63 Relativistic Dynamics
Transforming Energy and Momentum to a New Frame
An Invariant: the Rest Mass
We have shown (here) that
Notice we can write this last equation in the form
That is to say, depends only on the rest mass of the particle and the speed of light. It does not depend on the velocity of the particle, so it must be the samefor a particular particlein all inertial frames.
This is reminiscent of the invariance of the interval between two events, under the Lorentz transformations. One might guess from this that the laws governing the transformation from in one Lorentz frame to in another are similar to those for We can actually derive the laws for to check this out.
As usual, we consider all velocities to be parallel to the -axis.
We take the frame to be moving in the -direction at speed relative to
Consider a particle of mass (rest mass) moving at in the direction in frame and hence at along in where
The energy and momentum in are
It is straightforward (and you should do it!) to show that
Similarly, we can show that
These are the Lorentz transformations for particle energy and momentum.
It follows that
Energy, Momentum and Four-Velocity
Reminder from the previous lecture: the four velocity on a world line is in the nonrelativistic limit The magnitude
In the rest frame, where the incremental movement along the world line is purely in the time direction, and is just the four-velocity is
In general, it's (Recall )
Hence the energy (including rest energy) and momentum together can be written
so must transform just as does.
Photon Energies in Different Frames
For a zero rest mass particle, such as a photon, in all frames.
Since we also have
Notice that the ratios of photon energies in the two frames coincides with the ratio of photon frequencies found in the Doppler shiftand this result is independent of quantum mechanics, we didn’t need to say photon, we could just have imagined a classical electromagnetic long wave packet, approximating a plane wave. And, thinking of a classical plane wave, as Jackson points out (page 529) the phase at some point must be the same as at the transformed point in a Lorentz transformation, since it’s just a number, counting from some origin. That is,
is invariant. This can only be true if transforms in the same way as , notice the down indices. Writing this means (check the sign of from the low energy limit, or visualizing the wave)
This gives the familiar Doppler shift for light, and in particular, in contrast to sound, there is even a Doppler shift (from ) when
Particle Creation: Energy into Mass
Energy Necessary to Produce a Pion
The first "new" particle created in accelerators was the pion:
The neutral pion mass is 135 MeV, the charged pions have mass 140 MeV, where we follow standard high energy practice in calling mc2 the “mass”.
An incoming proton with 135 MeV of kinetic energy will not be able to create a neutral pion (rest mass 135 MeV) in a collision with a stationary proton. (Must also conserve momentum.)
Go to the center of mass frame, where initially two protons are moving towards each other with equal and opposite velocities, there being no total momentum. Obviously, in this frame the least possible K.E. must be just enough to create the with all the final state particles at rest.
we find the two incoming protons must both be traveling at 0.36c.
In the lab frame we must add a velocity of 0.36c + 0.36c = 0.64c. incoming proton has a relativistic mass of 1.3 times its rest mass, and thus a K.E. around 280 MeV.
To create a pion of rest energy 135 MeV, it is necessary to give the incoming proton at least 290 MeV of kinetic energy. This is called the “threshold energy”.
Doesn’t happenso charge conservation isn’t the only constraint on which particles can be produced. Experimentally, baryon number is also conserved.
A Machine Built to Produce One Particle
One of the first modern accelerators, built at Berkeley in the fifties, was designed specifically to produce the antiproton.
We can use to get lab frame information from the center of mass frame.
In the center of mass (CM) frame the momentum is zero, and in the lab frame the momentum is all in the incoming proton, so
where here m0 is the proton rest mass, and
At the antiproton production threshold, Ecm = 4m0c2, so
Therefore to create two extra particles, with total rest energy it is necessary for the incoming proton to have a kinetic energy of The Berkeley Gevatron had design energy 6.2 GeV.
As we go to higher energies, this “inefficiency” gets worseconsider energies such that the kinetic energy >> rest energy, and assume the incoming particle and the target particle have the same rest mass, with the incoming particle having relativistic mass
Comparing the center of mass energy with the lab energy at these high energies,
ultimately one must quadruple the lab energy to double the center of mass energy. So at higher energies, things get steadily worsethis is why colliders were built!