Laboratory method for measuring speed. Measuring the speed of light

Long before scientists measured the speed of light, they had to work hard to define the very concept of "light". One of the first to think about this was Aristotle, who considered light to be a kind of mobile substance that spreads in space. His ancient Roman colleague and follower Lucretius Car insisted on the atomic structure of light.

By the 17th century, two main theories of the nature of light were formed - corpuscular and wave. Newton belonged to the adherents of the first. In his opinion, all light sources emit the smallest particles. In the process of "flight" they form luminous lines - rays. His opponent, the Dutch scientist Christian Huygens, insisted that light is a form of wave motion.

As a result of centuries-old disputes, scientists have come to a consensus: both theories have the right to life, and light is the spectrum of electromagnetic waves visible to the eye.

A bit of history. How was the speed of light measured?

Most scientists of antiquity were convinced that the speed of light is infinite. However, the results of the studies of Galileo and Hooke admitted its limit, which was clearly confirmed in the 17th century by the outstanding Danish astronomer and mathematician Olaf Roemer.

He made his first measurements by observing the eclipses of Io, a satellite of Jupiter, at a time when Jupiter and the Earth were located on opposite sides of the Sun. Roemer recorded that as the Earth moved away from Jupiter at a distance equal to the diameter of the Earth's orbit, the delay time changed. The maximum value was 22 minutes. As a result of calculations, he received a speed of 220,000 km / s.

Fifty years later, in 1728, thanks to the discovery of aberration, the English astronomer J. Bradley "refined" this figure to 308,000 km / s. Later, the speed of light was measured by the French astrophysicists Francois Argo and Leon Foucault, having received 298,000 km / s at the “output”. An even more accurate measurement technique was proposed by the creator of the interferometer, the famous American physicist Albert Michelson.

Michelson's experiment to determine the speed of light

The experiments lasted from 1924 to 1927 and consisted of 5 series of observations. The essence of the experiment was as follows. A light source, a mirror and a rotating octagonal prism were installed on Mount Wilson near Los Angeles, and a reflecting mirror 35 km later on Mount San Antonio. First, light through a lens and a slit fell on a prism rotating with the help of a high-speed rotor (at a speed of 528 rpm).

The participants in the experiments could adjust the rotational speed so that the image of the light source was clearly visible in the eyepiece. Since the distance between the peaks and the frequency of rotation were known, Michelson determined the speed of light - 299796 km / s.

Scientists finally decided on the speed of light in the second half of the 20th century, when masers and lasers were created, which are distinguished by the highest radiation frequency stability. By the beginning of the 1970s, the measurement error had dropped to 1 km/sec. As a result, on the recommendation of the XV General Conference on Weights and Measures, held in 1975, it was decided to consider that the speed of light in vacuum is henceforth equal to 299,792.458 km/sec.

Can we reach the speed of light?

It is obvious that the development of the far corners of the universe is unthinkable without spaceships flying at great speed. Preferably at the speed of light. But is it possible?

The barrier of the speed of light is one of the consequences of the theory of relativity. As you know, an increase in speed requires an increase in energy. The speed of light would require virtually infinite energy.

Alas, the laws of physics are categorically against this. At a spacecraft speed of 300,000 km/sec, particles flying towards it, for example, hydrogen atoms, turn into a deadly source of powerful radiation equal to 10,000 sieverts/sec. It's about the same as being inside the Large Hadron Collider.

According to scientists at Johns Hopkins University, while in nature there is no adequate protection against such a monstrous cosmic radiation. Erosion from the impact of interstellar dust will complete the destruction of the ship.

Another problem with light speed is time dilation. At the same time, aging will become much longer. The visual field will also be distorted, as a result of which the ship's trajectory will pass as if inside a tunnel, at the end of which the crew will see a shining flash. Behind the ship will remain absolute pitch darkness.

So in the near future, humanity will have to limit its high-speed "appetites" to 10% of the speed of light. This means that it will take about 40 years to fly to the nearest star to the Earth - Proxima Centauri (4.22 light years).

The speed of light was first determined by the Danish astronomer Roemer in 1676. Until that time, there were two opposing opinions among scientists. Some believed that the speed of light is infinite. Others, although they considered it very large, nevertheless final. Roemer confirmed the second opinion. He correctly connected the irregularities in the time of eclipses of Jupiter's satellites with the time it takes for light to pass through the diameter of the Earth's orbit around the Sun. He was the first to draw a conclusion about the finite speed of light propagation and determined its magnitude. According to his calculations, the speed of light turned out to be 300870 km / s in modern units. (Data taken from the book: G. Lipson. Great experiments in physics.)

