# 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.

### 70 thoughts on “Mathematics and gravitation theory”

1. You guys are flailing.

Burt stated and asked: â€œThe binary system emits total energy and the universe at large absorbs it. We have a mechanism called GWs. What is yours?â€

As Iâ€™ve already written, 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 systemâ€™s gravitational field. And whereas an asymmetric oscillation of a binary system would produce a periodic, wavelike variation in the systemâ€™s field, a gradual change in its concentration due to an inspiral of the orbits would, in that aspect, be a continuous rather than wavelike function. The â€œmechanismâ€ is spacetime geometry, the same for the apple, the same for the planets, the same for the great-big twinkling stars.

Now answer my question. How does a change in a geometric relationship between two bodies produce an energy-bearing wave?

Burt would like to distinguish between the geodesic path of a stable orbit and a path that is inspiraling. Iâ€™ve pointed out elsewhere that an inspiraling orbit, like any relative acceleration in a gravitational field, is still a geodesic path. An object â€œfallingâ€ (and accelerating) toward a massive body, a body that is orbiting (and accelerating and decelerating) in a non-circular orbit, and a body that is spiraling (yes, accelerating) toward another â€“ all these are moving uniformly in their own frames of reference, i.e., along geodesic paths. There is no absolute change in their total energy, and hence, no energy to be radiated (or quantized).

So put it this way: How does a change in a geometric relationship between two bodies, each moving along geodesic paths (however irregular the paths may appear from another perspective), produce an energy-bearing wave?

Youâ€™ve all three expressed a strident distaste for ducking and dodging. Donâ€™t let yourselves down.

2. Fred notes in his most recent post;
Jim did not answer Burt’s question about the mechanism of energy transfer from the orbiting stars to the rest of the universe.

And Burt re-states his earlier question:
The binary system emits total energy and the universe at large absorbs it. We have a mechanism called GWs. What is yours?

This Gadfly notes that Jim continues to evade this key question. He simply rambles and asserts that gravitational waves do not exist, even though they fit the observations. Burt writes about perturbations to geodesics.

But the real perturbation for me is Jim’s refusal to address this question. For that he earns:

This bite of realism brought to you by “Gadfly.”

3. Hi Jim, you wrote:

You may be accurately describing the conventional view, but I want to point out that there can be no such thing as a â€œperturbed geodesic.â€ A geodesic is, by definition, a uniform path in spacetime.

So what do you call the movement of an orbit that does not follow a geodesic (like inspiraling binaries)? (Or our Moon, for that matter?)

JIm: “If we say simply that the loss of energy in the binary system is a loss of net kinetic/potential energy matched by an increase in net kinetic/potential energy between the binary system and the rest of the universe, there is a symmetry of cause and effect.

But this is precisely what the mainstream view on GWs is saying! The binary system emits total energy and the universe at large absorbs it. We have a mechanism called GWs. What is yours? And please don’t tell us it’s tidal gravity; we’ve been there, done it!

Burt Jordaan (www.Relativity-4-Engineers.com)

4. As Burt pointed out, the discussion seems to be repeating itself. The words may be different, but Jim is essentially repeating the same arguments that have not persuaded me in the past.

Most important, Jim has not noted any testable prediction other the nonexistence of gravitational waves that differs between his interpretation and the conventional one.

Without such a prediction, I see no evidence for or reason to consider an alternative to the conventional interpretation right now.

Burt may have a response to Jim’s denial of “perturbed geodesics,” but to me it looks like the same unproductive approach: Argue about the terminology when you can’t deny the evidence.

I think Jim is applying the theory incorrectly when he writes “test particle (or any test planet, or test pulsar) in a box,” since physicists consider the test particle massless (or has a mass that approaches zero). Once you get to something macroscopic, it is massive enough to have an effect on the rest of the universe.

Furthermore, Jim did not answer Burt’s question about the mechanism of energy transfer from the orbiting stars to the rest of the universe.
End edit

As for me, it’s time to head for the sidelines on this thread.

Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

5. Fred wrote: You need to persuade me that there is physical significance to the distinction [between relative, i.e., kinetic and potential energy, and non-relative energy] you are drawing.

