Moving objects in retarded gravitational potentials of an expanding spherical shell/Gravitational redshift

Gravitational redshift

Schwarzschild radius

Relation between the Schwarzschild radius $r_{S}$ and a redshift of an emitted photon $z$ in a distance $r$ caused by the gravitation of the massive sphere in the centre.

The Schwarzschild radius $r_{S}$ of any mass $M$ in a Schwarzschild sphere (typically a black hole) is given by:

$r_{S}={\frac {2\,G\,M}{c^{2}}}$

Apart from its mass, the Schwarzschild radius is only depending on two natural constants:

• Gravitational constant: $G=6.67\cdot 10^{-11}{\frac {{\text{m}}^{3}}{{\text{kg}}\,{\text{s}}^{2}}}$
• Speed of light: $c=299792458{\frac {\text{m}}{\text{s}}}$

The gravitational redshift z of photons emitted to the opposite direction of the Schwarzschild sphere and with a distance to the centre of the sphere $r$can be computed by the following formula:

$z={\frac {1}{\sqrt {1-{\frac {2\,G\,M}{r\,c^{2}}}}}}-1={\frac {1}{\sqrt {1-{\frac {r_{S}}{r}}}}}-1$

The redshift of very far objects such as the galaxy JADES.GS.z14-0 has a value of more than 14, and this is much larger than expected. The possible high speed of such a galaxy is not sufficient for such a high value. The distance of this galaxy is given by 13.5 billion light-years, and its age is assumed to be 290 million years after Big Bang.

The cosmic microwave background even has a redshift value of 1089, which is extremely high. It is associated with the first hydrogen atoms that occurred some 380,000 years after the Big Bang.

In a shell

The sum of all gravitational forces between the mass $m$ and a spherical arc (blue) with the line density $\lambda _{M}$ that represents a hollow spherical cap. It has the same effect as the gravitational force between the mass $m$ and an effective spherical mass $M_{eff}$ (black).

However, a gravitational redshift can not only occur outside of spheres, but also within in a hollow spherical cap. To estimate its gravitational redshift, the effective mass of such a cap can be integrated for any point within the cap. The corresponding effect can be described by a Schwarzschild sphere with the Schwarzschild radius.

For very fast-moving objects we can assume that they only experience the retarded gravitational potentials of the mass elements in front of them, since the backward potentials are even much more retarded, and therefore, contribute only weakly to the net gravity. In the following section only the arc of the shell cap is considered. The gravitational forces rectangular to the direction of movement are very small and can be neglected. This also holds for the gravitational forces behind the moving mass $m$. If the distance $d$ between the mass $m$ and the shell is given, we can estimate the fraction of the retarded forces from the opposite part of the shell in the distance $2\,R-d$, where $R$ is radius of the visible universe (cf. previous section "Retarded gravitational potentials"):

$F_{right}\propto {\frac {1}{d^{2}}}$

$F_{left}\propto {\frac {1}{(2\,R-d)^{2}}}$

If we assume that the distance is a fraction of the radius

$d={\frac {R}{n}}$

then the ratio of the two forces has the following relation:

${\frac {F_{right}}{F_{left}}}=\left({\frac {2\,R-{\frac {R}{n}}}{\frac {R}{n}}}\right)^{2}=(2\,n-1)^{2}$

This means if the distance $d$ is a tenth of the radius $R$ of the universe, the error by neglecting the left force would be less than 0.3 percent,.

Computation

Diagram for integrating the effective mass of an outer shell $M_{eff}$ acting on a fast moving mass $m$ by varying the centre angle $\alpha$ from $-\alpha _{R}$ to $\alpha _{R}$.
$R$ = radius of the visible universe
$d$ = distance of mass $m$ to the outer shell
$\lambda _{M}$ = line mass density of the outer shell

The bold arc on the right-hand side of the diagram is representing all mass elements of the outer shell of the universe in front of the mass that is moving to the right with high velocity. The outer shell is consisting of dark matter (mainly hydrogen), and in the central area of the universe this dark matter is only visible due to the cosmic microwave background.

We use the following constant values for the estimates derived from these premises:

• Light-year: $1\,{\text{ly}}=9.46\cdot 10^{15}{\text{m}}$
• Hubble length: $R=1.36\cdot 10^{26}{\text{m}}$ (equals 14,4 billion light-years)
• Mass of the universe: $M_{universe}=2.97\cdot 10^{53}{\text{kg}}$:

The mass of this arc can be computed by integrating the arc between the angles $-\alpha _{R}$ and $+\alpha _{R}$ with its line mass density $\lambda _{M}$:

$\alpha _{R}=\arcsin {\frac {x}{R}}$

$M_{arc}=\int _{-\alpha _{R}}^{+\alpha _{R}}\mathrm {dM} =\int _{-\alpha _{R}}^{+\alpha _{R}}R\,\lambda _{M}\,\mathrm {d\alpha }$

The mass of the whole shell $M_{S}$ is given by integrating a complete circle:

$M_{S}=\int _{-\pi }^{+\pi }\mathrm {dM} =2\pi \,R\,\lambda _{M}$

The vertical components of the gravitational forces are symetrical, and therefore, their net effect is zero.

