different states.  That means, in related path
integral formulations their probabilities have to be added.  No ghost
fields have to be introduced.  There is also no holography principle:
the number of degrees of freedom of the quantum theory is, for almost
homogeneous $\rho$, proportional to the volume.

The problems which have to be solved in this quantization program seem
to be not very hard in comparison with the problems of canonical GR
quantization.  This suggests, on the other hand, that quantization of
gravity following this scheme will not lead to non-trivial
restrictions of the parameter space of the standard model: There will
be a lot of quantum theories of gravity with very different material
models.

That a preferred frame is a way to solve the ``problem of time'' in
quantum gravity is well-known: ``... in quantum gravity, one response
to the problem of time is to `blame' it on general relativity's
allowing arbitrary foliations of spacetime; and then to postulate a
preferred frame of spacetime with respect to which quantum theory
should be written.''  \cite{Butterfield}.  This way to solve the
problem is rejected not for physical reasons, but because of
deliberate metaphysical preference for the standard
general-relativistic spacetime interpretation: ``most general
relativists feel this response is too radical to countenance: they
regard foliation-independence as an undeniable insight of
relativity.''  \cite{Butterfield}.  These feelings are easy to
understand. At ``the root of most of the conceptual problems of
quantum gravity'' is the idea that ``a theory of quantum gravity must
have something to say about the quantum nature of space and time''
\cite{Butterfield}.  The introduction of a Newtonian background
``solves'' them in a very trivial, uninteresting way.  It does not
tell anything about quantum nature of space and time, because space
and time do not have any quantum nature in this theory -- they have
the same classical nature of a ``stage'' as in non-relativistic
Schr\"odinger theory.  The hopes to find something new, very
interesting and fundamental about space and time would be dashed.  But
nature is not obliged to fulfill such hopes.

\section{Compatibility with Realism and Hidden Variables}
\label{appRealism}

An important advantage of a theory with preferred frame is related
with the violation of Bell's inequality \cite{Bell}.  It is widely
accepted that experiments like Aspect's \cite{Aspect} show the
violation of Bell's inequality and, therefore, falsify Einstein-local
realistic hidden variable theories.  Usually this is interpreted as a
decisive argument against hidden variable theories and the EPR
criterion of reality.  But it can as well turned into an argument
against Einstein locality.  Indeed, if we take classical realism (the
EPR criterion \cite{EPR}) as an {\em axiom\/}, the violation of Bell's
inequality simply proves the existence of superluminal causal
influences.  Such influences are compatible with a theory with
preferred frame and classical causality, but not with Einstein
causality.

This has been mentioned by Bell \cite{Bell1}: ``the cheapest
resolution is something like going back to relativity as it was before
Einstein, when people like Lorentz and Poincare thought that there was
an aether --- a preferred frame of reference --- but that our
measuring instruments were distorted by motion in such a way that we
could no detect motion through the aether. Now, in that way you can
imagine that there is a preferred frame of reference, and in this
preferred frame of reference things go faster than light.''

Note that at the time of the EPR discussion the situation was quite
different.  People have accepted the rejection of classical realism
under the assumtion that realistic hidden variable theories for
quantum theory do not exist, and that this is a theorem proven by von
Neumann.  Today we know not only that EPR-realistic, even
deterministic hidden variable theories exist -- we have Bohmian
mechanics (BM) \cite{Bohm}, \cite{BohmHiley} as an explicit example.
Thus, quantum theory alone does not give any argument against
classical realism, as was believed at that time.  Therefore, the only
argument against classical realism is its incompatibility with
Einstein causality --- thus, the reason why we mention it here as
support for our ether theory.

We use here the existence of Bohmian mechanics as an argument for
compatibility of classical realism with observation. Therefore, some
remarks about the problems of Bohmian mechanics are useful.  First,
the main ``problem'' of Bohmian mechanics is that it needs a preferred
frame.  This is already a decisive argument for many scientists, but
cannot count here because it is the reason why we mention it as
support for ether theory.  But there have been other objections.
Problems with spin, fermions, and relativistic particles are often
mentioned.  But these problems have been solved in modern
presentations of BM \cite{BohmHiley} \cite{Duerr}.  It seems useful to
mention here that BM is a quite general concept which works on quite
arbitrary configuration spaces.  Thus, it does not depend on a
particle picture.  Once a quantum theory starts with fields, the
natural ``Bohmianization'' of it is a Bohmian field theory.

