Thermal time and the Tolman-Ehrenfest effect: temperature as the "speed of time"

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by  Carlo Rovelli

Thermal time and Tolman-Ehrenfest effect:
“temperature as the speed of time”
Carlo Rovelli, Matteo Smerlak1
1Centre de Physique Th´eorique de Luminy∗, Case 907, F-13288 Marseille, EU
(Dated: January 19, 2011)
The notion of thermal time has been introduced as a possible basis for a fully general-relativistic
thermodynamics. Here we study this notion in the restricted context of stationary spacetimes.
We show that the Tolman-Ehrenfest effect (in a stationary gravitational field, temperature is not
constant in space at thermal equilibrium) can be derived very simply by applying the equivalence
principletoakeypropertyofthermaltime: atequilibrium,temperatureistherateofthermaltime
with respect to proper time – the ‘speed of (thermal) time’. Unlike other published derivations
of the Tolman-Ehrenfest relation, this one is free from any further dynamical assumption, thereby
1 illustrating thephysical import of the notion of thermal time.
1
0
2 I. INTRODUCTION from a theoretical persepective, nonetheless, it is very
significant,forit is constitutes abridge betweenthermo-
n
a Thenotionofthermaltime wasputforwardbyConnes dynamics and general relativity.)
J andoneoftheauthors[1,2]asabasisforafullygeneral- Tolman and Ehrenfest’s original derivation of this re-

8 relativistic thermodynamics – a problem which is still lationwasbasedonanumberofdynamicalassumptions,
1 open[2–5]. Inanutshell, thethermaltime ofasystemis including the validity of the Einstein equations [12, 13].

the natural flow induced by its statistical state on its al- Other authors later simplified their derivation by strip-
]
c gebraofobservables. Unlikeothernotionsoftime,which ping these down to what appeared their bare essential:
q are tied to the existence of a spacetime metric, thermal therelativisticequivalencebetweenmassandenergy[14–
r- time remains meaningful in the quantum gravitational 21]. (We recallsuchasimple derivationinAppendix A.)
g regime [6]. But the strength of thermodynamics as we usually know
[ So far, however, thermal time has remained a rather it is not to rely on few dynamical assumptions: it is to
5 abstract notion, with few concrete applications [7–11], rely on no dynamical assumption.
v andthedoubtislegitimatewhetherithasmuchphysical Here, we show that the Tolman-Ehrenfest law follows
5 content. Thepurposeofthisnoteistoshowthatitdoes, fromapplying the equivalence principle to thermal time,
8 and actually provides the most economical means to de- and nothing else. This entails in particular that the ge-
9
scribe the influence of gravity on temperature. We do ometric properties of the thermal time flow of an equi-
2
sobyderivingtheTolman-Ehrenfestlawfromastraight- librium state remain valid in a curved spacetime. The
.
5 forward application of the equivalence principle to the firstofthesepropertiesstatesthatthe thermaltime flow
0
thermal time of non-relativistic equilbrium states. generates a Killing symmetry of the metric. The second
0
RecallthattheTolman-Ehrenfesteffectisthefactthat isthattheratiobetweentheflowofthermaltimeandthe
1
: temperature is not constant in space at equilibrium, in flow of proper time – the ‘speed of (thermal) time’ – is
v thepresenceofgravity[12,13]. Inastationaryspacetime the temperature. In a stationary spacetime, the relation
i
X with timelike Killing vector field ξ, the temperature T (1) follows immediately from these two condtions.
satisfies instead the Tolman-Ehrenfest relation Besides demonstrating the truly thermodynamicalna-
r
a tureoftheTolman-Ehrenfesteffect,thisresultillustrates
Tkξk=const, (1) in our opinion the import and effectiveness of the notion
of thermal time – its physical content.
where kξk = pg ξaξb is the spacetime norm of ξ. In
ab
the Newtonian limit, this corresponds to a temperature
gradient
II. THERMAL TIME
∇T ~g
= , (2)
T c2 We start by recalling the mathematical definition of
thermaltime,inthesimplifiedsettingofclassicalHamil-
where ~g is the Galilean acceleration of gravity. In few tonian mechanics. (The full quantum version is recalled
simple words,a verticalcolumn of fluid atequilibrium is for completeness in Appendix B.) A general relativistic
hotter at the bottom. (Of course, this is a tiny 1/c2 rel- statistical system can be described by a Poisson algebra
ativistic effect, negligible in most practical situations;1 AofobservablesAonaphasespaceS. Statisticalstates
ρ arenormalized2 positive functions onS, interpretedas
probability densities on S.
Given a statistical state ρ, we define the thermal time
∗Unit´emixtederecherche(UMR6207)duCNRSetdesUniversit´es
de Provence (Aix-Marseille I), de la M´editerran´ee (Aix-Marseille
II)etduSud(Toulon-Var);laboratoireaffili´e`alaFRUMAM(FR
2291).
