What Is a Curve?

This post is simply an abridged version of Miles Mathis's essay http://milesmathis.com/pi.html

I see this essay as the theoretical foundation of Mathis's system, so it is a good place to start in order to avoid confusion when reading Mathis's other essays.

par. 10

Drawing a circle is a real event, not an abstract event. In fact, any possible circle must take time into consideration. This is true of orbits, bugs walking in circles, whirlwinds, and so on. When we apply mathematics to any of these situations, we must take time into account. That is why we find accelerations in all circular motion, the most famous of which is the centripetal acceleration. Centripetal acceleration can be due to gravity or to some other force, but in any circular motion there will always be a centripetal acceleration.

par. 14

Here is an equation that is used everyday, right now, by the smartest people alive:

v = C/t = 2πr/t

     where v is the orbital velocity, C is the circumference and t is the period of the orbit. Newton used this equation. ... We have C in the place of x, as if C is a simple distance. [However,] C is not a simple distance. There is no way to express C with just an x-dimension. In fact, as I have just shown, C is three-dimensional, if you include time. This equation is including time, as you can see by the denominator. You cannot have a t in the denominator and claim you are ignoring time. You cannot put a curve over a time and have it come out to be a simple velocity. Velocity is defined as x/ t. The variable x is one-dimensional and therefore cannot curve.

par. 18

[S]ay you are in a tiny spaceship at the center of the circle. You are instructed to fly at a thousand miles per hour for one hour, then turn left at a 90o angle and keep going, not pausing or changing your velocity. You will say, “I need some method for calculating velocity. What if the background changes in some weird way after I make the left turn?” I answer, “Just measure internally. Meaning, use your onboard clock and check your engine’s rpm. Whatever the rpm’s were as you were going a thousand miles per hour along the first straight line, keep them there after you turn left.” You do as I say and after exactly one hour you come to a rosebush and a sign that says, “left here.” Miraculously you make the sharp turn without slowing down at all. After some time you come to the rosebush again and you think, “Is that the same rosebush? What is going on?” What is going on is that I turned on a big magnet as soon as you got to the rosebush. My magnet and I, sitting at the center of the circle [with r = 1000 miles], are causing you to circle us.

      According to this set-up, your velocity out to the rosebush would be r/t. You were instructed to keep this velocity, by a method that would guarantee it was kept. Therefore your tangential velocity is also r/t.

par. 20

Now, the question is, what centripetal acceleration must I apply to you with my magnet to keep you moving in a circle? Surprisingly, the answer is always the same. It doesn’t matter what your speed is going out to the rosebush or how long it takes you to get there or how far away the rosebush is. As long as you keep your speed the same before and after you turn, the acceleration I must apply to you with my magnet is. . . . π.

par. 29

Pi only applies if the tangential velocity is equal to r/t. But in orbits and most physical problems, this will not be true. The centripetal acceleration and the tangential velocity are independent motions. They are not necessarily related, much less equal. That is why we don’t find the value of pi for the acceleration in gravitational fields.

par. 33

To be even more specific, a = v2/r works in experiment because v = 2πr/t works in experiment. The equation v = 2πr/t is a very useful number to us even though it does not really express the orbital velocity, or any velocity. It is more useful to us than the actual orbital velocity or the actual tangential velocity, both of which aren't really that interesting in experiment except as theoretical numbers. The number 2πr/t is a number we can use, and if we mislabel it as a velocity, well, who cares as long as we mislabel it the same way throughout the centuries?

    Engineers aren't paid or trained to care about such things, but theoretical scientists understand that such mistakes ultimately lead to ruin. In the short term they may lead to simple engineering failures, which is bad enough. But in the long term they always lead to theoretical dead-ends, since a sloppy equation is the surest of all possible ways to stop scientific progress. A correct equation is almost infinitely expandable, since its impedance is zero. Future scientists can develop it in all possible directions. But a false or imprecise equation can halt this development indefinitely, as we have ample proof. Mislabelling variables is not a semantic or metaphysical failure. Is it failure of science itself.

