# Causal mechanism for gravity

#### RJBeery

##### Natural Philosopher
Valued Senior Member
A Causal Mechanism for Gravity

Essay written for the Gravity Research Foundation 2020 Awards for Essays on Gravitation

rjbeery@gmail.com

Abstract
In this speculative paper, we show that electromagnetic (EM) mass and general relativistic time dilation are sufficient to predict gravitational attraction.

Time Dilation as Refraction
First, we consider light moving slowly through a local medium with a large refractive index; we then observe a remote light ray moving slowly in a large gravitational field due to relativistic time dilation, such that their respective apparent velocities are equal, and recognize the opportunity for a potential equivalence. Exploring this, we create a spherical refractive medium whose index varies with the distance from its center by the following:

where r is the distance from the center of the object and rs is the Schwarzschild radius of some gravitational object O with mass m which we are attempting to emulate.

What we discover is that light passing through such an object at a given radius r will behave identically as it would while passing by O at the same radius. This phenomenon, known as the optical-mechanical analogy (or more recently as F=ma optics), has been well-established and extensively studied over the last century. [ref 1-4]

As Sir Arthur Eddington wrote [ref 5] in his famous 1920 summary of General Relativity, “Space, Time and Gravitation”:

We can thus imitate the gravitational effect on light precisely, if we imagine the space round the sun filled with a refracting medium which gives the appropriate velocity of light. To give the velocity 1 − 2m/r, the refractive index must be 1/(1 − 2m/r), or, very approximately, 1 + 2m/r. At the surface of the sun, r = 697, 000 km., m = 1.47 km., hence the necessary refractive index is 1.00000424. At a height above the sun equal to the radius it is 1.00000212.

Any problem on the paths of rays near the sun can now be solved by the methods of geometrical optics applied to the equivalent refracting medium. It is not difficult to show that the total deflection of a ray of light passing at a distance r from the centre of the sun is (in circular measure)

whereas the deflection of the same ray calculated on the Newtonian theory would be

.

For a ray grazing the surface of the sun the numerical value of this deflection is

1”.75 (Einstein’s theory),

00”.87 (Newton’s theory).

The efficacy of the “F=ma optics” is without doubt, however, respective authors on the subject are careful to stress the purely analogous nature of the relationship. We would like to suggest that it isn’t an analogy at all, but rather a literal equivalence.

EM Mass
Let us envision an electromagnetic wave, with a wavelength of 2.43 * 10-12 m, moving in a periodic cycle which takes it back upon itself such that it becomes a self-reinforcing soliton. The complete orbital path length of this EM wave is equal to its wavelength but is such that it makes a double-loop. (see Fig 1)

Such a quasi-symmetrical object, if stable, would resemble an electron. It would have a physical radius of on the order of 2.43 * 10-12 m / 4, an electric field, a magnetic dipole, and a half-integral spin [ref 6]. It would also offer a physical manifestation of Einstein’s mass/energy equivalence (e.g. “releasing” the photon from its self-contained path would result in a burst equal to its “rest energy”).

Cosmic Speed Limit
Philosophically, many of us have been mystified by the limiting nature of c. EM mass might provide a straight-forward explanation -- a photon turning back upon itself does not follow the traditional geodesic between two points. As an EM mass particle is accelerated, a larger portion of its photon’s circuit is thus spent moving in the direction of its velocity; this percentage can be arbitrarily close to, but not quite, 1. (see Fig 2)

Transverse Waves
If we refer back to our sphere of graded refractive index, we would expect that the path of light moving radially to it would remain unaffected; only a light’s path with a transverse component would be altered. The photon of an EM mass particle moving in a closed circuit within the sphere would possess a transverse portion of its path relative to the center of the medium in a range between .5 and 1, depending upon the relative velocities of the sphere and the particle. This could manifest as relativistic mass.