First, I need to respond to Burt, who wrote: â€œOrbital motion is only on a geodesic for a particle with negligible size and mass, which are not subjected to gravitational wave (GW) radiation at all. Massive bodies in orbit around each other do not quite move on geodesics, but rather on perturbed geodesicsâ€

You may be accurately describing the conventional view, but I want to point out that there can be no such thing as a â€œperturbed geodesic.â€ A geodesic is, by definition, a uniform path in spacetime. If it is a path through varying shapes of spacetime (as is the case every real geodesic), it may appear perturbed from some other perspective, but for any test particle (or any test planet, or test pulsar) in a box, there is no indication of perturbation, no matter how wobbly its path may appear from a perspective outside. Leaving tidal effects aside, which may cause stresses to the molecular binding energies of an extended body due to the differential in the curvature of spacetime across its extension, you and others are positing an additional force-like effect that canâ€™t be consistently maintained â€“ is gravitation a geometric principle, or is it a force, or is it both? To justify saying itâ€™s both would require something more than an extrapolation from the field equations based on an analogy with the mathematics of electromagnetism. There is nothing force-like in the concept of gravitation as a geometric principle, there is no direct evidence of force-like effects, and in principle, any number of equations can (and have) described the dynamics of decaying orbits.

Back to Fredâ€™s question of the significance of my distinction between relative and non-relative energy, hereâ€™s one way of looking at the problem: Consider first the alleged production of gravitational waves in terms of a test particle (or extended test body) in a box in a mutual orbit with another body. If the orbit is decaying due to the loss of net kinetic/potential energy between the two, each would nonetheless be following its own geodesic, and there would be no indication of orbital decay inside the box. Orbital motion is relative – changes in the geometry of an orbit involve no indication of an exchange of energy except when viewed from another perspective, where the basis of the changes can be interpreted as the result of some kind of energy-exchange (pre-GR or contra-GR), or not (gravitation-as-geometry). Now consider the reception of the gravitational waves somewhere outside the binary system. Bondi proposed (it may have been Feynmanâ€™s original idea) the effect of beads on a stick as gravitational waves are passing by. The beads are hypothesized to vibrate on the stick, causing heat, an absolute, unambiguous indication that the waves are bearing energy. I would attribute any vibrations that might occur to tidal (geometric) fluctuations, but if weâ€™re considering the possibility of energy-bearing gravitational waves, hereâ€™s the problem: At the source, according to the GR interpretation of gravitation as a geometric principle, and according to the evidence inside the box, there is no indication of an energetic effect (itâ€™s purely geometric, and non-existent inside the box), in the distance there is supposed to be an absolute, unquestionable reception of energy. If we say simply that the loss of energy in the binary system is a loss of net kinetic/potential energy matched by an increase in net kinetic/potential energy between the binary system and the rest of the universe, there is a symmetry of cause and effect. But if we say the loss of relative energy between bodies that donâ€™t transmit and receive anything corresponds to an increase in absolute energy between a physical transmission and reception, donâ€™t we have a problem of asymmetry? Donâ€™t we have a non-intuitive and compound theory in place of a simple, intuitive theory, with no empirical evidence to explain by the counter-intuitiveness and complication?

6. Hi Fred, you asked: “So, Burt, do you see the need for a distinct form of energy called “relative” energy?

I read what Jim refers to as “relative energy” as the coordinate dependent (Lorentz variant) component of mechanical energy and I have no particular problem with that.

Whatever he calls it makes no difference to the fact that his arguments around it are somewhat flawed. Any component of mechanical energy can be transferred to some other place, provided you have a mechanism for its transfer.

Burt Jordaan (www.Relativity-4-Engineers.com)

7. So, Burt, do you see the need for a distinct form of energy called “relative” energy?

Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

8. Jim, the discussion is going in circles, partly because you are making at least one very invalid assumption, when you wrote:

Uniform motion is relative. The energy of relative motion is relative. Orbital motion is geodesic. Geodesic motion is uniform. The energy of orbital motion is relative.

Orbital motion is only on a geodesic for a particle with negligible size and mass, which are not subjected to gravitational wave (GW) radiation at all. Massive bodies in orbit around each other do not quite move on geodesics, but rather on perturbed geodesics. It is pretty certain that these perturbations of the geodesics cause GW radiation – a direct consequence of Einstein’s field equations (Einstein himself predicted GWs!)

We have already been through the fact that GWs and tidal gravity do not give the same effect – tidal gravity causes spinning, orbiting bodies to increase their separation, while GWs reduce their separation.

You have agreed before that there must be some energy transferred to the universe at large. Do you have a mechanism other than GWs in mind that can do that?

