Tests of the relativistic energy-momentum relationship

Kinetic energy according to the special theory of relativity (red) and Newtonian mechanics (green). The speed is plotted in units of the speed of light. The relativistic kinetic energy (red) increases to infinity when approaching the speed of light. A solid body can therefore not reach this speed.

Tests of the relativistic energy-momentum relationship are used to experimentally check statements of the special theory of relativity , which concern energy , kinetic energy , momentum and mass . According to this theory, the properties of very fast moving matter differ greatly from the properties known from classical mechanics . For example, the speed of light cannot be reached by solid bodies.

The relativistic energy-momentum relationship must be taken into account, for example, when designing and evaluating experiments in particle physics , and is routinely demonstrated for particles close to the speed of light in simple experiments as part of basic physics studies . See also tests of the special theory of relativity .

The energy-momentum relationship

In parallel to the kinetic energy, the relativistic impulse (red) also increases to infinity when approaching the speed of light. The speed is plotted in units of the speed of light.

Describes the mass and the speed of a body, according to classical mechanics it is the momentum and the kinetic energy . This would allow any given speed, including the speed of light, to be exceeded with a corresponding supply of energy. ${\displaystyle m}$ ${\displaystyle v}$ ${\displaystyle {\vec {p}}=m{\vec {v}}}$ ${\displaystyle E_{k}={\tfrac {mv^{2}}{2}}}$

On the other hand, the special theory of relativity says, among other things, that the speed of light in inertial systems represents a limit speed that cannot be reached for bodies with mass. This is expressed both by the Lorentz transformation and by the relativistic energy-momentum relationship ("relativistic Pythagoras"): ${\displaystyle c}$

${\displaystyle E^{2}-(pc)^{2}=(mc^{2})^{2}\,}$.

From this follow the relationships for the rest energy , relativistic energy (rest + movement) , kinetic energy and momentum of particles with mass : ${\displaystyle E_{0}}$ ${\displaystyle E}$

${\displaystyle E_{0}=mc^{2},\quad E=\gamma mc^{2},\quad E_{k}=(\gamma -1)mc^{2},\quad {\vec {p}}=\gamma m{\vec {v}}}$,

where . Relativistic energy and momentum rise beyond all boundaries when approaching the speed of light. Therefore, mass-laden particles cannot reach this speed. ${\displaystyle \gamma =1/{\sqrt {1-(v/c)^{2}}}}$

First experiments

→ Main article : Kaufmann-Bucherer-Neumann experiments

The first experiments that were able to prove such relationships were carried out by Walter Kaufmann , Alfred Heinrich Bucherer and others between 1901 and 1915. The deflection of beta radiation ( electrons ) in a magnetic field was measured in order to determine the charge-mass ratio . Since the constancy of the charge was known, changes could only affect the mass or the momentum of the electromagnetic field of the electrons. The term transverse electromagnetic mass was used earlier , equivalent to the relativistic mass mentioned above . Since the concept of "relativistic mass" is rarely used in modern texts, these experiments can be described as tests of relativistic momentum or energy according to the definitions above, because: ${\displaystyle m_{T}=m\gamma }$ ${\displaystyle M}$

${\displaystyle {\frac {M}{m}}={\frac {p}{mv}}={\frac {E}{mc^{2}}}=\gamma }$.

The results of the experiments by Bucherer and Neumann showed a decrease in the charge-mass ratio with increasing speed and consequently an increase in momentum, in quantitative agreement with the special theory of relativity. However, it was later shown that the measurements only matched qualitatively and were too imprecise to rule out certain competing models, such as the Max Abraham model .

However, as early as 1915 Arnold Sommerfeld was able to derive the fine structure of the hydrogen spectrum using the relativistic terms for momentum and energy (in the context of the Bohr – Sommerfeld theory ). Thereupon Karl Glitscher replaced the relativistic expressions in the derivation of the hydrogen spectrum with those of the theory of Abraham. He showed that Abraham's theory, unlike the theory of relativity, was incompatible with the observations.

Precision measurements

Measurement points from Rogers et al. in accordance with the theory of relativity.

Rogers et al . (1940) carried out the first deflection experiments with electrons with the necessary accuracy to conclusively refute the competing models. As in the Bucherer-Neumann experiments, the charge-to-mass ratio was measured, with electron velocities of up to 0.75c being achieved. They improved the measurement method, for example by using a Geiger counter . The possible deviations were only about one percent.

