TONY PADILLA: What's going on in London?
Well, I don't know just some small event known as the
Olympic Games is going on at London at the moment.
Last night.
In particular, there was a 100-meter final and Usain Bolt
winning it in 9.63 seconds.
Well, it was a little bit less, really, if you include
some of the relativistic effects from Einstein's theory
of relativity.
So I guess that's what we're going to talk about.
All right.
So do you want me to draw a picture of race?
BRADY HARAN: Yeah.
TONY PADILLA: OK.
I'll do my best.
So of course, you've got--
Here we are at the Olympic stadium in London.
There's my racetrack.
It's 100 meters long.
The guys start down here, right?
And then they run there along the tracks.
Let's say this is Tyson Gay or something.
And then somewhere out in front of him,
we've got Usain Bolt.
So Usain Bolt in front, and a bunch of the other guys sort
of en route.
So Usain Bolt went past the finishing line in 9.63
seconds, according to the stadium clock.
So let's make that clear, the stadium clock.
Now, what I want you to imagine is that Usain Bolt--
this is Usain here--
has got a little watch on.
So this is his watch.
So this is Usain Bolt's clock.
Well, what did Usain Bolt's clock read at the point moment
he crossed the finishing line?
Now, you might think, well, it's 9.63 seconds.
But it's not.
It's actually a little bit less than that.
It's 9.63 seconds less about more or less five millionths
of a nanosecond.
So he actually clocked according to his clock, five
millionths of a nanosecond less than the stadium clock,
less than this 9.63 seconds.
OK.
So a nanosecond is 10 to the minus 9 seconds.
So if you do it in the American billions, that's a
billionth of a second.
And then it's a millionth of that billionth.
Naught point naught naught, naught--
that's three naughts.
Naught, naught, naught, naught, naught.
We've got 12.
I've got four, five, six, seven, eight nine.
So Usain Bolt, if you actually looked at his watch, it would
have read this.
It would have read 9.629999999999995.
Well, this is all because of special relativity, Einstein's
theory of special relativity.
So what was the key thing about Einstein's theory?
A lot of people say it's the oh, you can't travel faster
than light.
And that's true, but that's not the most
important thing about it.
The most important thing about it is that everybody agrees
that light travels at about 300 million meters per second
in a vacuum.
That's the most important thing, that everybody agrees
on that, no matter whether they're moving
relative to it or not.
So this really changes how you think about velocities and how
you think about timing in a really fundamental way.
And so to illustrate that, imagine you're going along the
motor way, and you're going in a car at 70 miles an hour.
And The guy is going in the opposite direction at 70 miles
an hour relative to the road as well.
Then you would say, what's the relative speed of you
to the other car.
And you'd say, well, what would you say it is?
BRADY HARAN: 140.
TONY PADILLA: 140.
That's what most people would answer to that.
But actually, that's not quite right.
That's almost right.
It's a little bit less than 140.
And to see that it can't quite be the right answer, consider
what would happen if the car was going along at
the speed of light.
Then you would be saying what's our relative speed?
Well, is it the speed of light plus 70 miles per hour?
Well, no.
It can't be, because everybody agrees that something that
goes goes at the speed of light goes at the
speed speed of light.
So you've clearly got to change what you think about
velocities, what you think about time.
So what actually happens when somebody is moving relative to
something else is that their clock slows down, and it slows
down by a particular factor.
So if we were to work out what Usain Bolt's time is, tbolt.
And we were to compare it to tstadium.
This is a time measured by Usain Bolt on his clock.
This is the time measured by the stadium clock,
which is this one.
And they differ by a factor.
They're not the same.
They differ by this factor.
It's square root of 1 minus v squared over c squared.
Now, v here is Usain Bolt's speed relative to the track.
And c, of course, is the speed of light.
So this is a very small number, this v squared over c
squared, but it's not zero.
And these would only be equal if it were zero.
So what we can do is if we want to get this approximation
that we've got here, well, we do a little sort of bit of
mathematical trick here.
We do what's called a Taylor expansion on this thing,
because this system is very small.
It's v squared over c squared.
So we can do a Taylor expansion, and we can
approximate this by 1 minus a half v squared over c squared
times tstadium.
I'm going to have to look it up now, aren't I?
So I'm going to put that in.
So that's 299792458.
That's probably not exact values, but you're going to
have to make do with that.
4, 5, 8.
5.77 times 10 to the minus 15.
When I did this, I did this sort of essentially, very
quickly, approximating things.
But actually, maybe I should have done it a bit more
accurately and this should be a 7, 8 there.
So this is the number that Usain Bolt actually ran the
time that he clocked on his watch, which is 9.62999999--
how many 9's have I got--
422 seconds.
And that's the time that he actually would have
clocked on his clock.
Now, there is actually something even more remarkable
about this when you actually think about the implication.
You could ask the question the question.
So he's running it in a shorter time it would seem.
Does that mean he ran faster in some way?
Is that the implication?
And no, it doesn't mean that, because the laws of relativity
don't allow that.
Basically, Usain Bolt's running at roughly 10 meters
per second relative to the track.
And just by the laws of relativity, the track is
moving at about 10 meters per second relative to Usain Bolt,
which is completely equivalent.
So there is no change in their speeds.
So what's changing, then?
Well, actually, the answer is even more remarkable.
The answer is that from Usain Bolt's perspective, that track
is moving, and then because that track is moving, it
actually shrinks a little bit from his perspective by about
50 millionths of a nanometer.
So he clocks a shorter time basically because the track
shrunk a little bit for him.
So you could say that he didn't complete 100 meters.
They should take his metal off him.
JOHN HUNTON: --and each one goes outside one ring and
inside the other as you go around.