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Wednesday, October 04, 2006

Blinded With Science: Special relativity

I haven't really shown my science geek side on this blog yet, so it comes out a bit tonight.

Reading the mental_floss blog, I ran across an article mentioning special relativity. mental_floss (or rather the source they were quoting) stated:

When our instructor started talking about length dilation (how something’s length increases as it approaches the speed of light)...

Of course, as poster Garon Whited states, this is incorrect:

Actually, the length of an object appears to decrease, not increase.The faster the object moves, the shorter it appears. Relativistic speeds involve time dilation, mass gain, and length contraction.

I was originally going to post there to second his comment, but I found my comment going to a ridiculous length, so I'll make it an article here instead. Keep in mind that I'm drawing this from memory, so hopefully I'm remembering correctly.

If my memory serves me, the actual formula for spatial dilation in special relativity is:

x = x'((1 - v^2 / c^2) ^ 0.5)

Where x is the length of an object in the field that you view as stationary (the earth in this case), x' is the length of the object at relative rest, and v is the relative velocity of the object.

Since nothing can ever travel faster than (or, realistically, at the speed of) light:

0 <= v < c
0 <= v^2 < c^2
0 <= v^2 / c ^2 < 1
0 < 1 - v^2 / c^2 <= 1
0 < (1 - v^2 / c^2) ^ 0.5 <= 1

(Note: x ^ 0.5 is the same as "the square root of x".)

Of course, if you multiply the object's length by a number no greater than one, it will always shrink or stay the same.

Some interesting values:

If v = 0 (stationary object), then x = x' (no change)
If v = 0.25c (one-quarter the speed of light), then x = 0.968x (object loses about 3% of its length)
If v = 0.5c (half the speed of light), then x = 0.867x (object loses about 13% of its length)
If v = 0.75c (three-quarters the speed of light), then x = 0.661x (object loses about 34% of its length)
If v = 1 (traveling at the speed of light), then x = 0

The time-dilation formula is similar:

t = t'/((1 - v^2 / c^2) ^ 0.5)

Note that you divide in this case, making the number effectively larger.  What this means in practice is that, if you were watching people on an airplane moving near the speed of light, it would appear that time is passing far more slowly for them.  This is the basis for the twins paradox.

If v = 0 (stationary object), then t = t' (no change)
If v = 0.25c (one-quarter the speed of light), then t = 1.033x (it takes 62 seconds for the plane to experience a minute of time)
If v = 0.5c (half the speed of light), then t = 1.153t (one minute takes 69 seconds)
If v = 0.75c (three-quarters the speed of light), then t = 1.513t (one minute takes 91 seconds)
If v = 1 (traveling at the speed of light), then t is undefined (one moment takes eternity)

Despite what Isaac Newton and high school teach, the formula for calculating kinetic energy is:

E = m(c^2)/((1 - v^2 / c^2) ^ 0.5)

An object's mass increases with its kinetic energy, so (using the same formula as above for time dilation), an object traveling at three-quarters the speed of light is 1.513 times "heavier" than it would be otherwise, due to the excess kinetic energy.  Perhaps the most interesting (and well-known) consequence is how the formula simplifies at 0 velocity:

E = m(c^2)/((1 - 0^2 / c^2) ^ 0.5)
E = m(c^2)/((1 - 0 / c^2) ^ 0.5)
E = m(c^2)/((1 - 0) ^ 0.5)
E = m(c^2)/(1 ^ 0.5)
E = m(c^2)/1
E = m(c^2)

Yep.  There's good ol' E equals MC squared, which shows that even objects at rest have kinetic energy.  Of course, since this doesn't really fit the name "kinetic energy", we call it rest-mass energy instead.

Conversely, an object traveling at the speed of light would become infinitely massive.  Of course, since the energy required to push the increasingly massive object toward that velocity would increase exponentially as it approaches the speed of light, you can't actually get enough energy in the universe for it to hit that point.

Well, there it is.  Again, this was pretty much from memory, so you might take this with a grain or two of salt.  Maybe someday I'll post about some of the other intriguing aspects of special relativity, like the addition of velocities and the lack of a single stationery plane (since everything is relative), but we'll save that for later.

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