The Thermal Substructure of General Relativity

Manfred Requardt

Institut fuer Theoretische Physik

Universitaet Goettingen

Friedrich-Hund-Platz 1

37077 Goettingen Germany

(E-mail<EMAIL_ADDRESS>or
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###### Abstract

In a first step we will provide arguments for the understanding of quantum
space-time (QST), that means, the microscopic substructure which is assumed to
underly ordinary smooth classical space-time, as a thermal system at each
(macroscopic) point $x$ of the classical space-time manifold (ST). In this
context we exploit among other things some recent findings in the foundations
of quantum statistical mechanics.

We argue that the classical metric tensor field $g_{ij}(x)$ plays the role of
an order parameter field, signalling the existence of a primordial phase
transition in which both space-time and (quantum) matter emerged, accompanied
by the spontaneous symmetry breaking (SSB) of diffeomorphism invariance which
is induced by the existence of quantum excitations in QST.

We then analyze the role the Einstein equations (EEQ) are playing in this
context and relate them to the local thermodynamical heat influx into QST at
point $x$, thus defining the Einstein tensor or Ricci tensor as local heat
tensor. We also define local gravitational temperature, entropy and internal
energy.

Finally, we give a thermodynamic interpretation of the conserved Noether
current which derives from diffeomorphism invariance, separating it into heat
contributions and work terms. In this connection we discuss the role and
nature of tensor and non-tensor quantities and relate our investigation to the
socalled energy vector of Hilbert‘s classical analysis.

##  1 Introduction

Without question one of the most unexpected developements in general
relativity was the realisation that there exists a close relationship between
certain laws of classical black hole (BH) physics and the laws of
thermodynamics ([1]). This fundamental connection was first described by
Bekenstein ([2],[3]) and a little bit later supported by the fundamental
analysis of Hawking ([4]). A nice overview is given for example in [5].

That there might perhaps exist a closer and even more fundamental connection
between classical relativity and thermodynamics was, as far as we know,
emphasized by Jacobson ([6]) and further developed in e.g. [7] and many papers
of Padmanabhan (see, for example,[8],[9] or [10]). It was shown that one can
infer from the physics near horizons, in particular the event horizon of a
black hole, that the Einstein equation (EEQ) must hold. One important tool in
this context was the use of approximate Rindler-Unruh horizons and the
corresponding results of Fulling-Unruh ([11],[12]).

One should remark that, as far as we can see, all the many papers which do
exist in this context emphasize the importance of horizons and the role which
is played by the entanglement entropy of systems behind these horizons. That
is, from this view point, entropy and more generally thermodynamics appear to
be essentially located near these horizons and not e.g. in the bulk. These
properties seem to be suggested, for example, by the famous entropy area law
of black holes (BH).

Whereas horizons apparently seem to play such a fundamental role for the
thermal behavior of gravitation, we want to go a step further and will argue
in this paper that in our view the situation is a different one and that both
the Einstein equation (EEQ) and gravitation in general are of a thermal nature
being independent of the existence of horizons and which can be inferred by
analyzing the ordinary bulk situation in general relativity (GR). The special
behavior near horizons is then rather a particular consequence of this more
general scenario.

In a first step we will provide arguments for the understanding of quantum
space-time (QST), that is, the microscopic structure which underlies our
ordinary classical smooth space-time manifold ST, as a thermal system at each
macroscopic point $x\in ST$. This will be done in the next section (some
remarks in this direction have already been made in [13]. Early ideas can
already be found in the work of Sakharov but without a thermal connotation
(see e.g. [14] and [15]). We will exploit more recent findings in the
foundations of modern quantum statistical mechanics ([16],[17],[18]).
Furthermore we will introduce the notion of the classical space-time metric
$g_{ij}(x)$ as an order parameter field and classical smooth space-time as an
order parameter manifold.

By these notions we mean the following. As briefly described in [13], we
assume that (Q)ST emerged as the consequence of a primordial phase transition
PT. In this context a non-vanishing field of observables $g_{ij}(x)$ came into
existence which was zero or rather vanishing before the phase transition
happened in the primordial soup. That is, $g_{ij}(x)$ and ST represent
parameters which indicate that a certain phase transition took place. Such a
quantity is typically called an order parameter.

We start from the following assumption or observation: Classical space-time ST
remains to be smooth on all ordinary scales of the quantum matter regime. So
we conjecture that the typical scale where quantum gravity effects emerge is
the  Planck scale, that is

$l_{p}=\hbar G/c^{3}\quad,\quad t_{p}=\hbar G/c^{5}=l_{p}/c$ (1)

We hence assume that the typical quantum gravity degrees of freedom (DoF) live
on the same scale. This implies that, as e.g. in hydrodynamics, there exist
many quantum gravity DoF in the infinitesimal neighborhood of the macroscopic
point $x\in ST$ in the classical space-time manifold ST. That is, we can
assume that at each macroscopic point $x$ exists a microscopic subsystem in
QST, consisting of many DoF. That these subsystems behave thermally will be
proved in the next section.

In a second step we will interpret the EEQ (sign convention as in [19])

$G_{ab}:=R_{ab}-1/2\cdot R\cdot g_{ab}=\kappa\cdot
T_{ab}\quad,\quad\kappa=8\pi G/c^{4}$ (2)

in this thermodynamic context. That is, even if we assume that the global
state of the microscopic QST is pure, the local state around some macroscopic
point $x$ is a thermal ensemble (density matrix) due to its intricate
entanglement with its environment. From the findings in the papers, mentioned
above, we may then conclude that with overwhelming probability the local
states are thermal.

As the local states around the points of the space-time manifold ST are
thermal it seems reasonable to give also the EEQ a thermal interpretation. One
should note that already Lorentz or Levy-Civita in 1916 or 1917 tried to
interpret the EEQ as saying that the total energy of the universe is vanishing
(cf. [20])

$(T_{ab}-\kappa^{-1}G_{ab})\equiv
0\quad\text{hence}\quad\partial_{a}(T^{a}_{b}-\kappa^{-1}G^{a}_{b})\equiv
0\quad\partial_{a}:=\partial/\partial x^{a}$ (3)

Einstein provided an argument against such an interpretation (see [20]) which
seems to be sound at first glance, but one should take into account that at
that time quantum theory was not yet fully developed (cf. e.g. our arguments,
given in [13]).

In this paper we will only comment on the vast field of gravitational energy
and its localization in general relativity (GR) in so far as it concerns the
problems we want to analyze in the following. An interesting point of view is
developed for example in [21], where it is claimed that gravitational energy
should be located at places where $T_{ab}\neq 0$. However, one generally
assumes that non-vanishing curvature has something to do with gravitational
energy. The latter one can of course be non-zero where $T_{ab}$ vanishes.

If the microscopic substructure , i.e. QST, of the classical smooth space-time
ST is locally a thermal system, it should possess what one calls internal
energy in thermodynamics. As in [13] we assume the existence of an interaction
between matter and space-time on the quantum level, that is, between QST and
QM (quantum matter). The macroscopic ponderable matter and fields making up
the right side of the EEQ consist of many microscopic DoF on the quantum
scale. The same holds for the left side which describes a curvature property
of ST.

