Moving objects in retarded gravitational potentials of an expanding spherical shell/Printable version

Moving objects in retarded gravitational potentials of an expanding spherical shell

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Preface

Summary

• Retarded gravitational potentials of a very large spherical shell with mass.
• Acceleration forces on objects with mass within such a shell.
• Concept of the Schwarzschild distance from the outer invisible surface of such a shell ("black shell") to the visible universe.
• Gravitational redshift due to the mass of the black shell.
• Evolution of the Schwarzschild distance due to the continuously increasing mass of the black shell.

Abstract

The principles of retarded gravitational potentials are presented and explained. The application of retarded gravitational potentials to an expanding spherical shell of matter with a large mass (a "black shell") leads to acceleration forces on moving objects within such a huge shell.

The concept of a Schwarzschild distance from the inner surface of a hollow spherical cap is introduced. It can be used to determine equivalent spherical masses which can provide information about the evolution of the universe.

The unexpected high values of the cosmological redshift that are observed at extremely far astronomical objects including the cosmic microwave background could be related to the strong asymmetrical effect of the gravitation of the outer black shell of the universe. Due to the integrative effect of the retarded gravitational potentials of a concave shaped spherical black shell the gravitational forces of masses within the Schwarzschild distance are so enormous that the light from any atoms can no longer reach us. Exactly at the boundary to our visible universe, the gravitational redshift is infinite, just like on the surface of a convex shaped spherical black hole.

The evolvement of the Schwarzschild distance due to the continuously increasing mass of the outer invisible shell of our universe has two inflection points around the very points in time and space, where the cosmic microwave background as well as the youngest galaxies can be observed today. The strongly increasing Schwarzschild distance between these two points in time goes hand in hand with the postulated massive transformation of dark matter to "dark energy". Therefore, the effect of retarded gravitational potentials could also contribute to the explanation of the accelerated expansion of the universe, which often is related to the effect of "dark energy".

It seems that the hypothesis of relevant dark matter that continuously moves beyond the event horizon of our visible universe could open the gates for further interesting investigations and findings.

A shorter version of this Wikibook is available on Research Gate.

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Brief historical review

The finiteness of the propagation speed of gravity and its influence on gravitational forces was originally published by the Austrian astronomer Josef von Hepperger (1855–1928) in 1888 in Vienna.

The German physicist Paul Gerber (1854–1909) published 1898 his paper on "The spatial and temporal expansion of gravity", where he established that the perihelion precession of the planet Mercury is related to the propagation speed of gravity, which is quite close to that of electromagnetic radiation. His formula for the perihelion precession of the planet Mercury wasn't known to Albert Einstein (1879–1955), but six years after Paul Gerber's death, Einstein found an identical formula in his publication "Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie" (English: "Explanation of Mercury's perihelion motion from the general theory of relativity") by applying the laws of General Relativity. However, the contemporary scientists couldn't reproduce the derivation of Paul Gerber for the formula, and furthermore, they stated that some of the prerequisites used by him were wrong.

Shortly before his early death the German astronomer and physicist Karl Schwarzschild (1873–1916) published a paper on "Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie" (English: "On the gravitational field of a sphere of incompressible fluid according to Einstein's theory"), where he described how to compute the smallest possible radius for a sphere with a given mass. He found the radius for a sphere with the mass of the sun to be three kilometres. In recognition of his achievement, the corresponding radius is now called the Schwarzschild radius. It is interesting to notice that Schwarzschild also made important contributions to retarded potentials in electrodynamics already in 1903. According to the electrokinetic potential he claimed:

Es sind in jedem Raumelement die Werte der Dichte und der Geschwindigkeit zu benutzen,
welche dort zu einer um die Lichtzeit zurückliegenden Epoche galten.

In each spatial element, the values of density and velocity are to be used,
which were valid there at an epoch around light time ago.

The Belgian theologian and physicist George Lemaître SJ (1894–1966) is regarded as the founder of the Big Bang theory. In 1931, he introduced the term "atome primitif" (English: "primordial atom") to describe the hot initial state of the universe. Already in 1927 he wrote as the second conclusion of his publication about "a homogeneous universe of constant mass and increasing radius, accounting for the radial velocity of extra-galactic nebulae" with reference to the investigations of Edwin Hubble (1889–1953) in 1926:

Le rayon de l'univers croit sans cesse depuis une valeur asymptotique $R_{0}$ pour $t=-\infty$

The radius of the universe increases without limit from an asymptotic value $R_{0}$ for $t=-\infty$

In 1933 the Swiss astronomer Fritz Zwicky (1898–1974) obeserved a gravitational anomaly in the Coma galaxy cluster, and he coined the term dark matter (in German: "dunkle Materie") for the cause of this anomaly.

In his book Relativity, Gravitation and World Structure of the year 1935 the British astrophysicist and mathematician Edward Arthur Milne (1896–1950) introduced the special-relativistic cosmological Milne model. This model assumes an isotropic and for all time linearly expanding, but not homogeneous universe. It is independent of general theory of relativity, but nevertheless equivalent to a special case of the Friedmann–Lemaître–Robertson–Walker metric in the limit of zero energy density. The redshift is explained by the velocity caused by a hypothetical initial explosion of matter. The density of such a universe must be small in comparison to the critical density of the Friedmann equations that were published by Russian physicist and mathematician Alexander Friedmann (1888–1925) in 1924. Decades later, the Milne model fell almost completely into oblivion, whereas the cosmological models based on the general theory of relativity gained more and more attention.

In 1953 the German astrophysicist Erwin Finlay-Freundlich (1885–1964) derived a blackbody temperature for intergalactic space of 2.3 Kelvin according to his theory of tired light. The German-British mathematician and physicist Max Born (1882–1970) immediately recommended taking the problem seriously and pursuing it further.

• Portrait images of early contributing scientists
• 
  Josef Hepperger in the 1920s.
• 
  Paul Gerber around 1900.
• 
  Karl Schwarzschild around 1900.
• 
  Edward Arthur Milne in 1917.
• 
  Albert Einstein in 1921.
• 
  Alexander Friedmann around 1920.
• 
  George Lemaître in 1933.
• 
  Edwin Hubble in 1931.
• 
  Fritz Zwicky in the year 1947.
• 
  Erwin Freundlich in the 1930s.

Map of cosmic background radiation with artificial colours that show tiny temperature variations.

In 1965 the cosmic microwave background (CMB) was discovered by US-American radio astronomers Arno Allan Penzias (1933–2024) and Robert Woodrow Wilson (born 1936). It has a thermal black body spectrum at a very low temperature of about 2.7 Kelvin. Since the cosmic microwave background was mainly created in the visible area of the electromagnetic spectrum, it must have undergone a strong wavelength extension on its way to the observers. This can happen because of two main reasons:

• The origin of the radiation moves away from us very quickly, which will cause an increase of the wavelength according to the Doppler effect described by Christian Doppler (1803–1853) in 1842.
• There is a huge mass behind the origin of the radiation, which will cause an increase of the wavelength according to the gravitation and the relativity principle of Albert Einstein (1879–1955) of 1907.

In addition, it is discussed that an expansion of the spacetime during the propagation time of a light wave also would increase its wavelength to the same extent.

The redshift factor $z$ is defined as a relation between an emitted wavelength $\lambda _{0}$ and an observed wavelength $\lambda$ of electromagnetic radiation:

$z={\frac {\Delta \lambda }{\lambda _{0}}}={\frac {\lambda -\lambda _{0}}{\lambda _{0}}}={\frac {\lambda }{\lambda _{0}}}-1$

The redshift of the cosmic microwave background was found to be $z\approx 1089$ in 2003, which is an extremely high value. The age of the universe at the time this background radiation was created by hydrogen atoms has been estimated at around 379,000 years.

Observations of distant type Ia supernovae published by both the Supernova Cosmology Project as well as the High-Z Supernova Search Team in 1998 show that the relative expansion of the universe is accelerating. For the analysis the astronomers Saul Perlmutter, Brian P. Schmidt and Adam Riess were awarded the Nobel Prize in Physics in 2011.

From 2001 to 2010 the NASA spacecraft Wilkinson Microwave Anisotropy Probe (WMAP) was investigating the cosmic microwave background. Its measurements led to the current Standard Model of Cosmology. According to this model the universe currently consists of less than 5 percent ordinary baryonic matter; about 24 percent cold dark matter (CDM) that interacts only weakly with ordinary matter and electromagnetic radiation; and more than 70 percent of dark energy that is used to explain the accelerated expansion of the universe. These data were more or less confirmed by the Planck space observatory that was operated by the European Space Agency (ESA) from 2009 to 2013.

• Evolution of the universe investigated with the Wilkinson Microwave Anisotropy Probe (WMAP)
• 
  The evolution of the universe according to the interpretation of the measurements of the Wilkinson Microwave Anisotropy Probe (WMAP). The acceleration of the expansion leads to the concave outer rim surface of the evolution cone today.
• 
  The contents of of the universe today (top) and 380,000 years after the Big Bang (bottom) according to the interpretation of the Wilkinson Microwave Anisotropy Probe (WMAP) measurements.

