About the temperature of moving bodies

0.17 MB·English

by  T. S. Biró

ABOUT THE TEMPERATURE OF MOVING BODIES
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1 TAMA´SS.B´IRO´1 ANDPE´TERVA´N1,2
0
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Abstract. Relativisticthermodynamicsisconstructedfromthepointofview
n
ofspecialrelativistichydrodynamics. Arelativisticfour-currentforheatanda
a
generaltreatmentofthermalequilibriumbetweenmovingbodiesispresented.
J
The different temperature transformation formulas of Planck and Einstein,
8
Ott,LandsbergandDopplerappearuponparticularassumptionsaboutinter-
2
nalheatcurrent.

]
h
p
- 1. Introduction
s
s
Considering the temperature of moving bodies, the easier question is to answer,
a
l what is the apparent spectral temperature. In this case a spectral parameter is
c
transformed if the thermalized source is moving with respect to the observer (de-
.
s
tector system), andthe transformationrule canbe derivedfromthat of the energy
c
i and momentum in the co-moving system.
s
y This has been knownfrom the beginnings of the theory of special relativity and
h never has been seriously challenged. An essentially tougher problem is to under-
p stand the relativistic thermalization: what is the intensive parameter governing
[
the state with energy exchange equilibrium between two, relatively moving bodies

2 in the framework of special relativity. In particular how this general temperature
v shouldtransformandhow does it depend on the speed ofthe motion. Here several
0 answers has been historically offered, practically including all possibilities.
5
PlanckandEinsteinconcludedthatmovingbodiesarecoolerbyaLorentzfactor
6
[1, 2, 3], first Blanu˘sa then Ott has challenged this opinion [4, 5] by stating that
1
. on the contrary, such bodies are hotter by a Lorentz factor. During later disputes
5
several authors supported one or the other view (see e.g. [6, 7, 8, 9, 10, 11, 12,
0
9 13] and the references therein) and also some new opinions emerged. Landsberg
0 argued for unchanged values of the temperature [14, 15]. Other authors observed
: that for a thermometer in equilibrium with black body radiation the temperature
v
i transformation is related to the Doppler formula [16, 17, 18, 19, 20], therefore the
X
measured temperature seems to depend on the physical state of the thermometer.
r This problem is circumvented by the suggestion that thermal equilibrium would
a
have a meaning only in case of equal velocities [21, 22, 23].
Behindthesedifferentconclusionsthereare,inouropinion,differentviewsabout
the energytransferandmechanicalwork,andthe identificationofthe heat[12]. In
asimplifying mannerthe assumptionsandviewsaboutthe Lorentztransformation
properties of internal energy, work, heat, and entropy influence such properties
and the very definition of the absolute temperature. Coming to the era of fast
computers, a renewed interest emerged in such questions by modelling stochastic
phenomena at relativistic energy exchanges and relative speeds [24, 25, 26]. In
particular, dissipative hydrodynamics applied to high energy heavy ion collisions
requires the proper identification of temperature and entropy [27, 28, 29, 30, 31,
1
2 TAMA´SS.B´IRO´1 ANDPE´TERVA´N1,2
32, 33, 34, 35]. In this letter we show that our approach to replace the Israel-
Stewart theory of dissipative hydrodynamics, proposed earlier [36, 37, 38, 39], is
relatedtotheproblemofthermalizationofrelativelymovingbodieswithrelativistic
velocitiesandoursuggestioniscompatiblewiththefoundationsofthermodynamics
and guarantees causal heat propagation.
oø
By doing so we encounter the following questions in our analysis:
(1) What moves (or flows)? Total energy and momentum do flow correlated,
but further conserved charges (baryon number, electric charge, etc.) may
flow differently. In relativistic systems one has to deal with the possibility
that the velocity field is not fixed to either current, not being restricted to
the Landau-Lifshitz [40] or Eckart [41] frames.
(2) Whatisabody? Weexploit,howdointegralsoverextendedvolumesrelate
to the local theory of hydrodynamics, and what is a good local definition
for volume changeinrelativisticfluids. Incloserelationto this, we suggest
a four-vector generalization to the concept of heat.
(3) What is a proper equation of state? Here the functional dependency be-
tween entropy and the relativistic internal energy is fixed to a particular
form.
