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Mathematics and gravitation theory

June 13, 2008 by jarnold

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.

Comments

Re: A BIG Difference!

July 22, 2008 by jarnold, 1 year 18 weeks ago
Comment: 31180

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.

Re: Moving to a related discussion

July 20, 2008 by jarnold, 1 year 18 weeks ago
Comment: 31168

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?"

‘Round and ‘round you go

July 20, 2008 by jarnold, 1 year 18 weeks ago
Comment: 31162

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.

You said it yourself.

October 25, 2009 by Anonymous, 4 weeks 6 days ago
Comment: 45784

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,

Algae.

A BIG Difference!

July 21, 2008 by Burt, 1 year 18 weeks ago
Comment: 31176

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 (www.Relativity-4-Engineers.com)

Moving to a related discussion

July 20, 2008 by Fred Bortz, 1 year 18 weeks ago
Comment: 31163

This discussion continues in circles, so I suggest a tangent about the spectra of gravitational waves at my blog.

I hope that will be more productive.

A request for Jim. Please edit your last post. You didn't close the final italics html tag, and it is messing up all the text that follows.

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

My answer should be obvious – how about yours?

July 19, 2008 by jarnold, 1 year 18 weeks ago
Comment: 31153

You guys are flailing.

Burt stated and asked: “The binary system emits total energy and the universe at large absorbs it. We have a mechanism called GWs. What is yours?”

As I’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 system’s gravitational field. And whereas an asymmetric oscillation of a binary system would produce a periodic, wavelike variation in the system’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 “mechanism” 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. I’ve pointed out elsewhere that an inspiraling orbit, like any relative acceleration in a gravitational field, is still a geodesic path. An object “falling” (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 – 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?

You’ve all three expressed a strident distaste for ducking and dodging. Don’t let yourselves down.

Amen, Jim!

July 19, 2008 by Burt, 1 year 18 weeks ago
Comment: 31156

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 (www.Relativity-4-Engineers.com)

So what's the difference, and what other predictions are there?

July 19, 2008 by Fred Bortz, 1 year 18 weeks ago
Comment: 31154

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 (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

So - what need for the impasse?

July 17, 2008 by jarnold, 1 year 19 weeks ago
Comment: 31140

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: “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”

You may be accurately describing the conventional view, but I want to point out that there can be no such thing as a “perturbed geodesic.” 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 can’t be consistently maintained – is gravitation a geometric principle, or is it a force, or is it both? To justify saying it’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 Fred’s question of the significance of my distinction between relative and non-relative energy, here’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 Feynman’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 we’re considering the possibility of energy-bearing gravitational waves, here’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 (it’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 don’t transmit and receive anything corresponds to an increase in absolute energy between a physical transmission and reception, don’t we have a problem of asymmetry? Don’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?

Jim, your'e ducking the issue!

July 17, 2008 by Burt, 1 year 19 weeks ago
Comment: 31144

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 “perturbed geodesic.” 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)

And Gadfly makes three

July 17, 2008 by Gadfly, 1 year 19 weeks ago
Comment: 31145

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."

I am not persuaded

July 17, 2008 by Fred Bortz, 1 year 19 weeks ago
Comment: 31141

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.

Edit added:
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)

Re: The Impasse

July 14, 2008 by jarnold, 1 year 19 weeks ago
Comment: 31104

Fred,

The problem of getting a fair hearing for an idea that challenges a conventional belief is well-recognized. I don’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? I’ve made a living as a technical writer. When cherished beliefs aren’t at stake I seem to be able to communicate successfully enough to be “gotten.”

Let’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 “get” 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. That’s when they had difficulty being accepted – before they were accepted. But you countered with reference to Einstein’s success in popularizing his ideas after he was well-established, when he commanded almost universal respect and confidence. You’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 isn’t reduced to quibbling, there’s nowhere to go but to the problem of how relative energy becomes force-like in “gravitational waves.” We have yet to go to that problem. I would love to finally go there. Why don’t we – finally go there.

The Fred-Jim Impasse

July 15, 2008 by Burt, 1 year 19 weeks ago
Comment: 31122

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)

So, Burt...?

July 15, 2008 by Fred Bortz, 1 year 19 weeks ago
Comment: 31123

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)

Re: So, Burt...?

July 15, 2008 by Burt, 1 year 19 weeks ago
Comment: 31124

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)

Re 2: Impasse

July 14, 2008 by Fred Bortz, 1 year 19 weeks ago
Comment: 31109

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)

A failure to communicate or a failure to evaluate?

July 14, 2008 by jarnold, 1 year 19 weeks ago
Comment: 31098

Fred,

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

Fred: “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.)”

I don’t think that’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: “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.”

You haven’t been reading carefully, a common problem when someone is reading with a negative bias. In my most recent post, for example, I wrote “A loss of 'gravitational energy' (net kinetic/potential energy) to a system requires a transfer of kinetic/potential energy outside the system.”

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

Fred: “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).”

The geometric (“tidal”) 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 doesn’t make mine inconsistent with itself, it only makes it inconsistent with yours.