In 2013 the Indian researcher Chandrakant Raju proposed to apply the retarded gravitation theory (RGT) to explain the flyby anomaly of spacecrafts in the graviational field of the earth as well as the acceleration of masses in the retarded gravitational fields of spriral galaxies.

In 2016, researchers from the gravitational-wave observatory LIGO reported the first direct measurement of gravitational waves generated by the collision of two black holes (gravitational wave event GW150914). Already in 2017 the Nobel Prize in Physics was awarded for the first direct observation of gravitational waves.

On 17th August 2017 a merging binary neutron star was observed independently and simultanuously by the Advanced LIGO and Virgo detectors (gravitational wave event GW170817) and the Fermi Gamma-ray Burst Monitor as well as the Anticoincidence Shield for the Spectrometer for the International Gamma-Ray Astrophysics Laboratory (gamma ray burst event GRB 170817A). These observations prove that the propagation velocity of gravitational waves must be extremly close to that of electromagnetic waves.

JADES-GS-z14-0 taken with the Near-Infrared Spectrograph of the James Webb Space Telescope.

In January 2024 the very young and very far galaxis JADES-GS-z14-0 was found with the Near-Infrared Spectrograph (NIRSpec) of the James Webb Space Telescope (JWST). This galaxy was observed in a state 290 million years after the big bang. Its redshift measured with the well-known Lyman alpha break at a wavelength about 1.8 micometres has a very high value of a good 14, which is a very high value for an astronomical object, but much lower than that of the cosmic microwave background.

In February 2024, a study explained the anomalies in the rotation curves of more than hundred spiral galaxies of different types from the SPARC (Spitzer Photometry and Accurate Rotation Curves) Galaxy collection taken with the Spitzer Space Telescope, by accounting for retarded gravitational potentials and without taking into account dark matter or modifying general relativity.

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Classical approach

This Wikibook assumes a huge amount of dark matter that is in a huge shell behind the visible part of the universe. This matter is rather cold, and it can be assumed that it is in a in thermal equilibrium. Therefore, the electromagnetic radiation emitted by this shell can be represented by the Planck radiation law. The emission of such a cold source of electromagnetic radiation is not visible and additionally the radiation has experienced a strong redshift due to the relative velocity of the fast expanding emitting shell as well as due to the gravitation of the mass of this shell.

The gravitational force $F$ between two masses $m$ and $M$ in a distance of $s$ is given by Newton's law of universal gravitation with the gravitational constant $G$:

$F=G\,{\frac {m\cdot M}{s^{2}}}$

In a thought experiment it is possible to investigate the behaviour of a mass $m$ that is located within a spherical shell with the mass $M$. The approach described in 1687 by Isaak Newton (1643–1727 in Gregorian dates) is called Newton's shell theorem. According to this theorem a mass $m$ within a homogeneous and spherically symmetric shell with a mass $M$ experiences no net gravitational force. Additionally, this is regardless of the location of the mass $m$ within the shell as well as of its velocity.

This behaviour can be easily explained with the following consideration: let a mass $m$ be within a homogeneous and spherically symmetric shell with the radius $r$ and the areal density $\rho _{a}$. Then the area $A$ and the mass $M$ of the shell can be expressed as:

$A=4\pi \cdot r^{2}$

$M=\int \limits _{A}\mathrm {d} M=\int \limits _{A}\rho _{a}\cdot \mathrm {d} A=\rho _{a}\cdot 4\pi \cdot r^{2}$

About the equivalence of the gravitational force to a mass $m$ in a homogeneous and spherically symmetric shell with the two area elements $\mathrm {d} A_{1}$ and $\mathrm {d} A_{2}$.

If we take an axis symmetrical double cone where the tips of the two cones are located at the position of the mass $m$ within a homogeneous and spherically symmetric shell, we get the situation shown in the adjacent figure.

For the two infinitesimally small area elements $\mathrm {d} A_{1}$ and $\mathrm {d} A_{2}$ in the distances $s_{1}$ and $s_{2}$ of a mass $m$ with the same infinitesimally small angle elements $\mathrm {d} \alpha$:

$\mathrm {d} A_{1}={\frac {\pi }{4}}\,s_{1}^{2}\cdot \mathrm {d} \alpha ^{2}$

$\mathrm {d} A_{2}={\frac {\pi }{4}}\,s_{2}^{2}\cdot \mathrm {d} \alpha ^{2}$

For the appropriate masses $\mathrm {d} M_{1}$ and $\mathrm {d} M_{2}$ of the two area elements:

$\mathrm {d} M_{1}=\rho _{a}\cdot \mathrm {d} A_{1}=\rho _{a}\cdot {\frac {\pi }{4}}\,s_{1}^{2}\cdot \mathrm {d} \alpha ^{2}$

$\mathrm {d} M_{2}=\rho _{a}\cdot \mathrm {d} A_{2}=\rho _{a}\cdot {\frac {\pi }{4}}\,s_{2}^{2}\cdot \mathrm {d} \alpha ^{2}$

With Newton's law of universal gravitation, we can express the two infinitesimally small gravitational forces $\mathrm {d} F_{1}$ and $\mathrm {d} F_{2}$ due to the masses of the two area elements:

$\mathrm {d} F_{1}=G\,{\frac {m\cdot \mathrm {d} M_{1}}{s_{1}^{2}}}=G\,{\frac {m\cdot \rho _{a}\cdot {\frac {\pi }{4}}\,s_{1}^{2}\cdot \mathrm {d} \alpha ^{2}}{s_{1}^{2}}}={\frac {\pi }{4}}\,G\cdot m\cdot \rho _{a}\cdot \mathrm {d} \alpha ^{2}$

$\mathrm {d} F_{2}=G\,{\frac {m\cdot \mathrm {d} M_{2}}{s_{2}^{2}}}=G\,{\frac {m\cdot \rho _{a}\cdot {\frac {\pi }{4}}\,s_{2}^{2}\cdot \mathrm {d} \alpha ^{2}}{s_{2}^{2}}}={\frac {\pi }{4}}\,G\cdot m\cdot \rho _{a}\cdot \mathrm {d} \alpha ^{2}$

It is obvious that both forces have the same value, and since they must be exactly directed to opposite directions, they add to zero. In other words: a mass within a spherical shell does not experience any net gravitational force at all.

It is worth noting that this classical approach assumes the infinite propagation speed of gravitational waves, which is applicable for static but not for dynamic situations as well as huge systems.

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Retarded gravitational potentials

Already the German physicist and astronomer Karl Schwarzschild (1873–1916) described retarded potentials for elecrodynamic fields (he still used the term "electrokinetic potential") in 1903. These potentials with a delayed effect were adopted one year later with reference to Schwarzschild by the German mathematician Alexander Wilhelm von Brill (1842–1935) in Über zyklische Bewegung (English: "About cyclic movement"), where he coined the term "retardiertes Potential" (English: "retarded potential") in a footnote. Furthermore, he emphasized that these retarded potentials would cause a non-zero divergence in space, i.e. that there would be sources and sinks or that the medium would have the possibility of storing or releasing potential energy. Retarded potentials are a mathematical description of potentials in a field theory in which a field quantity propagates at a finite speed (speed of light) and not instantaneously. They occur in the investigation of time-dependent problems, such as the radiation of electromagnetic waves, but also in the propagation of gravitational waves.

Objects that are moving towards the outer rim of the universe will experience retarded gravitational potentials by the expanding dark matter that can be assumed at the very edge of the universe. Therefore, the appropriate gravitational forces will have delayed effects, and due to the large distances and the finite velocity of gravitational wave propagation they also will be weaker in the direction of the former location of the objects. As a result, the net gravitational force is directed in the direction of movement of these objects, and therefore, all objects that move outwards will be accelerated in the direction of their own movement, which would become observable as an accelerated expansion of the visible universe.

The gravitational potential $\Phi$ due to a mass $M$ in the distance $s$ is given by:

$\Phi (s)=-{\frac {G\cdot M}{s}}$

The gravitational force $F$ to another mass $m$ results as follows:

$F=-m\,{\frac {\mathrm {d} \Phi (s)}{\mathrm {d} s}}=G\,{\frac {m\cdot M}{s^{2}}}$

The time-depending potential $\Phi ({\vec {x}},t)$ is the solution of the inhomogeneous wave equation, where $v({\vec {x}},t)$ is the inhomogeneity, and $c$ is the speed of the propagation of the waves. For gravitational forces we can consider it as equal to the vacuum speed of electromagnetic waves:

${\frac {1}{c^{2}}}\,{\frac {\partial ^{2}\Phi ({\vec {x}},t)}{\partial t^{2}}}-\Delta \Phi ({\vec {x}},t)=\Box \Phi ({\vec {x}},t)=v({\vec {x}},t)$,

where $\Delta =\nabla ^{2}$ is the Laplace operator, $\Box$ is the D’Alembert operator.

