Tests of relativistic energy and momentum
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Tests of relativistic energy and momentum are aimed at measuring the relativistic expressions for
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
, momentum, and
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
. According to
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
, the properties of
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
s moving approximately at the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
significantly deviate from the predictions of Newtonian mechanics. For instance, the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
cannot be reached by
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
ive particles. Today, those relativistic expressions for particles close to the speed of light are routinely confirmed in
undergraduate Undergraduate education is education conducted after secondary education and before postgraduate education. It typically includes all postsecondary programs up to the level of a bachelor's degree. For example, in the United States, an entry-le ...
laboratories, and necessary in the design and theoretical evaluation of collision experiments in
particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...
s. See also Tests of special relativity for a general overview.


Overview

In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
,
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
and momentum are expressed as :E_=\tfracmv^ ,\quad p=mv . \, On the other hand,
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
predicts that the speed of light is constant in all inertial frames of references. The relativistic
energy–momentum relation In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is t ...
reads: :E^-(pc)^=(mc^)^ \,, from which the relations for rest energy E_, relativistic energy (rest + kinetic) E, kinetic energy E_, and momentum p of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
ive particles follow: :E_=mc^,\quad E=\gamma mc^,\quad E_=(\gamma-1)mc^,\quad p=\gamma mv, where \gamma=1/\sqrt. So relativistic energy and momentum significantly increase with speed, thus the speed of light cannot be reached by massive particles. In some relativity textbooks, the so-called "
relativistic mass The word "mass" has two meanings in special relativity: '' invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity o ...
" M=\gamma m\, is used as well. However, this concept is considered disadvantageous by many authors, instead the expressions of relativistic energy and momentum should be used to express the velocity dependence in relativity, which provide the same experimental predictions.


Early experiments

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 ray A beta particle, also called beta ray or beta radiation (symbol β), is a high-energy, high-speed electron or positron emitted by the radioactive decay of an atomic nucleus during the process of beta decay. There are two forms of beta decay, β ...
s within a magnetic field so as to determine the mass-to-charge ratio of electrons. Since the charge was known to be velocity independent, any variation had to be attributed to alterations in the electron's momentum or mass (formerly known as transverse
electromagnetic mass Electromagnetic mass was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy, is contributing to the mass of charged particles. It was first derived by J. J. Thomson in 1881 and was for ...
m_=m\gamma, equivalent to the "relativistic mass" M as indicated above). Since relativistic mass is not often used anymore in modern textbooks, those tests can be described of measurements of relativistic momentum or energy, because the following relation applies: :\frac=\frac=\frac=\gamma Electrons traveling between 0.25–0.75c indicated an increase of momentum in agreement with the relativistic predictions, and were considered as clear confirmations of special relativity. However, it was later pointed out that although the experiments were in agreement with relativity, the precision wasn't sufficient to rule out competing models of the electron, such as the one of
Max Abraham Max Abraham (; 26 March 1875 – 16 November 1922) was a German physicist known for his work on electromagnetism and his opposition to the theory of relativity. Biography Abraham was born in Danzig, Imperial Germany (now Gdańsk in Poland) t ...
. 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). Subsequently, Karl Glitscher simply substituted the relativistic expression's for Abraham's, demonstrating that Abraham's theory is in conflict with experimental data and is therefore refuted, while relativity is in agreement with the data.


Precision measurements

In 1940, Rogers ''et al.'' performed the first electron deflection test sufficiently precise to definitely rule out competing models. As in the Bucherer-Neumann experiments, the velocity and the charge-mass-ratio of beta particles of velocities up to 0.75c was measured. However, they made many improvements, including the employment of a Geiger counter. The accuracy of the experiment by which relativity was confirmed was within 1%. An even more precise electron deflection test was conducted by Meyer ''et al.'' (1963). They tested electrons traveling at velocities from 0.987 to 0.99c, which were deflected in a static homogenous magnetic field by which ''p'' was measured, and a static cylindrical electric field by which p^/(m\gamma) was measured. They confirmed relativity with an upper limit for deviations of ∼0.00037. Also measurements of the charge-to-mass ratio and thus momentum of protons have been conducted. Grove and Fox (1953) measured 385-MeV protons moving at ∼0.7c. 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. 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. Therefore, they conducted a more extensive measurement, in which protons of 660 MeV with mean velocity of 0.8112c were employed. The proton's momentum was measured using a Litz wire, and the velocity was determined by evaluation of Cherenkov radiation. They confirmed relativity with an upper limit for deviations of ∼0.0041.


