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130-408: The experiment compared the speed of light in perpendicular directions in an attempt to detect the relative motion of matter, including their laboratory, through the luminiferous aether, or "aether wind" as it was sometimes called. The result was negative, in that Michelson and Morley found no significant difference between the speed of light in the direction of movement through the presumed aether, and

260-442: A paper published in 1865, James Clerk Maxwell proposed that light was an electromagnetic wave and, therefore, travelled at speed c . In 1905, Albert Einstein postulated that the speed of light c with respect to any inertial frame of reference is a constant and is independent of the motion of the light source. He explored the consequences of that postulate by deriving the theory of relativity and, in doing so, showed that

390-417: A sodium flame (for alignment), or white light (for the actual observations), through a half-silvered mirror that was used to split it into two beams traveling at right angles to one another. After leaving the splitter, the beams traveled out to the ends of long arms where they were reflected back into the middle by small mirrors. They then recombined on the far side of the splitter in an eyepiece, producing

520-404: A "complete standstill" by passing it through a Bose–Einstein condensate of the element rubidium . The popular description of light being "stopped" in these experiments refers only to light being stored in the excited states of atoms, then re-emitted at an arbitrarily later time, as stimulated by a second laser pulse. During the time it had "stopped", it had ceased to be light. This type of behaviour

650-505: A further 4–24 minutes for commands to travel from Earth to Mars. Receiving light and other signals from distant astronomical sources takes much longer. For example, it takes 13 billion (13 × 10 ) years for light to travel to Earth from the faraway galaxies viewed in the Hubble Ultra-Deep Field images. Those photographs, taken today, capture images of the galaxies as they appeared 13 billion years ago, when

780-467: A letter to Lord Rayleigh in August 1887: The Experiments on the relative motion of the earth and ether have been completed and the result decidedly negative. The expected deviation of the interference fringes from the zero should have been 0.40 of a fringe – the maximum displacement was 0.02 and the average much less than 0.01 – and then not in the right place. As displacement is proportional to squares of

910-464: A light year is nearly 10 trillion kilometres or nearly 6 trillion miles. Proxima Centauri , the closest star to Earth after the Sun, is around 4.2 light-years away. Radar systems measure the distance to a target by the time it takes a radio-wave pulse to return to the radar antenna after being reflected by the target: the distance to the target is half the round-trip transit time multiplied by

1040-615: A material-dependent constant. The refractive index of air is approximately 1.0003. Denser media, such as water , glass , and diamond , have refractive indexes of around 1.3, 1.5 and 2.4, respectively, for visible light. In exotic materials like Bose–Einstein condensates near absolute zero, the effective speed of light may be only a few metres per second. However, this represents absorption and re-radiation delay between atoms, as do all slower-than- c speeds in material substances. As an extreme example of light "slowing" in matter, two independent teams of physicists claimed to bring light to

1170-409: A path difference. The path difference is zero only when the interferometer is aligned with or perpendicular to the aether wind, and it reaches a maximum when it is at a 45° angle. The path difference can be any fraction of the wavelength, depending on the angle and speed of the aether wind. To prove the existence of the aether, Michelson and Morley sought to find the "fringe shift". The idea was simple,

1300-461: A pattern of constructive and destructive interference whose transverse displacement would depend on the relative time it takes light to transit the longitudinal vs. the transverse arms. If the Earth is traveling through an aether medium, a light beam traveling parallel to the flow of that aether will take longer to reflect back and forth than would a beam traveling perpendicular to the aether, because

1430-424: A personal level, and the negative result haunted him for the rest of his life. If the same situation is described from the view of an observer co-moving with the interferometer, then the effect of aether wind is similar to the effect experienced by a swimmer, who tries to move with velocity c {\textstyle c} against a river flowing with velocity v {\textstyle v} . In

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1560-584: A result, if something were travelling faster than  c relative to an inertial frame of reference, it would be travelling backwards in time relative to another frame, and causality would be violated. In such a frame of reference, an "effect" could be observed before its "cause". Such a violation of causality has never been recorded, and would lead to paradoxes such as the tachyonic antitelephone . There are situations in which it may seem that matter, energy, or information-carrying signal travels at speeds greater than  c , but they do not. For example, as

1690-421: A rotating cogwheel with 720 notches that could be rotated at a variable speed several times a second. (Figure 1) Fizeau increased the rotation speed of the cogwheel until light passing through one notch of the cogwheel would be completely eclipsed by the adjacent tooth. At 12.6 rotations per second, the light was eclipsed. At twice this speed (25.2 rotations per second), it was again visible as it passed through

1820-402: A series of experiments from 1872 to 1876. The goal was to obtain a value for the speed of light accurate to one part in a thousand. Cornu's equipment allowed him to monitor high orders of extinction, up to the 21st order. Instead of estimating the intensity minimum of the light being blocked by the adjacent teeth, a relatively inaccurate procedure, Cornu made pairs of observations on either side of

1950-484: A speed of around 30 km/s (18.64 mi/s), or 108,000 km/h (67,000 mph). The Earth is in motion, so two main possibilities were considered: (1) The aether is stationary and only partially dragged by Earth (proposed by Augustin-Jean Fresnel in 1818), or (2) the aether is completely dragged by Earth and thus shares its motion at Earth's surface (proposed by Sir George Stokes, 1st Baronet in 1844). In addition, James Clerk Maxwell (1865) recognized

2080-412: A time dilation factor of γ  = 2 occurs at a relative velocity of 86.6% of the speed of light ( v  = 0.866  c ). Similarly, a time dilation factor of γ  = 10 occurs at 99.5% the speed of light ( v  = 0.995  c ). The results of special relativity can be summarized by treating space and time as a unified structure known as spacetime (with  c relating