The solution of the inhomogeneous wave equation is called retarded potential, and in three dimensions it can be given as:

$\Phi _{\text{ret}}({\vec {x}},t)={\frac {1}{4\pi }}\int {\frac {v({\vec {s}},t_{\text{ret}})}{|{\vec {x}}-{\vec {s}}|}}\,\mathrm {d} ^{3}s$

The retardation is to be interpreted in such a way that a source element $v({\vec {x}},t_{\text{ret}})$ at the point ${\vec {s}}$ and at the time $t_{\text{ret}}$ only influences the potential at the distant point of impact at ${\vec {x}}$ at a later time $t$:

$t=t_{\text{ret}}+{\frac {|{\vec {x}}-{\vec {s}}|}{c}}$

$t_{\text{ret}}=t-{\frac {|{\vec {x}}-{\vec {s}}|}{c}}$

$t_{\text{ret}}$ is called the retarded time. At the location ${\vec {x}}$ and at the time $t$ the retarded potential only depends on the inhomogeneity $v$ in the retarding back cone of the location. This inhomogeneity has a retarded effect to the solution, and it is delayed with the wave velocity $c$.

In a shell

Moving mass $m$ with the velocity $v$ in a spherical shell with the mass $M$, the radius $r$ and its centre $\bigcirc$ in a Euclidean plane.

In the simplified example in the adjacent figure the source term is the linear mass density $\lambda _{m}$ that is not time-dependant and only exists in the circle of the outer rim with the constant radius $r$ and its centre at $\bigcirc$:

$x_{\bigcirc }=0$

$y_{\bigcirc }=0$

$x^{2}+y^{2}=r^{2}\rightarrow \lambda _{m}={\text{ constans}}$

In all other locations within the plane the linear mass density $\lambda _{m}$ is zero:

$x^{2}+y^{2}\neq r^{2}\rightarrow \lambda _{m}=0$

All mass elements outside the regarded plane have an symmetrical effect to the mass, and therefore, in these locations the contribution of the mass elements to the potential can be neglected for the determination of inhomogeneity:

$z\neq 0\rightarrow \lambda _{m}=0$

Furthermore, the mass element $\mathrm {d} M$ on the homogenous circumference $C$ of a circle with the radius $r$ is given by:

$\mathrm {d} M=\lambda _{m}\cdot \mathrm {d} C=\lambda _{m}\cdot r\cdot \mathrm {d} \alpha$

$M=\int dM=\int \limits _{0}^{2\pi }\lambda _{m}\cdot r\cdot \mathrm {d} \alpha =\lambda _{m}\cdot 2\pi \cdot r=\lambda _{m}\cdot C$

The cosine formula gives us the relation between the location of the mass point $m$ in the horizontal distance $x$ of the origin $\bigcirc$ of the circle with the radius $r$, when the mass element on the circle $\mathrm {d} M$ is in the direction of the angle $\alpha$ and the distance $s$:

$r^{2}=x^{2}+s^{2}-2xs\,\cos(\pi -\alpha )=x^{2}+s^{2}+2xs\,\cos \alpha$

In the normalised standard form of the quadratic equation, we get:

$s^{2}+2xs\,\cos \alpha +x^{2}-r^{2}=0$

The solution for the distance $s$ is:

$s={\sqrt {x^{2}\,\cos ^{2}\alpha -x^{2}+r^{2}}}-x\,\cos \alpha$

It is obvious that the following simplifications are valid:

• $x=0\rightarrow s=r$
• $\alpha =0\rightarrow s=r-x$
• $\alpha =\pm {\frac {\pi }{2}}\rightarrow s={\sqrt {x^{2}+r^{2}}}$
• $\alpha =\pi \rightarrow s=r+x$

Moving mass $m$ with the velocity $v$ in a spherical shell with the mass $M$ and the radius $r$ in a Euclidean plane. The origin of the coordinate system is shifted to the mass $m$.

The origin of the coordinate system can be also shifted to the mass $m$:

$x_{\bigcirc }=-x(t)$

$y_{\bigcirc }=0$

The retarded time $t_{\text{ret}}$ and the retarded potential $\Phi _{\text{ret}}({\vec {x}},t)$ are given as follows, where $c$ represents the propagation speed of the potentials:

$t_{\text{ret}}=t-{\frac {s}{c}}$

$\Phi _{\text{ret}}({\vec {x}},t)={\frac {1}{4\pi }}\int \limits _{0}^{2\pi }{\frac {v(s,t_{\text{ret}})}{s}}\,\mathrm {d} \alpha$

Illustration

In a simplified example we only consider the infinitesimally small angles $\mathrm {d} \alpha$ in the origin of a spherical shell with the radius $r$ and the areal density $\rho _{A}$.

Symmetrical geometry

If the mass $m$ is in the centre of a spherical shell with the radius $r=s_{0}$ we have the following situation:

• Retarded gravitational potentials for a symmetrical situation
• 
  The retarded potentials of the two symmetric mass elements $\mathrm {d} M_{0}\propto \mathrm {d} A_{0}$ that simultaneously act to the mass $m$.

• Both angle elements $\mathrm {d} \alpha$ are equal.
• Both distances $s_{0}$ are equal to $r$.
• Both areal elements $\mathrm {d} A_{0}$ are equal.
• Both mass elements $\mathrm {d} M_{0}$ are equal.
• The retarded times of the gravitational potentials of both mass elements $t_{ret}={\frac {s_{0}}{c}}={\frac {r}{c}}$ are equal.

In this situation the mass $m$ does not experience any acceleration in the classical approach (see above) or if its velocity is zero.

Moving masses

The situation is changed, if the masses are moving starting at $t=t_{0}$ within a time span of $\Delta t$. The mass $m$ moves with the velocity $v_{m}>0$ to the right and the two mass elements $\mathrm {d} M_{1}\propto \mathrm {d} A_{1}$ and $\mathrm {d} M'_{1}\propto \mathrm {d} A'_{1}$ move with the radial velocity $v_{M}>0$:

• Retarded gravitational potentials for an asymmetrical situation
• 
  The retarded potentials of the two asymmetric mass elements $\mathrm {d} M_{1}\propto \mathrm {d} A_{1}$ and $\mathrm {d} M'_{1}\propto \mathrm {d} A'_{1}$ that don't act simultaneously to the mass $m$.

At the time $t_{1}$ the mass has moved with the velocity $v_{m}>0$ the distance $\Delta s$ to the right:

$t_{1}=t_{0}+\Delta t$

$\Delta s=v_{m}\cdot \Delta t$

The spherical shell has expanded with the velocity $v_{M}>0$ and gained an increased radius:

$\Delta r=v_{M}\cdot \Delta t$

$s_{1}=s_{0}+\Delta s+\Delta r=s_{0}+v_{m}\cdot \Delta t+v_{M}\cdot \Delta t=s_{0}+(v_{M}+v_{m})\,\Delta t$

$s'_{1}=s_{0}-\Delta s+\Delta r=s_{0}-v_{m}\cdot \Delta t+v_{M}\cdot \Delta t=s_{0}+(v_{M}-v_{m})\,\Delta t$

Therefore:

$s_{1}>s'_{1}$

This means that the distance of mass element $\mathrm {d} M_{1}$ to the mass $m$ is always greater than the distance of mass element $\mathrm {d} M'_{1}$ to the mass $m$.

For the two retarded times for these distances to the location of $m$ we get:

$t_{{\text{ret}},1}=t_{1}-{\frac {s_{1}}{c}}$

$t'_{{\text{ret}},1}=t_{1}-{\frac {s'_{1}}{c}}$

With $t_{0}=0$ and therefore $t_{1}=\Delta t$:

$t_{{\text{ret}},1}=\left(1-{\frac {v_{M}+v_{m}}{c}}\right)\cdot \Delta t-{\frac {s_{0}}{c}}$

$t'_{{\text{ret}},1}=\left(1-{\frac {v_{M}-v_{m}}{c}}\right)\cdot \Delta t-{\frac {s_{0}}{c}}$

Therefore:

$t'_{{\text{ret}},1}>t_{{\text{ret}},1}$

This means that the retarded time for the mass element $\mathrm {d} M'_{1}$ is always later than the retarded time for the mass element $\mathrm {d} M_{1}$.

Common case

A moving mass m with the velocity $\color {Blue}v_{m}$ with the position $x_{m}(t)$ at the time $t$. Two further moving mass elements with the velocities $v_{M}$ and $-v_{M}$ with the positions $x(t_{r})$ and $x'(t'_{r})$ at the retarded times $t_{r}$ and $t'_{r}$. The propagation velocity of the interacting waves is $\color {OliveGreen}c$. The effective distances between the two mass elements and the mass m are $\color {OliveGreen}s(t)$ and $\color {OliveGreen}s'(t)$.