Bertozzi experiment

Since the 1930s, relativity was needed in the construction of
particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...
s, 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). He employed the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
accelerator facility at
MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
in order to initiate five electron runs, with electrons of kinetic energies between 0.5 and 15
MeV In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacu ...
. These electrons were produced by a
Van de Graaff generator A Van de Graaff generator is an electrostatic generator which uses a moving belt to accumulate electric charge on a hollow metal globe on the top of an insulated column, creating very high electric potentials. It produces very high voltage ...
and traveled a distance of 8.4 m, until they hit an
aluminium Aluminium (aluminum in American and Canadian English) is a chemical element with the symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately one third that of steel. I ...
disc. First, the
time of flight Time of flight (ToF) is the measurement of the time taken by an object, particle or wave (be it acoustic, electromagnetic, etc.) to travel a distance through a medium. This information can then be used to measure velocity or path length, or as a w ...
of the electrons was measured in all five runs – the velocity data obtained were in close agreement with the relativistic expectation. However, at this stage the kinetic energy was only indirectly determined by the accelerating fields. Therefore, the heat produced by some electrons hitting the aluminium disc was measured by
calorimetry In chemistry and thermodynamics, calorimetry () is the science or act of measuring changes in ''state variables'' of a body for the purpose of deriving the heat transfer associated with changes of its state due, for example, to chemical reac ...
in order to directly obtain their kinetic energy - those results agreed with the expected energy within 10% error margin.


Undergraduate experiments

Various experiments have been performed which, due to their simplicity, are still used as
undergraduate Undergraduate education is education conducted after secondary education and before postgraduate education. It typically includes all postsecondary programs up to the level of a bachelor's degree. For example, in the United States, an entry-le ...
experiments. Mass, velocity, momentum, and energy of electrons have been measured in different ways in those experiments, all of them confirming relativity. They include experiments involving beta particles,
Compton scattering Compton scattering, discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a charged particle, usually an electron. If it results in a decrease in energy (increase in wavelength) of the photon ...
in which electrons exhibit highly relativistic properties and positron annihilation.


Particle accelerators

In modern
particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...
s 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 In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light . The expression for the relativistic energy of a particle with rest mass and momentum is given by :E^2 = m^2 c^4 + p^2 c^2. The energy of ...
. For instance,
time dilation In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
must be taken into account to understand the dynamics of particle decay, and the relativistic velocity addition theorem explains the distribution of synchrotron radiation. Regarding the relativistic energy-momentum relations, a series of high precision velocity and energy-momentum experiments have been conducted, in which the energies employed were necessarily much higher than the experiments mentioned above.


Velocity

Time of flight Time of flight (ToF) is the measurement of the time taken by an object, particle or wave (be it acoustic, electromagnetic, etc.) to travel a distance through a medium. This information can then be used to measure velocity or path length, or as a w ...
measurements have been conducted to measure differences in the velocities of electrons and light at the
SLAC National Accelerator Laboratory SLAC National Accelerator Laboratory, originally named the Stanford Linear Accelerator Center, is a United States Department of Energy National Laboratory operated by Stanford University under the programmatic direction of the U.S. Departme ...
. For instance, Brown ''et al.'' (1973) found no difference in the time of flight of 11-GeV electrons and visible
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
, setting an upper limit of velocity differences of \Delta v/c=(-1.3\pm2.7)\times10^. Another SLAC experiment conducted by Guiragossián ''et al.'' (1974) accelerated electrons up to energies of 15 to 20.5 GeV. They used a radio frequency separator (RFS) to measure time-of-flight differences and thus velocity differences between those electrons and 15-GeV
gamma ray A gamma ray, also known as gamma radiation (symbol γ or \gamma), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nuclei. It consists of the shortest wavelength electromagnetic waves, typically ...
s on a path length of 1015 m. They found no difference, increasing the upper limit to \Delta v/c=2\times10^. 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 Emission theory, also called emitter theory or ballistic theory of light, was a competing theory for the special theory of relativity, explaining the results of the Michelson–Morley experiment of 1887. Emission theories obey the principle of rel ...
. In this experiment, gamma rays were produced in the decay of 6-GeV pions traveling at 0.99975c. If Newtonian momentum p=mv were valid, those gamma rays should have traveled at superluminal speeds. However, they found no difference and gave an upper limit of \Delta v/c=10^.