Regards,

Burt Jordaan (www.Relativity-4-Engineers.com)

9. Jim,

I understand relative motion and relative position. And now I think I finally see what you mean by “relative energy.”

You distinguish between energy associated with a force and energy associated with gravity, which some people call a force and others do not. You do not consider it a force, so you call energy associated with it “relative.”

In my view, energy is associated with interactions, whether you call them forces or not and whether they are mediated by quanta or not. I do not see the need for the distinction that you are drawing between the two types of energy. As far as I am concerned, energy is energy.

You need to persuade me that there is physical significance to the distinction you are drawing.

For example, the distinction has no significance in the law of conservation of energy, which we both accept. The “relative” energy of the orbiting stars decreases with time, and that energy shows up as non-relative energy in the rest of the universe. How does that make the two forms different? It makes them seem the same to me.

What does that distinction produce in terms of insight into the physical universe, other than your claim that gravitational waves do not exist? That is a directly testable claim in principle, so we might as well wait until we have evidence that makes it directly testable in practice.

[There is certainly indirect evidence that gravitational waves exist. They successfully predict, within the measurements’ margin of error, the rate of in-spiraling of that double star–and the mathematics of general relativity does not distinguish “relative energy” from energy.]

Are there any other testable predictions from your interpretation that differ from the standard interpretations of general relativity? That, to me, is the key point of this whole discussion.

Why? Because those other predictions from your interpretation might lead us along interesting paths.

If there are no other predictions, then I see nothing to learn from your interpretation and thus no reason to consider it further. And likewise in that case, I see no need for a concept called “relative energy” unless evidence arises that casts doubt on gravitational waves.

For now, one form of energy seems to be enough, and the evidence is strongly suggestive that gravitational waves exist. That is the impasse.

Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

10. In struggling to make sense of what you are describing, I homed in on this:

“3: But gravitational energy, unlike electromagnetic energy, is relative (e.g., if the kinetic energy of a body increases as it accelerates toward the body itâ€™s orbiting, except for possible tidal effects, there is no detectible change to the accelerating body; itâ€™s a relative acceleration).”

That isn’t correct, even in classical physics. The bodies orbit a common center of mass. One does not orbit the other. If one accelerates, then the other also accelerates in such a way that momentum is conserved (namely zero momentum in the reference frame of the center of mass).

In Newtonian physics, energy of the pair is also conserved, but in General Relativity, there is a small decrease in the pair’s energy due to what most physicists call gravitational waves. Energy of the universe is, of course, conserved.

Also, you are using a nonstandard term here: “relative energy.” Total energy has an arbitrary constant, depending on how you define the zero point of potential energy. But otherwise, energy is absolute, not relative. Momentum is a relative quantity, and perhaps that’s where the confusion is arising.

Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

11. Fred,

The problem of getting a fair hearing for an idea that challenges a conventional belief is well-recognized. I donâ€™t think we need to treat it as a singular issue between you and me. Have I been less than marvelous in my communication skills? No doubt. But beyond comprehension? Iâ€™ve made a living as a technical writer. When cherished beliefs arenâ€™t at stake I seem to be able to communicate successfully enough to be â€œgotten.â€

Letâ€™s look at one of your interpretations of my points. I noted that Copernicus, Galileo, Einstein, and countless others would disagree with your conviction that the writer has the burden of making the readers â€œgetâ€ his point, and if he has difficulty persuading them, there must be, in your opinion, a problem with his presentation. I believe it should be obvious that I was referring to their attempts to change conventional ways of seeing the world, which were initially unavailing despite clear and persuasive arguments. Thatâ€™s when they had difficulty being accepted â€“ before they were accepted. But you countered with reference to Einsteinâ€™s success in popularizing his ideas after he was well-established, when he commanded almost universal respect and confidence. Youâ€™re an intelligent man. To miss my point, to counter with a point that deflects mine only by misinterpreting it, suggests to me a psychology of resistance as a more likely problem here than the inadequate communication of a challenging idea.

Your difficulty with the idea of relative energy is inexplicable to me as anything but more or less unconscious resistance. Uniform motion is relative. The energy of relative motion is relative. Orbital motion is geodesic. Geodesic motion is uniform. The energy of orbital motion is relative. What could be more clear – and conventional? But to acknowledge the difference between relative energy and the energy associated with force, which is communicated in waves and quantums, and detectible in absolute accelerations, goes to the crux of my argument. If the discussion isnâ€™t reduced to quibbling, thereâ€™s nowhere to go but to the problem of how relative energy becomes force-like in â€œgravitational waves.â€ We have yet to go to that problem. I would love to finally go there. Why donâ€™t we â€“ finally go there.