Foucault method

A method of measuring the speed of light, which consists in the successive reflection of a beam of light from a rapidly rotating mirror, then from a second fixed mirror located at an accurately measured distance, and then again from the first mirror, which has had time to turn through some small angle. The speed of light is determined (given the speed of rotation of the first mirror and the distance between the two mirrors) by changing the direction of the three times reflected light beam. Using this method, the speed of light in air was first measured by J. B. L. Foucault in 1862.

In 1878–82 and 1924–26 he made measurements of the speed of light, which for a long time remained unsurpassed in accuracy. In 1881, he experimentally proved and, together with E. W. Morley (1885–87), confirmed with great accuracy the independence of the speed of light from the speed of the Earth.

The operation of the corner reflectors of the optical range is based on the same principle, which is a small trihedral prism made of transparent glass, the edges of which are covered with a thin layer of metal. Such U. o. has a high Sef due to the large a/l ratio. For receiving omnidirectional U. about. use a system of several prisms. Optical U. about. became widespread after the advent of lasers. They are used in navigation, for measuring distances and the speed of light in the atmosphere, in experiments with the moon, and in other applications. in the form of colored glass with many recesses of a tetrahedral shape, they are used as a means of signaling in the road sector and in everyday life.

The famous American scientist Albert Michelson spent most of his life measuring the speed of light.

One day, a scientist was examining the supposed path of a light beam along a railroad track. He wanted to build an even better setup for an even more accurate method of measuring the speed of light. Prior to that, he had already worked on this problem for several years and achieved the most accurate values ​​\u200b\u200bfor that time. Newspaper reporters became interested in the behavior of the scientist and, perplexed, asked what he was doing here. Michelson explained that he was measuring the speed of light.

- What for? – followed question.

"Because it's devilishly interesting," Michelson replied.

And no one could have imagined that Michelson's experiments would become the foundation on which the majestic edifice of the theory of relativity would be built, giving a completely new idea of ​​the physical picture of the world.

Fifty years later, Michelson was still continuing his measurements of the speed of light.

Once the great Einstein asked him the same question:

"Because it's damn interesting!" Michelson and Einstein answered half a century later.

Fizeau method

In 1849, A. Fizeau set up a laboratory experiment to measure the speed of light. The light from the source 5 passed through the interrupter K (the teeth of the rotating wheel) and, reflected from the mirror 3, returned again to the gear wheel. Let us assume that the tooth and the slot of the gear wheel have the same width and the place of the slot on the wheel is occupied by the adjacent tooth. Then the light will be blocked by a tooth and it will become dark in the eyepiece. This will come under the condition that the time of light passing back and forth t=2L/c will be equal to the time of rotation of the gear by half of the slot t2=T/(2N)=1/(2Nv). Here L is the distance from the gear wheel to the mirror; T is the period of rotation of the gear-wheel; N is the number of teeth; v=1/T – rotation frequency. From the equality t1=t2 follows the calculation formula for determining the speed of light by this method:

c=4LNv

Using the rotating shutter method, Fizeau in 1849 obtained the value of the speed of light c = 3.13-10**5 km/s, which was not bad at all for those times. Subsequently, the use of various shutters made it possible to substantially refine the value of the speed of light. So, in 1950, the value of the speed of light (in vacuum) was obtained, equal to:

s = (299 793.1 ± 0.25) km/s.

An ingenious solution to the complex problem of determining the speed of light was found in 1676 by the Danish astronomer Olaf Roemer.

Olaf Roemer, observing the movement of Jupiter's satellites, noticed that during an eclipse, the satellite leaves the shadow region with a periodic delay. Remer explained this by the fact that by the time of the next observation, the Earth is at a different point in its orbit than the previous time, and, consequently, the distance between it and Jupiter is different. The maximum amount by which this distance increases is equal to the diameter of the earth's orbit. And just when the Earth is the most distant from Jupiter, the satellite leaves the shadow with the greatest delay.

Comparing these data, Roemer came to the conclusion that the light from the satellite travels a distance equal to the diameter of the earth's orbit - 299,106 thousand km in 1320 seconds. Such a conclusion not only convinces that the speed of propagation of light cannot be instantaneous, but also allows us to determine the magnitude of the speed; To do this, it is necessary to divide the diameter of the Earth's orbit by the delay time of the satellite.

According to Roemer's calculations, the speed of light propagation was found to be 215,000 km/sec.

Subsequent, more advanced methods for observing the delay time of Jupiter's satellites made it possible to refine this value. The speed of light propagation, according to modern data, is 299,998.9 km / s. For practical calculations, the speed of light in vacuum is assumed to be 300,000 km/sec. The enormous magnitude of the speed of light stunned not only Roemer's contemporaries, but also served as a pretext for denying the corpuscular theory of light.

If light is a stream of corpuscles, then at such a speed of motion their energy should be very high. Impacts of corpuscles when falling on bodies must be felt, i.e. Light must exert pressure!

Following Roemer, the speed of light was measured by James Bradley.

