A natural skeptical question: “What are the limits of applicability of Galileo’s transformations?” arose before humanity towards the end of the 19th and beginning of the 20th centuries. It arose in connection with the study of the paradoxical properties of the ether - a hypothetical absolutely elastic medium in which light propagates without attenuation, as in an absolutely solid medium.
Doubts about the infinite applicability of Galileo’s transformations, at least in part of the law of addition of velocities, arose when analyzing the results of the Michelson-Morley experiments to determine the speed of the “ethereal wind” from a comparison of the speed of light emitted by a source moving along the direction of the Earth’s movement in orbit and the speed of light along a direction perpendicular to the tangent to the orbit. The measurements were made using an extremely precise instrument - the Michelson interferometer. The Earth was ingeniously chosen as an object moving at a linear speed of 30 km/sec, practically unattainable by modern technology for massive objects.
Michelson's experiment, first performed in 1881 and giving a negative answer, was set up fundamentally: a plate up to 0.5 m thick on which the mirrors were mounted was made of granite, which expands slightly with heating, and floated in mercury for deformation-free rotation. The primary accuracy of the experiment made it possible to detect the “ethereal wind” at a speed of 10 km/s. Later it was repeated many times, the accuracy was increased to the ability to detect wind speeds of 30 m/s. But the answer was consistently zero.
Galileo's transformations were not confirmed when observing movements at high speeds. For example, there were no disturbances in the rhythm of the periodic motion of double stars, while the direction of the speed of their movement changes on the forward and backward paths of revolution. The speed of light thus turned out to be independent of the movement of the source.
From the time of the experiments by Michelson and Morley in 1881 until 1905 - before the development of the foundations of SRT - numerous attempts were made to develop hypotheses that would explain the results of the key experiment. And at the same time, everyone tried to preserve the ether, modifying only its properties.
The most famous are the curious attempts of the Irish physicist George Fitzgerald and the Dutch physicist Hendrik Lorentz. The first proposed the idea of reducing the length of the body in the direction of movement, the more, the higher the speed of movement. Lorenz suggested the possibility of a local flow of time (“local time”) in a moving system, according to laws that differ from the laws in a stationary system. Lorentz proposed modifying Galileo's coordinate transformations.
Einstein's postulates in the special theory of relativity
A decisive contribution to the creation of the special and then the general theory of relativity was made by Albert Einstein. In 1905, in the journal Annalen für Physik, a 26-year-old, unknown employee of the Swiss patent office, Albert Einstein, published a small 3-page article “On the electrodynamics of moving media.” According to historians of physics, he had not heard about the results of the Michelson-Morley experiments.
Einstein's concept allows us to abandon the existence of the ether and build a theory, now called the special theory of relativity (SRT) and confirmed by all experiments known today.
SRT is based on two postulates.
"The principle of the constancy of the speed of light."
The speed of light does not depend on the speed of movement of the light source, is the same in all inertial coordinate systems, and is equal to c = 3 in vacuum 10 8 m/s.
Later, the general theory of relativity (GTR), published in 1916, stated that the speed of light remains constant in non-inertial coordinate systems.
Special principle of relativity.
The laws of nature are the same (invariant, covariant) in all inertial coordinate systems.
Einstein later wrote:
“In all inertial coordinate systems, the laws of nature are in agreement. Physical reality is not possessed by a point in space or a moment in time when something happened, but only by the event itself. There is no absolute (independent of reference space) relationship in space, and there is no absolute relationship in time, but there is an absolute (independent of reference space) relationship. relationship in space and time" ( emphasized by Einstein).
Later, Einstein asserted the validity of this postulate for all, including non-inertial, reference systems.
The mathematical apparatus of STR uses the four-dimensional xyzt space-time continuum (Minkowski space) and Lorentz coordinate transformations as a mathematical reflection of facts objectively existing in the material world.
The assumption that the speed of light is absolute leads to a number of consequences that are unusual and not observed under the conditions of Newtonian mechanics. One of the consequences of the constancy of the speed of light is the rejection of the absolute nature of time, which was instilled in Newtonian mechanics. We must now assume that time flows differently in different reference systems - events that are simultaneous in one system will be non-simultaneous in another.
Let us consider two inertial frames of reference K And K", moving relative to each other. Let in a dark room moving with the system K", the lamp flashes. Since the speed of light in the system K" is equal (as in any frame of reference) c, then the light reaches both opposite walls of the room at the same time. This is not what will happen from the point of view of an observer in the system K. Speed of light in the system K also equal c, but since the walls of the room move relative to the system K, then the observer in the system K will detect that the light will touch one of the walls before the other, i.e. in system K these events are not simultaneous.
Thus, in Einstein's mechanics relative Not only properties of space, but also properties of time.
An attempt to interpret this result at the beginning of the 20th century resulted in a revision of classical concepts and led to the creation of the special theory of relativity.
When moving at near-light speeds, the laws of dynamics change. Newton's second law, relating force and acceleration, must be modified for bodies with velocities close to the speed of light. In addition, the expression for the momentum and kinetic energy of the body has a more complex dependence on speed than in the nonrelativistic case.
The special theory of relativity has received numerous experimental confirmations and is a correct theory in its field of applicability (see Experimental foundations of SRT). According to the apt remark of L. Page, “in our age of electricity, the rotating armature of every generator and every electric motor tirelessly proclaims the validity of the theory of relativity - you just need to be able to listen.”
The fundamental nature of the special theory of relativity for physical theories built on its basis has now led to the fact that the term “special theory of relativity” itself is practically not used in modern scientific articles; they usually only talk about the relativistic invariance of a separate theory.
Basic concepts and postulates of SRT
Special theory of relativity, like any other physical theory, can be formulated on the basis of basic concepts and postulates (axioms) plus the rules of correspondence to its physical objects.
Basic Concepts
Time synchronization
The STR postulates the possibility of determining a unified time within a given inertial reference system. To do this, a procedure is introduced to synchronize two clocks located at different points in the ISO. Let a signal (not necessarily light) be sent from the first clock at a moment in time to the second clock at a constant speed. Immediately upon reaching the second clock (according to its readings at time ) the signal is sent back at the same constant speed and reaches the first clock at time . The clocks are considered synchronized if the relation is satisfied.