12. Jim, as long as you continue to blame the reader, you will not reach the reader–not just me. Please stop making assumptions about how I’m reading this and just accept the fact that I was struggling to get your point and could not.

Here’s my last try at explaining why I do not get it. Blame me for being too dense, or recognize that you are not communicating–your choice.

Your definition of relative energy seems to apply to electromagnetic potential energy as well as gravitational potential energy. But once we define an arbitrary zero point of electromagnetic or gravitational potential energy, the changes from that are absolute. Are you calling “energy of position,” a term sometimes used for potential energy, “relative”? If so, why is just gravitational potential energy relative? Why not electromagnetic potential energy or, for that matter, the potential energy of the strong and weak nuclear interaction?

Without a clear definition to distinguish between “relative energy” and simply “energy” and without a clear explanation of why any physical phenomenon requires such a distinction, I am left with this conclusion.

You say you are not “objecting to the mathematics of general relativity despite the impressive agreement with observational evidence,” but rather the way the field equations are being interpreted.

To me, that remains a distinction without a difference, because no matter how people (including you) interpret the evidence and the theory, they conclude that some gravitational energy of the orbiting stars is transferred to the rest of the universe.

Your objection seems to be that others call that energy transfer radiation. Call it whatever you like, but don’t make such a big deal of it unless your perspective leads to testable predictions that differ from the established theory.

In the past, you have argued that gravitational waves will never be detected. The binary star system here is not a direct detection of gravitational waves, but it provides strong support for the notion.

When we discover a celestial event that would be expected to produce directly detectable gravitational waves, then you and I will be at last able to agree on what is happening here.

Until then, we are at an impasse.

The difference you cite between mass and rest mass is exactly the point I was making when I talked about gravitational waves as a “higher-order” effect. If you use “relative energy” as a synonym for a higher-order energy term that becomes significant as relative speeds approach c, then what do you call the higher order effects that show up where tidal effects are the non-relativistic equivalent? I am comfortable calling them gravitational waves.
End edit

As to your final point, Einstein is generally regarded as a brilliant writer and took his obligation to reach his audience quite seriously. His books about relativity for the educated layperson are excellent examples of how someone can take a challenging idea and meet his audience where they are sitting.

You might look at some of his writings for popular audiences before declaring that he didn’t care whether he reached non-expert readers.

I only hope I meet my obligation to communicate half as well when writing for my young readers or newspaper book review pages.

Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

13. Fred,

Itâ€™s always been difficult for people (even scientists) to listen to or read challenging views. You seem to believe Iâ€™ve been ambiguous on several points. To me itâ€™s clear youâ€™re trying to read while your head is shaking sideways (evidently producing some sort of perplexitational waves!).

Fred: â€œI am not even certain that you accept conservation of energy as a guiding principle here. (That principle treats all energy the same. There is no “relative” energy.)â€

I donâ€™t think thatâ€™s a position you want to be communicating. Like rest mass and relativistic mass, the energy of a body can have both relative and absolute components, depending on the frame of reference. The relative energy of a body depends entirely on its motion relative to another body, or in the case of potential energy, it depends on the relative position in a gravitational field.

Fred: â€œat times you seem to say that the energy lost in the orbital change is “only” relative and so it doesn’t need to be accounted for in the total energy of the universe. That implies you are willing to sacrifice conservation of energy in the service of your alternative theory.â€

You havenâ€™t been reading carefully, a common problem when someone is reading with a negative bias. In my most recent post, for example, I wrote â€œA loss of ‘gravitational energy’ (net kinetic/potential energy) to a system requires a transfer of kinetic/potential energy outside the system.â€

You continue with what amounts to an excellent example of how by insisting on a distinction which Iâ€™m denying, you accuse me of inconsistency, and/or of failing to appreciate the distinction:

Fred: â€œOr else you state that the energy must be due to a change in rotational energy of the stars due to tidal effects, even though tidal effects would lead to an out-spiraling (as we observe to a very small extent in the Earth-Moon system).â€

The geometric (â€œtidalâ€) fluctuations in a highly irregular system seems to be the culprit in orbital decay, by any interpretation of the results. My point – that the reduction in orbital (kinetic/potential) energy within the system produces an increase of energy of the same kind (kinetic/potential) outside the system – may conflict with the theory you endorse (the production of a special gravitational energy), but that doesnâ€™t make mine inconsistent with itself, it only makes it inconsistent with yours.