(4) What is the proper transformation of the temperature? As we have men-
tioned above prominent physicists expressed divergent opinions on this in
the past. This problemis intimately relatedto that ofthermalequilibrium
and to the proper description of internal energy.
2. Hydro- and thermodynamics
Inthisletterweconcentrateontheenergy-momentumdensityofaone-component
fluid, but the results can be generalized considering conserved currents in multi-
component systems easily. The energy-momentum tensor can be split into compo-
a
nents aligned to the fiducial four-velocity field, u (x), and orthogonalones:
ab a b a b a b ab
(1) T = eu u +u q +q u +P
with uaqa =0 and uaPab =Pabub =0. When considering complex systems, like a
quark-gluonplasma,thevelocityfieldcanbealignedonlywithoneoftheconserved
currents, unless several currents are parallel (i.e. different conserved charges are
fixed to the same carriers). In our present treatment the velocity field is general.
Relativisticthermodynamicsisobtainedbyintegratingthelocalenergy-momentum
conservation on a suitably defined extended and homogeneous thermodynamic
body. Therefore in the balance of energy-momentum we separate the terms per-
pendicular and parallel to the velocity field as
d
ab a a a a b
∂bT = (eu +q )+(eu +q )∂bu
dτ
d a a b dub a b ab
+ p( u +u ∂bu ) (u q +Π )
dτ − dτ
a a b ab a
(2) p+ b(u q +Π )=0 .
− ∇ ∇
Fromnowonthe propertime derivativeis denotedbya dotf˙=df/dτ =ua∂af
a a a c
foranarbitraryfunctionf(x). =∂ u uc∂ denotesaderivativeperpendicular
∇ −
ABOUT THE TEMPERATURE OF MOVING BODIES 3
to the velocity field and we also split the pressure tensor into a hydrostatic part
and a rest: Pab =p(uaub gab)+Πab.
Let us now assume, th−at ua is smooth and we may give a connected smooth
surfaceH thatisinitiallyperpendiculartothevelocityfieldandhasasmooth(two
-dimensional)boundary. Asafurthersimplificationwewillassumethatthevelocity
field is not accelerating u˙a = 0a, therefore ∂aua = aua and the hypersurface
∇
remains perpendicular to the four-velocity field. Hence the propagation of the
surface can be characterized by the proper time τ of any of its wordlines. We
refer to this hypersurface - a three dimensional spacelike set related to our fluid -
as a thermodynamic body. Considering homogeneous bodies we set ae = 0 and
ap=0. Itisimportantthatthevelocityfielditselfisnothomogeneou∇s, aub =0.
∇ ∇ 6
Now integration of (2) on H(τ) results in
a a a a b a b
(e˙u +q˙ +(eu +q )∂bu +pu ∂bu )dV =
Z
H(τ)
a b ab
(3) b(u q +Π )dV.
ZH(τ)∇
WiththeaboveconditionswaapplythetransporttheoremofReynoldstothel.h.s.
of eq.(3) and the Gauss-Ostrogradskytheorem to the r.h.s. of eq.(3) and obtain
(4) E˙u¯a+G˙a+pu¯aV˙ = uaqb+Πab dAb =δQa.
I
(cid:0) (cid:1)
∂H(τ)
Here u¯a = uadV/V is the average velocity field inside H, E = eV is the total
H
energy, Ga =R HqadV, and dAb is the two-formsurface measure circumventing the
homogeneousRbody in the region H(τ). The two-dimensional surface integral term
is the physical energy and momentum leak (dissipation rate) from the body under
study, we denote it by δQa. This is a four-vector generalization of the concept of
heat. Itdescribesbothenergyandmomentumtransferstoorfromthehomogeneous
body.
Thederivationofthetemperatureinthermodynamicsisrelatedtothemaximum
ofthetotalentropyofasystem(undervariousconstraints). Thiswayitsreciprocal,
1/T is anintegratingfactor to the heatin orderto obtaina totaldifferentialofthe
entropy[42,43]. Herewefollowthesamestrategyconsideringavectorialintegrating
factor Aa:
(5) δQa =E˙a+pu¯aV˙ =AaS˙ +Σa
with Ea = Eu¯a + Ga the energy-momentum vector of the body, u¯˙a = 0 and
Σa orthogonal to Aa. The decisive point is, that – according to the above – the
entropyofthehomogeneousbodyisafunctionoftheenergy-momentumvectorand
the volume: S = S(Ea,V). Multiplying eq.(5) by Aadτ/(AbAb) and utilizing that
du¯a =0 we obtain
Aau¯a Aa a Aau¯a
(6) dS = dE+ dG +p dV.