It is assumed that such a procedure in a given inertial reference frame can be carried out for any clocks that are motionless relative to each other, so the transitivity property is valid: if the clocks A synchronized with watch B, and the clock B synchronized with watch C, then the clock A And C will also be synchronized.
Coordination of units of measurement
To do this, it is necessary to consider three inertial systems S1, S2 and S3. Let the speed of system S2 relative to system S1 be equal to , the speed of system S3 relative to S2 equal to , and relative to S1, respectively, . By writing the sequence of transformations (S2, S1), (S3, S2) and (S3, S1), we can obtain the following equality:
Proof
Transformations (S2, S1) (S3, S2) have the form:
where, etc. Substitution from the first system to the second gives:
The second equality is a record of transformations between systems S3 and S1. If we equate the coefficients in the first equation of the system and in the second, then:
By dividing one equation by another, it is easy to obtain the desired relationship.
Since the relative velocities of the reference systems are arbitrary and independent quantities, this equality will be satisfied only in the case when the ratio is equal to some constant , common for all inertial reference systems, and, therefore, .
The existence of an inverse transformation between ISOs, which differs from the direct one only by changing the sign of the relative speed, allows us to find the function .
Proof
Postulate of the constancy of the speed of light
Historically, an important role in the construction of SRT was played by Einstein's second postulate, which states that the speed of light does not depend on the speed of the source and is the same in all inertial reference systems. It was with the help of this postulate and the principle of relativity that Albert Einstein in 1905 obtained the Lorentz transformation with a fundamental constant meaning the speed of light. From the point of view of the axiomatic construction of STR described above, Einstein’s second postulate turns out to be a theorem of the theory and directly follows from the Lorentz transformations (see relativistic addition of velocities). However, due to its historical importance, this derivation of Lorentz transformations is widely used in educational literature.
It should be noted that light signals, generally speaking, are not required when justifying SRT. Although the non-invariance of Maxwell's equations with respect to Galilean transformations led to the construction of SRT, the latter is more general in nature and is applicable to all types of interactions and physical processes. The fundamental constant that appears in Lorentz transformations has the meaning of the maximum speed of motion of material bodies. Numerically, it coincides with the speed of light, but this fact is associated with the masslessness of electromagnetic fields. Even if the photon had a non-zero mass, the Lorentz transformations would not change. It therefore makes sense to distinguish between fundamental speed and the speed of light. The first constant reflects the general properties of space and time, while the second is associated with the properties of a specific interaction. To measure the fundamental velocity, there is no need to perform electrodynamic experiments. It is enough, using, for example, the relativistic rule of adding velocities based on the velocity values of some object relative to two ISOs, to obtain the value of the fundamental velocity.
Consistency of the Theory of Relativity
The theory of relativity is a logically consistent theory. This means that from its initial provisions it is impossible to logically deduce a certain statement simultaneously with its negation. Therefore, many so-called paradoxes (like the twin paradox) are apparent. They arise as a result of incorrect application of the theory to certain problems, and not due to the logical inconsistency of STR.
The validity of the theory of relativity, like any other physical theory, is ultimately tested empirically. In addition, the logical consistency of STR can be proven axiomatically. For example, within the group approach it is shown that Lorentz transformations can be obtained based on a subset of the axioms of classical mechanics. This fact reduces the proof of the consistency of SRT to the proof of the consistency of classical mechanics. Indeed, if the consequences from a broader system of axioms are consistent, then they will be even more consistent when using only part of the axioms. From a logical point of view, contradictions can arise when a new axiom is added to existing axioms that does not agree with the original ones. In the axiomatic construction of STR described above, this does not happen, therefore SRT is a consistent theory.
Geometric approach
Other approaches to constructing a special theory of relativity are possible. Following Minkowski and Poincaré's earlier work, one can postulate the existence of a single metric four-dimensional spacetime with 4-coordinates. In the simplest case of flat space, the metric that determines the distance between two infinitely close points can be Euclidean or pseudo-Euclidean (see below). The latter case corresponds to the special theory of relativity. In this case, Lorentz transformations are rotations in such a space that leave the distance between two points unchanged.
Another approach is possible, in which the geometric structure of the velocity space is postulated. Each point of such space corresponds to some inertial reference system, and the distance between two points corresponds to the relative velocity module between the ISOs. By virtue of the principle of relativity, all points of such a space must be equal, and, therefore, the velocity space is homogeneous and isotropic. If its properties are given by Riemannian geometry, then there are three and only three possibilities: flat space, space of constant positive and negative curvature. The first case corresponds to the classical rule of adding velocities. The space of constant negative curvature (Lobachevsky space) corresponds to the relativistic rule for adding velocities and the special theory of relativity.
Different notations for the Lorentz transformation
Let the coordinate axes of two inertial reference systems S and S" be parallel to each other, (t, x,y, z) - the time and coordinates of some event observed relative to the system S, and (t",x",y",z") - time and coordinates the same events relative to the system S". If the system S" moves uniformly and rectilinearly with a speed v relative to S, then the Lorentz transformations are valid:
where is the speed of light. At speeds much less than the speed of light (), the Lorentz transformations transform into Galilean transformations:
Such a passage to the limit is a reflection of the correspondence principle, according to which a more general theory (STR) has as its limiting case a less general theory (in this case, classical mechanics).
Lorentz transformations can be written in vector form, when the speed of the reference frames is directed in an arbitrary direction (not necessarily along the axis):
where is the Lorentz factor, and are the radius vectors of the event relative to the systems S and S".
Consequences of Lorentz transformations
Speed addition
An immediate consequence of the Lorentz transformations is the relativistic rule for adding velocities. If some object has velocity components relative to the system S and - relative to S", then the following relationship exists between them:
In these relations, the relative speed of movement of the reference frames v is directed along the x axis. The relativistic addition of velocities, like the Lorentz transformation, at low velocities () transforms into the classical law of addition of velocities.
