**Pavlo ****Danylchenko**

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**Introduction**

Inherent
in the classical physics
ordinary notions about the absolute simultaneity of events and about
uniqueness of concept of time, as well as of determination of spatial volume
which moving body fills in, essentially hinder us from the formation of the
most perfect relativistic generalization of thermodynamics. Purely logically-mathematical
approach to the solution of the problems, which don’t permit to get the full interconsistency of thermodynamics special (SR) and general (GR)
relativities, can’t guarantee the positive result of theoretical research in
principle. To attain the aim we need to make philosophical remaking sense of many of
our physical notions, which are
only conceived as finally established and unshakable. This work is the attempt of construction
of relativistic thermodynamics, based on the rejection of some dogmata, inherent not only in classical physics but also in well-known
relativistic generalizations of thermodynamics.

It
is considered that SR itself doesn’t lead
to the unique concept of the temperature, attributed to moving body [1,2]. Therefore a few relativistic generalizations of thermodynamics with
lorentz-invariant pressure are known. First of all there are Planck-Hasenöhrl relativistic thermodynamics [3–5] and
Ott relativistic thermodynamics [6], which are equally
based on the lorentz-invariance of entropy and pressure but use essentially
different transformations of temperature and heat [2,7]. According to Planck and Hasenöhrl, moving body is “colder” than motionless [1].
On the contrary, according to Ott transformations, moving body
is “hotter” than motionless. Thermodynamics with lorentz-invariant
relativistic temperature [1,8] is attractive for the fact that in this thermodynamics temperatures of
phase transitions are the intrinsic properties of substances as they are in
classical thermodynamics. However the equations of this thermodynamics don’t lead
to such conversion of the energy of radiation that corresponds to the relativistic
Doppler shift of the frequency of radiation.

In
most relativistic generalizations of thermodynamics the linear momentum of moving body is used as an additional extensive parameter. However, in
contrast to mechanics, in relativistic thermodynamics this
linear momentum is considered to be proportional to the enthalpy *H* of
matter [1,9,10] but not to the internal energy *U*, which is equivalent to the
eigenvalue of the mass of matter. Therefore this linear momentum forms the four-vector
with hamiltonian of enthalpy but not with the hamiltonian of energy of the
matter [1,10]. And as D’Alembert pseudoforce of inertia is the value derived from the linear momentum, in fact it is proposed to use the enthalpy of matter
instead of its mass as the measure of inertness.

In some of the relativistic generalizations of
thermodynamics, along with invariant pressure, noninvariant forms of
relativistic pressure are proposed. In such noninvariant forms of relativistic
pressure the presence of mechanical linear momentum [9]
and heat exchange [11,12] for moving body is taking into account.

**1. ****Nonextensivity of relativistic molar volume**

Two “equal in rights” relativistic values of
molar volume of the matter, which is moving at the velocity *v*=*dx*/*dt* in the external frame of references of spatial coordinates and time
(FR), are possible in principle. These values are: Lagrangian volume v* _{R}* = v/

In classical thermodynamics eigenvalue of molar
volume v(*S*, *p*) is strictly extensive parameter and its change causes
the change of enthalpy, and consequently the change of rates of physical processes,
not directly, but via change of entropy *S* and eigenvalue of pressure *p*.
Relativistic values of molar volume v* _{R}* and v

All
this is the sufficiently strong reason for the use of only strictly extensive value of molar volume (v_{R}^{*}·v* _{R}*)

This gives the possibility to make the most
simple definition of lagrangian (according to [9], internal energy of matter *U*_{R}*L*=–*U*_{R}*e*v* _{R}*=–

**2. ****Noninvariance of the pressure**

According to Noether theorem [17] the law
of conversation of energy
is the consequence of the presence of symmetry for time and the fulfillment of
this law is possible only for homogeneity of time. This homogeneity of time is in
the invariance of physical laws relatively to the change of the start of time reading
and this homogeneity is guaranteed by the use of uniform scale, by which rates
of physical processes in matter in its identical thermodynamic states are equal
in any moment of time, for its measuring. According to this, mutual
complementarity of energy and time, which is declared by Bohr principle of
complementarity and becomes apparent in the presence of Heisenberg uncertainty
relation of these physical characteristics, takes place.