Conclusion
In this paper we have shown the connection of optics to the gravitational bending of light in a graded time dilation field. Additionally we have shown that if mass were to possess an electromagnetic nature moving in a cyclic fashion (i.e. “EM mass”) then we are able to precisely predict the gravitational behavior of that mass in the presence of such a time dilation field without invoking any other mechanism related to General Relativity. Lastly, we are able to show that this model may plausibly explain other aspects of Relativity, such as the limiting speed of light and relativistic mass. We feel that these aggregate theories provide ample potential to warrant further investigation.

References

[1]The optical-mechanical analogy for stationary metrics in general relativity; Paul M. Alsing; American Journal of Physics 66, 779 (1998); https://doi.org/10.1119/1.18957

[2] The optical-mechanical analogy in general relativity: Exact Newtonian forms for the equations of motion of particles and photons; James Evans, Kamal K. Nandi & Anwarul Islam; General Relativity and Gravitation volume 28, pages 413–439(1996)

[3] ‘‘F=ma’’ optics; James Evans and Mark Rosenquist; American Journal of Physics 54, 876 (1986); https://doi.org/10.1119/1.14861

[4] Sumana Bhadra, Electromagnetic Mass Models in General Theory of Relativity: https://arxiv.org/abs/0710.5619

[5] Space, Time and Gravitation; Sir Arthur Eddington; http://www.gutenberg.org/files/29782/29782-pdf.pdf

[6] Is the electron a photon with toroidal topology? J.G. Williamson and M.B. van der Mark, 1997; Annales de la Fondation Louis de Broglie, Volume 22, no.2, 133. http://home.claranet.nl/users/benschop/homepg2/electron.pdf

A Causal Mechanism for Gravity

Essay written for the Gravity Research Foundation 2020 Awards for Essays on Gravitation

rjbeery@gmail.com

Abstract
In this speculative paper, we show that electromagnetic (EM) mass and general relativistic time dilation are sufficient to predict gravitational attraction.

Time Dilation as Refraction
First, we consider light moving slowly through a local medium with a large refractive index; we then observe a remote light ray moving slowly in a large gravitational field due to relativistic time dilation, such that their respective apparent velocities are equal, and recognize the opportunity for a potential equivalence. Exploring this, we create a spherical refractive medium whose index varies with the distance from its center by the following:

where r is the distance from the center of the object and rs is the Schwarzschild radius of some gravitational object O with mass m which we are attempting to emulate.

What we discover is that light passing through such an object at a given radius r will behave identically as it would while passing by O at the same radius. This phenomenon, known as the optical-mechanical analogy (or more recently as F=ma optics), has been well-established and extensively studied over the last century. [ref 1-4]

As Sir Arthur Eddington wrote [ref 5] in his famous 1920 summary of General Relativity, “Space, Time and Gravitation”:

We can thus imitate the gravitational effect on light precisely, if we imagine the space round the sun filled with a refracting medium which gives the appropriate velocity of light. To give the velocity 1 − 2m/r, the refractive index must be 1/(1 − 2m/r), or, very approximately, 1 + 2m/r. At the surface of the sun, r = 697, 000 km., m = 1.47 km., hence the necessary refractive index is 1.00000424. At a height above the sun equal to the radius it is 1.00000212.

Any problem on the paths of rays near the sun can now be solved by the methods of geometrical optics applied to the equivalent refracting medium. It is not difficult to show that the total deflection of a ray of light passing at a distance r from the centre of the sun is (in circular measure)

whereas the deflection of the same ray calculated on the Newtonian theory would be

.

For a ray grazing the surface of the sun the numerical value of this deflection is

1”.75 (Einstein’s theory),

00”.87 (Newton’s theory).

The efficacy of the “F=ma optics” is without doubt, however, respective authors on the subject are careful to stress the purely analogous nature of the relationship. We would like to suggest that it isn’t an analogy at all, but rather a literal equivalence.

