Special theory of relativity. Special theory of relativity

After mathematicians created rules in the space of concepts and numbers, scientists were sure that they could only carry out experiments and, using logical constructions, explain the structure of all things. Within reasonable limits, the laws of mathematics work. But experiments that go beyond everyday concepts and ideas require new principles and laws.

Idea

In the middle of the 19th century, the convenient idea of ​​a universal ether spread everywhere, which suited most scientists and researchers. The mysterious ether became the most common model explaining the physical processes known at that time. But many inexplicable facts were gradually added to the mathematical description of the ether hypothesis, which were explained by various additional conditions and assumptions. Gradually, the coherent theory of the ether acquired “crutches”; there were too many of them. New ideas were needed to explain the structure of our world. The postulates of the special theory of relativity met all the requirements - they were concise, consistent and fully confirmed by experiments.

Michelson's experiments

The last straw that “broke the back” of the ether hypothesis was research in the field of electrodynamics and Maxwell’s equations explaining them. When bringing the results of experiments to a mathematical solution, Maxwell used the theory of the ether.

In their experiment, the researchers caused two beams coming in different directions to be emitted synchronously. Given that light moves in the "ether", one ray of light should have moved slower than the other. Despite numerous repetitions of the experiment, the result was the same - the light moved at a constant speed.

Otherwise it was impossible to explain the fact that, according to calculations, the speed of light in the hypothetical ether was always the same, regardless of how fast the observer was moving. But in order to explain the results of the research, it was required that the frame of reference be “ideal.” And this contradicted Galileo’s postulate about the invariance of all inertial frames of reference.

New theory

At the beginning of the twentieth century, a whole galaxy of scientists began to develop a theory that would reconcile the results of studies of electromagnetic oscillations with the principles of classical mechanics.

When developing the new theory, it was taken into account that:

Motion at near light speeds changes the formula of Newton's second law, which relates acceleration to force and mass;

The equation for the momentum of a body must have a different, more complex formula;

The speed of light remained constant, regardless of the chosen frame of reference.

The efforts of A. Poincaré, G. Lorentz and A. Einstein led to the creation of the special theory of relativity, which reconciled all the shortcomings and explained the existing observations.

Basic Concepts

The foundations of the special theory of relativity lie in the definitions with which this theory operates

1. Reference system - a material body that can be taken as the origin of the reference system and the time coordinate during which the observer will monitor the movement of objects.

2. An inertial frame of reference is one that moves uniformly and in a straight line.

3. Event. The special and general theories of relativity consider an event as a physical process localized in space with a limited duration. The coordinates of an object can be specified in three-dimensional space as (x, y, z) and a time period t. A standard example of such a process is a light flash.

The special theory of relativity considers inertial frames of reference in which the first frame moves near the second with constant speed. In this case, the search for relationships between object coordinates in these inertial systems is a priority for SRT and is included in its main tasks. The special theory of relativity was able to solve this issue using Lorentz's formulas.

Postulates of SRT

When developing the theory, Einstein discarded all the numerous assumptions that were necessary to support the theory of the ether. Simplicity and mathematical provability were the two pillars on which his special theory of relativity rested. Briefly, its premises can be reduced to two postulates that were necessary to create new laws:

1. All physical laws in inertial systems are carried out equally.
2. The speed of light in a vacuum is constant, it does not depend on the location of the observer and his speed.

These postulates of the special theory of relativity rendered the theory of the mythical ether useless. Instead of this substance, the concept of four-dimensional space was proposed, linking time and space together. When indicating the location of a body in space, it is necessary to take into account the fourth coordinate - time. This idea seems rather artificial, but it should be taken into account that the confirmation of this point of view lies within the limits of speeds commensurate with the speed of light, and in the everyday world the laws of classical physics do their job perfectly. Galileo's principle of relativity holds true for all inertial frames of reference: if the rule F = ma is observed in CO k, then it will be correct in another frame of reference k’. In classical physics, time is a definite quantity, and its value is unchanged and does not depend on the movement of the inertial reference.

Conversions to service stations

Briefly, the coordinates of the point and time can be denoted as follows:

x" = x - vt and t" = t.

This formula is given by classical physics. The special theory of relativity offers this formula in a more complicated form.