2210-459: A time interval of 1 ⁄ 299 792 458 of a second", fixing the value of the speed of light at 299 792 458  m/s by definition, as described below . Consequently, accurate measurements of the speed of light yield an accurate realization of the metre rather than an accurate value of c . Outer space is a convenient setting for measuring the speed of light because of its large scale and nearly perfect vacuum . Typically, one measures

2340-452: A vacuum, it was assumed that even a vacuum must be filled with aether. Because the speed of light is so great, and because material bodies pass through the aether without obvious friction or drag, it was assumed to have a highly unusual combination of properties. Designing experiments to investigate these properties was a high priority of 19th-century physics. Earth orbits around the Sun at

2470-484: Is a universal physical constant that is exactly equal to 299,792,458 metres per second (approximately 300,000 kilometres per second; 186,000 miles per second; 671 million miles per hour). According to the special theory of relativity , c is the upper limit for the speed at which conventional matter or energy (and thus any signal carrying information ) can travel through space . All forms of electromagnetic radiation , including visible light , travel at

2600-465: Is denoted by Δ λ {\displaystyle \Delta \lambda } because the beams are out of phase by a some number of wavelengths ( λ {\displaystyle \lambda } ). To visualise this, consider taking the two beam paths along the longitudinal and transverse plane, and lying them straight (an animation of this is shown at minute 11:00, The Mechanical Universe, episode 41 ). One path will be longer than

2730-424: Is described as a type of electromagnetic wave . The classical behaviour of the electromagnetic field is described by Maxwell's equations , which predict that the speed  c with which electromagnetic waves (such as light) propagate in vacuum is related to the distributed capacitance and inductance of vacuum, otherwise respectively known as the electric constant ε 0 and the magnetic constant μ 0 , by

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2860-421: Is discussed in the propagation of light in a medium section below, many wave velocities can exceed  c . The phase velocity of X-rays through most glasses can routinely exceed c , but phase velocity does not determine the velocity at which waves convey information. If a laser beam is swept quickly across a distant object, the spot of light can move faster than  c , although the initial movement of

2990-427: Is frame-independent, because it is impossible to measure the one-way speed of light (for example, from a source to a distant detector) without some convention as to how clocks at the source and at the detector should be synchronized. By adopting Einstein synchronization for the clocks, the one-way speed of light becomes equal to the two-way speed of light by definition. The special theory of relativity explores

3120-459: Is generally microscopically true of all transparent media which "slow" the speed of light. In transparent materials, the refractive index generally is greater than 1, meaning that the phase velocity is less than c . In other materials, it is possible for the refractive index to become smaller than   1 for some frequencies; in some exotic materials it is even possible for the index of refraction to become negative. The requirement that causality

3250-539: Is given by To find the path difference, simply multiply by c {\displaystyle c} ; Δ λ 1 = 2 L ( 1 1 − v 2 c 2 − 1 1 − v 2 c 2 ) {\displaystyle \Delta {\lambda }_{1}=2L\left({\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}-{\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\right)} The path difference

3380-505: Is important in determining how a light wave travels through a material or from one material to another. It is often represented in terms of a refractive index . The refractive index of a material is defined as the ratio of c to the phase velocity  v p in the material: larger indices of refraction indicate lower speeds. The refractive index of a material may depend on the light's frequency, intensity, polarization , or direction of propagation; in many cases, though, it can be treated as

3510-486: Is impossible for signals or energy to travel faster than  c . One argument for this follows from the counter-intuitive implication of special relativity known as the relativity of simultaneity . If the spatial distance between two events A and B is greater than the time interval between them multiplied by  c then there are frames of reference in which A precedes B, others in which B precedes A, and others in which they are simultaneous. As

3640-548: Is inserted into the above formula for T ℓ {\textstyle T_{\ell }} , then the light propagation time in the longitudinal direction becomes equal to that in the transverse direction: However, length contraction is only a special case of the more general relation, according to which the transverse length is larger than the longitudinal length by the ratio γ {\textstyle \gamma } . This can be achieved in many ways. If L 1 {\textstyle L_{1}}

3770-500: Is known as the Lorentz factor and is given by γ = (1 − v / c ) , where v is the speed of the object. The difference of γ from   1 is negligible for speeds much slower than  c , such as most everyday speeds – in which case special relativity is closely approximated by Galilean relativity  – but it increases at relativistic speeds and diverges to infinity as v approaches c . For example,

3900-491: Is not violated implies that the real and imaginary parts of the dielectric constant of any material, corresponding respectively to the index of refraction and to the attenuation coefficient , are linked by the Kramers–Kronig relations . In practical terms, this means that in a material with refractive index less than 1, the wave will be absorbed quickly. A pulse with different group and phase velocities (which occurs if

4030-559: Is observed, so information cannot be transmitted in this manner. Another quantum effect that predicts the occurrence of faster-than-light speeds is called the Hartman effect : under certain conditions the time needed for a virtual particle to tunnel through a barrier is constant, regardless of the thickness of the barrier. This could result in a virtual particle crossing a large gap faster than light. However, no information can be sent using this effect. So-called superluminal motion

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4160-473: Is possible for a particle to travel through a medium faster than the phase velocity of light in that medium (but still slower than c ). When a charged particle does that in a dielectric material, the electromagnetic equivalent of a shock wave , known as Cherenkov radiation , is emitted. The speed of light is of relevance to telecommunications : the one-way and round-trip delay time are greater than zero. This applies from small to astronomical scales. On

4290-401: Is seen in certain astronomical objects, such as the relativistic jets of radio galaxies and quasars . However, these jets are not moving at speeds in excess of the speed of light: the apparent superluminal motion is a projection effect caused by objects moving near the speed of light and approaching Earth at a small angle to the line of sight: since the light which was emitted when the jet