All this leads us to the following conjecture:

###### Conjecture 1.1

We assume that

$T_{ab}=\kappa^{-1}G_{ab}$ (4)

is the heat influx at point x into the local thermal state of QST coming from
the external QM system.

Note that this concerns only the gravitational effect of matter. Thus
$\kappa^{-1}G_{ab}$ represents a form of heat tensor which was discussed for
material systems e.g. in [22] or[23].

###### Remark 1.2

We will later discuss in this paper the work terms in this context,

To complete our analysis of the thermal substructure of Gravitation or QST, we
have firstly to deal with the non-tensorial character of gravitational energy.
Usually it holds

$\partial_{a}(\sqrt{-g}(T^{a}_{b}+t^{a}_{b}))\equiv 0$ (5)

with $t^{a}_{b}$ the pseudo tensor of gravitational energy-momentum (brief
discussions can be found in practically every book about GR, a particularly
nice discussion is e.g. given in [24]). In the canonical view this pseudo
tensor property is frequently considered to be a deficit of the theory. In
section (3) we reconsider the situation from a perhaps new vantage point.

It is a widespread belief that all concepts and notions which carry a physical
meaning should be of tensorial character. It is however reasonable to analyze
why tensorial behavior is considered to be of such a great relevance. If we
have a manifold with tangent and cotangent spaces at each point, tensor fields
can be built over tensor products of these spaces. Their importance is hence
that they carry a geometric meaning, that is, tensors may be said to represent
the same invariant object independent of the choice of coordinate systems. In
this context they have the important property that they cannot be transformed
away (transformed to zero) by an appropriate choice of coordinate system. In
particular, their transformation properties are relatively transparent.

On the other hand this philosphy cannot be exactly true as the wellknown
connection coefficients or Christoffel symbols $\Gamma^{I}_{jk}$ are of great
geometric significance and do not have a particularly simple transformation
behavior. In our view, Weinberg rightly remarks that there is nothing sacred
about the tensor transformation law ([25]). This holds the more so as physical
quantities are not automatically elements of the multilinear algebra over
tangent and/or cotangent space (which have essentially the meaning of velocity
or momentum vectors).

In older text books physical tensorial quantities are hence introduced as
tuples having simply a certain transformation behavior. We therefore should be
prepared that there may exist physical quantities which do not transform as
tensors. In the valuable and highly informative essay [26] there is an
interesting remark by Einstein, when asked by the Swiss student Humm, if
gravitational energy should be tensorial or not. He thinks not, referring for
example to the different behavior of kinetic energy $T$ and potential energy
$U$ under the Galilei group in

$\partial_{t}(T+U)=0$ (6)

Another argument, in our view, would be a pendulum clock with friction which
allows us to extract energy from the gravitational field in a non-inertial
reference system, which is not possible in a freely falling inertial system.
This latter example suggests the following conclusion:

###### Observation 1.3

The equivalence principle shows that gravitational energy cannot be tensorial.

We will discuss the deeper reasons for the possibility of non-tensorial
behavior of various physical quantities in section (3) in a more systematic
way, the main point being the effects of quantum theory on space-time and
gravitational behavior. These quantum effects will lead to a subtle
distinction between surface properties living over the smooth classical space-
time manifold ST and the behavior in the underlying microscopic substructure
denoted by QST. The former are expected to be tensorial, the latter typically
not. The spontaneous symmetry breaking of diffeomorphism invariance will play
an important role in this context.

In the last section we discuss in more detail the relevance of diffeomorphism
invariance and the corresponding conserved Noether current for our
investigation of the thermal character of the EEQ and GR in general. Wald
studied the Noether current of diffeomorphism invariance in the framework of
differential forms lying the main emphasis on integral representations
([27],[28],[29]). We will, on the other hand, rely more on the classical
tensor analysis and variational approach, using mainly the non-integrated
local point of view as it is e.g. also done in [8].

But before we will do this, we will add some remarks about the quite detailed
early work of Hilbert, Noether, Klein in this context (which, written in
German, is perhaps not so widely known and which is carefully discussed in
[26], for other aspects see also [30]). These papers dealt mainly with a
proper understanding of Hilbert’s famous socalled energy vector, which was
derived from a quite intricate variational analysis of some invariant action
functional and which seemed to puzzle not only the scientific community of
that time quite a lot ([31],[32],[33],[34]. We find it remarkable that these
long gone investigations anticipate in some sense more recent ones.

In Hilbert’s analysis an arbitrary vector field $\xi^{i}$ was introduced in
order to get a conserved current from diffeomorphism invariance. It was later
understood ([34]) to be a necessary consequence of E.Noether’s famous second
theorem. We will analyze the remarkable behavior of this conserved current,
give a physical interpretation of the vector field $\xi^{i}$, which is
technically the generator of the Lie derivative and interpret the role of
various terms occurring in the current as to their thermodynamic meaning and
implications. This winds up to our observation that what Hilbert used to call
the energy vector is rather an expression of the internal energy-momentum
tensor of the microscopic quantum gravitational substratum QST, underlying our
smooth space-time manifold ST.

##  2 Quantum Space Time, QST, as a Thermal System

As we have already remarked in the introduction, thermality usually enters the
field of (quantum) gravity or general relativity (GR) via the existence of
horizons, most notably in the paradigmatic case of black holes (BH). We want
to argue in this and the following sections that, in our view, classical space
time ST is the smooth, coarse grained hull, overlying QST, which is assumed to
be a system having quantum micro structure, that is, consists of a complex
network of a huge number of microscopic gravitational degrees of freedom
(DoF).

###### Assumption 2.1

We make the simplifying assumption that we may restrict ourselves to an
intermediate energy scale where all DoF can be treated as quantum.
Furthermore, we assume the gravitational DoF to live on a scale which is given
by the Planck units.

From this we may infer the following. Compared to the Planck scale, the
ordinary quantum scale on which our quantum matter is living, is many orders
of magnitude larger. This is the reason why on the level of ordinary quantum
theory space time can be treated as smooth and relatively invariable. On the
other hand, we have briefly described in e.g. [13] how the microscopic
gravitational DoF, living in QST cooperate so that as a consequence of a
primordial phase transition PT the classical space time manifold ST with its
points $x$ and the classical metric tensorfield $g_{ab}(x)$ do emerge, with
$g_{ab}(x)$ playing a double role as allowing, on the one hand, to measure
distances and time intervalls and, on the other hand, being the carrier of the
gravitational field.

As in [13] we introduce a Hilbert space $\mathcal{H}^{g}$, in which the global
quantum states, $\psi^{g}$, of QST are living. The microscopic elementary
gravitational DoF are represented by their corresponding local Hilbert spaces
$\mathcal{H}^{g}_{i},\,i\in\mathbb{N}$. The local bases in
$\mathcal{H}^{g}_{i}$ are denoted by $d_{i}^{\nu_{i}}$. Basis vectors in
$\mathcal{H}^{g}$ are then given by tensor product states

$\psi^{g}_{I}=\bigotimes_{i}\,d_{i}^{\nu_{i}}\,,\,I:=\\{\nu_{1},\nu_{2},\cdots\\}$
(7)

and a general pure quantum state in $\mathcal{H}^{g}$ is

$\psi^{g}=\sum_{I}\,c_{I}\psi_{I}^{g}$ (8)

###### Remark 2.2

We note that these assumptions are not really necessary prerequisites to reach
the following conclusions but will represent a convenient model to make the
derivations more concrete and precise.

