Charge-energy with heat returns back after the radial fall to gravitational center

TL;DR

This paper proposes a thermodynamic perspective on gravitational motion, suggesting heat transfer and energy exchange influence the dynamics of bodies falling into a gravitational center, with implications for understanding inertial energy and spacetime geometry.

Contribution

It introduces a thermodynamic framework for gravitational motion, linking heat transfer to geodesic paths and energy distribution in strong gravitational fields, offering new testable predictions.

Findings

Heat transfer affects the energy dynamics during gravitational fall.

Oscillatory behavior around energy equilibrium is predicted.

Inertial charges with heat influence motion in strong gravity.

Abstract

Heat transfer in the SR flow of power, rather than the Newton current of cold masses, provides proper referents for GR geodesic motion of inertial energy and for metric of non-empty space. GR can compare losses of internal energy under speed increases and can explain the attraction law by the body tendency toward equipartition of kinetic energies over internal and external degrees of freedom. Thermodynamic approach numerically describes the cyclic dynamics of the vertical fall to center with the final path deceleration followed by the accelerated rise as an oscillation around the kinetic energy equilibrium of geodetically moving body and pushes other testable predictions for inertial charges with heat in strong field gravitation.

Equations

$$E\equiv \frac{E_o\sqrt{g_{00}}}{\sqrt{1-{\beta }^2}}\equiv E_o\sqrt{1-{\beta }^2}+\left[\frac{E_o{\beta }^2}{\sqrt{1-{\beta }^2}}-\frac{E_o(1-\sqrt{g_{00}})}{\sqrt{1-{\beta }^2}}\right],$$

$$\sqrt{g_{oo}}\equiv (N+K+U)\frac{\sqrt{1-{\beta }^2}}{E_o}\equiv 1+\frac{U\sqrt{g_{oo}}}{E}\equiv \frac{1}{[1+(-U/E)]}.$$

$$N(r_{eq})\equiv E_o\sqrt{1-{\beta }^2(r_{eq})}=\frac{E_o{\beta }^2(r_{eq})}{\sqrt{1-{\beta }^2(r_{eq})}}\equiv K(r_{eq}),$$

$$\frac{g_{oo}(r)}{g_{oo}(R)}=1-\frac{v^2(r)}{c^2}=1-\frac{1}{g_{oo}(r)c^2}{\left(\frac{dr}{dt}\right)}^2.$$

$$\frac{d\mathbf{r}(t)}{dt}=\pm \hat{\mathbf{r}}c\sqrt{g_{oo}(r)\left[1-\frac{g_{oo}(r)}{g_{oo}(R)}\right]}.$$

$$\frac{d^2\mathbf{r}(t)}{d{t}^2}=\hat{\mathbf{r}}{c}^2\left(\frac{1}{2}-\frac{g_{oo}(r)}{g_{00}(R)}\right)\frac{d{g}_{oo}(r)}{dr}\Rightarrow \text{cases}-(1-\frac{4r_o}{r})\frac{c^2{r}_o\mathbf{r}}{r^3},g_{oo}^{1916}=1-\frac{2r_o}{r}\text{cr}\text{cr}-\frac{c^2{r}_o\mathbf{r}(r^2-2r_{o}r-r_o^2)}{(r+r_o{)}^5},g_{oo}^{1938}=\frac{r^2}{(r+r_o{)}^2}\text{cr}$$

Full text

**Charge-energy with heat returns back after the radial fall to gravitational center **

I.E. Bulyzhenkov, bulyzhenkov.ie@mipt.ru

*Lebedev Physics Institute RAS

Moscow Institute of Physics and Technology, Moscow, Russia *

Abstract. Heat transfer in the SR flow of power, rather than the Newton current of cold masses, provides proper referents for GR geodesic motion of inertial energy and for metric of non-empty space. GR can compare losses of internal energy under speed increases and can explain the attraction law by the body tendency toward equipartition of kinetic energies over internal and external degrees of freedom. Thermodynamic approach numerically describes the cyclic dynamics of the vertical fall to center with the final path deceleration followed by the accelerated rise as an oscillation around the kinetic energy equilibrium of geodetically moving body and pushes other testable predictions for inertial charges with heat in strong field gravitation.

Keywords: gravitation of thermodynamic bodies, inertia of energy, equipartition theorem, cooling by motion, geodetic fall deceleration, accelerated return from center

MSC: 83C15

PACS: 04.20.Cv

1

, , $v^{2}\ll c^{2}$. , ‘‘ ’’. , . ( ) .