While crossing the River Thames one day, Bradley noticed that while the boat was moving, the wind seemed to blow in a different direction than it really was. This observation probably gave him reason to explain by an analogous phenomenon the apparent movement of fixed stars, called aberration Sveta.

The light of a star reaches the Earth, just as drops of falling rain fall on the windows of a moving car. The motion of the beam of light and the motion of the earth add up.

Therefore, in order for the light from a star located perpendicular to the plane of motion of the Earth to enter the telescope, it must be tilted at a certain angle, which does not depend on the distance to the star, but only on the speed of light and the speed of the Earth (it was already at that time known - 30 km / sec).

By measuring the angle, Bradley found that the speed of light is 308,000 km/sec. Bradley's measurements, like Roemer's, did not resolve the controversial issue of the value of the constant in the law of refraction, since Bradley and Roemer determined the speed of the set not in any medium, but in outer space.

The idea of ​​a new method for measuring the speed of light was proposed by D. Arago. It was carried out in two different ways by I. Fizeau and L. Foucault.

Fizeau in 1849 carefully measured the distance between two points. In the bottom of them, he placed a source of light, and in the other - a mirror, from which the light should be reflected and again return to the source.

In order to determine the speed of propagation of light, it was necessary to very accurately measure the time interval that it takes for light to travel twice the path from the source to the mirror.

The distance from the source located on the outskirts of Paris, Surenay, to the mirror installed in Montmartre was 8633 m. This means that twice the distance was 17266 m. six hundred thousandths of a second.

There were no means for measuring such small time intervals then.

Hence, these measurements should be excluded from the experiment.

A spotting scope was installed in Suresnes, aimed at Paris. From the side, light came from a source through another tube. From the surface of a transparent glass plate placed in a tube at an angle of 45°, the light was partially reflected towards Paris.

In Paris, on Montmartre, another spotting scope was installed, into which light reflected by a transparent plate fell.

Looking through the eyepiece, one could see the light source located behind the side tube. The eyepiece of the tube installed in Montmartre was replaced by a mirror, thanks to which the light returned to Suresnes.

The light reflected by the mirror in Montmartre, meeting a transparent glass plate on the way back inside the tube, was partially reflected from its surface, and the sect, which passed through the plate and the eyepiece of the tube, fell into the observer's eye.

The telescope in Suresnes, in addition to the side tube through which light entered, had a slot in the place where the focus of the objective and the eyepiece was located. A gear wheel passed through the slot, which was set in motion by a clockwork. When the wheel was stationary and set so that the light passed between the teeth, then the eyepiece of the tube could see the light reflected from the mirror in Montmartre.

When the wheel was set in motion, the light disappeared. This happened at the moment when the light, passing between the teeth of the wheel towards Paris, met the tooth on the way back, and not the gap between the teeth.

In order for the light to reappear in the eyepiece, it was necessary to double the number of revolutions of the wheel.

With a further increase in the number of revolutions, the light disappeared again.

In Fizeau's experiments, the gear wheel had 720 teeth. The first disappearance of the set was observed when the wheel made 12.67 revolutions per second.

It made one revolution in a time equal to 1/12.67 sec. In this case, the gap between the teeth was replaced by a tooth. If there are 720 teeth, then there are also 720 gaps. Therefore, the change takes place in a time equal to 1/12.67*2*720 = 1/18245 sec.

During this time, the light traveled twice the distance from Suresnes to Montmartre.

Consequently, its speed was equal to 315 thousand km / s.

Such an ingenious method managed to avoid measurements of small time intervals and still determine the speed of light.

The relatively large distance between the light source and the mirror did not allow any medium to be placed in the path of the light. Fizeau determined the speed of light in air.

The speed of light in other media was determined by Foucault in 1862. In Foucault's experiments, the distance from the source to the mirror was only a few meters. This made it possible to place a tube filled with water in the path of light.

Foucault found that the speed of propagation of light in various media is less than in air. In water, for example, it is equal to the speed of light in air. The results obtained resolved a two-century dispute between the corpuscular and wave theories about the value of the constant in the law of refraction. The correct value in the law of refraction is given by the wave theory of light.

Measurements of the speed of light propagation in various media made it possible to introduce the concept of the optical density of a substance.

List of used literature

1. Simulation modeling. – [Electronic resource] – Access mode: webcache.googleusercontent.com – Access date: April 2014. - Zagl. from the screen.

The first experimental confirmation of the finiteness of the speed of light was given by Roemer in 1676. He discovered that the movement of Io, the largest satellite of Jupiter, does not occur quite regularly in time. It was found that the periodicity of eclipses of Io is violated by Jupiter. Over half a year of observation, the violation of the periodicity of the observed beginning of the eclipse increased, reaching a value of about 20 min. But this is almost equal to the time during which light travels a distance equal to the diameter of the Earth's orbit around the Sun (about 17 minutes).

