In Newtonian physics, the description
of a system interacting with its environment centers around the entity
called force. Newton’s program consists of trying to discover general force
laws in nature, so that one might know in advance the net force on a system.
From F = ma one can then find the acceleration; for given
initial conditions this might allow prediction of the subsequent motion.
In carrying out this program, momentum
and energy play important but auxiliary roles as quantities obeying conservation
laws.
According to Newton’s 3rd law any isolated
system has constant total momentum.
In many situations where the forces are
known as functions of the positions of the particles, one introduces the
kinetic and potential energies, the sum of which is also conserved.
Finally, it is assumed that "matter"
can neither be created nor destroyed, by which is meant that the total
mass of an isolated system is conserved.
There are thus three useful conserved
mechanical quantities: mass, momentum, and energy.
When we try to combine this approach
to mechanics with the properties of space and time described in Einstein’s
relativity, we find that important changes must be made, and that remarkable
new insights emerge.
Defining acceleration as usual by a
= dv/dt, one can work out the new transformation formulas for acceleration,
as we did for velocity. The result is complicated, and the simplicity of
the Galilean case (a’ = a) is lost.
If we use F = ma (or F
= dp/dt, with p = mv) then the transformation rules
for force are correspondingly complicated. Partly for this reason, force
does not play the central role in relativistic dynamics that it does in
Newtonian physics.
Fortunately, in the case of conservative
forces one can deal with the potential energy instead of the force; as
we will see, energy has relatively simple transformation laws.
In Galilean relativity, one shows easily
that if both the total momentum and the total mass of a system are conserved
in one inertial frame then both are also conserved in all other inertial
frames. One merely uses the velocity formulas.
But in Einstein’s relativity those formulas
are replaced by the more complicated ones, so the question must be reexamined.
We will now show by considering a simple
example that conservation of momentum (in its Newtonian form p =
mv) does not hold in different frames related by Lorentz transformations.
We consider an elastic collision of two identical particles,
A and B, observed in the usual two frames O and O'.
In frame O, A moves initially in the positive y direction
with speed v0, collides with B, and has its velocity simply
reversed by the collision.
In frame O', particle B moves initially along the negative
y axis with speed v0, and has its velocity reversed by the collision.
The situations are as shown:
Before the collision, the velocity of B in frame O' is
so, using the relativistic velocity
formulas, its velocity in frame O must be
After the collision, the velocity of
B in O' is
and in O
Now consider the total y-component of
the momentum as seen in frame O. Using p = mv for each particle,
we find this component before the collision to be
After the collision this
component is
These are equal only if v0=
0, that is, if there is no collision at all.
Conclusion: If we use p = mv
for the momentum of a particle then the Lorentz transformations are not
compatible with conservation of total momentum.
The new definition of the relativistic
momentum of a particle is thus:
where g
is evaluated using the particle's speed, v.
The velocity, of course, is v = dr/dt. It is
interesting that if we take the derivative with respect to proper time
(the time measured on a clock moving along with the particle) we find dr/dt
= gdr/dt =
gv.
Thus we can also define the particle's momentum by
Since proper time is an invariant (the
same in all inertial frames) this shows that under changes of reference
frame (Lorentz transformations) p changes the same way as r.
We take the Lorentz transformations for
the components of r and multiply by the mass, and then take derivatives
with respect to proper time t:
Here g
is calculated using the relative speed V of the two reference frames.
These are similar to what one would have
in Galilean relativity. One obvious difference is the appearance of g
in the first equation. The other difference is the (dt/dt)
factor, which in Galilean relativity is 1, since time is the same in all
frames.
We denote the new term by
Using the Lorentz transformation for time, we then find the
rule for this new quantity:
We have arrived at a curious situation. In order to transform
the momentum of a particle from one frame to another, we must know not
only the values of the momentum components, but also the value of this
new quantity p0.
Since dt = gdt
(where this g is
calculated using the particle's speed v) we can see that p0
= mg(v). Apart from
the relativistic factor g(v),
this is just the mass. In Galilean relativity (where g
= 1) this appearance of the mass of the particle is of no particular consequence;
it is conserved and is the same in all reference frames.
But in Einstein's relativity the quantity
is not the same in all frames. We must have some insight into what it means
physically.
where we have used the binomial expansion. The first
term in the series is the mass. The second term is the classical kinetic
energy, divided by c2.