Detail for integrating the infinitesimal mass element $\mathrm {dM}$ acting on a mass $m$ by varying the angle $\alpha$.
$\alpha$ = angle as seen from origin $\bigcirc$
$R$ = radius of the universe (Hubble length)
$h$ = height og the mass element $\mathrm {dM}$
$d$ = distance of mass $m$ to the outer shell
$s$ = distance between mass $m$ and mass element $\mathrm {dM}$
$e$ = auxiliary sagitta
$\beta$ = angle as seen from $m$

The distance $d$ between the mass $m$ and the outer shell is given, and we can compute the distances of the mass $m$ to any infinitesimal mass element $\mathrm {dM}$ on the arc depending on the angle $\alpha$:

$\alpha =\arcsin {\frac {h}{R}}$

$s(\alpha )={\sqrt {2R^{2}-2\,R\,d+d^{2}-2\,R\,(R-d)\,cos\,\alpha }}$

The angle $\beta$ as seen from the moving mass m to the infinitesimal mass element $\mathrm {dM}$ on the arc is given by the following expression:

$\beta =\arcsin \left({\frac {R}{s}}\sin \,\alpha \right)=\arcsin {\frac {h}{s}}$

With the auxiliary sagitta $e$ we get:

$e=R\,\left(1-\cos \alpha \right)$

$\cos \beta ={\frac {d-e}{s}}$

The horizontal force of this arc $F_{hor}$ that is accelerating the mass $m$ in horizontal direction can be achieved by integrating the arc with the correction factor $\cos \beta$:

$F_{hor}=G\,m\int _{-\alpha _{R}}^{+\alpha _{R}}{\frac {\cos \beta }{s^{2}}}\,\mathrm {dM} =G\,m\,R\,\lambda _{M}\,\int _{-\alpha _{R}}^{+\alpha _{R}}{\frac {\cos \beta }{s^{2}}}\,\mathrm {d\alpha }$

With the effective gravitational force $F_{hor}$ that acts on the mass $m$ by the gravitational force of the virtual effective mass $M_{eff}$ that is located in the intersection point of the trajectory of the mass $m$ with the outer shell, we have to consider the varying distances $s$ between $m$ and the infinitesimal mass elements $\mathrm {dM}$ along the arc:

$F_{hor}=G\,m{\frac {M_{eff}}{d^{2}}}=G\,m\,R\,\lambda _{M}\,\int _{-\alpha _{R}}^{+\alpha _{R}}{\frac {\cos \beta }{s^{2}}}\,\mathrm {d\alpha }$

We finally get the effective mass $M_{eff}$ in the distance $d$ from $m$:

$M_{eff}=R\,\lambda _{M}\,d^{2}\int _{-\alpha _{R}}^{+\alpha _{R}}{\frac {\cos \beta }{s^{2}}}\,\mathrm {d\alpha } ={\frac {M_{S}\,d^{2}}{2\pi }}\int _{-\alpha _{R}}^{+\alpha _{R}}{\frac {\cos \beta }{s^{2}}}\,\mathrm {d\alpha }$

Schwarzschild distance

The Schwarzschild distance $d_{S}$ of this effective mass is equal to the radius of a sphere with the effective mass:

$d_{S}={\frac {2\,G\,M_{eff}}{c^{2}}}$

The gravitational redshift $z$ of photons emitted to the centre of the universe that is caused by the effective mass in the distance $d$ of the outer shell is:

$z={\frac {1}{\sqrt {1-{\frac {2\,G\,M_{eff}}{d\,c^{2}}}}}}-1={\frac {1}{\sqrt {1-{\frac {d_{S}}{d}}}}}-1$

The equations above can be solved for the Schwarzschild distance of every shell with the mass $M_{S}$. With the following condition we can get the solutions with $d_{S}=d$, where $z(d_{S})=\infty$:

$2\,G\,M_{eff}(M_{S},d_{S})=d_{S}\,c^{2}$

The overall mass of the invisible outer spherical shell $M_{S}$ can be expressed in relation to the mass of the visible universe $M_{universe}$:

$q_{M}={\frac {M_{S}}{M_{universe}}}$

Results

It is assumed that the outer shell is expanding, spherical and consists of dark matter. The effective mass of the outer shell is computed only considering the geometrical region in front of the object. This assumption is based on the fact that the retarded gravitational potentials from the opposite border of the black shell can be neglected. The following two diagrams show the solution in two different representations over the relation of the mass of the invisible universe surrounding the visible universe in a spherical shell in units of the mass of the visible universe$q_{M}$:

• The Schwarzschild distance $d_{S}$ between the visible border of the universe and the invisible outer shell of the universe.
• The radius of the visible universe $R_{v}=R-d_{S}$.

