Speed of light

Speed of light The distance from the Sun to the Earth is shown as 150 million kilometers, an approximate average. Sizes to scale. Sunlight takes about 8 minutes 17 seconds to travel the average distance from the surface of the Sun to the Earth.

The speed of light in vacuum, c, is a universal physical constant, which, according to special relativity, is the maximum speed at which matter or information may travel. It is the speed of all massless particles and changes of the associated fields in a vacuum. Such particle/waves travel at c, regardless of the motion of the source or the inertial reference frame of the observer. The theory of relativity interrelates space and time using c, which also appears in the famous equivalence relation of mass and energy, E = mc².

CONTENT : A - F , G - L , M - R , S - Z , See also , External links

Quotes

Quotes are arranged alphabetically by author

A - F

• One of the problems has to do with the speed of light and the difficulties involved in trying to exceed it. You can't. Nothing travels faster than the speed of light with the possible exception of bad news, which obeys its own special laws.
  • Douglas Adams, Mostly Harmless (1992) from The Hitchhiker's Guide to the Galaxy series

• You will begin to touch heaven, Jonathan, in the moment that you touch perfect speed. And that isn't flying a thousand miles an hour, or a million, or flying at the speed of light. Because any number is a limit, and perfection doesn't have limits. Perfect speed, my son, is being there.
  • Richard Bach, Jonathan Livingston Seagull (1970)

• Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. … Fermat applied his method … and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite.
  • Carl B. Boyer, The Rainbow: From Myth to Mathematics (1959) p. 205

• The speed of light sucks.
  • John D. Carmack, as quoted in John Carmack Biography referring to network latency limitations.

• One of the funniest examples of these kinds of statistics comes from Evolution: Possible or Impossible by James F. Coppedge [who] cites an article by Ulric Jelinek … which claims that the odds are 1 in 10^243 against "two thousand atoms" (the size of one particular protein molecule) ending up in precisely that particular order "by accident." Where did Jelenik get that figure? From Pierre Lecompte du Nouy... who in turn got it from Charles-Eugene Guye, a physicist who died in 1942. Guye had merely calculated the odds of these atoms lining up by accident if "a volume" of atoms the size of the Earth were "shaken at the speed of light." In other words, ignoring all the laws of chemistry, which create preferences for the formation and behavior of molecules, and ignoring that there are millions if not billions of different possible proteins--and of course the result has no bearing on the origin of life, which may have begun from an even simpler protein. This calculation is thus useless for all these reasons, and is typical in that it comes to Coppedge third-hand (and thus to us fourth-hand), and is hugely outdated (it was calculated before 1942, even before the discovery of DNA), and thus fails to account for over half a century of scientific progress.
  • Richard Carrier, "Bad Science, Worse Philosophy", Addendum B, at The Secular Web (Internet Infidels: 2000)

• Hubble's law predicts that galaxies beyond ...the Hubble distance, recede faster than the speed of light ...this distance is about 14 billion light years. ...galaxies with a redshift of about 1.5—...150 percent longer than the laboratory reference value—are receding at the speed of light. Equivalently, we are receding from those galaxies...
  • Tamara Davis, Charles Lineweaver, "Misconceptions About the Big Bang," Scientific American (March, 2005)

• The radiation of the cosmic microwave background... has a red shift of about 1,000. ...the hot plasma of the early universe ...was receding from our location at about 50 times the speed of light.
  • Tamara Davis, Charles Lineweaver, "Misconceptions About the Big Bang," Scientific American (March, 2005)

• A light beam that is farther than the Hubble distance of 14 billion light-years ...cannot keep up with the stretching space.
  • Tamara Davis, Charles Lineweaver, "Misconceptions About the Big Bang," Scientific American (March, 2005)

• When you are next out of doors on a summer night, turn your head towards the zenith. Almost vertically above you will be shining the brightest star of the northern skies—Vega of the Lyre, twenty-six years away at the speed of light, near enough to the point of no return for us short-lived creatures. Past this blue-white beacon, fifty times as brilliant as our sun, we may send our minds and bodies, but never our hearts.
  For no man will ever turn homewards beyond Vega, to greet again those he knew and loved on Earth.
  • Arthur C. Clarke, We'll Never Conquer Space (1960)