Three Discrepancies Between Miles Mathis and Mainstream Physics

Here are the biggest discrepancies I have found between Miles Mathis’s theory of physics vs. the mainstream theory I learned growing up.

1. The Dark Matter theory. Mathis sees Dark Matter as a desperate attempt to make the numbers work for the mainstream model. 

2. The shape of space. Mathis agrees with Tesla's position, "space cannot be curved, for the simple reason that it can have no properties." Mathis's opinion is that Einstein made many important contributions, e=mc^2, relativity, but was mistaken about a couple of things, including the curvature of space. (See my previous post for more on Mathis's understanding of light and relativity.)

3.  In Mathis’s system, curved motion, such as the path a planet moves through, does not have the same units as straight line motion. (!!!) While this may sound far-fetched just from being so different from the historical mainstream, there is no reason that it could not be correct. For example, historically, we have defined the length of a curve to be the straight-line length of a thread that fits along that curve. Mathis claims that curved motion is fundamentally different than straight-line motion, and so curves must be treated different than lines

This is the most difficult assumption in Mathis's theory, as it forces the redefinition of many basic assumptions of the mainstream math and physics we have grown up with. But Miles Mathis’s system is consistent! Many of his papers seem to contain errors and contradictions to someone used to the mainstream model. There are multiple times in Mathis’s papers where I have come to what I thought was an obviously incorrect statement. Each time, I’ve found Mathis directly addresses my issue, either within the same essay or often another one. Neither Mathis's system nor the mainstream's have identifiable inconsistencies. It’s just a question of which system best fits the data.

The easiest target for critics of mainstream physics like  is Dark Matter. There is no empirical evidence for Dark Matter whatsoever--the whole theory is an attempt to make the numbers work for the mainstream’s theory of celestial motion. Another time I was reading Mathis’s essay on relativity. I thought that his transformations could not preserve the speed of light for different observers. I didn’t realize that Mathis had already written an entire essay addressing the exact question I had! Finally, this week (late July, 2013) I had a question about his analysis of circular motion. Again, a paper on the issue I had!

Mathis, a figure artist originally, has been writing these papers for over 10 years and over that time has written papers addressing most every issue that a reader will have. Keep this in mind whenever you think you’ve found an inconsistency reading him.

A Brief History of Light

What did people living before the 20th century know about light? Of course, from simple examples such as thunder and lightning, it is clear that light is much faster than the speed of sound. But was light instantaneous or just really, really fast? Philosophers disagreed, and for many centuries, no observations were accurate enough to decide. That is, until the invention of the telescope in the 1600s confirmed that light indeed did have a (really, really, really fast) finite speed.

How fast is light exactly?

Light travels at a speed of about 300,000 kilometers per second, which means it travels the distance of the equator 7.5 times in one second.

By comparison it takes sound (travelling through air) almost 33 hours to travel the distance of the equator just once!

How long have humans known the speed of light?

I was surprised to learn that we have known all of this since the 1670’s, when Dutch astrologer Ole Rømer used observations of eclipses of Jupiter’s moon Io to demonstrate that the speed of light was in fact finite, and even calculated the value with a fair degree of accuracy. Roemer’s publication provides the correct conception that light is virtually instantaneous for terrestrial measurements, but not fast enough to ignore for measurements within the solar system. [ ref. http://www.rundetaarn.dk/engelsk/observatorium/light.htm ]

Why does this matter?

Until the 20th century, the general consensus seemed to be “it doesn’t.” The concern was for terrestrial mechanics and nothing more. Since light travels so fast as to be virtually instantaneous on earth, no one really worried about it. For instance, Isaac Newton writes a side-note referencing that light takes approximately 8 minutes to travel from the sun to Earth, but in all of his theoretical work, he ignores light’s finite speed.