AbAb AbAb AbAb
Theconnectiontoclassicalthermodynamicsisbestestablishedbythe temperature
definition
1 Aau¯a
(7) := .
T AbAb
4 TAMA´SS.B´IRO´1 ANDPE´TERVA´N1,2
The intensive parameter associated to the change of the four-vector Ga is denoted
by
ga Aa
(8) := .
T AbAb
With these notations we arrive at the following form of the Gibbs relation:
a
(9) TdS =dE+gadG +pdV.
Due to the definitions eq. (7,8) gau¯a =1. Hence the Gibbs relationcan be written
in the alternative form
a
(10) TdS =gadE +pdV,
suggesting that the traditional change of the energy, dE, has to be generalized to
the change of the total energy-momentum four-vector, dEa =d(Eu¯a+Ga).
For the well-known Ju¨ttner distribution [44] ga = u¯a. This equality has been
postulated among others in the classical theory of Israel and Stewart [45]. Then
(10) reduces to
(11) TdS =dE˜+pdV,
where E˜ = u¯aEa. In this case the internal energy can be interpreted as E˜, but
its total differential contributes to the Gibbs relation. In the general case ga =u¯a
6
- considered below - there remains a term, related to momentum transfer. It is
reasonable to assume that the new intensive variable is timelike: gaga 0. Then
≥
we introduce
a a a
(12) w :=g u¯ .
−
Now wa 2 = wawa = gaga+1 1follows. Thespacelikefour-vectorwa hasthe
physikcal dkimen−sion of vel−ocity. Du≤e to 1 wawa 0 and u¯awa = 0 its general
form is given by wa = (γv w,γw). In−this≤case w≤2 1. We interpret w as the
| | | | ≤
velocity of the internal energy current.
Heresomeimportantphysicalquestionsarise: isitonlyasingleorseveraldiffer-
entialtermsdescribingthechangeofenergyandmomentum? Whentwo,relatively
moving bodies come into thermal contact what can be exchanged among them in
the evolution towards the equilibrium?
3. Two bodies in equilibrium
Let us now consider two different bodies with different average velocities and
energy currents. When all components of Ea and the total volume are kept con-
stant independently, i.e. dEa +dEa = 0 and dV +dV = 0 while dS(Ea,V )+
1 2 1 2 1 1
a
dS(E ,V )=0 in the entropy maximum, then from (10) we obtain the conditions
2 2
ga ga p p
(13) 1 = 2, 1 = 2.
T T T T
1 2 1 2
This, in general, does not mean the equality of temperatures.
In order to simplify the discussion we restrict ourselves to one-dimensional mo-
tions and consider u¯a =(γ,γv) with γ = 1/√1 v2 Lorentz-factors. The energy
a −a
current velocity is given by w =(γvw,γw) and q =(γ(1+vw),γ(v+w)). Here
ABOUT THE TEMPERATURE OF MOVING BODIES 5
wdescribesthespeedofinternalenergycurrent. Thethermalequilibriumcondition
(13) hence requires
γ (1+v w ) γ (1+v w )
1 1 1 2 2 2
= ,
T T
1 2
γ (v +w ) γ (v +w )
1 1 1 2 2 2
(14) = .
T T
1 2
Theratioofthesetwoequationsrevealsthatinequilibriumthecompositerelativis-
tic velocities are equal,
v +w v +w
1 1 2 2
(15) = ,
1+v w 1+v w
1 1 2 2
and the difference of their squares leads to
1 w2 1 w2
(16) − 1 = − 2.
p T p T
1 2
The equality of some other velocities were investigated by several authors [21, 22,
23, 46].
Onerealizesthatinthethermalequilibriumconditionfourvelocitiesareinvolved
for a generalobserver: v , v , w and w . By a Lorentztransformationonly one of
1 2 1 2
them can be eliminated. The remaining three (relative) velocities reflect physical
conditions in the system. According to eq.(15)
v+w
2
(17) w =
1
1+vw
2
withv =(v v )/(1 v v )relativevelocity. The associatedfactor, 1 w2 can
2− 1 − 1 2 − 1
be expressed and the temperatures satisfy p
√1 v2
(18) T =T − .
