Moving objects in retarded gravitational potentials of an expanding spherical shell/Retarded gravitational potentials
Retarded gravitational potentials
[edit  edit source]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.^{[1]} 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 nonzero 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.^{[2]} 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 timedependent 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 due to a mass in the distance is given by:
The gravitational force to another mass results as follows:
The timedepending potential is the solution of the inhomogeneous wave equation, where is the inhomogeneity, and 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:
 ,
where is the Laplace operator, 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:
The retardation is to be interpreted in such a way that a source element at the point and at the time only influences the potential at the distant point of impact at at a later time :^{[3]}
is called the retarded time. At the location and at the time the retarded potential only depends on the inhomogeneity 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 .
In a shell
[edit  edit source]In the simplified example in the adjacent figure the source term is the linear mass density that is not timedependant and only exists in the circle of the outer rim with the constant radius and its centre at :
In all other locations within the plane the linear mass density is zero:
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:
Furthermore, the mass element on the homogenous circumference of a circle with the radius is given by:
The cosine formula gives us the relation between the location of the mass point in the horizontal distance of the origin of the circle with the radius , when the mass element on the circle is in the direction of the angle and the distance :
In the normalised standard form of the quadratic equation, we get:
The solution for the distance is:
It is obvious that the following simplifications are valid:
The origin of the coordinate system can be also shifted to the mass :
The retarded time and the retarded potential are given as follows, where represents the propagation speed of the potentials:
Illustration
[edit  edit source]In a simplified example we only consider the infinitesimally small angles in the origin of a spherical shell with the radius and the areal density .
Symmetrical geometry
[edit  edit source]If the mass is in the centre of a spherical shell with the radius we have the following situation:

The retarded potentials of the two symmetric mass elements that simultaneously act to the mass .
 Both angle elements are equal.
 Both distances are equal to .
 Both areal elements are equal.
 Both mass elements are equal.
 The retarded times of the gravitational potentials of both mass elements are equal.
In this situation the mass does not experience any acceleration in the classical approach (see above) or if its velocity is zero.
Moving masses
[edit  edit source]The situation is changed, if the masses are moving starting at within a time span of . The mass moves with the velocity to the right and the two mass elements and move with the radial velocity :

The retarded potentials of the two asymmetric mass elements and that don't act simultaneously to the mass .
At the time the mass has moved with the velocity the distance to the right:
The spherical shell has expanded with the velocity and gained an increased radius:
Therefore:
This means that the distance of mass element to the mass is always greater than the distance of mass element to the mass .
For the two retarded times for these distances to the location of we get:
With and therefore :
Therefore:
This means that the retarded time for the mass element is always later than the retarded time for the mass element .
Common case
[edit  edit source]In the adjacent diagram there are three mass points that move in space. Their speed is given as follows:
 Outer mass element top (green):
 Mass point in between (blue):
 Outer mass element botton (green):
Their timedepending location is given by these three functions for their xcoordinates:
The timedepending effective distances and between the outer mass elements and the mass point in between them is the difference of the corresponding xcoordinates and linked to the propagation velocity of the interacting waves as follows:
As a result, the corresponding retarded times and for the two outer mass elemens are:
And therefore:
And:
These are the timedepending 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 .
For and and according to the diagram we can make the following assumption for the comparison of the effective distances:
 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:
 quod erat demonstrandum
Since is always greater than , 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 :
The moving mass experiences the corresponding retarded forces:
The net force to the mass is the sum of both:
For the acceleration of the mass we find:
Since the net force as well as the acceleration are positive, and the mass experiences an acceleration to positive xvalues, i.e. in the direction of its movement.
Special case
[edit  edit source]Let us have a look at the following special case:
Their timedepending location is given by these three functions for their xcoordinates:
The timedepending effective distances and between the outer mass elements and the mass point in between them is the difference of the corresponding xcoordinates:
As a result, the corresponding retarded times and for the two outer mass elemens are:
Therefore:
Thought experiment
[edit  edit source]In a thought experiment we look at the following situation, where is the propagation speed of the gravitational waves. The time line starts at , and the effect of the retarded potentials is synchronised with the occurence of the moving mass. The left moving mass element is regarded at and , the right moving mass element is regarded at and , whilst the moving mass is regarded at and , when the retarded gravitational potentials have their effect to the mass.
In the following diagram the velocity of the waves is normalised and used without unit:
And therefore, only for simplification and without any units, too:

Retarded potentials of the two expanding mass elements with the mass . These mass elements move on the left hand side from at to at ), and on the right hand side from at to at . They simultaneously act to the mass at (mass in grey, path for gravitational waves in black) and at (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 for the gravitational potential at are both equal:
At the time the mass experiences the retarded potentials on the lefthand side (distance to mass element is ) and at the righthand side (distance to mass element is ):
All inhomogeneities contribute to the retarded potential at the location of the mass with the value they had at the retarded times , and :
 , this corresponds to the effective time of the two area elements .
 , this corresponds to the effective time of the area element on the left.
 , this corresponds to the effective time of the area element on the right.
For a mass moving from the centre of a shell to the right the angle element to the left becomes smaller than the original angle element , and the angle element to the right becomes greater than the original angle element :
The following applies to the appropriate mass elements:
For the net force to the mass :
For the acceleration of the mass we find:
Since the net force is positive, and the mass experiences an acceleration 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 is proportional to the areal density of the expanding shell:
Nevertheless it should be noted that the areal density is decreasing with the expansion and the increasing radius of the outer shell.
References
[edit  edit source] ↑ Schwarzschild, Karl (1903). "Zur Elektrodynamik" [On electrodynamics]. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, MathematischPhysikalische Klasse. Göttingen: 127.
 ↑ von Brill, Alexander (1904). Written at Tübingen. Clebsch, Alfred; Neumann, Carl Gottfried; Klein, Felix; van Dyck, Walther; Hilbert, David (eds.). "Über zyklische Bewegung" [About cyclic movement]. Mathematische Annalen (in German). Leipzig: Benedictus Gotthelf Teubner. 58: 473.
 ↑ Dragon, Norbert (20160926). "Stichworte und Ergänzungen zu Mathematische Methoden der Physik" [Keywords and additions to Mathematical Methods in Physics] (PDF) (in German). pp. 222–224. Retrieved 20240225.