In 1886 Heinrich Hertz successfully confirmed James Clerk Maxwell’s theory published in 1865 that visible light was just a small part of a larger electro-magnetic wave spectrum. Hertz demonstrated that the signals generated by a spark gap transmitter could generate an electric field, proving that Maxwell’s electromagnetic radiation could be generated. Hertz also confirmed Maxwell’s prediction that this radiation travelled at 3.0 * 10^8 m/s, which is also the velocity for light.

At this time, interest in the precise nature of these strange new "radio waves" grew rapidly.

How does Light travel?

When you look up at the night sky, what is going on? How is the light getting from the stars to your eye?

According to Einstein, the relative velocity of the star to earth causes a bending of space-time, which preserves the velocity of light for all potential observers, whether they are traveling towards or away from the source of the light.

Tesla and Mathis, however, reject the bending of space. For them, both your eye and the distant star generate a charge field, in much the same way that all matter generates gravitational fields, and it is through this charge field that light travels.

Imagine the distant star is moving away from earth’s solar system at c/2, half the speed of light.

Einstein’s general relativity hypothesizes that the space-time between the two reference frames, earth’s and the star’s, will be bent such that both frames will calculate the speed of gravitational attraction (and of light) between the two frames to be c.

Here’s Wikipedia’s description of gravity in Einstein’s general relativity:

Formally, c is a conversion factor for changing the unit of time to the unit of space.[2] This makes it the only speed which does not depend either on the motion of an observer or a source of light and/or gravity. Thus, the speed of "light" is also the speed of gravitational waves and any massless particle. - http://en.wikipedia.org/wiki/Speed_of_gravity

So according to Einstein’s own theory of general relativity, the gravitational attraction between two objects does not depend either on the motion of an observer on earth or the star. Gravity is a mysterious field that each body contributes to, yet both observe as traveling at the speed c.

In the language of the mainstream physics' general relativity, only empty space is flat. Matter is a “disturbance” within “flat” space, which produces “perturbations.” Two bodies of matter will produce perturbations in just the right way to bend space-time so that the constancy of the speed of light (equivalently, the speed of gravity) is preserved. Notice what is happening here. Both objects are “bending” the field [space-time] through which the gravitational attraction between them occurs. 

Mathis is saying that light is generated by precisely the same kind of field as Einstein’s gravitational field in general relativity. That is, the light-producing charge field is generated by both the observer and the distant object in such a way that the speed c is preserved for both.

Mathis is saying that it is the light-producing charge field that is altered in order to preserve the constancy of c, and not space itself. Somehow Einstein missed this possibility, and thus we are left with curved space-time.

Mathis’s system is every bit as consistent as the Einstein/Lorentz system. The only question is which system best matches the empirical data. See my previous blog entry and the following excerpts from some of Mathis's papers for more on this comparison.

From http://milesmathis.com/tesla.html:  I hold that space cannot be curved, for the simple reason that it can have no properties. . . . Of properties we can only speak when dealing with matter filling the space. To say that in the presence of large bodies space becomes curved is equivalent to stating that something can act upon nothing. I, for one, refuse to subscribe to such a view. - Nikola Tesla

Tesla tells us that space can have no properties, since it is a “nothing”. Only matter can have properties, not space. I agree with him completely. And, although I accept the numerical findings of General Relativity, I do not accept curved space any more than Tesla. - Miles Mathis-

From http://milesmathis.com/rel4.html
You already know that great speed can make an image blurry, but Relativity is much more than that. Even if we have a very fast f-stop on our camera, and can get rid of any possible blur, great speed will still cause distortion. It causes distortion because the light we are seeing with must travel from the object to us. But since the object has size, different parts of the image reach us at slightly different times. If we give the object two ends, one end must be further away than the other end. All ends cannot be the same distance, unless the object is a point. And no object is a point, since a point is not an object. This means that we must get distortion, and that the distortion is due to size.

Now, according to this explanation, even an object at rest must be distorted, due to size. And this is also true. But the distortion of an object at rest is so small we may ignore it. To get any noticeable distortion due to Relativity on an object at rest, the object would have to be exceedingly large, so that light traveling from one end would arrive late. Normally, Relativity is not applied to objects at rest, and that is why.

