S4.7 Relativity: Mass, Momentum and Energy
This section is included for general interest. It is not part of the examinable programme of the course.
[A] Next steps
Working from just these two postulates:
- the laws of physics are the same in all inertial reference frames
- the speed of light in empty space always has the same value, c
Einstein was able to show that we have to radically revise our notions of space and time. In particular he discovered:
- moving clocks run slow
- moving objects shrink
- our definition of simultaniety depends on velocity
So what happens to our ‘commonsense’ (Newtonian) notions of mass, momentum and energy? Do these need revision too?
[B] The ‘ultimate speed’ experiment
- An electron can easily be accelerated to extrememly high speeds using a linear particle accelerator (a linac).
- Consider the following experiment:
- An electrons is accelerated to a very high speed towards a target.
- The energy of the electron is known (an electron accelerated through an electric potential difference of V has kinetic energy K=eV, where e is the electric charge.)
- The electrons are timed over a fixed distance, d; this gives their speed v=d/t
- The energy of the electrons at the target is measured from the rate of heating of the target. This can be used as a check on the kinetic energy of the accelerated electrons.
- What do we find from the experiment?
- What conclusions do we draw?
- The normal Newtonian relationships between kinetic energy and
speed break down! (i.e.
)
- Maximum speed of an electron is bounded by the speed of light, c; the energy of the electron is unbounded.
- Our notion of work remains intact (the work we do in accelerating electrons is recovered in the heating of the target).
- The normal Newtonian relationships between kinetic energy and
speed break down! (i.e.
- Can these results be reconciled with our common sense view
of energy?
- If No - we have to abandon the energy principle or the relativity postulate.
- if Yes - we have to modify our definition of energy in such a way as to recover commonsense Newtonian mechanics at low velocities.
[C] Energy and momentum of photons: an aside
- Photons carry energy and momentum
The energy of a photon is
E=hνThe momentum of a photon is
E=cpThis result can be verified by measuring the radiation pressure of a beam of light.
Commentary
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This radiation pressure is of the order of pN (10-12N !!). So the Star Trek tractor beam that can hold entire ships immobile is a little unrealistic. However, the radiation pressure can be used to trap very small particlesand atoms and is exploited in an experimental micromanipulation technique called Optical Tweezers, which you can find out more about here
[D] A gendanken experiment: Einstein’s box
- Consider a closed, isolated, box of mass M, length L.
- Imagine that one end of the box emits a photon.
- This photon carries momentum E/c.
- Since the total momentum of the system is zero, the box must recoil, and acquire a momentum equal to -E/c.
Hence the box recoils with speed v where
After travelling for a short time Δt(∼L/c), the radiation collides with the other end of the box, bringing the box to rest. The result of this process is to move the box
- So either the centre of mass of the box has moved, or the radiation carried some mass with it!!!
We postulate that the radiation carried mass m such that
mL+MΔx=0Putting the last two equations together we see that we get
E=mc2This is the all too famous law relating energy and mass. Buy the T shirt.
[E] Energy, momentum and mass
Let’s try and synthesise all of these results
- Energy carries momentum (radiation pressure)
E=cp
- Mass and energy are interlinked (Einstein’s box)
E=mc2
- Particles have an ultimate speed c (ultimate speed experiment)
- Work and energy relationships still seem to be valid (ultimate speed experiment)
There are two ways of combining these results:
We could write
and view this as a special case of the Newtonian results m=p/v for photons which move at speed c.
Alternatively, guided by the ‘ulimate speed’ experiment, we could try and combine them into a single general result expressing the relationship between mass, energy and momentum. Using p=mv and E=mc2 let’s write
- Let’s now use the fact that work and energy are still related correctly: the increase in kinetic energy of the electrons in the ultimate speed experiment (dE) equals the work done by external forces (Fdx).
- We will use this to redefine kinetic energy.
The kinetic energy change for a small displacemnt
so that
dE=vdpMultiplying by E=c2p/v:
EdE=c2pdpand integrating we get
where
is a constant of integration.
[F] A new type of energy
- Substituting for cp=Ev/c, we get
- Lets look at the low velocity behaviour of this expression. If v<<c we can approximate this (arbitrarily well) as
For this to fit in with our Newtonian picture, E0/c2 must be the (Newtonian) inertial mass of the particle. We denote this by m0. Then we have
We have discovered a new contribution to the total energy of a particle - the rest energy E=m0c2.
[G] Kinetic energy
The kinetic energy is the difference between the total energy of the particle and the rest energy
- Doing work on a particle moving close to the speed of light increases its mass rather than its speed.
[H] Inertial mass
If E=mc2 is generally true, we must redefine inertial mass so that
- m0, the inertial mass in Newtonian mechanics, assumes the role of the rest mass in special relativity. Increasing velocity increases the inertial mass of the body.
[I] A summary
If we define a new quantity
![\[ \gamma = \frac{1}{(1-v^2/c^2)^{1/2}} \]](mastermathpng-13.png)
we can define forms for mass, energy and momentum that are relativistically correct, and reduce to the correct Newtonian forms for small v
[J] Is it correct?
In deriving the above, we
- appealed to an experiment using electrons
- used a thought experiment concerning photons
- assumed that we could extend the use of energy and momentum to relativistic velocties
Are such conclusions justified on the basis of such flimsy evidence? The answer is a resounding yes. We have faith in these results because
- the same results can be derived using only Einsteins two postulates (and a lot more work).
- they can be tested experimentally; they work.
Learning Resources
![]() | The presentation here is taken mainly from ‘Special Relativity’ by A.P. French (Van Nostrand Reinhold (UK) Ltd: Wokingham) |