In this equation, the quantities (x,x’ y,y’ z,z’ t,t’) denote the coordinates of the object and the passage of time in the observed frames of reference, v is the speed of the object, and c is the speed of light in vacuum.

The velocities of objects in this case must correspond to non-standard Galilean

formula v= s/t, and this Lorentz transformation:

As can be seen, at a negligibly low speed of the body, these equations degenerate into the well-known equations of classical physics. If we take the other extreme and set the speed of the object to be equal to the speed of light, then in this limiting case we still get c. From here, the special theory of relativity concludes that not a single body in the observable world can move at a speed exceeding the speed of light.

Consequences of STO

Upon further consideration of Lorentz transformations, it becomes clear that non-standard things begin to happen to standard objects. The consequences of special relativity are changes in the length of an object and the passage of time. If the length of a segment in one reference system is equal to l, then observations from another OS will give the following value:

Thus, it turns out that an observer from the second frame of reference will see a segment shorter than the first.

Amazing transformations also affected such a quantity as time. The equation for the t coordinate will look like this:

As you can see, time flows slower in the second frame of reference than in the first. Naturally, both of these equations will give results only at speeds comparable to the speed of light.

Einstein was the first to derive the formula for time dilation. He also proposed to solve the so-called “twin paradox.” According to the conditions of this problem, there are twin brothers, one of whom remained on Earth, and the second flew on a rocket into space. According to the formula written above, brothers will age differently since time passes slower for the traveling brother. This paradox has a solution if we take into account that the stay-at-home brother was always in an inertial frame of reference, and the restless twin was traveling in a non-inertial frame of reference, which was moving with acceleration.

Mass change

Another consequence of STR is the change in the mass of the observed object in different FRs. Since all physical laws operate equally in all inertial frames of reference, the fundamental laws of conservation - momentum, energy and angular momentum - must be observed. But since the speed for an observer in a stationary CO is greater than in a moving one, then, according to the law of conservation of momentum, the mass of the object should change by the amount:

In the first frame of reference, the object must have a greater body mass than in the second.

Taking the speed of the body equal to the speed of light, we get an unexpected conclusion - the mass of the object reaches an infinite value. Of course, any material body in the observable universe has its own finite mass. The equation only says that no physical object can move at the speed of light.

Mass/Energy Relationship

When the speed of an object is much less than the speed of light, the equation for mass can be reduced to the form:

The expression m 0 c represents a certain property of an object that depends only on its mass. This quantity is called rest energy. The sum of the energies of rest and motion can be written as follows:

mc 2 = m 0 c + E kin.

It follows that the total energy of an object can be expressed by the formula:

The simplicity and elegance of the body energy formula gave completeness,

where E is the total energy of the body.

The simplicity and elegance of Einstein's famous formula gave completeness to the special theory of relativity, making it internally consistent and not requiring many assumptions. Thus, the researchers explained many contradictions and gave impetus to the study of new natural phenomena.

Einstein's special theory of relativity (STR) expands the boundaries of classical Newtonian physics, which operates in the region of non-relativistic speeds, small compared to the speed of light c, to any, including relativistic ones, i.e. comparable to c, speeds. All results of relativistic theory at transform into results of classical non-relativistic physics (correspondence principle).

Postulates of SRT. The special theory of relativity is based on two postulates:

The first postulate (Einstein's principle of relativity): all physical laws - both mechanical and electromagnetic - have the same form in all inertial frames of reference (IRS). In other words, no experiments can single out any one frame of reference and call it at rest. This postulate is an extension of Galileo's principle of relativity (see Section 1.3) to electromagnetic processes.

Einstein's second postulate: the speed of light in a vacuum is the same for all ISOs and is equal to c. This postulate contains two statements at once:

a) the speed of light does not depend on the speed of the source,

b) the speed of light does not depend on the ISO in which the observer with instruments is located, i.e. does not depend on the speed of the receiver.

The constancy of the speed of light and its independence from the movement of the source follow from Maxwell's equations of the electromagnetic field. It seemed obvious that such a statement could only be true in one frame of reference. From the point of view of classical ideas about space-time, any other observer, moving with speed, must obtain speed for an oncoming ray, and for a forward-emitted ray - speed. Such a result would mean that Maxwell's equations are satisfied only in one ISO, filled with a stationary ether, relative to which light waves propagate. However, an attempt to detect a change in the speed of light associated with the movement of the Earth relative to the ether gave a negative result (Michelson-Morley experiment). Einstein suggested that Maxwell's equations, like all laws of physics, have the same form in all ISOs, i.e. that the speed of light in any ISO is equal to c (second postulate). This assumption led to a revision of the basic concepts of space and time.