Thus, we can use BM here as support for our ether theory.  Reversely,
our ether theory defines a program for Bohmian gravity.  Indeed,
following the quantization program described in
app. \ref{appQuantization} we end with canonical quantization of a
discrete theory, which fits into the domain of applicability of the
classical scheme of Bohmian mechanics.

\section{Cosmological Predictions}
\label{appPredictions}

It is not the purpose of this paper to consider the various
experimental differences between GR and our theory in detail.
Nonetheless, we want to give here a short description of some
differences to support the claim that such differences exist and lead
to observable effects.  It is clear that the predictions have to be
worked out in detail.

In general, because the Einstein equations appear as the limit
$\Xi,\Upsilon\to 0$, the overwhelming experimental evidence in favor
of the EEP and the Einstein equations (Solar system tests, binary
pulsars and many others) as described by Will \cite{Will} can give
only upper bounds for the parameters $\Xi, \Upsilon$.  It is not the
purpose of this paper to obtain numerical values for these bounds.

Some qualitative differences have been already mentioned: non-trivial
topologies and closed causal loops are forbidden.  There should exist
a global time-like harmonic coordinate.  Additional predictions
require reasonable guesses about the preferred coordinates.  But this
is easy.  For the global universe, we obtain such natural guesses
based on obvious symmetry considerations.  For local situations
(gravitational collapse) we have the requirement that the preferred
coordinates have to fit the global solution and need reasonable
initial values.  Together with the harmonic equation, this is enough
to define them.

Note that all the differences between GLET and GR considered below
depend on the assumptions about the special geometric nature of the
preferred coordinates.  Especially the GLET solutions have the boundary
conditions appropriate for coordinates, but not for scalar matter
fields.  Therefore, the difference between GR with scalar fields and
GLET is not only a purely theoretical one, it leads to really
observable differences.

\subsection{The Homogeneous Universe}

The background-dependent terms of GLET lead to interesting observable
effects.  Let's consider at first the homogeneous universe solutions
of the theory.  Because of the Newtonian background frame, only a flat
universe may be homogeneous.  The usual FRW-ansatz for the flat
universe

\begin{equation} ds^2 = d\tau^2 - a^2(\tau)(dx^2+dy^2+dz^2) \end{equation}

leads to ($p=k\varepsilon$, $8\pi G = c = 1$):

\begin{eqnarray*}
3(\dot{a}/a)^2  &=&
        - \Upsilon/a^6 + 3 \Xi/a^2 + \Lambda + \varepsilon\\
2(\ddot{a}/a) + (\dot{a}/a)^2 &=&
        + \Upsilon/a^6 +   \Xi/a^2 + \Lambda - k \varepsilon
\end{eqnarray*}

The $\Upsilon$-term influences only the early universe, its influence
on later universe may be ignored.  But, if we assume $\Upsilon >0$,
the qualitative behavior of the early universe changes in a
remarkable way.  We obtain a lower bound $a_0$ for $a(\tau)$ defined
by

\begin{equation} 
\Upsilon/a_0^6 = 3 \Xi/a_0^2 + \Lambda + \varepsilon 
\end{equation}

The solution becomes symmetrical in time -- a big bounce.  Note that
this solves two problems of cosmology which are solved in standard
cosmology by inflation theory: the flatness problem and the
cosmological horizon problem.\footnote{About these problem, see for
example Primack \cite{Primack}: First, ``the angular size today of the
causally connected regions at recombination ($p^+ + e^-\to H$) is only
$\Delta\theta\sim 3^o$.  Yet the fluctuation in the temperature of the
cosmic background radiation from different regions is very small:
$\Delta T/T\sim 10^{-5}$. How could regions far out of causal contact
have come to temperatures that are so precisely equal?  This is the
`horizon problem'.''  (p.56) Even more serious seems the following
problem: In the standard hot big bang picture, ``the matter that
comprises a typical galaxy, for example, first came into causal
contact about a year after the big bang.  It is hard to see how
galaxy-size fluctuations could have formed after that, but even harder
to see how they could have formed earlier'' (p.8).}  The question what
the theory predicts about the distribution of fluctuations of the CMBR
radiation requires further research: It depends on reasonable initial
conditions before the bounce.  The bounce itself is too fast to allow
the establishment of a global equilibrium, thus, the equilibrium
should exist already before the bounce. 

