| A human being is part of a whole, called by us the "Universe," a part limited in time and space. He experiences himself, his thoughts and feelings, as something separated from the rest -- a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest us. Our task must be to free ourselves from this prison by widening our circles of compassion to embrace all living creatures and the whole of nature in its beauty. |
The other day I was looking through an old closet and found a dusty little book, written by Einstein, about his Theory of Relativity.
He had a knack for simplifying complex ideas, so I thought I'd mention the stuff I found in an appendix, on the Special Theory. It's quite interesting.
>The Theory of Relativity? You're kidding eh?
Pay attention. It's just high school math.
We're going to talk about Sam and Sally, twins, each 30 years old.
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We consider two moving axes, x and x', where Sam lives on the x-axis and Sally lives on the x'-axis. Sam sees Sally, on the x' axis, moving past at a velocity v. What we want is to determine the relationship between any space and time measurements made by Sam and by Sally. Now one of Einstein's fundamental assumptions is that they will each get the SAME value for the speed of light: c. |  |
Suppose a light signal is moving by (in the direction of the positive x and x' axes) and they each measure the speed of that signal as c.
Sam says: The location of the signal is x = ct measuring the location, x, at time t
Sally says: The location of the signal is x' = ct' measuring the location, x', at time t'
In particular, Sam says:
[1a] x - ct = 0
and Sally says:
[1b] x' - ct' = 0
Following Einstein's appendix, we note that [1a] and [1b] would be satisfied if
[2a] x' - ct' = A(x - ct) for some number A so, when one is zero, so is the other
If the light signal were travelling in the opposite direction (hence with velocity -c) we'd have:
[2b] x' + ct' = B(x + ct) for some number B.
If we add [2a] and [2b] then t' is eliminated and we can get x' expressed in terms of x and t. The relation has the form:
[3a] x' = ax - bct
If we subtract [2a] from [2b] we get t' in terms of x and t, in the form:
[3b] ct' = act - bx
Equations [3a / b] give the relationship between any space and time measurements that Sally makes ... and Sam's measurements.
Alas, we don't know "a" and "b".
>You got [3a] and [3b] by considering just that light signal. How about ...?
Another measurement? You've been reading that appendix, eh? Yes, we need two more situations in order to detemine both "a" and "b".
Okay, suppose that Sally sits permanently at the origin of her axis, at x' = 0.
Then [3a] would read: 0 = ax - bct so x = (bc/a)t.
>Huh?
[3a / b] give the relationship between measurements that Sam makes and those that Sally makes ...
>And Sally sees herself at x' = 0, right?
Yes. That's her measurement of her own position.
>Uh ... so when Sam says x = (bc/a)t, what is Sam measuring?
Sally's velocity.
>But you said that Sally is moving past Sam at the velocity v.
Right! So v = bc/a. We can then write b = av/c and that'd change our equations [3a / b] to:
[4a] x' = ax - avt
[4b] ct' = act - (av/c)x
We still have one parameter to evaluate. That's 'a".
However, another of Einstein's fundamental assumptions (which gives rise to the term "relativity") goes like so:
- Sally is sitting with Sam, on his axis. (She's not moving on the x'-axis ... not yet!).
- They're each holding a stick of length L.
- Sally gets onto her x' axis and moves off at velocity v.
- Sally measures Sam's stick length and Sam measures Sally's.
- According to Einstein, they'd each get the same value!
No, but they DO get the same value. That's "relativity", eh?
Sally, if she measures the length of her stick, will still find it to be L.
To determine Sam's value we let Sam take a snapshot of Sally's (moving) stick at time t = 0.
From [4a], putting t = 0, we get: x' = ax so distances measured by Sam are Sally's distances divided by "a".
Sam, viewing his "snapshot" of Sally's (moving) stick, then measures the distance between the endpoints of Sally's stick as L/a.
That's important so we give it a place of importance:
| [A] If Sally measures a length on her x' axis as L, Sam will measure that same length (moving past at velocity v) as: L/a. |
Similarly, Sally takes a snapshot of Sam's stick at time t' = 0.
Putting t' = 0 in [4a / b] we get a relation between what Sally measures (at time t'=0) and what Sam measures, namely:
[5] x' = a (1 - v2/c2)x.
That means that, if Sam measures a stick on his x axis as having a length L, Sally will measure the length as a (1 - v2/c2)L.
That's important so we give it a place of importance:
| [B] If Sam measures a length on his x axis as L, Sally will measure that same length (moving past at velocity v) as: a(1 - v2/c2)L. |
Uh ... yes, you're quite right. Fortunately, [B] doesn't change if we replace c by -c.
>But you said they'd be the SAME! They sure don't look ...
I didn't say that. Einstein did!
Anyway, we set them equal and determine "a". Neat eh?
We set them equal: L/a = a(1 - v2/c2)L
and get a2 = 1/(1 - v2/c2).
>Doesn't "a" have a place of importance?
Yes. We can now write [4a / b] as:
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[6]
If two axes are moving parallel to each other with relative velocity v,
then space and time measurements in each are related by the Lorentz transformation  |
>zzzZZZ
Can't you see? If Sam measures Sally's stick length (as she passes at velocity v), he measures the length as L/a and, now that we know "a",
we see that he measures L SQRT[1 - v2/c2] so, if Sally is whizzing by at 95% of the speed of light (so v/c = 0.95) then he gets a measurement of 0.31L.
See? Space measurements of some moving object are reduced ... according to how rapidly the object is moving.
A yardstick whizzing by at 0.95c appears to be about 11 inches long.
>zzzZZZ
Time measurements have similar characteristics.
Suppose that Sally is sitting at her origin, x' = 0, with a clock. Two successive clicks of her clock occur at t' = 0 and t' = 1.
In the meantime, Sam is measuring the time between the clicks on Sally's clock (according to his clock).
The relationship between their measurements is given by the equations [6].
Let x' = 0 and get x = vt (from the first of [6]). That's Sam, measuring the position of Sally at her origin. She's moving past at velocity v.
Now stick x = vt into the second of equations [6] and get t' = t SQRT[1 - v2/c2]
or, to get Sam's measurement: t = t' / SQRT[1 - v2/c2].
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That's a larger measurement. In fact, if Sally is whizzing by at 0.95c and she measures the time between clicks on her clock as 20 years, Sam will measure the same time as 20 / SQRT[1-0.952] = 64 years. When Sally is ... >Wait! Sally and Sam were 30 years old so when Sally is 30 + 20 = 50 years old, then ...
>So you understand all this stuff?
>That's disgusting. |  |
>zzz
In fact, Einstein has the mass as
where mo is the mass when at rest; that's v = 0.
Not only that, but a mass moving with velocity v has energy
>zzz
If we expand as a series: [1-(v/c)2]-1/2 = 1 + (1/2)(v/c)2 + ...
we'd get E = mc2 + (1/2) m v2 + ...
That (1/2) m v2 we recognize as the kinetic energy associated with a moving body.
BUT (and this is the really neat part), even if the mass is at rest (so v = 0 and m = mo) we'd still have energy: E = moc2
Do you recognize that ... the most famous equation in the whole universe?
>zzzZZZ