If an object moves at the speed of light along the x axis relative to the system S, then it will have the same speed relative to S": This means that the speed is invariant (the same) in all ISOs.
Time dilation
If the clock is stationary in the system, then for two consecutive events . Such clocks move relative to the system according to the law, so time intervals are related as follows:
It is important to understand that in this formula the time interval is measured alone moving clock. It is compared with the readings several different, synchronously running clocks located in the system, past which the clock moves. As a result of this comparison, it turns out that moving clocks go slower than stationary clocks. Associated with this effect is the so-called twin paradox.
If a clock moves at a variable speed relative to an inertial reference frame, then the time measured by this clock (the so-called proper time) does not depend on acceleration, and can be calculated using the following formula:
where, using integration, time intervals in locally inertial reference systems (the so-called instantly accompanying ISO) are summed up.
The relativity of simultaneity
If two spatially separated events (for example, flashes of light) occur simultaneously in a moving reference frame, then they will be non-simultaneous relative to the “stationary” frame. When from the Lorentz transformations it follows
If , then and . This means that, from the point of view of a stationary observer, the left event occurs before the right one. The relativity of simultaneity makes it impossible to synchronize clocks in different inertial reference frames throughout space.
From the point of view of the system S
From the point of view of the S system"
Let there be clocks in two reference systems along the x axis, synchronized in each system, and at the moment the “central” clocks coincide (in the figure below), they show the same time.
The left figure shows how this situation looks from the point of view of an observer in frame S. Clocks in a moving frame show different times. The clocks located in the direction of travel are behind, and those located against the direction of movement are ahead of the “central” clock. The situation is similar for observers in S" (right figure).
Reduction of linear dimensions
If the length (shape) of a moving object is determined by simultaneously fixing the coordinates of its surface, then from the Lorentz transformations it follows that the linear dimensions of such a body relative to the “stationary” reference system are reduced:
,where is the length along the direction of movement relative to the stationary reference frame, and is the length in the moving reference frame associated with the body (the so-called proper length of the body). At the same time, the longitudinal dimensions of the body (that is, measured along the direction of movement) are reduced. The transverse dimensions do not change.
This size reduction is also called Lorentz contraction. When visually observing moving bodies, in addition to the Lorentz contraction, it is necessary to take into account the time of propagation of the light signal from the surface of the body. As a result, a fast-moving body appears rotated, but not compressed in the direction of movement.
Doppler effect
Let a source moving with speed v emits a periodic signal with frequency . This frequency is measured by an observer associated with the source (the so-called natural frequency). If the same signal is recorded by a “stationary” observer, then its frequency will differ from its natural frequency:
where is the angle between the direction to the source and its speed.
There are longitudinal and transverse Doppler effects. In the first case, that is, the source and receiver are on the same straight line. If the source moves away from the receiver, then its frequency decreases (red shift), and if it approaches, then its frequency increases (blue shift):
The transverse effect occurs when , that is, the direction towards the source is perpendicular to its speed (for example, the source “flies over” the receiver). In this case, the effect of time dilation is directly manifested:
There is no analogue of the transverse effect in classical physics, and this is a purely relativistic effect. In contrast, the longitudinal Doppler effect is due to both the classical component and the relativistic time dilation effect.
Aberration
remains valid also in the theory of relativity. However, the time derivative is taken from the relativistic impulse, and not from the classical one. This leads to the fact that the relationship between force and acceleration differs significantly from the classical one:
The first term contains the “relativistic mass”, equal to the ratio of force to acceleration if the force acts perpendicular to the speed. In early work on the theory of relativity it was called "transverse mass." It is its “growth” that is observed in experiments on the deflection of electrons by a magnetic field. The second term contains the “longitudinal mass”, equal to the ratio of force to acceleration if the force acts parallel to the speed:
As noted above, these concepts are outdated and associated with an attempt to preserve Newton's classical equation of motion.
The rate of change of energy is equal to the scalar product of force and the speed of the body:
This leads to the fact that, as in classical mechanics, the component of the force perpendicular to the velocity of the particle does not change its energy (for example, the magnetic component in the Lorentz force).
Energy and momentum conversions
Similar to the Lorentz transformations for time and coordinates, relativistic energy and momentum, measured relative to various inertial reference systems, are also related by certain relations:
where the components of the momentum vector are equal to . The relative speed and orientation of the inertial reference systems S, S" are determined in the same way as in the Lorentz transformations.
Covariant formulation
Four-dimensional space-time
Lorentz transformations leave the following quantity invariant (unchanged), called interval:
where , etc. are the differences in times and coordinates of two events. If , then they say that the events are separated by a time-like interval; if , then spacelike. Finally, if , then such intervals are called light-like. The light-like interval corresponds to events associated with a signal that travels at the speed of light. The invariance of an interval means that it has the same value relative to two inertial reference frames:
In its form, an interval resembles a distance in Euclidean space. However, it has a different sign for the spatial and temporal components of the event, so they say that the interval specifies the distance in pseudo-Euclidean four-dimensional space-time. It is also called Minkowski spacetime. Lorentz transformations play the role of rotations in such a space. Rotations of the basis in four-dimensional space-time, mixing the time and spatial coordinates of 4-vectors, look like a transition to a moving reference frame and are similar to rotations in ordinary three-dimensional space. In this case, the projections of four-dimensional intervals between certain events onto the time and spatial axes of the reference system naturally change, which gives rise to relativistic effects of changes in time and spatial intervals. It is the invariant structure of this space, specified by the postulates of STR, that does not change when moving from one inertial reference system to another. Using only two spatial coordinates (x, y), four-dimensional space can be represented in coordinates (t, x, y). Events associated with the origin event (t=0, x=y=0) by a light signal (light-like interval) lie on the so-called light cone (see figure on the right).
Metric tensor
The distance between two infinitely close events can be written using the metric tensor in tensor form:
where , and over repeating indices implies summation from 0 to 3. In inertial reference systems with Cartesian coordinates, the metric tensor has the following form:
Briefly, this diagonal matrix is denoted as follows: .