Fred: â€œYou insist that the mechanism [of orbital decay] is an undefined something else–whatever that something else is.â€

I have no problem with the conventional explanation, which refers to orbital decay resulting from the intensity, proximity, and asymmetry of the binary system.

Fred: â€œIn other words, my interpretation is that you are either (1) denying conservation of energy, (2) objecting to the mathematics of general relativity despite the impressive agreement with observational evidence, or (3) arguing semantics over the use of the term â€˜gravitational wavesâ€™ in an interpretation of that mathematical analysis.â€

As to (1) of course not; as to (2) Iâ€™m objecting to a derivative mathematics based on electromagnetic analogy, not the field equations of GR, and Iâ€™ve pointed out that mathematics can be physically ambiguous (and in fact there are, I think, at least 3 competing formulations of gravitational waves, each of which is in close approximation with observation); as to (3) I have no idea how you can reduce a distinction between relative energy and radiant energy to semantics.

Fred: â€œAs a writer, I never blame my audience for not “getting” what I am trying to explain. It is my burden to communicate it.â€

Copernicus, Galileo, and Einstein (and innumerable others) would object and disagree. You can lead a horse, a scientist, and a layman to whatever….

14. “The distinction here is irreconcilable.”

Perhaps, but the problem may be a failure to communicate. As someone with considerable experience in reaching audiences through words, let me make one more try to help you communicate your points in a persuasive way.

We have now returned to the point of previous discussions in which you are insisting that I want you to accept the theory before you can refute it, which is not true.

My point is not about the theory but rather effective written communication. The burden of effective communication is always the writer’s, not that of the readers the writer is trying to reach. And that means using standard terminology the readers understand and precisely and clearly defining nonstandard terms, such as “relative energy.”

When I read your posts, for example, I am not even certain that you accept conservation of energy as a guiding principle here. (That principle treats all energy the same. There is no “relative” energy.)

You seem to accept the fact that the gravitational energy of the pair of stars decreases in the specific situation we are discussing. But then at times you seem to say that the energy lost in the orbital change is “only” relative and so it doesn’t need to be accounted for in the total energy of the universe. That implies you are willing to sacrifice conservation of energy in the service of your alternative theory.

Or else you state that the energy must be due to a change in rotational energy of the stars due to tidal effects, even though tidal effects would lead to an out-spiraling (as we observe to a very small extent in the Earth-Moon system).

At other times, you seem to agree that the lost energy is transferred to the rest of the universe, thereby preserving conservation of energy, but you are unable to postulate a mechanism for that transfer.

The only thing you have to say about the mechanism is that it can’t possibly be gravitational waves, even though the mathematics that describes gravitational radiation predicts the observed decrease in orbital energy. You insist that the mechanism is an undefined something else–whatever that something else is.

In other words, my interpretation is that you are either (1) denying conservation of energy, (2) objecting to the mathematics of general relativity despite the impressive agreement with observational evidence, or (3) arguing semantics over the use of the term “gravitational waves” in an interpretation of that mathematical analysis.

If it is one of the first two, we are indeed at an impasse, since I see no evidence to overturn either of those well-established principles.

If it is the third, then I return to the phrase “distinction without a difference.”

If it is something else, then you are, unfortunately, failing to communicate–and I have tried mightily to understand you.

As a writer, I never blame my audience for not “getting” what I am trying to explain. It is my burden to communicate it.

Jim, with all due respect, you seem to be falling into the trap of blaming the readers when you refuse to use terminology that they consider fundamental to the discussion.

Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

15. On terminology:

Fred: â€œbodies orbit a common center of mass. One does not orbit the other.â€

Both statements are true, depending on whether you take the perspective of one of the binaries, or a perspective outside the system.

Fred: â€œâ€¦ you are using a nonstandard term here: â€˜relative energy.â€™ Total energy has an arbitrary constant, depending on how you define the zero point of potential energy. But otherwise, energy is absolute, not relative. Momentum is a relative quantity, and perhaps that’s where the confusion is arising.â€

Uniform motion is relative. Geodesic, i.e., orbital motion is uniform. The energy of uniform motion relative to another body (â€œkinetic energyâ€) is relative. The potential energy of a body is a potential, and only a potential, relative to another body.

