E=mc at LYRATEK.COM

"That fellow Einstein suits his convenience. Every year he retracts what he wrote the year before."
~ Albert Einstein, speaking of himself, December 1915

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The Unraveling of E=mc

In 1905 Albert Einstein formulated his famous theory that says that energy and mass are both different expressions of the same thing. Then, almost as an afterthought during the process of writing out the explanation of his theory for fellow scientists, he scribbled what is now the world's most famous equation:

E = m c

Most learned individuals are able to recite this as "Energy equals mass times the speed of light squared", but I believe this leads to a grand misconception that does not get sufficiently challenged. Can it be that those who believe that the speed of light represents a speed in this equation have not transcended Newtonian Physics as intended?

As much as his ideas may have been a break-away from then-traditional "Newtonian physics", E=mc by itself does not transcend Newtonian Physics. In fact, the equation can easily be derived from other equations basic to Newtonian Physics, where its meaning seems to be rather less than what was intended.

Of course the intended meaning is to show the relationship between energy and mass, and how much of one you can get from the other. I've always wondered exactly what the speed of light had to do with the conversion. That has also led me to question exactly what kind of energy Einstein sees the mass converting into through this equation. The tendency is to assume kinetic energy: the energy of movement, and to wonder whether the mass has "disappeared" by accelerating itself to the speed of light.

Let's dig into the math for a moment, to see where we end up if we don't transcend the easier-to-understand Newtonian Physics. What light it can shed on this equation? As you will see, it looks promising for a while, but then all falls apart....

(Or click on this link to skip the math and continue with the concepts).


A Newtonian Derivation

Newtonian Physics states that Energy is measured in Joules. Work is also measured in Joules. Therefore:

E = W

We also know that Work is a Force applied through a distance. Since W = F x d, we can substitute this into our equation:

E = F d

Force = mass times acceleration, the second law of Newtonian physics. When we substitute this understanding of Force into our equation we get:

E = m a d

Now energy equals mass accelerating over a certain distance. Starting to sound familiar, isn't it? Let's now break acceleration into its components. Remember that acceleration is measured in metres per second squared. This is distance over time squared:
a =
d
t t
Plugging this into our master formula, it gives us:
E = m
d
t t
d
That's a bit messy. Let's rearrange it a bit:
E = m
d
t
d
t
Notice how we've now got d/t in there twice. What is d/t? Distance over time? Metres per second. Or kilometres per hour. Speed! Also known as velocity in physics, and given the symbol "v". So that means:

E = m v v

Or, since we're multiplying velocity by itself, it's neater to write:

E = m v

Now energy equals mass times velocity squared. Think about that. This formula looks to be true in Newtonian physics for ANY velocity you want to substitute as "v". Einstein chose to be specific in his formula to name the speed of light. Why? Well, that frustrated me for years. It seemed I could never encounter a reason that could be explained as being more than arbitrary. Perhaps experiments with particle accelerators simply weren't yet able to prove it wrong, so it was accepted.

But finally I stumbled upon what may be a good or partial answer, which I will talk about below. However, it really does require you to look beyond Newtonian Physics, specifically the speed involved in kinetic energy, for the more we truly understand the meaning of relativity, the less meaning speed has at all.

So what did we manage to wring out of Newtonian Physics? What is E=mv? The kinetic energy of a mass moving AT a certain velocity? Not quite. To understand it as we've derived it, we have to back up a bit:
E = m
d
t t
d
It is essential that we be able to square the d and t values separately, in order to simplify this equation to E=mv. The t values are simple. Both values are the same, and typically 1 second each, because we derived them from acceleration which is measured in metres per second squared. The t values came to us squared already. But what about the d values? The first d describes the value of the acceleration, the second the distance through which the force necessary for that acceleration must operate. These two values are not necessarily the same, and if they're not the same, we can't square them.

Of course, we could arbitrarily make them the same, as Einstein seems to have done. In that case we could square them, and get E=mv. If our identical d values were 10 metres, let's say, what would our equation describe? The energy required to accelerate a mass at 10 metres per second per second, over a distance of 10 metres.

Note that we're not actually describing a speed here. In fact, if our object started at rest, what speed would it reach after 10 metres? Well, it's not going to cover 10 metres in the first second, because it's still accelerating from zero up to 10 metres per second in the first second. So this acceleration through this distance will last more than 1 second, therefore the velocity will increase more than 10 m/s.

But what if our object doesn't begin at rest. What if it's already moving at 20 m/s when it enters the area of distance that we're measuring? It could cover our 10 metres in half a second at that speed alone, and it's accelerating on top of that. If it spends less than half a second in our measuring zone, it won't have time to increase its speed even by as much as 5 metres per second.