But motion increases this effect greatly, and very fast motion increases it to a point where it becomes measurable. The reason is that very fast motion can make the farthest end of an object seem closer than it is. A small object passing you very fast will seem even smaller, since any part of the object traveling away from you will seem to be compressed. This is called length contraction.

Also from http://milesmathis.com/rel4.html
Light, like sound, has a wave. The analogy to sound is not perfect or complete, but light does have a wave. A train approaching us will have its light waves compressed and a train departing will have its waves stretched, for the same reason as we saw with the sound waves. We see the train at 100 feet, and then the train at 99 feet, and so on. We don’t see a continuous image, we create one from the still images we receive. Since the later light has less distance to travel, it makes up time on earlier light, and the wave we see gets compressed. In reverse the same thing happens as the train recedes.

Many will think this must make the receding train look longer--since waves that are stretched must be longer--but this is not what happens. The longer waves only make the train look redder. We read longer waves as redder and shorter waves as bluer, so a larger wavelength will cause a redshift.

The reason the receding train looks shorter is that the length of the train is determined by a single image. Unlike the wave, which is built of a series of images, the length is determined by one image only. In other words, we could take a picture with a real camera, and using that one image, we could determine the apparent length of the train. [And, yes, that one image would be distorted by Relativity. That real picture, taken by a real camera, would be distorted by Relativity.] Now, that one image is made up of all the light reaching us at the same instant, from all the points on the train. Since all the light is moving the same speed, the light from more distant points on the train must be earlier light. To say it another way, all the light is reaching US at the same time, to make the image, so it can’t have left all points on the train at the same time. If we work backwards from our eye, and go the speed of light for x seconds, we can reach some points on the train, but not others. This means that our image is made up of older and newer light. For instance, if the light from the nearest parts of the train was emitted at t = .0002s, then the light from the farthest parts of the train might have been emitted at t = .0001s. The light has farther to go, so to reach us at the same time, it had to be emitted earlier. If it was emitted earlier, then it was emitted when the object was not quite as far away. Therefore, the far end of the object will appear closer than it is. Therefore, the object will appear smaller or shorter than it really is.

From http://milesmathis.com/ether.html
Light, when seen or measured, is always local: meaning, it is always right in your eye or your instrument. Furthermore, it is always moving right at you when you see it or measure it. You cannot measure tangent light or light at any angle or light at any distance. You cannot measure light moving parallel to you, perpendicular to you, or moving away from you. Any attempt to measure the speed of light will be the attempt to measure light that is already impinging on the eye or instrument. ... [T]he emission and reception are both relatively instantaneous. Light is so small and moving so fast, that any motion of the emitter or receiver becomes negligible. ... This is the reason all receivers measure the speed of light to be c, and for no other reason.

Tom's note: This exact same argument is used by Einstein's general relativity to explain gravitational attraction between objects moving very fast towards or away from each other. Mathis just takes the argument away from gravitational fields, and applies it to a "foundational E/M field" that generates light, thus avoiding Einstein's warping of space.

Reality Is Infinite, measurement is finite

I hold that space cannot be curved, for the simple reason that it can have no properties. . . . Of properties we can only speak when dealing with matter filling the space. To say that in the presence of large bodies space becomes curved is equivalent to stating that something can act upon nothing. I, for one, refuse to subscribe to such a view. - Nikola Tesla

Tesla tells us that space can have no properties, since it is a “nothing”. Only matter can have properties, not space. I agree with him completely. And, although I accept the numerical findings of General Relativity, I do not accept curved space any more than Tesla. - Miles Mathis- http://milesmathis.com/tesla.html

I first stumbled onto Miles Mathis’s website when I googled the words [ “variable acceleration” newton leibniz ]. I found an essay proposing that Newton’s use of infinitesimal length intervals is improper for describing kinematic situations. How would you measure an “infinitesimal interval”? It seems that Newton realized that the concept of an “instantaneous velocity” leads to obvious questions, so he just pushed the problem up to a higher level abstraction! Thus “infinitesimal intervals.” This idea had never occurred to me before, but made sense. In particular, I thought the following paragraph in his paper on “time” demonstrated a broad base knowledge and thoughtfulness:

Whenever we measure time, we measure movement. ... Every clock measures movement: the vibration of a cesium atom, the swing of pendulum, the movement of a second hand.