At this point we multiply by c2
and define the relativistic energy
of the particle:
At low speeds we see that
This is a constant plus the kinetic energy. Since in
classical physics mass is always conserved, the constant has no particular
significance. We will see that in Einstein's relativity this constant term
has great importance.
We can now write out the four transformation rules for momentum-energy:
These were arrived at by considering the momentum and energy
of a single particle of mass m, but they apply equally to the total momentum
and energy of a system of particles.
They also apply to the difference between the total momentum
and energy initially and finally for a system. In the usual notation, we
have
Now suppose in frame O that the system is isolated so that
total momentum is conserved, i.e., that Dp
= 0. Then we see that momentum will be conserved in O' (Dp'
= 0) only if DE =
0. That is, momentum conservation applies to both frames only if
energy conservation also holds in O.
Conversely, if DE
= 0 then DE' = 0
only if Dp
= 0. Energy conservation applies in both frames only if momentum
conservation also holds.
This is a remarkable new situation. We
see that momentum and (relativistic) energy are tied together, so that
for conservation of either to hold in all inertial frames conservation
of the other must also hold. We speak of the conservation
of momentum-energy in an isolated system.
In Newtonian physics an isolated system
indeed has conservation of momentum, regardless of the internal forces.
But energy (kinetic plus potential) is conserved only if all forces, including
internal ones, are conservative.
In Einstein's relativity, however, an
isolated system always conserves both momentum and energy (defined to be
the sum of the relativistic energies of the particles) regardless
of the nature of the internal forces.
The relativistic energy of a particle contains a constant
term proportional to the mass, and terms which go to zero if the particle
is at rest. The latter thus represent "energy of motion" and are defined
to be the particle's kinetic energy:
The constant term mc2 is called the rest
energy.
In an isolated system of particles the total energy is conserved.
There are two ways this can occur:
The total kinetic energy can be conserved, and the total
rest energy also conserved. The latter means that the total mass remains
fixed.
Neither the total kinetic energy nor the total rest energy
remain constant, but the sum is conserved. Some kinetic energy is converted
to rest energy, or vice versa.
An example of Case 1 is an elastic collision of two particles.
Case 2 shows the new possibility of interchanging mass with
kinetic energy, where the energy equivalent of mass m is mc2.
Consider a situation where two identical particles move toward
each other along a straight line, with equal speeds. They collide and stick
together.
Conservation of momentum gives
from which we conclude that V = 0, so the final object
is at rest.
Conservation of total relativistic energy gives
since V = 0. We thus find
Since g >
1, this shows that the mass of the final object is larger than the sum
of the original mases. The lost kinetic energy has been converted to
rest energy (mass).
The classical explanation for the loss
of kinetic energy attributes it to conversion into thermal energy (heat):
the final object will have a higher temperature, or more specifically a
larger internal energy.
This suggests that the mass of a compound
object is a measure of its total energy content, including thermal energy
and the binding energies that hold its atoms or molecules in place.
Consider an unstable particle of mass
M which decays while at rest into two identical particles of mass m. Conservation
of momentum-energy gives
The first of these shows that the particles
move in opposite directions with equal magnitude of momentum. Since their
masses are equal, this means they have the same speed.
Then they will also have the same energy.
Calling it E, we have E = Mc2/2. Then the kinetic energy of
each particle is easily obtained:
If M > 2m, this is positive, and thedifference
between initial and final mass has been converted to kinetic energy.
Since momentum-energy obeys essentially
the same transformation rules as space-time, it is easy to show that there
is an analog to the invariant interval. It called the energy
invariant and is usually defined by
This quantity is the same in all reference
frames.
Here E and p can be the total
energy and momentum of any system.
For a single particle of mass m we can
use the formulas for E and p to obtain
This important formula allows us to treat
E and p as the useful variables in describing the behavior of a
particle, simplifying algebraic calculations.
allow us to determine the emergy
and momentum of a particle if we know its velocity.
Since g
becomes infinite as v approaches c, it requires infinite energy to accelerate
a particle to the speed of light. This is another reason why that is impossible.
But light itself travels at speed c,
and it carries energy and momentum. In quantum theory it is given many
particle attributes as well. How is this possible, given the above formulas?
The answer is that the mass of a light
particle (photon) is zero. The above formulas then become indeterminate.
But the energy invariant formula says (for m = 0) that
This is all one can say about the energy
and momentum of a photon. They do not depend on the speed, which is always
c.
In quantum theory we learn that the energy
of a photon is proportional to its frequency.
There may be other massless particles
(the neutrino, perhaps) which would also travel at the speed of light.