So, not only does E=mv not describe a speed, it cannot either describe a specific change of speed independently of its "measuring zone", or frame of reference. Doesn't sound like E=mv is a very useful equation, does it?

It does, however, prove one thing. If v is substituted with the speed of light, and we write E=mc, this now refers to the energy required to accelerate a mass either BEYOND the speed of light, if it began at rest at the beginning of our measuring zone, or a mass that was going faster than the speed of light to begin with and has gone yet faster still. Fascinating....



But we've still got an unchanging mass, and we're measuring the energy as a function of that mass's movement: kinetic energy. The Newtonian law of conservation of mass is satisfied. And the Newtonian way of satisfying the law of conservation of energy is to say that, well, if it wasn't actually moving with that amount of kinetic energy, it could be, therefore when it's at rest, it has an equal amount of potential energy.

What's the maximum amount of potential energy any old lump of matter could have? Depends on how fast it can go, according to Newtonian physics. It could be anything. It could be infinite. And E=mv has not given us any reason to limit this; quite the reverse in fact.

But Einstein tackled this in his conversion theory, then gave us E=mc. And perhaps a finite mass, if it is converted into energy, should produce a finite amount of energy. Does that then limit its top speed to the speed of light?

It shouldn't. Remember, E=mv is not about converting mass into energy. It's about measuring the amount of energy required to accelerate it smoothly over a given distance.


If E=mc is true as a mass-energy conversion principle, then we must recognize that it's not really about the KINETIC energy of anything moving at the speed of light. But the speed of light has since become a bizarre lightning rod for theories and scientific thought for the past century, and is long overdue for a lot of extra scrutiny.


The Perception of the Speed of Light

Books about Physics are most often full of explanations of why nothing can travel faster than the speed of light, or worse, why going faster than the speed of light would send you backwards in time.

My first encounter with these ideas was very vivid. It was 1980, and I was initiated into the universe of Battlestar Galactica with a three episode arc in which the space travelers find present day Earth. But present day 1980 was not enough; they soon sped their way into the 1940's, proudly explaining that this was achieved by traveling faster than light, and giving the viewers some very memorable imagery of them doing so.

My 10 year old mind could not rest the night after I saw that. It just didn't make sense. Surely the speed of light could not be fast enough to induce reverse time travel. Surely you would have to be going fast enough to be completely instantaneous, AND THEN go faster yet to experience time in reverse. The speed of light is too slow and mundane.

My quest to understand the concept had begun, and nothing I've ever seen or heard or read since has satisfied me that this commonly spouted theory holds the truth.

Funhouse Mirror Explanations

This is the big trap that most attempts at explaining the speed of light fall into. Our latest model of science, based on Quantum Physics, has as one of its tenets that we cannot ever truly know the universe as an arbitrary construct. We can only know it through our perception of it. And our perception of it at the same time alters it. Don't worry too much while trying to wrap your head around that one. Just know that perception is an irremovable part of the quantum explanation of the universe.

Fine. Does that mean any perception of the universe is correct or helpful in understanding how it operates? Is there no longer any room in the universe for the idea of complete misperception? I don't think so.

Think of a funhouse mirror, where the glass bends inwards and outwards. What happens when you look at yourself in that mirror? You might look taller or shorter than you really are. You probably look ultra fat in one part of your body, and ultra skinny in another part. Does that mean that your weight and height and shape have actually changed? Of course not. You look at yourself directly without the mirror, and you can see your true shape.

Simple enough concept. Now pick up a science book, and read explanations about the ultimate limit of the speed of light, especially if it delves into an alteration or bending or stretching of the time/space continuum. What do these explanations have in common?

All the ones I've encountered invariably "prove their point" by describing what an observer would perceive as the speed of light is approached or surpassed. By perceive, we should actually say "see", because these observers invariably depend on their eyesight for these observations.

What's wrong with this? Putting aside the limitations of our minds to respond quickly enough to notice anything traveling at the speed of light, our eyesight depends on light to carry the information. Speeds approaching the speed of light will mess with the flow of that information, compressing or stretching it, perhaps reversing or upsetting the sequence in which it reaches us. In short:

When nearing or surpassing the speed of light, we can no longer trust our eyesight to give us an accurate perception of reality than if we looked into a funhouse mirror.

And yet, book after book, writer after writer, scientist after scientist, teacher after teacher, have tried to say that because the hypothetical observer would SEE a change in mass, or a reverse sequence of events, mass and time have ACTUALLY altered.

Now, I do like to think that matter is energy locked in pattern, and that the two can "convert" so to speak. And I do like to think that travel in time is possible, along with a lot of other neat ideas.

But I don't see that speed has anything to do with it. Can anyone prove me wrong without simply resorting to another funhouse mirror? (Or "Can I prove myself wrong?" I say to myself as I read this over...)



