Abstract
To describe flux of gravitational energy by using an analog to the Maxwell-Heaviside equations for electrodynamics, the Liénard–Wiechert potentials and fields are derived for gravitation along with radiation patterns and corresponding Larmor formulae for total radiated power. Due to attraction of like gravitational charges (masses) as opposed to repulsion of like electrical charges, the mass-density and current-density terms pick up a negative sign. This results in a sign change of the Poynting vector, indicating energy is gained by the field as opposed to energy being lost by the field in the case of electromagnetic radiation. The gravitational and co-gravitational fields, analogous to the electric and magnetic fields, respectively, behave as described by Heaviside and Lorentz. Like an electric charge, a gravitic charge (mass) in uniform motion is found to produce a spherical field which contracts as its velocity approaches the speed of propagation. The speed of gravitational propagation is assumed to be equivalent to that of light, though this may not necessarily be true. For an accelerated mass, the resulting gravitational radiation mirrors the dipole pattern produced by an electric charge, similarly contracting at relativistic speeds. These results seek to further inquire on the nature of gravitational fields and the true speed of gravity.
During my final year as a graduate student, I wrote my thesis on an extension of an analogy between gravitation and electromagnetism written by Oliver Heaviside in 1893. This analogy was formulated around James Clerk Maxwell's equations for electromagnetism. In fact, it was Oliver Heaviside who, using his own vector notation, reduced Maxwell's original set of twenty equations down to the standard four. From Heaviside's analogy, I followed the same procedure used to develop classical electromagnetic radiation to instead derive equations for gravitational radiation
The analogy can be explained by comparing the behavior of masses and charges. Charges can be positive, negative, or neutral while masses are considered* to be positive. If all masses are alike and positive, then all masses attract other masses. Like charges repel while opposites attract and, conversely, like masses attract. The interaction between like charges and like masses are, then, opposites. An illustration for both cases is shown:
The lines shown in Figure 1 and 2 are the lines of force. These should be fairly intuitive! For two positive masses, say the "parent" and the "child", the two will be attracted towards each other. For two positive charges, they will be repelled away from each other. The direction of these forces is represented mathematically by the divergence equation. The divergence of a vector field describes the flux of a differential volume. If the divergence is positive, then the volume is said to act as "source" of the field. If the divergence is negative, then the volume acts as a "sink". In the case of zero divergence, the field is said to be "solenoidal" and is neither a source nor a sink.
From the divergence equation for the electric field in Maxwell's equations for electromagnetism, it can seen that the electric field has a positive divergence and thus acts as a source. From the equations for the gravitational analogy, the gravitic field has a negative divergence and thus acts as a sink. This is simply the less intuitive mathematic expression of the magnitude and direction of the more intuitive lines of force. Curiously, the known equations for representing the force between two masses and two charges are essentially identical in their formalism. Newton's law of gravitation states that the force between two masses is given by:
F = -GmM/R^2
while Coulomb's force law states that the force between two charges is given by:
F = kqQ/R^2.
The analogy becomes more clear by understanding the relationships between sources, fields, and forces. A source (or sink) produces a field that exerts a force. The force exerted on a mass, m, in a gravitational field, g, is given by:
F = mg,
while the force exerted on a charge, q, in an electric field, E, is given by:
F = qE.
The source of the gravitational field, g, is produced by a parent source and the force is experienced by a child within the field. Likewise for the scenario in electromagnetism. The dimensionality of these parameters will be explored in a later post. Now, considering the dynamics of such a system, as the force acts on the child, another component must be considered to fully describe gravity. In electromagnetism, a moving charge produces a rotating magnetic field. By analogy, Heaviside reasons that a gravitational analog to this field must exist which behaves like to the magnetic field. Although he did not use the term, this field is sometimes referred to as the "cogravitational" field or, occasionally, the Heaviside field.
A set of four equations can be derived for gravity, which closely resemble those for electromagnetism, using four assumptions as shown in my paper (link below). From these four vector equations, scalar and vector potentials can be determined. These potentials are necessary to then identify the wave equations for gravity. Following the same procedure to obtain the Liénard–Wiechert potentials and fields in classical electromagnetism, it is possible to find the LW potentials and fields for gravitation.
The gravitational and cogravitational (LW) fields were shown to be:
and depend on the relativistic velocity beta. A plot of the gravitational field is shown below.
[Insert picture of gravitational field.]
These fields are seen to contract along the direction of motion as velocity increases, as one would expect by the behavior of Lorentz contraction.
Electromagnetic radiation is produced by an accelerated charge. Analogously, gravitational radiation is produced by an accelerated mass. However, just as the direction of the force is reversed (which, mathematically, is denoted by the sign change of the source terms) so too is the direction of energy flux. By rederiving the Poynting vector for gravitation, the Larmor formula for total power radiated can be determined.
Like an accelerated charge, an accelerated mass also produces a dipole radiation pattern that is oriented about the axis of acceleration and contracts along the axis of motion at relativistic speeds.
In the case of a charge confined to a circular orbit, the force (acceleration) exerted on the charge causes the charge to radiate energy away, indicated mathematically by the positive Poynting vector, and thus the radius of the orbiting charge decays. The resulting Poynting vector for gravitation is negative. This means that an accelerated mass radiates away negative energy. Strangely, this implies that masses are continuously absorbing energy. And if so, where does this energy come from? This "negative field energy" discouraged Maxwell from further pursuing the analogy. Rather than coming closer to understanding gravity, Heaviside remarked in his paper that the analogy "only serves to further illustrate the mystery".
*If I remember correctly, some masses are actually considered to be negative (anti-particles have negative mass?) so this absolute is not actually absolute.