• Diagrams for the early evolvement of the universe considering the effects of the retarded gravitational potentials of an invisible outer shell
• 
  Scheme with the outer black shell of the invisible universe with dark matter (right), the shell of the origin of the cosmic microwave background (CMB, dark red) as well as a far galaxy (light red). The matter in the vicinity of the black shell is accelerated towards the shell due to the effect of retarded gravitational forces and may disappear into it by crossing the event horizon at the Schwarzschild distance. During the era of transforming dark matter into "dark energy" the Schwarzschild distance was fastly increasing.
• 
  Evolvement of the effective mass of the black shell of the universe with the decreasing radii of the visible universe and the increasing effective masses of the black shell.
• 
  Schwarzschild distance $d_{S}$ between the visible border of the universe and the invisible outer sphere of the universe as a function of the mass of the black shell in units of the mass of the visible universe. The two dots indicate the Schwarzschild distances in light-years.
• 
  Radius of the visible universe $R_{v}=R-d_{S}$ as a function of the mass of the invisible universe surrounding it in a spherical black shell in units of the mass of the visible universe. The two dots indicate the Schwarzschild thicknesses of the invisible black shell in light-years.

Linear mass density $\lambda _{m}$ in black shell in kilogram per metre	Mass $M_{S}$ of black shell in kilograms	Ratio ${\frac {M_{S}}{M}}$ of mass of black shell to mass of universe	Schwarzschild distance $d_{S}$ in metres	Effective mass $M_{eff}$ of sphere at any point of black shell in kilograms	Ratio ${\frac {d_{S}}{R}}$ of Schwarzschild distance to radius of universe	Schwarzschild distance $d_{S}$ in light-years	Ratio ${\frac {R-d_{S}}{R}}$ of radius of the visible universe to radius of the universe
1,000E+26	8,545E+52	0,288	2,7E+19	1,8E+46	2,0E-07	2,89E+03	0,9999998
1,500E+26	1,282E+53	0,432	1,0E+20	6,7E+46	7,3E-07	1,06E+04	0,9999993
1,600E+26	1,367E+53	0,460	1,5E+20	1,0E+47	1,1E-06	1,63E+04	0,9999989
1,650E+26	1,410E+53	0,475	2,3E+20	1,5E+47	1,7E-06	2,42E+04	0,999998
1,675E+26	1,431E+53	0,482	3,6E+20	2,5E+47	2,7E-06	3,85E+04	0,999997
1,680E+26	1,436E+53	0,483	4,7E+20	3,2E+47	3,5E-06	5,02E+04	0,999997
1,683E+26	1,438E+53	0,484	3,6E+21	2,4E+48	2,6E-05	3,80E+05	0,99997
1,685E+26	1,440E+53	0,485	6,9E+22	4,6E+49	5,1E-04	7,29E+06	0,9995
1,690E+26	1,444E+53	0,486	3,3E+23	2,2E+50	2,4E-03	3,49E+07	0,998
1,700E+26	1,453E+53	0,489	9,5E+23	6,4E+50	7,0E-03	1,00E+08	0,993
1,710E+26	1,461E+53	0,492	1,7E+24	1,1E+51	1,3E-02	1,80E+08	0,988
1,720E+26	1,470E+53	0,495	2,5E+24	1,7E+51	1,8E-02	2,64E+08	0,982
1,724E+26	1,473E+53	0,496	2,7E+24	1,8E+51	2,0E-02	2,90E+08	0,980
1,727E+26	1,476E+53	0,497	3,0E+24	2,0E+51	2,2E-02	3,19E+08	0,978
1,729E+26	1,477E+53	0,497	3,2E+24	2,1E+51	2,3E-02	3,35E+08	0,977
1,730E+26	1,478E+53	0,498	3,3E+24	2,2E+51	2,4E-02	3,49E+08	0,976
1,792E+26	1,532E+53	0,516	9,5E+24	6,4E+51	7,0E-02	1,00E+09	0,930

In this model the mass of the invisible outer shell is increasing continuosly. This might be due to the fact that more and more matter is moving from the visible universe behind the event horizon, where it becomes invisible and unaccessible.

It is very noteworthy to recapitulate that the age of the cosmic microwave background with its afterglow light pattern is about 380,000 years (shell mass $M_{S}=1.438\cdot 10^{53}\,{\text{kg}}=0.484\cdot M_{S}$, Schwarzschild distance $d_{S}=3.6\cdot 10^{21}\,{\text{m}}=2.6\cdot 10^{-5}\cdot R$), exactly where the radius of the visible universe had significantly begun to decrease (see right diagram). The so-called "dark ages" begun, and they lasted for several hundreds of million years.

The age of the oldest known galaxy JADES.GS.z14-0 represents the youngest stars that emit light, and therefore, the end of the dark ages. Its age is about 290 million years (shell mass $M_{S}=1.473\cdot 10^{53}\,{\text{kg}}=0,496\cdot M_{S}$, Schwarzschild distance $d_{S}=2.7\cdot 10^{24}\,{\text{m}}=0.020\cdot R$), where the size of the visible universe had begun to decelerate (see left diagram).

For comparison: the Schwarzschild radius of the universe can be computed by its mass, too:

$r_{S,universe}={\frac {2\,G\,M_{universe}}{c^{2}}}\approx 4.4\cdot 10^{26}{\text{m}}\approx 46.6\cdot 10^{9}{\text{ly}}$