EM Mass
Let us envision an electromagnetic wave, with a wavelength of 2.43 * 10-12 m, moving in a periodic cycle which takes it back upon itself such that it becomes a self-reinforcing soliton. The complete orbital path length of this EM wave is equal to its wavelength but is such that it makes a double-loop. (see Fig 1)

Such a quasi-symmetrical object, if stable, would resemble an electron. It would have a physical radius of on the order of 2.43 * 10-12 m / 4, an electric field, a magnetic dipole, and a half-integral spin [ref 6]. It would also offer a physical manifestation of Einstein’s mass/energy equivalence (e.g. “releasing” the photon from its self-contained path would result in a burst equal to its “rest energy”).

Cosmic Speed Limit
Philosophically, many of us have been mystified by the limiting nature of c. EM mass might provide a straight-forward explanation -- a photon turning back upon itself does not follow the traditional geodesic between two points. As an EM mass particle is accelerated, a larger portion of its photon’s circuit is thus spent moving in the direction of its velocity; this percentage can be arbitrarily close to, but not quite, 1. (see Fig 2)

Transverse Waves
If we refer back to our sphere of graded refractive index, we would expect that the path of light moving radially to it would remain unaffected; only a light’s path with a transverse component would be altered. The photon of an EM mass particle moving in a closed circuit within the sphere would possess a transverse portion of its path relative to the center of the medium in a range between .5 and 1, depending upon the relative velocities of the sphere and the particle. This could manifest as relativistic mass.

Conclusion
In this paper we have shown the connection of optics to the gravitational bending of light in a graded time dilation field. Additionally we have shown that if mass were to possess an electromagnetic nature moving in a cyclic fashion (i.e. “EM mass”) then we are able to precisely predict the gravitational behavior of that mass in the presence of such a time dilation field without invoking any other mechanism related to General Relativity. Lastly, we are able to show that this model may plausibly explain other aspects of Relativity, such as the limiting speed of light and relativistic mass. We feel that these aggregate theories provide ample potential to warrant further investigation.

References

[1]The optical-mechanical analogy for stationary metrics in general relativity; Paul M. Alsing; American Journal of Physics 66, 779 (1998); https://doi.org/10.1119/1.18957

[2] The optical-mechanical analogy in general relativity: Exact Newtonian forms for the equations of motion of particles and photons; James Evans, Kamal K. Nandi & Anwarul Islam; General Relativity and Gravitation volume 28, pages 413–439(1996)

[3] ‘‘F=ma’’ optics; James Evans and Mark Rosenquist; American Journal of Physics 54, 876 (1986); https://doi.org/10.1119/1.14861

[4] Sumana Bhadra, Electromagnetic Mass Models in General Theory of Relativity: https://arxiv.org/abs/0710.5619

[5] Space, Time and Gravitation; Sir Arthur Eddington; http://www.gutenberg.org/files/29782/29782-pdf.pdf

[6] Is the electron a photon with toroidal topology? J.G. Williamson and M.B. van der Mark, 1997; Annales de la Fondation Louis de Broglie, Volume 22, no.2, 133. http://home.claranet.nl/users/benschop/homepg2/electron.pdf

Speaking of relativistic-mass...
Wouldn't my relativistic-spaceship get ever more massive , as I approached lightspeed ? Doesn't that mean that at a certain point , it reaches several solar-masses , and collapses into a black-hole ?
D.H.

Wouldn't my relativistic-spaceship get ever more massive , as I approached lightspeed ?
From the point of view of somebody watching your spaceship fly past, yes (if you want to look at it that way). From your point of view on the spaceship, no; it would only have its rest mass. Relativistic mass is a reference frame-dependent quantity.

Doesn't that mean that at a certain point , it reaches several solar-masses , and collapses into a black-hole ?
No. If it doesn't collapse to a black hole in one reference frame (the frame co-moving with the spaceship, for instance), then it can't do so in any other frame either.