An even more precise experiment was carried out by Meyer et al. (1963) by. They observed electrons with velocities from 0.987 to 0.99c. The deflections took place in a static-homogeneous magnetic field, with which p was measured, and in a static-cylindrical electric field, with which measurements were made. They confirmed the theory of relativity with an upper limit for deviations of ∼0.00037. ${\displaystyle p^{2}/(m\gamma )}$

Measurements of the charge-to-mass ratio and consequently the momentum of protons were also made. Grove and Fox (1953) observed 385 MeV protons moving at ∼0.7 c. By determining the angular frequencies and the magnetic field, the charge-mass ratio could be determined. This and the measurement of the magnetic center allowed the relativistic prediction to be confirmed with a precision of ∼0.0006.

However, Zrelov et al. (1958) suggest that the information given by Grove and Fox is too sparse and that such experiments are subject to considerable difficulties due to the complex motion of protons. Therefore, Zrelov et al. carried out an extended measurement with 660 MeV protons, which reached an average velocity of 0.8112 c. The speed of the protons was measured by evaluating the Cherenkov radiation , the impulse with the method of the current-carrying thin wire, which in the magnetic field assumes the same shape as a corresponding particle path. The relativistic calculation was confirmed with an upper limit for deviations of ∼0.0041.

Bertozzi experiment

Data from the Bertozzi experiment show a close agreement with the special theory of relativity. Velocity of the five electron measurements: 0.752, 0.828, 0.922, 0.974, 1.0 in v / c (or 0.867, 0.910, 0.960, 0.987, 1 in (v / c) ²). Kinetic energy: 0.5, 1, 1.5, 4.5, 15 MeV (or 1, 2, 3, 9, 30 in mc²).

The energy-momentum relationship of the special theory of relativity has been required in particle accelerators since its introduction in the 1930s, and the above measurements of momentum and velocity also confirmed the energy-momentum relationship of the relativity theory with high precision, so that there was no longer any doubt as to its correctness. However, the determination of momentum and speed in deflection curves is also dependent on additional factors and effects that must be taken into account together. Therefore William Bertozzi (1964) carried out an experiment to demonstrate the relativistic effects particularly clearly by directly measuring the speed and kinetic energy of electrons.

He used MIT's electron accelerator for five experimental runs in which electrons with energies between 0.5 and 15 MeV were generated by a Van de Graaff accelerator and traveled 8.4 meters until they hit an aluminum disk. First the flight time and thus the speed of the electrons was measured in all five passes - these data were in close agreement with the special theory of relativity (see picture). At this stage, the kinetic energy was initially only determined indirectly through the accelerating fields. Therefore, Bertozzi measured the heat ( calorimetry ) that electrons between 1.6 and 4.8 MeV generated on the aluminum disk and found agreement within an error limit of 10%.

Experiments for undergraduate studies

In the meantime, measurements of the relativistic energy or the momentum can be carried out in simple form in university laboratories that are suitable for undergraduate studies . Essentially three methods are used: a) experiments with beta radiation , for example to detect the impulse during deflection in a magnetic field or the kinetic energy upon impact with the detector; b) Compton effect , whereby the electrons can be brought to relativistic speed; c) Positron annihilation , where the energy and momentum of the resulting radiation can be checked.

Beta radiation Marvel et al. 2011 Lund et al. 2009 Luetzelschwab 2003 Couch et al. 1982 Geller et al. 1972 Parker 1972 Bartlett et al. 1965

Compton effect Jolivette et al. 1994 Hoffman 1989 Egelstaff et al. 1981 Higbie 1974

Positron annihilation Dryzek et al. 2006

High-energy experiments in particle accelerators

In modern particle accelerators , the predictions of the special theory of relativity are routinely confirmed at high energies and are necessary for the design and evaluation of collision experiments. For example, the time dilation of moving particles is observed during the decay of unstable particles, and the relativistic addition theorem of the velocities is necessary to understand the distribution of synchrotron radiation . The relativistic energy-momentum relationship was also confirmed in speed measurements and numerous high-energy experiments .

speed

Far beyond the energy values ​​of the Bertozzi experiment, time-of-flight measurements to determine the speed differences between electrons and light were carried out by the Stanford Linear Accelerator Center (SLAC). Brown et al. (1973) found no difference and determined an upper limit for speed differences between 11-GeV electrons and visible light from . Guiragossián et al. (1974) accelerated the electrons to 15-20.5 GeV in a further experiment. They used a radio frequency separator (RFS) to measure time-of-flight differences between these electrons and 15 GeV gamma rays over a distance of 1015 m. Again, no difference was found, with a maximum upper limit of . ${\displaystyle \Delta v/c=(-1{,}3\pm 2{,}7)\times 10^{-6}}$ ${\displaystyle \Delta v/c=2\times 10^{-7}}$