In
the system of units of measurment of physical magnitude, which is
based on dimensionless Planck constant *h* and, thus, reflects the presence of mutual complementarity of
energy and time, the dimension of pressure [sec^{–1}m^{–3}] denotes the
following. In not comoving with matter FR of exterior observer the value of pressure, as well
as the value of energy (dimension [sec^{–1}]), must depend on the rate of time
in this FR. According to GR in rigid FR only proportional synchronization of clocks, time rates by which in
different points of space with different values of gravitational potential are
not equal, is possible. In accordance with this, value of pressure in any point
*j* of such physically inhomogeneous space is determined by rates of physical
processes not only in this point but also in the point *i*, from which observation
takes place [16]: * _{j}^{i}p*=

**3. ****Main differential equations of relativistic
thermodynamics in contravariant representation**

Equations of relativistic thermodynamics are usually determined in coordinates of
contravariant world space (Minkowski space) and, therefore, covariant
relativistic value of energy (hamiltonian) is being expressed not via covariant
relativistic value of enthalpy *H*_{R}* ^{*}*=

*U*_{R}^{*} = *U*_{R}* **P*^{*}*v* =(*H** _{R}*– v

Here, taking into account all stated above, total relativistic energy *U*_{R}^{*}=*U**G*
(haniltonian) and covariant value of mechanical linear momentum of matter *P*^{*}=–(∂*U** _{R}*/∂

*dU*_{R}* ^{*}* =(

and no
work on relativistic “selfcontraction” of matter is executed. Here: *w*=*vP*^{*}=*U**G**v*^{2}/*c*^{2} – external energy
(energy of tranfer [9]) of matter, *T** _{R}*=

In contrast to Planck linear momentum,
mechanical linear momentum (which doesn’t depend on pressure directly) *P*^{*}=* **U*_{R}^{*}*v*/*c*^{2} together with hamiltonian *U*_{R}* ^{*}* form four-momentum. And any
limitations of the dependence of coordinate (gravibaric [15]) value of
the velocity of light

**F*** _{in}*=– (∂

as well
as gravitational pseudoforce:

**F*** _{g}*=–(∂

where: *U*_{R}_{g}* ^{*}*=

The increment of contravariant relativistic
value of enthalpy:

*dH** _{R}*=(

as well as increment of ordinary enthalpy *H*, is determined by increments of only intensive parameters (except, of course,
the increment of entropy).

**4. ****Main differential equations of relativistic
thermodynamics in covariant representation**

In FR, in which the motion of matter is
observed, differental equations of relativistic thermodynamics may be also determined
in coordinates of covariant world space:

*U*_{R}*dt*^{*}+*Pdx* =*U*_{R}^{*}*dt*^{*}=(*H*_{R}^{*}–v*p*_{R}^{*})*dt*^{*}*=**Udt*.

At that, contravariant relativistic value of energy (lagrangian) *U*_{R}* ^{ }*=

*U*_{R}* *=*UP*/*P ^{*}*=

where: *G** ^{*}*=(1+

*P*^{*}=*Uv** ^{*}*/

and: *U*_{R}^{*}*–**U*_{R}* *=* **P*^{*}*P*/*U *= *–**P*(∂*U** _{R}*/∂

Main differential equations of relativistic
thermodynamics in their contravariant representation have
the following form:

*dH*_{R}* ^{*}*=(

*dU** _{R}* =(

Here: *T*_{R}* ^{*}*=

*U _{c}*=

*j*_{0 }and (*j*_{0}+p) – angles in
comoving with matter FR between directions of propagation of photons and direction of matter motion;* **j*
and *j**' *– corresponding to them angles in not comoving with matter FR, in which directions of propagation
of correlated photons are not parallel in general case.

This, as well as numerous works [2,6,7,10], which confirmed the validity of consideration of
Ott relativistic temperature *T*_{R}* ^{*}*=

**Conclusions**

Examined here relativistic generalization of thermodynamics with strictly extensive molar volume and lorentz-invariant entropy
and energy of density is devoid of many disadvantages of relativistic generalizations with lorentz-invarisnt pressure and allows us to newly interpret perception of course of physical processes in moving body from not
comoving with it FR. Possibility of two complementary representations of differential equations of
relativistic thermodynamics (contravariant and covariant) solves
the problem of the presence of two alternative relativistic temperatures –
Planck temperature and Ott temperature.

Full version of the article PDF ( 156 kb), DOC ( 40 kb).

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