4420-450: Is some number of wavelengths, and λ {\displaystyle \lambda } which is a single wavelength. As can be seen by this relation, fringe shift n is a unitless quantity. Since L  ≈ 11 meters and λ ≈ 500 nanometers , the expected fringe shift was n  ≈ 0.44. The negative result led Michelson to the conclusion that there is no measurable aether drift. However, he never accepted this on

4550-424: Is the moving longitudinal length and L 2 {\textstyle L_{2}} the moving transverse length, L 1 ′ = L 2 ′ {\textstyle L'_{1}=L'_{2}} being the rest lengths, then it is given: φ {\textstyle \varphi } can be arbitrarily chosen, so there are infinitely many combinations to explain

4680-1609: Is true (if the velocity of the aether is small relative to the speed of light), then the expression can be simplified using a first order binomial expansion; ( 1 − x ) n ≈ 1 − n x {\displaystyle (1-x)^{n}\approx {1-nx}} So, rewriting the above in terms of powers; Δ λ 1 = 2 L ( ( 1 − v 2 c 2 ) − 1 − ( 1 − v 2 c 2 ) − 1 / 2 ) {\displaystyle \Delta {\lambda }_{1}=2L\left(\left({1-{\frac {v^{2}}{c^{2}}}}\right)^{-1}-\left(1-{\frac {v^{2}}{c^{2}}}\right)^{-1/2}\right)} Applying binomial simplification; Δ λ 1 = 2 L ( ( 1 + v 2 c 2 ) − ( 1 + v 2 2 c 2 ) ) = 2 L v 2 2 c 2 {\displaystyle \Delta {\lambda }_{1}=2L\left((1+{\frac {v^{2}}{c^{2}}})-(1+{\frac {v^{2}}{2c^{2}}})\right)={2L}{\frac {v^{2}}{2c^{2}}}} Therefore; Δ λ 1 = L v 2 c 2 {\displaystyle \Delta {\lambda }_{1}={L}{\frac {v^{2}}{c^{2}}}} It can be seen from this derivation that aether wind manifests as

4810-559: The Deep Space Network determine distances to the Moon, planets and spacecraft, respectively, by measuring round-trip transit times. There are different ways to determine the value of c . One way is to measure the actual speed at which light waves propagate, which can be done in various astronomical and Earth-based setups. It is also possible to determine c from other physical laws where it appears, for example, by determining

4940-439: The Lorentz factor . This hypothesis was partly motivated by Oliver Heaviside 's discovery in 1888 that electrostatic fields are contracting in the line of motion. But since there was no reason at that time to assume that binding forces in matter are of electric origin, length contraction of matter in motion with respect to the aether was considered an ad hoc hypothesis . If length contraction of L {\textstyle L}

5070-615: The United States Naval Academy in Annapolis, Michelson conducted his first known light speed experiments as a part of a classroom demonstration. In 1881, he left active U.S. Naval service while in Germany concluding his studies. In that year, Michelson used a prototype experimental device to make several more measurements. The device he designed, later known as a Michelson interferometer , sent yellow light from

5200-472: The electromagnetic nature of light and developed what are now called Maxwell's equations , but these equations were still interpreted as describing the motion of waves through an aether, whose state of motion was unknown. Eventually, Fresnel's idea of an (almost) stationary aether was preferred because it appeared to be confirmed by the Fizeau experiment (1851) and the aberration of star light . According to

5330-400: The geometrized unit system where c = 1 . Using these units, c does not appear explicitly because multiplication or division by   1 does not affect the result. Its unit of light-second per second is still relevant, even if omitted. The speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of the inertial frame of reference of

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5460-515: The impedance of free space . This article uses c exclusively for the speed of light in vacuum. Since 1983, the constant c has been defined in the International System of Units (SI) as exactly 299 792 458  m/s ; this relationship is used to define the metre as exactly the distance that light travels in vacuum in 1 ⁄ 299 792 458 of a second. By using the value of c , as well as an accurate measurement of

5590-459: The local speed of light is constant and equal to  c , but the speed of light can differ from  c when measured from a remote frame of reference, depending on how measurements are extrapolated to the region. It is generally assumed that fundamental constants such as  c have the same value throughout spacetime, meaning that they do not depend on location and do not vary with time. However, it has been suggested in various theories that

5720-430: The printed circuit board refracts and slows down signals. Processors must therefore be placed close to each other, as well as memory chips, to minimize communication latencies, and care must be exercised when routing wires between them to ensure signal integrity . If clock frequencies continue to increase, the speed of light may eventually become a limiting factor for the internal design of single chips . Given that

5850-400: The quantum states of two particles that can be entangled . Until either of the particles is observed, they exist in a superposition of two quantum states. If the particles are separated and one particle's quantum state is observed, the other particle's quantum state is determined instantaneously. However, it is impossible to control which quantum state the first particle will take on when it

5980-467: The speed of light in air. Subsequent experiments performed by Marie Alfred Cornu from 1872 to 1876 improved the methodology and made more accurate measurements. In 1848–49, Hippolyte Fizeau determined the speed of light using an intense light source at the bell tower of his father's holiday home in Suresnes , and a mirror 8,633 meters away on Montmartre . The light source was interrupted by

6110-414: The speed of light may have changed over time . No conclusive evidence for such changes has been found, but they remain the subject of ongoing research. It is generally assumed that the two-way speed of light is isotropic , meaning that it has the same value regardless of the direction in which it is measured. Observations of the emissions from nuclear energy levels as a function of the orientation of