    ...

Time is just a second, comparative, measurement of distance... When we measure distance, we measure movement.  We measure the change in position.  When we measure time, we measure the same thing, but give it another name.   Why would we do this?  Why give two names and two concepts to the same thing?  Distance and Time.  I say, in order to compare one to the other.

Over the last few months, I’ve slowly become convinced that there is something to Miles Mathis’s alternative to the Lorentz equations. I went through periods of doubt and denial, but all my misgivings were covered in one of Mathis’s papers, found by a simple google search of his website.

Mathis’s basic claim is that there is an alternative approach to describing the motion of light than the Lorentz equations, used by Einstein. I realize there are plenty of "Einstein was wrong" / conspiracy theorist websites where the author is clearly on a big ego trip looking for more attention -- see http://www.crank.net/einstein.html -- so you may wonder “why would I think Miles Mathis is any different?”

For a few of reasons:
1. He's not saying Einstein was wrong. Mathis agrees with relativistic effects, and the necessary implications of relativity, as is correctly explained in this youtube video: https://www.youtube.com/watch?v=wteiuxyqtoM . Mathis just disagrees over the precise rules that relativity follows.
2. He's interdisciplinary. Most "crank" websites love targeting the "big names" of the field like Einstein, Bohr, Lorentz, Newton, but they will seldom give credit for what the big names got correct and NEVER look to philosophy for help. The "crank" author is on an ego trip, not trying to uncover the truth, and so is not interested in the reasons why obviously intelligent theorist would allow mistakes. Mathis, on the other hand, explores the philosophy of science, citing George Berkeley's and Karl Popper's perspectives of Newton and Einstein, respectively.
3. He offers alternatives that are fully developed and consistent. The presentation of Mathis's theory needs help, but it's all there.

Within technical fields such as mathematics and physics, every generation has a Newton or an Einstein; they just happened to be their generations’ best at the time when human history started caring about motion and light. Without the inventions of the telescope, itself a product of the increasing availability of tools due to mining, and the proliferation of newletters made possible by the printing press, Isaac Newton would have been forgotten long ago.

Isaac Newton beat out his contemporaries, and the same is true for Einstein. However, it was History that chose Newton’s generation to be the discoverers of the laws of motion, and Einstein’s generation to discover the laws for the motion of light. Had Newton never published or never lived, it is quite likely Leibniz and perhaps a few others would have filled in all of Newton’s discoveries within a couple of decades at most. The same is true regarding Einstein. The “Great Man” theory of history applies more to art than to science. Inspiration can be involved in both art and science, but science is objective and can be altered, taken apart, and added to in a way that a work of art currently cannot be (although I believe open-source and public copyright licenses will get art to that point eventually, hopefully sooner rather than later!). The mythicization of a Newton and Einstein promotes a false view of science as requiring the type of genius that before was correctly reserved for artists and spiritual leaders.

As a quick summary:

Mainstream:	What I’ve learned:
The Lorentz factor (used by Einstein) is the only way to preserve a constant speed of light as a constant for all observers	Mathis’s system preserves c also, by hypothesizing that all matter generates an electric field, in much the same way that all matter generates a gravitational field.
The Lorentz equations have been verified experimentally many times over the past few decades.	The purpose of mass media is to prop up existing power structures. Funding for a department increases when positive media coverage is generated. The media likes articles with keywords that consumers recognize. Thus in the 20th century, the fastest way to academic credentials is to latch into the existing models and accepted theories. Original theoretical work of science departments was largely abandoned, and the game became to get the “right” numbers in your experiments--that is, ones that agreed with the accepted theory’s predictions.