The speed of light measured by Römer was 2

c Römer = 214300 km/s.

(4)

Römer's method was not very accurate, but it was his calculations that showed astronomers that in order to determine the true movement of the planets and their satellites, it is necessary to take into account the propagation time of the light signal.

Aberration of starlight

In 1725, James Bradley discovered that the star γ The dragon, located at the zenith (i.e., directly overhead), makes an apparent movement with a period of one year in an almost circular orbit with a diameter of 40.5 arc seconds. For stars seen elsewhere in the firmament, Bradley also observed a similar apparent movement - generally elliptical.

The phenomenon observed by Bradley is called aberration. It has nothing to do with the star's own motion. The reason for the aberration is that the value of the speed of light is finite, and the observation is carried out from the Earth moving in orbit at a certain speed v.

Knowing the angle α and the Earth's orbital speed v, you can determine the speed of light c.

Measuring methods based on the use of gears and rotating mirrors

See Berkeley Course in Physics (BCF), Mechanics, p. 337.

Resonant cavity method

It is possible to very accurately determine the frequency at which a certain number of half-wavelengths of electromagnetic radiation fits in a cavity resonator of known dimensions. The speed of light is determined from the ratio

where λ is the wavelength, and ν - frequency of light (see BKF, mechanics, p. 340).

Shoran Method

See BKF, Mechanics, p. 340.

Application of modulated light indicator

See BKF, Mechanics, p. 342.

Methods based on the independent determination of the wavelength and frequency of laser radiation

In 1972, the speed of light was determined from independent measurements of the wavelength λ and frequencies of light ν . The light source was a helium-neon laser ( λ = 3.39 µm). Received value c = λν = 299792458± 1.2 m/s. (see D.V. Sivukhin, Optics, p. 631).

Independence of the speed of light from the movement of the source or receiver

In 1887, the famous experiment of Michelson and Morley finally established that the speed of light does not depend on the direction of its propagation with respect to the Earth. Thus, the then-existing theory of the ether was thoroughly undermined (see BKF, Mechanics, p. 353).

Ballistic hypothesis

The negative result of the experiments of Michelson and Morley could be explained by the so-called ballistic the hypothesis that the speed of light in vacuum is constant and equal to c only relative to the source. If the light source is moving at a speed v relative to any reference system, then the speed of light c " in this frame of reference is vectorially summed from c and v , i.e. c " = c + v (as it happens with the speed of the projectile when firing from a moving gun).

This hypothesis is refuted by astronomical observations of the motion of double stars (Sitter, a Dutch astronomer, 1913).

Indeed, let us assume that the ballistic hypothesis is correct. For simplicity, let's assume that the components of a binary star revolve around their center of mass in circular orbits in the same plane as the Earth. Let's follow the movement of one of these two stars. Let the speed of its movement in a circular orbit be equal to v. In that position of the star, when it moves away from the Earth along the straight line connecting them, the speed of light (relative to the Earth) is equal to c–v, and in the position when the star is approaching, it is equal to c+v. If we count the time from the moment when the star was in the first position, then the light from this position will reach the Earth at the moment t 1 = L/(c–v), where L is the distance to the star. And from the second position, the light will reach at the moment t 2 = T/2+L/(c+v), where T- period of revolution of a star

(7)

With a large enough L, t 2 <t 1 , i.e. the star would be visible in two (or more) positions at the same time, or even rotate in the opposite direction. But this has never been observed.

Sade experience

Sade performed a beautiful experiment in 1963 showing that the speed γ -rays is constant regardless of the speed of the source (see BKF, Mechanics, p. 372).

In his experiments, he used annihilation during the run of positrons. During annihilation, the center of mass of a system consisting of an electron and a positron moves at a speed of about (1/2) c, and as a result of annihilation, two γ -quantum. In the case of annihilation in a stationary state, both γ -quanta are emitted at an angle of 180° and their speed is c. In the case of runaway annihilation, this angle is less than 180° and depends on the speed of the positron. If the speed γ -quantum was added with the speed of the center of mass according to the classical rule of vector addition, then γ -quantum moving with a certain velocity component in the direction of the positron path, should have had a velocity greater than c, and that γ -quantum, which has a velocity component in the opposite direction, must have a velocity less than c. It turned out that for the same distances between the counters and the annihilation point, both γ -quanta reach the counters at the same time. This proves that for a moving source both γ -quanta propagate at the same speed.

Top speed

Bertozzi experiment 1964

The following experiment illustrates the statement that it is impossible to accelerate a particle to a speed exceeding the speed of light c. In this experiment, the electrons were accelerated successively by increasingly strong electrostatic fields in a Van de Graaff accelerator, and then they moved at a constant speed through the field-free space.

Their flight time at a known distance AB, and hence their speed, was measured directly, and the kinetic energy (turned into heat when hitting the target at the end of the path) was measured using a thermocouple.