• Another topic deserving discussion is Einstein’s modification of Newton’s law of gravitation. In spite of all the excitement it created, Newton’s law of gravitation is not correct! It was modified by Einstein to take into account the theory of relativity. According to Newton, the gravitational effect is instantaneous, that is, if we were to move a mass, we would at once feel a new force because of the new position of that mass; by such means we could send signals at infinite speed. Einstein advanced arguments which suggest that we cannot send signals faster than the speed of light, so the law of gravitation must be wrong. By correcting it to take the delays into account, we have a new law, called Einstein’s law of gravitation. One feature of this new law which is quite easy to understand is this: In the Einstein relativity theory, anything which has energy has mass—mass in the sense that it is attracted gravitationally. Even light, which has an energy, has a “mass.” When a light beam, which has energy in it, comes past the sun there is an attraction on it by the sun. Thus the light does not go straight, but is deflected. During the eclipse of the sun, for example, the stars which are around the sun should appear displaced from where they would be if the sun were not there, and this has been observed.
  • Richard Feynman, The Feynman Lectures on Physics, Vol. I, Ch. 7. The Theory of Gravitation

• For over 200 years the equations of motion enunciated by Newton were believed to describe nature correctly, and the first time that an error in these laws was discovered, the way to correct it was also discovered. Both the error and its correction were discovered by Einstein in 1905.

  Newton’s Second Law, which we have expressed by the equation $F={\frac {d\left({mv}\right)}{dt}}$ was stated with the tacit assumption that m is a constant, but we now know that this is not true, and that the mass of a body increases with velocity. In Einstein’s corrected formula m has the value $m={\frac {{m}_{0}}{\sqrt {1-v^{2}/c^{2}}}}$ where the “rest mass” m₀ represents the mass of a body that is not moving and c is the speed of light, which is about 3×10⁵ km⋅sec⁻¹ or about 186,000 mi⋅sec⁻¹.

  • Richard Feynman, The Feynman Lectures on Physics, Vol. I, Ch. 15. The Special Theory of Relativity

G - L

• The strangest explanation was put forth by an Irish physicist, George Francis Fitzgerald. Perhaps, he said, the ether wind puts pressure on a moving object, causing it to shrink a bit in the direction of motion. To determine the length of a moving object, its length at rest must be multiplied by the following simple formula, in which $\scriptstyle v^{2}$ is the velocity of the object multiplied by itself, $\scriptstyle c^{2}$ is the velocity of light multiplied by itself: $\scriptstyle {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$. ...The speed of light in an unobtainable limit; when this is reached the formula becomes $\scriptstyle {\sqrt {1-{\frac {c^{2}}{c^{2}}}}}$ which reduces to 0. ...In other words, if an object could obtain the speed of light, it would have no length at all in the direction of its motion!
  • Martin Gardner, Relativity Simply Explained (1962) Ch. 2 The Michelson–Morley experiment

• In the special theory of relativity, the speed of light becomes... a new absolute. ...Regardless of the motion of its source, light always moves through space with the same constant speed.
  • Martin Gardner, Relativity Simply Explained (1962) Ch. 3 The Special Theory of Relativity, Part I

• Two ships are passing each other with uniform speed close to that of light. As they pass, a beam of light on the other ship is sent from the ceiling to the floor. There it strikes a mirror and is reflected back to the ceiling again. You will see the path of this light as a V [shape]. If you had sufficiently accurate instruments (of course no such instrument exists), you could clock the time it takes this light beam to traverse the V-shaped path. By dividing the length of the path by the time, you obtain the speed of light. ...an astronaut inside the other ship is doing the same thing [measuring his light beam's speed]. From his point of view... the light simply goes down and up along the same line, obviously a shorter distance than along the V that you observed. ...he also obtains the speed of light. ...But his light path is shorter. ...There is only one possible explanation: his clock is slower. Of course, the situation is perfectly symmetrical. If you send a beam down and up inside your ship, he will see its path as V-shaped. He will deduce that your clock is slower.
  • Martin Gardner, Relativity Simply Explained (1962) Ch. 3 The Special Theory of Relativity, Part I