1 OnthesurfaceoftheEarth, ∇TT =10−18cm−1. 2 InthesensethatRSdsρ(s)=1.
2
flow αρ : A → A as the Poisson flow of (−lnρ) in A. these trajectories. Among them arethe local observables
τ
That is A , which depend on the positions and momenta of the
x
particles in a neighborhood of the spacetime point x =
dαρ(A)
τ =−{A,lnρ}. (3) (~x,t) (for instance the density).
dτ Let ρ be an equilibrium state and αρ its thermal time
τ
where the r.h.s. is the Poissonbracket. flow. We now make the physical hypothesis that the
The import of this definition is that it relies only on equivalenceprinciplecanbeappliedtothermaltimeand
the statistical state and on the Poisson structure of the the two properties (i,ii). The first then becomes
system, and makes no reference to a kinematical time
(i’) The thermal time flow αρ has a geometric action
variable. For this reason,it continues to make sense in a τ
on local observables A according to
x
fully general-relativisticcontext,wherepreferrednotions
of time may be absent. dαρ(A)
Considernowanon-relativisticBoltzmann-Gibbsequi- τ =LξρAx, (7)
dτ
librium state ρ , describing thermal equilibrium at tem-
T
perature T in the canonical ensemble, where the Lie derivative Lξρ acts on Ax seen as
a function on spacetime, and the vector field ξρ
ρT =Z−1e−kHT, (4) generatesatimelikesymmetryofspacetime,thatis,
is a timelike Killing field for the stationary metric.
where H is the energy, k the Botzmann constant and
Z =RSds e−kHT. The thermal time flow of ρT is The second, which gives the temperature, now reads
(ii’) At every space point, that is along every station-
dαρT(A) 1 1 dA
τ =−{A,lnρ }= {A,H}= (5) ary timelike curve, temperature is the ratio of the
T
dτ kT kT dt thermal time flow to proper time s,
Hencethethermaltimeτ ofanequilibriumstateattem-
1 d
perature T and the Newtonian mechanical time t are re- ξρ = . (8)
kT ds
lated by
d 1 d Now,observethatateachpointatimelikeKillingfield
= . (6) is tangent to d/ds along a stationary timelike curve, but
dτ kT dt
in general the ratio of their norms is not constant in
In other words, the thermal time of a non-relativistic space. By taking the norm of the last equation, we have
equilibrium state satisfies the following two properties: indeed
(i) Theflow d hasageometricactiononspacetime: it 1
dτ kξρk= . (9)
actsonobservablesinthe samemannerasavector kT
field ξ = 1 d. Furthermore ξ is a Killing sym-
kT dt Replacing ξρ by any other tangent Killing vector (which
metry of the (Galilean) spacetime: it generates a
mustbegloballyproportionaltoξρ),wegettheTolman-
one-parameter group of timelike isometries.
Ehrenfestlaw(1). Thus,theinfluenceofgravityonther-
(ii) The ratio between the flow of thermal time τ and malequilibriumcanbeobtaineddirectlybyapplyingthe
the flow of mechanical time t is the temperature equivalenceprinciple to the twoproperties(i) and(ii) of
kT. thermal time.
Property (i) states that, for an equilibrium state, the
thermaltimeflowisproportionaltothekinematicaltime
IV. DISCUSSION
flow. Property(ii)providesanewandremarkabledefini-
tion of temperature: temperature is the “speed” of ther-
1. Thermal equilibrium states are the final states of
mal time, namely the ratio between the flow of thermal
irreversible processes. In standard statistical mechan-
time and the flow of mechanical time.
ics, such states can be characterized in a number of
A crucial fact pointed out in [7, 11] is that (ii) can be
ways: stochastically, by the condition of detailed bal-
interpreted locally: at any given spacetime point, tem-
ance of microscopic probability fluxes; dynamically, by
perature is given by the ratio between the two flows at
a condition of stability under small perturbations of the
that point. In the next section, we use this observation
Hamiltonian; thermodynamically, by a condition of pas-
to derive the Tolman-Ehrenfest law. sivity3;information-theoretically,bythemaximization of
entropy;quantummechanically,bytheperiodicityofcor-
relation functions in imaginary time, aka the KMS con-
III. STATIONARY SPACETIMES
dition; and so on. When moving to curved spacetimes,
Consider now more generally a macroscopic system,
say a gas, in a stationary spacetime. The phase space
S of such a system can be thought of covariantly as the 3 Passivity refers to the impossibilityto extract work from cyclic
set of solutions ~x (t) of the equations of motion of all
n processesinasystematthermalequilibrium–Kelvin’sformula-
the particules of the gas. Observables are functions of tionofthesecondlaw.