We have now to discuss what consequences can be inferred from the existence of
the nonvanishing classical metric tensor field $g_{ab}(x)$ which lives over
the classical space-time manifold ST and by the same token over the underlying
microscopic substructure QST. Usually it is assumed that a quantum observable
$\hat{g}_{ab}(x)$ exists on QST with

$g_{ab}(x)=<\hat{g}_{ab}(x)>:=(\psi^{g}\mid\hat{g}_{ab}(x)\psi^{g})$ (9)

for some global state $\psi^{g}$ which corresponds as a quantum correlate to
the classical ST.

This, at first glance, natural assumption leads however to a number of highly
non-trivial consequences and has now to be discussed in more detail. We
assumed for example in [13] that the existence respectively non-vanishing of
the metric tensor field $g_{ab}(x)$ is the result of a primordial phase
transition PT. In other words, the existence of a stable space-time distance
concept is a non-trivial consequence of a particular physical process. Hence,
following the tradition of e.g. quantum statistical mechanics as a paradigm,
dealing with a great number of quantum DoF, we define:

###### Definition 2.3

We call $g_{ab}(x)$ an order parameter field and the space-time manifold ST an
orderparameter manifold, where its non-vanishing signals the existence of a
transition to a greater structural order in the underlying microscopic
substratum (see e.g. [13] or [35],[36] and earlier references given there).

In our context, where we have a double structure of an overlying classical
smooth space-time manifold ST and an underlying microscopic quantum structure
QST, both being correlated in a subtle way, we have to introduce some
particular structural elements, that is, for example, the concept of macro
observables. As far as we know, this notion was for the first time introduced
by v.Neumann in an ingenious but perhaps little kown paper (see [39], the
second part of [38] or [40]).

Due to the vastly different scales of, on the one hand, ordinary quantum
matter QM and, on the other hand, quantum space-time QST, we can assume that
to a macroscopic point $x$ do belong a huge number of DoF in QST or local
Hilbert spaces $\mathcal{H}_{i}$ which live in the infinitesimal neighborhood
of $x$. We denote this situation by

$\mathcal{H}_{x}:=\bigotimes_{x}\,\mathcal{H}_{i}\sim[x]$ (10)

where $[x]$ denote the gravitational DoF in the infinitesimal neighborhood of
$x$. Furthermore, to a macroscopic value, $g_{ab}(x)$, we can choose a
corresponding subspace $[g_{ab}(x)]$ in $\bigotimes_{x}\,\mathcal{H}_{i}$
containing the local vectors $\psi^{g}(x)$ with

$<\psi^{g}\mid\hat{g}_{ab}(x)\psi^{g}>=g_{ab}(x)$ (11)

###### Definition 2.4

Following the tradition in quantum statistical mechanics, we call
$[g_{ab}(x)]$ a phase cell and which is spanned by certain basis vectors
$\psi_{i}^{g}(x)$ so that

$\psi^{g}(x)=\sum\,c_{i}\psi^{g}_{i}(x)$ (12)

Furthermore, in the class of global states $\psi^{g}$, we may select the
subclass $[g_{ab}]$ so that

$<\psi^{g}\mid\hat{g}(x)\psi^{g}>=g_{ab}(x)\quad\text{for all}\;x\in ST$ (13)

###### Remark 2.5

One should note that the various sets, $[x]$, or $[g_{ab}(x)]$, are not
necessarily disjoint in QST. There may be a certain overlap.

We now come to a central point in our analysis. Given a global state
$\psi^{g}$ in $\mathcal{H}^{g}$, we can test it by sufficiently localized
observables, that is, observables localized in an infinitesimal neighborhood
of some macroscopic point $x$. That is, we may concentrate ourselves on state
vectors from $\mathcal{H}_{x}$ or $[g_{ab}(x)]$. More specifically, if we have
a pure global state $\psi^{g}$ and test it with observables taken from
$\mathcal{B}(\mathcal{H}_{x})$, the bounded operators operating on
$\mathcal{H}_{x}$, we can represent the global state $\psi^{g}$ by a density
matrix $\rho_{\psi}$, living over $\mathcal{H}_{x}$ or $[g_{ab}(x)]$.

###### Observation 2.6

It exists a density matrix $\rho_{\psi}$ over $\mathcal{H}_{x}$ or
$[g_{ab}(x)]$ with

$<\psi^{g}\mid A(x)\psi^{g}>=Tr(A(x)\cdot\rho_{\psi})$ (14)

for observables $A(x)\in\mathcal{B}(\mathcal{H}_{x})$

It is a relatively recent observation that much stronger results can in fact
be derived which lead to the thermalization results, which we mentioned in the
introduction. To our knowledge, early results in this direction were already
derived by v.Neumann in [39]. More recently, these phenomena were analyzed in
e.g. [16],[17], or [18]. In the context of decoherence by environment see also
[40]. Important tools in this connection are, on the one hand, the socalled
concentration of measure phenomenon and the Levy estimates (see for example
[16], a more systematic discussion can be found in [41]) and, on the other
hand, the use of random (unit) vectors (that is, quantum mechanical states
lying on high-dimensional unit spheres) and socalled typical states. We will
not go into the complex technical details but restrict ourselves to an
application of the results to our particular situation.

That is, we are given the classical order parameter field $g_{ab}(x)$,
defining the class of microscopic quantum states $\psi^{g}\in[g_{ab}]$ with

$<\psi^{g}\mid\hat{g}(x)\psi^{g}>=g_{ab}(x)\quad\text{for all}x\in ST$ (15)

We observed above that locally $\psi^{g}$ can be replaced by a trace operator
or density matrix $\rho_{\psi}(x)$.

The results derived in the above cited papers, in particular [16], now yields
the following in our scenario.

###### Conclusion 2.7

For a global pure state $\psi^{g}\in[g_{ab}]$ it hold with very high
probability that for local observables it is equivalent to the micro canonical
ensemble, i.e.

$\Omega:=d^{-1}\cdot Tr(\mathbf{1}\cdot)$ (16)

with d the dimension of $[g_{ab}]$ and $\mathbf{1}$ the identity operator. The
local state $\rho_{\psi}(x)$ at some arbitrary macroscopic point is equivalent
to this $\Omega$ and is called the generalized canonical ensemble. In other
words, this holds for a typical or randomly selected state. One should however
note that $\Omega$ is a global state, $\rho_{\psi}(x)$ a local one.

###### Remark 2.8

In case the generalized canonical ensemble can be associated with some form of
energy, the local states have the form of the true canonical ensemble known
from thermodynamics. This is the reason why the corresponding local state can
rightly be called a generalized canonical ensemble. That is, the global micro
canonical ensemble behaves locally as a canonical ensemble.