• If the ships could attain a relative speed equal to that of light, observers on each ship would think the other ship had shrunk to zero in length, acquired an infinite mass, and that time on the other ship had slowed to a full stop! If inertial mass did not vary in this way, then the steady application of force, such as the force supplied by rocket motors, could keep increasing a ship's velocity until it passed the speed of light. ...When the ship has contracted to one-tenth its rest length, its relativistic mass has become ten times as great. ...ten times as much force is required to produce the same increase in speed.
  • Martin Gardner, Relativity Simply Explained (1962) Ch. 4 The Special Theory of Relativity, Part II

• The speed of light can never be reached. If it were reached, the outside observer would find that the ship had shrunk to zero length, had acquired an infinite mass, and was exerting an infinite force with its rocket motors. Astronauts inside the ship would observe no changes in themselves, but they would find the cosmos hurtling backward with the speed of light, cosmic time at a standstill, every star flattened to a disk and infinitely massive.
  • Martin Gardner, Relativity Simply Explained (1962) Ch. 4 The Special Theory of Relativity, Part II

• Nothing in nature or the cosmos is ever completely still — as I write this, several wild Mallards have returned to the Museum courtyard and are creating a frantic spectacle of water and wings as they dive and attack in their annual spring ritual. Further from home, a supermassive black hole at the center of a galaxy 56 million light years from Earth has recently been observed to be spinning at close to the speed of light.
  • Evalyn Gates, "Letter from the director", Explore magazine, Cleveland Museum of Natural History (Spring 2013) p. 4

• The false report that measuring one of the photons immediately affects the other leads to all sorts of unfortunate conclusions. ...the alleged effect ...would violate the requirement of relativity theory that no signal... can travel faster than the speed of light. If it were to do so, it would appear to observers in some states of motion that the signal were traveling backward in time.
  • Murray Gell-Mann, The Quark and the Jaguar (1994) Ch. 12 : Quantum Mechanics and Flapdoodle, pp. 172-173 see EPR paradox.

• [A]fter a close study of the experimental work of Michael Faraday,... James Clerk Maxwell succeeded in uniting electricity and magnetism in the framework of the electromagnetic field. ...Beyond uniting... all... electric and magnetic phenomena in one mathematical framework, Maxwell's theory showed—quite unexpectedly, that electromagnetic disturbances travel at a fixed and never-changing speed that turns out to equal that of light. From this, Maxwell realized that visible light itself is nothing but a particular kind of electromagnetic wave... Maxwell's theory also showed that all electromagnetic waves—visible light among them—are the epitome of the peripatetic traveler. They never stop. They never slow down. Light always travels at light speed.
  • Brian Greene, The Elegant Universe (1999, 2003) Ch. 1 Tied Up with a String.

• The alternative physics is a physics of light. Light is composed of photons, which have no antiparticle. This means that there is no dualism in the world of light. The conventions of relativity say that time slows down as one approaches the speed of light, but if one tries to imagine the point of view of a thing made of light, one must realize that what is never mentioned is that if one moves at the speed of light there is no time whatsoever. There is an experience of time zero. … The only experience of time that one can have is of a subjective time that is created by one's own mental processes, but in relationship to the Newtonian universe there is no time whatsoever. One exists in eternity, one has become eternal. The universe is aging at a staggering rate all around one in this situation, but that is perceived as a fact of this universe—the way we perceive Newtonian physics as a fact of this universe. One has transited into the eternal mode. One is then apart from the moving image; one exists in the completion of eternity.
  • Terence McKenna, "New Maps of Hyperspace" (1984) given at the Berkeley Institute for the Study of Consciousness

• But here is my mark, and there is where I'm supposed to look, and believe me, the power and the pleasure and the emotion of this moment is as constant as the speed of light. It will never be diminished, nor will my appreciation.
  • Tom Hanks, 67th Academy Award Speech (1995)

• Einstein had drawn attention to nonlocality in 1935 in an effort to show that quantum mechanics must be flawed. ...Einstein proposed a thought experiment—now called the EPR experiment—involving two particles that spring from a common source and fly in opposite directions.
  According to the standard model of quantum mechanics, neither particle has a definite position or momentum before it is measured; but by measuring the momentum of one particle, the physicist instantaneously forces the other particle to assume a fixed position... Deriding this effect as "spooky action at a distance," Einstein argued that it violated both common sense and his own theory of special relativity, which prohibits the propagation of effects faster than the speed of light; quantum mechanics must therefore be an incomplete theory. In 1980, however, a group of French physicists carried out a version of the EPR experiment and showed that it did indeed give rise to spooky action. (The reason that the experiment does not violate special relativity is that one cannot exploit nonlocality to transmit information.)
  • John Horgan, The End of Science (1996)