Energy and Calorimetry

The intrusion of particles into
particle detector In experimental and applied particle physics, nuclear physics, and nuclear engineering, a particle detector, also known as a radiation detector, is a device used to detect, track, and/or identify ionizing particles, such as those produced by nu ...
s is connected with
electron–positron annihilation Electron–positron annihilation occurs when an electron () and a positron (, the electron's antiparticle) collide. At low energies, the result of the collision is the annihilation of the electron and positron, and the creation of energetic photo ...
, Compton scattering, Cherenkov radiation etc., so that a cascade of effects is leading to the production of new particles (photons, electrons,
neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s, etc.). The energy of such
particle shower In particle physics, a shower is a cascade of secondary particles produced as the result of a high-energy particle interacting with dense matter. The incoming particle interacts, producing multiple new particles with lesser energy; each of these t ...
s corresponds to the relativistic kinetic energy and rest energy of the initial particles. This energy can be measured by calorimeters in an electrical, optical, thermal, or acoustical way. Thermal measurements in order to estimate the relativistic kinetic energy were already carried out by Bertozzi as mentioned above. Additional measurements at SLAC followed, in which the heat produced by 20-GeV electrons was measured in 1982. A beam dump of water-cooled
aluminium Aluminium (aluminum in American and Canadian English) is a chemical element with the symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately one third that of steel. I ...
was employed as calorimeter. The results were in agreement with special relativity, even though the accuracy was only 30%. However, the experimentalists alluded to the fact, that calorimetric tests with 10-GeV electrons were executed already in 1969. There,
copper Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkis ...
was used as beam dump, and an accuracy of 1% was achieved. In modern calorimeters called electromagnetic or
hadron In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...
ic depending on the interaction, the energy of the particle showers is often measured by the
ionization Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecul ...
caused by them. Also excitations can arise in scintillators (see scintillation), whereby light is emitted and then measured by a scintillation counter. Cherenkov radiation is measured as well. In all of those methods, the measured energy is proportional to the initial particle energy.


Annihilation and pair production

Relativistic energy and momentum can also be measured by studying processes such as
annihilation In particle physics, annihilation is the process that occurs when a subatomic particle collides with its respective antiparticle to produce other particles, such as an electron colliding with a positron to produce two photons. The total energy ...
and
pair production Pair production is the creation of a subatomic particle and its antiparticle from a neutral boson. Examples include creating an electron and a positron, a muon and an antimuon, or a proton and an antiproton. Pair production often refers specifi ...
. For instance, the rest energy of electrons and positrons is 0.51 MeV respectively. When a photon interacts with an
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...
, electron-positron pairs can be generated in case the energy of the photon matches the required
threshold energy In particle physics, the threshold energy for production of a particle is the minimum kinetic energy that must be imparted to one of a pair of particles in order for their collision to produce a given result. If the desired result is to produce a t ...
, which is the combined electron-positron rest energy of 1.02 MeV. However, if the photon energy is even higher, than the exceeding energy is converted into kinetic energy of the particles. The reverse process occurs in electron-positron annihilation at low energies, in which process photons are created having the same energy as the electron-positron pair. These are direct examples of E_0=mc^2 (
mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physici ...
). There are also many examples of conversion of relativistic kinetic energy into rest energy. In 1974,
SLAC National Accelerator Laboratory SLAC National Accelerator Laboratory, originally named the Stanford Linear Accelerator Center, is a United States Department of Energy National Laboratory operated by Stanford University under the programmatic direction of the U.S. Departme ...
accelerated electrons and positrons up to relativistic velocities, so that their relativistic energy \gamma mc^ (i.e. the sum of their rest energy and kinetic energy) is significantly increased to about 1500 MeV each. When those particles collide, other particles such as the J/ψ meson of rest energy of about 3000 MeV were produced. Much higher energies were employed at the
Large Electron–Positron Collider The Large Electron–Positron Collider (LEP) was one of the largest particle accelerators ever constructed. It was built at CERN, a multi-national centre for research in nuclear and particle physics near Geneva, Switzerland. LEP collided elect ...
in 1989, where electrons and positrons were accelerated up to 45 GeV each, in order to produce
W and Z bosons In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
of rest energies between 80 and 91 GeV. Later, the energies were considerably increased to 200 GeV to generate pairs of W bosons. Such bosons were also measured using proton-
antiproton The antiproton, , (pronounced ''p-bar'') is the antiparticle of the proton. Antiprotons are stable, but they are typically short-lived, since any collision with a proton will cause both particles to be annihilated in a burst of energy. The exis ...
annihilation. The combined rest energy of those particles amounts to approximately 0.938 GeV each. The Super Proton Synchrotron accelerated those particle up to relativistic velocities and energies of approximately 270 GeV each, so that the center of mass energy at the collision reaches 540 GeV. Thereby, quarks and antiquarks gained the necessary energy and momentum to annihilate into
W and Z bosons In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
. Many other experiments involving the creation of a considerable amount of different particles at relativistic velocities have been (and still are) conducted in
hadron In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...
colliders such as
Tevatron The Tevatron was a circular particle accelerator (active until 2011) in the United States, at the Fermi National Accelerator Laboratory (also known as ''Fermilab''), east of Batavia, Illinois, and is the second highest energy particle collider ...
(up to 1 TeV), the
Relativistic Heavy Ion Collider The Relativistic Heavy Ion Collider (RHIC ) is the first and one of only two operating heavy-ion colliders, and the only spin-polarized proton collider ever built. Located at Brookhaven National Laboratory (BNL) in Upton, New York, and used by a ...
(up to 200 GeV), and most recently the Large Hadron Collider (up to 7 TeV) in the course of searching for the Higgs boson.


References


External links

* Physics FAQ
List of SR tests
{{Tests of special relativity Momentum Physics experiments Special relativity