On the issue:

Fred: â€œif you truly want to persuade me, please use standard physics terminology to make your point.â€

I donâ€™t think the problem is a non-standard terminology. Iâ€™ve described the transfer of energy between a binary system and the rest of the universe in standard terms of kinetic and potential energy, without need of energy-bearing â€œgravitational wavesâ€. What I read you (and others) as insisting is that I accept the standard interpretation of gravitational waves before I can criticize the standard interpretation of gravitational waves. How would that be possible? When I write that the standard interpretation is incoherent, the standard response is that it canâ€™t be incoherent because itâ€™s a standard â€“ i.e., it canâ€™t be incoherent because itâ€™s coherent. Please use standard, direct, and non-tautological logic to refute my argument.

Fred: â€œIt is common to describe higher-order effects using terminology that is different from the zeroth or first order effect. The important higher order effect here is what transfers energy from the orbiting pair to the rest of the universe. That higher order non-classical effect is commonly called gravitational waves.â€

Again, my issue isnâ€™t one of terminology. Iâ€™m questioning not just the terminology, not just the coherence, but the need and justification for positing a second-order effect.

A loss of electromagnetic energy to a system requires the transfer of electromagnetic energy outside the system in the form of a quantizable energy-bearing wave, with a corresponding loss of rest-mass and, in principle, an absolute and detectible effect on some element of the system. Thatâ€™s because electromagnetism is a force. A loss of â€œgravitational energyâ€ (net kinetic/potential energy) to a system requires a transfer of kinetic/potential energy outside the system. But there is no loss of rest-mass, nor detectible, non-relative effect on any element of the system. The only change is in the relationship of the elements of the system, and the shape of spacetime. Thatâ€™s because gravitation is not a force. So I hope you wonâ€™t â€œcontinue to see [my] discussion as a distinction without a difference.â€ The distinction here is irreconcilable.

16. Jim, I can’t discuss this using your terminology since I don’t see a “physical incoherence” here. General relativity is remarkably coherent with all evidence seen to date.

So, if you truly want to persuade me, please use standard physics terminology to make your point.

We agree on everything here, except the use of the term “gravitational waves.” To summarize:

We agree that we are discussing a phenomenon that arises from a gravitational interaction, and that interaction is described by the spacetime geometrical analysis of General Relativity.

I think we also agree that the classical limit of GR is what we call Newtonian physics, and that Newtonian physics describes the tides we see in everyday life.

I think we both agree also that GR is consistent with conservation of energy.

I think we also agree that it is a classical approximation to isolate the pair of stars from the rest of the universe.

What we are trying to get at is the difference between classical physics and GR.

It is common to describe higher-order effects using terminology that is different from the zeroth or first order effect. The important higher order effect here is what transfers energy from the orbiting pair to the rest of the universe. That higher order non-classical effect is commonly called gravitational waves.

It seems to me that the only thing you are arguing about is that particular terminology, which is why I continue to see your discussion as a distinction without a difference.

Science is often served by taking a different perspective. But in this case, I don’t see that your different perspective is leading to any new insight.

That doesn’t reflect poorly on you. It shows you are probing the universe. That’s to your credit.

All I am asking is for you to be content that after all your probing, you have come up with the same thing that others have, even though you prefer a different terminology.

Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

17. Fred wrote: â€œit seems to me that you are right when you say the two effects [â€˜tidal gravityâ€™ and â€˜gravitational wavesâ€™] arise from the same mathematics.â€œ

Iâ€™m saying they are physically the same thing, and should therefore arise from the same mathematics.

Fred: â€œBut you are oddly denying the higher order effects when other people choose to call them gravitational waves, even as you seem to acknowledge them in the rest of your discussion.â€

If you were to consider my argument in my terms, youâ€™d have to begin by reading it in my terms, not yours.

1: An orbital relationship between two bodies involves kinetic and potential energy (â€œgravitational energyâ€).

2: If there is a loss of gravitational energy between them, gravitational energy must be gained between them as a single system and the rest of the universe.

3: But gravitational energy, unlike electromagnetic energy, is relative (e.g., if the kinetic energy of a body increases as it accelerates toward the body itâ€™s orbiting, except for possible tidal effects, there is no detectible change to the accelerating body; itâ€™s a relative acceleration).