The choice of a non-Cartesian coordinate system (for example, transition to spherical coordinates) or consideration of non-inertial reference systems leads to a change in the values of the components of the metric tensor, but its signature remains unchanged. Within the framework of special relativity, there is always a global transformation of coordinates and time that makes the metric tensor diagonal with components. This physical situation corresponds to the transition to an inertial reference system with Cartesian coordinates. In other words, the four-dimensional space-time of special relativity is flat (pseudo-Euclidean). In contrast, the general theory of relativity (GTR) considers curved spaces in which the metric tensor cannot be brought to a pseudo-Euclidean form in the entire space by any coordinate transformation, but the signature of the tensor remains the same.
4-vector
SRT relations can be written in tensor form by introducing a vector with four components (the number or index at the top of a component is its number, not its degree!). The zero component of a 4-vector is called temporal, and the components with indices 1,2,3 are called spatial. They correspond to the components of an ordinary three-dimensional vector, so a 4-vector is also denoted as follows: .
The components of a 4-vector, measured relative to two inertial frames moving with a relative velocity , are related to each other as follows:
Examples of 4-vectors are: a point in pseudo-Euclidean space-time characterizing an event, and energy-momentum:
.Using the metric tensor, you can introduce the so-called covectors, which are denoted by the same letter, but with a subscript:
For a diagonal metric tensor with signature , a covector differs from a 4-vector by the sign in front of the spatial components. So, if , then . The convolution of a vector and a covector is an invariant and has the same meaning in all inertial frames of reference:
For example, the convolution (square - 4-vector) of energy-momentum is proportional to the square of the particle mass:
.Experimental foundations of SRT
The special theory of relativity underlies all modern physics. Therefore, there is no separate experiment that “proves” SRT. The entire body of experimental data in high-energy physics, nuclear physics, spectroscopy, astrophysics, electrodynamics and other fields of physics is consistent with the theory of relativity within the limits of experimental accuracy. For example, in quantum electrodynamics (a combination of special relativity, quantum theory and Maxwell's equations), the value of the anomalous magnetic moment of an electron coincides with the theoretical prediction with relative accuracy.
In fact, SRT is an engineering science. Its formulas are used in the calculation of particle accelerators. Processing of huge amounts of data on collisions of particles moving at relativistic speeds in electromagnetic fields is based on the laws of relativistic dynamics, deviations from which have not been detected. Corrections resulting from SRT and GTR are used in satellite navigation systems (GPS). SRT is the basis of nuclear energy, etc.
All this does not mean that SRT has no limits of applicability. On the contrary, as in any other theory, they exist, and their identification is an important task of experimental physics. For example, Einstein's theory of gravity (GTR) considers a generalization of the pseudo-Euclidean space of STR to the case of space-time with curvature, which allows us to explain most of the astrophysical and cosmological observable data. There are attempts to detect the anisotropy of space and other effects that can change the STR relations. However, it is necessary to understand that if they are discovered, they will lead to more general theories, the limiting case of which will again be STR. In the same way, at low speeds, classical mechanics, which is a special case of the theory of relativity, remains correct. In general, due to the principle of correspondence, a theory that has received numerous experimental confirmations cannot turn out to be incorrect, although, of course, the scope of its applicability may be limited.
Below are just some experiments illustrating the validity of STR and its individual provisions.
Relativistic time dilation
The fact that time flows more slowly for moving objects is constantly confirmed in experiments carried out in high energy physics. For example, the lifetime of muons in the ring accelerator at CERN increases with precision in accordance with the relativistic formula. In this experiment, the speed of muons was equal to 0.9994 times the speed of light, as a result of which their lifetime increased by 29 times. This experiment is also important because with a 7-meter radius of the ring, the muon acceleration reached values equal to the acceleration of gravity. This, in turn, indicates that the effect of time dilation is due only to the speed of the object and does not depend on its acceleration.
Measurements of the magnitude of time dilation were also carried out with macroscopic objects. For example, in the Hafele-Keating experiment, a comparison was made of the readings of a stationary atomic clock and an atomic clock flying on an airplane.
Independence of the speed of light from the motion of the source
At the dawn of the theory of relativity, Walter Ritz's ideas that the negative result of Michelson's experiment could be explained using ballistic theory gained some popularity. In this theory, it was assumed that light is emitted with speed relative to the source, and the speed of light and the speed of the source are added in accordance with the classical rule of speed addition. Naturally, this theory contradicts SRT.
Astrophysical observations provide a convincing refutation of such an idea. For example, when observing double stars rotating about a common center of mass, in accordance with the Ritz theory, effects would occur that are not actually observed (de Sitter argument). Indeed, the speed of light (“image”) from a star approaching the Earth would be higher than the speed of light from a star moving away during rotation. At a greater distance from the binary system, the faster “image” would significantly outperform the slower one. As a result, the apparent motion of double stars would look quite strange, which is not observed.
Sometimes the objection is raised that Ritz's hypothesis is "in fact" correct, but light, when moving through interstellar space, is re-emitted by hydrogen atoms, which have an average zero velocity relative to the Earth, and quickly acquires speed.
However, if this were so, there would be a significant difference in the image of double stars in different spectral ranges, since the effect of “entrainment” with the medium of light depends significantly on its frequency.
In the experiments of Tomaszek (1923), using an interferometer, interference patterns from terrestrial and extraterrestrial sources (Sun, Moon, Jupiter, stars Sirius and Arcturus) were compared. All of these objects had different speeds relative to the Earth, but no shift in the interference fringes expected in the Ritz model was detected. These experiments were subsequently repeated several times. For example, in the experiment of Bonch-Bruevich A.M. and Molchanov V.A. (1956), the speed of light from various edges of the rotating Sun was measured. The results of these experiments also contradict the Ritz hypothesis.
