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

Moving objects in retarded gravitational potentials of an expanding spherical shell

The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at
https://en.wikibooks.org/wiki/Moving_objects_in_retarded_gravitational_potentials_of_an_expanding_spherical_shell

Permission is granted to copy, distribute, and/or modify this document under the terms of the Creative Commons Attribution-ShareAlike 3.0 License.

This Wikibook explores the concept of gravitational forces within an expanding shell of mass. It describes the acceleration of moving objects because of retarded gravitational forces within a huge expanding shell with a large mass. For this purpose, the principles of retarded and gravitational potentials are presented and applied.

The concept of a Schwarzschild distance from the inner surface of a hollow spherical cap is introduced, which can be used to deliver insights into the evolution of the universe. The unexpected high values of redshift that are observed at extremely far astronomical objects including the cosmic microwave background could be related to the extremely strong effect of the gravitation of the outer shell of the universe. The gravitational effect for atoms within the Schwarzschild distance is so enormous due to the integrative and asymmetric effect of the retarded gravitational potentials that the light from this area can no longer reach us. Exactly at the boundary to us, the redshift is infinite, just like at the surface of a black hole.

The evolvement of the Schwarzschild distance due to the 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 contribute to the explanation of the accelerated expansion of the universe, which often is related to the term called "dark energy".

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

---

Brief historical review

The finiteness of the propagation speed of gravity and its influence to gravitational forces was originally stated by Paul Gerber (1854–1909) in 1898.

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.

In 1933 the Swiss astronomer Fritz Zwicky 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 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.

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.

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

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

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 400,000 years. In January 2024 the very young and very far galaxis JADES-GS-z14-0 was found with the Near-Infrared Spectrograph (NIRSpec) 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 much lower than that of the cosmic microwave background.

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.

The evolution of the universe according to the interpretation of the WMAP measurements. 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 WMAP measurements.

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.

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 electrodynamic waves.

---

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 $\mathrm {d} \alpha$:

$\mathrm {d} A_{1}=s_{1}^{2}\cdot \mathrm {d} \alpha ^{2}$

$\mathrm {d} A_{2}=s_{2}^{2}\cdot \mathrm {d} \alpha ^{2}$

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

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

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

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:

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

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

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

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.

---

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. 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 out rim 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 $O$ 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 $O$:

$x_{O}=0$

$y_{O}=0$

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

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 $O$ 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 $m$:

$x_{O}=-x(t)$

$y_{O}=0$

The retarded time $t_{\text{ret}}$ and the retarded potential $\Phi _{\text{ret}}({\vec {x}},t)$ are given as follows:

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

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

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 $\mathrm {d} M'_{1}\propto \mathrm {d} A'_{1}$ 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 of mass element $\mathrm {d} M'_{1}$ is always later than the retarded time of 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

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.

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 $M_{e}ff$ (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,.

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 $m$:

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

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 $M_{universe}$:

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

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 outer 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
• 
  Schwarzschild distance $d_{S}$ between the visible border of the universe and the invisible outer shell of the universe as a function of the mass of the invisible universe surrounding it in a spherical shell in units of the mass of the visible 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 shell in units of the mass of the visible universe.

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:

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

---

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 shell of the universe would consist of expanding dark matter with a huge mass. However, this would also hold in a steady state universe and even if the outer dark shell of the universe would not move at all. But in the latter case any moving mass in that shell would reach the shell in a 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 therefore, the closer they are to the spherical shell of dark matter. Furthermore, also objects within smaller spherical shells that not only are surrounded by dark matter, but also by visible objects will experience the retarded gravitational potentials of these objects, and therefore, will experience an additional acceleration which is directed outwards.

Computing the corresponding Schwarzschild distances results in two remarable points in time where we today can observe the cosmic microwave background at a cosmic age of 380,000 years as well as the farest known galaxy at a cosmic age of 290 million years.

Due to Newton's third law (actio = reactio) the 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 shell are to it.

The high velocity of visible objects near the outer dark rim of the universe as well as the huge mass of the dark matter in front of them seem to be the reason for the extreme redshift of their light that can be observed. 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. One reason for this could be found in the effects of the retarded gravitational potentials of a massive dark outer 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.

However, the computation of the accelerations of moving objects due to retarded forces becomes extremely expensive, if the outer shell is neither homogeneous, nor spherical or symmetrical. 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 take into account the loss of mass in the visible universe over the time, since visible matter may traverse the event horizon that is caused by the huge mass of the outer shell of the universe.

---