Instead, the $\Xi$-term gives a dark energy term. For $\Xi >0$ it
behaves like homogeneously distributed dark matter with
$p=-(1/3)\varepsilon$.  The additional term has the same influence on
$a(\tau)$ as the classical curvature term.  On the other hand, the
universe should be flat.
\footnote{This makes the comparison with observation quite easy: as
long as $a(\tau)$ is considered, the prediction of a $\Xi$CDM theory
with $\Lambda=0, \Xi>0$ is the same as for OpenCDM.  Therefore, the
SNeIa data which suggest an accelerating universe are as problematic
for $\Xi$CDM as for OpenCDM. The 2000 review of particle physics
\cite{RPP2000}, 17.4, summarizes: ``the indication of
$\Omega_\Lambda\ne 0$ from the SNeIa Hubble diagram is very
interesting and important, but on its own the conclusion is
susceptible to small systematic effects.  On the other hand small
scale CBR anisotropy confirming a nearly flat universe ... strongly
suggests the presence of $\Lambda$ or other exotic (highly negative
pressure) form of dark mass-energy.''  The fit with the farthest SN
1997ff can be seen in in figure 11 of \cite{Riess} -- the figure for
$\Omega_M=0.35,\Omega_\Xi=0.65$ is the same as for the
$\Omega_M=0.35,\Omega_\Lambda=0$ open universe given in this figure.
Thus, while not favored by the SNeIa results, $\Xi$CDM may be
nonetheless a viable dark energy candidate, better than OpenCDM
because it requires a flat universe.}

\subsection{Gravitational Collapse}

Another domain where $\Upsilon>0$ leads to in principle observable
differences to GR is the gravitational collapse.  Let's consider for
this purpose a stable spherically symmetric metric in harmonic
coordinates.  For a function $m(r), 0<m<r, r=\sqrt{\sum (X^i)^2}$, the
metric

\begin{equation} ds^2 = (1-\frac{m \partial m/\partial r }{ r})
                (\frac{r-m}{ r+m}dt^2-\frac{r+m}{ r-m}dr^2)
        - (r+m)^2 d\Omega^2 \end{equation}

is harmonic in $X^i$ and $T=t$.  For constant m this is the
Schwarzschild solution in harmonic coordinates.  For time-dependent
$m(r,t)$ the $X^i$ remain harmonic but $t$ not.  $T$ has to be
computed by solving the harmonic equation for Minkowski initial values
before the collapse.

Now, a non-trivial term $\Upsilon>0$ becomes important near the
horizon.  Indeed, let's consider as a toy ansatz for the inner part of
a star the distribution $m(r)=(1-\Delta)r$.  This gives

\begin{eqnarray*}
ds^2 &=& \Delta^2dt^2 - (2-\Delta)^2(dr^2+r^2d\Omega^2) \\
0    &=& -\Upsilon \Delta^{-2} +3\Xi(2-\Delta)^{-2}+\Lambda+\varepsilon\\
0    &=& +\Upsilon \Delta^{-2} + \Xi(2-\Delta)^{-2}+\Lambda-p
\end{eqnarray*}

We can ignore the cosmological terms $\Xi, \Lambda$ in comparison to
the matter terms $\varepsilon, p$, but for very small $\Delta$ even a
very small $\Upsilon>0$ becomes important.  Our ansatz gives a stable
``frozen star'' solution for ultra-relativistic $p=\varepsilon$ and
$\Xi=\Lambda=0$.  The surface time dilation is
$\Delta^{-1}=\sqrt{\varepsilon/\Upsilon} \sim M^{-1}$ for mass M.  In
general, it seems obvious that the term $\Upsilon>0$ prevents
$g^{00}\sqrt{-g}$ from becoming infinite in harmonic coordinates.
Therefore, the gravitational collapse stops and leads to some stable
frozen star with a size very close to the Schwarzschild radius.