Historical sketch
Connection to other theories
Gravity
Classical mechanics
The theory of relativity is in significant conflict with some aspects of classical mechanics. For example, Ehrenfest's paradox shows the incompatibility of STR with the concept of an absolutely rigid body. It should be noted that even in classical physics it is assumed that the mechanical effect on a solid body propagates at the speed of sound, and not at all at an infinite speed (as it should be in an imaginary absolutely solid medium).
Quantum mechanics
Special relativity (as opposed to general relativity) is completely compatible with quantum mechanics. Their synthesis is relativistic quantum field theory. However, both theories are completely independent of each other. It is possible to construct both quantum mechanics, based on Galileo’s non-relativistic principle of relativity (see Schrödinger’s equation), and theories based on SRT, which completely ignore quantum effects. For example, quantum field theory can be formulated as a non-relativistic theory. At the same time, such a quantum mechanical phenomenon as spin, sequentially cannot be described without invoking the theory of relativity (see Dirac equation).
The development of quantum theory is still ongoing, and many physicists believe that the future complete theory will answer all questions that have a physical meaning, and will provide within the limits of both STR in combination with quantum field theory and GRT. Most likely, SRT will face the same fate as Newtonian mechanics - the limits of its applicability will be precisely outlined. At the same time, such a maximally general theory is still a distant prospect.
see also
Notes
Sources
- Ginzburg V.L. Einstein collection, 1966. - M.: Nauka, 1966. - P. 363. - 375 p. - 16,000 copies.
- Ginzburg V.L. How and who created the theory of relativity? V Einstein collection, 1966. - M.: Nauka, 1966. - P. 366-378. - 375 p. - 16,000 copies.
- Satsunkevich I. S. Experimental roots of special relativity. - 2nd ed. - M.: URSS, 2003. - 176 p. - ISBN 5-354-00497-7
- Misner C., Thorne K., Wheeler J. Gravity. - M.: Mir, 1977. - T. 1. - P. 109. - 474 p.
- Einstein A. “Zur Elektrodynamik bewegter Korper” Ann Phys.- 1905.- Bd 17.- S. 891. Translation: Einstein A. “On the electrodynamics of a moving body” Einstein A. Collection of scientific works. - M.: Nauka, 1965. - T. 1. - P. 7-35. - 700 s. - 32,000 copies.
- Matveev A. N. Mechanics and theory of relativity. - 2nd edition, revised. - M.: Higher. school, 1986. - pp. 78-80. - 320 s. - 28,000 copies.
- Pauli W. Theory of relativity. - M.: Science, 3rd Edition, revised. - 328 p. - 17,700 copies. - ISBN 5-02-014346-4
- von Philip Frank und Hermann Rothe“Über die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme” Ann. der Physik, Ser. 4, Vol. 34, No. 5, 1911, pp. 825-855 (Russian translation)
- Fok V. A. Theory of space-time and gravity. - 2nd edition, supplemented. - M.: State Publishing House. physics and mathematics lit., 1961. - pp. 510-518. - 568 p. - 10,000 copies.
- "Lorentz Transformations" in the book "The Relativistic World".
- Kittel C., Nait U., Ruderman M. Berkeley Physics Course. - 3rd edition, revised. - M.: Nauka, 1986. - T. I. Mechanics. - pp. 373,374. - 481 p.
- von W.v. Ignatowsky“Einige allgemeine Bemerkungen zum Relativitätsprinzip” Verh. d. Deutsch. Phys. Ges. 12, 788-96, 1910 (Russian translation)
- Terletsky Ya. P. Paradoxes of the theory of relativity. - M.: Nauka, 1966. - P. 23-31. - 120 s. - 16,500 copies.
- Pauli W. Theory of relativity. - M.: Science, 3rd Edition, revised. - P. 27. - 328 p. - 17,700 copies. - ISBN 5-02-014346-4
- Landau, L. D., Lifshits, E. M. Field theory. - 7th edition, revised. - M.: Nauka, 1988. - 512 p. - (“Theoretical Physics”, volume II). - ISBN 5-02-014420-7
The special theory of relativity, created by Einstein in 1905, in its main content can be called the physical doctrine of space and time. Physical because the properties of space and
time in this theory are considered in close connection with the laws
physical phenomena occurring in them. The term "special"
emphasizes the fact that this theory considers phenomena only in inertial frames of reference.
Before moving on to its presentation, let us formulate the basic principles
Newtonian mechanics:
1) Space has 3 dimensions; Euclidean geometry is valid.
2) Time exists independently of space in the sense in which
three spatial dimensions are independent.
3) Time intervals and sizes of bodies do not depend on the reference system
4) The validity of the Newton-Galileo law of inertia is recognized (I law
5) When moving from one ISO to another, the Galilean transformations for coordinates, velocities and time are valid.
6) Galileo's principle of relativity is fulfilled: all inertial frames of reference are equivalent to each other with respect to mechanical phenomena.
7) The principle of long-range action is observed: the interactions of bodies propagate instantly, that is, with infinite speed.
These ideas of Newtonian mechanics were fully consistent with all
the totality of experimental data available at that time.
However, it was discovered that in a number of cases Newtonian mechanics did not work. The law of addition of velocities was the first to be tested. Galileo's principle of relativity stated that all ISOs are equivalent in their mechanical properties. But they can probably be distinguished by electromagnetic or some other properties. For example,
You can do experiments on the propagation of light. In accordance with
existing at that time wave theory there was a certain absolute
reference system (the so-called “ether”), in which the speed of light was equal
With. In all other systems, the speed of light had to obey
the law c’ = c - V. Michelson and then Morley undertook to test this assumption. The purpose of the experiment was to discover the “true”
motion of the Earth relative to the ether. The movement of the Earth along
orbit at a speed of 30 km per second.
travel time SAS
As the starting points of the special theory of relativity, Einstein
accepted two postulates, or principles, in favor of which the whole
experimental material (and primarily Michelson's experience ):
1) the principle of relativity,
2) independence of the speed of light from the speed of the source.
The first postulate is a generalization of the principle of relativity
Galileo for any physical processes:
all physical phenomena proceed in the same way in all inertial
reference systems; all the laws of nature and the equations that describe them,
invariant, i.e., do not change when moving from one inertial
frame of reference to another.