Lorentz transformations. The Lorentz transformations connect the coordinates and time of an event, measured in two ISOs, one of which moves relative to the other with a constant speed V. With the same choice of coordinate axes and time reference as in the Galilean transformations (formula (7)), the Lorentz transformations have view:

It is often convenient to use transformations for the difference between the coordinates and times of two events:

where for brevity the notation is introduced

The Lorentz transformations transform into the Galilean transformations at . They are derived from the second postulate of SRT and from the requirement of linearity of transformations, expressing the condition of homogeneity of space. Inverse transformations from to K can be obtained from (42), (43) by replacing V with -V:

Length reduction. The length of a moving segment is defined as the distance between the points where the ends of the segment were located simultaneously (i.e. Consider a rigid body that moves translationally with speed and associate a reference system with it. From equation (43) (in which we must put we obtain that the longitudinal dimensions of the moving bodies contract:

where is the own longitudinal size, i.e. measured in the reference frame K, in which the body is motionless. The transverse dimensions of a moving body do not change.

Example 1. If a square moves with speed along one of its sides, then it turns into a rectangle with an angle between the diagonals equal to .

Relativity of the passage of time. From the Lorentz transformations it is clear that time flows differently in different ISOs. In particular, events occurring in system K simultaneously but

at different points in space, in K may not be simultaneous: it can be both positive and negative (the relativity of simultaneity). A clock moving with the reference frame (i.e., stationary relative to or showing the proper time of this ISO. From the point of view of an observer in frame A, these clocks lag behind his own (time slowdown). Considering two readings of a moving clock as two events, from (45) we get:

where is the proper time of the moving clock (more precisely, the associated equality of all ISOs is manifested in the fact that from the point of view of observer K, clocks stationary relative to , will lag behind his own. (Note that in order to control a moving clock, a stationary observer at different moments time uses different clocks.) The twin paradox is that SRT predicts a difference in age between two twins, one of whom remained on Earth and the other of whom traveled in deep space (the astronaut will be younger); this would seem to violate the equality of their frames of reference. In fact, only the earthly twin was in the same ISO all the time, while the astronaut changed the ISO to return to Earth (his own frame of reference is non-inertial).

Example 2. Average proper lifetime of an unstable muon, i.e. Due to the effect of time dilation, from the point of view of an earthly observer, a cosmic muon, flying at a speed close to the speed of light (7 1), lives on average and flies from its birthplace in the upper atmosphere a distance of the order of magnitude, which allows it to be recorded on the surface of the Earth.

Addition of speeds in the service station. If a particle moves with a speed relative to then its speed relative to K can be found by expressing from (45) and substituting in

At c there is a transition to the non-relativistic law of addition of velocities (formula). An important property of formula (48) is that if V and is less than c, then it will be less than c. For example, if we accelerate a particle to and then, moving to its reference frame, Let's accelerate it again until the resulting speed turns out to be no. It can be seen that it is not possible to exceed the speed of light. The speed of light is the maximum possible speed of transmission of interactions in nature.

Interval. Causality. Lorentz transformations do not preserve either the value of the time interval or the length of the spatial segment. However, it can be shown that under Lorentz transformations the quantity

where is called the interval between events 1 and 2. If then the interval between events is called timelike, since in this case there is an ISO in which i.e. events take place in one place, but at different times. Such events may be causally related. If, on the contrary, then the interval between events is called space-like, since in this case there is an ISO in which, i.e. events occur simultaneously at different points in space. There cannot be a causal relationship between such events. The condition means that a ray of light emitted at the moment of an earlier event (for example, from a point does not have time to reach the point by the moment of time. Events separated from event 1 by a time-like interval represent in relation to it either the absolute past or the absolute future, the sequence of these events is the same in all ISOs.The sequence of events separated by a space-like interval may be different in different ISOs.

Lorentz 4-vectors. Four quantities that, when moving from system K to system K, are transformed in the same way as i.e. (see (42)):

is called a Lorentz four-dimensional vector (or, for short, a Lorentz -vector). The quantities are called the spatial components of the vector, and its time component. The sum of two -vectors and the product of a -vector and a number are also -vectors. When changing the ISO, a value similar to the interval is preserved: as well as the scalar product. Physical equality, written as the equality of two -vectors, remains true in all ISOs.