From the point of view of somebody watching your spaceship fly past, yes (if you want to look at it that way). From your point of view on the spaceship, no; it would only have its rest mass. Relativistic mass is a reference frame-dependent quantity.

No. If it doesn't collapse to a black hole in one reference frame (the frame co-moving with the spaceship, for instance), then it can't do so in any other frame either.

Yeah , but ...
That seems to imply that at a certain point , my ship will grow an "event-horizon" , as experienced by the pass -by observer . Could he even see me , would he get spaghettified ?
D.

Well...since we're already in Alternative Theories I'll provide my answer: take two ships of equal mass passing each other at velocities high enough to cause their respective relativistic masses to produce "black holes" according to the opposite observer. Locally, for each observer, they are not experiencing any spaghettification or any discomfort whatsoever. Additionally, they can each slow down, turn around, and meet up for beers afterward, completely contradicting the currently accepted theories surrounding black holes. The reason for this, I believe, is that black holes are never formed and do not exist.

Yeah , but ...
That seems to imply that at a certain point , my ship will grow an "event-horizon" , as experienced by the pass -by observer . Could he even see me , would he get spaghettified ?
D.
" ...it would only have its rest mass... " So no.

Recall, there's nothing to say that the spaceship is moving past you - you can just as easily say the spaceship is stationary and you are moving past it.
If you accelerated to .999999c and flew past the Moon, would you expect it to suddenly turn into a back hole?

And we can't say that these "relativistic masses" are illusory. They cause gravitation, but only when the relativistic masses exist from the perspective of the attracted bodies.

https://authors.library.caltech.edu/1544/

That would be the perspective of the observer not moving , in relation to the CMB .
So...when relativistic-mass DOES get larger than black-hole mass , does my ship attract the observer , and in what manner ?
D.

That would be the perspective of the observer not moving , in relation to the CMB .
CMB has nothing to do with it.

So...when relativistic-mass DOES get larger than black-hole mass , does my ship attract the observer , and in what manner ?
D.
As pointed out: it's relative. There is nothing to say that A is stationary and B is moving, any more than A is moving and B is stationary. Thus, a mass moving relativistically past you is no different than you moving relativistically past it.

And there are particles entering our atmosphere us right now at relativistic speeds. Do you feel your mass change every time a particle flies past? In the particle's frame of reference it is stationary and Earth is moving at relativistic speed. That is a perfecty valid frame of reference.

Do you think every time this happens, the Earth is in danger of collapsing into a black hole?

Mass under general relativity is not so simply defined as described by the posts in this thread. It seems based on the stress-energy tensor, which is beyond my ability to field questions. But I can quote http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html
on the subject at hand here:
If you go too fast, do you become a black hole?
When an object approaches the speed of light, its mass increases without limit, and its length contracts towards zero. Thus its density increases without limit. Sometimes people think that this implies it should form a black hole; and yet, they reason, since its mass and volume haven't changed in its rest frame, it should not form a black hole in that frame—and therefore not in any other frame either. So does a black hole form or not?

The answer is that a black hole does not form. The idea that "if enough mass is squeezed into a sufficiently small space it will form a black hole" is rather vague. Crudely speaking, we might say that if an amount of mass, M, is contained within a sphere of radius 2GM/c2 (the Schwarzschild radius), then it must be a black hole. But this is based on a particular static solution to the Einstein field equations of general relativity, and ignores momentum and angular momentum as well as the dynamics of spacetime itself. In general relativity, gravity does not only couple to mass as it does in the newtonian theory of gravity. Gravity also couples to momentum and momentum flow; the gravitational field is even coupled to itself. It is actually quite difficult to determine the correct conditions for a black hole to form. Hawking and Penrose proved a number of useful singularity theorems about the formation of black holes. But even these theorems do assume certain conditions which we cannot be sure are true "out there".