The following quote gets to the heart of the issue. Einstein, following the general trend of his time, prioritizes measurement over reality. “If we can’t measure it, it does not exist.” This summarizes the dogma of the Western university system as it is today.

[T]he distinction between finite and infinite appears to me to be an ultimate distinction between measurement and reality. Reality is infinite. Measurement is finite. ... Although we live at infinity we cannot calculate at infinity. What this means is that real bodies do in fact converge to the limit. They reach the limit. Achilles passes the tortoise, etc. ... A mathematical term that expresses the motion of a body is logically a different sort of entity than the body itself. The body reaches the limit. The term, however, does not. - http://www.milesmathis.com/cant.html

What does this have to do with relativity? It has to do with the idea of velocity. Specifically, whether or not velocity is a measurement or is inherent. The mainstream treats velocity as being inherent, so that an observer would measure the velocity of an object the same as the object would measure its velocity itself. Mathis points out there is an alternative that is just as consistent.

Motion does not require a medium, it only requires a background. That background is automatically created relative to previous positions. You don’t need a medium to describe the motion of quanta. You only need a mathematical or diagrammed background, and previous positions give you that. - http://milesmathis.com/tesla.html

I’ve come to realize that my thoughts on the topic are still not organized enough to lay out his whole theory in a straightforward manner yet, but I feel I’ve developed a good grasp for Mathis’s arguments going through Mathis’s papers and connecting the dots with google searches of Mathis’s website. So instead of trying to include all the relevant derivations, what I’d like to do is give a general outline of the concepts, and open the comments up for questions that anyone has.

	Sound Waves	Old Ether Theory of Light	Einstein’s Theory of Light	Mathis’s Theory of Light
What creates the wave?	vibration of air molecules	The structure of the ether	The motion of many photons travelling together	Each individual messenger “B-photon” creates its own wave
What is the medium?	Air molecules	Ether as the medium of all motion (in the same way that air is the medium for sound.)	Curvature (of space-time), which changes with velocity. This preserves the constancy of the speed of light for all observers, no matter how fast they may be moving relative to one another.	no physical medium. Light moves in a wave pattern without the need of any medium. ... Light is made up of many photons, but each photon moves as a wave. This is not true of air or water waves, where each molecule moves up and down.

Mathis and Einstein both agree that light waves are not analogous to sound waves--there is no “ether” through which light moves. They just disagree on the mechanics of how. Einstein bends space-time in order to preserve the constancy of c. Mathis adopts an electric field generated by all matter, in much the same way that all matter generates a gravitational field. Both systems are consistent. Both systems preserve the constancy of “c” for all observers. It is just a question of which system best fits our observed data.

I don’t expect to fully convince anyone of Mathis’s theories in this entry.. I had to check and recheck the math many times myself before gaining enough confidence that I’m not wasting everyone’s time. Now, I am confident that his work deserves consideration.

A quick google of “evidence for time dilation” or “evidence for length contraction” will show that experiments to confirm the precise values of the Lorentz equations. are (unsurprisingly) quite hard to set up! So while relativity is certainly experimentally confirmed to exist, the exact values of relativistic have a far less solid footing. The following chart compares Mathis’s equations to the Lorentz equations.

So even at 10% of c, Mathis predicts a far greater distortion of the image. However, 10% of c is still an incredibly fast speed (30,000 km / s) and so these measurements are incredibly hard to confirm to the necessary degree of accuracy.

I won’t go through all of the proposed confirmations of einstein’s equations, because Mathis has already done some of this, and my own prejudice is that 20th century physics abandoned the project of scientific objectivity, and became co-opted by the market for academic credentials. Ask me and I’ll post the links to Mathis’s responses to the common claims of confirmations of Einstein’s relativity equations in the comments. Also, please ask for further clarifications in the comments, and I will try to answer promptly.
Thanks, Tom