In this episode several different mathematical topics are mentioned, but we're going to focus on LIGO, plasma cutters, and Benford's law.

LIGO is the place where the fictional character Larry works, but it is also
a scientific project in real life. It is a joint project by Caltech and MIT
which is designed to detect gravitational waves. These can be understood by
analogy to waves in a pond. (Look
here
for more information about waves.) Imagine a very still pond on a windless
day. The
top of pond will be completely flat. We can describe this mathematically by
assigning each point of the pond a number H(x,y,t) corresponding to its
height, and in this case, each point in the pond has the same height. This is
described by the equation H(*x,y,t*) = *c*, where *c* is
a constant. Now if a rock is dropped into the middle of the pond, the
function H(*x,y,t*) will no longer be a constant, but will vary as
a function of time and position. The function in general is difficult to
describe because it depends on the boundary of the pond as well as how you
choose to model dropping the rock, but if you fix the position and just
look at the height of one point as a function of time, (i.e.
f(*t*) = H(*x,y,t*)), then you get something like f(*t*) =
cos(*t*).

Now to understand gravitational waves, we have to describe gravity in a way
similar to our description of the height of the pond. To do this, we have
a function G(*x,y,z,t*) which to each point in space *x,y,z* and
time *t* assigns some kind of mathematical object to describe the
gravity. Let's think for a bit about what we need to describe gravity.
We can feel a force from gravity, and the strength (or magnitude)
of that force depends
on where we are. However, the direction of that force also varies depending
on where we are. A mathematical object which describes a direction and
magnitude is called a
*vector*, so our function G must assign a vector
to each point in space and time.

Actually, if you're careful you'll realize that at any point on the Earth
the gravitational force isn't a constant as a function of time, even without
gravitational waves. Do you know why? A partial answer is that the moon exerts
a gravitational force on the earth that changes as the earth
rotates. This is the main cause of the
tides in the ocean.

Now without any waves, the function G won't be
a constant for us on the Earth since the Earth exerts a gravitational force
that we can feel that varies as a function of where we are.
However, at any particular point on the Earth, G will be
a constant as a function of time if there aren't any gravitational waves.
However, some very large cosmic events like
supernova are capable of
creating gravitational waves, which change G so that at a fixed point,
the vector describing the gravitation oscillates as a function of time
(i.e. looks something like cos( Now the goal of LIGO is to detect such waves. The basic idea of how this
is done is as follows. The scientists built two mile-long tubes that are
perpendicular to each other. Then a laser beam is split so that it travels
down each othe tubes at exactly the same time. At the end of each tube are
mirrors that reflect the beam back to its starting point. The tubes
are built with very precise lengths so that if there are no gravitational
waves, then the two beams are exactly out of phase so they cancel out (this
means that if you write the equation for the intensity of the two beams
at the point where they meet, they look something like cos(*t*) and
cos(*t + &pi *), so when they add together you get 0, which means there's
no laser light). However, if there
is a gravitational wave that is oriented in the right direction, it will change
the length of one of the tubes but not the length of the other, and this
will make it so that the two beams aren't out of phase anymore. This means
there will be a brief flash of light where the two beams meet.

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