In the adjacent diagram there are three mass points that move in space. Their speed is given as follows:

• Outer mass element top (green): $v_{M}$
• Mass point in between (blue): $\color {Blue}v_{m}$
• Outer mass element botton (green): $-v_{M}$

Their time-depending location is given by these three functions for their x-coordinates:

$\color {Blue}x_{m}(t)=v_{m}\cdot t$

$\color {OliveGreen}x(t_{r})=-v_{M}\cdot t_{r}$

$\color {OliveGreen}x'(t'_{r})=v_{M}\cdot t'_{r}$

The time-depending effective distances $\color {OliveGreen}s(t)$ and $\color {OliveGreen}s'(t)$ between the outer mass elements and the mass point in between them is the difference of the corresponding x-coordinates and linked to the propagation velocity of the interacting waves $\color {OliveGreen}c$ as follows:

$\color {OliveGreen}s(t)=x_{m}(t)-x(t_{r})=(t_{r}-t)\cdot c$

$\color {OliveGreen}s'(t)=x'(t'_{r})-x_{m}(t)=(t'_{r}-t)\cdot c$

As a result, the corresponding retarded times $t_{r}$ and $t'_{r}$ for the two outer mass elemens are:

$t_{r}={\frac {c-v_{m}}{c+v_{M}}}\cdot t$

$t'_{r}={\frac {c+v_{m}}{c+v_{M}}}\cdot t$

And therefore:

$x(t_{r})=-v_{M}\cdot {\frac {c-v_{m}}{c+v_{M}}}\cdot t$

$x'(t'_{r})=v_{M}\cdot {\frac {c+v_{m}}{c+v_{M}}}\cdot t$

And:

$\color {OliveGreen}s(t)=\left(v_{M}\cdot {\frac {c-v_{m}}{c+v_{M}}}+v_{m}\right)\cdot t$

$\color {OliveGreen}s'(t)=\left(v_{M}\cdot {\frac {c+v_{m}}{c+v_{M}}}-v_{m}\right)\cdot t$

These are the time-depending effective distances for the gravitational potentials of the outer mass elements at their retarded times that have a simultaneous effect to the mass in between them at the time $t$.

For $t>0$ and $c>v_{M}>v_{m}>0$ and according to the diagram we can make the following assumption for the comparison of the effective distances:

$t'_{r}>t_{r}$

${\frac {c+v_{m}}{c+v_{M}}}\cdot t>{\frac {c-v_{m}}{c+v_{M}}}\cdot t$

$v_{m}>-v_{m}$ quod erat demonstrandum

This means that the retarded time of the upper mass element is always later than the retarded time of the lower mass element.

As well as:

$s(t)>s'(t)$

$v_{M}\cdot {\frac {c-v_{m}}{c+v_{M}}}+v_{m}>v_{M}\cdot {\frac {c+v_{m}}{c+v_{M}}}-v_{m}$

$v_{M}\cdot {\frac {-v_{m}}{c+v_{M}}}+v_{m}>v_{M}\cdot {\frac {v_{m}}{c+v_{M}}}-v_{m}$

$2\cdot v_{m}>2\cdot v_{M}\cdot {\frac {v_{m}}{c+v_{M}}}$

$1>{\frac {v_{M}}{c+v_{M}}}$ quod erat demonstrandum

Since $c+v_{M}$ is always greater than $v_{M}$, the assumption is proven.

This means that the effective distance of the moving mass in between them to the lower mass element is always greater than to the moving upper mass element. Finally, it can be stated that the absolute value of the retarded gravitational potential at the location of the moving mass in between them is always greater for the upper mass element than for the lower mass element, if both mass elements have the same value $M$:

$\Phi (t_{r})=-{\frac {G\cdot M}{s(t)}}$

$\Phi '(t'_{r})=-{\frac {G\cdot M}{s'(t)}}$

$|\Phi (t_{r})|<|\Phi '(t'_{r})|$

The moving mass $m$ experiences the corresponding retarded forces:

$F=-G\,{\frac {m\cdot M}{s(t)^{2}}}$

$F'=+G\,{\frac {m\cdot M}{s'(t)^{2}}}$

The net force to the mass $m$ is the sum of both:

$F_{net}=F'+F=G\cdot m\cdot M\cdot \left({\frac {1}{s'(t)^{2}}}-{\frac {1}{s(t)^{2}}}\right)$

For the acceleration $a$ of the mass $m$ we find:

$a={\frac {F_{net}}{m}}=G\cdot M\cdot \left({\frac {1}{s'(t)^{2}}}-{\frac {1}{s(t)^{2}}}\right)$

Since $s'(t)<s(t)$ the net force $F_{net}$ as well as the acceleration $a$ are positive, and the mass experiences an acceleration to positive x-values, i.e. in the direction of its movement.

Special case

Diagram for two retarded gravitational potentials of two moving masses with the positions $\color {Gray}x'(t)$ (grey) and $\color {Red}x(t)$ (red) that have an effect to another moving mass at the position $\color {DarkBlue}x_{m}(t)$ (dark blue). The positions of the two outer masses at the retarded times $\color {CornflowerBlue}t_{r}$ and $\color {Dandelion}t'_{r}$ are indicated in light blue ($\color {CornflowerBlue}x'(t'_{r})$) and orange ($\color {Dandelion}x(t_{r})$). The path of the gravitational waves is indicated by the diagonal dashed green lines.

Let us have a look at the following special case:

$c=1$

$v_{M}={\frac {1}{2}}$

$v_{m}={\frac {1}{4}}$

Their time-depending location is given by these three functions for their x-coordinates:

$\color {DarkBlue}x_{m}(t)=v_{m}\cdot t={\frac {t}{4}}$

$\color {Red}x(t)=-v_{M}\cdot t=-{\frac {t}{2}}$

$\color {Gray}x'(t)=+v_{M}\cdot t=+{\frac {t}{2}}$

The time-depending effective distances $\color {Green}s(t)\color {Black}$ and $\color {Green}s'(t)\color {Black}$ between the outer mass elements and the mass point in between them is the difference of the corresponding x-coordinates:

$\color {Green}s(t)=x_{m}(t)-x(t_{r})={\frac {t}{4}}+{\frac {t_{r}}{2}}\color {Black}$

$\color {Green}s'(t)=x'(t'_{r})-x_{m}(t)={\frac {t'_{r}}{2}}-{\frac {t}{4}}\color {Black}$

As a result, the corresponding retarded times $\color {Dandelion}t_{r}$ and $\color {CornflowerBlue}t'_{r}$ for the two outer mass elemens are:

$\color {Dandelion}t_{r}={\frac {\frac {3}{4}}{\frac {3}{2}}}\cdot t={\frac {1}{2}}\cdot t$

$\color {CornflowerBlue}t'_{r}={\frac {\frac {5}{4}}{\frac {3}{2}}}\cdot t={\frac {5}{6}}\cdot t$

Therefore:

$\color {Dandelion}x(t_{r})=-{\frac {1}{2}}\cdot {\frac {\frac {3}{4}}{\frac {3}{2}}}\cdot t=-{\frac {1}{4}}\cdot t$

$\color {CornflowerBlue}x'(t'_{r})={\frac {1}{2}}\cdot {\frac {\frac {5}{4}}{\frac {3}{2}}}\cdot t={\frac {5}{12}}\cdot t$

$\color {Dandelion}s(t)=\left({\frac {1}{2}}\cdot {\frac {\frac {3}{4}}{\frac {3}{2}}}+{\frac {1}{4}}\right)\cdot t={\frac {1}{2}}\cdot t$

$\color {CornflowerBlue}s'(t)=\left({\frac {1}{2}}\cdot {\frac {\frac {5}{4}}{\frac {3}{2}}}-{\frac {1}{4}}\right)\cdot t={\frac {1}{6}}\cdot t$

Thought experiment

In a thought experiment we look at the following situation, where $c$ is the propagation speed of the gravitational waves. The time line starts at $t=0$, and the effect of the retarded potentials is synchronised with the occurence of the moving mass. The left moving mass element is regarded at $t=0$ and $t=1$, the right moving mass element is regarded at $t=0$ and $t=2$, whilst the moving mass $m$ is regarded at $t=8$ and $t=11$, when the retarded gravitational potentials have their effect to the mass.

$v_{M}=c$

$v_{m}={\frac {v_{M}}{3}}$

$s=c\cdot t$

In the following diagram the velocity of the waves is normalised and used without unit:

$c=1$

And therefore, only for simplification and without any units, too:

$s=t$

• Retarded gravitational potentials at different points in time
• 
  Retarded potentials of the two expanding mass elements with the mass $\mathrm {d} M$. These mass elements move on the left hand side from $\mathrm {d} A_{0}$ at $t=0$ to $\mathrm {d} A_{1}$ at $t=1$), and on the right hand side from $\mathrm {d} A_{0}$ at $t=0$ to $\mathrm {d} A_{2}$ at $t=2$. They simultaneously act to the mass $m$ at $t_{0}=8$ (mass in grey, path for gravitational waves in black) and at $t'=11$ (mass in red, path for gravitational waves in blue). The numbers in the figure give the time and the length in steps without units.