Fred: “You insist that the mechanism [of orbital decay] is an undefined something else--whatever that something else is.”

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: “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.”

As to (1) of course not; as to (2) I’m objecting to a derivative mathematics based on electromagnetic analogy, not the field equations of GR, and I’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: “As a writer, I never blame my audience for not "getting" what I am trying to explain. It is my burden to communicate it.”

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

Impasse reached

July 14, 2008 by Fred Bortz, 1 year 19 weeks ago
Comment: 31100

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.

Edit added:
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)

That's not the problem, Fred

July 13, 2008 by jarnold, 1 year 19 weeks ago
Comment: 31092

On terminology:

Fred: “bodies orbit a common center of mass. One does not orbit the other.”

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

Fred: “… 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.”

Uniform motion is relative. Geodesic, i.e., orbital motion is uniform. The energy of uniform motion relative to another body (“kinetic energy”) is relative. The potential energy of a body is a potential, and only a potential, relative to another body.

On the issue:

Fred: “if you truly want to persuade me, please use standard physics terminology to make your point.”

I don’t think the problem is a non-standard terminology. I’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 “gravitational waves”. 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 can’t be incoherent because it’s a standard – i.e., it can’t be incoherent because it’s coherent. Please use standard, direct, and non-tautological logic to refute my argument.

Fred: “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.”

Again, my issue isn’t one of terminology. I’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. That’s because electromagnetism is a force. A loss of “gravitational energy” (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. That’s because gravitation is not a force. So I hope you won’t “continue to see [my] discussion as a distinction without a difference.” The distinction here is irreconcilable.

What we have here is a failure to communicate :)

July 14, 2008 by Fred Bortz, 1 year 19 weeks ago
Comment: 31095

"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)

re: Tidal effects vs. gravitational waves

July 11, 2008 by jarnold, 1 year 20 weeks ago
Comment: 31072

Fred wrote: “it seems to me that you are right when you say the two effects [‘tidal gravity’ and ‘gravitational waves’] arise from the same mathematics.“

I’m saying they are physically the same thing, and should therefore arise from the same mathematics.

Fred: “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.”

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

1: An orbital relationship between two bodies involves kinetic and potential energy (“gravitational energy”).

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 it’s orbiting, except for possible tidal effects, there is no detectible change to the accelerating body; it’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 don’t believe it’s overstating the point to say it’s incoherent to posit an absolute change arising from a purely relative change. And mathematics cannot rescue a physical incoherence.

Perhaps this is the problem, Jim

July 11, 2008 by Fred Bortz, 1 year 20 weeks ago
Comment: 31076

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 it’s orbiting, except for possible tidal effects, there is no detectible change to the accelerating body; it’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)

You've lost me again, Jim

July 11, 2008 by Fred Bortz, 1 year 20 weeks ago
Comment: 31074

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)

    • re: Gravitational waves as simply changes in curvature

    July 7, 2008 by jarnold, 1 year 20 weeks ago
    Comment: 30978

    Seems I wasn't logged in when making that comment... The most recent "Anonymous" was "jarnold" before he had coffee...

    Gravitational waves as simply changes in curvature

    July 7, 2008 by Anonymous, 1 year 20 weeks ago
    Comment: 30976

    SL, Burt, & Fred,

    I’m sure you’ll agree: Science values economy of explanation. You’ve argued for two principles, "gravitational waves" and "tidal gravity"; I’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 – please don’t respond that there are two because important physicists have developed mathematics which interprets them as two. That’s begging the question, and mathematics can be physically ambiguous.)

    Here’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 – 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 “gravitational energy” 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 “voila!”, I would say “presto!”) 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.

    Tidal effects vs. gravitational waves

    July 7, 2008 by Fred Bortz, 1 year 20 weeks ago
    Comment: 30981

    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)

    Re: Why complicate something simple

    July 2, 2008 by jarnold, 1 year 21 weeks ago
    Comment: 30915

    Fred wrote: Astronomers view a double star system and determine the orbital parameters. From those parameters, they can compute the total orbital energy (kinetic + potential).... Some time later, they observe the system again and note that the orbit has gotten smaller. They compute the new orbital energy, which is now less than before. This is not loss of "relative" energy but of actual mechanical energy. Unless you are prepared to throw out the law of conservation of energy, that loss must be accounted for.

    I'll make it as simple as I can: Geodesics are uniform, i.e., unaccelerated in their own frame of reference. Uniform motion is relative. Kinetic and potential energy are relative. Loss of kinetic+potential energy within a system equals gain of kinetic+potential energy between the system and the rest of the universe.

    O.K., that's simple, and it leads to a simple question

    July 3, 2008 by Fred Bortz, 1 year 21 weeks ago
    Comment: 30924

    Loss of kinetic+potential energy within a system equals gain of kinetic+potential energy between the system and the rest of the universe.

    That leads to this simple question: What accounts for this transfer of energy from the system to the rest of the universe?

    As Burt notes, one explanation is that it pops out of the theory of General Relativity as gravitational waves.

    And as SL notes, you are already talking about waves that propagate the energy outward.