1 2
1+vw
2
This includes the general Doppler formula [16, 17, 18, 19, 20, 47].
Itisenlighteningtoinvestigatethisformulawithdifferentassumptionsaboutthe
energy current speed in the observed body, w . The induced energy current speed
2
inanidealthermometer,w andthe temperatureitshows,T ,arenowdetermined
1 1
by eqs.(17) and (18). Figure 1 plots temperature ratios T /T for a body closing
1 2
with v = 0.6 as a function of the energy current speed, w .
2
−
(1) w = 0: the current stands in the observed body. In this case w =
2 1
v, the measured energy current speed is that of the moving body, and
T =T √1 v2 <T , the moving body appears cooler by a Lorentz factor
1 2 2
−
[1, 3, 2] (see Fig 2).
(2) w = 0: the current stands in the thermometer. In this case we must
1
havew = v andT =T /√1 v2 >T ,themovingbodyappearshotter
2 1 2 2
− −
[4, 5, 7, 12] (see Fig 3).
(3) w +w = 0: the current is standing in the total system of moving body
1 2
and thermometer, the individual contributions exactly compensate each
other. This is achieved by a special value of the energy current velocities,
w = w,w =wwithw=(1 √1 v2)/v. Inthiscaseeventheapparent
2 1
− − −
temperatures are equal, T =T [14, 15] (see Fig 4).
1 2
6 TAMA´SS.B´IRO´1 ANDPE´TERVA´N1,2
(4) w = 1: a radiating body (e.g. a photon gas) is moving. In this case
2
w = 1, and one obtains T = T 1−v. It means that T < T for v > 0,
1 1 2q1+v 1 2
a Doppler red shifted temperature is measured for an aparting body (see
Fig 5) - quite common for astronomical objects - and T >T for v <0, a
1 2
Dopplerblueshiftedtemperatureappearsforclosingbodies-morecommon
in high energy accelerator experiments.
On figures (2)-(5) we fix the reference frame to the thermometer, therefore
ua = (1,0) (the vertical axis is time). The energy current velocity four-vectors
1
are perpendicular to the corresponding four-velocities, therefore they are on lines
symmetrical to the light cones. The four-velocity vectors end on the timelike hy-
perbolasand the spacelikeenergy currentvelocities end inside the spacelikehyper-
bolas. The temperature ratios are determined by the magnitudes of the ga-s as
T /T = ga / ga (see [48].
1 2 k 1k k 2k
4. Lorentz scalar temperature
According to the classical ansatz ga = u¯a, the total entropy has to depend on
the total energy E = u¯aEa. Then the equilibrium conditions (17) and (18) result
in zero relative velocity v =0 and the temperatures are equal T =T .
1 2
However, thermodynamic and generic stability considerations are favoring an
other Lorentz-scalar combination E = √EaEa [36, 37, 38, 39]. Denoting the
k k
partialderivative of entropy S( E ,V) with respect to its first argumentby 1/θ =
∂S k k
, one re-writes the total differential,
∂kEk
1 p˜ ga a p
(19) dS = d E + dV = dE + dV,
θ k k θ T T
and comparing to the general Gibbs relation (10) one obtains the correspondence
ga 1 Ea p p˜
(20) = , = .
T θ E T θ
k k
a
It follows that the length of the intensive four-vector, g , is the ratio of the tradi-
tional(energyassociated)andscalar(energy-momentumfour-vectorlength associ-
ated)temperatures: √gaga =T/θ. Ontheotherhanditsprojectiontotheaverage
velocity reveals the value in the comoving system:
a T Eau¯a
(21) gau¯ = =1.
θ E
k k
This equationrelates the energy-momentum-associatedscalartemperature, θ to
the energy-associatedone, T. As a consequence we obtain
a Ea a Ea u¯a(Ebu¯b)
(22) g = , w = − .
Ebu¯b Ecu¯c
a
The later formula clearly interpret w as the quotient of the comoving, average
velocity related energy current (momentum) and energy of the thermodynamic
body, that is the energy current velocity.
