Mathematics and gravitation theory

The mathematics of gravitation theory is remarkable for its expansibility and physical ambiguity. To a large extent it applies equally well to an interpretation of gravitation as a force and as a geometric distortion of spacetime. But given the pre-relativistic association of gravitation with force, that ambiguity, combined with the current primacy of mathematics in the interpretation of physical phenomena, has led to an overextension of the mathematics and resulted in theoretical misdirection.

Einstein’s heuristic insight leading to the General Theory derived from a consideration that the ratio of circumference to diameter of a rotating disk will deviate from pi with relativistic accelerations at the radius. In his original pursuit of a generalization of relativity, where he hypothesized the equivalence of inertial acceleration and gravitation, the similarity of the inertial effect on the rotating disk and the gravitational pressure we experience at the earth’s surface suggested that gravitation might be explicable as a geometric principle. Experimentation has confirmed the validity of that seminal insight, and the service of the mathematical analogy. But in the kinematical similarity between objects on a rotating disk and in a gravitational orbit there is a distinct physical difference. A test particle in a box fixed at the edge of a rotating disk presses against the radial wall of the box, manifesting a centrifugal “force”, derivative of the actual force that is rotating the disk; in contrast, a test particle in a box orbiting a massive body floats freely, following its geodesic in spacetime, and gives no indication of the presence of a force or acceleration. There is thus a mathematical analogy due to the similar kinetics of the rotating disk and the orbiting body, but not a physical equivalence.

The subsequent development of the Field Equations was based on another mathematical analogy, formalizing the behavior of bodies being accelerated or pressured toward an attractive or determinant center, as in a field of force or field of gravity. The analogy holds in this case because gravity, like a field of force, produces a typically curved, actually concentric form to the relative motion of affected bodies. But again, the mathematical analogy is not a physical equivalence. A neutral test particle inside a charged box accelerating toward the vertex of a field of force presses against the wall opposite the direction of force, and a non-neutral particle of different mass than the box accelerates at a different rate than the box; in contrast, a particle in a box falling in a gravitational field floats freely, following its geodesic in spacetime in parallel with the box, and gives no indication of the presence of a force or acceleration.

In each case, the rotating disk or orbiting body and the attractive or determinant field, there is a discernable difference in the empirical behavior of test particles being acted upon by a force and those moving in a gravitational field. The mathematical analogy between gravitation and force is limited to the trajectories of idealized, dimensionless particles.

Empirical gravitational phenomena consist in the distortion or compression of spacetime in the presence of mass, the relative curvature of geodesic motion in the presence of a gravitational distortion, and the static acceleration of bodies when their geodesics are resisted at the surfaces of large masses. In these descriptions there is no indication that gravitation might somehow produce energy or manifest a force.

We are left to question how the predictions of gravitational waves and “gravitomagnetic” effects can be justified, as both are based on the supposition of a gravitational energy. They are mathematical extrapolations from the Field Equations, drawn from electromagnetic analogies. It is here that the physical ambiguity and indifference of mathematics has been misleading gravitation theory, and consequently, it is here that the derivative predictions of General Relativity remain unconfirmed. In the absence of a coherent physical theory that could somehow link non-energetic phenomena with the production of energy, there is no reason to expect such predictions will ever be confirmed.


  1. You said it yourself. Resistance, of whatever form it may take (geometric, math etc), are only factors that negate gravitational force. Geometry (with mass) is a component of both resistance and attraction. I believe you are also right when you say the math is secondary, it is the physical that describes math, not the other way around. My only concern is that without the math, there are no suitable symbols to describe and therefore no common language. I truly wish I had the mathematics to describe such a general theory of gravity. Yes, myself and an infinite number of physicists :) The problem remains so obvious as it’s shown us on a daily basis yet no one sees it, or at least can adequately describe it. Please feel free to deduce and reduce. In all sincerity,


  2. Burt,

    A more accurate description of our difference regarding the instability of the binary system and its orbital decay is the question of whether it’s a transfer of intrinsic kinetic/potential energy to extrinsic kinetic/potential energy or to force-like gravitational waves. But a big difference – yes.

  3. Jim, you wrote: “To argue, as you and others do, that gravitational waves aren’t the same as “tidal gravity” because they’re different is a circular argument.

    You have never answered the conflict of your argument with observation: tidal gravity makes the orbiting bodies spiral AWAY from each other (observed), while GWs make them spiral TOWARD each other (observed). That’s a rather significant (non-circular) difference. :-)

    Burt Jordaan (

  4. Gosh, you’re moving on without answering my question. I’m reminded of a scene from Monty Python: “Is there someone else we can talk to?”

  5. Fred,

    You might stop going “’round and ’round” if you’d read more carefully. Or maybe you could read more carefully if you weren’t going ‘round and ‘round’.

    You write: “You don’t dispute the math.”

    Again, I do dispute the math that extrapolates from the field equations to represent gravitational waves by analogy with electromagnetism. Gravitation is geometry, not a force. A body influenced by a gravitational field behaves differently than a body influenced by a force. Anyone who builds theories based on the assumption that gravitation is force-like (positing gravitons, gravity strings, gravitational waves, etc.) is ignoring the simple empirical fact that no force can be detected in association with gravitation except when gravitation is being resisted, as at the surface of a massive body.