    So, Jim, it seems as if you are talking about gravitational waves, though you call the hypothesis that they exist "groundless." Perhaps, as Burt notes, you are on the verge of grasping the distinction between tidal gravity and gravitational waves. It seems that all you need is to realize that gravitational waves are the waves that you have been describing in other terminology.

    It seems we talking about a distinction without a difference here.

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

    Re2: Why complicate something simple

    July 3, 2008 by Burt, 1 year 21 weeks ago
    Comment: 30923

    Jim, when you wrote: "Loss of kinetic+potential energy within a system equals gain of kinetic+potential energy between the system and the rest of the universe", you were, as SL wrote "almost there"! I have just one concern: that you still view tidal gravity and gravitational waves as the same thing, which they are not.

    A good, but imperfect analogy that I can think of is a pebble dropped onto a smooth pond, causing two distinct effects: (i) a splash, by which most of its kinetic energy is dissipated into the water in the immediate vicinity of the impact; (ii) ripples spreading out, carrying some of that energy to the edges of the pond.

    The splash is like the tidal gravity, extremely local and transferring energy into the spacetime medium. The ripples are like gravitational waves, carrying some of that energy away to the "far" region.

    In gravitational physics, this same "near" region and "far" region effects pop out of the theory.

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

    Re4: The relativity of orbits and collisions

    July 2, 2008 by jarnold, 1 year 21 weeks ago
    Comment: 30907

    Burt,

    I agree with… almost everything you say!

    But consider this: a binary star system, viewed from a vantage close enough to observe their mutual orbit in 3 dimensions. Each seems to wobble under the influence of the changing warp of the other’s gravitational field. From our common experience, we tend to associate a wobbling motion with inertial acceleration, analogous to the effect we feel when riding in a car on a mountain road, for example. But we know from direct evidence that when a body is being accelerated solely by changes in spacetime geometry, no acceleration is detectible, no force is apparent, however variable the body’s relative motion when viewed from another perspective. Think of an object falling from some elevation above the earth: Although a falling body may appear to be accelerating toward the surface, when considered from its own reference frame (e.g., a test particle in a box or a scientist in an elevator with a broken cable), there is no indication of being accelerated or acted upon by a force. It’s a purely relative acceleration.

    So whether in a gravitational wobble or in the cyclic transition between perihelion and aphelion, between orbital acceleration and deceleration, the motion of a body in its own reference frame remains uniform (i.e., a geodesic); its apparent accelerations are just that – relative to other bodies in the orbital system, and to the universe at large.

    We agree that orbital energy is a composite of kinetic and potential energy. In the binaries in question, orbital energy is the fluctuating combination of kinetic and potential energy relative to each other’s companion. What, therefore, is the nature of the orbital energy lost to the system when the orbits decay? Whether as kinetic or potential, there is nothing but relative energy involved, and therefore, nothing but relative energy lost. In terms of relativistic gravitation, there are changes in local spacetime geometry, just as there are with stable orbital relations. Gravitational accelerations are commonly thought of in terms of electromagnetic analogy, probably because we observe the behavior of binary systems from a distance where they appear to move in two dimensions, much like the alternating current in a radio transmitter. But we have local and direct experience of the effects of changes in a gravitational field – most dramatically, in the orbit of our moon. Energy is unleashed by such changes only where geodesics are obstructed at or beneath massive surfaces, and there only because their geodesics are reoriented, allowing in some cases a conversion of some formerly obstructed potential energy to kinetic energy. There is no reason to think such changes in spacetime geometry are any different with the binaries, where relative motion is unobstructed, and therefore, there is no reason to posit energy-bearing gravitational waves.

    It comes down to this: When distant gravitational phenomena are interpreted by electromagnetic analogy, we get energy-bearing gravitational waves, gravitomagnetism, and gravitons. When interpreted based on local, empirical, testable gravitational phenomena, we get changing shapes of spacetime geometry which can be expected to propagate as waves, but waves that produce only relative changes in geodesic motion (except where geodesics are obstructed, in or on massive bodies). The former interpretation is based on a hypothesis that cannot be correlated with local phenomena. The latter interpretation is based on direct, local, empirical evidence and is simpler, involving no hypostatization of an additional form of energy. It may be potentially disruptive to some projects and careers, but the latter is superior by consensus scientific standards.

    Re5: The relativity of orbits and collisions

    July 2, 2008 by Scruffy, 1 year 21 weeks ago
    Comment: 30918

    Jarnold writes:

    When interpreted based on local, empirical, testable gravitational phenomena, we get changing shapes of spacetime geometry which can be expected to propagate as waves, but waves that produce only relative changes in geodesic motion (except where geodesics are obstructed, in or on massive bodies).

    Jim, I think you are almost there! Those waves that you are talking about do transfer energy to remote locations of the universe - energy is conserved, but the binary pair have lost some. Viola!

    SL: Your Aerospace Watchdog

    Why complicate something simple

    July 2, 2008 by Fred Bortz, 1 year 21 weeks ago
    Comment: 30910

    Jim, you're losing me here. Perhaps Burt can reinterpret.

    But it seems that you are making a simple story complicated.

