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MARKUS KLUTE: Welcome back
to 8.20, Special Relativity.

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In this section, we're
going to further investigate

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the energy momentum for
vector, which we introduced

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in the previous sections.

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But here, we focus on the zeroth
component, the first component

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of this vector, where
we find mass in A

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for particle A times
the proper velocity

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of the zeroth component is equal
to mA times c times 1 over 1

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minus uA square
over c square, which

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is the energy of this
particle A over c.

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Or in other words, the energy
is equal to the mass times c

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square over 1 square root of
1 minus uA square c square.

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So let's discuss or look
at-- let's have a look

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at this a little bit more.

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The first question we can ask--

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how does this now look like
for particles which travel

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with reasonably low velocity?

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So uA-- much smaller than c.

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So we can Taylor expand
this following this equation

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here, which we
discussed earlier.

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And we find that the energy
is equal to mA c square.

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That's the first
term, which we call

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rest mass, the energy given--

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just the rest mass by the
mass of the particle times c

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square, plus 1/2 mA c square
times uA square over c square.

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The c squares cancel.

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And we find what we know as the
kinetic energy, 1/2 m v square,

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or in this case,
1/2 mA uA square.

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That looks very familiar.

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So the energy of a particle
is given by its rest

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mass plus its kinetic energy.

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All right, now investigating
this for vector,

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then we can ask, how does the
invariant interval look like?

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How does this property,
which is invariant,

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and the Lorentz transformation
look like when we multiply

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the vector with itself?

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Here, we find minus E square
over c square plus the 3

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momentum squared is equal
to minus m0 c square.

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Or in other words, we find this
energy momentum, energy mass

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relation, energy
momentum mass relation,

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where the energy is given by
the momentum square times c

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square plus the rest mass
square times c to the 4th power.

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Again, we can unroll this
now and ask, how does this

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look for a particle at rest?

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And again, we find the
energy is equal to mc square.

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No surprise.

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That's how we started
the definition of this.

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In general, we can
find that the energy

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is equal to a
relativistic mass times c

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square, which is equal to the
rest mass times gamma times c

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square.

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And that's equal to the rest
mass times c square plus k,

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the kinetic energy, square.

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All right, does
this definition--

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I can tell you that this
confused me as a student

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quite a bit.

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This understanding that
the mass becomes heavier

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for a part of this-- really,
one I didn't quite like.

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I just like to think about the
fact that the kinetic energy--

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there's a relativistic component
to the kinetic energy, which

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is owned by the
particle in addition

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to the rest mass of this
particle times c square.

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Also interesting to note is
that for particles at rest--

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particles which are
massless, like a photon,

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the energy is equal to
the momentum times c.

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If you want to know what
the energy is of a photon,

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you need to know
what the momentum is

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of the photon multiplied by c.