##  3 The Thermal Role of the Einstein Equation

In section 2 we have argued that in an infinitesimal neighborhood of a
macroscopic point $x\in ST$ quantum space-time (Q)ST has the characteristics
of a local thermal system. We now want to introduce and discuss the various
physical parameters which define and characterize such a thermal system. We
begin with a new conceptual understanding of the EEQ. Originally the EEQ are
viewed or conceived as a dynamic or evolution equation of the space-time
continuum under the influence of (ponderable) matter and classical fields.

As mentioned in the introduction, much later Jacobson, Padmanabhan (and
perhaps some other workers in the field) argued that the EEQ must hold due to
the observed or conjectured thermal behavior near horizons, most notably, near
the event horizon of a BH (see e.g. [6],[7],[8],[9],[10]). In the following we
want to argue that the EEQ carry a direct thermal meaning in the bulk of ST,
i.e., not necessarily near horizons and independent of the existence of
horizons or singularities. This holds also for the concept of entropy, which
is primarily understood in this context as a form of entanglement entropy,
typically arising near horizons. In our context it will have the ordinary
thermodynamical meaning of number of configurational alternatives.

As we said in the introduction, already in the early days (1916,1917) e.g.
Lorentz or Levi-Civita tried to give the EEQ a slightly different
interpretation, which was however rejected by Einstein by an argument which
was perhaps convincing at that time with quantum (field) theory still in its
infancies (cf. [20]). Lorentz and Levi-Civita speculated that by bringing the
energy-momentum tensor, occurring on the rhs of the EEQ to the lhs, we get an
expression which vanishes identically (see equation 3 in the introduction).
They then conjectured that this expression can then be identified with the
total (vanishing) energy of the universe, consisting of gravitational and
matter energy.

Einstein argued that the vanishing of the total energy of the universe would
not prevent the material systems and the gravitational energy to annihilate
each other. However, meanwhile we know that exactly the opposite process might
have happened in various models of the inflationary scenario (creation from
nothing, huge quantum fluctuations etc.). We discussed this zero energy
universe idea in more detail in section 2 of [13].

In [13] we introduced and described a similar but slightly different scenario.
We argued that our universe, i.e. (quantum) matter, (Q)M, and (quantum) space-
time, (Q)ST, emerged both as a result of a primordial phase transition PT from
a more primordial phase, QX:

$QX\underset{PT}{\rightarrow}(Q)ST+(Q)M$ (17)

While the internal energy of QST is lowered compared to the original phase QX
as the result of an order transition by an amount $\Delta E$, this same amount
is transferred to QM. That is,

$E(QX)=E(QST)+E(QM)$ (18)

In the language of Hilbert spaces we describe the structure of our universe as
a pure state in the tensor product

$\psi\in\mathcal{H}=\mathcal{H}^{g}\otimes\mathcal{H}^{M}$ (19)

with

$\psi=\psi^{g}\otimes\psi^{M}$ (20)

being a state where the gravitational and the matter state do exist
independently from each other. This may be an approximation of a situation
where e.g. BH’s are absent. The BH situation was discussed in [13].

A thermal system should have an internal energy. In the introduction we
formulated the following conjecture:

###### Conjecture 3.1

We assume that

$T_{ab}(x)=\kappa^{-1}G_{ab}(x)$ (21)

is the heat influx at point x into the local thermal state of QST coming from
the external QM system. Note that this concerns only the gravitational effect
of matter. Thus $\kappa^{-1}G_{ab}$ represents a form of heat tensor which was
discussed for material systems e.g. in [22] or[23].

That is, the EEQ is essentially a statement about the total heat influx at
point $x\in ST$ into the infinitesimal neighborhood of $x$ in QST. We now want
to provide arguments why we think this is indeed the case.

In [42] we studied thermodynamics in the regime of special relativity. While
in this field there do exist a variety of different approaches, in the
approach we favored, i.e. with temperature being the zero component of a
contravariant four-vector,

$T=\gamma\cdot T_{0}\quad,\quad\gamma=(1-u^{2}/c^{2})^{-1/2}$ (22)

with $\gamma$ the Lorentz factor, $T_{0}$ the rest temperature in a comoving
coordinate system, $u$ the 3-velocity, $T$ the temperature in the laboratory
frame, we get the following interesting transformation property for the heat
influx into the system:

$\delta Q=\gamma\cdot\delta Q_{0}$ (23)

that is, in contrast to internal energy and work, the heat influx transforms
as the zero component of a contravariant 4-vector. That is, it transforms
covariantly (cf. in particular section 4.4 of [42]).

Another paradigm is BH physics and the first law of BH-thermodynamics. In
geometric units (c=G=1) it reads:

$\delta M=(8\pi)^{-1}\cdot\kappa\delta A+\Omega\delta J$ (24)

with $\kappa$ the surface gravity, $\delta A$ the change of area, $\Omega$
angular velocity of horizon, $\delta J$ change of angular momentum, $\delta M$
the change of mass in the center. The identification reads:

$\delta S_{BH}=\delta A/4\,,\,T=\kappa/2\pi\,,\,\delta M\,\text{change of
internal energy}$ (25)

It is perhaps surprising that heat and entropy occur at such a prominent place
in this field while the, at first glance, more natural work contributions seem
to be missing. We now try to explain this on a perhaps more fundamental level.
We already briefly mentioned in the introduction that, in our view, space-time
consists of at least two levels, a smooth surface structure, ST, with its
tangential and cotangential structure and an underlying microscopic quantum
structure, QST. This compound system is assumed to have emerged via a
primordial phase transition, PT, together with a (quantum) matter component,
QM.

This phase transition was a transition to a greater order, that is, (Q)ST
carries an extra space-time structure compared to the more primitive phase,
QX, given, for example, by the existence (or non-vanishing) of the metric
tensor field, $g_{ab}(x)$, having the character of an order parameter field,
and in the underlying QST the existence of certain phase cells, $[g_{ab}(x)]$,
containing the quantum states of the universe, leading to the same macroscopic
metric tensor field in ST, i.e. $g_{ab}(x)$. Therefore we can conclude:

###### Conclusion 3.2

The primordial phase transition, PT, is accompanied by a spontaneous symmetry
breaking, SSB, of diffeomorphism invariance (or covariance). That is, the more
symmetric but less ordered phase, QX, goes over into the phase (Q)ST, which
has lesser symmetry but higher order.

###### Remark 3.3

These points were already discussed in [35] and [36].

.

What are the consequences of these observations for our question of non-
covariance of e.g. gravitational energy and related questions? As we said
above, we have a two storey structure of space-time. We argue that the smooth
surface structure ST (i.e. the macroscopic part) supports observables which
are diffeomorphism invariant. The SSB, on the other hand, is located in the
micro structure of the quantum regime QST. That is, we assume that the
existence of quantum excitations, vacuum fluctuations and the emergence of the
space-time structure $[g_{ab}(x)]$ is responsible for the breaking of
diffeomorphism invariance. We conjecture the following:

###### Conjecture 3.4

We assume that quantities which live over the smooth macroscopic manifold ST
are diffeomorphism covariant, i.e. behave tensorial. On the other hand,
quantities which rather express properties of the microscopic underlying
quantum structure QST we assume to behave non-tensorial in general.