Alväger et al. (1964) carried out a time-of-flight measurement at the CERN Proton Synchrotron in order to test the Newtonian momentum relationship , as it is valid, for example, in emission theory . This resulted in gamma rays from the decay of 6 GeV pions at a speed of 0.99975 c. If Newton's relation were valid, the gamma rays should have been significantly faster than light. However, no such effect was found, with a maximum cap of . ${\displaystyle p=mv}$ ${\displaystyle \Delta v/c=10^{-5}}$

Energy and calorimetry

The penetration of sufficiently fast particles into a particle detector goes hand in hand with electron-positron annihilation , Compton scattering, Cherenkov radiation , etc., so that a cascade of effects leads to the formation of new particles (photons, electrons, neutrinos, etc.). The energy of this shower of particles corresponds to the relativistic kinetic energy and the rest energy of the penetrating particles. Based on the interactions with the detector, this energy can be measured, for example, by specially designed calorimeters , with this measurement being able to be carried out electrically, optically, thermally or acoustically.

Calorimetric measurements of the relativistic energy on a thermal basis were already carried out by Bertozzi, as described above. Further measurements by SLAC followed, in which in 1982 the heat of electrons accelerated to 20 GeV was measured; A water-cooled absorber (beam dump) made of aluminum served as the calorimeter. Agreement was found with the relativistic energy-momentum relationship, but only to an accuracy of 30%. However, the experimenters pointed out that calorimetric measurements with 10 GeV electrons had already been carried out in 1969. Copper was used as the beam absorber, and the theory was confirmed with a much greater accuracy of 1%.

In modern calorimeters (which are called either electromagnetic or hadronic depending on the type of interaction ), the energy of the particle showers is often determined by measuring the ionization they cause . Excitations ( scintillation ) also occur in the detector, which lead to the emission of light, which is measured by scintillation counters . Cherenkov radiation can also be evaluated. These methods show that the measured energy is proportional to the original particle energy.

Annihilation and pairing

Relativistic energy and momentum appear directly in processes such as annihilation and pairing . For example, the rest energy of electrons and positrons is 0.51 MeV each. If a photon interacts with an atomic nucleus , electron-positron pairs can be formed if the photon has the necessary threshold energy of 1.02 MeV. When the photon energy is greater, the excess energy is converted into kinetic energy of the particles. The reverse process occurs, for example, with electron-positron annihilation at low energies, where photons are created whose total energy and momentum correspond to those of the starting particles. These are direct examples of the equivalence of mass and energy according to . ${\displaystyle E_{0}=mc^{2}}$

These relationships appear much more clearly at much larger energies, where relativistic kinetic energy is converted into rest energy. In 1974, the SLAC accelerator accelerated both electrons and positrons to relativistic speeds, where their relativistic energy (the sum of rest energy and kinetic energy) was approximately 1500 MeV each. When these particles collided (see colliding beam experiment ), J / ψ mesons with a rest energy of about 3000 MeV were created. Much higher energies were achieved from 1989 onwards at the Large Electron-Positron Collider , where electrons and positrons were accelerated to 45 GeV each, whereby W bosons and Z bosons with resting energies of 80 to 91 GeV could arise. Later energies up to 200 GeV were reached, so that these particles could arise in pairs. These energetic bosons were generated earlier (1984) in the super proton synchrotron by proton - antiproton collisions. The rest energy of these particles is 0.938 GeV each. They were now accelerated to about 270 GeV, so that the center of gravity energy in the collision was 540 GeV. This energy was necessary so that its constituents, quarks and antiquarks, received the necessary energy and momentum to generate W and Z bosons. ${\displaystyle \gamma mc^{2}}$

Besides these examples, a large number of experiments were conducted where both stable particles such as protons and electrons and a large number of unstable particles were generated. In addition to the institutions mentioned, hadron accelerators in particular achieve enormous energies: HERA (up to 920 GeV), the Tevatron (up to 1 TeV), the Relativistic Heavy Ion Collider (up to 200 GeV) and above all the Large Hadron Collider (up to 6.5 TeV) .

