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.