    Astronomers view a double star system and determine the orbital parameters. From those parameters, they can compute the total orbital energy (kinetic + potential).

    Some time later, they observe the system again and note that the orbit has gotten smaller. They compute the new orbital energy, which is now less than before. This is not loss of "relative" energy but of actual mechanical energy. Unless you are prepared to throw out the law of conservation of energy, that loss must be accounted for.

    The astronomers apply the gravitational equations of General Relativity, which can be considered a purely geometric computation in spacetime (not a force calculation), and predict a certain rate of radiation of gravitational energy. Their prediction of net radiated energy agrees with the decease [EDIT: that should be decRease, of course :)] that is observed.

    Meanwhile, you continue to insist that it makes no sense to have gravitational waves, despite the agreement of computation and observation. To me, that is the sum total of your convoluted statement. If there is more to it than that, you are, unfortunately, failing to communicate it.

    As I noted, mere insistence that gravitational waves can't exist is not an argument. To persuade others that your viewpoint has any value, you need to make an argument that has the potential of leading to an alternate explanation of the evidence.

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

    Re2: The relativity of orbits and collisions

    June 27, 2008 by Anonymous, 1 year 22 weeks ago
    Comment: 30838

    Burt,

    That’s a beautiful diagram. You should sell prints! And I take your point - another factor in unstable orbits is that both bodies must be relatively massive.

    I’ve tried to be specific about my argument, that grav waves and gravitomagnetism (and gravitons) are unwarranted extrapolations from the field equations, because they postulate an absolute form of energy arising from changes in relative motion. I’ve pointed out that mathematics can be misleading because it's physically indifferent. I’ve offered a general alternative hypothesis on what happens to energy lost in orbital systems (it’s a transfer of kinetic energy to the universe at large), although an alternative isn't needed to demonstrate an unwarranted theory.

    Maybe you can offer a conceptual justification for the mathematics of grav waves – how (presumably) either 1) “orbital energy” is just kinetic energy in orbit, and it somehow gets transformed into a gravitational form of radiant energy when orbits decay, or 2) “orbital energy” is something in addition to kinetic energy.

    Re3: The relativity of orbits and collisions

    June 27, 2008 by Burt, 1 year 22 weeks ago
    Comment: 30845

    Thanks Jim. The picture is just educational, so we'll leave it at that.

    You wrote: "I’ve offered a general alternative hypothesis on what happens to energy lost in orbital systems (it’s a transfer of kinetic energy to the universe at large), although an alternative isn't needed to demonstrate an unwarranted theory."

    But this is essentially what gravitational waves (GWs) are supposed to do! The process of transferring that energy is by means of 'ripples' in the curvature of spacetime propagating at the speed of light.

    It is analogues to EM radiation, although not the same. In EM radiation, a charged particle 'jumping up-and-down' here, causes a charged particle to 'jump down-and-up' there. In GW radiation, a mass 'jumping up-and-down' here, causes a mass to 'jump down-and-up' there - not quite, but close!

    Jim: " ... how (presumably) either 1) “orbital energy” is just kinetic energy in orbit, and it somehow gets transformed into a gravitational form of radiant energy when orbits decay, or 2) “orbital energy” is something in addition to kinetic energy."

    We have said many times that orbital energy is defined as the total mechanical energy of an orbit, i.e., kinetic + potential energy, leaving out pressure, thermal, electrical charge and so on. It is useless to only consider kinetic energy, e.g., at the perihelion and aphelion of a planet's orbit, it has identical orbital energies, yet totally different kinetic energies.

    It is obviously true that robbing an orbiting body of kinetic energy without a compensating increase in potential energy will reduce the total orbital energy, but the same can be said for potential energy. In the end it is a loss of total orbital energy that causes tight binary orbits to spiral in.

    That lost energy (being it kinetic, potential, or both) is transferred to the universe at large and the consensus today is that the medium is spacetime and the agents are ripples in the medium (GWs, not gravitons, yet...)

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

    The relativity of orbits and collisions

    June 23, 2008 by jarnold, 1 year 22 weeks ago
    Comment: 30783

    Burt,

    On point (1) I agree in part - tidal effects, the cumulative effects of a gravitational gradient stressing binding energies, can affect orbits, but that's not exactly what you're talking about, and I think it's secondary to your point (2).

    You ask where orbital energy goes when an orbit inspirals. Consider a body that approaches another with a trajectory so steep it doesn't complete a full orbit before colliding. Its trajectory can be the same as the final trajectory of one member of an inspiraling binary, it has a brief and rapidly decreasing "orbital energy", and yet its path is essentially the same as a free-fall: potential energy is lost, kinetic energy is gained in its (geodesic) approach.

    Is there a "radiation" of "gravitational energy"? To a relatively insignificant extent, whenever two bodies combine, the rest of the universe gains kinetic energy toward their concentrated masses, as the combined masses gain kinetic energy toward the slightly contracted universe. So beyond the conversion of potential to kinetic energy between the colliding bodies, that's where additional energy goes - to a more general conversion of potential to kinetic.