In principle, this leads to observable effects.  The surface of the
objects considered as black holes in GR should be visible but highly
redshifted.  Astronomical observations like described by Menou et
al. \cite{Menou} would be in principle able to observe such effects.
Unfortunately, surface time dilation should be very large for
cosmological reasons: To preserve the standard big bang scenario (see
above), we obtain very small upper bounds for $\Omega_\Upsilon$ today.
For small enough $\Upsilon$ we obtain a large enough redshift and the
surface remains invisible in comparison to the background radiation.

Are there effects which may distinguish a stable frozen star with very
small $\Upsilon\ll 1$ from a GR black hole?  At least in principle,
yes.  A GR black hole will Hawking radiate, a stable frozen star (as
every stable star in semiclassical theory) not. This leads to
observable differences for very small black holes.  A difference
appears also if we look at radiation going through a transparent star
\footnote{Note in this context that for neutrinos all usual stars are
transparent.}.  For a GR black hole we will see nothing (exponentially
redshifted radiation), while in the case of a stable frozen star we
see non-redshifted radiation.  In above cases this is caused by the
same difference: For an external observer, the GR black hole always
looks like a collapsing, non-static solution.  Instead, a collapsing
star in our theory becomes a stable frozen star after quite short
time.  This time depends on $\Upsilon$ only in a logarithmic way,
therefore, the prediction does not depend on the value of $\Upsilon$.

\subsection{Ether Sound Waves and Gravitational Waves}

In fluids described by continuity and Euler equations there are sound
waves.  This seems to be in contradiction with the situation in
general relativity.  They cannot correspond to gravitational waves as
predicted by general relativity, which are quadrupole waves.

Now, in the Lorentz ether these waves are gravity waves.  Because
general-relativistic symmetry is broken by the additional
background-dependent terms, the gravitational field has more degrees of
freedom in the Lorentz ether.  In some sense, these additional degrees
of freedom are the four fields which appear in GRCF discussed in
app. \ref{GRCF}.  Now, these additional degrees of freedom interact with
the other degrees of freedom only via the background-dependent term.
That means, they interact with matter only as dark matter.  Second, in
the limit $\Xi,\Upsilon\to 0$ they no longer interact with usual
matter at all.  Thus, in this limit they become unobservable for
internal observers.  That's why in this limit they may be removed from
considerations as hidden variables.  

On the other hand, for non-zero $\Xi,\Upsilon$ they nonetheless
interact with matter.  And this interaction leads, in principle, to
observable effects.  Nonetheless, we have not considered such effects,
because we believe that the restrictions which follow from
cosmological considerations are much stronger.  This belief is
qualitatively justified by our considerations about the GR limit: We
have argued that the GR limit is relevant for small distance high
frequency effects, and if the total variation of the $g_{\mu\nu}$
remains small.  This is exactly the situation for gravitational waves.

On the other hand, there may be other factors which override these
simple qualitative considerations.  For example the ability of gravity
waves to go over large distances which leads to integration of these
small effects over large distances, or for measurements of energy loss
as done for binary pulsars, which allows integration over many years.
For the range of gravity waves in theories with massive graviton such
considerations have been made in \cite{Logunov}, and their results
agree with our expectation that the upper bounds coming from
cosmological considerations are nonetheless much stronger.

