Cherenkov radiation (pronunciation: /tʃɛrɛnˈkɔv/) is an electromagnetic radiation emitted when a charged particle (such as an electron) passes through a dielectric medium at a speed greater than the phase velocity of light in that medium. The characteristic blue glow of an underwater nuclear reactor is due to Cherenkov radiation.
In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.
In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, which are all believed to have the same charge. Another charged particle may be an atomic nucleus devoid of electrons, such as an alpha particle.
The electron is a subatomic particle, symbol
, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ. As it is a fermion, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.
The radiation is named after the Soviet scientist Pavel Cherenkov, the 1958 Nobel Prize winner, who was the first to detect it experimentally under the supervision of Sergey Vavilov at the Lebedev Institute in 1934. Therefore it is also known as Vavilov–Cherenkov radiation.Cherenkov saw a faint bluish light around a radioactive preparation in water during experiments. His doctorate thesis was on luminescence of uranium salt solutions that were excited by gamma rays instead of less energetic visible light, as done commonly. He discovered the anisotropy of the radiation and came to the conclusion that the bluish glow was not a fluorescent phenomenon.
The Soviet Union, officially the Union of Soviet Socialist Republics (USSR), was a socialist state in Eurasia that existed from 30 December 1922 to 26 December 1991. Nominally a union of multiple national Soviet republics, its government and economy were highly centralized. The country was a one-party state, governed by the Communist Party with Moscow as its capital in its largest republic, the Russian Soviet Federative Socialist Republic. Other major urban centres were Leningrad, Kiev, Minsk, Alma-Ata, and Novosibirsk. It spanned over 10,000 kilometres east to west across 11 time zones, and over 7,200 kilometres north to south. It had five climate zones: tundra, taiga, steppes, desert and mountains.
Pavel Alekseyevich Cherenkov was a Soviet physicist who shared the Nobel Prize in physics in 1958 with Ilya Frank and Igor Tamm for the discovery of Cherenkov radiation, made in 1934.
The Nobel Prize in Physics is a yearly award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions for humankind in the field of physics. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others being the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and Nobel Prize in Physiology or Medicine.
A theory of this effect was later developed in 1937 within the framework of Einstein's special relativity theory by Cherenkov's colleagues Igor Tamm and Ilya Frank, who also shared the 1958 Nobel Prize.
Albert Einstein was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics. His work is also known for its influence on the philosophy of science. He is best known to the general public for his mass–energy equivalence formula E = mc2, which has been dubbed "the world's most famous equation". He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect", a pivotal step in the development of quantum theory.
In physics, special relativity is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time. In Albert Einstein's original pedagogical treatment, it is based on two postulates:
Igor Yevgenyevich Tamm was a Soviet physicist who received the 1958 Nobel Prize in Physics, jointly with Pavel Alekseyevich Cherenkov and Ilya Mikhailovich Frank, for their 1934 discovery of Cherenkov radiation.
Cherenkov radiation as conical wave front had been theoretically predicted by the English polymath Oliver Heaviside in papers published between 1888 and 1889and by Arnold Sommerfeld in 1904 , but both had been quickly forgotten following the relativity theory's restriction of super-c particles until the 1970s. Marie Curie observed a pale blue light in a highly concentrated radium solution in 1910, but did not investigate its source. In 1926, the French radiotherapists Lucien Mallet described the luminous radiation of radium irradiating water having a continuous spectrum.
England is a country that is part of the United Kingdom. It shares land borders with Wales to the west and Scotland to the north-northwest. The Irish Sea lies west of England and the Celtic Sea lies to the southwest. England is separated from continental Europe by the North Sea to the east and the English Channel to the south. The country covers five-eighths of the island of Great Britain, which lies in the North Atlantic, and includes over 100 smaller islands, such as the Isles of Scilly and the Isle of Wight.
A polymath is a person whose expertise spans a significant number of subject areas, known to draw on complex bodies of knowledge to solve specific problems.
Oliver Heaviside FRS was an English self-taught electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations, reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science for years to come.
While electrodynamics holds that the speed of light in a vacuum is a universal constant (c), the speed at which light propagates in a material may be significantly less than c. For example, the speed of the propagation of light in water is only 0.75c. Matter can be accelerated beyond this speed (although still to less than c) during nuclear reactions and in particle accelerators. Cherenkov radiation results when a charged particle, most commonly an electron, travels through a dielectric (electrically polarizable) medium with a speed greater than that at which light propagates in the same medium.
