Part Two: Quantum Relativity

Catch up on Part 1 here.

Intriguing as a supposition might be,
It is typically not as relevant as the question that follows.


Suppose that Gravitons exist and that they experience Gravitational Lensing in the same manner as light, would this actually account for the requisite quantity of Dark Matter expected in the universe?

This is actually an Oops Moment for me. I had an idea and I did not follow through to the end. In this essay I will try to make amends and address the possible significance of the idea.

This article is part of the series called Quantum Relativity. If you view my Lists and About data, all the other articles can be seen.

Chapter 1 — Background

In my previous essay Quantum Relativity And Dark Matter, I proposed a thought experiment linking the quantum world with General Relativity and showing how the Graviton quantum particle might be the missing link to the mystery of Dark Matter. Summarizing, could hypothetical Gravitons experience Gravitational Lensing in the same manner as light and increase gravitation attraction in clusters of stellar bodies.

This essay expands on that concept. If we assume the thought experiment is true, can we account for the actual quantity of Dark Matter observed in the universe?

Here are the links to the pertinent concepts.

Wikipedia Dark Matter (23 August 2022, at 01:26 UTC.)

Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe.Various astrophysical observations – including gravitational effects which cannot be explained by currently accepted theories of gravity unless more matter is present than can be seen – imply dark matter's presence.In the standard Lambda-CDM model of cosmology, the total mass-energy content of the universe contains 5% ordinary matter and energy, 27% dark matter, and 68% of a form of energy known as dark energy.[1960’s] Early radio astronomy observations, performed by Seth Shostak, later SETI Institute Senior Astronomer, showed a half-dozen galaxies spun too fast in their outer regions – pointing to the existence of dark matter.[1970’s] They showed most galaxies must contain about six times as much dark as visible mass…

Wikipedia Galaxy Rotation Curve (21 August 2022, at 06:23 UTC.)

The rotation curve of a disc galaxy (also called a velocity curve) is a plot of the orbital speeds of visible stars or gas in that galaxy versus their radial distance from that galaxy's centre.The rotational/orbital speeds of galaxies/stars do not follow the rules found in other orbital systems such as stars/planets and planets/moons that have most of their mass at the centre. Stars revolve around their galaxy's centre at equal or increasing speed over a large range of distances. In contrast, the orbital velocities of planets in planetary systems and moons orbiting planets decline with distance according to Kepler’s third law. This reflects the mass distributions within those systems. The mass estimations for galaxies based on the light they emit are far too low to explain the velocity observations.The galaxy rotation problem is the discrepancy between observed galaxy rotation curves and the theoretical prediction, assuming a centrally dominated mass associated with the observed luminous material. When mass profiles of galaxies are calculated from the distribution of stars in spirals and mass-to-light ratios in the stellar disks, they do not match with the masses derived from the observed rotation curves and the law of gravity. A solution to this conundrum is to hypothesize the existence of dark matter and to assume its distribution from the galaxy's center out to its halo.

This article gets it right, but you will often hear the term Dark Matter Halo. If you think there is some irony here as halos are generally made of light, it is not the case. We call it a “halo” out of habit.

See the Wikipedia article on Galactic Halos (31 May 2022, at 15:27 UTC.)

A galactic halo is an extended, roughly spherical component of a galaxy which extends beyond the main, visible component. Several distinct components of galaxies comprise the halo: the stellar halo; the galactic corona (hot gas, i.e. a plasma); [and] the dark matter halo…

If someone is unwise enough to let me rename it, I think it would make more sense to call it a Dark Matter Shroud, but who in their right mind would let me do that?

Chapter 2 — A Primer for Gravity

I added this chapter after most of the essay was written because I am focusing on an odd aspect of gravity which is important to the discussion and I had misgivings about folks actually understanding that aspect, since no one really explains it fully in the textbooks. If you are comfortable with the concept of Gravitation Density, i.e. you actually know why gravitational attraction drops off with the inverse square of the distance, then please feel free to skip this section.

We start with a (very) simple model of gravity. Imagine that we replace the concept of gravity with long rods which represent the trajectory of gravitons leaving an object. Instead of mass determining the force of gravity, we will let mass determine how many rods that we can attach to an object. For our model, we will allow one rod for each kilogram of mass. Suppose we have a 6 Kg sphere and a 36 Kg sphere and we connect them via the these rods.

