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The 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 substitue 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 componenents. Remember that acceleration
is measured in metres per second squared. This is distance
over time squared:
Plugging this into our master formula, it gives us:
That's a bit messy. Let's rearrange it a bit:
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 v²
Now energy equals mass times velocity squared. Think about that.
This formula is 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, I can't tell you.
I've never encountered a reason that could be explained as being
more than arbitrary. Perhaps experiments with
particle accelerators have not yet been able to prove it wrong.
So what is E=mv²? 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:
It is essential that we be able to square the d and t values
separately, in order to simplify this equation to E=mv².
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=mv². 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=mv² 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=mv² 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=mc², 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=mv² 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=mc². 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=mv² 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=mc² 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.
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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. Present day 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.
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 unremovable 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 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.
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=mc², and the speed of light.
Lo and behold, I seem to have independantly 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".
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So don't let anyone tell you that Einstein said that traveling
faster than light was impossible, certainly not writer David Fisher
through his character Emilia Rumford in
Doctor Who: The Stones of Blood. (She contradicts herself
with her next line anyway.) Einstein's work actually proves
the opposite.
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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 graviational 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.
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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=mc² 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.
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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.
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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 solidly 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 independantly, 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
proposeed 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 eletromagnetic
radiation (aka 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.
In the best tradition of Stephen Covey's principles and 7 habits,
we don't like this early draft view of the universe either,
because it doesn't yet include your input. We want to hear
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