Editing Virial theorem (section)

== Relativistic uniform system ==
In case when in the physical system the pressure field, the electromagnetic and gravitational fields are taken into account, as well as the field of particles’ acceleration, the virial theorem is written in the relativistic form as follows:<ref>{{cite journal |last=Fedosin|first=S. G.|s2cid=53692146|title=The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept|journal=Continuum Mechanics and Thermodynamics|volume=29|issue=2|pages=361–371| date=2016| doi=10.1007/s00161-016-0536-8|arxiv=1801.06453|bibcode=2017CMT....29..361F}}</ref>

<math display="block"> \left\langle W_k \right\rangle \approx - 0.6 \sum_{k=1}^N\langle\mathbf{F}_k\cdot\mathbf{r}_k\rangle ,</math>

where the value {{math|''W<sub>k</sub>'' ≈ ''γ<sub>c</sub>T''}} exceeds the kinetic energy of the particles {{mvar|T}} by a factor equal to the Lorentz factor {{math|''γ<sub>c</sub>''}} of the particles at the center of the system. Under normal conditions we can assume that {{math|''γ<sub>c</sub>'' ≈ 1}}, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient {{sfrac|1|2}}, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the field of particles’ acceleration inside the system, while the derivative of the scalar {{mvar|G}} is not equal to zero and should be considered as the [[material derivative]].

An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature:<ref>{{Cite journal |last=Fedosin |first=Sergey G. |s2cid=125180719 |date=2018-09-24 |title=The integral theorem of generalized virial in the relativistic uniform model |url=http://em.rdcu.be/wf/click?upn=lMZy1lernSJ7apc5DgYM8f7AyOIJlVFO4uFv7zUQtzk-3D_DUeisO4Ue44lkDmCnrWVhK-2BAxKrUexyqlYtsmkyhvEp5zr527MDdThwbadScvhwZehXbanab8i5hqRa42b-2FKYwacOeM4LKDJeJuGA15M9FWvYOfBgfon7Bqg2f55NFYGJfVGaGhl0ghU-2BkIJ9Hz4M6SMBYS-2Fr-2FWWaj9eTxv23CKo9d8nFmYAbMtBBskFuW9fupsvIvN5eyv-2Fk-2BUc7hiS15rRISs1jpNnRQpDtk2OE9Hr6mYYe5Y-2B8lunO9GwVRw07Y1mdAqqtEZ-2BQjk5xUwPnA-3D-3D |journal=Continuum Mechanics and Thermodynamics |volume=31|issue=3|pages=627–638|language=en |doi=10.1007/s00161-018-0715-x |issn=1432-0959 |bibcode=2019CMT....31..627F |arxiv=1912.08683 }}</ref>

<math display="block"> v_\mathrm{rms} = c \sqrt{1- \frac {4 \pi \eta \rho_0 r^2}{c^2 \gamma^2_c \sin^2 \left( \frac {r}{c} \sqrt {4 \pi \eta \rho_0} \right) } } ,</math>

where <math>~ c </math> is the speed of light, <math>~ \eta </math> is the acceleration field constant, <math>~ \rho_0 </math> is the mass density of particles, <math>~ r </math> is the current radius.

Unlike the virial theorem for particles, for the electromagnetic field the virial theorem is written as follows:<ref>{{cite journal|last=Fedosin |first=S.G. |url= https://dergipark.org.tr/en/pub/gujs/issue/45480/435567 |title=The Integral Theorem of the Field Energy |journal= Gazi University Journal of Science |volume=32 |issue= 2 |pages= 686–703 |year=2019 |doi=10.5281/zenodo.3252783|s2cid= 197487015 |doi-access=free }}</ref>
<math display="block">~ E_{kf} + 2 W_f =0 , </math>
where the energy <math display="inline">~ E_{kf} = \int A_\alpha j^\alpha \sqrt {-g} \,dx^1 \,dx^2 \,dx^3 </math> considered as the kinetic field energy associated with four-current <math> j^\alpha </math>, and
<math display="block">~ W_f = \frac {1}{4 \mu_0 } \int F_{\alpha \beta} F^{\alpha \beta} \sqrt {-g} \,dx^1 \,dx^2 \,dx^3 </math>
sets the potential field energy found through the components of the electromagnetic tensor.