In other words, all inertial frames of reference are equivalent
(indistinguishable) in their own way, physical properties; no experience is possible in
in principle, highlight none of them as preferable.
The second postulate states that the speed of light in a vacuum does not depend on
movement of the light source and is the same in all directions.
This means that the speed light in a vacuum is the same in all ISOs. So
way , the speed of light occupies a special position in nature. Unlike
all other velocities that change during the transition from one frame of reference to
another, the speed of light in vacuum is an invariant quantity. Like us
we will see that the presence of such a speed significantly changes the idea of
space and time.
It also follows from Einstein's postulates that the speed of light in a vacuum is
ultimate: no signal, no influence of one body on another
can propagate at speeds exceeding the speed of light in a vacuum.
It is the limiting nature of this speed that explains the sameness
speed of light in all reference systems. In fact, according to the principle
relativity, the laws of nature must be the same in all
inertial reference systems. The fact that the speed of any signal is not
may exceed the limit value, there is also a law of nature.
Consequently, the value of the limiting speed - the speed of light in vacuum -
Must be the same in all inertial frames of reference: otherwise
In this case, these systems could be distinguished from each other.__
Lorentz transformations
Let us be given two reference systems k and k`. At the moment t = O both of these coordinate systems coincide. Let the system k` (let's call it mobile) move in such a way that the x` axis slides along the x axis, the y` axis is parallel to the y axis, the speed v- the speed of movement of this coordinate system (Fig. 109).
Point M has coordinates in the k system - x, y, z, and in the k` system - x`, y`, z`.
Galileo's transformations in classical mechanics have the form:
Coordinate transformations that satisfy the postulates of the special theory of relativity are called Lorentz transformations.
They were first proposed (in a slightly different form) by Lorentz to explain the negative Michelson-Morley experiment and to give Maxwell's equations the same form in all inertial frames of reference.
Einstein derived them independently based on his theory of relativity. We emphasize that not only the formula for transforming the coordinate x has changed (compared to the Galilean transformation), but also the formula for transforming time t. From the last formula you can directly see how spatial and temporal coordinates are intertwined.
Consequences from Lorentz transformations
Length of the moving rod.
Let us assume that the rod is located along the x` axis in the k` system and moves together with the k` system at a speed v.
The difference between the coordinates of the end and beginning of a segment in the reference system in which it is stationary is called own segment length. In our case l 0 = x 2 ` - x 1 `, where x 2 ` is the coordinate of the end of the segment in the k` system and x/ is the coordinate of the beginning. The rod moves relative to the system k. The length of a moving rod is taken to be the difference between the coordinates of the end and beginning of the rod at the same moment in time according to the clock of the k system.
Where l - moving rod length, l 0 - own length of the rod. The length of the moving rod is less than its own length.
The pace of a moving clock.
Let at point x 0 ` of the moving coordinate system k ` two events occur sequentially at moments t/ and t 2 . In a fixed coordinate system k, these events occur at different points at moments t 1 and t 2. The time interval between these events in a moving coordinate system is delta t` = t 2 ` - t 1 `, and in a stationary system delta t = t 2 - t 1 .
Based on the Lorentz transformation we obtain:
The time interval delta t` between events, measured by a moving clock, is less than the time interval delta t between the same events, measured by a stationary clock. This means that the pace of moving clocks is slower than that of stationary clocks.
Time that is measured by a clock connected to a moving point is called own time this point.
The relativity of simultaneity.
From the Lorentz transformations it follows that if in system k at a point with coordinates x 1 and x 2 two events occurred simultaneously (t 1 = t 2 = t 0), then in system k` the interval
the concept of simultaneity is a relative concept. Events that were simultaneous in one coordinate system turned out to be non-simultaneous in another.
Relativity of simultaneity and causation.
From the relativity of simultaneity it follows that the sequence of the same events in different coordinate systems is different.
Could it happen that in one coordinate system the cause precedes the effect, and in another, on the contrary, the effect precedes the cause?
In order for the cause-and-effect relationship between events to be objective and independent of the coordinate system in which it is considered, it is necessary that no material influences that carry out the physical connection of events occurring at different points can be transmitted at a speed greater than the speed of light.
Thus, the transfer of physical influence from one point to another cannot occur at a speed greater than the speed of light. Under this condition, the causal relationship of events is absolute: there is no coordinate system in which cause and effect change places.
Interval between two events
All physical laws of mechanics must be invariant under Lorentz transformations. The invariance conditions in the case of four-dimensional Minkowski space are a direct analogue of the invariance conditions when rotating the coordinate system in real three-dimensional space. For example, an interval in STR is invariant under Lorentz transformations. Let's look at this in more detail.
Any event is characterized by the point where it occurred, which has coordinates x, y, z and time t, i.e. each event occurs in four-dimensional space-time with coordinates x, y, z, t.
If the first event has coordinates x 1, y 1, z 1, t 1, the other with coordinates x 2, y 2, z 2, t 2, then the value
Let's find the value of the interval between two events in any ISO.
where t=t 2 - t 1, x=x 2 - x 1, у=y 2 - y 1, z=z 2 - z 1.
Interval between events in a moving ISO K *
(S *) 2 =c 2 (t *) 2 - (x *) 2 - (у *) 2 - (z *) 2 .
According to Lorentz transformations, we have for ISO K *
; у * =у; z * =z; .
With this in mind
|
(S *) 2 =c 2 t 2 - x 2 - у 2 - z 2 =S 2. |
Consequently, the interval between two events is invariant to the transition from one ISO to another.
RELATIVISTIC IMPULSE
The equations of classical mechanics are invariant with respect to Galilean transformations, but they turn out to be non-invariant with respect to Lorentz transformations. From the theory of relativity it follows that the equation of dynamics, invariant with respect to Lorentz transformations, has the form:
where is invariant, i.e. a quantity that is the same in all reference systems is called the rest mass of a particle, v is the speed of the particle, and is the force acting on the particle. Compare with the classical equation
We come to the conclusion that the relativistic momentum of the particle is equal to
Energy in relativistic dynamics.
