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Before
we derive any equations, we must understand the concept of a light
clock. A light clock looks like this:
We can use
it to measure time because we know that time = distance / rate.
(If you went 10 miles at 20 miles per hour, it took you 0.5 hours,
right?) The distance the light travels is twice the length of the
clock (this clock is on end, so it's the height here) and the rate
at which it travels is the speed of light. So with a light clock,
time = 2Length / c (remember that c is always the speed of light,
or 3 x 10^8 m/sec).
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The
Lorentz Time Dilation Equation
Jim is in a train car. He's looking at his light clock.
Bob can see Jim's light clock too. This is what Bob sees. Bob has
his own light clock, which disagrees with Jim's.
Let's say:
- L is the height of
the light clocks
- c is the speed of
light
- t is Jim's time, the
time it takes for one tick of his light clock, or t = 2L/c.
- t' (read "t prime")
is Bob's time, the time is takes for one tick of Bob's clock.
- v is Jim's velocity.
Let's figure out the
distance the light traveled, (according to Bob), using the Pythagorean
Theorem (remember a2 + b2 = c2):
= 
= 
= 
=
Since
we can also say distance = rate x time (2 hours driving 60 miles
per hour takes you 120 miles), we can say distance = ct'.
They are
two different ways of saying the same thing: ct' = .
Let's work it out.
Square both sides:
c2t´2 = 4L2 + v2t´2
Move term to the other side: c2t´2
- v2t´2 = 4L2
Factor out: t´2(c2 - v2)
= 4L2
Bring term to the other
side: t´2 = 
Factor out a term on
the bottom: t´2 = 
Get rid of Squares: t´
= 
Simplify: t´
= 
Since
we know that, on a time clock, 2L/c is the time for each tick,
we can replace 2L/c with t.
For each
tick, t, on Jim's clock, Bob observes t' ticks on his own clock.
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The
Lorentz Length Contraction Equation
There
are methods that use reasoning similar to the proof for the Lorentz
Time Dilation Equation to prove the Length Contraction Equation.
We won't look at them, but merely reason through discussion to
come up with the equation for length contraction.
If you
look at the time dilation equation (shown
to the right) you can see that the bottom term (the part under
the square root) approaches zero as the velocity approaches c.
This means that when you divide t by that number the result gets
larger and larger. What would you do if, instead of getting larger
and larger, you wanted the result to get smaller and smaller?
You, of course, would multiply by the term that goes to zero.
The closer to zero it is, the smaller the result.
Since
you know that length decreases to zero as velocity increases to
c, you can reason that it decreases at the same rate the flow
of time increases. This rate is what physicists call gamma. In
the time dilation equation, you multiply t by gamma to get t',
so t' = t(gamma). In the length contraction equation, you divide
the length by gamma, so L' = L/(gamma). The equation looks like
this:
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Why
Mass Increases with Velocity
Because
it can? Not quite. Suppose Jim takes off in a spaceship leaving
Bob behind. Jim has his spaceship set on constant thrust mode, so
that the force of the engines pushing his spaceship forward is always
the same. This makes Jim accelerate; he is always going faster and
faster. But as we all know, the fastest Jim can ever go is the speed
of light. So how can he always be going faster and faster?
Jim approaches
the speed of light asymptotically, meaning he always is moving toward
it, but never quite reaches it. His acceleration decreases the faster
he goes, even though he is always accelerating. His acceleration
will eventually be very small, but it will never be zero.
"But",
says you, "physics tells us the Force = Mass x Acceleration,
and you said Jim's spaceship's engines were always exerting the
same force on the spaceship." And you are correct to point
this out. Since f = ma, and f stays the same while a decreases,
m, Mass, must increase. And it does. Look at the following graph:
 Since
Mass approaches infinity, and Jim's thrust is constant, his acceleration
will approach zero. If Jim wanted his acceleration to be constant,
he would have to increase his thrust at the same rate at which his
mass was increasing. This is very hard, since the thrust required
to accelerate him would also approach infinity.
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Adding
Velocities in Relativity
In netwonian
physics, you just added velocities.
For example,
let's say Jim is on a train and Bob is on the side of the track.
The train is going 50 miles per hour. Jim walks forward inside the
train at 3 miles per hours relative to the train. In Newton's model
of the universe, Bob would observe Jim moving at 53 miles per hour,
because 50 + 3 = 53.
Newtonian
physics breaks down at very high velocities, because if one simply
added velocities, the result might very well exceed the speed of
light. If the train in the previous example we moving at 75% the
speed of light, and Jim walks forward at 50% the speed of light
(a very rare feat indeed), Newtonian physics says he's going 125%
the speed of light.
The trick
is to realize that Bob see's Jim in slow motion. When Jim thinks
he's running along the inside of the train at 50% the speed of light,
Bob sees him as moving much slower.
Let's call
- w the velocity observed
by Bob.
- u the velocity of
the train.
- v the velocity of
Jim walking inside the train.
- c, of course, the
velocity of light.
Here's the
formula: w = 
If you plug
in all the numbers, you calculate that Bob sees Jim moving at 90.91%
the speed of light. Of course, the difference between Newtonian
physics and relativistic physics is only significant at very high
velocities.
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Where
e=mc2 Comes From
Everyone
knows the equation, but few know why it comes from relativity or
how to derive it. Let's take a look at the equation for mass:
We can re-write
this equation like this (replacing M' for Observed Mass and M for
the mass of the object at rest):
We can expand
using the binomial theorem just like we can expand (a+b)2
into a2 + 2ab + b2. As it turns out, when
you expand
you end up with an infinite series. But the infinite series is such
that each term is less significant than the term before it. So we
just need to look at the significant parts to get an aproximation:
the first two terms. We end up with something like this:
= 
We can put
the expanded version back into our mass equation:
Distribute
the M: 
A really
cool thing happened, (maybe you noticed). In physics, we use 
all the time to represent the energy of a moving body. So,
e = .
(Piece of cake.) Now we can substitute it in our equation.
You can
express the difference between something's rest mass and its observed
mass with (M' - M). We'll call this difference in mass m. The rest
of the derivation becomes obvious, and you have no other choice
than to come up with  .
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The
Math Behind General Relativity
The math
behind general relativity is called Einstein Field Equations. They
are equations of the coupled hyperbolic-elliptic nonlinear partial
differential type, which, in plain English, means that they are
really, really hard. Einstein himself recognized the mathematical
difficulties of general relativity as "very serious."
He predicted it as being the primary hindrance of general relativity's
development. The equation can be stated in a "symbolic form"
that isn't very useful. Here it is:
It doesn't
mean much to us, but you can see on the left of the equal sign the
stuff that describes the curvature of space-time. On the right is
the matter within space-time, and how it behaves.
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