3
these characterizations tend to become problematic, be- the Killing field, and hence the thermal time flow, varies
cause of the effect of gravity. In the case of stationary from point to point. While the global scale of thermal
spacetimes, several generalizations of the condition of time remains arbitrary, the ratio between its flow in dif-
thermal equilibrium have been studied [14–19, 21]. Con- ferent spacetime points (with respect to proper time) is
ditions (i’,ii’) can be taken as a generalcharacterization physically meaningful.
of thermal equilibrium in a stationary spacetime. They Thus, the Tolman-Ehrenfest law can be stated very
reduce to the standard Boltzmann-Gibbs ansatz in the simply as: the stronger the gravitational potential, the
non-relativistic context,but remainvalid on generalsta- fasterthe thermaltime flowwithrespectto propertime,
tionary spacetimes. and hence the higher the temperature. It is amusing in
2. Noticethatcondition(i)consistsoftwoparts. First, thisrespectto noticethatthe expression“thermaltime”
itstatesthatthethermaltimeinducedbyathermalstate is also used in biology to indicate the linear relation-
ρ is geometric, that is, there exist a vector field ξρ for shipbetweendevelopmentrateandtemperaturewhichis
which(7)holds. Thisisahighlynon-trivialconditionon widely observed among plants and ectotherms [24]. For
thestate,andunderstandingtheunderlyingmathematics them too – the hotter, the faster.
isaninterestingopenproblem[22,23]. (Inthecontextof
algebraicquantumfieldtheory,ithasevenbeenproposed
that this condition of geometric modular action can be Appendix A: A simple derivation of the
Tolman-Ehrenfest effect
used as a “criterion for selection physically interesting
states on generalspacetimes”[23].) Second, condition (i)
statesthatξρ isKilling. Thismeansthatthestateshares Severalintuitive argumentscanbe usedtomakephys-
the same time-translation symmetry as the underlying ical sense of the Tolman-Ehrenfest effect. Here is one,
space-time, giving overall stationarity. Having the ther- which makes use of E = mc2 and the equivalence of in-
maltime flow proportionalto a timelike Killing fieldis a ertial and gravitational mass. Equilibrium between two
physical property which we propose captures the condi- systems happens when the total entropy is maximized
tion of equilibrium.
dS =dS1+dS2 =0. (A1)
3. In this note we have studied the notion of thermal
time in the context of stationary spacetimes. The mo- If a heat quantity dE1 leaves the first system, and the
tivation for the introduction of this notion, though, was samequantitydE2 =dE1 entersthesecondsystem,then
to provide a notion of time flow in the physicalsituation
where the gravitational field itself is thermalized (or is dS1/dE1−dS2/dE2 =0. (A2)
in a quantum state) and therefore the thermal (or quan-
tum)superpositionofgeometriesmakesconventionalno- Since T :=dS/dE, we obtain T2 =T1.
However, if the two systems are at different gravita-
tions of time meaningless. For statistical states at, or
tionalpotentials(intheNewtonianlimit),theamountof
approaching, equilibrium on a fixed geometry, the two
notionsmatchandthis matching is partofthe condition energydE1 leaving,say,theupperone,isnottheamount
of equilibrium itself, since the gravitational field must of energy dE2 entering the lower one. Indeed, E = mc2
and the equality of inertial and gravitational mass im-
be stationary with respect to the thermal time in order
ply that any form of energy has a gravitational mass,
for the entire system to be stationary. But the notion of
thermaltimeretainsitsvalidityalsointhecontextwhere and“falls”. Hence dE2 is dE1 increasedby the potential
energy m∆Φ, where Φ is the gravitationalpotential:
aclassicalbackgroundmetricisnotavailable. The“ther-
maltimehypothesis”of[1,2],indeed,postulatesthatthe 2
dE2 =dE1(1+∆Φ/c ), (A3)
thermal time governing the thermodynamics of a macro-
scopic system described by a given statistical state is this which yields immediately (2).
flow. This might allow to define thermodynamics also in
the context where space-time is not defined, because of
thermalorquantumfluctuationsofthegeometry. We do Appendix B: Thermal time hypothesis for general
not address this issue here. relativistic quantum systems
A generalcovariantquantum system can be described
by an algebra A of observables A (a von Neumann alge-
V. CONCLUSION
bra) and a states on A. For instance, in quantum grav-
ity the pure states can be given by the solutions ψ of
Summarizing, we have shown that the non-relativistic
the Wheeler-DeWitt equation, and observables by self-
properties of thermal time can be directly generalized
adjointoperatorson a Hilbert space defined by these so-
to stationary spacetimes, leading immediately to the
lutions (see for instance [6]). Statistical states are inco-
Tolman-Ehrenfesteffect. Thekeyfeatureofthermaltime
herent superpositions of such states, and can be defined
used in this derivation is the fact that the ratio between
as positive linear functionals on A.
the the flow of thermal time and the flow of proper time
Given a (faithful, normal) state ρ, the thermal time
is the temperature. This fact appears somewhat tauto- flow αρ is the Tomita flow of the state ρ in A [1]. It
τ
logical in flat space, as it amounts to fixing an arbitrary
satisfies in particular
unobservable scale for the thermal time. But it becomes
consequential in a curved spacetime, where the norm of ρ(αρA)=ρ(A). (B1)
τ
4
Thisflowdependsonthestate,buttheflowsgenerated outer thermal time flow β , as the flow αρ up to inner
τ τ
by different states are equivalent up to inner automor- automorphisms. Remarkably, this is state independent.
phisms in A [25, 26]. Therefore we can further define an
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