6240-774: The y direction (assuming equal-length arms) and v T 3 {\textstyle vT_{3}} in the x direction. This inclined travel path follows from the transformation from the interferometer rest frame to the aether rest frame. Therefore, the Pythagorean theorem gives the actual beam travel distance of L 2 + ( v T 3 ) 2 {\textstyle {\sqrt {L^{2}+\left(vT_{3}\right)^{2}}}} . Thus c T 3 = L 2 + ( v T 3 ) 2 {\textstyle cT_{3}={\sqrt {L^{2}+\left(vT_{3}\right)^{2}}}} and consequently

6370-499: The Earth with speeds proportional to their distances. Beyond a boundary called the Hubble sphere , the rate at which their distance from Earth increases becomes greater than the speed of light. These recession rates, defined as the increase in proper distance per cosmological time , are not velocities in a relativistic sense. Faster-than-light cosmological recession speeds are only a coordinate artifact. In classical physics , light

6500-632: The Latin celeritas (meaning 'swiftness, celerity'). In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used c for a different constant that was later shown to equal √ 2 times the speed of light in vacuum. Historically, the symbol V was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1894, Paul Drude redefined c with its modern meaning. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to c , which by then had become

6630-534: The Michelson–Morley null result. For instance, if φ = 1 {\textstyle \varphi =1} the relativistic value of length contraction of L 1 {\textstyle L_{1}} occurs, but if φ = 1 / γ {\textstyle \varphi =1/\gamma } then no length contraction but an elongation of L 2 {\textstyle L_{2}} occurs. This hypothesis

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6760-434: The advantage which radio waves travelling at near to the speed of light through air have over comparatively slower fibre optic signals. Similarly, communications between the Earth and spacecraft are not instantaneous. There is a brief delay from the source to the receiver, which becomes more noticeable as distances increase. This delay was significant for communications between ground control and Apollo 8 when it became

6890-591: The amount necessary to compensate for the angle discrepancy of the two beams. A first step to explaining the Michelson and Morley experiment's null result was found in the FitzGerald–Lorentz contraction hypothesis , now simply called length contraction or Lorentz contraction, first proposed by George FitzGerald (1889) in a letter to same journal that published the Michelson-Morley paper, as "almost

7020-544: The apparatus was assembled in a closed room in the basement of the heavy stone dormitory, eliminating most thermal and vibrational effects. Vibrations were further reduced by building the apparatus on top of a large block of sandstone (Fig. 1), about a foot thick and five feet (1.5 m) square, which was then floated in a circular trough of mercury. They estimated that effects of about 0.01 fringe would be detectable. Michelson and Morley and other early experimentalists using interferometric techniques in an attempt to measure

7150-428: The beam travel time T 3 {\textstyle T_{3}} as mentioned above. The classical analysis predicted a relative phase shift between the longitudinal and transverse beams which in Michelson and Morley's apparatus should have been readily measurable. What is not often appreciated (since there was no means of measuring it), is that motion through the hypothetical aether should also have caused

7280-587: The consequences of this invariance of c with the assumption that the laws of physics are the same in all inertial frames of reference. One consequence is that c is the speed at which all massless particles and waves, including light, must travel in vacuum. Special relativity has many counterintuitive and experimentally verified implications. These include the equivalence of mass and energy ( E = mc ) , length contraction (moving objects shorten), and time dilation (moving clocks run more slowly). The factor  γ by which lengths contract and times dilate

7410-410: The development of the Lorentz transformation and special relativity . After the "failed" experiment Michelson and Morley ceased their aether drift measurements and started to use their newly developed technique to establish the wavelength of light as a standard of length . The beam travel time in the longitudinal direction can be derived as follows: Light is sent from the source and propagates with

7540-399: The device to turn with close to zero friction, so that once having given the sandstone block a single push it would slowly rotate through the entire range of possible angles to the "aether wind", while measurements were continuously observed by looking through the eyepiece. The hypothesis of aether drift implies that because one of the arms would inevitably turn into the direction of the wind at

7670-409: The direction and the speed of the motion. At any given point on the Earth's surface, the magnitude and direction of the wind would vary with time of day and season. By analyzing the return speed of light in different directions at various different times, it was thought to be possible to measure the motion of the Earth relative to the aether. The expected relative difference in the measured speed of light

7800-454: The distance between two objects in a frame of reference with respect to which both are moving (their closing speed ) may have a value in excess of  c . However, this does not represent the speed of any single object as measured in a single inertial frame. Certain quantum effects appear to be transmitted instantaneously and therefore faster than c , as in the EPR paradox . An example involves

7930-479: The emitting nuclei in a magnetic field (see Hughes–Drever experiment ), and of rotating optical resonators (see Resonator experiments ) have put stringent limits on the possible two-way anisotropy . According to special relativity, the energy of an object with rest mass m and speed v is given by γmc , where γ is the Lorentz factor defined above. When v is zero, γ is equal to one, giving rise to

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8060-418: The equation In modern quantum physics , the electromagnetic field is described by the theory of quantum electrodynamics (QED). In this theory, light is described by the fundamental excitations (or quanta) of the electromagnetic field, called photons . In QED, photons are massless particles and thus, according to special relativity, they travel at the speed of light in vacuum. Extensions of QED in which

8190-512: The equatorial circumference of the Earth is about 40 075  km and that c is about 300 000  km/s , the theoretical shortest time for a piece of information to travel half the globe along the surface is about 67 milliseconds. When light is traveling in optical fibre (a transparent material ) the actual transit time is longer, in part because the speed of light is slower by about 35% in optical fibre, depending on its refractive index n . Straight lines are rare in global communications and

8320-412: The expected velocity of the Earth's motion in orbit and "certainly less than one-fourth". Although this small "velocity" was measured, it was considered far too small to be used as evidence of speed relative to the aether, and it was understood to be within the range of an experimental error that would allow the speed to actually be zero. For instance, Michelson wrote about the "decidedly negative result" in

