Einstein Velocity Addition

The relative velocity of any two objects never exceeds the velocity of light. Applying the Lorentz transformation to the velocities, expressions are obtained for the relative velocities as seen by the different observers. They are called the Einstein velocity addition relationships.

Atomic Fine Structure

One of the great successes of the quantum theory was the prediction of the energy levels of the hydrogen atom. When attempts were made to explain the fine structure of the hydrogen spectral lines, it was found that the splitting of the lines was in error by a factor of two. It was realized that relativistic time dilation must be used in calculating the frequencies, and calculations showed that this relativistic correction, called Thomas precession, was the factor of two which was needed for agreement with experiment.

Cerenkov Radiation

When highly radioactive objects are observed under water, such as in "swimming pool" reactors and in the underwater temporary spent fuel storage areas at nuclear reactors, they are seen to be bathed in an intense blue light called Cerenkov radiation. It is caused by particles entering the water at speeds greater than the speed of light in the water. As the particles slow down to the local speed of light, they produce a cone of light roughly analogous to the bow wave of a boat which is moving through water at a speed greater than the wave speed on the surface of the water. Another analogy statement is to say that the Cerenkov cone is like a sonic boom except that it is done with light.

One of the valuable applications of Cerenkov radiation is in the detection of neutrinos and distintinguishing between different types of neutrinos. An energetic muon remains intact while slowing down and its Cerenkov cone paints out a well-defined circular ring on the detector array. A high energy electron on the other hand will produce a diffuse ring on the detectors because it will produce a shower of electrons, each with its own Cerenkov cone. 契克夫辐射的一个有价值的应用是用于探测中微子以及区分不同类型的中微子。一个高能缪子在减速过程中保持完整，其契克夫锥在探测器阵列上形成一个清晰的圆形环。而高能电子则会在探测器上产生一个模糊的环，因为它会产生一系列具有各自契克夫锥的电子。

Cerenkov Applications

Cerenkov radiation can be used to detect the occurrence of certain nuclear interactions. Such interactions can release large amounts of energy and eject particles at highly relativistic speeds. If these interactions take place in water or another clear substance, then the Cerenkov radiation emitted as the reaction products travel through the water can be detected by photomultiplier tubes. This kind of detection is to be used in the Sudbury Neutrino Observatory to detect neutrino interactions.

The Super-Kamiokande neutrino detector facility in Japan has 11,000 photomuliplier tubes in place to detect Cerenkov radiation and is able to detect and distinguish electron and muon neutrinos.

Measurements of particle speeds can be made by measuring the angle of the Cerenkov cone, like photographing ship wakes to measure ship speeds. A portion of the light emitted by the decelerated particle is coherent and is emitted at a characteristic angle

The total amount of energy appearing in Cerenkov radiation is small compared to the total energy loss by ionization as the particle enters the medium. According to Rohlf, a relativistic particle near the speed of light will lose energy at the rate of about 200 MeV/m, and of that loss only about 40 keV/m will be in Cerenkov radiation, about 1/5000 of the total.

Although the example given here is for electrons in water, applicable to the use of Cerenkov radiation for neutrino detection, Cerenkov radiation occurs for any charged particle which enters a material medium at a speed greater than the speed of light in that medium. The Cerenkov radiation is independent of the mass of the particle, depending only upon its charge and speed. Cerenkov radiation is emitted at all frequencies in the visible if it occurs in an optically transparent medium, but the energy per unit wavelength is proportional to the inverse cube of the wavelength. Short wavelengths are then preferred, and the visible color is described by Evans as "bluish white".

Early Cerenkov detectors used glass, lucite and mica as detector media. They all had indices of refraction around n=1.5, so the limiting Cerenkov angle was about 48°. Although the Cerenkov radiation can be produced at all forward angles less than the limiting angle as the particle slows down, in practice the emission is seen as a narrow cone with only a few degrees width. The particles slow down very quickly, so the radiation comes from a very short path length in the medium and a very short time interval (<10^-10 sec, Richtmyer, et al). The emission comes from discrete atoms at steps along the path, and is emitted coherently. As early as 1951, Mather reported the use of Cerenkov radiation for determining the energy of 340 MeV protons with an uncertainty of only +/- 0.8 MeV.

Analytical treatments of Cerenkov radiation were carried out by Frank & Tam and by Fermi.

Cyclotron Frequency

One of the early tests of the relativistic mass expression was the behavior of cyclotrons when the speed of the accelerated particles approached the speed of light. The non-relativistic expression for the cyclotron frequency indicated that the particles could continue to be accelerated by the constant angular frequency

but in fact that did not prove to be the case. Large charge oscillations occurred at high energies and the particles did not continue to accelerate. When the relativistic mass was included, giving the frequency

it was clear that the frequency must be changed as the particles accelerated. The success of the accelerators which used the above frequency was a test for the expression for relativistic mass.