In this experiment, the magnitude of the accelerating potential was determined with great accuracy φ . The kinetic energy of an electron is

If flies through the beam section N electrons per second, then the power transferred to the aluminum target at the end of their path should be equal to 1.6 10 -6 N erg/sec This exactly coincided with the directly determined (using a thermocouple) power absorbed by the target. Thus, it was confirmed that the electrons gave the target all the kinetic energy received during their acceleration.

From these experiments it follows that the electrons received from the accelerating field an energy proportional to the applied potential difference, but their speed could not increase indefinitely and approached the speed of light in vacuum.

Many other experiments, like the one described above, indicate that c is the upper limit of particle velocity. Thus, we are firmly convinced that c is the maximum speed of signal transmission both with the help of particles and with the help of electromagnetic waves; c is the top speed.

Conclusion:

1. Value c is invariant for inertial frames of reference.

2. c- the maximum possible signal transmission rate.

Relativity of time

Already in classical mechanics, space is relative, i.e. the spatial relationships between different events depend on the frame of reference in which they are described. The statement that two events of different times occur in the same place in space or at a certain distance relative to each other becomes meaningful only when it is indicated to which frame of reference this statement refers. Example: a ball bouncing on a table in a compartment of a train car. From the point of view of the passenger in the compartment, the ball hits the table at approximately the same place on the table. From the point of view of the observer on the platform, each time the coordinate of the ball is different, since the train moves along with the table.

On the contrary, time is absolute in classical mechanics. This means that time flows in the same way in different frames of reference. For example, if any two events are simultaneous for one observer, then they will be simultaneous for any other. In general, the time interval between two given events is the same in all frames of reference.

One can, however, be convinced that the concept of absolute time is in deep contradiction with Einstein's principle of relativity. To this end, let us recall that in classical mechanics, based on the concept of absolute time, the well-known law of addition of velocities takes place. But this law, when applied to light, says that the speed of light c" in the frame of reference K", moving at a speed V regarding the system K, related to the speed of light c in system K ratio

those. The speed of light turns out to be different in different frames of reference. This, as we already know, contradicts the principle of relativity and experimental data.

Thus the principle of relativity leads to the result that time is not absolute. It flows differently in different frames of reference. Therefore, the statement that a certain period of time has passed between two given events makes sense only if it is indicated at the same time to which frame of reference this refers. In particular, events that are simultaneous in some frame of reference will not be simultaneous in another frame.

Let's explain this with a simple example.

Consider two inertial coordinate systems K and K" with coordinate axes xyz and x " y " z" , and the system K"moves relative to the system K right along the axes x and x" (Fig. 8). Let from some point A on axle x"Signals are sent simultaneously in two mutually opposite directions. Since the signal propagation speed in the system K" , as in any inertial frame, is (in both directions) c, then the signals will reach equidistant from A points B and C at the same moment in time (in the system K ").

It is easy, however, to make sure that these two events (the arrival of signals at B and C) will not be simultaneous for an observer in the system K. For him, too, the speed of light is c in both directions, but dot B moves towards the light, so that its light reaches earlier, and the point C moves away from the light and therefore the signal will come to it later.

Thus, Einstein's principle of relativity introduces fundamental changes in basic physical concepts. Based on everyday experience, our ideas about space and time turn out to be only approximate, related to the fact that in everyday life we ​​deal only with speeds that are very small compared to the speed of light.

1 An interaction propagating from one particle to another is often referred to as a "signal" sent from the first particle and "letting know" to the second of the change that has occurred to the first. The speed of propagation of interactions is often referred to as "signal speed".

2 The period of revolution of Jupiter around the Sun is approximately 12 years, the period of revolution of Io around Jupiter is 42 hours.

LECTURE 2

Interval. Geometry of Minkowski. Interval invariance.

· Timelike and spacelike intervals.

Absolutely future events, absolutely past events,

completely removed events.

Light cone.

Interval

In the theory of relativity, the concept is often used developments. An event is defined by the place where it happened and the time when it happened. Thus, an event that happened to some material particle is determined by the three coordinates of this particle and the moment in time when this event happened: x, y, z and t.

In what follows, for reasons of clarity, we will use an imaginary four-dimensional space, on the axes of which three spatial coordinates and time are plotted. In this space, any event is represented by a dot. These points are called world points. Each particle corresponds to a certain line - world line in this four-dimensional space. The points of this line determine the coordinates of the particle at all times. If a particle is at rest or moves uniformly and rectilinearly, then a straight world line corresponds to it.

We now express the principle of invariance of the value of the speed of light 1 mathematically. To do this, consider two inertial frames of reference K and K" , moving relative to each other at a constant speed. We choose the coordinate axes so that the axes x and x" coincided, and the axes y and z would be parallel to the axes y" and z". Time in systems K and K" denoted by t and t".