Speed is Relative

Any object that spends a measurable amount of time traveling near, at, or beyond the speed of light, in a straight line, will be in outer space quite quickly. Which begs the question: If the speed of light is a limit that no object can travel faster than, how do we know when we've reached that speed? What stationary post do we measure that speed against?

We have speed limits on our roads and highways. We measure those speeds against the road itself. The road rests on the Earth, and for purposes of highway speed, we regard the road and the Earth as stationary objects.

But if we want to measure the speed of light, we really are forced into looking to outer space for our stationary markers. And the Earth is not stationary. It rotates daily on its axis, and is nipping along at 29 km/s through its yearly orbit around the sun. That's 29 kilometres per second, not per hour, so it's quite a good speed. The sun isn't stationary either; it travels in an orbit around the centre of our galaxy, the Milky Way. The galaxies are moving too, with the most popular theory being that they are all moving outward from the centre of the universe where the big bang started it all. However, astronomy is digging deeper into that theory, and it may be under revision as we speak.

Where in all of that can we find a stationary marker against which we can measure any astronomical speed? The center of the universe is still a theoretical thing that we can't yet quite find. It's certainly not commonly accessible yet. And for all we know now, it too may be moving.

Out of Sight Speed

There are a lot of speed of light analogies that begin by sticking a person in a hot-rod rocket ship and blasting them off of the Earth. We're going to imagine one now, except that for this example, the Earth is not a stationary object. It is in fact traveling at 60% of the speed of light "to the east" for want of a better reference. That's much less than the speed of light, so the strict lightspeed scientist-cops out there won't give it a ticket.

Now we'll put our dude in a rocket ship. He's going to blast off in his rocket "towards the west", and basically accelerate until he's no longer going east with the Earth at such a clip. In fact he's going to come to rest in outer space, and let the Earth continue to fly to the east at 0.6 of the speed of light.

That was too much fun. Our dude wants to accelerate some more. He increases his velocity to the west until he himself is moving at a good clip. He opens her up to 60% of the speed of light, going westward.

Notice what we have here. The Earth going at 60% of the speed of light to the east, and hot-rod rocket dude going 60% of the speed of light to the west. The scientist-cops can't give either of them a ticket for exceeding the speed of light.

But what is their speed relative to each other? Hot-rod rocket dude is now traveling away from the Earth at 120% of the speed of light. As far as the Earth is concerned, he has exceeded the speed of light.

So remember, speed is relative to whatever "stationary" markers you want to measure it against. If the speed of light cannot be exceeded, we better be sure we define exactly what that speed is measured against. If we are allowed to change our minds about what we measure speed against any time we like, we may never know any limits at all.

And in fact, is this not the real meaning of "Relativity", which Einstein was trying to impart?


Think of hot-rod rocket dude looking out from his spaceship, traveling at 0.6 of the speed of light to the west. He can't see the Earth anymore; no light photons coming from it can catch up with him. But he can see some other stars and planets which are also traveling at 0.6 of the speed of light to the west with him. Since they're both traveling at the same speed as he is, it looks to him as if he and they are sitting still. So he makes a little paradigm shift. They become his new stationary markers. He accelerates a further 90% of the speed of light to the west, relative to those objects, and finds he has caught up with yet other celestial objects that no one on stationed on Earth had ever remotely detected before, since there was no way that light photons emitted from them could ever have caught up with the Earth. Cool. Perhaps they could become his new "stationary markers", while he accelerates yet another 90% of the speed of light to the west.

The whole point I'm trying to make here is that the speed of light isn't necessarily a limit to the speed you can actually travel at. What it really is, is a limit to our perception, specifically our sight.

Anyone who wants to challenge this had better either come up with a good absolute stationary marker to measure all speeds against, or a better understanding of the electromagnetic radiation spectrum of which light is a part and which all travels at "the speed of light". And no funhouse mirrors please.

Now, what's interesting is that I wrote down the above theory of mine that I've had for some time, supposedly as an argument against one of the popular interpretations of Einstein's Theory of Relativity, then I went and visited http://leiwen.tripod.com/theory.htm to further my understanding of Einstein, relativity, E=mc, and the speed of light. Lo and behold, I seem to have independently come up with one of the two fundamental postulates upon which Einstein's Theory of Relativity is based. In fact, this is the idea behind the name "Relativity".

See also Einstein Online - Principle of Relativity, this is a nice and succinctly written piece insisting that all space time motion is relative, and that there are no absolutes. Notice however, how the next page Einstein Online - The Relativity of Space and Time completely contradicts this by saying that moving clocks move slower than stationary ones - even after establishing that we have no way of knowing which ones would be stationary and which would be moving - we could only know that they all move relative to each other. Also grossly neglected are limitations of perception, where the time we think things happened is affected by the speed of light bringing sight to our eyes, versus the speed of sound bringing noise to our ears. (Note how many times the page uses the phrase "appears to be" instead of "is", which is quite typical of most in-depth explanations of relativity.) Then there's the whole mystery surrounding the speed of a nerve impulse traveling from the sense organ to the brain, and whether or not some kind of quantum superposition or time reversal occurs allowing the brain to perceive a sensation before the nerve impulse could possibly have transmitted it. These are factors. Special Relativity, however, ignores them to equate perception with fact.