###### Corollary 3.5

As we already explained in the introduction, since the scale on which QST is
living is many orders of magnitude finer than even the regime of ordinary
quantum matter, the latter should also behave covariantly in general.

###### Conclusion 3.6

As the energy-momentum tensor, $T_{ab}(x)$, lives essentially over the
classical manifold ST, the heat influx, $\kappa^{-1}\cdot G_{ab}(x)$, is by
the same token also tensorial or covariant. On the other hand, the part of the
relativistic energy-momentum (pseudo) tensor, $t_{ab}(x)$, describing the
gravitational energy contained in QST, is non-tensorial because it describes
the contributions to gravitational energy which are contained in the
underlying network of gravitational DoF. making up the microscopic structure
of QST.

##  4 The Concept of Gravitational Temperature

On general grounds we know that our local systems must have a temperature.
From what we have said above, the local gravitational system around the point
$x\in ST$ has an internal energy $U(x)$ and an entropy $S(x)$. In section 2 we
showed that the system at $x\in ST$ can be considered as a generalized
canonical ensemble. This entails that we can define its v.Neumann entropy

$S(x):=-Tr\,(\rho_{x}\cdot\ln\,\rho_{x})=-\sum_{i}\,p_{i}\ln\,p_{i}$ (26)

with $p_{i}$ the probabilities of the individual states $\psi_{i}(x)$.
Therefore we can (in principle) calculate

$T(x)=\partial U(x)/\partial S(x)\;\text{with}\;V(x)\;\text{held fixed}$ (27)

($V(x)$ some infinitesimal volume element around $x$).

We will complete this somewhat abstract definition of gravitational
temperature by two different more concrete approaches in this context.
Firstly, we discuss the Tolman-Ehrenfest approach (cf. e.g. [43],[44]) which
was originally introduced for thermal material systems which live in ST. For
convenience of the reader we will briefly recapitulate our own approach which
we gave in [45].

###### Remark 4.1

In order that we can apply the Tolman-Ehrenfest results to purely
gravitational systems, that is, systems not consisting of ordinary material
constituents, we must at first motivate that these systems have a thermal
character and consist of microscopic constituents. This is what we have done
in this paper up to now.

The Tolman-Ehrenfest result is the following:

###### Observation 4.2

In thermal equilibrium in a static gravitational field we have for an isolated
system

$T(x)\cdot\sqrt{-g_{00}(x)}=const$ (28)

I.e., in contrast to the non-relativistic regime (cf. sect.2), there exists in
general a temperature gradient in a system being in thermal equilibrium in the
relativistic regime.

To derive this result we use the entropy maximum principle. We shall use, for
reasons of simplicity, the weak field expansion of the gravitational field.
With $\phi(x)$ the Newtonian gravitational potential we have

$\sqrt{-g_{00}}=(1+2\phi/c^{2})^{1/2}\approx 1+\phi/c^{2}$ (29)

###### Remark 4.3

Note that the gravitational potential is negative and is usually assumed to
vanish at infinity.

We assume an isolated macroscopic system to be in thermal equilibrium in such
a static weak gravitational field. Its total entropy and internal energy
depend on the gravitational field $\phi(x)$. We now decompose the large system
into sufficiently small subsystems so that the respective thermodynamic
variables can be assumed to be essentially constant in the small subsystems.
As the entropy is an extensive quantity, we can write

$S(\phi)=\sum_{i}S_{i}(E^{0}_{i},V_{i},N_{i})$ (30)

where $E^{0}_{i}$ is the thermodynamical internal energy, not including the
respective potential energy.

###### Observation 4.4

It is important that in the subsystems the explicit dependence on the
gravitational potential has vanished. The entropy in the subsystems depends
only on the respective thermodynamical variables, the values of which are of
course functions of the position of the respective subsystem in the field
$\phi(x)$.

At its maximum the total entropy is constant under infinitesimal
redistribution of the internal energies $E^{0}_{i}$ with the total energy and
the remaining thermodynamic variables kept constant. We now envisage two
neighboring subsystems, denoted by (1) and (2). To be definite, we take
$\phi(2)\geq\phi(1)$. We now transfer an infinitesimal amount of internal
energy $dE_{2}^{0}$ from (2) to (1) (note, it consists of pure heat as for
example the particle numbers remain unchanged by assumption!). As heat has
weight relativistically it gains on its way an extra amount of potential
energy.

###### Observation 4.5

By transferring $dE_{2}^{0}$ from (2) to (1) we gain an additional amount of
gravitational energy

$dE_{2}^{0}\cdot\Delta\phi/c^{2}\quad,\quad\Delta\phi=\phi_{2}-\phi_{1}$ (31)

It is important to realize that this gravitational energy has to be
transformed from mechanical energy into heat energy or, rather, internal
energy and reinjected in this form into system (1) (for example by a stirring
mechanism acting on system (1) and being propelled by the quasistatic fall of
the energy $dE_{2}^{0}$).

The energy balance equation now reads

$dE_{1}^{0}=dE_{2}^{0}+dE_{2}^{0}\cdot\Delta\phi/c^{2}=dE_{2}^{0}(1+\Delta\phi/c^{2})$
(32)

We then have (with $dS_{1}=-dS_{2}$ in equilibrium)

$T_{1}^{-1}=dS_{1}/dE_{1}^{0}=-dS_{2}/-(dE_{2}^{0}(1+\Delta\phi/c^{2}))=T_{2}^{-1}\cdot(1+\Delta\phi/c^{2})^{-1}$
(33)

that is

###### Conclusion 4.6

It holds

$T_{1}=T_{2}(1+\Delta\phi/c^{2})\quad,\quad\Delta\phi:=\phi_{2}-\phi_{1}\geq
0$ (34)

###### Observation 4.7

The subsystem (1), having a potential energy being lower than (2), has a
higher temperature.

We can give the above relation another more covariant form. In the
approximation we are using it holds:

$\frac{\sqrt{1+2\phi_{2}/c^{2}}}{\sqrt{1+2\phi_{1}/c^{2}}}=\frac{1+\phi_{2}/c^{2}}{1+\phi_{1}/c^{2}}=\frac{1+(\phi_{1}+\Delta\phi)/c^{2}}{1+\phi_{1}/c^{2}}=1+\Delta\phi/c^{2}$
(35)

######  Conclusion 4.8 (Covariant form)

It holds

$T_{1}\cdot\sqrt{-g_{00}(1)}=T_{2}\cdot\sqrt{-g_{00}(2)}=const$ (36)

The non-infinitesimal result follows by using a sequence of infinitesimal
steps.

