Tests of relativistic energy and momentum

According to special relativity, the properties of particles moving approximately at the speed of light significantly deviate from the predictions of Newtonian mechanics.

In classical mechanics, kinetic energy and momentum are expressed as On the other hand, special relativity predicts that the speed of light is constant in all inertial frames of references.

First experiments capable of detecting such relations were conducted by Walter Kaufmann, Alfred Bucherer and others between 1901 and 1915.

These experiments were aimed at measuring the deflection of beta rays within a magnetic field so as to determine the mass-to-charge ratio of electrons.

However, it was later pointed out that although the experiments were in agreement with relativity, the precision was not sufficient to rule out competing models of the electron, such as the one of Max Abraham.

[4][5] Already in 1915, however, Arnold Sommerfeld was able to derive the Fine structure of hydrogen-like spectra by using the relativistic expressions for momentum and energy (in the context of the Bohr–Sommerfeld theory).

[6] In 1940, Rogers et al. performed the first electron deflection test sufficiently precise to definitely rule out competing models.

Determination of the angular frequencies and of the magnetic field provided the charge-to-mass ratio.

This, together with measuring the magnetic center, allowed to confirm the relativistic expression for the charge-to-mass ratio with a precision of ~0.0006.

[9] However, Zrelov et al. (1958) criticized the scant information given by Grove and Fox, emphasizing the difficulty of such measurements due to the complex motion of the protons.

The proton's momentum was measured using a Litz wire, and the velocity was determined by evaluation of Cherenkov radiation.

[10] Since the 1930s, relativity was needed in the construction of particle accelerators, and the precision measurements mentioned above clearly confirmed the theory as well.

But those tests demonstrate the relativistic expressions in an indirect way, since many other effects have to be considered in order to evaluate the deflection curve, velocity, and momentum.

So an experiment specifically aimed at demonstrating the relativistic effects in a very direct way was conducted by William Bertozzi (1962, 1964).

These electrons were produced by a Van de Graaff generator and traveled a distance of 8.4 m, until they hit an aluminium disc.

First, the time of flight of the electrons was measured in all five runs – the velocity data obtained were in close agreement with the relativistic expectation.

Therefore, the heat produced by some electrons hitting the aluminium disc was measured by calorimetry in order to directly obtain their kinetic energy - those results agreed with the expected energy within 10% error margin.

Mass, velocity, momentum, and energy of electrons have been measured in different ways in those experiments, all of them confirming relativity.

[13] They include experiments involving beta particles, Compton scattering in which electrons exhibit highly relativistic properties and positron annihilation.

In modern particle accelerators at high energies, the predictions of special relativity are routinely confirmed, and are necessary for the design and theoretical evaluation of collision experiments, especially in the ultrarelativistic limit.

[3] For instance, time dilation must be taken into account to understand the dynamics of particle decay, and the relativistic velocity addition theorem explains the distribution of synchrotron radiation.

[26] Another SLAC experiment conducted by Guiragossián et al. (1974) accelerated electrons up to energies of 15 to 20.5 GeV.

[27] Already before, Alväger et al. (1964) at the CERN Proton Synchrotron executed a time of flight measurement to test the Newtonian momentum relations for light, being valid in the so-called emission theory.

This energy can be measured by calorimeters in an electrical, optical, thermal, or acoustical way.

[29] Thermal measurements in order to estimate the relativistic kinetic energy were already carried out by Bertozzi as mentioned above.

[31] In modern calorimeters called electromagnetic or hadronic depending on the interaction, the energy of the particle showers is often measured by the ionization caused by them.

[29] Relativistic energy and momentum can also be measured by studying processes such as annihilation and pair production.

can be tested in nuclear reactions, as the percent differences between the masses of the reactants and the products are big enough to measure; the change in total mass should account for the change in total kinetic energy.

Einstein proposed such a test in the paper where he first stated the equivalence of mass and energy, mentioning the radioactive decay of radium as a possibility.

[35] The first test in a nuclear reaction, however, used the absorption of an incident proton by lithium-7, which then breaks into two alpha particles.