Momentum and energy in service stations. The velocity components transform differently than the 4-vector components (compare equations (48) and (50)) because both the numerator and the denominator are transformed in the expression. Therefore, the value corresponding to the classical definition of momentum cannot be conserved in

all ISOs. The relativistic momentum vector is defined as

where is the infinitesimal change in the particle’s own time (see (47)), i.e. measured in an ISO whose speed is equal to the speed of the particle at a given moment does not depend on from which ISO we observe the particle.) The spatial components of the -vector form the relativistic impulse

and the time component turns out to be equal to where E is the relativistic energy of the particle:

Relativistic energy includes all types of internal energy.

Example 3. Let the energy of a body at rest increase by Find the momentum of this body in a reference frame moving with speed .

Solution. In accordance with the relativistic transformation formulas (54), the momentum is equal to It can be seen that the increase in mass corresponds to formula (58).

Basic law of relativistic dynamics. The force applied to the particle is equal, as in classical mechanics, to the derivative of momentum:

but the relativistic impulse (51) differs from the classical one. Under the action of an applied force, the momentum can increase without limit, but from definition (51) it is clear that the speed will be less than c. Work of force (59)

equal to the change in relativistic energy. Here the formulas were used (see (56)) and .

In 1905, Albert Einstein published his special theory of relativity (STR), which explained how to interpret motions between different inertial reference frames—simply put, objects that move at a constant speed relative to each other.

Einstein explained that when two objects are moving at constant speed, one should consider their motion relative to each other, rather than taking one of them as an absolute frame of reference.

So if two astronauts, you and, say, Herman, are flying on two spacecraft and want to compare your observations, the only thing you need to know is your speed relative to each other.

The special theory of relativity considers only one special case (hence the name), when the motion is rectilinear and uniform. If a material body accelerates or turns to the side, the laws of STR no longer apply. Then the general theory of relativity (GTR) comes into force, which explains the movements of material bodies in the general case.

Einstein's theory is based on two main principles:

1. The principle of relativity: physical laws are preserved even for bodies that are inertial frames of reference, that is, moving at a constant speed relative to each other.

2. Speed ​​of Light Principle: The speed of light remains the same for all observers, regardless of their speed relative to the light source. (Physicists designate the speed of light as c).

One of the reasons for Albert Einstein's success is that he valued experimental data over theoretical data. When a number of experiments revealed results that contradicted the generally accepted theory, many physicists decided that these experiments were wrong.

Albert Einstein was one of the first who decided to build a new theory based on new experimental data.

At the end of the 19th century, physicists were in search of the mysterious ether - a medium in which, according to generally accepted assumptions, light waves should propagate, like acoustic waves, the propagation of which requires air, or another medium - solid, liquid or gaseous. Belief in the existence of the ether led to the belief that the speed of light should vary depending on the speed of the observer in relation to the ether.

Albert Einstein abandoned the concept of the ether and assumed that all physical laws, including the speed of light, remain unchanged regardless of the speed of the observer - as experiments showed.

Homogeneity of space and time

Einstein's SRT postulates a fundamental connection between space and time. The material Universe, as we know, has three spatial dimensions: up-down, right-left and forward-backward. Another dimension is added to it - time. Together these four dimensions make up the space-time continuum.

If you are moving at a high speed, your observations of space and time will be different from those of other people moving at a slower speed.

The picture below is a thought experiment that will help you understand this idea. Imagine that you are on a spaceship, in your hands you have a laser, with which you send rays of light to the ceiling on which a mirror is mounted. The light, reflected, falls on the detector, which registers them.

From above - you sent a beam of light to the ceiling, it was reflected and fell vertically onto the detector. Bottom - For Herman, your beam of light moves diagonally to the ceiling, and then diagonally to the detector

Let's say your ship is moving at a constant speed equal to half the speed of light (0.5c). According to Einstein's SRT, this doesn't matter to you; you don't even notice your movement.

However, Herman, watching you from a resting starship, will see a completely different picture. From his point of view, a beam of light will pass diagonally to the mirror on the ceiling, be reflected from it and fall diagonally onto the detector.

