Thermodynamic potentials and parameters of matter as well as the temperatures of its phase transitions are purely internal properties of matter [Bazarov, 1964] and, therefore, fundamentally should not be changed during relativistic transformations of increments of spatial coordinates and time. One more thing that denotes it is the presence of two absolutely opposite relativistic generalizations of thermodynamics, according to one of which [Hasenöhrl, 1907; Planck, 1907] moving body is colder than resting body, while according to another one [Ott, 1963], moving body is hotter than motionless body. Moreover, in spite of the declared in SR relativistic shrinkage of the size of body along the direction of its motion the molar volume of moving matter also should not be changed during the relativistic transformations of increments of spatial coordinates and time [Danylchenko, 2008: 60; 2020: 5]. In order to fulfill the general covariance of equations of not only thermodynamics but also mechanics in SR and in GR there should be a principle of unobservability of deformation and metrical inhomogeneity of matter on the level of its microobjects. Indeed, instead of metrically inhomogeneous background Euclidean space the intrinsic spaces of matter that have gravitational curvature are used in GR. And, therefore, of course, the local kinematic “curvature” of intrinsic space of the observer of moving body should be introduced in SR instead of relativistic length shrinkage.

If body moves at velocity  and taking this into account its limit velocity in background regular space of the FR_out of external observer is , then in commoving with it FR₀ and in FR_out the increments of coordinates (and, thus, of metrical segments) of its moving objects and time will be as follows:

where:  (); ; ,  (, );  and  are values of the limit velocity of motion in the background regular space and imaginary Lorentz dilatation of intrinsic time of mobile object m (matter) in FR_out; , , ; , , ,  and  are values of the limit velocity of motion and imaginary Lorentz dilatation of intrinsic time of stationary and mobile objects in FR₀ correspondingly; , ,  are the increments of metrical segment projections of mobile object in FR₀, and: , ,  are the increments of coordinates of mobile object in FR_out of observer of the motion of the whole body and its objects;  and ,  are the real and limit velocities of motion of object m in FR_out correspondingly;  and  are the all-side shrinkages of the size of moving body in background regular space in FR_out and in FR₀ correspondingly;  and  are the functions of index p and the acceleration G of forced motion;  is a parameter that characterizes the curvature of intrinsic space of observer in the rest initial state of moving body;  and  are the kinematic scale factors in the FR_out and in the FR₀ correspondingly;  is a degree of shrinkage of the size of moving body in background regular space_. According to this, the transformations of the velocities projections of motion will have the following form:

where the real velocities of motion of observed object and of FR₀ are equal to , , , , ,  and  correspondingly.

When , ,  (, , , ) and  (that corresponds to the identical collective spatial-temporal Gibbs microstate of the whole matter that moves at  velocity) there is a relativistic invariance of longitudinal metrical sizes of moving body (), independently from the values of index p. Due to the isotropy of kinematic self-contraction of the sizes of moving body in background regular space the transversal metrical segments will also be relativistically invariant. And this, of course, corresponds to the accepted in GR principle of unobservability of deformation of matter on the level of its microobjects (de facto to the principle of metrical homogeneity of the space of observer of matter motion). Inequality of increments of transversal metrical segments in different FRs, when relativistically invariant increments of metrical time are the same, is caused by the difference of transversal components of velocities of motion in those FRs.

According to the increment of times  and  ( and , and: , , , ), as well as of conformal interval when body moves inertially ():

takes place not only the invariance of the rate of the count of inertially moving clock, but also the invariance in relation to relativistic transformations of counted by them unified gravithermodynamic time of the Universe () which, of course, can correspond only to the unprompted motion (inertial motion or chaotic motion) of any objects. From the condition of conservation in FR₀ of the Hamiltonian of bodies that are free falling it follows that . In case of forced motion (when  and , ) the dilatation of intrinsic time of moving objects can indeed take place, which is confirmed by the increasing of lifespan of unstable microobjects that are formed in experiments on accelerators.

According to this the Hamiltonian and momentum of the inertially moving body are correspondingly equal to:  

Conclusion

Thermodynamic internal energy of matter is equal in all FRs of bodies that move inertially relatively to it. And exactly this is the guarantee of Lorentz-invariance of all thermodynamic potentials and parameters of matter. Since matter motion is accompanied by the all-sided conformally-gauge self-contraction of its size in background Euclidean space of the Universe the rate of the intrinsic time of inertially moving body is not dilated but, quite the contrary, remains invariant, despite the presence of gravitational decreasing of the rate of intrinsic time for nearby static objects. De facto the motion of the matter as well as its gravitational self-contraction in background Euclidean space of the Universe leads to its advance over unobservable in people’s world evolutionary self-contraction of the conventionally motionless matter in the Universe.

Reference

Bazarov, I.P.: 1964, Thermodynamics. New York: Pergamon Press, [distributed in the Western Hemisphere by Macmillan, New York] (1964).

Danylchenko, Pavlo: 2008, Introduction to Relativistic Gravithermodynamics. Vinnitsa: Nova knyga, 60-94. http://pavlo-danylchenko.narod.ru/docs/RelativisticGeneralization_Rus.html.

Danylchenko, Pavlo: 2020, Foundations of Relativistic Gravithermodynamics. Foundations and consequences of Relativistic Gravithermodynamics (in Ukrainian), Vinnytsia: Nova knyga, 5-84. http://pavlo-danylchenko.narod.ru/docs/FoundationRGTDUkr.pdf (in Ukrainian).

Hasenöhrl, Friedrich: 1904, Zur Theorie der Strahlung in bewegten Körpern. Annalen der Physik. 320 (12): 344–370; Wien. Ber., 116, 1391.

Ott, Heinrich Z.: 1963, Lorentz-Transformation der Wärme und der Temperatur. Zeitschrift für Physik, Springer Nature, 175, 70-104.

Planck, Max: 1907, On the Dynamics of Moving Systems. Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften, Berlin. Erster. Halbband (29): 542-570.

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