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.
The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299,792,458 metres per second. It is exact because by international agreement a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity, c is the maximum speed at which all conventional matter and hence all known forms of information in the universe can travel. Though this speed is most commonly associated with light, it is in fact the speed at which all massless particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. In the special and general theories of relativity, c interrelates space and time, and also appears in the famous equation of mass–energy equivalence E = mc2.
Vacuum is space devoid of matter. The word stems from the Latin adjective vacuus for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss ideal test results that would occur in a perfect vacuum, which they sometimes simply call "vacuum" or free space, and use the term partial vacuum to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is lower than atmospheric pressure. The Latin term in vacuo is used to describe an object that is surrounded by a vacuum.
A common analogy is the sonic boom of a supersonic aircraft. The sound waves generated by the supersonic body propagate at the speed of sound itself; as such, the waves travel slower than the speeding object and cannot propagate forward from the body, instead forming a shock front. In a similar way, a charged particle can generate a light shock wave as it travels through an insulator.
A sonic boom is the sound associated with the shock waves created whenever an object travelling through the air travels faster than the speed of sound. Sonic booms generate enormous amounts of sound energy, sounding similar to an explosion or a thunderclap to the human ear. The crack of a supersonic bullet passing overhead or the crack of a bullwhip are examples of a sonic boom in miniature.
An aircraft is a machine that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines. Common examples of aircraft include airplanes, helicopters, airships, gliders, and hot air balloons.
In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.
Moreover, the velocity that must be exceeded is the phase velocity of light rather than the group velocity of light. The phase velocity can be altered dramatically by employing a periodic medium, and in that case one can even achieve Cherenkov radiation with no minimum particle velocity, a phenomenon known as the Smith–Purcell effect. In a more complex periodic medium, such as a photonic crystal, one can also obtain a variety of other anomalous Cherenkov effects, such as radiation in a backwards direction (see below) whereas ordinary Cherenkov radiation forms an acute angle with the particle velocity.
In their original work on the theoretical foundations of Cherenkov radiation, Tamm and Frank wrote,"This peculiar radiation can evidently not be explained by any common mechanism such as the interaction of the fast electron with individual atom or as radiative scattering of electrons on atomic nuclei. On the other hand, the phenomenon can be explained both qualitatively and quantitatively if one takes in account the fact that an electron moving in a medium does radiate light even if it is moving uniformly provided that its velocity is greater than the velocity of light in the medium.".However, some misconceptions regarding Cherenkov radiation exist: for example, it is believed that the medium becomes electrically polarized by the particle's electric field. If the particle travels slowly then the disturbance elastically relaxes back to mechanical equilibrium as the particle passes. When the particle is traveling fast enough, however, the limited response speed of the medium means that a disturbance is left in the wake of the particle, and the energy contained in this disturbance radiates as a coherent shockwave. Such conceptions do not have any analytical foundation, as electromagnetic radiation is emitted when charged particles move in a dielectric medium at subluminal velocities which are not considered as Cherenkov radiation.
In the figure on the geometry, the particle (red arrow) travels in a medium with speed such that
where is speed of light in vacuum, and is the refractive index of the medium. If the medium is water, the condition is , since for water at 20 °C.
We define the ratio between the speed of the particle and the speed of light as
The emitted light waves (blue arrows) travel at speed
The left corner of the triangle represents the location of the superluminal particle at some initial moment (t = 0). The right corner of the triangle is the location of the particle at some later time t. In the given time t, the particle travels the distance
whereas the emitted electromagnetic waves are constricted to travel the distance
So the emission angle results in
Cherenkov radiation can also radiate in an arbitrary direction using properly engineered one dimensional metamaterials. ), reversing or steering Cherenkov emission at arbitrary angles given by the generalized relation:The latter is designed to introduce a gradient of phase retardation along the trajectory of the fast travelling particle (
Note that since this ratio is independent of time, one can take arbitrary times and achieve similar triangles. The angle stays the same, meaning that subsequent waves generated between the initial time t=0 and final time t will form similar triangles with coinciding right endpoints to the one shown.