8450-495: The famous E = mc formula for mass–energy equivalence. The γ factor approaches infinity as v approaches  c , and it would take an infinite amount of energy to accelerate an object with mass to the speed of light. The speed of light is the upper limit for the speeds of objects with positive rest mass, and individual photons cannot travel faster than the speed of light. This is experimentally established in many tests of relativistic energy and momentum . More generally, it

8580-560: The first crewed spacecraft to orbit the Moon : for every question, the ground control station had to wait at least three seconds for the answer to arrive. The communications delay between Earth and Mars can vary between five and twenty minutes depending upon the relative positions of the two planets. As a consequence of this, if a robot on the surface of Mars were to encounter a problem, its human controllers would not be aware of it until approximately 4–24 minutes later. It would then take

8710-414: The frame of reference in which their speed is measured. In the theory of relativity , c interrelates space and time and appears in the famous mass–energy equivalence , E = mc . In some cases, objects or waves may appear to travel faster than light (e.g., phase velocities of waves, the appearance of certain high-speed astronomical objects , and particular quantum effects ). The expansion of

8840-453: The fringes of the interference pattern should shift when rotating it by 90° as the two beams have exchanged roles. To find the fringe shift, subtract the path difference in first orientation by the path difference in the second, then divide by the wavelength , λ {\displaystyle \lambda } , of light; Note the difference between Δ λ {\displaystyle \Delta \lambda } , which

8970-435: The fundamental tests of special relativity . Physics theories of the 19th century assumed that just as surface water waves must have a supporting substance, i.e., a "medium", to move across (in this case water), and audible sound requires a medium to transmit its wave motions (such as air or water), so light must also require a medium, the " luminiferous aether ", to transmit its wave motions. Because light can travel through

9100-416: The group velocity to become infinite or negative, with pulses travelling instantaneously or backwards in time. None of these options allow information to be transmitted faster than c . It is impossible to transmit information with a light pulse any faster than the speed of the earliest part of the pulse (the front velocity). It can be shown that this is (under certain assumptions) always equal to c . It

9230-421: The incorrect expression because he overlooked the increase in path length in the rest frame of the aether. This was corrected by Alfred Potier (1882) and Hendrik Lorentz (1886). The derivation in the transverse direction can be given as follows (analogous to the derivation of time dilation using a light clock ): The beam is propagating at the speed of light c {\textstyle c} and hits

9360-407: The increase in elapsed time from traveling against the aether wind is more than the time saved by traveling with the aether wind. Michelson expected that the Earth's motion would produce a fringe shift equal to 0.04 fringes—that is, of the separation between areas of the same intensity. He did not observe the expected shift; the greatest average deviation that he measured (in the northwest direction)

9490-429: The individual crests and troughs of a plane wave (a wave filling the whole space, with only one frequency ) propagate is called the phase velocity   v p . A physical signal with a finite extent (a pulse of light) travels at a different speed. The overall envelope of the pulse travels at the group velocity   v g , and its earliest part travels at the front velocity   v f . The phase velocity

9620-463: The intense work of Michelson during the preparation of the experiments. In 1886, Michelson and Morley successfully confirmed Fresnel's drag coefficient – this result was also considered as a confirmation of the stationary aether concept. This result strengthened their hope of finding the aether wind. Michelson and Morley created an improved version of the Michelson experiment with more than enough accuracy to detect this hypothetical effect. The experiment

9750-416: The intensity minima, averaging the values obtained with the wheel spun clockwise and counterclockwise. An electric circuit recorded the wheel rotations on a chronograph chart, which enabled precise rate comparisons against the observatory clock. A telegraph key arrangement allowed Cornu to mark the precise moments when he judged that extinction had been entered on this same chart or exited. His final experiment

9880-726: The intensity minimum of the light being blocked by the adjacent teeth. Other sources of error include the measurement of the distance from the wheel to the mirror, and the measurement of the speed of rotation of the wheel. Fizeau's paper appeared in Comptes Rendus Hebdomadaires de séances de l’Academie de Sciences (Paris, Vol. 29 [July–December 1849], pp. 90–92). At the behest of the Paris Observatory under Urbain Le Verrier , Marie Alfred Cornu repeated Fizeau's 1848 toothed wheel measurement in

10010-497: The interferometer was set up in a basement. Because the fringes would occasionally disappear due to vibrations caused by passing horse traffic, distant thunderstorms and the like, an observer could easily "get lost" when the fringes returned to visibility. The advantages of white light, which produced a distinctive colored fringe pattern, far outweighed the difficulties of aligning the apparatus due to its low coherence length . As Dayton Miller wrote, "White light fringes were chosen for

10140-482: The longitudinal direction the swimmer first moves upstream, so his velocity is diminished due to the river flow to c − v {\textstyle c-v} . On his way back moving downstream, his velocity is increased to c + v {\textstyle c+v} . This gives the beam travel times T 1 {\textstyle T_{1}} and T 2 {\textstyle T_{2}} as mentioned above. In

10270-502: The massive photon is described by Proca theory , the experimental upper bound for its mass is about 10 grams ; if photon mass is generated by a Higgs mechanism , the experimental upper limit is less sharp, m ≤ 10   eV/ c   (roughly 2 × 10  g). Another reason for the speed of light to vary with its frequency would be the failure of special relativity to apply to arbitrarily small scales, as predicted by some proposed theories of quantum gravity . In 2009,

10400-428: The mirror at time T 1 {\textstyle T_{1}} and thus travels the distance c T 1 {\textstyle cT_{1}} . At this time, the mirror has traveled the distance v T 1 {\textstyle vT_{1}} . Thus c T 1 = L + v T 1 {\textstyle cT_{1}=L+vT_{1}} and consequently