The effective distances $s_{0}$ for the gravitational potential at $t_{0}=8$ are both equal:

$s_{0}=t_{0}-t_{\text{ret,0}}=8$

At the time $t'=11$ the mass experiences the retarded potentials on the left-hand side (distance to mass element is $s_{1}$) and at the right-hand side (distance to mass element is $s_{2}$):

$s_{1}=t'-t_{\text{ret,1}}=10$

$s_{2}=t'-t_{\text{ret,2}}=9$

All inhomogeneities contribute to the retarded potential at the location of the mass with the value they had at the retarded times $t_{\text{ret,0}}$, $t_{\text{ret,1}}$ and $t_{\text{ret,2}}$:

$t_{\text{ret,0}}=t_{0}-{\frac {s_{0}}{c}}=8-{\frac {8}{1}}=0$, this corresponds to the effective time of the two area elements $\mathrm {d} A_{0}$.

$t_{\text{ret,1}}=t'-{\frac {s_{1}}{c}}=11-{\frac {10}{1}}=1$, this corresponds to the effective time of the area element $\mathrm {d} A_{1}$ on the left.

$t_{\text{ret,2}}=t'-{\frac {s_{2}}{c}}=11-{\frac {9}{1}}=2$, this corresponds to the effective time of the area element $\mathrm {d} A_{2}$ on the right.

For a mass $m$ moving from the centre of a shell to the right the angle element $\mathrm {d} \alpha _{1}$ to the left becomes smaller than the original angle element $\mathrm {d} \alpha _{0}$, and the angle element $\mathrm {d} \alpha _{2}$ to the right becomes greater than the original angle element $\mathrm {d} \alpha _{0}$:

$\mathrm {d} \alpha _{2}>\mathrm {d} \alpha _{0}>\mathrm {d} \alpha _{1}$

The following applies to the appropriate mass elements:

$\mathrm {d} M_{0}=\mathrm {d} M_{1}=\mathrm {d} M_{2}=\rho _{a}\cdot \mathrm {d} A_{0}=\rho _{a}\cdot 4\pi \cdot s_{0}^{2}\cdot \mathrm {d} \alpha _{0}^{2}$

For the net force $\mathrm {d} F$ to the mass $m$:

$\mathrm {d} F=\mathrm {d} F_{2}+\mathrm {d} F_{1}=G\cdot m\cdot \mathrm {d} M_{0}\cdot \left({\frac {1}{s_{2}^{2}}}-{\frac {1}{s_{1}^{2}}}\right)$

For the acceleration $\mathrm {d} a$ of the mass $m$ we find:

$\mathrm {d} a={\frac {\mathrm {d} F}{m}}=G\cdot \mathrm {d} M_{0}\cdot \left({\frac {1}{s_{2}^{2}}}-{\frac {1}{s_{1}^{2}}}\right)$

Since $s_{2}<s_{1}$ the net force $\mathrm {d} F$ is positive, and the mass experiences an acceleration $\mathrm {d} a$ to the right, i.e. in the direction of its movement. This result is absolutely inline with the findings above in the section "Common case" above.

Furthermore, it is noteworthy to state that the acceleration $\mathrm {d} a$ is proportional to the areal density $\rho _{a}$ of the expanding shell:

$\mathrm {d} a\propto \rho _{a}$

Nevertheless it should be noted that the areal density is decreasing with the expansion and the increasing radius of the outer shell.

---

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 linear mass 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 $M_{eff}$ 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 in first approximation for the gravitational forces behind the very fast-moving moving mass $m$. If the distance $d$ between the mass $m$ and the shell is given, we can carry out an estimation for 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 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:

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

The ratio of the retarded gravitational forces in the distance of oberserved objects versus the corresponding redshifts

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. This distance is corresponding to a redshift of about 7.5.

In the diagram on the left the ratio of the two forces is plotted against the redshift which is observed at corresponding distances. The high values of redshift at far distant objects seem to be dominated by the gravitaional redshift, whereas at shorter distances the redshift is dominated by the Doppler effect.

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
$\lambda _{M}$ = linear 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 with the linear mass density $\lambda _{M}$ in front of the mass $m$ 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 might only be 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: $R=1.36\cdot 10^{26}{\text{m}}=14,4\cdot 10^{9}{\text{ly}}$ (14.4 billion light-years)

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

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

$M_{arc}=\int _{-\alpha _{R}}^{+\alpha _{R}}\mathrm {dM} =\int _{-\alpha _{R}}^{+\alpha _{R}}R\,\lambda _{M}\,\mathrm {d\alpha } =2\,\int _{0}^{\alpha _{R}}R\,\lambda _{M}\,\mathrm {d\alpha } =2\,R\,\lambda _{M}\,\alpha _{R}$

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 Pythagorean theorem gives the following result for the half chord of the circle:

$x={\sqrt {2\,R\,d-d^{2}}}$

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 $m$

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

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

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

The angle $\beta$ as seen from the moving mass m to the infinitesimal mass element $\mathrm {dM}$ on the arc can be derived by applying the law of sines and 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 vertical components of the gravitational forces are symetrical, and therefore, their net effect is zero. The net horizontal force of this arc $F_{hor}$ that is accelerating the mass $m$ in horizontal direction can be achieved by integrating a semi-circle with the correction factor $\cos \beta$:

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

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:

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

We finally get the effective mass $M_{eff}$ that is acting via gravitational forces on the mass $m$ in the distance $d$:

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

Schwarzschild distance

The Schwarzschild distance $d_{S}$ of this effective mass $M_{eff}$ is equal to the Schwarzschild 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 $d_{S}$ of every shell with the mass $M_{S}$. With the following condition we can get the solutions with $d_{S}=d$, where $z(d_{S})=\infty$:

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

Therefore $d_{S}$ can be determined for a given $\lambda _{M}$:

$d_{S}(\lambda _{M})={\frac {2\,G}{c^{2}}}M_{eff}(M_{S},d_{S})={\frac {2\,G\,R\,\lambda _{M}\,d_{S}^{2}}{c^{2}}}\int _{-{\frac {\pi }{2}}}^{+{\frac {\pi }{2}}}{\frac {d_{S}-R\,(1-\cos \alpha )}{{\left(2R^{2}-2\,R\,d_{S}+d_{S}^{2}-2\,R\,(R-d_{S})\,\cos \,\alpha \right)}^{\frac {3}{2}}}}\,\mathrm {d\alpha }$

The Schwarzschild distance $d_{S}$ can be interpreted as the distance between the surface of the Hubble sphere with the Hubble radius $R$ and the inner Schwarzschild sphere of the black shell, which is the visible limit of the universe. The Hubble radius $R$ would be the distance between an observer in the centre of the universe and all objects that are receding from him at the speed of light. It can be expressed by the Hubble time $t_{H}$:

$R=t_{H}\cdot c=14.4\cdot 10^{9}\,{\text{s}}\cdot {c}=14.4\cdot 10^{9}\,{\text{ly}}=1.36\cdot 10^{26}\,{\text{m}}$

• Redshift z as a function of the distance
• 
  Redshift z as a function of the distance of the object from the centre of the universe within the Schwarzschild limit, which asymptotically approaches the Hubble radius for large redshifts.
• 
  Circles with a given redshift z within the Hubble circle (dashed line) with the Hubble radius.

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 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 $R_{v}=R-d_{S}$.
• Mass of the universe: $M_{universe}=2.97\cdot 10^{53}{\text{kg}}$:

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

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

→ See appendix: table with results for different values of $\lambda _{M}$ numerically computed by a Java program.

In this model the mass $M_{S}$ of the invisible outer black shell continuously increases by absorbing mass from the visible universe, due to the net forces of the retarded gravitational potentials and the corresponding acceleration in direction of the black shell. More and more matter is moving from the visible universe behind the event horizon, where it becomes invisible and unaccessible, but leaves its gravitational action to the visible part of the universe. Furthermore, the mass of the black shell is in the same magnitude as the mass of the visible universe even in the earlier cosmical ages.

• 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 Schwarzschild distances as well as 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 Schwarzschild distance was grewing much faster than the mass of the black shell around $M_{S}\approx 0.96865\cdot M_{universe}$.
• 
  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 size of the visible universe began to shrink very suddenly at the time around $M_{S}\approx 0.96865\cdot M_{universe}$.