In other words, the path of the light beam will look different for you and for Herman and its length will be different. And therefore, the length of time it takes for the laser beam to travel the distance to the mirror and to the detector will seem different to you.

This phenomenon is called time dilation: time on a starship moving at high speed flows much more slowly from the point of view of an observer on Earth.

This example, as well as many others, clearly demonstrates the inextricable connection between space and time. This connection clearly appears to the observer only when we are talking about high speeds, close to the speed of light.

Experiments conducted since Einstein published his great theory have confirmed that space and time are indeed perceived differently depending on the speed of objects.

Combining mass and energy

According to the theory of the great physicist, when the speed of a material body increases, approaching the speed of light, its mass also increases. Those. The faster an object moves, the heavier it becomes. If the speed of light is reached, the mass of the body, as well as its energy, become infinite. The heavier the body, the more difficult it is to increase its speed; Accelerating a body with infinite mass requires an infinite amount of energy, so it is impossible for material objects to reach the speed of light.

Before Einstein, the concepts of mass and energy were considered separately in physics. The brilliant scientist proved that the law of conservation of mass, as well as the law of conservation of energy, are parts of the more general law of mass-energy.

Thanks to the fundamental connection between these two concepts, matter can be turned into energy, and vice versa - energy into matter.

The content of the article

RELATIVITY SPECIAL THEORY – modern theory of space and time, in the most general form establishing a connection between events in space-time and determining the form of recording physical laws that does not change when moving from one inertial reference system to another. The key to the theory is a new understanding of the concept of simultaneity of events, formulated in the seminal work of A. Einstein On the electrodynamics of moving media(1905) and based on the postulate of the existence of a maximum speed of signal propagation - the speed of light in a vacuum. The special theory of relativity generalizes the ideas of classical Galileo-Newton mechanics to the case of bodies moving at speeds close to the speed of light.

Disputes about broadcasting.

Since the wave nature of light was established, physicists have been confident that there must be a medium (it was called the ether) in which light waves propagate. This point of view was confirmed by all the experience of classical physics, examples of acoustic waves, waves on the surface of water, etc. When J.C. Maxwell proved that there must be electromagnetic waves that travel through empty space at the speed of light c, he had no doubt that these waves must propagate in some medium. G. Hertz, who was the first to register the radiation of electromagnetic waves, adhered to the same point of view. Since electromagnetic waves turned out to be transverse (this follows from Maxwell’s equations), Maxwell had to build an ingenious mechanical model of a medium in which transverse waves could propagate (this is only possible in very elastic solids) and which at the same time would be completely permeable and did not interfere with the movement of bodies through it. These two requirements contradict each other, but until the beginning of this century it was not possible to propose a more reasonable theory of the propagation of light in vacuum.

The hypothesis about the existence of the ether entails a number of obvious consequences. The simplest of them: if the receiver of a light wave moves towards the source at a speed v relative to the ether, then according to the laws of classical physics, the speed of light relative to the receiver should be equal to the speed of light relative to the ether (which is naturally considered constant) plus the speed of the receiver relative to the ether (Galileo’s law of addition of velocities): Withў = c + v. Similarly, if the source moves at a speed v towards the receiver, then the relative speed of light should be equal to Withў = c - v. Thus, if the ether exists, then there is a certain absolute frame of reference, relative to which (and only relative to it) the speed of light is equal With, and in all other reference systems uniformly moving relative to the ether, the speed of light is not equal With. Whether this is true or not can only be decided with the help of a direct experiment, which consists in measuring the speed of light in different reference frames. It is clear that it is necessary to find such reference frames that move with maximum speed, especially since it can be proven that all observed effects of the deviation of the speed of light from the value With, associated with the movement of one reference system relative to another, must be of the order v 2/c 2. A suitable object seems to be the Earth, which revolves around the Sun with linear speed v~ 10 4 m/s, so the corrections should be of the order of ( v/c) 2 ~ 10 –8 . This value seems extremely small, but A. Michelson managed to create a device - the Michelson interferometer, which was capable of recording such deviations.

In 1887, A. Michelson, together with his colleague Yu. Morley, measured the speed of light in a moving frame of reference. The idea of ​​experience is reminiscent of measuring the time a swimmer spends crossing a river across the current and back, and swimming the same distance along and against the current. The answer was stunning: the movement of the reference system relative to the ether does not have any effect on the speed of light.

