Moving objects in retarded gravitational potentials of an expanding spherical shell

This Wikibook decribes the acceleration of a moving object because of retarded gravitational forces within a huge expanding shell with a large mass. For this purpose, the principles of gravitational potentials and retarded potentials are presented and applied by way of example. The effect of retarded gravitational potentials could contribute to the explanation of the accelerated expansion of the universe, which often is related to a term called "dark energy".

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.

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 relativistic 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 red shift 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 red shift of the cosmic microwave background was found to be $z\approx 1089$ in 2003, which is a very high value.

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.

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.

Classic 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 it can be represented by the Planck radiation law. The emission of such a cold source of electromagnetic radiation is not visible and additionally it will have experienced red shift due to the relative velocity of the emitting shell and its mass.

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.

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.

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.

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 expanding dark matter, but also by expanding visible objects will experience the retarded gravitational potentials of these visible objects, and therefore, will experience an additional acceleration which is directed outwards.

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 expanding dark matter in front of them seem to be the reason for the extreme red shift of their light that can be observed. The observation that this red shift 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 huge, massive dark outer shell.

However, the computation of the accelerations of moving objects due to retarded forces become extremely expensive, if the outer shell is neither homogeneous, nor spherical or symmetrical. Furthermore, relativistic effects could cause time dilatation and mass increase that would have to be considered, too.

Summary of the project

• Target audience: Astronomers, astrophysicists, physicists, mathematicians
• Learning objectives: Application of retarded gravitational potentials on large scales.
• Contact person: User:Bautsch
• Are co-authors currently desired? Yes. Corrections of obvious errors directly in the text; content please after discussion.
• Guidelines for co-authors: Wikimedia-like.