• What Einstein actually said was that nothing can accelerate to the speed of light because its mass would become infinite. Einstein said nothing about entities already traveling at the speed of light or faster.
  • K-PAX (2001)

• In 5 billion years, the expansion of the universe will have progressed to the point where all other galaxies will have receded beyond detection. Indeed, they will be receding faster than the speed of light, so detection will be impossible. Future civilizations will discover science and all its laws, and never know about other galaxies or the cosmic background radiation. They will inevitably come to the wrong conclusion about the universe......We live in a special time, the only time, where we can observationally verify that we live in a special time.
  • Lawrence M. Krauss, "A Universe From Nothing" AAI (2009)

M - R

• The great attraction of cultural anthropology in the past was precisely that it seemed to offer such a richness of independent natural experiments; but unfortunately it is now clear that there has been a great deal of historical continuity and exchange among those "independent" experiments, most of which have felt the strong effect of contact with societies organized as modern states. More important, there has never been a human society with unlimited resources, of three sexes, or the power to read other people's minds, or to be transported great distances at the speed of light. How then are we to know the effect on human social organization and history of the need to scrabble for a living, or of the existence of males and females, or of the power to make our tongues drop manna and so to make the worse appear the better reason? A solution to the epistemological impotence of social theory has been to create a literature of imagination and logic in which the consequences of radical alterations in the conditions of human existence are deduced. It is the literature of science fiction. … [S]cience fiction is the laboratory in which extraordinary social conditions, never possible in actuality, are used to illumine the social and historical norm. … Science fiction stories are the Gedanken experiments of social science.
  • Richard Lewontin "The Last of the Nasties?" New York Review of Books (Feb 29, 1996)

• The laws of economics are subject to the laws of physics. The physical processes that govern this planet and the continued life upon it place as stringent an upper limit on economic growth as the speed of light does on our knowledge of the universe.
  • Ian McDonald River of Gods (2006) Ch. 9 (p. 93).

• Everybody tends to merge his identity with other people at the speed of light. It's called being mass man.
  • Marshall McLuhan, The Education of Mike McManus, TVOntario (28 December 1977).

• Attention spans get very weak at the speed of light, and that goes along with a very week identity.
  • Marshall McLuhan, The Education of Mike McManus, TVOntario (28 December 1977).

• It is hard to understand how this infinitely dense singularity can evaporate into nothing. For matter inside the black hole to leak out into the universe requires that it travel faster than the speed of light.
  • John Moffat, Reinventing Gravity (2008) Ch. 5, Conventional Black Holes, p. 85

• A much faster speed of light in the infant universe solved the horizon problem and therefore explained the overall smoothness of the temperatures of the CMB radiation, because light now traveled extremely quickly between all parts of the expanding but not inflating universe.
  • John Moffat, Reinventing Gravity (2008) Ch. 6, Inflation And Variable Speed Of Light (VSL), p. 100

• Inflation itself proceeds at a speed faster than the measured speed of light.
  • John Moffat, Reinventing Gravity (2008) Ch. 6, Inflation And Variable Speed Of Light (VSL), p. 102

• Minkowski's idea and the solution of the twin paradox can best be explained by means of an analogy between space and spacetime... Time as a fourth dimension rests vertically on the other three—just as in space the vertical juts out of the two-dimensional plane as a third dimension. Distances through spacetime comprise four dimensions, just as space has three. The more you go in one direction, the less is left for the others. When a rigid body is at rest and does not move in any of the three dimensions, all of its motion takes place on the time axis. It simply grows older. ...The faster he moves away from his frame of reference... and covers more distance in the three dimensions of space, the less of his motion through spacetime as a whole is left over for the dimension of time. ...Whatever goes into space is deducted from time. ...In comparison with the distances light travels, all distances in the dimensions of space, even those involving airplane travel, are so very small that we essentially move only along the time axis, and we age continually. Only if we are able to move away from our frame of reference very quickly, like the traveling twin... would the elapsed time shrink to near zero, as it approached the speed of light. Light itself... covers its entire distance through spacetime only in the three dimensions of space... Nothing remains for the additional dimension... the dimension of time... Because light particles do not move in time, but with time, it can be said that they do not age. For them "now" means the same thing as "forever." They always "live" in the moment. Since for all practical purposes we do not move in the dimensions of space, but are at rest in space, we move only along the time axis. This is precisely the reason we feel the passage of time. Time virtually attaches to us.
  • Jürgen Neffe, Einstein: A Biography (1956)