Let the first event be that from a point with coordinates x 1 , y 1 , z 1 at time t 1 (in reference frame K) sends a signal that travels at the speed of light. We will observe from the frame of reference K for the propagation of this signal. Let the second event be that this signal arrives at the point x 2 , y 2 , z 2 at time t 2. Because the signal travels at the speed of light c, the distance traveled is c(t 2 –t one). On the other hand, this distance is equal to:

As a result, the following relation between the coordinates of both events in the system turns out to be valid K

If a x 1 , y 1 , z 1 , t 1 and x 2 , y 2 , z 2 , t 2 are the coordinates of any two events, then the value

Geometry of Minkowski

If two events are infinitely close to each other, then for the interval ds between them we have

ds 2 = c 2 dt 2 –dx 2 –dy 2 –dz 2 .

(4)

The form of expressions (3) and (4) allows us to consider the interval, from a formal mathematical point of view, as a "distance" between two points in an imaginary four-dimensional space (on the axes of which the values ​​are plotted x, y, z and work ct). There is, however, a significant difference in the rule for compiling this quantity compared to the rules of ordinary Euclidean geometry: when the square of the interval is formed, the square of the difference in coordinates along the time axis enters with a plus sign, and the squares of differences in spatial coordinates enter with a minus sign. Such a four-dimensional geometry, defined by the quadratic form (4), is called pseudo-Euclidean in contrast to the usual, Euclidean, geometry. This geometry in connection with the theory of relativity was introduced by G. Minkowski.

Interval invariance

As we showed above, if ds= 0 in some inertial frame of reference, then ds" = 0 in any other inertial frame. But ds and ds" are infinitesimal quantities of the same order of smallness. Therefore, in the general case, these two conditions imply that ds 2 and ds"2 must be proportional to each other:

ds 2 = a ds" 2 .

(5)

Proportionality factor a can depend only on the absolute value of the relative velocity V both inertial systems. It cannot depend on coordinates and time, since then different points of space and moments of time would be unequal, which contradicts the homogeneity of space and time. It cannot also depend on the direction of the relative velocity V , since this would contradict the isotropy of space.

Consider three inertial frames of reference K, K 1 and K 2. Let V 1 and V 2 - movement speeds of systems K 1 and K 2 regarding the system K. Then we have

But the speed V 12 depends not only on the absolute values ​​of the vectors V 1 and V 2 but also from the corner α between them. 2 Meanwhile, the latter does not enter the left-hand side of relation (8) at all. Therefore, this relation can be satisfied only if the function a(V) = const = 1.

In this way,

We have thus arrived at a very important result:

This invariance is the mathematical expression for the constancy of the speed of light.

There are various methods for measuring the speed of light, including astronomical and using various experimental techniques. Quantity measurement accuracy With is constantly increasing. This table provides an incomplete list of experimental work on the determination of the speed of light.

	Experiment	Experimental Methods	Measurement results, km/s	experimental error,
	Weber-Kohlrausch Maxwell Michelson Perrotin Rose and Dorsey Mittelyptedt Pease and Pearson Anderson	Jupiter moon eclipse light aberration moving bodies rotating mirrors Electromagnetic constants Electromagnetic constants rotating mirrors rotating mirrors Electromagnetic constants rotating mirrors rotating mirrors Electromagnetic constants Kerr gate cell rotating mirrors Kerr gate cell Microwave interferometry

The figure graphically presents the numerical values ​​of the speed of light obtained in different years (Figure Olimpusmicro.com).

One can trace how the accuracy of measurements has changed with progress in the field of science and technology.

The first successful measurement of the speed of light dates back to 1676.

The drawings show a reproduction of a drawing by Römer himself, as well as a schematic interpretation.

Römer's astronomical method is based on the measurement the speed of light according to observations from Earth of the eclipses of the moons of Jupiter. Jupiter has several satellites that are either visible from Earth near Jupiter or hidden in its shadow. Astronomical observations of the satellites of Jupiter show that the average time interval between two successive eclipses of any particular satellite of Jupiter depends on how far apart the Earth and Jupiter are at the time of observation. In the figure: Roemer's method. S - sun, Yu - jupiter, Z - earth

Let at a certain moment of time Earth Z1 and Jupiter Yu1 be in opposition, and at this moment one of Jupiter's satellites, observed from the Earth, disappears in the shadow of Jupiter (the satellite is not shown in the figure). Then, if denoted by R and r are the radii of the orbits of Jupiter and the Earth and throughc - the speed of light in the coordinate system associated with the Sun C, on Earth, the departure of the satellite into the shadow of Jupiter will be registered on ( R- r)/s seconds later than it occurs in the time frame associated with Jupiter.