It is very noteworthy to recapitulate that the age of the cosmic microwave background with its afterglow light pattern is about 380,000 years (the corresponding Schwarzschild distance belongs to a shell mass $M_{S}=2.8769\cdot 10^{53}\,{\text{kg}}=0.96865\cdot M_{universe}$, Schwarzschild distance $d_{S}=377\cdot 10^{3}\,{\text{ly}}=3.56\cdot 10^{21}\,{\text{m}}=2.62\cdot 10^{-5}\cdot R$), exactly where the radius of the visible universe had significantly begun to decrease in relation to the Hubble radius (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 (the corresponding Schwarzschild distance belongs to a shell mass $M_{S}=2.92\cdot 10^{53}\,{\text{kg}}=0,984\cdot M_{universe}$, Schwarzschild distance $d_{S}=293\cdot 10^{6}\,{\text{ly}}=2.78\cdot 10^{24}\,{\text{m}}=0.020\cdot R$), where the conversion from dark matter to "dark energy" during the so-called "dark ages" was finished.

At the break-even point, where the masses of the visible universe and the black shell have the same value, the Schwarzschild distance is about 860 million light-years (it belongs to a shell mass $M_{S}=2.97\cdot 10^{53}\,{\text{kg}}=M_{universe}$).

Schwarzschild radius of the universe

Simplified diagram of the radius $r$ of the universe over time from origin at $t=0$ to time of universe $t_{U}$. The particle horizon is constantly expanding with the speed of light $c$:

Radius of end of inflationary period (Schwarzschild radius of universe):
$r_{S,universe}=t_{U}\cdot c={\frac {2\,G\,M_{universe}}{c^{2}}}\approx 14283{\text{ Mpc}}\approx 46,6\cdot 10^{9}{\text{ ly}}$

Co-moving radius of the cosmic microwave background (CMB):
$r_{CMB}\approx 14000{\text{ Mpc}}\approx 45,7\cdot 10^{9}{\text{ ly}}$

Hubble radius at Hubble time $t_{H}$:
$r_{H}=t_{H}\cdot c\approx 4480{\text{ Mpc}}\approx 14,6\cdot 10^{9}{\text{ ly}}$

Radius of visible universe as seen from observer:
$r_{V}\approx 4230{\text{ Mpc}}\approx 13,8\cdot 10^{9}{\text{ ly}}$

Time of the observer: $t_{observer}=2\cdot t_{H}$
Time of the visible event horizon: $t_{v}=2\cdot t_{H}-{\frac {r_{V}}{c}}>t_{H}$

The Schwarzschild radius of the visible universe $r_{S,universe}$ can be computed by its mass, too:

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

This value fits very well with the radius of the invisible particle horizon at the end of inflationary period of the universe today. The question is whether this coincidence is the case by accident or due to varying values of the mass of the visible universe $M_{universe,visible}$ or the gravitational constant $G$. Of course it is disputable, whether the mass of the visible universe and the gravitational constant are constant on large time scales. However, the British mathematician and astrophysicist Edward Arthur Milne (1896–1950) published already in 1935 in his book on "Relativity, Gravitation and World-Structure" the possibility that the gravitational constant could be proportional to the time since the creation of the universe $t$ and reciprocal to the mass of the visible universe $M_{universe,visible}$.

$G={\frac {c^{3}}{M_{universe,visible}}}\,t$

According to Edward Arthur Milne the cosmological constant $\Lambda$ is in this model:

$\Lambda ={\frac {3}{c^{2}\,t^{2}}}$

Today this value is approximately $\Lambda =1.7\cdot 10^{-53}{\text{ m}}^{-2}$, which is about 6.6 times smaller than the current value of the standard model of cosmology known as the ΛCDM model (CDM = cold dark matter).

The time-dependent relation for the gravitational constant would mean that the gravitational constant is grewing with the following rate, if the mass of the universe keeps constant:

${\frac {\mathrm {d} G}{\mathrm {d} t}}={\frac {c^{3}}{M_{universe,visible}}}\approx 9.1\cdot 10^{-29}{\frac {{\text{m}}^{3}}{{\text{kg s}}^{3}}}$

As a result the gravitational constant would change in the following magnitudes, which are much smaller than the uncertainty of the measurements of the gravitational constant of around $2.2\cdot 10^{-5}$:

${\frac {\frac {\mathrm {d} G}{\mathrm {d} t}}{G}}\approx 1.36\cdot 10^{-18}{\frac {1}{\text{s}}}\approx 4.3\cdot 10^{-11}{\frac {1}{\text{a}}}$

Only after half a million of years the corresponding change would be in the magnitude of the uncertainty.

If we substitute the expression of Milne for the gravitational constant in the formula for the Schwarzschild radius of the universe, we get:

$r_{S,universe}={\frac {2\,c^{3}\,M_{universe}}{c^{2}\,M_{universe,visible}}}\,t$

Assuming that the whole universe is visible, and therefore, $M_{universe}=M_{universe,visible}$, this value would be two times larger than the value of the Schwarzschild radius computed by the mass of the universe:

$r_{S,universe}=2\,c\,t$

It is noteworthy to mention that according to the results the following estimates apply for Schwarzschild distances of around 858 million light-years, where the ratio of the mass of black shell to the mass of visible universe was equal to one:

$M_{universe,visible}\approx M_{universe,invisible}=M_{S}$

This Schwarzschild distance is similar to the differences of the particle horizon and the plane of the cosmic microwave background (CMB) that is determined by the ΛCDM model that is preferred by most cosmologists today:

$r_{S,universe}-r_{CMB}\approx 0.9\cdot 10^{9}{\text{ ly}}$

$r_{H}-r_{V}\approx 0.8\cdot 10^{9}{\text{ ly}}$

Therefore:

$M_{universe,total}=M_{universe,visible}+M_{universe,invisible}\approx 2\,M_{universe,visible}\approx 2\,M_{S}$

This assumption is supported by the fact that the mass of ordinary matter within radius of the visible universe $r_{visible,universe}$ has the following value that can be computed by its density $\rho _{ordinary\,matter}$ and the corresponding volume $V_{visible,universe}$ yielded by the Planck Survey of 2013:

$\rho _{ordinary\,matter}=4.08\cdot 10^{-28}{\frac {\text{kg}}{{\text{m}}^{3}}}$

$r_{visible,universe}=4.3\cdot 10^{26}{\text{ m}}$

$V_{visible,universe}={\frac {4}{3}}\,\pi \,r_{visible,universe}^{3}=3.3\cdot 10^{80}{\text{ m}}^{3}$

$M_{ordinary\,matter}=\rho _{ordinary\,matter}\cdot V_{visible,universe}=1.4\cdot 10^{53}{\text{ kg}}\approx {\frac {1}{2}}\,M_{universe}={\frac {1}{2}}\,2.97\cdot 10^{53}{\text{ kg}}\approx 1.5\cdot 10^{53}{\text{ kg}}$

However, if the gravitational constant would be given by the following relation using the total mass of the universe $M_{universe,total}$ including the black shell, the values would match much better:

$G={\frac {c^{3}}{M_{universe,total}}}\,t\approx {\frac {c^{3}}{2\,M_{universe,visible}}}\,t$

$r_{S,universe}\approx {\frac {2\,c^{3}\,M_{universe,visible}}{2\,c^{2}\,M_{universe,visible}}}\,t=c\,t$

This value is identical to the radius of the particle horizon of our universe. In this case the change of the gravitational constant would be in the magnitude of the uncertainty only after a million years. The total mass of the universe would not change or influence the gravitational constant even if any mass flow from the visible part of the universe to the invisible outer black shell would decrease the mass of the visible part of the universe.

The following applies for the hypothetical case that the Schwarzschild radius of the universe is only determined by the ordinary mass of the visible part of the universe, which is only half of the total mass of the universe. The Schwarzschild radius of the visible universe with the mass $M_{ordinary\,matter}$ that is completely visible to an observer in the centre of the universe, causes any light originating from the centre of the universe being invisible to any observer outside of this sphere, i.e. in the outer black shell with the similar mass $M_{S}$. Vice versa, any light originating from the outer black shell is invisible to any observer in the centre of the universe. If the these two regions share the very same spherical boundary surface, this surface would separate two worlds that obviously cannot exchange any information. According to this assumption, the Schwarzschild radius of our universe would define this spherical boundary surface and would be around 860 million light-years smaller than the radius of the particle horizon respectively the outer radius of the black shell.

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Conclusion

The considerations described above show that retarded gravitational potentials can contribute to the explanation of the accelerated expansion of the visible part of the universe, if the outer spherical black shell of the universe would consist of expanding matter with a huge mass. This would also hold in a steady state universe and even if the black shell would not move at all. In the latter case any moving masses within the black shell would reach the black shell in finite time.