10530-420: The mirror at time T 3 {\textstyle T_{3}} , traveling the distance c T 3 {\textstyle cT_{3}} . At the same time, the mirror has traveled the distance v T 3 {\textstyle vT_{3}} in the x direction. So in order to hit the mirror, the travel path of the beam is L {\textstyle L} in

10660-499: The most famous failed experiment in history. Instead of providing insight into the properties of the aether, Michelson and Morley's article in the American Journal of Science reported the measurement to be as small as one-fortieth of the expected displacement (Fig. 7), but "since the displacement is proportional to the square of the velocity" they concluded that the measured velocity was "probably less than one-sixth" of

10790-406: The next notch. At 3 times the speed it was again eclipsed. Given the rotational speed of the wheel and the distance between the wheel and the mirror, Fizeau was able to calculate a value of 2 × 8633m × 720 × 25.2/s = 313,274,304 m/s for the speed of light. Fizeau's value for the speed of light was 4.5% too high. The correct value is 299,792,458 m/s. It was difficult for Fizeau to visually estimate

10920-499: The observation of gamma-ray burst GRB 090510 found no evidence for a dependence of photon speed on energy, supporting tight constraints in specific models of spacetime quantization on how this speed is affected by photon energy for energies approaching the Planck scale . In a medium, light usually does not propagate at a speed equal to c ; further, different types of light wave will travel at different speeds. The speed at which

11050-411: The observations because they consist of a small group of fringes having a central, sharply defined black fringe which forms a permanent zero reference mark for all readings." Use of partially monochromatic light (yellow sodium light) during initial alignment enabled the researchers to locate the position of equal path length, more or less easily, before switching to white light. The mercury trough allowed

11180-421: The observer. This invariance of the speed of light was postulated by Einstein in 1905, after being motivated by Maxwell's theory of electromagnetism and the lack of evidence for motion against the luminiferous aether . It has since been consistently confirmed by many experiments. It is only possible to verify experimentally that the two-way speed of light (for example, from a source to a mirror and back again)

11310-517: The only hypothesis that can reconcile" the apparent contradictions. It was independently also proposed by Hendrik Lorentz (1892). According to this law all objects physically contract by L / γ {\textstyle L/\gamma } along the line of motion (originally thought to be relative to the aether), γ = 1 / 1 − v 2 / c 2 {\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}} being

11440-408: The other hand, some techniques depend on the finite speed of light, for example in distance measurements. In computers , the speed of light imposes a limit on how quickly data can be sent between processors . If a processor operates at 1   gigahertz , a signal can travel only a maximum of about 30 centimetres (1 ft) in a single clock cycle – in practice, this distance is even shorter since

11570-498: The other hand, the much more precise Michelson–Morley experiment (1887) apparently confirmed complete aether dragging and refuted the stationary aether. In addition, the Michelson–Morley null result was further substantiated by the null results of other second-order experiments of different kind, namely the Trouton–Noble experiment (1903) and the experiments of Rayleigh and Brace (1902–1904). These problems and their solution led to

11700-452: The other, this distance is Δ λ {\displaystyle \Delta \lambda } . Alternatively, consider the rearrangement of the speed of light formula c Δ T = Δ λ {\displaystyle c{\Delta }T=\Delta \lambda } . If the relation v 2 / c 2 << 1 {\displaystyle {v^{2}}/{c^{2}}<<1}

11830-439: The parameter c had relevance outside of the context of light and electromagnetism. Massless particles and field perturbations, such as gravitational waves , also travel at speed c in vacuum. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer . Particles with nonzero rest mass can be accelerated to approach c but can never reach it, regardless of

11960-420: The parameter  c is ubiquitous in modern physics, appearing in many contexts that are unrelated to light. For example, general relativity predicts that  c is also the speed of gravity and of gravitational waves , and observations of gravitational waves have been consistent with this prediction. In non-inertial frames of reference (gravitationally curved spacetime or accelerated reference frames ),

12090-403: The phase velocity is not the same for all the frequencies of the pulse) smears out over time, a process known as dispersion . Certain materials have an exceptionally low (or even zero) group velocity for light waves, a phenomenon called slow light . The opposite, group velocities exceeding c , was proposed theoretically in 1993 and achieved experimentally in 2000. It should even be possible for

12220-412: The photon has a mass have been considered. In such a theory, its speed would depend on its frequency, and the invariant speed  c of special relativity would then be the upper limit of the speed of light in vacuum. No variation of the speed of light with frequency has been observed in rigorous testing, putting stringent limits on the mass of the photon. The limit obtained depends on the model used: if

12350-438: The properties of the luminiferous aether, used (partially) monochromatic light only for initially setting up their equipment, always switching to white light for the actual measurements. The reason is that measurements were recorded visually. Purely monochromatic light would result in a uniform fringe pattern. Lacking modern means of environmental temperature control , experimentalists struggled with continual fringe drift even when

12480-411: The refractive index of glass is typically around 1.5, meaning that light in glass travels at ⁠ c / 1.5 ⁠ ≈ 200 000  km/s ( 124 000  mi/s) ; the refractive index of air for visible light is about 1.0003, so the speed of light in air is about 90 km/s (56 mi/s) slower than c . The speed of light in vacuum is usually denoted by a lowercase c , for "constant" or

12610-422: The relative velocities it follows that if the ether does slip past the relative velocity is less than one sixth of the earth’s velocity. From the standpoint of the then current aether models, the experimental results were conflicting. The Fizeau experiment and its 1886 repetition by Michelson and Morley apparently confirmed the stationary aether with partial aether dragging, and refuted complete aether dragging. On

12740-526: The relativity theory as a (halfway) redemption." Michelson–Morley type experiments have been repeated many times with steadily increasing sensitivity. These include experiments from 1902 to 1905, and a series of experiments in the 1920s. More recently, in 2009, optical resonator experiments confirmed the absence of any aether wind at the 10 level. Together with the Ives–Stilwell and Kennedy–Thorndike experiments , Michelson–Morley type experiments form one of