Generally speaking, two conclusions can be drawn from this. Perhaps the ether exists, but when bodies move through it, it is completely carried away by the moving bodies, so that the speed of the bodies in relation to the ether is zero. This entrainment hypothesis was tested experimentally in the experiments of Fizeau and Michelson himself and turned out to contradict the experiment. John Bernal called the famous Michelson–Morley experiment the most outstanding negative experiment in the history of science. The second possibility remained: no ether that could be experimentally detected exists, in other words, there is no distinguished absolute frame of reference in which the speed of light is equal to With; on the contrary, this speed is the same in all inertial frames of reference. It was this point of view that became the foundation of the new theory.

The special (particular) theory of relativity (STR), which successfully resolved all the contradictions associated with the problem of the existence of the ether, was created by A. Einstein in 1905. An important contribution to the development of SRT was made by H.A. Lorenz, A. Poincaré and G. Minkowski.

The special theory of relativity had a revolutionary impact on physics, marking the end of the classical stage of development of this science and the transition to modern physics of the 20th century. First of all, the special theory of relativity completely changed the views on space and time that existed before its creation, showing the inextricable connection of these concepts. Within the framework of SRT, the concept of simultaneity of events was clearly formulated for the first time and the relativity of this concept and its dependence on the choice of a specific reference system was shown. Secondly, STR completely resolved all the problems associated with the hypothesis of the existence of the ether, and made it possible to formulate a harmonious and consistent system of equations of classical physics, which replaced the Newtonian equations. Thirdly, STR became the basis for the construction of fundamental theories of interactions of elementary particles, primarily quantum electrodynamics. The accuracy of experimentally verified predictions of quantum electrodynamics is 10 –12, which characterizes the accuracy with which we can talk about the validity of STR.

Fourthly, SRT has become the basis for calculating the energy release in nuclear decay and fusion reactions, i.e. the basis for the creation of both nuclear power plants and atomic weapons. Finally, the analysis of data obtained from particle accelerators, as well as the design of the accelerators themselves, are based on SRT formulas. In this sense, SRT has long become an engineering discipline.

Four-dimensional world.

A person does not exist in a three-dimensional spatial world, but in a four-dimensional world of events (an event is understood as a physical phenomenon at a given point in space at a given moment in time). An event is characterized by specifying three spatial coordinates and one time coordinate. Thus, every event has four coordinates: ( t; x, y, z). Here x, y, z– spatial coordinates (for example, Cartesian). To determine the coordinates of an event, you should set (or be able to set): 1) the origin of the coordinates; 2) an infinite rigid lattice of mutually perpendicular rods of unit length filling the entire space; further, you should: 3) place an identical clock at each lattice node (i.e., a device capable of counting equal periods of time; the specific device does not matter); 4) synchronize clocks. Then any point in space located near a lattice node has as spatial coordinates the number of nodes along each of the axes from the origin and a time coordinate equal to the clock readings at the nearest node. All points with four coordinates fill a four-dimensional space called space-time. The key question for physics is the question of geometry this space.

To describe events in space-time, it is convenient to use space-time diagrams, which depict the sequence of events for a given body. If (for illustration) we restrict ourselves to two-dimensional ( x,t)-space, then a typical space-time diagram of events in classical physics looks as shown in Fig. 1.

Horizontal axis x corresponds to all three spatial coordinates ( x, y, z), vertical – time t, and the direction from the “past” to the “future” corresponds to the movement from bottom to top along the axis t.

Any point on a horizontal line intersecting an axis t below zero, corresponds to the position of some object in space at a moment in time (in the past relative to an arbitrarily chosen point in time t= 0). So, in Fig. 1 body was at the point A 1 space at a time t 1. Points of a horizontal line coinciding with the axis x, depict the spatial position of bodies at a given moment in time t= 0 (dot A 0). A straight line drawn above the axis x, corresponds to the position of bodies in the future (point A 2 – the position that the body will occupy at the moment of time t 2). If you connect the dots A 1, A 0, A 2, you get a world line bodies. Obviously, the position of the body in space does not change (spatial coordinates remain constant), so this world line represents a body at rest.