\end{appendix}

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\bibitem{BohmHiley} D.~Bohm, B.J. Hiley, The undivided universe: an
ontological interpretation of quantum theory, Routledge, London 1993

\bibitem{Butterfield} J.~Butterfield, C.J.~Isham, Spacetime and the
philosophical challenge of quantum gravity, gr-qc/9903072

\bibitem{Choquet-Bruhat}Y.~Choquet-Bruhat,
Theoreme d'existence pour certain systemes d'equations aux derivees
partielles non lineaires, Acta Math., 88, 141 (1952)

\bibitem{Deser} S.~Deser, Gen. Rel. and Grav 1, 9, 1970

\bibitem{Duerr} D. D\"urr, S. Goldstein, N. Zanghi, Bohmian mechanics
as the foundation of quantum mechanics, quant-ph/9511016, (1995)

\bibitem{EPR} A.~Einstein, B.~Podolsky, N.~Rosen, Phys. Rev. 47,
777-780, 1935

\bibitem{Feynman} R.P.~Feynman, Lectures on gravitation,
Calif. Inst. of Technol. 1963

\bibitem{Fock}
V.~Fock, Theorie von Raum, Zeit und Gravitation, Akademie-Verlag
Berlin, 1960

\bibitem{RPP2000} D.E. Groom et al., 2000 Review of Particle Physics,
The European Physical Journal C15 (2000) 1, available on the PDG WWW
pages (URL: http://pdg.lbl.gov/)

%\bibitem{Isham}        C.~Isham, gr-qc/9210011

%\bibitem{Jacobson} T.~Jacobson, Phys Rev D 44 1731 (1991)

\bibitem{Jacobson95} T. Jacobson, Thermodynamics of spacetime: the
Einstein equation of state, gr-qc/9504004

\bibitem{Jacobson00} T. Jacobson, Spontaneously broken Lorentz
symmetry and gravity, gr-qc/0007031

%\bibitem{Jegerlehner} F.~Jegerlehner, hep-th/9803021

\bibitem{Kretschmann}
E. Kretschmann, Ann. Phys. 53, 575, 1917

\bibitem{Kuchar}
K.V.~Kuchar, C.G.~Torre, Harmonic gauge in canonical gravity, Physical
Review D, vol. 44, nr. 10, pp. 3116-3123, 1991

%\bibitem{Lightman} A.P. Lightman, D.L. Lee, New two-metric theory of
%gravity with prior geometry, Phys.Rev. D 8, 10, 3293-3302 (1973)

\bibitem{Logunov}
S.S.~Gershtein, A.A.~Logunov, M.A.~Mestvirishvili, The upper limit on
the graviton mass, hep-th/9711147

\bibitem{Malyshev} C. Malyshev, cond-mat/9901316

\bibitem{Menou} K. Menou, E.Quataert, R. Narayan, Astrophysical
evidence for black hole event horizons, gr-qc/9803057

\bibitem{MTW} C. Misner, K. Thorne, J. Wheeler, Gravitation, Freedman,
San Francisco, 1973

\bibitem{Nelson} E. Nelson, Derivation of the {S}chr\"odinger Equation
from Newtonian Mechanics, Phys.Rev. 150, 1079--1085, 1966

%\bibitem{Ni} W. Ni, A new theory of gravity, Phys Rev D 7 (10)
%2880-2883 (1972)

\bibitem{Primack} J.R.~Primack, astro-ph/9707285

\bibitem{Polchinski} J. Polchinski, String theory, Cambridge
University Press, Cambridge (1998)

\bibitem{Riess} 
A.G. Riess et al, The farthest known supernova: support for an
accelerating universe and a glimpse of the epoch of deceleration,
astro-ph/0104445

\bibitem{Rosen}
N. Rosen, Ann. of Phys. vol. 22, nr. 1, pp. 1-11, 1963

%\bibitem{Rosen}
%N. Rosen, A theory of gravitation, Ann. Phys. 84, 45-473 (1974)

\bibitem{Rosu} H. C. Rosu, Hawking-like effects and Unruh-like
effects: toward experiments? gr-qc/9406012

\bibitem{Sakharov}
A.D.~Sakharov, Sov.Phys.Doklady 12, 1040 (1968)

\bibitem{Schmelzer} I.~Schmelzer, General ether theory -- a metric
theory of gravity with condensed matter interpretation, Proc. of the
XXII international workshop on high energy physics and field theory,
Protvino, June 23-25, 1999