Finally we remark,thatin the simple two dimensionalparticularcase we obtain
that θ = T/√1 w2. Therefore the equilibrium condition (16) gives equal scalar
−
temperatures: θ = θ . This is a stronger reflection of Landsberg’s view, and his
1 2
physical arguments in [14, 15] than the assumption of zero total energy current
velocity.
ABOUT THE TEMPERATURE OF MOVING BODIES 7
2.0
Dopplerblueshift
1.5
Blanusa-Ott
(cid:144)TT121.0 Planck-Einstein Landsberg
0.5
Dopplerredshift
0.0
-1.0 -0.5 0.0 0.5 1.0
w2
Figure 1. Ratio of the temperatures of the observedbody in its
rest frame, T to that shown by an ideal thermometer, T as a
2 1
function ofthe the speedofthe heatcurrentinthe body, w while
2
approaching with the relative velocity v = 0.6.
−
v=-0.6
u2
g2 g1 u1
w1 w2
Figure 2. The space-time figure for the Planck-Einstein rule of
two thermodynamic bodies in equilibrium. There is no energy
current in the observed body (wa = 0), therefore the ua four-
2 2
a a
velocity (solid arrow) is parallel to the vectors (g ,g ). The ratio
1 2
of the temperatures is T /T <1.
1 2
5. Summary
We investigated the possible derivation of basic thermodynamical laws for ho-
mogeneous bodies from relativistic hydrodynamics. The dependence of entropy
on internal energy is replaced by a dependence on the energy-momentum four-
a
vector, E . As a novelty a relativistic heat four-vector has been formulated. For
the traditional, energy exchange related temperature, T, a universal transforma-
tion formula is obtained. For a general observer four velocities are involved in the
equilibrium condition of two thermodynamic bodies in equilibrium. One of them
can be eliminated by choosing the observing frame, the physical relation depends
onlyonthe relativevelocity. Anotherconditionconnectsthe internalheatcurrents
in the bodies in thermal contact. So there remains two velocity like parameters
to describe thermal equilibrium: the energy current speed (the velocity related to
the integratedinternalheat currentdensity) in one of the bodies and their relative
8 TAMA´SS.B´IRO´1 ANDPE´TERVA´N1,2
v=-0.6
u2 u1 g1
g2
w1
w2
Figure 3. The space-time figure for the Blanuˇsa-Ott rule. The
energy current stands in the thermometer (wa =0), therefore the
1
uafour-velocity(solidarrow)isparalleltothevectors(ga,ga). The
1 1 2
ratio of the temperatures is T /T >1.
1 2
v=-0.6
u2g2g1 u1
w1
w2
Figure 4. The space-time figure for the Landsberg rule. There
is no energy current in the composed system, therefore the four-
vectors (ga, ga, dotted arrows) are equal. The ratio of tempera-
1 2
tures is T /T =1. Here w = 0.33, w =0.33.
1 2 1 2
−
velocity. The traditional temperature transformation formulas belong to corre-
sponding particular choices on the energy current speeds. This can be the reason
that no agreement could be achieved historically. For most common cases there is
no heat current in the observed body but it flows in the thermometer. This leads
to the Planck-Einstein transformation formula.
The closer relationto dissipative hydrodynamicsfavorsa particular dependence
of entropy on energy-momentum and leads to a Lorentz scalar temperature.
Ourapproachiscovariant,andthecompatibilitytohydrodynamicsclarifiesthat
the Planck-Ottimbrogliois nota problemofsynchronizationasitwassupposedin
[49,25]. ItmakespossibletointerprettheclassicalparadoxicalresultsofPlanckand
Einstein, Ott, Landsberg and Doppler in a unified treatment. Our investigations
reveal that despite of the apparent paradoxes related to Lorentz transformations,
ABOUT THE TEMPERATURE OF MOVING BODIES 9
v=-0.6
u2 u1
g1
g2
w1
w2
Figure 5. The space-time figure for the Doppler red shift rule
of two thermodynamic bodies in equilibrium. The energy cur-
rent speed (dashed arrows) in the observed body is that of the
light w = 1, therefore w = 1, and the four-vectors of energy-
2 1
momentum intensives (ga, ga, dotted arrows) are light-like.
1 2
there is a covariant relativistic thermodynamics with proper absolute temperature
in full agreement with relativistic hydrodynamics.
6. Acknowledgement
The authors thank to L. Csernai for his enlighting remarks.
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