    I lean toward a hypothesis that all orbits between bodies of unequal mass are inherently unstable, ranging in degree from insignificant to severe, to the extent that there is an imbalance of mass and a close proximity. I don't want to get drawn into discussing an alternative explanation before gravitational waves are disposed, but it's the only way I know to answer your question directly. In this view, orbits (with the exception of binaries of equal mass in ideal mutual orbit) are just more or less rapid inspirals, and all gravitational collisions are just more or less inadequate orbits, all explicable in terms of conversions between kinetic and potential energy. Maybe someone could do the math. In any case, whatever the merit of this hypothesis, the idea of gravitational energy fails to explain the conversion at-a-distance of relative energy to radiant energy. Please, let's not orbit the issue.

    Re: The relativity of orbits and collisions

    June 23, 2008 by Burt, 1 year 22 weeks ago
    Comment: 30785

    Hi Jim, you wrote: "Please, let's not orbit the issue."

    I'm sorry, but I can't comply ;)

    I've done the math and here is a plot of the orbit of a single particle coming very close to a black hole. Without gravitational waves (particles around black holes can't generate them), there is no evidence of "all orbits between bodies of unequal mass are inherently unstable".

    Stunning picture, don't you agree? The colors are my own doing by changing the color for every new orbit, just for effect. The parameters of the orbit were chosen to allow a slow (apparent) precession of the orbit. Actually, the precession is more than
    360 degrees, as the orbit comes to within 3 times the event horizon radius of the black hole. If not disturbed, the particle can orbit like this forever.

    When the particle is replaced by another massive body, the orbit "spirals" into the hole, due to the radiation of gravitational waves, using the proper differential geometric math.

    BTW, two massive bodies on a near-straight collision course only start to radiate gravitational waves after they have collided and the shape of the coalesced single body vibrates violently. The bodies actually maintain their "orbital energy" until they make contact. It is not true that an infalling body loses energy, unless it spirals in and is massive enough to create the "wiggling" of the massive bodies required for gravitational wave radiation. That's the standard GR view.

    Regards,

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

    Orbits, collisions, and "lost" energy

    June 24, 2008 by Fred Bortz, 1 year 22 weeks ago
    Comment: 30795

    Burt, you write:
    When the particle is replaced by another massive body, the orbit "spirals" into the hole, due to the radiation of gravitational waves, using the proper differential geometric math.

    Isn't this the place where Jim parts company? He claims the in-spiraling is due to something other than gravitational waves.

    Consider the orbit of the earlier discussed binary system.

    You've given me a satisfactory explanation of why the orbit at time T2 is smaller than the earlier orbit at T1. Jim hasn't even acknowledged that the smaller orbit has less total orbital energy.

    Jim keeps talking about an exchange of kinetic and potential energy, but no net loss of orbital energy. Either he doesn't understand orbital mechanics enough to know that there is a simple relationship between orbit shape and energy, or he is trying to avoid answering the question that you and I have both asked: What explains the loss of orbital energy between T1 and T2?

    Since energy is conserved, the orbital energy has to appear somewhere else. The mathematics of GR says that it appears in gravitational waves and predicts the amount of energy radiated.

    Given your detailed explanation, I would say that Jim can no longer call gravitational waves "a groundless hypothesis," which means he needs to answer our question about where the orbital energy goes.

    As Gadfly suggested, it appears that Jim needs to "put up or shut up." Don't you agree, Burt?

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

    Re: Orbits, collisions, and "lost" energy

    June 24, 2008 by Burt, 1 year 22 weeks ago
    Comment: 30801

    Hi Fred.

    Your summary is correct, but I'm mostly more lenient than to say: "As Gadfly suggested, it appears that Jim needs to "put up or shut up." ". Even if Jim cannot put something concrete on the table for explaining the energy loss, it is normally educational for all of us to read/think/write about the opposing arguments.

    Regards,

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

    Re2: Orbits, collisions, and "lost" energy

    June 25, 2008 by Fred Bortz, 1 year 22 weeks ago
    Comment: 30809

    Burt, you write:
    Even if Jim cannot put something concrete on the table for explaining the energy loss, it is normally educational for all of us to read/think/write about the opposing arguments.

    You are a kinder man than I, and most people usually consider me a nice guy.

    Of course I think it is educational to consider opposing viewpoints.

    The problem is that Jim has, in my view, failed to give an opposing argument.

    He just calls the prediction of gravitational waves "a groundless hypothesis," despite its grounding in a well-developed theory and its observational support. A denial like that is not an argument, just semantics.

    So when I echo Gadfly's call to put up or shut up, I guess I am telling Jim that his mere denial of gravitational waves has not risen to the level of "an argument."

    My challenge to Jim all along is this: Make a real argument. That means you have to go beyond saying, "There is no way that gravitational waves can possibly exist, no matter where the evidence seems to be pointing."

    Whenever I posed that challenge, he failed to provide anything more that could be "educational." That's what has made this discussion so frustrating for me.

    The closest analogy I can see to this is the argument some people make for Intelligent Design. Their theory does not consider the possibility that the "designer" does not exist, even in the face of evidence that has discredited their key hypothesis of "irreducible complexity." (I'll be posting a review of a book about that issue after it appears in print on Sunday.)