A blue sphere with a few spikes touches a green sphere with numerous spikes.
Gravity Rods connecting a Large and Small Object

As indicated in the illustration above, we note that very few rods actually touch the opposing spheres. This is because a gravitation field is uniform around a contributing body. Thus the model demonstrates that very little of the available gravitons are really responsible for a typical act of attraction. This dispersion of the gravitons also explains why gravitation affects fall off as quickly as they do. If you look at the number of gravitons that penetrate a sphere at 2 meters distance it will be the exact same number that penetrate a sphere at 4 meters, e.g. 6 rods or 36 rods from the illustration. Gravitation Density is different though. At 2 meters the surface of the sphere is 4 × π × r² = 4 × π × 2² ≈ 50.265 square meters and at 4 meters we have 4 × π × 4² ≈ 201.062 square meters, so the densities for the 36 rod object are 0.716 rods per square meter at 2 meters and 0.179 rods per square meter at 4 meters. Thus the density drops to only 25% in a mere 2 meters, which by the way, is exactly proportional to the inverse of the square of the distance, i.e 2² / 4² = 4/16 = ¼ or 25%, something Newton figured out a few hundred years ago.

Not so obvious from the illustration, basically because I ran out of space, but the gravitons that do not find a target mass immediately will continue to travel at the speed of light until they do encounter one. That mass may be a few hundred or even a few billion light-years away. When they do finally interact with the mass, they behave like (friction-less) bullets, they react as if they traveled no distance at all. The result is of course negligible, but only because there are very few gravitons on that trajectory, not because gravity can not reach that far. If more were focused to this point, we would expect a greater strength of the gravitation field in this location.

Chapter 3 — The Galaxy Rotation Problem

To continue our original examination, we first need to understand the problem. Astronomers have observed our neighboring galaxies for decades with sophisticated instruments. The results that they found did not agree with their theories of orbital mechanics.

Since we are familiar with satellites that orbit the earth, let’s review the basics from that perspective. First though, understand that without propulsion, we fall straight down if we are in the gravitational field of a planet, and toward the end, we are falling very fast. The field creates a force that accelerates us toward the surface.

See the howstuffworks article on How Satellites Work (Gary Brown & William Harris.)

Orbital velocity is the velocity needed to achieve balance between gravity's pull on the satellite and the inertia of the satellite's motion – the satellite's tendency to keep going.The orbital velocity of the satellite depends on its altitude above Earth. The nearer to Earth, the faster the required orbital velocity. At an altitude of 124 miles (200 kilometers), the required orbital velocity is a little more than 17,000 mph (about 27,400 kph). To maintain an orbit that is 22,223 miles (35,786 kilometers) above Earth, the satellite must orbit at a speed of about 7,000 mph (11,300 kph).

Thus in a low orbit, we must travel faster than what would be required in a higher orbit. Be aware that changing your speed works the opposite of how you might think. If you fire retro-rockets and slow down, you drop to a lower orbit. You do not go higher. If you fire thrusters and speed up, you go higher. How this works is this, when we slow down, we stop orbiting momentarily and we fall. When we fall a little while, the force of gravity increases and we accelerate and gain the extra speed we need for a lower orbit.

When we increase our thrust, we climb higher, but we are fighting gravity all the way and we are likely losing speed as we climb. After we climb a while, we reach a point where the force of gravity is weaker and we can coast at the higher orbit with the speed we have left. Note that if we have unlimited propulsion we simple keep climbing until we reach an Escape Velocity and we leave the planet completely. (There’s some fun math here. Check the Wired article Changing Orbits And Changing Speed by Rhett Allain, Nov 23, 2010 7:28 PM.)

Getting back to the point of the essay, what astronomers found was that at distances farther from the galactic center, i.e. a higher orbit, the speed of the stars was much faster than expected. More intriguing, the speed leveled out and stayed more or less constant well beyond the limits of observations of the galactic boarders. The solution was simple, a halo of Dark Matter must surround most galaxies. True to form the supposition leaves us with a more difficult question. What exactly is Dark Matter?

I will not dwell on the frustration to astronomers that discovering the nature of Dark Matter has presented. If you want to explore this further, the Wikipedia article from above is a good place to start. I can state though, that at the time of this writing, I am not aware of any published answer to the question that has satisfied the scientific community. In following this tradition of failure, I have no confidence that this essay will be any more successful, but I am always hopeful that what I create might give someone the inspiration to move forward.