For the energy of a particle in the theory of relativity, the following expression is obtained:
This quantity is called the rest energy of the particle. The kinetic energy is obviously equal to
From the last expression it follows that the energy and mass of a body are always proportional to each other. Any change in body energy is accompanied by a change in body weight
and, conversely, any change in mass is accompanied by a change in energy. This statement is called the law of relationship or the law of proportionality of mass and energy.
Mass and Energy
If a constant resultant force acts on a body with rest mass m 0, then the speed of the body increases. But the speed of a body cannot increase indefinitely, since there is a limiting speed c. On the other hand, with increasing speed comes an increase in body weight. Consequently, the work performed on the body leads not only to an increase in speed, but also to the body mass.
From the law of conservation of momentum, Einstein derived the following formula for the dependence of mass on velocity:
where m 0 is the mass of the body in the frame of reference in which the body is motionless (rest mass), m is the mass of the body in the frame of reference relative to which the body moves at speed v.
The momentum of a body in the special theory of relativity will have the following form:
Newton's second law will be valid in the relativistic region if it is written in the form:
Where R - r elativistic impulse.
Usually work done on the body increases its energy. This aspect of relativity led to the idea that mass is a form of energy, a defining feature of Einstein's theory of special relativity.
According to the law of conservation of energy, the work done on a particle is equal to its kinetic energy (KE) in the final state, since in the initial state the particle was at rest:
The quantity mс 2 is called total energy (we assume that the particle has no potential energy).
Based on the idea of mass as a form of energy, Einstein called m 0 c 2 the rest energy (or self-energy) of a body. This is how we get Einstein's famous formula
E = mс 2 .
If the particle is at rest, then its total energy is equal to E = m 0 s 2 (rest energy). If the particle is in motion and its speed is comparable to the speed of light, then its kinetic energy will be equal to: E k = mс 2 - m 0 с 2.
You sit facing the direction of the spaceship and look at the light bulb that is located in its bow. Light from a light bulb, regardless of its movement, moves relative to the stars at a speed of C = 300,000 km/s. You are moving towards the light with a speed, therefore, relative to you, the light must have a speed
You measure this speed, compare it with the known value of C and come to the conclusion that you are moving at a speed of 50,000 km/s, thus electromagnetic phenomena seem to allow you to distinguish rest from uniform rectilinear motion. That is, a paradox arises: on the one hand, the speed of light of 300,000 km/s should not depend on whether the light source is moving or at rest, on the other hand, according to the classical law of addition of velocities, it should depend on the choice of the reference system.
Different solutions were proposed, one of the opinions, of which Lorentz was a supporter, said: inertial frames of reference, equal in mechanical phenomena, are not equal in the laws of electrodynamics.
That is, in electrodynamics there is a certain privileged, main, absolute frame of reference, which scientists associated with the so-called ether.
American scientists Michelson and Morley tried to verify the validity of the presence of a reference system associated with the ether and the presence of this ether itself. They checked whether there was a so-called absolute reference system associated with the ether, and all other reference systems moving relative to it, that is, the so-called ethereal wind, which could influence the speed of light. And, as you have just seen, there is no ethereal wind. Physics of that time was faced with an insoluble paradox: what is true - classical mechanics, Maxwellian electrodynamics, or something else.
At the time of the publication of his work, Albert Einstein was not a recognized world scientist; the ideas that he expressed seemed so revolutionary that at first they had practically no supporters. Nevertheless, a huge number of experiments and measurements that were carried out after this showed the validity of Albert Einstein’s point of view.
Let us formulate once again the problems that physics faced at that time and talk about the solutions that Einstein proposed.
It is not possible to detect a privileged frame of reference associated with the motionless world ether.
Does this mean that it does not exist at all, this privileged absolute frame of reference does not exist? Albert Einstein expanded the action of Galileo's principle in mechanics to the whole of physics, and this is how Einstein's principle of relativity turned out: every physical phenomenon under the same initial conditions proceeds the same way in any inertial frame of reference.
That is, not any mechanical phenomenon, but any physical phenomenon.
The next difficulty: electrodynamics contradicts mechanics in that Maxwell’s equations are not invariant under Galilean transformations, that is, this is precisely the difficulty associated with the speed of light.
Maybe Maxwell is wrong? Nothing of the kind, Maxwell's electrodynamics is quite fair. Does this mean that all other areas of physics are unfair, the Galilean transformations that connect these parts of physics are incorrect? After all, from them follows the classical law of addition of speeds, which we use when solving problems, such as: a train travels at a speed of 40 km/h, and a passenger walks along the carriage at a speed of 5 km/h and relative to an observer on the ground, this passenger will move with speed of 45 km/h (Fig. 2).
Rice. 2. Example of classic addition of velocities ()
Einstein actually states: since Galileo’s transformations are unfair, then this law of addition of velocities is also unfair. A complete breakdown of foundations, an absolutely obvious life example, an absolutely obvious life law turns out to be unfair, what is the problem here? The problem is deep within the foundations of classical mechanics, which were laid by Newton. It turns out that the main problem with classical mechanics is that all interactions within mechanics are assumed to propagate instantaneously. Consider, for example, the gravitational attraction of bodies.
If you move one of the bodies to the side, then, according to the law of universal gravitation, the second body will feel this fact instantly as soon as the distance from it to the first body changes, that is, the interaction is transmitted at infinite speed. In reality, the interaction mechanism is as follows: changing the position of the first body changes the gravitational field around it. This change in the field begins to travel at some speed to all points in space, and when it reaches the point at which the second body is located, the interaction of the first and second bodies changes accordingly. That is, the speed of propagation of interaction has some finite value. But if interactions are transmitted at some finite speed, then in nature there must be some maximum permissible speed of propagation of these interactions, a maximum speed with which the interaction can be transmitted. This is stated by the second postulate, which assigns an exclusive role to the speed of light, the principle of invariance of the speed of light: in each inertial frame of reference, light moves in a vacuum at the same speed. The magnitude of this speed does not depend on whether the light source is at rest or moving.
