WEBVTT
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MARKUS KLUTE: Welcome back
to 8.20 Special Relativity.
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In this short section,
we want to introduce
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a new notation, four-vectors.
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And if you look at
previous discussions,
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this is actually not that new.
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We have seen that we need
to treat time and space
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in a consistent manner.
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And you have often applied
Lorentz's transformation,
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for example, to a vector of
time and the next component
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of space.
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Now you just want to do
this with x, y, and z here
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and not treat the y component
and z component as 0.
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So as a starting point, you
can just simply say, OK,
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we have this new four-vector.
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And the 0's component
is the time or time
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times the speed of light.
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And then the first component,
second and third component
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are the spatial
component, x, y, and z.
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Now I wrote a
vector Xi mew here,
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with the mew being
the upper index.
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I can also introduce
Xi with a lower index.
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And you see little y
and little y is useful.
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Where the 0's component
is not t but minus ct--
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but minus ct.
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As a reminder for
three-vectors, you
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learned about the dot
product, which is just
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a multiplication of
two, three-vectors
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where all vectors with n
components, where you multiply
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the same component
of each vector
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and add those results together.
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So the dot product of
vector a and vector b
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is the sum of all
indices for ai and bi.
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Now for our four-vector,
we do the very same thing.
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We just sum over
all four components.
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And we treat the
vectors as a product
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of the vector with the lower
index and the upper index.
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And you find here then
we get minus c squared t
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squared plus x squared,
y squared, and z squared.
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More generally, this is for
two vectors of the same-- two
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of the same vectors.
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More generally for
two different vectors,
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you can write in this way.
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Or in short, you can
define a new notation
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in which you basically sum over
all indices which are equal.
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So here we have an upper
and lower indices together.
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So you sum over
this case here where
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there's the same index,
mew, for both vectors.
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And one is lower
and one is upper.
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And we can continue
the introduction
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and just introduce a few tools
to work with those vectors.
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For example, if you wanted
to bring the component
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mew from the bottom
to the top, you
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can do this with multiplying
the vector with a matrix.
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And the matrix here is
also called a metric.
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And simply what
you have to do is
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multiply the first
component with the minus 1
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and the rest with 1.
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You see this here
on the diagonal
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and on other
components later on.
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What this does-- you can
check this if you want--
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is bringing the
index of the vector
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from a lower to an upper one.
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An interesting
example is the product
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of a four-vector with itself.
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And we have already seen
this because we saw this
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as our invariant interval.
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Here, the four-vector is the
distance in space and time
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between two events.
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So we looked at delta Xi
mew times delta Xi mew.
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And delta Xi mew is the
difference between event A
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and B. And so we have
seen this already
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and calculated the
invariant and showed
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that this squared over
a distance of two events
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is actually invariant in
the Lorentz transformation.
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But there's other
examples for vectors.
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The first one we'll investigate
some more in the next sections
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to come.
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It's the energy
momentum four-vector,
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where we place in the first
component the energy--
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in the 0's component the energy,
and then the first, second,
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and third components the
three-vector of the momentum.
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But there's others, for
example, the four-potential,
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where in the 0's
component, you have
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the potential-- the
electric potential.
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And then the first, second,
and third component,
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you have this new
field A, which is
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related to the magnetic
and electric field.
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So E and M is not
part of this course,
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but we'll come back to
this in the last week
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and discuss the consequences
and ideas a little bit more.
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But if you then look at the
invariant four-vector, which
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is a product of the
energy momentum vector,
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you find that the first
component, the energy square
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or minus the energy
square over c
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square plus the three-component
vector of the momentum squared.
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And that's constant, we can just
here name this mass or minus
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mass square times c square.
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So if you write this, you
find this energy momentum mass
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relation E squared is equal
to p squared c squared, plus m
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squared c to the fourth power.
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And if you look at
this four particles
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of 0 momentum, in which case
this component here is 0,
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you find the equation E
is equal to mc square.