It is perhaps useful to derive the above result in yet another, slightly
different, way. In case $\phi(x)$ vanishes at infinity we can apply the
entropy-maximum principle as follows. We extract the energy (bringing it to
infinity)

$dE_{2}=dE_{2}^{0}+dE_{2}^{0}\cdot\phi_{2}/c^{2}$ (37)

from subsystem (2) and reinject the energy

$dE_{2}=dE_{1}=dE_{1}^{0}+dE_{1}^{0}\cdot\phi_{1}/c^{2}$ (38)

from infinity into (1). We get

$dE_{1}^{0}=dE_{2}^{0}\cdot\frac{1+\phi_{2}/c^{2}}{1+\phi_{1}/c^{2}}$ (39)

and

$T_{1}^{-1}=dS_{1}/dE_{1}^{0}=T_{2}^{-1}\cdot\left(\frac{1+\phi_{2}/c^{2}}{1+\phi_{1}/c^{2}}\right)^{-1}$
(40)

hence

$T_{1}\cdot(1+\phi_{1}/c^{2})=T_{2}\cdot(1+\phi_{2}/c^{2})$ (41)

As an example one may mention the Rindler/Unruh space-time. We have in Rindler
coordinates

$g_{00}=-\xi^{2}\;,\;a=\xi^{-1}\;,\;T=a/2\pi$ (42)

i.e.

$T\sqrt{-g_{00}}=2\pi^{-1}=const$ (43)

###### Remark 4.9

One should mention that this method to go to $\infty$ does make only sense if
the thermal system extends to $\infty$ as well.

In a second approach we will employ the Fulling-Unruh observation of
thermalisation in accelerated frames of reference (local Rindler frames were
introduced by for example Padmanabhan in e.g. [9]). We will however use it in
a slightly different way compared to Jacobson or Padmanabhan. For reasons of
convenience we wil choose a static space-time ST and restrict ourselves to a
sufficiently small neighborhood of some arbitrary point $x\in ST$.

We choose a local inertial frame (LIF) at $x$ with Lorentz-orthonormal
coordinates $(X,T)$ so that $x$ has the coordinates $(0,0)$. We assume that
the LIF moves along a certain geodesic through $x$. We place a thermometer
system at the point $x$ which is gauged so that it yields the result zero in a
LIF. However, we assume that the thermometer is in principle sensitive against
the thermal excitations of the underlying microscopical gravitational system.
The thermometer now experiences the gravitational field $g_{ab}(x)$ at the
macroscopic point $x\in ST$.

Now, relative to the LIF (moving on a geodesic through $x$) and employing the
inertial coordinates $(X,T)$ an observer at the point $x$ (and the
thermometer) experiences an acceleration $a(X,T)$ (the detailed numerical
calculations, which are a little bit intricate, can be found in [22] exercise
in section 9.6). One should however note that everything we have said holds
only locally. But locally, near the origin, we can switch from inertial
coordinates $(X,T)$ to socalled Rindler coordinates $(x_{R},t_{R})$.

$T=x_{R}\cdot\sinh\,(\kappa t_{R})\;,\;X=x_{R}\cdot\cosh(\kappa t_{R})$ (44)

with $\kappa$ the corresponding proper acceleration.

In full Rindler space, i.e. the right wedge $W_{R}$, we know from the work of
Fulling ([11]) and Unruh ([12]) that the thermometer will detect thermal
Rindler modes. In our case the situation is only locally, i.e. in a
neighborhood of $x$ or $(X,T)=(0,0)$, Rindler like. Therefore we cannot expect
to have such fully developed Rindler modes. However we know already that ST at
$x$ or $(X,T)=(0,0)$ behaves microscopically as a thermal system, that is, as
a socalled generalized canonical ensemble, hence carrying a distribution of
local thermal excitations. Therefore we make the conjecture:

###### Conjecture 4.10

The observer at $x$ or $(X,T)=(0,0)$ will detect a certain thermal excitation
spectrum, consisting of certain quasi particles (which may be approximations
of corresponding Rindler modes), leading to a corresponding local temperature.

##  5 Diffeomorphism Invariance and its Conserved Noether Current

In section 3 we discussed the consequences of the possible SSB of
diffeomorphism invariance due to the emergence of quantum DoF in the
underlying micro structure of ST, i.e. QST as a result of the primordial phase
transition PT. We argued that we may have two classes of observables, the one
class of covariant observables which live above the macroscopic space-time
manifold ST and another class, which we assume to consist of quantities which
rather describe properties of the underlying micro structure within QST. We
argued that the elements of this latter class need not behave covariantly
under geometric transformations of the surface structure ST. A prominent
example in this context is the concept of gravitational energy. Another
important role will be played by the work terms which occur in the complete
conserved Noether current we will derive below. Their covariance comes about
by exploiting the vector field $\xi^{i}(x)$ which enters via the Lie
derivative, which is a covariant operation.

But befor we enter into the technical details, we want to recapitulate what we
said in the introduction concerning the work of Hilbert, Noether, Klein etc.
([31],[32],[33],[34]). We were surprised to see that Hilbert essentially got
after a long and intricate calculation a conservation law for what he
considered to be the gravitational energy vector. This energy vector contained
an arbitrary vector field $\xi^{i}$, the deeper role of which was not really
understood and appreciated at that time, while technically it is a consequence
of E.Noether’s second theorem. In our view Hilbert already performed
calculations which were much later repeated in a similar context, being
apparently unaware of the earlier results. Below we will explain the physical
role of the arbitrary vector field $\xi^{i}$ while its mathematical role is
clear, it is simply the vector field which generates the diffeomorphism group
and occurs in the Lie derivative. In traditional physicist’s notation:

$x^{i}\to\overline{x}^{i}=x^{i}+\epsilon\xi^{i}(x)$ (45)

with $\epsilon$ infinitesimal. These intensive investigations and discussions
performed in the Hilbert group were carefully studied in two beautiful essays
by Rowe and Sauer ([26],[30]).

We give now a brief description of the sequence of necessary steps which lead
to the form of conserved Noether current deriving from the assumption of
diffeomorphism invariance. We will perform most of these steps in an appendix.
Our main motivation for this is a consequence of the observation that in many
representations of this stuff for example important and nonvanishing boundary
terms are frequently dropped which then results in only partial results.

###### Remark 5.1

If for example certain boundary terms are dropped one gets only the contracted
Bianchi identity instead of the full conserved current. This is in our view
dangerous because in contrast to ordinary variations diffeomorphisms typically
need not vanish at infinity. Furthermore, we will show that variations, having
a local support, lead to results which differ in some important respects.

In the following we restrict ourselves to a variation of the Hilbert-Einstein
action

$S[g_{ab}]=\int\,R(g_{ab})\cdot\sqrt{-g}\,d^{4}x$ (46)

with $R$ the scalar curvature

$R:=R_{a}^{a}\;\text{with}\;R_{ab}\;\text{the Ricci
tensor}\;R_{ac}:=R_{abc}^{b}$ (47)

the rhs being the Riemann curvature tensor. $g$ is the determinant of the
metric tensor. $R$ is a scalar and $\sqrt{-g}\,d^{4}x$, the canonical volume
element, is an invariant under coordinate transformations or diffeomorphisms.
Hence $S$ is invariant under diffeomorphisms

$\phi_{\lambda}:\,x\to\,x(\lambda)\;\text{or
infinitesimal:}\;x\to\bar{x}=x+d\lambda\,\xi(x)$ (48)

where the vector field $\xi(x)$ induces the flow of the diffeomorphism group
$\phi_{\lambda}$.