    Without offering an alternative theory, or at least the beginnings of one, the mere claim that gravitational waves do not exist has no educational content or value.

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

    Jim Learning Relativity?

    June 25, 2008 by Burt, 1 year 22 weeks ago
    Comment: 30813

    Fred, you wrote: "My challenge to Jim all along is this: Make a real argument. That means you have to go beyond saying, "There is no way that gravitational waves can possibly exist, no matter where the evidence seems to be pointing."

    Whenever I posed that challenge, he failed to provide anything more that could be "educational." That's what has made this discussion so frustrating for me."

    Frustrating or not, Jim has (or had) his idea of what's going on with the inspiraling neutron stars and he stated it. I like to think that our efforts have helped him to see that his ideas were not quite tenable and perhaps Jim has learned something.

    If this is the case and, even better, if any other reader has learned something out of the exchange, I think it was worthwhile. ;-)

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

    Gadflies and watchdogs

    June 22, 2008 by jarnold, 1 year 22 weeks ago
    Comment: 30760

    Such an excitable boy is “Gadfly.” Let me try to soothe his savage breast.

    He and others insist that a criticism of a hypothesis must be accompanied by a positive alternative. But it’s enough to identify a problem with an existing hypothesis, however well-established, without offering an alternative. If a person making a criticism doesn’t have a definite alternative to offer, challenging the established hypothesis might nonetheless encourage the development by others of fruitful alternative interpretations that wouldn’t otherwise be explored. It's okay. Don't let it disturb you.

    A defender of the faith might seek to divert attention from an issue by making aggressive ad hominem comments, by appealing to the sacrosanct status for an established hypothesis and its author, by nitpicking terminology or side-issues, or by insisting that an alternative hypothesis be discussed instead. Who would consciously want to do such things?

    There’s “Scruffy,” who is at least clear about the role he seeks to play by calling himself a “watchdog” rather than a “gadfly”:

    “Scruffy”: I think Burt has stated repeatedly that the equations of GR show exactly [how a relative phenomenon becomes absolute] - what more do you want? It is no excuse if you are ignorant of the mathematics of gravitational waves (which I must confess is a bit above my head as well!)…. Try to understand what greater minds than yours and mine are trying to tell us before challenging.

    That’s a remarkable statement for someone whose interests seem to incline to science rather than religion. Who would ever challenge "greater minds"?

    I’ve pointed out that mathematics is ultimately indifferent to physics, that a close correspondence of mathematics to evidence means nothing if the suggested physics is incoherent. Put simply, in this case, a change in the relative kinetic energy between bodies doesn’t produce radiant energy. Gravitation is relative - testable gravitational phenomena do not involve absolute exchanges of energy unless geodesics happen to conflict in collisions. Think about it, stop relying on “greater minds.” You’ll feel better about yourself.

    Re: Gadflies and watchdogs

    June 23, 2008 by Burt, 1 year 22 weeks ago
    Comment: 30767

    Hi Jim, you wrote:

    "Gravitation is relative - testable gravitational phenomena do not involve absolute exchanges of energy unless geodesics happen to conflict in collisions."

    I understand what you try to say here, but technically you are wrong on two counts:

    1. Take the Earth and Moon: tidal gravity is transferring an enormous amount of rotational energy from the Earth to the Moon's orbital energy, causing the separation between Earth and Moon to become larger. The total energy of the 'closed system', including the rotational energies of the Earth and Moon, are not changed by this; it is just a conversion. [Edit: not exactly true either: some of that energy is wasted as heat and is radiated away, but the amount is negligible compared to what is transferred to the orbit]
    2. Gravitational waves are robbing the Earth-Moon of a very tiny amount of that total energy (some 300 W continuously). This energy is radiated away, so it is not a conversion, but a loss.

    The 300W is not observable (too tiny), but in binary star systems, orbiting at close quarters, the net loss of total mechanical energy is severe and very observable. That is despite the fact that the binary systems also convert rotational energy into orbital energy by means of tidal gravity, trying to increase the orbital radius. Despite this, the orbital radius decreases[1] and hence the orbital energy reduces remarkably.

    Where do you think that energy goes?

    Regards,

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

    Note:
    [1] If an orbit is circular its total orbital energy decreases with its radius. If it is elliptical, its energy decreases with the square root of the product of the semi-major and semi-minor axes. In an inspiral, both semi-major and semi-minor axes decrease in size - it's not a simple exchange of energy. [Edit2: in relativity this only holds for orbits that never enter the "near-region" (defined as r <= 6GM/c^2). Inside that radius, orbits actually need more energy the lower they get and the orbits are unstable. A tiny loss of energy causes a rapid inspiral.]

    No problem identified, as far as this gadfly can tell

    June 22, 2008 by Gadfly, 1 year 22 weeks ago
    Comment: 30763

    Jim continues his lackluster bluster:

    But it’s enough to identify a problem with an existing hypothesis, however well-established, without offering an alternative.

    Jim, you claim to have identified a problem with the hypothesis (gravitational waves) and insist it is groundless.

