Special relativity
In a television cathode-ray tube the electrons travel at 30 % of the speed of light, and GPS would drift out of sync by eleven kilometers per day if it did not correct for relativistic effects. Relativity is not an exotic topic: it is inside the device.
01Two postulates and everything else
Einstein did not add complications: he removed an assumption. Faced with the null result of Michelson-Morley, instead of inventing mechanisms to save the ether, he declared it unnecessary and postulated two things.
| Postulate | It says | What it forces us to abandon |
|---|---|---|
| Relativity | The laws of physics are the same in every inertial frame | The idea of a privileged frame |
| Constancy of \( c \) | Light travels at \( c \) for all observers | Absolute time and the simple addition of velocities |
The second one is the hard one. If an observer chases a pulse of light at 0.9 c, he should see it moving away at 0.1 c; what he measures is \( c \). For that to be possible, what has to give way is the measurement of time and space. Everything else follows from there, with no new physics.
You move the speed and all the effects appear at once. The cases in the selector are real situations, and some are much closer to home than you would expect.
02Where it shows up in a device
Relativity is usually presented as an astrophysics topic, and it is inside several everyday devices.
| Case | The effect | Consequence |
|---|---|---|
| Cathode-ray tube | Electrons at 30 % of \( c \) | The deflection has to be calculated with \( \gamma \) |
| GPS | The satellite clocks run at a different rate | 11 km of error per day if not corrected |
| Atmospheric muons | They reach the ground when they should not | The most accessible experimental proof |
| Magnetism | It is electrostatics seen from another frame | They are not two phenomena: it is one |
This is the nicest result in the whole unit. Take a current-carrying wire, neutral: the positive charges of the lattice are at rest and the electrons drift. An external charge also at rest feels nothing. But if that charge moves parallel to the wire, from its frame the two charge distributions contract differently —because they have different velocities relative to it— and the wire stops being neutral. A net electric force appears. That is exactly what in the laboratory we call magnetic force. Magnetism is not a separate phenomenon: it is what electrostatics looks like when there is relative motion, and \( c \) appears in Maxwell's equations precisely for that reason.
03Mass and energy
The classical kinetic energy \( \tfrac12 mv^2 \) is the first term in the expansion of the relativistic one, and that is why it works at low speed. As \( c \) is approached the difference becomes enormous: the classical expression keeps growing like a polynomial and the relativistic one shoots up, which is another way of saying that you cannot reach \( c \) with finite energy.
The last relation, \( E^2 = (pc)^2 + (m_0c^2)^2 \), includes the photon: with \( m_0 = 0 \) it becomes \( E = pc \), and from there comes the fact that light has momentum without having mass. That is the radiation pressure of the waves unit, and also the link that the next unit will use for the photoelectric effect.
Many books speak of a “mass that increases with speed”, \( m = \gamma m_0 \). It is a way of writing things that works for rectilinear motion and confuses in everything else, and today it is preferred to say that mass is invariant and that what changes with speed are energy and momentum. If a text uses relativistic mass it is not wrong, but you should know that it is an old convention.
04In the lab
Calculate the speed of the electrons in a tube accelerated through 5, 15 and 25 kV, first with the classical expression and then with the relativistic one. Tabulate the percentage error of the classical one in each case.
Calculate the daily delay due to orbital speed and the daily advance due to gravity for a GPS satellite. Obtain the net value and translate it into position error, knowing that the signal travels at \( c \).
With a mean proper lifetime of 2.2 µs and a speed of 0.998 c, calculate how far a muon would travel without time dilation and how far it travels with it. Compare with the 10 km it actually crosses.
05Common mistakes
- Adding velocities as if nothing were different. It holds at low speed; near \( c \) you have to use the relativistic composition.
- Believing the effect is “apparent”. The measured time really is different, and muons prove it.
- Forgetting which one is the proper time. \( \Delta t_0 \) is the one measured by the clock that travels with the event.
- Using \( \tfrac12 mv^2 \) with accelerated electrons. Above a few kV the error is already noticeable.
- Thinking relativity only matters in astronomy. The GPS in your phone corrects for it every day.
- Confusing mass with rest energy. \( m_0c^2 \) is energy; mass is invariant.
06Self-assessment
What are the two postulates?
That the laws of physics are the same in every inertial frame, and that the speed of light is the same for all observers.
What is \( \gamma \) at 50 % of \( c \)?
1.155. It only reaches 2 near 87 % of \( c \).
What error would GPS accumulate without relativistic correction?
About 38 microseconds per day, which at the speed of light is more than eleven kilometers.
What is the rest energy of the electron?
511 keV. That is why accelerating it through 25 kV already puts it 4.9 % above that energy.
Which relation holds for the photon?
\( E = pc \), which follows from \( E^2 = (pc)^2 + (m_0c^2)^2 \) with zero mass.
What is the relationship between magnetism and relativity?
The magnetic force is the electrostatic force seen from a moving frame: the charge densities contract differently and the wire no longer looks neutral.
07Further reading
- Robert Eisberg and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd ed., John Wiley & Sons, 1985 (Spanish edition, “Física cuántica”, Limusa). The relativity appendix, oriented toward what quantum physics needs afterward.
- Edward M. Purcell and David J. Morin. Electricity and Magnetism. 3rd ed., Cambridge University Press, 2016 (Spanish edition, “Electricidad y magnetismo”). Chapter 5 derives the magnetic field from electrostatics plus relativity. It is the best treatment of that idea there is.
- Neil Ashby. Relativity in the Global Positioning System. Living Reviews in Relativity, vol. 6, 2003. All the relativistic corrections that a GPS receiver applies, in detail and with the numbers.