So don't let anyone tell you that Einstein said that traveling faster than light was impossible. His work on relativity actually proves the opposite.


Gravity Holes

One of the more interesting concepts that Einstein came up with is that gravity actually bends the space/time continuum. Would it then not also affect the speed of electromagnetic radiation passing through its field? In other words, does gravity change the speed of light?

The whole concept of a black hole in space seems to depend on this idea. A star that becomes so massive that it collapses in on itself, concentrates its gravitational pull so much that light can no longer escape it, that's what a black hole is. The speed of light basically slowed down so much within it that it reversed itself. No light can come out of it, or pass through from behind it. It sucks it all in. Spooky. Dark. Completely black. A hole in the backdrop of space.

All gravitational bodies must be doing this to a small extent then. Not stopping light completely and more, but at least slowing it down a little. All stars are doing this now. Ours must be doing it.

I don't know how they're measuring the speed of light these days, but the first attempt to do so is neatly chronicled in the book E=mc by David Bodanis, under the section "c is for celeritas". Basically they measured differences in the time it took for light to reach the Earth from the moons of Jupiter at different times of the year. In other words, that difference was based on a distance from one side of the Earth's orbit around the sun to the other. Their point of reference, or stationary marker, was the center of our solar system, the sun itself.
David Bodanis' "Biography" of the world's most famous equation is a great read about the history of many of the physicists, ideas, and other contributions leading to the formation of the equation, and its subsequent uses.

How much was the sun's gravitational pull affecting that speed? Have we ever been able to measure the speed of light without the gravitational pull of our own sun affecting that speed? Could we ever be able to measure the speed of light without the gravitational pull of the center of our galaxy affecting it? Are scientists taking these factors into account?

Perhaps the speed of light can be infinite, when free of gravity. Perhaps there is no real environment totally free of all gravity. Who knows. But these are the questions and concepts that fascinate me.


The Hyperspace Factor

We're still not sure what light, or any other wavelength of electromagnetic radiation, really is. A particle? A wave? A photon that has properties of particles and waves, and is sort of both and neither at the same time? I'm not sure I understand energy anymore either, as a concept that can take in all its forms: heat, light, electricity, magnetism, potential stored chemically. It's most obvious form - movement - seems to have been rendered impotent by relativity, since for any object in motion there is a frame of reference that says it is standing still. Plus, kinetic energy is, I think, easily confused with momentum. Right now, the only form of energy I think I really understand is mass. I can see it, touch it, measure it accurately, and not be fooled by its apparent solidity at the same time.

The truly bizarre phenomenon that has been observed and must be accounted for, is that no matter what speed you travel at, light can always be found traveling at "c" relative to your frame of reference. The explanation offered that I don't buy is to suppose that the time/space continuum stretches or contracts to compensate.

Has anyone ever thought this: Perhaps light is always moving around us at every imaginable speed, but we can only perceive, with our eyes and instruments, the light that travels at c relative to ourselves. Anything else won't trigger our sensors and receiving apparatuses. (And if a black hole has gotten in the way, perhaps it's already "eaten" those photons that would have been traveling at the right speed to allow you to see it.)

Perhaps this sheds some light on what seems to be the real reason for the inclusion of "c" in Einstein's equation. The speed of light may operate like something of a 90-degree right angle in interspatial geometry, allowing one dimension to interact with another. "C" seems to have been an important element in the "Lorentz transformations" dealing with trans-dimensional geometry. Three or four physicists came up with these same transformations independently, including both Lorentz and Einstein even though neither of them were the first to do so, so they must have some validity.

Einstein's theories soon insisted on looking at more than just the three dimensions of space and one dimension of time that we all experience daily. Personally, I'm always keen to add one or more dimensions of choice (leading to parallel universes) to my own space-time continuum. Was Einstein laying the basis for these in physics when he proposed that there were actually nine dimensions in space, of which we only perceive three? His new physics takes into account movements and forces operating through all those dimensions, the ones we can't see being referred to as hyperspace. And perhaps you need the speed of electromagnetic radiation (a.k.a. light) in the Lorentz transformations to find out how a force or movement in a hyperspace dimension affects a force or movement or electromagnetic aggregation of energy into matter in the dimensions we're accustomed to. That's my best understanding of it at the moment. It feels unproven, and I'm not totally convinced yet.


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