    In fact, not only is there no problem with that hypothesis, which arises from the well-grounded theory of GR, but also a growing body of evidence, such as Burt's example binary star system, is supporting that hypothesis.

    With both mathematics and observation supporting the prediction of gravitational waves, the "problem" that you have "identified" simply does not exist. The problem is your stubborn refusal to accept both the math and the data.

    This bite of realism brought to you by "Gadfly."

    Gravitation: Key answers dodged

    June 20, 2008 by jarnold, 1 year 23 weeks ago
    Comment: 30736

    I’ll respond to several recent comments in the following:

    Fred - If you had read my article more closely, your evident frustration and disdain might have been avoided.

    “I tried to get Jim to show how his idea is a worthy challenger to General Relativity, but he just dismisses the consensus because he will not accept gravitational waves.”

    I don’t challenge GR, I’ve pointed out that the extrapolations on GR by electromagnetic analogy are without foundation. I don’t “just dismiss” the consensus, I show that the direct evidence we have of gravitational phenomena is without energy-exchange, whereas the hypothesis you’re defending presumes an energy exchange.

    Fred: He offers no explanation for the loss of orbital energy in the observed systems, but continues to insist that there is no mechanical energy lost in the in-spiraling--a manifest misunderstanding of orbital mechanics, in which potential and kinetic energy are continuously being interchanged in non-circular orbits.

    The interchange between potential and kinetic energy is zero-sum (the increase of kinetic equals the decrease of potential, and vice versa). You should know that.

    Fred: Merely calling the hypothesis that gravitational waves exist "groundless" does not make it so. It is the consensus hypothesis because, to the scientific community, it is solidly grounded in a rich and broadly applicable theory.

    Again, I didn’t “merely” call the hypothesis groundless. Read the article again. (Or not.) I showed that we have no direct evidence for energy exchange in testable gravitational phenomena. On the contrary, the evidence shows no exchange of anything but relative energy. The hypothesis is therefore groundless.

    Fred: Especially since it is the consensus, the burden is on you to show that your alternative proposal is at least as good.

    Actually, as you should know, it is sufficient to demonstrate that a hypothesis is incorrect, without offering an alternative explanation. Michelson & Morley showed that light has the same velocity in all directions; they didn’t need to formulate Special Relativity or any alternative explanation. I’m sure you can come up with numerous such examples in the history of science.

    Fred: … you need to present your theory in sufficient detail to show how well it matches the observations that Burt cites, as well as all other observations where General Relativity has demonstrated its validity.

    You may have somehow overlooked the testable observations I provided where gravitation involves no exchange of energy, other than purely relative potential and kinetic energies.

    Fred: What you call "regular and manifest gravitational waves" are called tides in my book.

    They are, nonetheless, wavelike - in any book.

    Fred: Also, there is nothing "exotic" about the gravitational waves predicted by General Relativity...

    Exotic means “of foreign origin.”

    SL wrote: Gravitational wave energy loss it is precisely the deviation from this simplistic view [potential + kinetic is a zero-sum] that is apparent in observation.

    As I wrote in response to Fred, the zero-sum is the net change in potential + kinetic energy. More to the point:

    SL: … when particles or even spacecraft orbit any planet or other massive body, there is negligible gravitational wave energy being radiated. In such a case Jim is right - geodesics do not radiate energy…. It is when two massive bodies orbit each other that the phenomenon is predicted by relativity theory and it agrees with the inspiral of binary star systems in tight orbits.

    Here you’ve come to the crux. You seem to be saying (I believe anyone who believes in gravitational waves has to be saying) that gravitational wave radiation is negligible except in cases where mass and proximity are significant. Presumably, the same principle applies in all such relationships, but the measure may be vanishingly small. So if, as I’ve pointed out, there is no energy exchange when a body orbits another, as each follows its geodesic (however variable the geodesics may appear from another reference frame) a coherent hypothesis for gravitational waves has to explain how a relative phenomenon becomes absolute, generating energy-bearing waves outside a purely relational system.

    SL: [effects like the orbit of the moon] are very different from gravitational waves. For one, tidal gravity energy falls off with the inverse cube of distance, while gravitational wave energy falls off with the inverse square of distance. I think it's time Jim gets a handle on this.

    You want me to acknowledge what I deny – either, in your opinion, I have to accept your premise before I can challenge it, or I can’t, in principle, challenge it. Request to Fred: Look up “dogmatism” while you’re looking up “exotic.”

    Burt wrote: The gravitational wave equations say that [unless the bodies are massive enough and close enough] … the energy radiated will be utterly negligible. If both are very massive, the energy radiated will become noticeable, provided that the separation (R) between the bodies is relatively small.

    Thanks to Burt’s explicit description of the mathematics of the principle, it’s clear that there is something non-zero in any orbital relationship that would generate gravitational waves. So my question is: How is it that any two bodies following their geodesics in a mutual orbit, undergoing periodic changes in their purely relative kinetic energies, can radiate a “gravitational energy”?

    Fred wrote: If gravitational energy is quantized, the system emits gravitons as it drops to a lower energy level, just as an atom emits a photon when an electron drops to a lower energy.

