Take the gravitation-electromagnetism analogy seriously and derive the radiation patterns. The Poynting vector flips sign — energy flows INTO the field rather than out of it. What does that tell us about gravitational radiation?

Theoretical Foundation

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.

Sources and Fields: The Fundamental Asymmetry

The analogy can be explained by comparing the behavior of masses and charges. Charges can be positive, negative, or neutral, while masses are conventionally considered to be positive (though some exotic particles may exhibit negative mass properties). If all masses are alike and positive, then all masses attract other masses. Like charges repel while opposites attract; conversely, like masses attract. The interaction between like charges and like masses is therefore opposite.

The lines of force for these two cases illustrate this fundamental difference. For two positive masses ("parent" and "child"), the two are attracted toward each other. For two positive charges, they are 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. A positive divergence indicates a "source" of the field, while negative divergence indicates a "sink". Zero divergence indicates a "solenoidal" field, neither source nor sink.

From Maxwell's equations for electromagnetism, the electric field has positive divergence and thus acts as a source. From the gravitational analogy, the gravitic field has negative divergence and thus acts as a sink. This is the mathematical expression of the intuitive fact that like masses attract while like charges repel.

Remarkably, the equations for force between two masses and two charges are essentially identical in formalism:

Newton's Law of Gravitation: F = -GmM/R²

Coulomb's Law: F = kqQ/R²

The analogy deepens when considering relationships between sources, fields, and forces. A source (or sink) produces a field that exerts a force. The force exerted on a mass in a gravitational field is F = mg, while the force exerted on a charge in an electric field is F = qE. The source produces the field, and the field exerts the force—the same structural relationship in both cases.

Field Dynamics and the Cogravitational Field

In electromagnetism, a moving charge produces a rotating magnetic field. By analogy, Heaviside reasoned that a gravitational analog to this field must exist, behaving similarly to the magnetic field. 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 that closely resemble those for electromagnetism, using four foundational assumptions detailed in my thesis. From these four vector equations, scalar and vector potentials can be determined. These potentials are necessary to identify the wave equations for gravity. Following the same procedure used to obtain the Lienard-Wiechert potentials and fields in classical electromagnetism, it is possible to derive the LW potentials and fields for gravitation. These fields depend on relativistic velocity and are shown to contract along the direction of motion as velocity increases, as expected from Lorentz contraction.

Radiation Patterns

Electromagnetic radiation is produced by an accelerated charge. Analogously, gravitational radiation is produced by an accelerated mass. However, just as the direction of force is reversed (mathematically denoted by the sign change of 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 produces a dipole radiation pattern oriented about the axis of acceleration. This pattern contracts along the axis of motion at relativistic speeds, mirroring the electromagnetic case.

The Energy Paradox

In the electromagnetic case, a charge confined to a circular orbit experiences force (acceleration) that causes it to radiate energy away. Mathematically, this is indicated by a positive Poynting vector, causing the orbit to decay. The resulting Poynting vector for gravitation, however, is negative. This means that an accelerated mass appears to radiate away negative energy—which strangely implies that masses are continuously absorbing energy. Where does this energy come from?

This "negative field energy" paradox 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". This work extends the analogy and leaves open the fundamental questions: What is the true nature of gravitational fields? What is the actual speed of gravitational propagation? And how should we interpret negative energy flow in a gravitational system?

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