Version 1
: Received: 18 May 2022 / Approved: 19 May 2022 / Online: 19 May 2022 (08:57:52 CEST)
Version 2
: Received: 6 August 2022 / Approved: 8 August 2022 / Online: 8 August 2022 (05:27:29 CEST)
Version 3
: Received: 23 January 2023 / Approved: 25 January 2023 / Online: 25 January 2023 (11:44:00 CET)
V. B. Belayev; The gravitational mass of the rarefied cloud of the relativistic massive particles. AIP Conf. Proc. 28 September 2023; 2872 (1): 100001. https://doi.org/10.1063/5.0162971
V. B. Belayev; The gravitational mass of the rarefied cloud of the relativistic massive particles. AIP Conf. Proc. 28 September 2023; 2872 (1): 100001. https://doi.org/10.1063/5.0162971
V. B. Belayev; The gravitational mass of the rarefied cloud of the relativistic massive particles. AIP Conf. Proc. 28 September 2023; 2872 (1): 100001. https://doi.org/10.1063/5.0162971
V. B. Belayev; The gravitational mass of the rarefied cloud of the relativistic massive particles. AIP Conf. Proc. 28 September 2023; 2872 (1): 100001. https://doi.org/10.1063/5.0162971
Abstract
A cloud of relativistic material particles is considered, for which the distortions of length and time caused by the presence of the Lorentz factor an order of magnitude greater than curvature of space-time by gravity. The gravitational mass of the cloud is found in the region where its size is insignificant. It is established that the coefficient of dependence of the cloud mass on the total rest mass of its particles includes a complete elliptic integral of the 2nd kind.
Keywords
Relativistic material particles; active gravitational mass; Lorentz transformations; isotropic Schwarzschild metric; geodesics equations
Subject
Physical Sciences, Particle and Field Physics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received:
8 August 2022
Commenter:
Wladimir Belayev
Commenter's Conflict of Interests:
Author
Comment:
1) In part 1, the dot in the penultimate line has been removed and put at the end of the last sentence. 2) In part 2 a) In the 3rd line $K'$ is replaced by $K' = (t',x',y',z')$ b) In the 4th line “distance” is replaced by “distance”. c) Part of the 1st sentence of the 2nd paragraph after word “directions” is deleted. 3) In part 3 in equation 6 “\bar r' ” is replaced by “r' ”. 4) In part 4 a) In the 1st sentence “in the corresponding frame” is deleted. b) In the 2nd sentence of the 2nd paragraph “substitutions” is replaced by “substitutions of velocities”. c) In the last sentence of part “disregarding small quantities of a larger order” moved to the end of the sentence “. 5) In part 5 a) The 1st sentence is replaced with “The absolute value of acceleration \begin{equation}a' = \sqrt {({\ddot x'})^2 + ({\ddot y'})^2 + ({\ddot z'})^2}\label{eqn141}\end{equation} imparted to the test particle by the two-body system will be \begin{equation}a' = \frac{1 + {\beta ^2}}{2(1 - {\beta ^2})}\frac{c^2\alpha _1}{({ r'})^3}\sqrt {\frac{(x')^2}{1 - {\beta ^2}} + {({y'})^2} + {({z'})^2}} . \label{eqn15}\end{equation} provided that the size of the system is insignificant compared to the distance to the test particle.” b) The 3rd sentence is deleted. c) In the 4th sentence “of a system consisting of a pair of particles:” is deleted. d )In the 2nd paragraph (new version) the 2ndt sentence is replaced with “Assuming a uniform distribution of the directions of their motion over the corners, the average gravitational mass of a pair of particles in the gas cloud will be”. 6) In Conclusions a) In the 1st sentence “An erroneous” is replaced by “The fallacy of”. b) In the 2nd sentence “the idea” is replaced by “concept”.
Commenter: Wladimir Belayev
Commenter's Conflict of Interests: Author
2) In part 2
a) In the 3rd line $K'$ is replaced by $K' = (t',x',y',z')$
b) In the 4th line “distance” is replaced by “distance”.
c) Part of the 1st sentence of the 2nd paragraph after word “directions” is deleted.
3) In part 3 in equation 6 “\bar r' ” is replaced by “r' ”.
4) In part 4
a) In the 1st sentence “in the corresponding frame” is deleted.
b) In the 2nd sentence of the 2nd paragraph “substitutions” is replaced by “substitutions of velocities”.
c) In the last sentence of part “disregarding small quantities of a larger order” moved to the end of the sentence “.
5) In part 5
a) The 1st sentence is replaced with “The absolute value of acceleration
\begin{equation}a' = \sqrt {({\ddot x'})^2 + ({\ddot y'})^2 + ({\ddot z'})^2}\label{eqn141}\end{equation}
imparted to the test particle by the two-body system will be
\begin{equation}a' = \frac{1 + {\beta ^2}}{2(1 - {\beta ^2})}\frac{c^2\alpha _1}{({ r'})^3}\sqrt {\frac{(x')^2}{1 - {\beta ^2}} + {({y'})^2} + {({z'})^2}} . \label{eqn15}\end{equation}
provided that the size of the system is insignificant compared to the distance to the test particle.”
b) The 3rd sentence is deleted.
c) In the 4th sentence “of a system consisting of a pair of particles:” is deleted.
d )In the 2nd paragraph (new version) the 2ndt sentence is replaced with “Assuming a uniform distribution of the directions of their motion over the corners, the average gravitational mass of a pair of particles in the gas cloud will be”.
6) In Conclusions
a) In the 1st sentence “An erroneous” is replaced by “The fallacy of”.
b) In the 2nd sentence “the idea” is replaced by “concept”.