A3.11 − RELATIVISTIC MASS-ENERGY EQUIVALENCE
The derived and experimentally verified mass-transformation equation has
important consequences, one of which is a new expression for the kinetic energy
to replace T=½mv2. Suppose we increase the kinetic energy of a body an amount
dT by exerting a force through a distance. The change of kinetic energy equals
the work done, or
dT = F ∙ ds.
This is the classical expression relating a change in kinetic energy to the work,
defined as F∙ds. We choose to retain this expression in our development of a
relativistic mechanics. Similarly, we choose to retain Newton's second law, which
can be considered the definition of force:
F=
dp d ( mv )
=
dt
dt
where we have as before written p=mv, but with m now being the relativistic
(increased) mass.
The change of kinetic energy then is
dT =
d ( mv )
⋅ ds
dt
(A3-38)
But since v=ds/dt, we can write the preceding equation as
dT = v ∙ d(mv) = v2 dm + mv ∙ dv
(A3-39)
This expression can be simplified by using the mass-transformation equation, Eq.
(A3-37), which after squaring and rearranging becomes
m2c2 = m2v2 + m02c2
(A3-40)
We now differentiate, noting that v2=v∙v and that both m0 and c are constant. We
obtain
Tratto da Enge, Wehr, Richards, INTRODUCTION TO ATOMIC PHYSICS
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2mc2 dm = 2mv2 dm + 2m2 v dv
(A3-41)
When 2m is divided out of this equation, the right side becomes identical with our
expression for dT in Eq. (A3-39), so we have, finally,
dT = c2 dm
(A3-42)
This famous equation shows that in relativity a change in kinetic energy can be
expressed in terms of mass as the variable. This equation is valid for a change of
energy in any form whatever, although it was derived here for a change of kinetic
energy.
Since the kinetic energy is zero when v=0, then it is also zero when m=m0.
Therefore we integrate, and obtain
T
T = ∫ dT = c
0
2
∫
m
m0
dm = c 2 ( m − m0 )
or
T = mc 2 − m0 c 2
(A3-43)
This is the relativistic expression for kinetic energy. The classical relation T=½mv2
does not give the correct value of the kinetic energy even when the relativistic
values of the mass and the velocity are used.
Like the other transformation equations, this expression for T should reduce to
the classical expression for v≪c. If we transform m in Eq. (A3-43), we obtain
T=
m0 c 2
2
1− v c
2
− m0 c 2
(A3-44)
Now, since v≪c, we can say
⎛ v2 ⎞
= ⎜ 1− 2 ⎟
1− v 2 c 2 ⎝ c ⎠
1
− 12
⎛
⎞
v 2 3v 4
= ⎜ 1+ 2 + 4 + ...⎟
⎝ 2c 8c
⎠
Tratto da Enge, Wehr, Richards, INTRODUCTION TO ATOMIC PHYSICS
(A3-45)
2
where we have carried three terms of the binomial expansion. We then have,
from Eq. (A3-44),
⎛
⎞ 1
v 2 3v 4
3 v4
2
T = m0 c ⎜ 1+ 2 + 4 + ...− 1⎟ = m0 v + m0 2 + ...
8
c
⎝ 2c 8c
⎠ 2
2
(A3-46)
The first term of this is the classical expression for the kinetic energy. Obviously,
it is the only significant term for low velocities.
In our derivation of a relativistic mechanics, we have used:
1. The relativistic transformation equation for distance, time, and velocity;
2. The law of conservation of momentum which, together with point 1, led to the
concept of a mass increasing with the velocity of a body relative to the
observer;
3. The definition of force by Newton's second law;
4. The classical definition of work and the relationship between work and change
in kinetic energy.
By retaining the classical conservation laws and concepts, we have constructed a
relativistic mechanics that is as simple as possible and also as close to classical
mechanics as possible. The important changes that we need to remember when
solving problems in relativistic mechanics (i.e., when v/c is not small) are the
variable mass and the new formula for kinetic energy.
Equation (A3-42) or (A3-43) shows that there is a very intimate relationship
between mass and energy. Einstein suggested that there also is energy
associated with the rest mass of a body. According to this view, Eq. (A3-43) can
be written as
E = mc2 = m0c2 + T
(A3-47)
where E or mc2 is the total energy, m0c2 is the rest energy and T is the kinetic
energy of a body. All relevant experimental data show that Einstein was right.
Potential energy could also be included in Eq. (A3-47), but not without including
other interacting bodies. Potential energy is a result of conservative forces acting
between two or more bodies. For such a system of bodies, the total energy mc2 is
the sum of all rest energies, kinetic energies, and potential energies. An
interesting consequence of this statement is that the rest mass of the hydrogen
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atom is less than the sum of the rest mass of the two constituents, the proton
and the electron. The difference is the mass equivalent of 13.6 eV or, stated as a
fraction: Δm/m = 1.45 x 10-8.
We can now obtain an important relation between the total mass-energy E of a
body and its momentum p from Eq. (A3-40). If we multiply through this equation
by c2 and make the appropriate substitutions from E=mc2 and p=mv, we get
E2 = (pc)2 + m0c4
(A3-48)
This equation gives the relationship between the total energy, the rest energy,
and the momentum. By substituting E from Eq. (A3-47) and solving for the
momentum, we obtain
p = 2m0T + T 2 c 2
(A3-49)
Except for the second term under the square root, this equation is identical with
the classical formula for momentum. The last term is therefore called the
relativistic-correction term.
As an aid for the memory, Eq. (A3-48) is represented graphically in Fig. A3-5. The
figure has no real meaning, but is merely given as a mnemonic device.
Fig. A3-5 Mnemonic triangle for the
relationship (mc2)2 = (m0c2)2 + (pc)2
The equivalence of mass and energy brings a satisfying consequence. The broad
principle of conservation of energy now takes unto itself another broad
conservation principle, the conservation of mass. The identity of mass and energy
resulting in a unified concept of mass-energy is certainly the most "practical"
consequence of relativity. In Chapter 12, it is shown how nuclear reactions and
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nuclear disintegration processes demonstrate the fact that neither mass nor
energy as conceived classically is conserved separately. In relativity we see that it
is mass-energy that is conserved. In any interaction the total mass, m (which
includes any kinetic energy in mass units), is the sane before and after the
interaction. Similarly, in any interaction, the total energy E (which includes any
rest mass in energy units), is the same before and after the interaction. We are
now witnessing the growth of the whole new field of nuclear engineering which is
based on E equals mc2.
Most of the mass-energy we use comes from the sun, where rest mass-energy is
converted into thermal energy. Today mankind is converting rest mass-energy
into thermal energy here on the earth. Nuclear fission and fusion are the
techniques for this conversion.
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