A reverse Cherenkov effect can be experienced using materials called negative-index metamaterials (materials with a subwavelength microstructure that gives them an effective "average" property very different from their constituent materials, in this case having negative permittivity and negative permeability). This means, when a charged particle (usually electrons) passes through a medium at a speed greater than the phase velocity of light in that medium, that particle will emit trailing radiation from its progress through the medium rather than in front of it (as is the case in normal materials with, both permittivity and permeability positive).One can also obtain such reverse-cone Cherenkov radiation in non-metamaterial periodic media where the periodic structure is on the same scale as the wavelength, so it cannot be treated as an effectively homogeneous metamaterial.
The frequency spectrum of Cherenkov radiation by a particle is given by the Frank–Tamm formula:
The Frank-Tamm formula describes the amount of energy emitted from Cherenkov radiation, per unit length traveled and per frequency . is the permeability and is the index of refraction of the material the charge particle moves through. is the electric charge of the particle, is the speed of the particle, and is the speed of light in vacuum.
Unlike fluorescence or emission spectra that have characteristic spectral peaks, Cherenkov radiation is continuous. Around the visible spectrum, the relative intensity per unit frequency is approximately proportional to the frequency. That is, higher frequencies (shorter wavelengths) are more intense in Cherenkov radiation. This is why visible Cherenkov radiation is observed to be brilliant blue. In fact, most Cherenkov radiation is in the ultraviolet spectrum—it is only with sufficiently accelerated charges that it even becomes visible; the sensitivity of the human eye peaks at green, and is very low in the violet portion of the spectrum.
There is a cut-off frequency above which the equation can no longer be satisfied. The refractive index varies with frequency (and hence with wavelength) in such a way that the intensity cannot continue to increase at ever shorter wavelengths, even for very relativistic particles (where v/c is close to 1). At X-ray frequencies, the refractive index becomes less than unity (note that in media the phase velocity may exceed c without violating relativity) and hence no X-ray emission (or shorter wavelength emissions such as gamma rays) would be observed. However, X-rays can be generated at special frequencies just below the frequencies corresponding to core electronic transitions in a material, as the index of refraction is often greater than 1 just below a resonant frequency (see Kramers-Kronig relation and anomalous dispersion).
As in sonic booms and bow shocks, the angle of the shock cone is directly related to the velocity of the disruption. The Cherenkov angle is zero at the threshold velocity for the emission of Cherenkov radiation. The angle takes on a maximum as the particle speed approaches the speed of light. Hence, observed angles of incidence can be used to compute the direction and speed of a Cherenkov radiation-producing charge.
Cherenkov radiation can be generated in the eye by charged particles hitting the vitreous humour, giving the impression of flashes,as in cosmic ray visual phenomena and possibly some observations of criticality accidents.
Cherenkov radiation is widely used to facilitate the detection of small amounts and low concentrations of biomolecules.Radioactive atoms such as phosphorus-32 are readily introduced into biomolecules by enzymatic and synthetic means and subsequently may be easily detected in small quantities for the purpose of elucidating biological pathways and in characterizing the interaction of biological molecules such as affinity constants and dissociation rates.
More recently, Cherenkov light has been used to image substances in the body.These discoveries have led to intense interest around the idea of using this light signal to quantify and/or detect radiation in the body, either from internal sources such as injected radiopharmaceuticals or from external beam radiotherapy in oncology. Radioisotopes such as the positron emitters 18F and 13N or beta emitters 32P or 90Y have measurable Cherenkov emission and isotopes 18F and 131I have been imaged in humans for diagnostic value demonstration. External beam radiation therapy has been shown to induce a substantial amount of Cherenkov light in the tissue being treated, due to the photon beam energy levels used in the 6 MeV to 18 MeV ranges. The secondary electrons induced by these high energy x-rays result in the Cherenkov light emission, where the detected signal can be imaged at the entry and exit surfaces of the tissue.
Cherenkov radiation is used to detect high-energy charged particles. In pool-type nuclear reactors, beta particles (high-energy electrons) are released as the fission products decay. The glow continues after the chain reaction stops, dimming as the shorter-lived products decay. Similarly, Cherenkov radiation can characterize the remaining radioactivity of spent fuel rods. This phenomenon is used to verify the presence of spent nuclear fuel in spent fuel pools for nuclear safeguards purposes.