%\bibitem{SchmelzerWave} I. Schmelzer,
%http://relativity.ilja-schmelzer.net/mathnotions.html, see also
%http://groups.google.com/groups?as_umsgid=%3Ci3g4sthiygk.fsf%40fermi.wias-berlin.de%3E (1998)

\bibitem{SchmelzerHypocrystalline} I. Schmelzer, hep-th/0209167

\bibitem{UnruhAcoustic} W.G.~Unruh, Phys Rev Lett {\bf46} 1351  (1981)

\bibitem{VisserAcoustic} M.~Visser, Acoustic black holes: horizons,
ergospheres, and Hawking radiation, gr-qc/9712010

\bibitem{Volovik} G.E.~Volovik, Induced gravity in superfluid $^3He$,
cond-mat/9806010

\bibitem{Volovik98}
G.E.~Volovik, Field theory in superfluid $^3He$, cond-mat/9812381

\bibitem{Wagner}
H.-J. Wagner, Das inverse Problem der Lagrangeschen Feldtheorie in
Hydrodynamik, Plasmaphysik und hydrodynamischem Bild der
Quantenmechanik, Universit\"at Paderborn, 1997

\bibitem{Weinberg65} S.~Weinberg, Phys.Rev. B138, 988 (1965)

\bibitem{Weinberg} S.~Weinberg, What is quantum field theory, and what
did we think it is, hep-th/9702027

%\bibitem{Wilczek} F.~Wilczek, hep-th/9803075

\bibitem{Will} C.M. Will, The confrontation between general relativity
and experiment, gr-qc/9811036

\end{thebibliography}
\end{document}















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J.S.~Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1, 195-200, 1964,
also in \cite{Bell}, 14-21

\bibitem{Bell1} J.S.~Bell, in Davies, P.C.W. and Brown,J.R., eds., The
Ghost in the Atom, Cambridge: Cambridge University Press 1986

\bibitem{Broer} L.F.J. Broer, J.A. Kobussen, Conversion from material
to local coordinates as a canonical transformation, Appl. Sci.Res. 29,
419-429, 1974  %% cited by Wagner, 8

\bibitem{Bohm} D.~Bohm, Phys. Rev. 85, 166-193, 1952

\bibitem{BohmHiley} D.~Bohm, B.J. Hiley, The undivided universe: an
ontological interpretation of quantum theory, Routledge, London 1993

\bibitem{Butterfield} J.~Butterfield, C.J.~Isham, Spacetime and the
philosophical challenge of quantum gravity, gr-qc/9903072

\bibitem{Choquet-Bruhat}Y.~Choquet-Bruhat,
Theoreme d'existence pour certain systemes d'equations aux derivees
partielles non lineaires, Acta Math., 88, 141 (1952)

\bibitem{Deser} S.~Deser, Gen. Rel. and Grav 1, 9, 1970

\bibitem{Duerr} D. D\"urr, S. Goldstein, N. Zanghi, Bohmian mechanics
as the foundation of quantum mechanics, quant-ph/9511016, (1995)

\bibitem{EPR} A.~Einstein, B.~Podolsky, N.~Rosen, Phys. Rev. 47,
777-780, 1935

\bibitem{Feynman} R.P.~Feynman, Lectures on gravitation,
Calif. Inst. of Technol. 1963

\bibitem{Fock}
V.~Fock, Theorie von Raum, Zeit und Gravitation, Akademie-Verlag
Berlin, 1960

\bibitem{RPP2000} D.E. Groom et al., 2000 Review of Particle Physics,
The European Physical Journal C15 (2000) 1, available on the PDG WWW
pages (URL: http://pdg.lbl.gov/)

%\bibitem{Isham}        C.~Isham, gr-qc/9210011

%\bibitem{Jacobson} T.~Jacobson, Phys Rev D 44 1731 (1991)

\bibitem{Jacobson95} T. Jacobson, Thermodynamics of spacetime: the
Einstein equation of state, gr-qc/9504004

\bibitem{Jacobson00} T. Jacobson, Spontaneously broken Lorentz
symmetry and gravity, gr-qc/0007031