Thus, we will not be able to carry out the example described above with a light bulb in a starship in reality; this will contradict this postulate of Einstein’s theory. The speed of light relative to the observer in the spaceship will be equal to C, and not C + V, as we said before, and the observer will not be able to notice the fact that the spaceship is moving. The classical law of adding speeds in relation to the speed of light does not work, strange as it may seem to us, but the speed of light for an observer on Earth and for an astronaut will be exactly the same and equal to 300,000 km/s. It is this position that underlies the theory of relativity and has been quite successfully proven by a huge number of experiments.
The mechanics that was built on the basis of these two postulates is called relativistic mechanics (from the English relativity - “relativity”). It may seem that relativistic mechanics cancels classical Newtonian mechanics, since it is based on different postulates, but the fact is that classical Newtonian mechanics is a special case of Einstein's relativistic mechanics, which manifests itself at speeds much lower than the speed of light. In the world around us we live at such speeds; the speeds we encounter are much less than the speed of light. Therefore, classical Newtonian mechanics is sufficient to describe our life.
For small speeds, significantly less than the speed of light, we quite successfully use classical mechanics, but if we work with speeds close to the speed of light, or want great accuracy in describing phenomena, we must use the special theory of relativity, that is, relativistic mechanics.
Bibliography
- Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - M.: Mnemosyne, 2012.
- Gendenshtein L.E., Dick Yu.I. Physics 10th grade. - M.: Mnemosyne, 2014.
- Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.
- Pppa.ru ().
- Sfiz.ru ().
- Eduspb.com ().
Homework
- Define Einstein's principle of relativity.
- Define Galileo's principle of relativity.
- Define Einstein's principle of invariance.
In September 1905 A. Einstein’s work “On the Electrodynamics of Moving Bodies” appeared, in which the main provisions of the Special Theory of Relativity (STR) were outlined. This theory meant a revision of the classical concepts of physics about the properties of space and time. Therefore, this theory in its content can be called a physical doctrine of space and time . Physical because the properties of space and time in this theory are considered in close connection with the laws of the physical phenomena occurring in them. The term " special"emphasizes the fact that this theory considers phenomena only in inertial frames of reference.
As the starting points of the special theory of relativity, Einstein accepted two postulates, or principles:
1) the principle of relativity;
2) the principle of independence of the speed of light from the speed of the light source.
The first postulate is a generalization of Galileo's principle of relativity to any physical processes: all physical phenomena proceed in the same way in all inertial frames of reference. All laws of nature and the equations that describe them are invariant, i.e. do not change when moving from one inertial reference system to another.
In other words, all inertial frames of reference are equivalent (indistinguishable) in their physical properties. No amount of experience can single out any of them as preferable.
The second postulate states that the speed of light in a vacuum does not depend on the movement of the light source and is the same in all directions.
It means that the speed of light in vacuum is the same in all inertial frames of reference. Thus, the speed of light occupies a special position in nature.
From Einstein's postulates it follows that the speed of light in a vacuum is limiting: no signal, no influence of one body on another can propagate at a speed exceeding the speed of light in a vacuum. It is the limiting nature of this speed that explains the same speed of light in all reference systems. The presence of a limiting speed automatically implies a limitation of the particle speed by a value of “c”. Otherwise, these particles could transmit signals (or interactions between bodies) at a speed exceeding the limit. Thus, according to Einstein’s postulates, the value of all possible speeds of movement of bodies and the propagation of interactions is limited by the value “c”. This rejects the principle of long-range action of Newtonian mechanics.
Interesting conclusions follow from SRT:
1) LENGTH REDUCTION: The movement of any object affects the measured value of its length.
2) TIME SLOW DOWN: with the advent of SRT, the statement arose that absolute time has no absolute meaning, it is only an ideal mathematical representation, because in nature there is no real physical process suitable for measuring absolute time.
The passage of time depends on the speed of movement of the reference frame. At a sufficiently high speed, close to the speed of light, time slows down, i.e. relativistic time dilation occurs.
Thus, in a rapidly moving system, time flows more slowly than in the laboratory of a stationary observer: if an observer on Earth were able to follow a clock in a rocket flying at high speed, he would come to the conclusion that it was running slower than his own. The time dilation effect means that the inhabitants of the spaceship age more slowly. If one of the two twins made a long space journey, then upon returning to Earth he would find that his twin brother left at home was much older than him.
In some system we can only talk about local time. In this regard, time is not an entity independent of matter; it flows at different speeds under different physical conditions. Time is always relative.
3) INCREASE IN WEIGHT: The mass of a body is also a relative quantity, depending on the speed of its movement. The greater the speed of a body, the greater its mass becomes.
Einstein also discovered the connection between mass and energy. He formulates the following law: “the mass of a body is a measure of the energy contained in it: E=mс 2 ". If we substitute m=1 kg and c=300,000 km/s into this formula, then we get a huge energy of 9·10 16 J, which would be enough to burn an electric light bulb for 30 million years. But the amount of energy in the mass of a substance is limited by the speed of light and the amount of mass of the substance.
The world around us has three dimensions. SRT argues that time cannot be considered as something separate and unchanging. In 1907, the German mathematician Minkowski developed the mathematical apparatus of SRT. He suggested that three spatial and one temporal dimensions are closely related. All events in the Universe occur in four-dimensional space-time. From a mathematical point of view, SRT is the geometry of four-dimensional Minkowski space-time.
SRT has been confirmed on extensive material, by many facts and experiments (for example, time dilation is observed during the decay of elementary particles in cosmic rays or in high-energy accelerators) and underlies theoretical descriptions of all processes occurring at relativistic speeds.
So, the description of physical processes in SRT is essentially connected with the coordinate system. Physical theory does not describe the physical process itself, but the result of the interaction of the physical process with the means of research. Therefore, for the first time in the history of physics, the activity of the subject of cognition, the inseparable interaction of the subject and object of cognition, was directly manifested.