• Einstein's famous theory of relativity states that while phenomena appear different to someone close to a black hole, traveling close to the speed of light, or in a falling elevator here on earth, scientists in profoundly different environments will nevertheless always discover the same underlying laws of nature.
  • F. David Peat, From Certainty to Uncertainty (2002)

• Communication becomes the defining characteristic of homo sapiens; we are the species that speaks. We utter the words that create our world, and have learned to take our words and translate them into the ethereal play of zeros and ones, lay them out, at the speed of light, first on a wire, then a radio wave, and lately, on a beam of light, so that the voice, once constrained by mouth and ear, now straddles the entire planet in thirty millionths of a second, messages pinging back and forth, not unlike the meeting points of a synaptic gap, using photons as neurotransmitters, and each network router the equivalent of a synapic junction, deciding whether to activate or extinguish each message that crosses the continents, connected now in a seamless, endless web of knowledge, more than two billion pages, more than any one of us could ever read or know, the collected and collective intelligence of a species that seems to have made information the central mystery of culture, the project of civilization, and the goal of being.
  • Mark Pesce, from the video Becoming Transhuman (2001)

• * The principle of the limiting character of the velocity of light. This statement... is not an arbitrary assumption but a physical law based on experience. In making this statement, physics does not commit the fallacy of regarding absence of knowledge as evidence for knowledge to the contrary. It is not absence of knowledge of faster signals, but positive experience which has taught us that the velocity of light cannot be exceeded. For all physical processes the velocity of light has the property of an infinite velocity. In order to accelerate a body to the velocity of light, an infinite amount of energy would be required, and it is therefore physically impossible for any object to obtain this speed. This result was confirmed by measurements performed on electrons. The kinetic energy of a mass point grows more rapidly than the square of its velocity, and would become infinite for the speed of light.
  • Hans Reichenbach, The Philosophy of Space and Time (1928, tr. 1957) § 32

S - Z

• Once we overcome our fear of being tiny, we find ourselves on the threshold of a vast and awesome Universe that utterly dwarfs—in time, in space, and in potential—the tidy anthropocentric proscenium of our ancestors. We gaze across billions of light-years of space to view the Universe shortly after the Big Bang, and plumb the fine structure of matter. We peer down into the core of our planet, and the blazing interior of our star. We read the genetic language in which is written the diverse skills and propensities of every being on Earth. We uncover hidden chapters in the record of our origins, and with some anguish better understand our nature and prospects. We invent and refine agriculture, without which almost all of us would starve to death. We create medicines and vaccines that save the lives of billions. We communicate at the speed of light, and whip around the Earth in an hour and a half. We have sent dozens of ships to more than seventy worlds, and four spacecraft to the stars. We are right to rejoice in our accomplishments, to be proud that our species has been able to see so far, and to judge our merit in part by the very science that has so deflated our pretensions.
  • Carl Sagan, Pale Blue Dot: A Vision of the Human Future in Space (1994) p. 53

• The idea that as I walk in this direction my watch goes slightly slower and I am contracted in the direction of motion and my mass has increased slightly does not correspond to everyday experience. ...the reason that it does not correspond to common sense is that we are not in the habit of traveling close to the speed of light. We may one day be in that habit, and then the Lorentz transformations will be natural, intuitive.
  • Carl Sagan, The Varieties of Scientific Experience: A Personal View of the Search for God (2006)

• Some of the effects predicted by the theory [of loop quantum gravity] appear to be in conflict with one of the principles of Einstein's special theory of relativity... that the speed of light is a universal constant. ...Photons of higher energy travel slightly slower than low-energy photons. ...the principle of [general] relativity is preserved but Einstein's special theory of relativity requires modification. ...A photon can have an energy-dependent speed without violating the principle of [general] relativity!
  • Lee Smolin, "Loop Quantum Gravity," The New Humanists: Science at the Edge (2003)