It can also be considered that the effect of the retarded gravitational potentials will accelerate moving objects the more, the faster they move and the closer they are to the black shell. Furthermore, also objects within smaller spherical shells that not only are surrounded by the black shell, but also by other visible objects, will experience the retarded gravitational potentials of these objects, and therefore, they will experience an additional acceleration which is directed outwards. This all leads to an inhomogeneous evolution of the mass density in particular during the early development stages of an inflating universe:

• Evolvement of the mass density in an expanding spherical universe with constant mass
• 
  Evolvement of the universe with instant gravitational potentials in a homogeneous sphere.
• 
  Evolvement of the universe with retarded gravitational potentials. Due to the acceleration of matter in the direction of the outer rim of the universe, a shell is growing with a thickness corresponding to its Schwarzschild distance (bluish). This matter within this shell cannot emit any electromagnetic radiation or send any information back to the centre of the universe, since the strong gravitational forces keep the radiation and all masses within the shell.

The agglomeration of mass in the outer areas of our universe not only leads to a shell with increased mass density, but the high mass density could also be the seed for primordial black holes, which could be a reason for the formation of very early galaxies observed by the James Webb Space Telescope (JWST).

Computing the corresponding and permanently increasing Schwarzschild distances gives three remarkable points:

1. Today we can observe the cosmic microwave background at a cosmic age of 380,000 years, where the Schwarzschild distance was increasing much faster than the mass of the black shell.
2. The most distant known galaxy JADES-GS-z14-0 has a cosmic age of 290 million years, where the increase of the Schwarzschild distance with the mass of the black shell became much smaller.
3. The Schwarzschild distance for the equality of the masses of the visible universe and the black shell is 860 million light-years. This value is rather close to the distances that are computed by the standard ΛCDM model between the particle horizon that expands with speed of light and the visible horizon that can be observed as the cosmic microwave background.

Cosmologists call the period in between the creation of the cosmic microwave background and the first stars as the dark ages, where dark matter is supposed to be transformed into "dark energy". So far, the nature of "dark energy" is unclear, but the increase of the mass of an invisible black shell behind the visible universe could contribute to find explanations.

Due to Newton's third law (actio = reactio) the black shell experiences the same retarded gravitational forces as the moving masses within the shell. These forces will cause a deceleration of the shell which is the stronger the closer the masses within the visible universe are to it.

The high velocity of visible objects in the vincity of the black shell as well as the huge mass of the matter in front of them seem to be the reasons for the extreme redshift of their light that can be observed by us. The observation that this redshift is greater than expected for distant objects led to the assumption that the further away these objects are, the more they are accelerated. Another reason for this could be found in the effects of the retarded gravitational potentials of a massive black shell. Furthermore, the additional gravitational redshift leads to a higher value than expected only by the relativistic Doppler effect. This is in accordance with the observations of very far and young galaxies such as JADES-GS-z14-0 or of the cosmic microwave background (CMB).

However, the computation of the equations of motion based on retarded gravitational forces becomes very expensive, if the black shell is neither homogeneous, nor spherical or symmetrical, or if the concept of spacetime has a structure based on a non-Euclidean geometry. Furthermore, relativistic effects (including the transverse Doppler effect) could cause time dilatation or mass increase that would have to be considered, too. Finally, is would be necessary to consider the loss of mass in the visible universe over the time, since a significant amount of visible matter may have traversed the event horizon that is caused and built by the huge mass in the black shell of the universe.

If matter and antimatter were created in equal parts during the Big Bang, it would be conceivable that the visible universe consists of the remaining matter and the black shell of the remaining antimatter. This could be a reason for the observed baryon asymmetry. Since we cannot obtain any information from the black shell, antimatter would not be observable for us. Furthermore, any electromagnetic radiation from any source including the annihilation of matter and antimatter that leaves the visible universe by passing the limit of the Schwarzschild distance into the black shell cannot return to us.

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Appendix

Table with computed results

In the following table you will find different computed values based on the linear mass density of the black shell $\lambda _{M}$. The values had been numerically computed by a Java program.

The Schwarzschild distance that is corresponding to the Hubble radius of the universe during the generation of the cosmic microwave background (CMB) at around 380,000 light-years (ly) is marked in bold.

The same applies to a Schwarzschild distance of 858 million light-years, where the ratio between the mass of visible universe to the mass of the black shell is equal to one. This distance corresponds to the distance between the particle horizon and the visible horizon of the cosmic microwave background (CMB) according to the ΛCDM model.

Note that to 380,000 years after the Big Bang, the slope of the Schwarzschild radius in relation to the slightly increasing linear mass density was very high. The Schwarzschild distance was grewing much faster than the mass of the black shell during this period.

Linear mass density $\lambda _{M}$ in ${\frac {\text{kg}}{\text{m}}}$	Mass of black shell $M_{S}$ in ${\text{kg}}$	Mass of black shell in units of mass of visible universe ${\frac {M_{S}}{M}}$	Effective mass of black shell $M_{eff}$ in ${\text{kg}}$	Schwarzschild distance $d_{S}$ in ${\text{m}}$	Schwarzschild distance in units of Hubble radius ${\frac {d_{S}}{R}}$	Schwarzschild distance $d_{S}$ in ${\text{ly}}$	Radius of the visible universe in units of the Hubble radius ${\frac {R-d_{S}}{R}}$
3.36663E+26	2.876830E+53	0.968630	1.67E+48	2.48E+21	1.83E-05	262,000	0.999982
3.36664E+26	2.876838E+53	0.968632	1.78E+48	2.64E+21	1.94E-05	279,000	0.999981
3.36665E+26	2.876847E+53	0.968635	1.90E+48	2.83E+21	2.08E-05	299,000	0.999979
3.36666E+26	2.876855E+53	0.968638	2.02E+48	3.00E+21	2.21E-05	317,000	0.999978
3.36667E+26	2.876864E+53	0.968641	2.15E+48	3.19E+21	2.35E-05	337,000	0.999977
3.36668E+26	2.876873E+53	0.968644	2.28E+48	3.39E+21	2.49E-05	358,000	0.999975
3.36669E+26	2.876881E+53	0.968647	2.40E+48	3.56E+21	2.62E-05	377,000	0.999974
3.3667E+26	2.876890E+53	0.968650	2.53E+48	3.76E+21	2.77E-05	398,000	0.999972
3.3668E+26	2.876975E+53	0.968678	3.83E+48	5.69E+21	4.19E-05	602,000	0.99996
3.3669E+26	2.877060E+53	0.96871	5.19E+48	7.71E+21	5.67E-05	815,000	0.99994
3.367E+26	2.877146E+53	0.96874	6.60E+48	9.80E+21	7.20E-05	1,040,000	0.99993
3.368E+26	2.878000E+53	0.96902	2.24E+49	3.32E+22	2.44E-04	3,510,000	0.9998
3.369E+26	2.878855E+53	0.96931	4.03E+49	5.98E+22	4.40E-04	6,320,000	0.9996
3.37E+26	2.879709E+53	0.96960	5.96E+49	8.85E+22	6.51E-04	9,360,000	0.9993
3.38E+26	2.88826E+53	0.9725	3.05E+50	4.53E+23	3.33E-03	47,900,000	0.997
3.39E+26	2.89680E+53	0.9754	6.17E+50	9.16E+23	6.74E-03	96,800,000	0.993
3.40E+26	2.9053E+53	0.9782	9.84E+50	1.46E+24	1.07E-02	154,000,000	0.989
3.41E+26	2.9139E+53	0.9811	1.40E+51	2.08E+24	1.53E-02	220,000,000	0.985
3.42E+26	2.9224E+53	0.9840	1.87E+51	2.78E+24	2.04E-02	293,000,000	0.980
3.43E+26	2.9310E+53	0.9869	2.39E+51	3.55E+24	2.61E-02	375,000,000	0.974
3.44E+26	2.9395E+53	0.990	2.96E+51	4.39E+24	3.23E-02	464,000,000	0.968
3.45E+26	2.9481E+53	0.993	3.58E+51	5.32E+24	3.91E-02	563,000,000	0.961
3.46E+26	2.9566E+53	0.995	4.27E+51	6.34E+24	4.66E-02	670,000,000	0.953
3.47E+26	2.9652E+53	0.998	5.01E+51	7.44E+24	5.47E-02	787,000,000	0.945
3.47566E+26	2.97000E+53	1.00000	5.46E+51	8.11E+24	5.97E-02	858,000,000	0.940
3.48E+26	2.9737E+53	1.001	5.83E+51	8.65E+24	6.36E-02	915,000,000	0.936
3.49E+26	2.9823E+53	1.004	6.71E+51	9.97E+24	7.33E-02	1,050,000,000	0.927

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Java program

The following Java program can be used to numerically compute the Schwarzschild distances for different linear mass densities. It implements a numerical integration for the effective masses as well as the numerical solvation for the Schwarzschild distances.