%\bibitem{Jegerlehner} F.~Jegerlehner, hep-th/9803021

\bibitem{Kretschmann}
E. Kretschmann, Ann. Phys. 53, 575, 1917

\bibitem{Kuchar}
K.V.~Kuchar, C.G.~Torre, Harmonic gauge in canonical gravity, Physical
Review D, vol. 44, nr. 10, pp. 3116-3123, 1991

%\bibitem{Lightman} A.P. Lightman, D.L. Lee, New two-metric theory of
%gravity with prior geometry, Phys.Rev. D 8, 10, 3293-3302 (1973)

\bibitem{Logunov}
S.S.~Gershtein, A.A.~Logunov, M.A.~Mestvirishvili, The upper limit on
the graviton mass, hep-th/9711147

\bibitem{Malyshev} C. Malyshev, cond-mat/9901316

\bibitem{Menou} K. Menou, E.Quataert, R. Narayan, Astrophysical
evidence for black hole event horizons, gr-qc/9803057

\bibitem{MTW} C. Misner, K. Thorne, J. Wheeler, Gravitation, Freedman,
San Francisco, 1973

\bibitem{Nelson} E. Nelson, Derivation of the {S}chr\"odinger Equation
from Newtonian Mechanics, Phys.Rev. 150, 1079--1085, 1966

%\bibitem{Ni} W. Ni, A new theory of gravity, Phys Rev D 7 (10)
%2880-2883 (1972)

\bibitem{Primack} J.R.~Primack, astro-ph/9707285

\bibitem{Polchinski} J. Polchinski, String theory, Cambridge
University Press, Cambridge (1998)

\bibitem{Riess} 
A.G. Riess et al, The farthest known supernova: support for an
accelerating universe and a glimpse of the epoch of deceleration,
astro-ph/0104445

\bibitem{Rosen}
N. Rosen, Ann. of Phys. vol. 22, nr. 1, pp. 1-11, 1963

%\bibitem{Rosen}
%N. Rosen, A theory of gravitation, Ann. Phys. 84, 45-473 (1974)

\bibitem{Rosu} H. C. Rosu, Hawking-like effects and Unruh-like
effects: toward experiments? gr-qc/9406012

\bibitem{Sakharov}
A.D.~Sakharov, Sov.Phys.Doklady 12, 1040 (1968)

\bibitem{Schmelzer} I.~Schmelzer, General ether theory -- a metric
theory of gravity with condensed matter interpretation, Proc. of the
XXII international workshop on high energy physics and field theory,
Protvino, June 23-25, 1999

%\bibitem{SchmelzerWave} I. Schmelzer,
%http://relativity.ilja-schmelzer.net/mathnotions.html, see also
%http://groups.google.com/groups?as_umsgid=%3Ci3g4sthiygk.fsf%40fermi.wias-berlin.de%3E (1998)

\bibitem{SchmelzerHypocrystalline} I. Schmelzer, hep-th/0209167

\bibitem{UnruhAcoustic} W.G.~Unruh, Phys Rev Lett {\bf46} 1351  (1981)

\bibitem{VisserAcoustic} M.~Visser, Acoustic black holes: horizons,
ergospheres, and Hawking radiation, gr-qc/9712010

\bibitem{Volovik} G.E.~Volovik, Induced gravity in superfluid $^3He$,
cond-mat/9806010

\bibitem{Volovik98}
G.E.~Volovik, Field theory in superfluid $^3He$, cond-mat/9812381

\bibitem{Wagner}
H.-J. Wagner, Das inverse Problem der Lagrangeschen Feldtheorie in
Hydrodynamik, Plasmaphysik und hydrodynamischem Bild der
Quantenmechanik, Universit\"at Paderborn, 1997

\bibitem{Weinberg65} S.~Weinberg, Phys.Rev. B138, 988 (1965)

\bibitem{Weinberg} S.~Weinberg, What is quantum field theory, and what
did we think it is, hep-th/9702027

%\bibitem{Wilczek} F.~Wilczek, hep-th/9803075

\bibitem{Will} C.M. Will, The confrontation between general relativity
and experiment, gr-qc/9811036

\end{thebibliography}