If $T(x)$ is some arbitrary tensor field, we construct an $\lambda$-dependent
tensor field in the following way. We shift back the tensor at $x(\lambda)$ to
the point $x$

$T_{\lambda}(x):=\phi^{*}_{-\lambda}(T(x_{\lambda}))$ (49)

with $\phi^{*}_{-\lambda}=(\phi^{*}_{\lambda})^{-1}$ the map, induced by
$\phi_{\lambda}:x\to x(\lambda)$. Now all these $T_{\lambda}(x)$ are defined
at the same point $x$ and we can take the derivative with respect to $\lambda$
at $\lambda=0$. We get

$0=d\,S[g_{ab}]/d\,\lambda=\int\,d/d\lambda\,\mathcal{L}(g_{ab}(x;\lambda)\,d^{4}\,x$
(50)

where in the following the derivative is taken always at $\lambda=0$ and
$\mathcal{L}$ is the scalar density $R(g_{ab})\cdot\sqrt{-g}$.

###### Remark 5.2

Technical details can be found, for example, in [46], Appendix C.

###### Observation 5.3

The derivative with respect to $\lambda$ at $\lambda=0$ is nothing but the Lie
derivative, $\mathcal{L}_{\xi}$, of the tensor $T$ or more general geometric
objects. As with ordinary derivatives, the Lie derivative obeys the Leibniz
rule.

###### Definition 5.4

In the following we abreviate

$\mathcal{L}_{\xi}(T)(x):=\delta T(x)$ (51)

that is, in particular

$d/d\lambda\,g_{ab}(x;\lambda)=\mathcal{L}_{\xi}\,g_{ab}(x)=\delta g_{ab}(x)$
(52)

It is an important observation that, as the Lie derivative is defined for
general differentiable manifolds, it is independent of the concept of
covariant derivative. It can hence be expressed with the help of, on the one
hand, partial derivatives, on the other hand, expressed by means of an
arbitrary covariant derivative operator. For example with the help of the
covariant derivative, induced by $g_{ab}(x)$:

$\mathcal{L}_{\xi}\,g_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=\xi^{c}\partial_{c}g_{ab}+g_{cb}\partial_{a}\xi^{c}+g_{ac}\xi^{c}$
(53)

While the Lie derivative for tensor fields is canonically given via
$\phi^{*}_{-\lambda}(T(x_{\lambda}))$ as

$\mathcal{L}_{\xi}(T)(x):=\lim_{\lambda\to
0}\;(\phi^{*}_{-\lambda}(T(x_{\lambda}))-T(x))/\lambda$ (54)

one has to say some words in the case of e.g. densities.

###### Remark 5.5

A scalar density like $\mathcal{L}(x)$ becomes an invariant if multiplied by
the volume element $d^{4}x$. If we have a scalar density at point
$x(\lambda)$, its translate back to $x$ has to be again a scalar density so
that e.g. $\mathcal{L}\,d^{4}(x_{\lambda})$ remains invariant.

What we have said in the remark, allows us to fix the necessary transformation
properties and define the Lie derivative of a scalar density.

###### Lemma 5.6

We have

$\mathcal{L}_{\xi}(\sqrt{-g}\cdot
R)=-\sqrt{-g}\nabla_{a}(R\cdot\xi^{a})=-\partial_{a}(R\cdot\xi^{a}\cdot\sqrt{-g})$
(55)

We are now going to describe the strategy which will lead to the derivation of
a conserved current, following from diffeomorphism invariance. We shall
however not follow the perhaps more obvious strategy to exploit the above
integral expression for $S[g_{ab}]$ but will use a local approach as it is for
example done in [47] in the case of the ordinary energy-momentum tensor
conservation law following from 4-translation invariance in quantum field
theory and also used in [8]. Our approach will consist of essentially two
steps.

i) We take the Lagrange density $\mathcal{(}[g_{ab}(x)]$ which is a lengthy
expression in $g_{ab}(x)$ and its first and second partial derivatives and,
using the Leibniz rule and that partial derivatives and Lie derivative do
commute, represent its Lie derivative as a long expression consisting
ultimately terms containing the Lie derivative of the basic building blocks
$g_{ab}(x)$ which we derived above. Note that we use the abbreviation

$d/d\lambda\,g_{ab}(x;\lambda)=\mathcal{L}_{\xi}(g_{ab}(x)=:\delta g_{ab}(x)$
(56)

Diffeomorphism invariance enters in the way that $\mathcal{L}[g_{ab}]$ does
not explicitly depend on the coordinates but only via the field $g_{ab}$ and
its derivatives. That is, this approach exploits the structural form of the
Lagrange density.

ii) In a second step we simply directly calculate the Lie derivative of the
density $R\sqrt{-g}$ as described in formula (55). We then equate the
numerically identical expressions and bring them on the same side, thus
getting a vanishing expression which can be written as a conserved current.

Now, using formula 74 in the appendix and the contracted Bianch identity, we
get for the variation of $\mathcal{L}[g_{ab}]$:

$\delta\mathcal{L}[g_{ab}]=\sqrt{-g}\cdot\nabla^{a}(v_{a}+2G_{ab}\xi^{b})$
(57)

with

$v^{a}=\nabla^{b}(\nabla^{a}\xi_{b}+\nabla_{b}\xi^{a})-g^{cd}\nabla^{a}(\nabla_{c}\xi_{d}+\nabla_{d}\xi_{c})$
(58)

(some technical remarks:

$\nabla^{a}:=g^{ab}\nabla_{b}\quad\text{and}\quad\nabla^{b}g^{ac}=0=\nabla_{b}g_{ac}$
(59)

for the Levi-Civita connection).

Now using in the second step the direct Lie derivative of the Lagrange density
$R\cdot\sqrt{-g}$, which we already wrote down above (see formula 55), we
arrive at the conserved Noether current.

###### Theorem 5.7

The Lie derivative of $R\cdot\sqrt{-g}$, derived in the two ways, described
above, yields the conserved Noether current

$0=\nabla_{a}(R\xi^{a}+2G^{ab}\xi_{b}+v^{a})$ (60)

with

$v^{a}=\nabla^{b}(\nabla^{a}\xi_{b}+\nabla_{b}\xi^{a})-g^{cd}\nabla^{a}(\nabla_{c}\xi_{d}+\nabla_{d}\xi_{c})$
(61)

We can rewrite this formula to get a slightly different result. With

$2G^{ab}\xi_{b}=2R^{ab}\xi_{b}-Rg^{ab}\xi_{b}\quad,\quad
g^{ab}\xi_{b}=\xi^{a}$ (62)

we get

###### Corollary 5.8

A variant of the above result is

$0=\nabla_{a}j^{a}\quad,\quad j^{a}=(2R^{ab}\xi_{b}+v^{a})$ (63)

##  6 The Thermal Meaning of the Conserved Noether Current

We want now to come back to the interpretation of the conserved Noether
current we have derived above. We mentioned in the introduction that Hilbert
and his colleagues had great difficulties to understand the role the arbitrary
vector field $\xi^{i}(x)$ is playing in this expression. As we have an energy-
momentum tensor $T_{ab}$, playing a fundamental role in the theory, the
occurrence of a conserved energy vector (as Hilbert liked to call it) like our
$j^{i}(x)$ was puzzling.