/*
  Source file: SchwarzschildDistance.java
  Program: Computation of the Schwarzschild Distance within the Hubble sphere
  Autor: Markus Bautsch
  Location: Berlin, Germany
  Licence: public domain
  Date: 18th July 2024
  Version: 1.3
  Programming language: Java
 */

/*
  This Java-Programm computes Schwarzschild distances for any black shell masses.
  The black shell is at the outer rim of the Hubble space.
 */

public class SchwarzschildDistance
{
	// Class constants
	final static double G = 6.67430e-11; // Gravitational constant in cube metres per kilogram and square second
	final static double c = 2.99792458e8; // Speed of light in metres per second
	final static double TropicalYear = 365.24219052 * 24 * 3600; // Tropical year in seconds
	final static double LightYear = c * TropicalYear; // Light-year in metres
	final static double R = 1.36e26; // Hubble length in metres
	final static double MUniverse = 2.97e53; // Mass of the visible universe in kilograms

	final static double lambdaMfirst = 3.36663e26; // start value for linear mass density of black shell
	final static double lambdaMlast  = 3.36670e26; // startop value for linear mass density of black shell
	final static double deltaLambda  = 0.00001e26; // step size for linear mass density of black shell

	// This method computes and returns the distance s of a mass m to a mass point dM in the black shell
	// distance is the closest distance of the mass m to to the mass point dM in the black shell in m
	// alpha is the angle beweteen the mass m and the mass point dM in the black shell as seen from the centre of the universe in rad
	private static double distanceToMassElementShell (double distance, double alpha)
	{
		double cosAlpha = java.lang.Math.cos (alpha);	
		double squareDistance = distance * distance;
		double argument = 2*R*R - 2*R*distance + squareDistance - 2*R*(R-distance)*cosAlpha;
		double s;			
		if (argument <= squareDistance) // because of possible rounding errors
		{
			s = distance;
		}
		else
		{
			s = java.lang.Math.sqrt (argument);
		}
		return s;
	}

	// This method computes and returns the angle of a half chord x in m as seen from the centre of the universe in rad
	private static double alpha (double x)
	{
		double alpha = java.lang.Math.asin (x / R);
		return alpha;
	}

	// This method computes and returns the integrand of the integral
	// distance is the closest distance of the mass m to to the mass point dM in the black shell in m
	// alpha is the angle beweteen the mass m and the mass point dM in the black shell as seen from the centre of the universe in rad
	private static double integrand (double distance, double alpha)
	{
		double s = distanceToMassElementShell (distance, alpha);
		double e = R * (1 - java.lang.Math.cos (alpha)); // the auxiliary sagitta e
		double cosBeta = (distance - e) / s;
		double integrand = cosBeta / s / s;
		return integrand;
	}

	// This method computes and returns the effective mass from 0 to alphaR for a given lambdaM and a given distance
	// lambdaM is the linear mass density of the black shell in kg/m
	// distance is the closest distance of the mass m to to the mass point dM in the black shell in m
	private static double integrateEffectiveMass (double lambdaM, double distance)
	{
		final long steps = 1000000; // number of steps for numerical integration
		double squareDistance = distance * distance;
		double xMax = java.lang.Math.sqrt (2*R*distance - squareDistance); // maximum half chord for a given maximum angle alphaR
		double x = xMax; // x runs from xMax to 0 in steps of deltaX
		double alpha = alpha (x); // alpha runs from alphaR to 0
		double integrand = integrand (distance, alpha);
		double integral = 0;
		long step = 0;
		while (step <= steps)
		{
			x = xMax * (1 - (double) step / steps);
			double alphaNext = alpha (x);
			double integrandNext = integrand (distance, alphaNext);
			double deltaAlpha = alpha - alphaNext;
			double averageIntegrand = (integrandNext + integrand) / 2;
			double area = averageIntegrand * deltaAlpha; // numerical integration
			integral = integral + area;
			alpha = alphaNext;
			integrand = integrandNext;
			step++;
		}
		double effectiveMass = R * lambdaM * squareDistance * (2*integral);
		return effectiveMass;
	}

	// This method computes and returns the Schwarzschild distance d_S for a given effective mass mEff
	private static double schwarzschildDistance (double effectiveMass)
	{
		double schwarzschildDistance = 2 * G * effectiveMass / c / c;
		return schwarzschildDistance;
	}

	// This method computes and returns the effective mass mEff for a given Schwarzschild distance d_S
	private static double effectiveMass (double schwarzschildDistance)
	{
		double effectiveMass = schwarzschildDistance * c * c / 2 / G;
		return effectiveMass;
	}

	// This method solves and returns the Schwarzschild distance d_S for a given lambdaM
	// lambdaM is the linear mass density of the black shell in kg/m
	private static double solve (double lambdaM)
	{
		final double limit = 5e-16; // for relative precision of determination of Schwarzschild distance
		double lowerDistance = 1e20; // first low guess for Schwarzschild distance
		double upperDistance = 1e25; // first high guess for Schwarzschild distance
		double delta; // delta shall become smaller than limit
		double distance;
		double schwarzschildDistance;
		double deltaDistance;
		do
		{
			distance = (lowerDistance + upperDistance) / 2;
			double effectiveMass = integrateEffectiveMass (lambdaM, distance);
			schwarzschildDistance = schwarzschildDistance (effectiveMass);
			deltaDistance = (distance - schwarzschildDistance) / schwarzschildDistance;
			if (deltaDistance < 0) // Schwarzschild distance too large
			{
				lowerDistance = (upperDistance + lowerDistance) / 2;
			}
			else // Schwarzschild distance too small
			{
				upperDistance = (upperDistance + lowerDistance) / 2;
			}
			delta = (upperDistance - lowerDistance) / lowerDistance;
		} while (delta > limit);
		if (java.lang.Math.abs (deltaDistance) > 1e-6)
		{
			java.lang.System.out.println ("unstable numerical result:");
			java.lang.System.out.println ("Computed      distance = " + distance);
			java.lang.System.out.println ("Schwarzschild distance = " + schwarzschildDistance);
		}
		return schwarzschildDistance;
	}

	// This method outputs the table header
	private static void printHeader ()
	{
		java.lang.System.out.print ("lambda_M in kg/m;");
		java.lang.System.out.print ("   M_S  in kg   ;");
		java.lang.System.out.print ("   M_S/M        ;");
		java.lang.System.out.print ("   M_eff in kg  ;");
		java.lang.System.out.print ("   d_S im m     ;");
		java.lang.System.out.print ("   d_S/R        ;");
		java.lang.System.out.print ("   d_S in ly    ;");
		java.lang.System.out.print ("   (R-d)/R      ");
		java.lang.System.out.println ();
	}

	// This method outputs a table result line
	// lambdaM is the linear mass density of the black shell in kg/m
	// distance is the closest distance of the mass m to to the mass point dM in the black shell in m
	// mEff is the effective mass of a sphere in the black shell
	private static void printResults (double lambdaM, double distance, double effectiveMass)
	{
		double mShell = lambdaM * R * 2 * java.lang.Math.PI; // mass of black shell
		double mRatio = mShell / MUniverse;
		double dRatio = distance / R;
		double dLightYears = distance / LightYear;
		double deltaRatio = (R - distance) / R;
		java.lang.System.out.printf ("%15.7e", lambdaM);
		java.lang.System.out.print (" ;");
		java.lang.System.out.printf ("%15.6e", mShell);
		java.lang.System.out.print (" ;");
		java.lang.System.out.printf ("%15.6f", mRatio);
		java.lang.System.out.print (" ;");
		java.lang.System.out.printf ("%15.2e", effectiveMass);
		java.lang.System.out.print (" ;");
		java.lang.System.out.printf ("%15.2e", distance);
		java.lang.System.out.print (" ;");
		java.lang.System.out.printf ("%15.2e", dRatio);
		java.lang.System.out.print (" ;");
		java.lang.System.out.printf ("%15.0f", dLightYears);
		java.lang.System.out.print (" ;");
		java.lang.System.out.printf ("%15.9f", + deltaRatio);
		java.lang.System.out.println ();
	}

	// This method has a loop for computing a sequence of effective masses and Schwarzschild distances for different lambdaM values
	// lambdaM is iterated from lambdaMfirst to lambdaMlast in steps of deltaLambda
	private static void computeScharzschildRadii (double lambdaMfirst, double lambdaMlast, double deltaLambda)
	{
		double lambdaM = lambdaMfirst; // linear mass density of black shell at Hubble radius in kg/m
		lambdaMlast = lambdaMlast * 1.00000001; // for the reason of numerical rounding errors
		printHeader ();
		do
		{
			double schwarzschildDistance = solve (lambdaM);
			double effectiveMass = effectiveMass (schwarzschildDistance);
			printResults (lambdaM, schwarzschildDistance, effectiveMass);
			lambdaM = lambdaM + deltaLambda;
		} while (lambdaM <= lambdaMlast);
	}

	// Main program
	public static void main (java.lang.String [] arguments)
	{
		computeScharzschildRadii (lambdaMfirst, lambdaMlast, deltaLambda);
	}
}

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