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SPECIAL THEORY OF RELATIVITY |
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all rights reserved 2002
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Energy
Probably
the most famous scientific equation of all time, first derived by Einstein is
the relationship E = mc2. This
tells us the energy corresponding to a mass m at rest. What this means is that
when mass disappears, for example in a nuclear fission process, this amount of
energy must appear in some other form. It also tells us the total energy of a
particle of mass m sitting at rest. Einstein
also showed that the correct relativistic expression for the energy of a
particle of mass m with momentum p is E2 = m2c4
+ p2c2. This is a key equation for any real particle,
giving the relationship between its energy (E), momentum ( p),
and its rest mass (m). If
we substitute the equation for p into the equation for E above, with a
little algebra, we get E = gamma mc2, so energy is gamma
times rest energy. (Notice again that if we call the quantity M =gamma m
the mass of the particle then E = Mc2 applies for any
particle, but remember, particle physicists don't do that.) Let's
do a calculation. The rest energy of an electron is 0.511 MeV.
As we saw earlier, when an electron has gone about 10 feet along the SLAC linac,
it has a speed of 0.99c and a gamma of 7.09. Therefore, using the equation E
= gamma x the rest energy, we can see that the electron's energy after ten
feet of travel is 7.09 x 0.511 MeV = 3.62 MeV. At the end of the linac, where
gamma = 100,000, the energy of the electron is 100,000 x 0.511 MeV = 51.1 GeV. The
energy E is the total energy of a freely moving particle.
We can define it to be the rest energy plus kinetic energy (E = KE + mc2)
which then defines a relativistic form for kinetic energy. Just as the equation
for momentum has to be altered, so does the low-speed equation for kinetic
energy (KE = (1/2)mv2). Let's make a guess based on what we
saw for momentum and energy and say that relativistically KE = gamma(1/2)mv2.
A good guess, perhaps, but it's wrong. Now
here is an exercise for the interested reader. Calculate the quantity KE = E
- mc2 for the case of v very much smaller than c,
and show that it is the usual expression for kinetic energy (1/2 mv2)
plus corrections that are proportional to (v/c)2 and higher
powers of (v/c). The complicated result of this exercise points out why it is
not useful to separate the energy of a relativistic particle into a sum of two
terms, so when particle physicists say "the energy of a moving
particle" they mean the total energy, not the kinetic energy. Another
interesting fact about the expression that relates E and p
above (E2 = m2c4 + p2c2),
is that it is also true for the case where a particle has no mass (m=0).
In this case, the particle always travels at a speed c, the speed of
light. You can regard this equation as a definition of momentum for such a
mass-less particle. Photons have kinetic energy and momentum, but no mass! In
fact Einstein's relationship tells us more, it says Energy and mass are
interchangeable.
Or, better said, rest mass is just one form of energy. For a compound object,
the mass of the composite is not just the sum of the masses of the constituents
but the sum of their energies, including kinetic, potential, and mass energy.
The equation E=mc2 shows how to convert between energy units
and mass units. Even a small mass corresponds to a significant amount of energy.
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In
the case of an atomic explosion, mass energy is released as kinetic energy of
the resulting material, which has slightly less mass than the original material.
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In
any particle decay
Even
in chemical processes there are tiny changes in mass which correspond to the
energy released or absorbed in a process. When chemists talk about conservation
of mass, they mean that the sum of the masses of the atoms involved does not
change. However, the masses of molecules are slightly smaller than the sum of
the masses of the atoms they contain (which is why molecules do not just fall
apart into atoms). If we look at the actual molecular masses, we find tiny mass
changes do occur in any chemical reaction. At
SLAC, and in any particle physics facility, we also see the reverse effect --
energy producing new matter. In the presence of charged particles a photon
(which only has kinetic energy) can change into a massive particle and its
matching massive antiparticle. The extra charged particle has to be there to
absorb a little energy and more momentum, otherwise such a process could not
conserve both energy and momentum. This process is one more confirmation of
Einstein's special theory of relativity. It also is the process by which antimatter
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process, some of the initial mass energy becomes kinetic energy of the products.