When a high-energy (TeV) gamma photon or cosmic ray interacts with the Earth's atmosphere, it may produce an electron-positron pair with enormous velocities. The Cherenkov radiation emitted in the atmosphere by these charged particles is used to determine the direction and energy of the cosmic ray or gamma ray, which is used for example in the Imaging Atmospheric Cherenkov Technique (IACT), by experiments such as VERITAS, H.E.S.S., MAGIC. Cherenkov radiation emitted in tanks filled with water by those charged particles reaching earth is used for the same goal by the Extensive Air Shower experiment HAWC, the Pierre Auger Observatory and other projects. Similar methods are used in very large neutrino detectors, such as the Super-Kamiokande, the Sudbury Neutrino Observatory (SNO) and IceCube. Other projects operated in the past applying related techniques, such as STACEE, a former solar tower refurbished to work as a non-imaging Cherenkov observatory, which was located in New Mexico.
Astrophysics observatories using the Cherenkov technique to measure air showers are key to determine the properties of astronomical objects that emit Very High Energy gamma rays, such as supernova remnants and blazars.
Cherenkov radiation is commonly used in experimental particle physics for particle identification. One could measure (or put limits on) the velocity of an electrically charged elementary particle by the properties of the Cherenkov light it emits in a certain medium. If the momentum of the particle is measured independently, one could compute the mass of the particle by its momentum and velocity (see four-momentum), and hence identify the particle.
The simplest type of particle identification device based on a Cherenkov radiation technique is the threshold counter, which gives an answer as to whether the velocity of a charged particle is lower or higher than a certain value (, where is the speed of light, and is the refractive index of the medium) by looking at whether this particle does or does not emit Cherenkov light in a certain medium. Knowing particle momentum, one can separate particles lighter than a certain threshold from those heavier than the threshold.
The most advanced type of a detector is the RICH, or Ring-imaging Cherenkov detector, developed in the 1980s. In a RICH detector, a cone of Cherenkov light is produced when a high speed charged particle traverses a suitable medium, often called radiator. This light cone is detected on a position sensitive planar photon detector, which allows reconstructing a ring or disc, the radius of which is a measure for the Cherenkov emission angle. Both focusing and proximity-focusing detectors are in use. In a focusing RICH detector, the photons are collected by a spherical mirror and focused onto the photon detector placed at the focal plane. The result is a circle with a radius independent of the emission point along the particle track. This scheme is suitable for low refractive index radiators—i.e. gases—due to the larger radiator length needed to create enough photons. In the more compact proximity-focusing design, a thin radiator volume emits a cone of Cherenkov light which traverses a small distance—the proximity gap—and is detected on the photon detector plane. The image is a ring of light, the radius of which is defined by the Cherenkov emission angle and the proximity gap. The ring thickness is determined by the thickness of the radiator. An example of a proximity gap RICH detector is the High Momentum Particle Identification Detector (HMPID),a detector currently under construction for ALICE (A Large Ion Collider Experiment), one of the six experiments at the LHC (Large Hadron Collider) at CERN.
The Cherenkov effect can occur in vacuum [ further explanation needed ] the phase velocity decreases and the velocity of charged particles can exceed the phase velocity while remaining lower than . In such a system, this effect can be derived from conservation of the energy and momentum where the momentum of a photon should be ( is phase constant) rather than the de Broglie relation . This type of radiation (VCR) is used to generate high power microwaves.. In a slow-wave structure,
The photon is a type of elementary particle, the quantum of the electromagnetic field including electromagnetic radiation such as light, and the force carrier for the electromagnetic force. The photon has zero rest mass and always moves at the speed of light within a vacuum.
Compton scattering, discovered by Arthur Holly Compton, is the scattering of a photon by a charged particle, usually an electron. It results in a decrease in energy of the photon, called the Compton effect. Part of the energy of the photon is transferred to the recoiling electron. Inverse Compton scattering occurs when a charged particle transfers part of its energy to a photon.
In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.
Cyclotron radiation is electromagnetic radiation emitted by accelerating charged particles deflected by a magnetic field. The Lorentz force on the particles acts perpendicular to both the magnetic field lines and the particles' motion through them, creating an acceleration of charged particles that causes them to emit radiation as a result of the acceleration they undergo as they spiral around the lines of the magnetic field.
The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted due to an atom or molecule making a transition from a high energy state to a lower energy state. The photon energy of the emitted photon is equal to the energy difference between the two states. There are many possible electron transitions for each atom, and each transition has a specific energy difference. This collection of different transitions, leading to different radiated wavelengths, make up an emission spectrum. Each element's emission spectrum is unique. Therefore, spectroscopy can be used to identify the elements in matter of unknown composition. Similarly, the emission spectra of molecules can be used in chemical analysis of substances.
Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is just the low-energy limit of Compton scattering: the particle kinetic energy and photon frequency do not change as a result of the scattering. This limit is valid as long as the photon energy is much smaller than the mass energy of the particle: , or equivalently, if the wavelength of the light is much greater than the Compton wavelength of the particle.
Transition radiation (TR) is a form of electromagnetic radiation emitted when a charged particle passes through inhomogeneous media, such as a boundary between two different media. This is in contrast to Cherenkov radiation, which occurs when a charged particle passes through a homogeneous dielectric medium at a speed greater than the phase velocity of electromagnetic waves in that medium.
Relativistic beaming is the process by which relativistic effects modify the apparent luminosity of emitting matter that is moving at speeds close to the speed of light. In an astronomical context, relativistic beaming commonly occurs in two oppositely-directed relativistic jets of plasma that originate from a central compact object that is accreting matter. Accreting compact objects and relativistic jets are invoked to explain the following observed phenomena: x-ray binaries, gamma-ray bursts, and, on a much larger scale, active galactic nuclei (AGN).
A Cherenkov detector is a particle detector using the speed threshold for light production, the speed-dependent light output or the speed-dependent light direction of Cherenkov radiation.
The ring-imaging Cherenkov, or RICH, detector is a device for identifying the type of an electrically charged subatomic particle of known momentum, that traverses a transparent refractive medium, by measurement of the presence and characteristics of the Cherenkov radiation emitted during that traversal. RICH detectors were first developed in the 1980s and are used in high energy elementary particle-, nuclear- and astro-physics experiments.
An air shower is an extensive cascade of ionized particles and electromagnetic radiation produced in the atmosphere when a primary cosmic ray enters the atmosphere. When a particle, which could be a proton, a nucleus, an electron, a photon, or (rarely) a positron, strikes an atom's nucleus in the air it produces many energetic hadrons. The unstable hadrons decay in the air speedily into other particles and electromagnetic radiation, which are part of the shower components. The secondary radiation rains down, including x-rays, muons, protons, antiprotons, alpha particles, pions, electrons, positrons, and neutrons.
Particle identification is the process of using information left by a particle passing through a particle detector to identify the type of particle. Particle identification reduces backgrounds and improves measurement resolutions, and is essential to many analyses at particle detectors.
ALICE is one of seven detector experiments at the Large Hadron Collider at CERN. The other six are: ATLAS, CMS, TOTEM, LHCb, LHCf and MoEDAL.
IACT stands for Imaging AtmosphericCherenkov Telescope or Technique. It is a device or method to detect very-high-energy gamma ray photons in the photon energy range of 50 GeV to 50 TeV. There are four operating IACT systems: High Energy Stereoscopic System (H.E.S.S.), Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC), First G-APD Cherenkov Telescope (FACT), and Very Energetic Radiation Imaging Telescope Array System (VERITAS). Set to be the world's largest telescope at the highest altitude, the Major Atmospheric Cerenkov Experiment Telescope (MACE) is being built at Hanle, Ladakh, India. Also under design is the Cherenkov Telescope Array (CTA).
In particle detectors a detection of internally reflected Cherenkov light (DIRC) detector measures the velocity of charged particles and is used for particle identification. It is a design of a ring imaging Cherenkov detector where Cherenkov light that is contained by total internal reflection inside the solid radiator has its angular information preserved until it reaches the light sensors at the detector perimeter.
The Frank–Tamm formula yields the amount of Cherenkov radiation emitted on a given frequency as a charged particle moves through a medium at superluminal velocity. It is named for Russian physicists Ilya Frank and Igor Tamm who developed the theory of the Cherenkov effect in 1937, for which they were awarded a Nobel Prize in Physics in 1958.
Cherenkov luminescence imaging (CLI) is an emerging imaging modality, similar to bioluminescence imaging, that captures visible photons emitted by Cherenkov radiation. It basically is the optical imaging of radiotracers that emit charged particles traveling faster than the phase velocity of light in that particular medium. It can be used to quickly evaluate radio tracers in preclinical research but also to obtain clinical images in patients. While radioactivity itself can not be modified, the emitted light provides an opportunity to generate radioactivity-based activatable or "smart" imaging agents that sense for example enzymatic activity.
Tests of relativistic energy and momentum are aimed at measuring the relativistic expressions for energy, momentum, and mass. According to special relativity, the properties of particles moving approximately at the speed of light significantly deviate from the predictions of Newtonian mechanics. For instance, the speed of light cannot be reached by massive particles.
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