    This is a third aspect of the force-analogy used in post GR physics that I haven’t touched upon. Like the others, it hasn’t been confirmed by direct evidence. Although it’s at least explicit and internally coherent, it’s the most remarkable of the force-analogies (i.e., gravitational waves and gravitomagnetism) because it contradicts the geometric concept of gravitation (and all the force-free direct evidence) without attempting to refute it.

    Re: Gravitation: Key answers dodged

    June 21, 2008 by Scruffy, 1 year 22 weeks ago
    Comment: 30747

    Jim wrote:

    So if, as I’ve pointed out, there is no energy exchange when a body orbits another, as each follows its geodesic (however variable the geodesics may appear from another reference frame) a coherent hypothesis for gravitational waves has to explain how a relative phenomenon becomes absolute, generating energy-bearing waves outside a purely relational system.

    I think Burt has stated repeatedly that the equations of GR show exactly that - what more do you want? It is no excuse if you are ignorant of the mathematics of gravitational waves (which I must confess is a bit above my head as well!)

    Jim wrote:

    You want me to acknowledge what I deny – either, in your opinion, I have to accept your premise before I can challenge it, or I can’t, in principle, challenge it.

    It's not my premise that you have to accept. Try to understand what greater minds than yours and mine are trying to tell us before challenging. This is the "handle" that I was talking about. Your ignorant claim of flaws in a theory of gravitational waves that you apparently do not understand is getting you nowhere.

    SL: Your Aerospace Watchdog

    Re2: Gravitation: Key answers dodged

    June 22, 2008 by Burt, 1 year 22 weeks ago
    Comment: 30758

    Hi SL.

    Jim wrote:

    "So if, as I’ve pointed out, there is no energy exchange when a body orbits another, as each follows its geodesic (however variable the geodesics may appear from another reference frame) a coherent hypothesis for gravitational waves has to explain how a relative phenomenon becomes absolute, generating energy-bearing waves outside a purely relational system."

    To which you replied:

    "I think Burt has stated repeatedly that the equations of GR show exactly that - what more do you want?"

    Jim might argue that there may be other ways for binary systems to lose orbital energy and spiral inwards. It is known that the accretion disks of black holes emit enormous amounts of electromagnetic radiation, caused by the friction and compression of gas and dust outside the black hole. This continuously robs each particle of some total orbital energy and hence they spiral inwards.

    Astrophysicists did consider every other known form of energy loss in binary star systems, but came up far short of the observed loss rates in binary neutron star orbits. Only GR's gravitational waves were found to 'balance the books' of the observed inspiral.

    This does not completely rule out other mechanisms, but it is very unlikely for GR, which is so right about so many things, to be wrong about this one. Here we are not working with the very small or the super energetic, so no quantum effects (where GR may come up short) are involved.

    Regards,

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

    jarnold - Put up or shut up

    June 20, 2008 by Gadfly, 1 year 23 weeks ago
    Comment: 30738

    Jim,

    You are challenging a prediction of General Relativity.

    Burt has given you some evidence that shows a binary star system is inspiraling. Burt and Fred have pointed out that the inspiraling means the orbital energy of the system is decreasing.

    Burt notes that the decrease is consistent with GR's prediction of gravitational waves. It isn't a direct observation of gravitational waves, but it is consistent with the hypothesis that they exist. In addition, the rate of decrease agrees with the rate predicted by GR. The agreement is quantitative as well as qualitative.

    That consistency counters your claim that the existence of gravitational waves is a "groundless" hypothesis.

    You don't have to accept that hypothesis, Now is your chance to show how your theory explains the same phenomenon without gravitational waves.

    That's the only way your theory can gain credibility. Stop responding to others and simply show us how your theory explains the evidence.

    Put up or shut up!

    These bites of realism brought to you by "Gadfly."

    Re: Interpretation and gravitation theory

    June 17, 2008 by Burt, 1 year 23 weeks ago
    Comment: 30674

    Jarnold wrote:

    "And what exactly is “orbital energy”? According to the geometric theory of gravitation, a body in orbit around another is moving uniformly along its geodesic. The only energy involved in this is a purely relative kinetic energy."

    Perhaps having to ask this question indicates a certain lack of knowledge about the issue. I would recommend reading just about any technical book on GR to lay a foundation. Briefly, orbital energy is the sum of the potential and kinetic energies of the orbiting body, a la Newton, but modified by the components of curved spacetime in GR.

    Other relevant comments made to Fred's questions.

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

    PS: I tried to post the equation for (mechanical) orbital energy in GR (because it 'speaks' more than many words), but without success so far. The equation can be viewed here.

    The symbols used are:

    E~ is the orbital energy per unit orbiting mass.

    g_tt is the time-time coefficient and g_rr the space-space coefficient of the Schwarzschild metric. Both are functions of the gravitating mass (M) and the distance from the mass (r): g_tt = 1-2GM/(rc^2) and g_rr = 1/g_tt, with G and c having their usual meanings.

    v_r is the radial and v_t the tangential orbital velocity components respectively.



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