    “If it’s not due to waves, what is carrying the gravitational energy outward?”

    The “mechanism” that carries the influence of a gravitational field is the same whether it’s due to the motion of our moon or the orbits of binary pulsars. We go “’round and ’round” when you and others insist that I’m confusing “tidal gravity” with “gravitational waves.” No, Fred, THAT’S MY POINT. I’m saying they are one and the same. And unless you’re willing and able to CONSIDER my point for a moment, and move beyond your presumption that I’m confused, you will continue to go ‘round and ‘round. To argue, as you and others do, that gravitational waves aren’t the same as “tidal gravity” because they’re different is a circular argument.

    There is no transfer of energy when the orbit of our moon causes the earth’s oceans to move in tides. The changing shape of spacetime geometry due to the moon’s motion relative to the earth reorients the geodesics of molecules on earth, causing massive dislocations in large fluid bodies. There is no energy transfer from moon to earth. Unless a body’s geodesic is already bound or accelerated at or below the earth’s surface, the effect of the relative motion of the moon is nonexistent, except relative to the moon. For example, the geodesic of a body in orbit around the earth will appear to be perturbed from our perspective by the motion of the moon, but it remains geodesic.

    An asymmetrical binary star system generates wavelike fluctuations in spacetime geometry in the same way as our moon, and in the same way, if the orbits are stable, there’s no transfer of energy involved. The effect of changes in spacetime geometry needn’t even be wavelike. Imagine a large asteroid colliding with the earth: The asteroid’s gravitational field increases continuously until the collision – there is no gravitational “wave” unless there is a recurrent relationship. In other words, waves are entirely incidental to gravitation.

    And where there is no force, there is no radiation of energy. Relative energy may be transferred, as when a binary star system loses net kinetic/potential energy, but the transfer is from a relationship within the system to a relationship with the rest of the universe. And the “mechanism” is just a change in the shape of spacetime, which is communicated much the same way as the indentation on a mattress moves when a heavy ball is rolled across it.

    “So we still have what I have repeatedly described as ‘a distinction without a difference.’”

    If you can’t see the difference between a force, with a corresponding transfer of energy, and a geometric phenomenon, with no transfer of energy, you’re just not looking.

    “Furthermore, you continue to dodge my question about other predictions of your interpretation that might cause people to sit up and take notice.”

    Here’s the irony in your statement, Fred. I’m predicting no gravitational waves due to “gravitational radiation.” You’re predicting gravitational waves due to “gravitational radiation.” In a science based strictly on evidence, my theory carries the day unless and until there is direct evidence to the contrary. In a science based also on simplicity and coherence, my theory explains gravitation with one (geometric) principle; yours requires two, and invokes the second only to explain phenomena that have not been observed. As for the math, mathematics is physically ambiguous. The orbit of the moon can be described mathematically as if it is force-like or as if it is geometric. To invoke mathematics to justify, rather than confirm, a physical concept is to go ‘round and ‘round. When used to defend a theory, it’s a dodge. And unfortunately, “going ‘round and ‘round” is a pretty good description of the state of gravitation theory in almost a hundred years.

  6. Jim, you wrote: “… if orbital (net kinetic/potential) energy is lost to a system the net kinetic/potential energy between the system and the universe at large increases accordingly, and is manifested in a change in the geometry of the system’s gravitational field.

    That effect is called “gravitational waves”! ;-) This is because “a change in the geometry of the system’s gravitational field” propagates at the speed of light through space and if the change is periodic, it constitutes a wave. Amen!

    Burt Jordaan (

  7. Jim,

    Who’s flailing? It’s ’round and ’round we go:

    How does a change in a geometric relationship between two bodies produce an energy-bearing wave?

    Burt has repeatedly pointed out that the mathematics of general relativity predicts precisely that. I don’t see that the mechanism you describe as spacetime geometry or a change in the geometry of the system’s gravitational field is different from the conventional mathematical description. You don’t dispute the math. You just dispute the conventional interpretation of the resulting phenomena as gravitational waves.

    How does the distinction you try to describe, namely
    a gradual change in [the field’s] concentration due to an inspiral of the orbits would, in that aspect, be a continuous rather than wavelike function,
    preclude a wavelike mechanism for the energy transfer to the rest of the universe? It’s the mechanism that we are asking about. We’re not disputing that it arises from a changing spacetime geometry. If it’s not due to waves, what is carrying the gravitational energy outward?

    So we still have what I have repeatedly described as “a distinction without a difference.”

    Furthermore, you continue to dodge my question about other predictions of your interpretation that might cause people to sit up and take notice. Without other predictions that differ from the conventional interpretation, all you have is the argument over the existence of gravitational waves.

    And at the present time, the prediction of gravitational waves is favored by a growing body of evidence. I’m open to new evidence, of course; but without such evidence, your argument remains unpersuasive.

    Having read this, I wonder why I am bothering to leave the sidelines, since I am going ’round and ’round, repeatedly asking questions that are never answered.

    Distinction with a difference

    No new predictions

    Distinction with a difference

    No new predictions

    Distinction with a difference

    No new predictions

    Okay, I’m done now.

    Fred Bortz — Science and technology books for young readers ( and Science book reviews (


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