If the world line is straight, inclined at a certain angle (straight IN 1IN 0IN 2 in Fig. 1), this means that the body moves at a constant speed. The smaller the angle between the world line and the horizontal plane, the greater the speed of the body. Within the framework of classical physics, the inclination of the world line can be anything, since the speed of a body is not limited by anything.

This statement about the absence of a limit to the speed of motion of bodies is implicitly contained in Newtonian mechanics. It allows us to give meaning to the concept of simultaneity of events without reference to a specific observer. Indeed, moving at a finite speed, from any point WITH 0 on a surface of equal time one can get to a point WITH 1, corresponding to a later time. Possible from an earlier point WITH 2 get to the point WITH 0. However, it is impossible, moving with finite speed, to move from the point WITH 0 to any points A, IN,...on the same surface. All events on this surface are simultaneous (Fig. 2). You can put it another way. Let there be identical clocks at each point in three-dimensional space. Ability to transmit signals With infinitely high speed means that it is possible to simultaneously synchronize all clocks, no matter how far from each other they are and no matter how fast they move (indeed, the exact time signal reaches all clocks instantly). In other words, within the framework of classical mechanics, the progress of a clock does not depend on whether it is moving or not.

The concept of simultaneity of events according to Einstein.

Within the framework of Newtonian mechanics, all simultaneous events lie in the “plane” of fixed time t, completely occupying three-dimensional space (Fig. 2). Geometric relationships between points in three-dimensional space obey the laws of ordinary Euclidean geometry. Thus, the space-time of classical mechanics is divided into space and time independent from each other.

The key to understanding the foundations of STR is that it is impossible to imagine space-time as independent of each other. The course of clocks at different points of a single space-time is different and depends on the speed of the observer. This amazing fact is based on the fact that signals cannot propagate at infinite speed (failure to operate at a distance).

The following thought experiment allows us to better understand the meaning of the concept of simultaneity. Suppose that at two opposite walls of a train car moving at a constant speed v, flashes of light were simultaneously produced. For an observer located in the middle of the car, flashes of light from the sources will arrive simultaneously. From the point of view of an external observer standing on the platform, a flash will come earlier from the source that is approaching the observer. All these considerations imply that light travels at a finite speed.

Thus, if we abandon long-range action, in other words, the possibility of transmitting signals at an infinitely high speed, then the concept of simultaneity of events becomes relative, dependent on the observer. This change in the view of simultaneity is the most fundamental difference between STR and pre-relativistic physics.

To define the concept of simultaneity and synchronization of clocks located at different spatial points, Einstein proposed the following procedure. Let from the point A a very short light signal is sent in a vacuum; when sending a signal, the clock is at the point A show time t 1 . The signal arrives at the point IN at the moment when the clock is at the point IN show time t". After reflection at a point IN the signal returns to the point A, so that at the moment the clock arrives A show time t 2. By definition, hours in A And IN synchronized if at point IN the clock is set so that t" = (t 1 + t 2)/2.

Postulates of the special theory of relativity.

1. The first postulate is the principle of relativity, which states that from all conceivable movements of bodies one can distinguish (without reference to the movement of other bodies) a certain class of movements called non-accelerated, or inertial. The reference frames associated with these motions are called inertial reference frames. In the class of inertial systems there is no way to distinguish a moving system from a stationary one. The physical content of Newton's first law is a statement about the existence of inertial frames of reference.

If there is one inertial system, this means that there are infinitely many of them. Any reference system moving relative to the first one with constant speed is also inertial.

The principle of relativity states that all equations of all physical laws have the same form in all inertial frames of reference, i.e. physical laws are invariant with respect to the transition from one inertial frame of reference to another. It is important to establish what formulas determine the transformation of coordinates and time of an event during such a transition.

In classical Newtonian physics, the second postulate is an implicit statement about the possibility of signals propagating at an infinitely high speed. This leads to the possibility of simultaneous synchronization of all clocks in space and to the independence of the clock from the speed of their movement. In other words, when moving from one inertial system to another, time does not change: tў = t. Then the formulas for transforming coordinates when moving from one inertial reference system to another (Galilean transformations) become obvious:

xў = x – vt, yў = y, zў = z, tў = t.

The equations expressing the laws of classical mechanics are invariant under Galilean transformations, i.e. do not change their shape when moving from one inertial frame of reference to another.