Mass under general relativity is not so simply defined as described by the posts in this thread. It seems based on the stress-energy tensor, which is beyond my ability to field questions. But I can quote http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html
on the subject at hand here:
Baez claims that "the idea...is rather vague" but it isn't at all. Derek H's original comment is pretty succinct. What is truly vague is Baez's rambling answer. He's basically saying that he doesn't know, but he's certain that others do. (Exactly what Halc did, if we're being honest...)

Well, I do know the answer due to having performed a simple empirical test. I just don't have a sufficient grasp on GR theory to explain it in proper terms.

The only line in that short article about not knowing something seems to be the last one about assumptions about certain conditions under which black holes might form, but an ordinary object moving fast obviously can't be one of them because it's trivial to empirically test. I personally have enough relativistic mass in some frame to fit within my own Schwarzschild radius, and yet I don't collapse. I can have a causal effect on any passing inertial observer. That's the easy and obvious answer.

Baez claims that "the idea...is rather vague" but it isn't at all.
What Baez is saying is that without the mathematics of general relativity, the concept that a lot of people have in mind is rather vague. There's nothing vague when you do the maths properly.

Derek H's original comment is pretty succinct. What is truly vague is Baez's rambling answer. He's basically saying that he doesn't know, but he's certain that others do.
No he isn't. He's trying to answer a complicated question in a way that laymen can understand.

Well, I do know the answer due to having performed a simple empirical test. I just don't have a sufficient grasp on GR theory to explain it in proper terms.

The only line in that short article about not knowing something seems to be the last one about assumptions about certain conditions under which black holes might form, but an ordinary object moving fast obviously can't be one of them because it's trivial to empirically test. I personally have enough relativistic mass in some frame to fit within my own Schwarzschild radius, and yet I don't collapse. I can have a causal effect on any passing inertial observer. That's the easy and obvious answer.
I agree with your logic. The fact that there are existing frames that would apparently give you, Halc, enough kinetic energy to form a black hole (which obviously does not exist) proves that black holes would not be created in this manner. It looks like the consensus for physicists and laymen alike is that they would not form. I have yet, though, to see an explanation as to why this kinetic energy is apparently immune from gravitational collapse.

James R said:
He [Baez] is trying to answer a complicated question in a way that laymen can understand.
I'm sorry James but I call BS when I see it. Do YOU understand Baez's explanation?
Baez said:
Crudely speaking, we might say that if an amount of mass, M, is contained within a sphere of radius 2GM/c2 (the Schwarzschild radius), then it must be a black hole. But this is based on a particular static solution to the Einstein field equations of general relativity, and ignores momentum and angular momentum as well as the dynamics of spacetime itself. In general relativity, gravity does not only couple to mass as it does in the newtonian theory of gravity. Gravity also couples to momentum and momentum flow; the gravitational field is even coupled to itself.
He basically says "Crudely speaking, such and such must be a black hole...but this is ignoring the other elements of general relativity such as momentum, angular momentum and the dynamics of spacetime itself...gravity couples to mass AND momentum AND momentum flow AND the gravity field itself."
So his argument is that gravity it complex in GR, fine. But he provided no explanation whatsoever of why this thought experiment does not produce a black hole. There is literally nothing there. If anything he just restates that relativistic mass (i.e. kinetic energy) affects gravity -- something that we all already understand, which is why we are discussing the subject.

If Baez simply said that general relativity claims that momentum energy only affects external bodies...or something of substance, even if I didn't understand it...then my BS meter would have remained quiet.

I'm sorry James but I call BS when I see it. Do YOU understand Baez's explanation?

He basically says "Crudely speaking, such and such must be a black hole...but this is ignoring the other elements of general relativity such as momentum, angular momentum and the dynamics of spacetime itself...gravity couples to mass AND momentum AND momentum flow AND the gravity field itself."
So his argument is that gravity it complex in GR, fine. But he provided no explanation whatsoever of why this thought experiment does not produce a black hole.
Without doing the maths, we already know why there's no black hole. The explanation has already been given, and say you agree with it. If there's no black hole in one frame of reference, there won't be one in any frame moving relative to that frame, even one going at relativistic speed. What happens in spacetime happens in spacetime, and what doesn't doesn't. Changing frames of reference doesn't change spacetime events.