12870-408: The same time that another arm was turning perpendicularly to the wind, an effect should be noticeable even over a period of minutes. The expectation was that the effect would be graphable as a sine wave with two peaks and two troughs per rotation of the device. This result could have been expected because during each full rotation, each arm would be parallel to the wind twice (facing into and away from

13000-451: The second, one can thus establish a standard for the metre. As a dimensional physical constant , the numerical value of c is different for different unit systems. For example, in imperial units , the speed of light is approximately 186 282 miles per second, or roughly 1 foot per nanosecond. In branches of physics in which c appears often, such as in relativity, it is common to use systems of natural units of measurement or

13130-409: The speed at right angles. This result is generally considered to be the first strong evidence against some aether theories , as well as initiating a line of research that eventually led to special relativity , which rules out motion against an aether. Of this experiment, Albert Einstein wrote, "If the Michelson–Morley experiment had not brought us into serious embarrassment, no one would have regarded

13260-413: The speed of light c {\textstyle c} in the aether. It passes through the half-silvered mirror at the origin at T = 0 {\textstyle T=0} . The reflecting mirror is at that moment at distance L {\textstyle L} (the length of the interferometer arm) and is moving with velocity v {\textstyle v} . The beam hits

13390-424: The speed of light fixes the ultimate minimum communication delay . The speed of light can be used in time of flight measurements to measure large distances to extremely high precision. Ole Rømer first demonstrated in 1676 that light does not travel instantaneously by studying the apparent motion of Jupiter 's moon Io . Progressively more accurate measurements of its speed came over the following centuries. In

13520-470: The speed of light. A Global Positioning System (GPS) receiver measures its distance to GPS satellites based on how long it takes for a radio signal to arrive from each satellite, and from these distances calculates the receiver's position. Because light travels about 300 000  kilometres ( 186 000  miles ) in one second, these measurements of small fractions of a second must be very precise. The Lunar Laser Ranging experiment , radar astronomy and

13650-469: The speed of light. As pointed out by Maxwell (1878), only experimental arrangements capable of measuring second order effects would have any hope of detecting aether drift, i.e., effects proportional to v / c . Existing experimental setups, however, were not sensitive enough to measure effects of that size. Michelson had a solution to the problem of how to construct a device sufficiently accurate to detect aether flow. In 1877, while teaching at his alma mater,

13780-487: The speed of light. For many practical purposes, light and other electromagnetic waves will appear to propagate instantaneously, but for long distances and very sensitive measurements, their finite speed has noticeable effects. Much starlight viewed on Earth is from the distant past, allowing humans to study the history of the universe by viewing distant objects. When communicating with distant space probes , it can take minutes to hours for signals to travel. In computing ,

13910-509: The spot is delayed because of the time it takes light to get to the distant object at the speed  c . However, the only physical entities that are moving are the laser and its emitted light, which travels at the speed  c from the laser to the various positions of the spot. Similarly, a shadow projected onto a distant object can be made to move faster than  c , after a delay in time. In neither case does any matter, energy, or information travel faster than light. The rate of change in

14040-444: The standard symbol for the speed of light. Sometimes c is used for the speed of waves in any material medium, and c 0 for the speed of light in vacuum. This subscripted notation, which is endorsed in official SI literature, has the same form as related electromagnetic constants: namely, μ 0 for the vacuum permeability or magnetic constant, ε 0 for the vacuum permittivity or electric constant, and Z 0 for

14170-416: The stationary and the partially dragged aether hypotheses, Earth and the aether are in relative motion, implying that a so-called "aether wind" (Fig. 2) should exist. Although it would be theoretically possible for the Earth's motion to match that of the aether at one moment in time, it was not possible for the Earth to remain at rest with respect to the aether at all times, because of the variation in both

14300-536: The time needed for light to traverse some reference distance in the Solar System , such as the radius of the Earth's orbit. Historically, such measurements could be made fairly accurately, compared to how accurately the length of the reference distance is known in Earth-based units. Fizeau wheel From 1848 to 1849, Hippolyte Fizeau used a toothed wheel apparatus to perform absolute measurements of

14430-413: The transverse direction, the swimmer has to compensate for the river flow by moving at a certain angle against the flow direction, in order to sustain his exact transverse direction of motion and to reach the other side of the river at the correct location. This diminishes his speed to c 2 − v 2 {\textstyle {\sqrt {c^{2}-v^{2}}}} , and gives

14560-770: The travel time T 1 = L / ( c − v ) {\textstyle T_{1}=L/(c-v)} . The same consideration applies to the backward journey, with the sign of v {\textstyle v} reversed, resulting in c T 2 = L − v T 2 {\textstyle cT_{2}=L-vT_{2}} and T 2 = L / ( c + v ) {\textstyle T_{2}=L/(c+v)} . The total travel time T ℓ = T 1 + T 2 {\textstyle T_{\ell }=T_{1}+T_{2}} is: Michelson obtained this expression correctly in 1881, however, in transverse direction he obtained

14690-505: The travel time T 3 = L / c 2 − v 2 {\textstyle T_{3}=L/{\sqrt {c^{2}-v^{2}}}} , which is the same for the backward journey. The total travel time T t = 2 T 3 {\textstyle T_{t}=2T_{3}} is: The time difference between T ℓ {\displaystyle T_{\ell }} and T t {\displaystyle T_{t}}

14820-471: The travel time increases when signals pass through electronic switches or signal regenerators. Although this distance is largely irrelevant for most applications, latency becomes important in fields such as high-frequency trading , where traders seek to gain minute advantages by delivering their trades to exchanges fractions of a second ahead of other traders. For example, traders have been switching to microwave communications between trading hubs, because of