, $E_{o}{\sqrt{g_{oo}}}/{\sqrt{1-v^{2}c^{-2}}}$ [1], 1908 : $Q=Q_{o}{\sqrt{1-v^{2}c^{-2}}}$. , . . , $\hbar\omega=\hbar\omega_{o}{\sqrt{1-v^{2}c^{-2}}}$. , , - , . , , $E_{o}=M_{o}c^{2}$ , , $M_{o}c^{2}{\sqrt{1-v^{2}c^{-2}}}-M_{o}c^{2}\approx-M_{o}v^{2}/2$. $M_{o}v^{2}$ , $M_{o}v^{2}-M_{o}v^{2}/2=M_{o}v^{2}/2\ll M_{o}c^{2}$. , $M_{o}v^{2}/2$ $M_{o}v^{2}$ . ‘‘ ’’ ( ) ( ) ?

2

, , . - . $M_{o}$ $E_{o}$ , ‘‘ ’’ $E_{o}$. , . ‘‘ ’’ .

c $E_{o}$ [1] ,

[TABLE]

( ) $q\equiv E/\varphi_{o}$ ( $\varphi_{o}\equiv c^{2}/{\sqrt{G}}=1.04\times 10^{27}$ B). $N\equiv{E_{o}}{\sqrt{{1-\beta^{2}}}}>0$, , $K\equiv{E_{o}v^{2}}/{c^{2}\sqrt{1-\beta^{2}}}>0$, $U\equiv$ $-{E_{o}(1-{\sqrt{g_{00}}})}/{\sqrt{1-\beta^{2}}}$ $<0$, $0<g_{oo}\leq 1$ $\beta^{2}\equiv v^{2}/c^{2}$ $<1$. , / $q$ $E/\varphi_{o}$ .

(1) , $M\equiv K+U=E-N=E_{o}(\beta^{2}+{\sqrt{g_{00}}}-1)/{\sqrt{1-\beta^{2}}}.$ , ( ) , $T\equiv N+K={E_{o}}/{\sqrt{1-\beta^{2}}}$, 1905 . , ‘‘ ’’ $E_{o}\beta^{2}/2$ , , . ${E_{o}}{\sqrt{{1-\beta^{2}}}}$ , - . 1908 , . , , , , . , .

3

, . , $g_{oo}(x)$ (1) $U/E$ $E$, :

[TABLE]

(1)-(2) $g_{oo}$- : ‘‘ - $\varphi\equiv U/E$ ${g_{oo}}=(1-\varphi)^{-2}$, $-\infty<\varphi\leq 0$ ’’.

$\varphi=-GE_{2}/c^{4}r$ (2) $g_{oo}(r)=[c^{4}r/(c^{4}r+GE_{2})]^{2}$. [2] $g_{oo}(r)=(c^{2}r-2GM_{2})/c^{2}r$ (1) . . 1938 , [3]. 4- ‘ ’ [4] ( ‘‘the reluctant father of black holes’’ - ).

, 1913 4- - , , [5]. , - $ds={\sqrt{d\tau^{2}-dl^{2}}}$ , $d\tau^{2}\equiv{{g_{oo}}}[dt+(g_{oi}dx^{i}/g_{oo})]^{2}$ $dl^{2}\equiv(g_{oi}g_{oj}g^{-1/2}_{oo}-g_{ij})dx^{i}dx^{j}$. [6] , . 1938-1939 1913-1916 ?

4

. 3- 1916 [2] , . , , , - . 1916 80 . , [7] - 1908 . ‘‘ ’’, .

, , . , , , , : ‘‘ - , , - , . , . , . , .’’

• . , . , ( ).

1938 1916 ( - - ) ( - , [7]), , - - - . . , 1916 . - , (1) (2) $g_{oo}(r)=r^{2}/(r+r_{o})^{2}$ - $dl^{2}=\delta_{ij}dx^{i}dx^{j}$ , $G_{oo}=0$. , $R_{oo}=g_{oo}R/2\neq 0$, $r_{o}\varphi_{o}^{2}$ .

5

1938 (1)-(2) . , $E_{o}$ . (1) $N(\infty)=E_{o}$ : $M(\infty)=K(\infty)+U(\infty)=0$ $K(\infty)=0,U(\infty)=0$. $R$ , $\beta(R)=0$ $K(R)=0$, $N(R)=E_{o}$ $M(R)=U(R)=-E_{o}[1-{\sqrt{g_{oo}(R)}}]<0$. , $E(R)=N(R)+M(R)=E_{o}{\sqrt{g_{oo}(R)}}>0$, (1) .