In this context we remind the reader what we said in section 3 concerning the
role of tensorial or covariant quantities compared to non-tensorial or non-
covariant quantities, the former referring to properties of the smooth
classical surface structure ST, the latter ones referring to the underlying
quantum mechanical micro structure QST. If we assume the vector field
$\xi^{i}(x)$ is chosen to be time like, we can associate it with the orbits of
observers or measuring devices. Then expressions like

$T_{ab}(x)\xi^{b}(x)\quad\text{or}\quad G_{ab}(x)\xi^{b}(x)$ (64)

represent (energy) flows as observed or measured by the respective moving
observers. That is, they have an objective (geometric) quality. In this sense
they should have a covariant tensorial character and therefore can create a
covariant conserved current.

###### Observation 6.1

The (time like) vector fields $\xi^{i}(x)$ can be assumed to be the orbits of
observers, floating through space-time.

In section 3 we argued that

$G_{ab}(x)=\kappa\cdot T_{ab}(x)$ (65)

is a statement about gravitational heat energy influx at macroscopic space-
time point $x$. Correspondingly the first part of the conserved Noether
current describes the flow of heat energy contributing to the internal energy
of the gravitational system. We now analyze the physical meaning of the second
part $v^{i}(x)$.

$v^{a}=\nabla^{b}(\nabla^{a}\xi_{b}+\nabla_{b}\xi^{a})-g^{cd}\nabla^{a}(\nabla_{c}\xi_{d}+\nabla_{d}\xi_{c})$
(66)

If $\xi^{i}(x)$ is a Killing vector field, that is, if it induces a symmetry
of the metric tensor,

$\mathcal{L}_{\xi}g_{ab}(x)=0\quad\text{or}\quad\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=0$
(67)

and using

$\nabla_{b}\xi^{a}=\nabla_{b}(g^{ac}\xi_{c})\;,\;\nabla^{a}\xi_{b}=g^{ac}\nabla_{c}\xi_{b}\;,\;\nabla_{b}g^{ac}=0$
(68)

we get:

###### Theorem 6.2

If $\xi^{i}(x)$ is a Killing vector field we have $v^{i}(x)=0$.

Since in the preceding sections we argued that QST is a thermal system,
possessing at each macroscopic space-time point $x$ the local state functions
internal energy and entropy as well as a notion of local heat influx given by
$T_{ab}(x)$ or $G_{ab}(x)$, it suggests itself to regard the vector field
$v^{i}(x)$, which contains mainly contributions built from the metric,
$g_{ab}(x)$, and the vector field $\xi^{i}(x)$ as work terms.

###### Conjecture 6.3

We assume that the vector field $v^{i}(x)$ contains the work terms of our
gravitational system QST. It contains essentially the effects of 4-volume
changes (compressions and decompressions).

###### Observation 6.4

This interpretation is supported by the above result that $v^{i}(x)=0$ for
Killing vector fields inducing geometric symmetries,
$\mathcal{L}_{\xi}g_{ab}=0$, i.e., essentially no 4-volume changes.

In the introduction we mentioned the work of Sakharov. Induced Gravity is
assumed to result from the deformation of the structure of vacuum fluctuations
by curvature. Our above working philosophy is a related one. Locally
compressing or decompressing QST affects the local level structure of the
thermal system at point $x$ and thus is a kind of work, done at the system.

We would like to make some remarks concerning a point which irritates or
irritated many researchers. It is frequently argued that mere coordinate
transformations can completely alter the numerical values of quantities like
energy or work, even make them vanish in case we are dealing with non-tensor
quantities. We dealt already with such problems in [40] in the context of
special relativity (SR), in particular if volume changes due to Lorentz
contraction have to be included in thermodynamic work terms.

Some researchers have the attitude to consider Lorentz contraction as not
being real, whatever that actually means. We think Pauli in [20] sect.5 made
this point particularly clear. He argues that the atomic physics, underlying
the contraction of a measuring rod is complicated but has to obey Lorentz
covariance. Therefore Lorentz contraction, in his view, is, on the one hand,
an objective process, but, as it is at the same time a result of Lorentz
symmetry, it can as well be explained with the help of the general Lorentz
invariance of SR.

The same is true, in our view, in the case of curvature effects on the micro
structure of QST. Furthermore, we can choose at each space-time point $x$ a
local geodesic coordinate system in which SR does hold, thus establishing the
close relatedness of SR and GR. What concerns pure coordinate transformations,
there exist basically two possibilities. On the one hand, they may be related
to concrete changes of reference systems to which our above remarks do apply.
On the other hand, they may not be implementable by concrete reference
systems. In that case we may assume that transformation behavior should be
regarded as a consequence following from consistency reasons.

##  7 Appendix: The Conserved Noether Current

We begin with the calculation of the variation of $R[g_{ab}]\cdot\sqrt{-g}$,
that is, reducing $\delta(R[g_{ab}]\cdot\sqrt{-g})$ to an expression which
contains only terms like $\delta g_{ab}$, remembering that $\delta$ denotes
the Lie derivative or $d/d\lambda$ at $\lambda=0$. Furthermore we use the
above expression for the Lie derivative

$\mathcal{L}_{\xi}\,g_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=\xi^{c}\partial_{c}g_{ab}+g_{cb}\partial_{a}\xi^{c}+g_{ac}\xi^{c}$
(69)

As $R$ is a relative complex expression in the variables $g_{ab}$ and its
first and second partial derivatives, the calculations are, as is often the
case in this context, lengthy and a little bit intricate.

We have (see [46] p.453)

$\delta(R\cdot\sqrt{-g})=\sqrt{-g}(\delta R_{ab})g^{ab}+\sqrt{-g}R_{ab}\delta
g^{ab}+R\delta(\sqrt{-g})$ (70)

with $R_{ab}$ the Ricci tensor. Furthermore ([46] p.185)

$g^{ab}\delta R_{ab}=\nabla^{a}v_{a}\quad v_{a}=\nabla^{b}(\delta
g_{ab})-g^{cd}\nabla_{a}(\delta g_{cd})$ (71)

and (see [46] p.453, [20] section 23)

$\delta(\sqrt{-g})=-1/2\cdot\sqrt{-g}g^{ab}\delta
g_{ab}=+1/2\cdot\sqrt{-g}g_{ab}\delta g^{ab}$ (72)
$(0=\delta(g^{ab}g_{ab})=g^{ab}\delta g_{ab}+\delta g^{ab}g_{ab})$ (73)

###### Conclusion 7.1

The above formulas yield

$\delta\mathcal{L}(g_{ab})=(R_{ab}-1/2Rg_{ab})\delta
g^{ab}\sqrt{-g}+\nabla^{a}v_{a}\sqrt{-g}$ (74)

###### Observation 7.2

$G_{ab}=(R_{ab}-1/2Rg_{ab})$ is called Einstein tensor with $G_{ab}=8\pi\cdot
T_{ab}$. It fulfills the contracted Bianchi identity $\nabla^{a}G_{ab}=0$
which can also be derived from diffeomorphism invariance if one neglects
boundary terms (see [48] p.138).

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