Individual evidence

1. ^ ^{A b} Edwin F. Taylor, John Archibald Wheeler : Spacetime Physics: Introduction to Special Relativity . WH Freeman, New York 1992, ISBN 0-7167-2327-1 .
2. ↑ ^{a b} Plettner, Tomas; Byer, Robert L .; Siemann, Robert H .: The impact of Einstein's theory of special relativity on particle accelerators . In: Journal of Physics B . 38, No. 9, 2005, pp. S741-S752. doi : 10.1088 / 0953-4075 / 38/9/020 .
3. ^ Zahn, CT and Spees, AA: A Critical Analysis of the Classical Experiments on the Variation of Electron Mass . In: Physical Review . 53, 1938, pp. 511-521. bibcode : 1938PhRv ... 53..511Z . doi : 10.1103 / PhysRev.53.511 .
4. ^ PS Faragó and L. Jánossy: Review of the experimental evidence for the law of variation of the electron mass with velocity . In: Il Nuovo Cimento . 5, No. 6, 1957, pp. 379-383. doi : 10.1007 / BF02856033 .
5. ^ Glitscher, Karl: Spectroscopic comparison between the theories of the rigid and the deformable electron . In: Annals of Physics . 357, No. 6, 1917, pp. 608-630. doi : 10.1002 / andp.19173570603 .
6. ↑ Rogers, Marguerite M .; McReynolds, AW; Rogers, FT: A Determination of the Masses and Velocities of Three Radium B Beta-Particles: The Relativistic Mass of the Electron . In: Physical Review . 57, No. 5, 1940, pp. 379-383. bibcode : 1940PhRv ... 57..379R . doi : 10.1103 / PhysRev.57.379 .
7. ↑ Meyer, V.; Reichart, W.; Staub, HH: Experimental investigation of the mass-momentum relation of the electron . In: Helvetica Physica Acta . 36, 1963, pp. 981-992. doi : 10.5169 / seals-113412 .
8. ↑ Grove, DJ; Fox, JC: e / m for 385-MeV protons (UA7) . In: Physical Review . 90, 1953, p. 378.
9. ↑ Zrelov, VP; Tiapkin, AA; Farago, PS: Measurement of the mass of 600 MeV protons . In: Soviet Physics JETP . 7, No. 3, 1958, pp. 384-387.
10. ^ Bertozzi, William: Speed ​​and Kinetic Energy of Relativistic Electrons . In: American Journal of Physics . 32, No. 7, 1964, pp. 551-555. doi : 10.1119 / 1.1970770 .
11. ↑ ^{a b} Marvel, Robert E .; Vineyard, Michael F .: Relativistic Electron Experiment for the Undergraduate Laboratory . In: Department of Physics and Astronomy . 2011. arxiv : 1108.5977 .
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16. ^ Parker, Sherwood: Relativity in an Undergraduate Laboratory-Measuring the Relativistic Mass Increase . In: American Journal of Physics . 40, No. 2, 1972, pp. 241-244. doi : 10.1119 / 1.1986498 .
17. ↑ Bartlett, AA; Correll, Malcolm: An Undergraduate Laboratory Apparatus for Measuring e / m as a Function of Velocity. I . In: American Journal of Physics . 33, No. 4, 1965, pp. 327-339. doi : 10.1119 / 1.1971493 .
18. ↑ Jolivette, PL; Rouze, N .: Compton scattering, the electron mass, and relativity: A laboratory experiment . In: American Journal of Physics . 62, No. 3, 1994, pp. 266-271. doi : 10.1119 / 1.17611 .
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22. ↑ Dryzek, Jerzy; Singleton, Douglas; Suzuki, Takenori; Yu, Runsheng: An undergraduate experiment to test relativistic kinematics using in flight positron annihilation . In: American Journal of Physics . 74, No. 1, 2006, pp. 49-53. doi : 10.1119 / 1.2142624 .
23. ^ Zhang, Yuan Zhong: Special Relativity and Its Experimental Foundations . World Scientific, 1997, ISBN 9789810227494 .
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25. ↑ Guiragossián, ZGT; Rothbart, GB; Yearian, MR; Gearhart, RA; Murray, JJ: Relative Velocity Measurements of Electrons and Gamma Rays at 15 GeV . In: Physical Review Letters . 34, No. 6, 1974, pp. 335-338. doi : 10.1103 / PhysRevLett.34.335 .
26. ↑ Alväger, T .; Farley, FJM; Kjellman, J .; Wallin, L .: Test of the second postulate of special relativity in the GeV region . In: Physics Letters . 12, No. 3, 1964, pp. 260-262. bibcode : 1964PhL .... 12..260A . doi : 10.1016 / 0031-9163 (64) 91095-9 .
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30. ^ Burton Richter : From the Psi to Charm - The Experiments of 1975 and 1976 . Nobel lecture 1976. 1976. Retrieved June 2, 2012.
31. ↑ LEP collaborations: Electroweak parameters of the Z0 resonance and the standard model . In: Physics Letters B . 276, No. 12, 1992, pp. 247-253. doi : 10.1016 / 0370-2693 (92) 90572-L .
32. ^ Carlo Rubbia : Experimental Observation of the Intermediate Vector Bosons W +, W- and Z0 . Nobel lecture 1984. 1984. Retrieved June 2, 2012.

Web links

• Physics FAQ: List of tests on SRT