After 0.545 years, Earth Z2 and Jupiter U2 are in conjunction. If at this time there isn-th eclipse of the same satellite of Jupiter, then on Earth it will be registered with a delay of ( R+ r)/s seconds. Therefore, if the period of revolution of a satellite around Jupitert, then the time intervalT1 flowing between the first andnth eclipse observed from Earth is

After another 0.545 years, Earth 33 and Jupiter 33 will again be in opposition. During this time, it tookn-1) revolutions of the satellite around Jupiter and (n-1) eclipses, of which the first took place when the Earth and Jupiter occupied positions 32 and 102, and the last - when they occupied positions 33 and 33. The first eclipse was observed on Earth with a delay ( R+ r)/s, and the latter with a delay ( R- r)/ c with respect to the moments of the satellite's departure into the shadow of the planet Jupiter. Therefore, in this case we have

Römer measured the time intervals T1 and T2 and found that T1-T2=1980 s. But from the formulas written above it follows that Т1-Т2=4 r/s, so c=4 r/1980 m/s. Takingr, the average distance from the Earth to the Sun, equal to 1500000000 km, we find the value of 3.01 * 10 for the speed of light 6 m/s.

Determination of the speed of light by observing aberration in 1725-1728. Bradley undertook an observation in order to find out whether there is an annual parallax of stars, i.e. the apparent displacement of stars in the firmament, reflecting the movement of the Earth in orbit and associated with the finiteness of the distance from the Earth to the star.

Bradley indeed discovered such a shift. He explained the observed phenomenon, which he called aberration of light, the final value of the speed of light propagation and used it to determine this speed.

Knowing the angle α and the speed of the Earth's orbit v, we can determine the speed of light c.

He got the value of the speed of light equal to 308,000 km / s.

It is important to note that the aberration of light is associated with a change in the direction of the Earth's speed during the year. A constant speed, no matter how great it may be, cannot be detected with the help of aberration, because with such a movement the direction to the star remains unchanged and there is no way to judge the presence of this speed and what angle it makes with the direction to the star. The aberration of light makes it possible to judge only about the change in the speed of the Earth.

In 1849, A. Fizeau performed the first determination of the speed of light under laboratory conditions. His method was called the cogwheel method. A characteristic feature of his method is the automatic registration of the moments of starting and returning the signal, carried out by regularly interrupting the light flux (gear wheel).

Fig 3. Scheme of the experiment to determine the speed of light by the gear wheel method.

The light from the source passed through the interrupter (the teeth of the rotating wheel) and, reflected from the mirror, returned again to the gear wheel. Knowing the distance between the wheel and the mirror, the number of teeth of the wheel, the speed of rotation, you can calculate the speed of light.

Knowing the distance D, the number of teeth z, angular speed of rotation (rpm)v, you can determine the speed of light. He got it equal to 313,000 km / s.

Many methods have been developed to further improve the measurement accuracy. Soon it even became necessary to take into account the refractive index in air. And soon, in 1958, Frum obtained the value of the speed of light equal to 299792.5 km / s, using a microwave interferometer and an electro-optical shutter (Kerr cell).

Doppler effect in optics

Experimental Foundations of Special Relativity

Modern methods for measuring the speed of light

Propagation of light in moving media

Classical experiments on measuring the speed of light

The problem of determining the speed of light is one of the most important problems in optics and physics in general. The solution of this problem was of great fundamental and practical importance. The establishment that the speed of propagation of light is finite, and the measurement of this speed, have made more concrete and clear the difficulties facing various optical theories. The first methods for determining the speed of light, based on astronomical observations, contributed in their turn to a clear understanding of purely astronomical questions. Accurate laboratory methods for determining the speed of light, developed later, are used in geodetic surveys.

The main difficulty that an experimenter encounters in determining the speed of light propagation is associated with the enormous value of this quantity, which requires completely different scales of experience than those that take place in classical physical measurements. This difficulty made itself felt in the first scientific attempts to determine the speed of light, undertaken by Galileo (1607). Galileo's experiment was as follows: two observers at a great distance from each other

others are equipped with lockable lanterns. Observer BUT opens the lantern; after a certain time interval, the light will reach the observer AT, who at the same moment opens his lantern; after a certain time, this signal will reach BUT, and the latter can thus mark the time τ elapsed from the moment he gave the signal until the moment he returned. Assuming that the observers respond to the signal instantly and that light has the same speed in the direction AB and VA, get that way AB+VA=2D light travels in time τ , i.e. speed of light With=2D/τ . The second of the assumptions made can be considered very plausible. The modern theory of relativity raises even this assumption into a principle. But the assumption that it is possible to instantly respond to a signal does not correspond to reality, and therefore, at a huge speed of light, Galileo's attempt did not lead to any results; in essence, it was not the propagation time of the light signal that was measured, but the time taken by the observer to react. The situation can be improved if the observer AT be replaced by a mirror that reflects light, thus freeing itself from the error introduced by one of the observers. This measurement scheme has remained, in essence, in almost all modern laboratory methods for determining the speed of light; however, later excellent techniques were found for recording signals and measuring time intervals, which made it possible to determine the speed of light with sufficient accuracy even at relatively small distances.

a) Roemer's method.