14950-461: The two beams to diverge as they emerged from the interferometer by about 10 radians. For an apparatus in motion, the classical analysis requires that the beam-splitting mirror be slightly offset from an exact 45° if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed. In the relativistic analysis, Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely

15080-475: The units of space and time), and requiring that physical theories satisfy a special symmetry called Lorentz invariance , whose mathematical formulation contains the parameter  c . Lorentz invariance is an almost universal assumption for modern physical theories, such as quantum electrodynamics , quantum chromodynamics , the Standard Model of particle physics , and general relativity . As such,

15210-481: The universe is understood to exceed the speed of light beyond a certain boundary . The speed at which light propagates through transparent materials , such as glass or air, is less than c ; similarly, the speed of electromagnetic waves in wire cables is slower than c . The ratio between c and the speed v at which light travels in a material is called the refractive index n of the material ( n = ⁠ c / v ⁠ ). For example, for visible light,

15340-529: The universe was less than a billion years old. The fact that more distant objects appear to be younger, due to the finite speed of light, allows astronomers to infer the evolution of stars , of galaxies , and of the universe itself. Astronomical distances are sometimes expressed in light-years , especially in popular science publications and media. A light-year is the distance light travels in one Julian year , around 9461 billion kilometres, 5879 billion miles, or 0.3066 parsecs . In round figures,

15470-436: The values of the electromagnetic constants ε 0 and μ 0 and using their relation to c . Historically, the most accurate results have been obtained by separately determining the frequency and wavelength of a light beam, with their product equalling c . This is described in more detail in the "Interferometry" section below. In 1983 the metre was defined as "the length of the path travelled by light in vacuum during

15600-542: The wavelength of light as a standard of length . At this time Michelson was professor of physics at the Case School of Applied Science, and Morley was professor of chemistry at Western Reserve University (WRU), which shared a campus with the Case School on the eastern edge of Cleveland. Michelson suffered a mental health crisis in September 1885, from which he recovered by October 1885. Morley ascribed this breakdown to

15730-435: The wind giving identical readings) and perpendicular to the wind twice. Additionally, due to the Earth's rotation, the wind would be expected to show periodic changes in direction and magnitude during the course of a sidereal day . Because of the motion of the Earth around the Sun, the measured data were also expected to show annual variations. After all this thought and preparation, the experiment became what has been called

15860-498: Was farther away took longer to reach the Earth, the time between two successive observations corresponds to a longer time between the instants at which the light rays were emitted. A 2011 experiment where neutrinos were observed to travel faster than light turned out to be due to experimental error. In models of the expanding universe , the farther galaxies are from each other, the faster they drift apart. For example, galaxies far away from Earth are inferred to be moving away from

15990-428: Was intended to detect interferometric fringe shifts due to speed differences of oppositely propagating light waves through water at rest. The results of such experiments were all negative. This could be explained by using Fresnel's dragging coefficient , according to which the aether and thus light are partially dragged by moving matter. Partial aether-dragging would thwart attempts to measure any first order change in

16120-467: Was later extended by Joseph Larmor (1897), Lorentz (1904) and Henri Poincaré (1905), who developed the complete Lorentz transformation including time dilation in order to explain the Trouton–Noble experiment , the Experiments of Rayleigh and Brace , and Kaufmann's experiments . It has the form It remained to define the value of φ {\textstyle \varphi } , which

16250-476: Was only 0.018 fringes; most of his measurements were much less. His conclusion was that Fresnel's hypothesis of a stationary aether with partial aether dragging would have to be rejected, and thus he confirmed Stokes' hypothesis of complete aether dragging. However, Alfred Potier (and later Hendrik Lorentz ) pointed out to Michelson that he had made an error of calculation, and that the expected fringe shift should have been only 0.02 fringes. Michelson's apparatus

16380-452: Was performed in several periods of concentrated observations between April and July 1887, in the basement of Adelbert Dormitory of WRU (later renamed Pierce Hall, demolished in 1962). As shown in the diagram to the right, the light was repeatedly reflected back and forth along the arms of the interferometer, increasing the path length to 11 m (36 ft). At this length, the drift would be about 0.4 fringes. To make that easily detectable,

16510-482: Was possible with the accuracy required. For instance, the Fizeau wheel could measure the speed of light to perhaps 5% accuracy, which was quite inadequate for measuring directly a first-order 0.01% change in the speed of light. A number of physicists therefore attempted to make measurements of indirect first-order effects not of the speed of light itself, but of variations in the speed of light (see First order aether-drift experiments ). The Hoek experiment , for example,

16640-409: Was quite small, given that the velocity of the Earth in its orbit around the Sun has a magnitude of about one hundredth of one percent of the speed of light. During the mid-19th century, measurements of aether wind effects of first order, i.e., effects proportional to v / c ( v being Earth's velocity, c the speed of light) were thought to be possible, but no direct measurement of the speed of light

16770-507: Was shown by Lorentz (1904) to be unity. In general, Poincaré (1905) demonstrated that only φ = 1 {\textstyle \varphi =1} allows this transformation to form a group , so it is the only choice compatible with the principle of relativity , i.e., making the stationary aether undetectable. Given this, length contraction and time dilation obtain their exact relativistic values. Speed of light The speed of light in vacuum , commonly denoted c ,

16900-577: Was subject to experimental errors far too large to say anything conclusive about the aether wind. Definitive measurement of the aether wind would require an experiment with greater accuracy and better controls than the original. Nevertheless, the prototype was successful in demonstrating that the basic method was feasible. In 1885, Michelson began a collaboration with Edward Morley , spending considerable time and money to confirm with higher accuracy Fizeau's 1851 experiment on Fresnel's drag coefficient, to improve on Michelson's 1881 experiment, and to establish

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