4: A change in gravitational energy between two bodies, or two systems, involves a change in spacetime geometry, not an exchange of quantums of energy or energy-bearing waves.

5: The transfer of gravitational energy cannot therefore be described, or expressed mathematically, in the same way as, or by analogy with, a transfer of electromagnetic energy.

I donâ€™t believe itâ€™s overstating the point to say itâ€™s incoherent to posit an absolute change arising from a purely relative change. And mathematics cannot rescue a physical incoherence.

18. SL, Burt, & Fred,

Iâ€™m sure youâ€™ll agree: Science values economy of explanation. Youâ€™ve argued for two principles, “gravitational waves” and “tidal gravity”; Iâ€™ve said they are one and the same. The question is: Are two needed and justified, or is one sufficient and exclusive? (I understand that the consensus is that there are two â€“ please donâ€™t respond that there are two because important physicists have developed mathematics which interprets them as two. Thatâ€™s begging the question, and mathematics can be physically ambiguous.)

Hereâ€™s an attempt at a simplified explanation of a singular (exclusively geometric) principle: Gravitational energy is an expression of spacetime geometry, the measure of local curvature relative to some other locality within the same concentration of spacetime. A location with relatively less curvature has relatively greater potential energy, less kinetic energy. A location with relatively more curvature has relatively greater kinetic, less potential energy. A binary system (two masses in mutual orbit) forms a dynamically irregular geometry, generating undulations of curvature relative to other bodies â€“ hence fluctuations in their kinetic/potential energies relative to the binary system. If the binaries are inspiraling, they create an increasingly concentrated geometry, producing less net energy (curvature differential) between them, and more net energy (curvature differential) between their combined concentration and the rest of the universe. The effect is wavelike (until the binaries reach singularity), but the waves produced are no different in kind from any changes in the shape of spacetime geometry, such as the fluctuation we experience with the orbit of our moon. In every case, a transfer of â€œgravitational energyâ€ is just a change in the shape of spacetime due to the relative movement of mass, whether recurring or not.

A consistently geometric description of gravitational phenomena is not only simpler, it avoids an inexplicable leap (Scruffy says â€œvoila!â€, I would say â€œpresto!â€) from relative energy to an absolute, force-like, presumably quantizable form that has not been observed, either as a special kind of wave or as a quantum.

Gravitation was once thought of as a kind of force; the idea of an energy-bearing gravitational wave is an extraneous legacy of that outmoded association. The time and money spent searching for force-like gravitational waves and gravitons is time and money il-spent.

19. Seems I wasn’t logged in when making that comment… The most recent “Anonymous” was “jarnold” before he had coffee…

20. Jim,

I’m trying to view this discussion from the position of an outsider who hasn’t grappled with general relativity in detail. Here’s what I am seeing.

Tidal effects show up in classical Newtonian interpretations, whereas both gravitational waves and tidal effects come from general relativity.

In other words, GR predicts both phenomena, but gravitational waves are a higher order effect that we can observe only in extreme cases.

Describing tidal effects with GR is similar to describing momentum and kinetic energy in a two-body collision using special relativity even at relative speeds much less than c. It works, but the math is unduly complicated. (Why worry about changes in mass and use KE=mc^2-m0c^2, where m0 is the rest mass? Use KE=(mv^2)/2 instead.)

To use your terminology, tidal energy is “relative” energy and is “local” to a two-body system. Gravitational radiation involves the rest of the universe.

Gravitational radiation is a higher-order effect in the GR math. As Burt points out, it is only 300 watts for the Earth-Moon system–obviously negligible compared to the tidal energy within the two-body system.

The energy exchange we can see in the Earth-Moon gravitational interplay–the tides–is leading to a slow recession of the Moon and a gradual lengthening of Earth’s day. If we add up all the translational and rotational energies involved and could measure them perfectly, we’d see a net decrease of 300 joules every second.

That decrease could not be accounted for by Newtonian mechanics. It is a higher-order effect that comes into play when we use GR instead. Tidal energy is the classical limit of tidal energy plus gravitational waves.

So it seems to me that you are right when you say the two effects arise from the same mathematics. But you are oddly denying the higher order effects when other people choose to call them gravitational waves, even as you seem to acknowledge them in the rest of your discussion.

In other words, I still think this is a case of a distinction without a difference. Computing tides using GR is possible but far more complicated mathematically than using Newtonian mechanics.

Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)