Jupiter has several satellites that are either visible from Earth near Jupiter or hidden in its shadow. Astronomical observations of the satellites of Jupiter show that the average time interval between two successive eclipses of any particular satellite of Jupiter depends on how far apart the Earth and Jupiter are at the time of observation.

Roemer's method (1676), based on these observations, can be explained with the help of Fig. 9.1. Let at a certain point in time the Earth W 1 and Jupiter YU 1 are in confrontation and at this point in time, one of Jupiter's moons, as seen from Earth, disappears into Jupiter's shadow. Then, if denoted by R and r the radii of the orbits of Jupiter and the Earth and through With- the speed of light in the coordinate system associated with the Sun, on Earth, the departure of the satellite into the shadow of Jupiter will be registered seconds later than it occurs in the time frame associated with Jupiter.

After 0.545 years the Earth W 2 and Jupiter YU 2 are in connection. If at this time there is n th eclipse of the same satellite of Jupiter, then on Earth it will be registered with a delay of seconds. Therefore, if the period of revolution of a satellite around Jupiter t, then the time interval T 1 , flowing between the first and n th eclipse observed from Earth is

After another 0.545 years, the Earth W 3 and Jupiter YU 3 will be back in confrontation. During this time, there have been ( n–1) revolutions of the satellite around Jupiter and ( n-1) eclipses, of which the first took place when the Earth and Jupiter occupied positions W 2 and YU 2 , and the last - when they occupied the position W 3 and YU 3 . The first eclipse was observed on Earth with a delay, and the last with a delay in relation to the moments of the satellite's departure into the shadow of the planet Jupiter. Therefore, in this case we have:

Römer measured time intervals T 1 and T 2 and found that T 1 –T 2 =1980 s. But from the above formulas it follows that T 1 –T 2 =, therefore . Taking r, the average distance from the Earth to the Sun, equal to 150 10 6 km, we find the value for the speed of light: With\u003d 301 10 6 m / s.

This result was historically the first measurement of the speed of light.

b) Determining the speed of light by observing aberration.

In 1725-1728. Bradley undertook observations in order to find out whether there is an annual parallax of stars, i.e. the apparent displacement of stars in the firmament, reflecting the movement of the Earth in orbit and associated with the finiteness of the distance from the Earth to the star. A star in its parallactic motion must describe an ellipse, the angular dimensions of which are the greater, the smaller the distance to the star.

For stars lying in the plane of the ecliptic, this ellipse degenerates into a straight line, and for stars near the pole, into a circle. Bradley indeed discovered such a shift. But the major axis of the ellipse turned out to be for all stars having the same angular dimensions, namely 2 α \u003d 40 ", 9. Bradley explained (1728) the observed phenomenon, which he called aberration of light, the finiteness of the speed of light propagation and used it to determine this speed. The annual parallax was established more than a hundred years later by V.Ya. Struve and Bessel (1837, 1838).

For simplicity, instead of a telescope, we will use a sighting device consisting of two small holes located along the axis of the pipe. When the Earth's speed is in the same direction as SE, the pipe axis points to the star. When the speed of the Earth (and the pipe) makes an angle j with the direction to the star, then in order for the light beam to remain on the axis of the pipe, the pipe must be rotated through an angle a(Fig. 9.2), because during the time t as long as the light goes the way SE, the pipe itself moves a distance E "E=u 0 t. From fig. 9.2 can define a turn a. Here SE determines the direction of the pipe axis without taking into account aberration, SE"- offset direction of the axis, ensuring the passage of light along the axis of the pipe during the entire time t. Taking advantage of that angle a very small because u 0 <<с (пренебрегая членами порядка ), можно считать, чтоj =0 или p.

If the star is at the pole of the ecliptic, then j=90° throughout the year, i.e. the angular deviation of the star remains unchanged in magnitude (); but since the direction of the vector u 0 changes during the year by an angle of 2 p, then the angular displacement of the star also changes in direction: the star describes an apparent circular orbit with an angular radius .

In general, when the star is located at an angular distance d from the plane of the ecliptic, the aberrational trajectory of the star is an ellipse, the major semiaxis of which has angular dimensions a 0 , and small - a 0 sin d. It was precisely this character that the apparent displacement of the stars according to Bradley's observation bore. Determining from observations a 0 and knowing u 0 , can be found from. Bradley found With\u003d 308,000 km / s. V. Ya. Struve (1845) significantly improved the accuracy of observations and obtained a 0 =20",445. The most recent definitions give a 0 \u003d 20", 470, which corresponds to With\u003d 299 900 km / s.

It should be noted that the aberration of light is associated with a change in the direction of the Earth's velocity during the year.