Chapter 4 — The Graviton Lensing Solution

If we are assuming that gravitons behave in the same manner as light with regard to gravitational lensing, perhaps it would be a good time to review the subject.

See the Wikipedia article Gravitational Lens (26 July 2022, at 14:12 UTC.)

A gravitational lens is a distribution of matter (such as a cluster of galaxies) between a distant light source and an observer that is capable of bending the light from the source as the light travels toward the observer.Unlike an optical lens, a point-like gravitational lens produces a maximum deflection of light that passes closest to its center, and a minimum deflection of light that travels furthest from its center. Consequently, a gravitational lens has no single focal point, but a focal line.If the (light) source, the massive lensing object, and the observer lie in a straight line, the original light source will appear as a ring around the massive lensing object (provided the lens has circular symmetry). If there is any misalignment, the observer will see an arc segment instead.Gravitational lenses act equally on all kinds of electromagnetic radiation, not just visible light, and also in non-electromagnetic radiation, like gravitational waves.

When you read this Wiki piece you have to wonder, how many people read this and thought, “Wonder if it could bend gravity as well?”

Returning to our subject, the largest impact of Graviton Lensing would occur if we use the Black Hole at the center of a galaxy as our lens. We will assume that unlike light, the gravitons are not absorbed appreciably by dust clouds and consequently we have good transmission of most gravitons emitted in any region of the galaxy. We note that many types of Electromagnetic Radiation like infrared are not greatly effected by dust, so this seems a fair assumption.

Galaxy back ground showing a black sphere with a pink ring suspeded between two yellow spheres.
Graviton Lensing in a Galaxy

In the illustration above, we see a Just Right (pink) zone where any line of sight graviton beam can be deflected in a favorable way. Note that if this illustration were to scale, we would see basically a flat line from one end of the galaxy to the other with an infinitesimal bump in the middle. Beams approaching the zone that are too low become part of the gravitational field of the Black Hole. Beams that are too high may get slightly deflected, but they will most likely just travel out into intergalactic space more of less in the direction they originally were taking. Even the Just Right beams will only appear in a single linear cylinder on the opposite side of the galaxy before they continue down (or up) passing through the galactic disk and back into open space. If they fail to locate a mass in that cylinder they will have no affect.

The important aspect here is the Gravitational Density transmission. If you have a stellar object ten thousand light years from the galactic center, your effective Gravitational Density will never be any better than what arrives at the galactic center, i.e. 1/10,000². That being said, you do get to count every graviton that arrives at the Just Right zone. The size of the zone depends on the size of the Black Hole and the distance of the stellar object from the galactic center, so there is no single solution. In this case, one can devote a few months to making a ton of calculations or one can use a little speculation. Anyone guess where I intend to take this?

If we look at astronomical photos of actual Gravitational Lensing we see either a ring or four to six separate images, so I am throwing out the number six as the minimum equivalents of the original stellar object. Six times is also the estimated ratio of Dark Matter to ordinary matter. You are of course free to prove me wrong and you have my total encouragement for what it is worth.

Using this model we can see two interesting things. First, the effective distance of the lensing would extend well past the boundaries of the galaxy, which fits nicely with observations. The flat trajectory would create longer cylinders at greater distances, possibly even in nearby galaxies, which again fits observations.

Second, a spiral galaxy would tend to be symmetrical about the galactic center, which fits nicely with many astronomical images. We would expect objects to gather in the remote cylinder zones and project their gravitons back to the other side of the galaxy to the original stellar objects and thus create the ultimate long distance relationships.

Chapter 5 — Conclusions

You are free to draw your own conclusions about the validity of this argument.

I look at this like a criminal investigation. All that I am trying to do is establish a Probable Cause. Any burden of proof must fall on the investigators that decide to actually work the case.

There are several areas that raise concerns. Do gravitons actually exist? They are still problematic. If they exist would they be subject to Gravitational Lensing? Maybe. And finally, if we are good to this point, could the mechanism proposed here actually account of all Dark Matter? Given the history of the subject, we would have to place our bets with the caveat “Maybe, but most likely less than half.”

If we work under the assumption that this is “Too good to be true,” then we have to be ready to accept that this is probably all just whimsical thinking.



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JB Johnson

JB Johnson


I am a science and technology junky and this is my place where I can share my ideas.