If you want the details, you could start with the Einstein equation of gravity and read up on the Stress-Energy (or Energy-Momentum) tensor. For instance, see here: https://en.wikipedia.org/wiki/Stress–energy_tensor

This might be a good chance to brush up on your tensor calculus.

Photons have momentum, and that goes in the stress-energy pot for curving spacetime. So, why do photons already travelling at c, not become black holes?

Without doing the maths, we already know why there's no black hole. The explanation has already been given, and say you agree with it. If there's no black hole in one frame of reference, there won't be one in any frame moving relative to that frame, even one going at relativistic speed. What happens in spacetime happens in spacetime, and what doesn't doesn't. Changing frames of reference doesn't change spacetime events.

If you want the details, you could start with the Einstein equation of gravity and read up on the Stress-Energy (or Energy-Momentum) tensor. For instance, see here: https://en.wikipedia.org/wiki/Stress–energy_tensor

This might be a good chance to brush up on your tensor calculus.
Baez made no mention of frames of reference. And mentioning stress-energy tensor doesn't resolve anything -- it's literally the stress-energy tensor that's the problem! We can "create" as much energy as we want, without limit, in an area as compact as we want, by accelerating a mass in our thought experiment. If black holes were created by mass alone this would be a non-issue.

Also, the frames-of-reference explanation is "simply" a logical contradiction, not an explanation of why general relativity doesn't predict a black hole in those circumstances.

Love that last paragraph/confession ! I am a firm believer in an physical substrate of space ; one that reflects an actual "true" frame-of-reference . To perceive this , one must step outside of light-transmitted appearances . IF we assume a means of tracking/recieving spaceborne objects instantaneously , then we can actually measure the temporal-distortions caused by travel through space . If we launch two ships in opposite directions from a point in space which is unmoving , relative to the CMB , and have them accelerate to the same velocity , relative to that point , they should reflect the same amount of time-contraction . If they don't , that shows that the "substrate" is not static , but is moving .
Likewise , if two ships so equipped pass each other in the deep , then comparing their clock-rates will also reveal which one is passing through the substrate at a higher rate of speed . This would correspond to a higher relativistic-mass , compared to the slower ship . Should the faster ship approach lightspeed , it should generate such a virtual-mass , as to BE a black-hole .
This is where the original question becomes germain ; Will this "Black-hole-ship" manifest an event-horizon , and will it absorb the approaching ship ?
Keep in mind , no particle ever created has gone fast enough to manifest this much "total-energy" .
D.H.

Keep in mind , no particle ever created has gone fast enough to manifest this much "total-energy" .
Your complete lack of understanding of relativity theory aside, it seems they've accelerated an electron to some pretty impressive speeds, enough to get its mass up to the level you indicate. What's the Schwarzschild radius of something with the mass of an electron going at 0.999999999988c? Now what's the contracted length of it? Gamma is over 200,000 at that speed.

Clue: Trick question.

Your complete lack of understanding of relativity theory aside, it seems they've accelerated an electron to some pretty impressive speeds, enough to get its mass up to the level you indicate. What's the Schwarzschild radius of something with the mass of an electron going at 0.999999999988c? Now what's the contracted length of it? Gamma is over 200,000 at that speed.

Clue: Trick question.
I'm a generalist , not a physicist ...
However , I don't actually see a real answer to my question . That is formulaic-obfuscation up above ; where it should say "the electron's virtual-mass/total-energy is paltry , so it can't answer your question ." . You see , if it can't be translated into simple-math and plain-english , then it's usually "Dustin Hoffman !" .
D.