so the symbol i'm going to talk about is
gamma little gamma
which gets used for lots of different
things but in this particular context
i'm going to talk about its use in
special relativity relativity is
something that everyone is actually
familiar with it's the fact that the
laws of physics are kind of the same in
whatever reference frame you're in so
for example when you're sitting on an
airplane unless you look out the window
you're really not aware of the fact that
you're traveling from like 500 miles an
hour that if you were to you know
sitting there at your table to throw
something up in the air ball up in the
air it would just fall back down again
and so on the laws of physics are
basically the same even though you're
moving relative to the earth at high
speed
and what special relativity says
is that all the laws of physics are
basically the same in any reference
frame that's moving at constant speed
and when you work through the mass of
all that and what that actually implies
one of the quantities that sort of comes
out all over the place is is the same
mathematical expression comes out all
over the place which is one over the
square root of one minus v squared over
c squared where v is the speed at which
the reference frame is moving and sees
the speed of light and this comes up so
often that it's actually given its own
symbol it's called gamma it is a it's a
symbol that occurs all over the place in
special relativity for example one of
the places it occurs most blatantly is a
thing called time dilation which is that
if you're in a reference frame which
moving along at speed v it turns out
that your clocks move more slowly than
things that are stationary relative to
you so in fact if you were to just watch
the speed at which the time was ticking
away on a clock in a in something that's
moving like a rocket moving at some
velocity v it actually moves more slowly
than the clocks that are stationary and
the factor by which it moves more slowly
is one over gamma and so for example if
you put the numbers in for something
traveling at ninety percent of the speed
of light when you plug them into that
formula for gamma gamma comes out at
about two bit over two and so that means
that something moving at ninety percent
of the speed of light the clocks there
will be traveling at about a factor of
two slower than those that are
stationary
this sounds very bizarre right because
people are used to the idea that clocks
are just clocks and as long as they're
behaving themselves they should all tick
at one second per second and this idea
that the rate at which time progresses
depends on what reference frame you're
in is pretty bizarre but there's very
strong experimental evidence that it
actually really works um one of the
simplest experiments is it turns out
that there are particles crashing into
the earth's atmosphere crashing into the
top of the earth's atmosphere sort of 40
or 50 kilometers up
and when they crash into the earth's
atmosphere these cosmic rays smashing
into the atmosphere turn into muons you
create muons now it turns out muons as a
particle have very short lifetimes an
elementary particle that decays very
quickly and so they don't actually in
the time between them being created they
actually decay before they get to the
ground even traveling at close to the
speed of light they don't have time to
get to ground level
and so in the absence of special
relativity you would predict that you
should never detect any of these muons
if you have a muon detector on the
ground because although they're being
created up there they don't actually
have time to travel all the way to the
ground but because they're traveling at
close to the speed of light that means
that their internal clocks are going
slower and when your clocks are going
slower everything goes slower including
this process of radioactive decay that
these particles actually decay more
slowly than they would do if they were
just sitting stationary in the lab which
means they actually do have time to get
all the way from where they're created
tens of kilometers up down to ground
level so if you have a muon detector in
the lab you can actually detect these
cosmic ray muons even though in the
absence of special relativity none of
them should ever have made it all the
way from the top of the atmosphere to
the ground and
nine
we have ignition sequence so there's
probably the most famous story of this
associated with this effective time
dilation is this thing called the twins
paradox
which is that if you were you start with
a pair of identical twins and you send
one of them off on a rocket ship to
alpha centauri um and they go off at
some fraction of a high you know high
fraction of the speed of light so
they're traveling sufficiently close to
the speed of light you have to worry
about these things
the one who travels away and goes off to
alice and tori and then comes back again
um because they're moving the person who
stays at home would see this person's
clock travelling more slowly and so
therefore by the time the twin who's
gone to all centauri and then come back
again gets back again they'll actually
be younger than the twin who stayed at
home
now the reason why this is a paradox is
because everything is relative which
means now let's look at it from the
twins point of view who's on the rocket
from the twins point of view who's on
the rocket
he just stayed in the rocket the entire
time and from his perspective the earth
shot off in the opposite direction for a
while and then when he turned around the
earth shot back again so from his
perspective he just stayed stationary
and the other twin the one who stayed on
earth did all the moving moved backwards
and came back again and so by exactly
the same argument the twin who'd
actually gone all the way to alpha
centauri and back again would argue
actually i've seen my twin doing
traveling at high speed so i would see
my twins clocks traveling more slowly
and therefore my twins should be younger
than me so that you end up with this
paradox that each of them by this
argument of special relativity will
claim that they should then be younger
than their their twin
and the resolution to this paradox is
because there isn't actually a symmetry
in this situation which is because the
twin who actually did the journey at
some point had to turn around and come
back and that means that the twin who
did the journey didn't stay in the same
reference frame the whole time he was
initially in one reference frame
traveling one way and then later on was
in another reference frame traveling the
other way which means at some point he
must have accelerated decelerated to a
stop and then accelerated back the other
way and as soon as you introduce
acceleration into the mix of things then
special relativity doesn't work anymore
and if you really want to resolve this
twin paradox properly you have to
consider general relativity which is the
the version of relativity you have to
use as soon as you have accelerations so
the one who would end up being younger
is actually the one who went to alpha
centauri and back the ultimate
resolution when you do the job properly