• 'New Science' must demolish the Constants, supposedly Absolute,
  ... for example: Speed of Light must be increased a Billionfold,
  in order to achieve Extraterrestrial Communication.
  • B. B. Stoller, Biologizing the Universe (1983)

• One thing leads to another, and soon you are searching for answers to basic questions.
  Another time during lectures on Classical Logic, we were introduced to an “experimentum crucis”. It was illustrated by the deciding experiment of Fizeau on the speed of light in water as compared to its speed in air. Since wave theory predicts that speed in water is less, and corpuscular theory (with point particles) predicts it would be faster, this is supposed to have selected the wave theory is correct. But then how would one accommodate the photoelectric effect? Then it turns out that if the “corpuscle” of light had a finite size, corpuscular theory also predicts lower speed of light in water. But then one can ask how come photoelectric emission being prompt even in feeble light, how could the energy of a photon spread over π(λ/2)2 act as a whole and liberate a single photoelectron! This leads us to question the square of the amplitude being interpreted as the probability of the particle being formed in the immediate vicinity. How do probabilities enter quantum mechanics? Thus the questions (and the quest) go on.
  • George Sudarshan, A Glance Back at Five Decades of Scientific Research, published in Particles and Fields: Classical and Quantum, Journal of Physics: Conference Series 87 (2007), IOP Publishing, p. 1-2.

• The more I learn about light the more I realize, man, we don't know anything about light... It's just bizarre... a particle has its own proper time which slows down as you speed up. But at the speed of light... there's no time. That's bizarre … that we can, right now, as you know, see — interact with the light that has come from the birth of the universe. So … from our point of view, that light traveled for 14 billion years but from the point of view of the light it's the moment of creation.
  • Brian Swimme interview, MeaningofLife.tv (2007)

• Nature may reach the same result in many ways. Like a wave in the physical world, in the infinite ocean of the medium which pervades all, so in the world of organisms, in life, an impulse started proceeds onward, at times, may be, with the speed of light, at times, again, so slowly that for ages and ages it seems to stay, passing through processes of a complexity inconceivable to men, but in all its forms, in all its stages, its energy ever and ever integrally present. A single ray of light from a distant star falling upon the eye of a tyrant in bygone times may have altered the course of his life, may have changed the destiny of nations, may have transformed the surface of the globe, so intricate, so inconceivably complex are the processes in Nature. In no way can we get such an overwhelming idea of the grandeur of Nature than when we consider, that in accordance with the law of the conservation of energy, throughout the Infinite, the forces are in a perfect balance, and hence the energy of a single thought may determine the motion of a universe.
  • Nikola Tesla, "On Light And Other High Frequency Phenomena" A lecture delivered before the Franklin Institute, Philadelphia (24 February 1893), and before the National Electric Light Association, St. Louis (1 March 1893), published in The Electrical review (9 June 1893), p. Page 683; also in The Inventions, Researches And Writings of Nikola Tesla (1894)

• Changing electric fields produce magnetic fields, and changing magnetic fields produce electric fields. Thus the fields can animate one another in turn, giving birth to self-reproducing disturbances that travel at the speed of light. Ever since Maxwell, we understand that these disturbances are what light is.
  • Frank Wilczek, The Lightness of Being – Mass, Ether and the Unification of Forces (2008)

• E = mc² really applies only to isolated bodies at rest. In general, when you have moving bodies, or interacting bodies, energy and mass aren't proportional. E = mc² simply doesn't apply. ...For moving bodies, the correct mass-energy equation is
  $E={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$
  where $v$ is the velocity. For a body at rest $(v=0)$, this becomes E = mc². ...we must consider the special case of particles with zero mass... examples include photons, color gluons, and gravitons. If we attempt to put m = 0 and $v$ = c in our general mass-energy equation, both the numerator and denominator on the right-hand-side vanish, and we get the nonsensical relation E = 0/0. The correct result is that the energy of a photon can take any value. ...The energy E of a photon is proportional to the frequency f of the light it represents. ...they are related by the Planck-Einstein-Schrödinger equation E = hf, where h is Plank's constant.
  • Note: when the velocity $v$ approaches the speed of light c, the denominator approaches 0 thus E approaches infinity, unless m = 0.
  • The Lightness of Being – Mass, Ether and the Unification of Forces (2008) Ch. 3, p. 19 & Appendix A