$R$ , :

[TABLE]

$v^{2}(r_{eq})=c^{2}/2$ $r_{eq}<R$.

( ) , . . $K$ (1) $N$. . , , , .

$R$ $E_{2}\gg E_{o}$, $E(R)=E_{o}{\sqrt{g_{oo}(R)}}$ $g_{oo}(r)$. (1) , $E[r(t)]/E_{o}={\sqrt{g_{oo}(r)/[1-\beta^{2}(r)]}}={\sqrt{g_{oo}(R)}}$. $g_{oo}(r)$ , $0\leq v^{2}(r)<c^{2}$, , $dr/dt$ $dt\equiv dx^{o}/c$ . :

[TABLE]

(4) ${\hat{\bf r}}\equiv{\bf r}/r$:

[TABLE]

, , ${\bf 0}\rightarrow d{\bf r}/dt\rightarrow-{\hat{\bf r}}c{\sqrt{g_{oo}(R)}}/2$, ${{g_{oo}(R)}}\geq g_{oo}(r\rightarrow r_{eq})\geq{{g_{oo}(R)}}/2$. , ${{g_{oo}(R)}}/2\geq g_{oo}(r\rightarrow 0)/\geq 0$, , $-{\hat{\bf r}}c{\sqrt{g_{oo}(R)}}/2$ $\rightarrow d{\bf r}/dt\rightarrow{\bf 0}$. $|{\bf v}|=|d{\bf r}/{\sqrt{g_{oo}(r)}}dt|$ , $N(0)=0$. . . $v\rightarrow c$, , $dr/dt\rightarrow 0$, .

$N(0)=0,M(0)=K(0)+U(0)=E_{o}{\sqrt{g_{oo}(R)}}$ $N(r_{eq})=K(r_{eq})=E_{o}{\sqrt{2}}/2$ $v^{2}=dr^{2}/g_{oo}dt^{2}=c^{2}/2$, ( ) $dr/dt=c{\sqrt{g_{oo}(R)}}/2$ $g_{oo}(r_{eq})={{g_{oo}(R)}}/2$. , $r(t)=r_{eq}$, $N(r)$ $R$, $E=E_{o}{\sqrt{g_{oo}(R)}}$, $N(R)=E_{o},K(R)=0,U(R)/E_{o}{\sqrt{g_{oo}(R)}}=1-g^{-1/2}_{oo}(R).$

( ) (5) $t$ $dg_{oo}(r[t])/dt$ $=$ $(dg_{oo}/dr)dr/dt$:

[TABLE]

, $R/r_{o}\Rightarrow\infty$, $g_{oo}(r_{eq})=1/2$ . (6) 1916 , $g_{oo}=1-2r_{o}/r,$ $r_{o}\equiv GE_{2}/c^{4}$, $g_{oo}=r^{2}/(r+r_{o})^{2}$ (2) $U/E=-r_{o}/r$ 1938 . ${-c^{2}r_{o}{{\bf r}}}/{r^{3}}$ $r>2r_{o}$, . . .

6

1916 $r=2r_{o}\equiv 2GE_{2}/c^{4}$ $E_{2}=r_{o}\varphi^{2}_{o}$. 1938 $g_{oo}(r)$ $r=0$. 1938 , : [8].

(6) , , . .

, . ( - ) - . . , , .

, . ( ) . , , .

, , . . . 1908 .

. , .

Список литературы

• [1] . . . . , , , 2, ( .: , 2006, . 8).
• [2] K. Schwarzschild, Sitzungsber. Deut. Akad. Wiss., Berlin 189 (1916).
• [3] . , . , ( : , 1965, .3), . 3.
• [4] A. Einstein, Annals of Mathematics ${\bf 40}$, 922 (1939).
• [5] A. Einstein and M. Grossmann, Zs. Math. und Phys. $\bf 62$, 225 (1915).
• [6] A. Einstein, Annalen der Physik ${\bf 49}$, 769 (1916).
• [7] A.A. , ( : ‘‘ ’’, 2012).
• [8] . . . , 2016, B.4, 37 (2016).

**- **

. . , bulyzhenkov.ie@mipt.ru

*. .

• , , *

Аннотация

, , . . .

: , , , , ,