In the special theory of relativity, the principle of relativity applies to all physical phenomena and can be expressed as follows: no experiments (mechanical, electrical, optical, thermal, etc.) make it possible to distinguish one inertial frame of reference from another, i.e. There is no absolute (observer-independent) way to know the speed of an inertial reference frame.

2. The second postulate of classical mechanics about the unlimited speed of propagation of signals or the movement of bodies is replaced in STR by the postulate about the existence of a limiting speed of propagation of physical signals, numerically equal to the speed of propagation of light in vacuum

With= 2.99792458·10 8 m/s.

More precisely, the STR postulates the independence of the speed of light from the speed of movement of the source or receiver of this light. After this it can be proven that With is the maximum possible speed of signal propagation, and this speed is the same in all inertial reference frames.

What will space-time diagrams look like now? To understand this, we should turn to the equation that describes the propagation of the front of a spherical light wave in vacuum. Let in the moment t= 0 there was a flash of light from a source located at the origin ( x, y, z) = 0. At any subsequent time t> 0 the front of the light wave will be a sphere with a radius l = ct, expanding evenly in all directions. The equation of such a sphere in three-dimensional space has the form:

x 2 + y 2 + z 2 = c 2t 2 .

On the space-time diagram, the world line of the light wave will be depicted as straight lines inclined at an angle of 45° to the axis x. If we take into account that the coordinate x If the diagram actually corresponds to the set of all three spatial coordinates, then the equation of the light wave front defines a certain surface in the four-dimensional space of events, which is usually called the light cone.

Each point on the space-time diagram is an event that occurred in a certain place at a certain point in time. Let the point ABOUT in Fig. 3 corresponds to some event. In relation to this event, all other events (all other points on the diagram) are divided into three areas, conventionally called the cones of the past and future and the space-like area. All events within the cone of the past (for example, the event A on the diagram) occur at such moments in time and at such a distance from ABOUT so that you can reach the point ABOUT, moving at a speed not exceeding the speed of light (from geometric considerations it is clear that if v > c, then the inclination of the world line to the axis x decreases, i.e. the angle of inclination becomes less than 45°; and vice versa if v c, then the angle of inclination to the axis x becomes more than 45°). Likewise, the event IN lies in the cone of the future, since this point can be reached by moving at speed v c.

A different situation with events in a space-like region (for example, the event WITH). For these events, the relationship between the spatial distance to the point ABOUT and the time is such that to get to ABOUT is possible only by moving at superluminal speed (the dotted line in the diagram depicts the world line of such prohibited motion; it can be seen that the inclination of this world line to the x axis is less than 45°, i.e. v > c).

So, all events in relation to a given one are divided into two nonequivalent classes: those lying inside the light cone and outside it. The first events can be realized by real bodies moving at speed v c, the second - no.

Lorentz transformations.

The formula describing the propagation of the front of a spherical light wave can be rewritten as:

c 2t 2 – x 2 – y 2 – z 2 = 0.

Let s 2 = c 2t 2 – x 2 – y 2 – z 2. Magnitude s called an interval. Then the equation for the propagation of a light wave (the equation of a light cone on a space-time diagram) will take the form:

From geometric considerations in the areas of the absolute past and absolute future (otherwise they are called time-like areas) s 2 > 0, and in the space-like region s 2 s is invariant with respect to the transition from one inertial reference frame to another. According to the principle of relativity, the equation s 2 = 0, which expresses the physical law of light propagation, must have the same form in all inertial frames of reference.

Magnitude s 2 is not invariant under Galilean transformations (checked by substitution) and we can conclude that there must be other transformations of coordinates and time when moving from one inertial system to another. At the same time, taking into account the relative nature of simultaneity, it is no longer possible to consider tў = t, i.e. consider time absolute, moving independently of the observer, and generally separate time from space, as could be done in Newtonian mechanics.

Transformations of the coordinates and time of an event during the transition from one inertial reference system to another, without changing the value of the interval s 2, are called Lorentz transformations . In the case when one inertial reference system moves relative to another along the axis x with speed v, these transformations look like:

Here they are written as Lorentz transformations from an unprimed coordinate system TO(conventionally it is considered to be a stationary, or laboratory system) to a hatched system TOў and back. These formulas differ in the speed sign v, which corresponds to Einstein’s principle of relativity: if TOў moves relative to TO with speed v along the axis x, That TO moves relative to TOў with speed – v, and in other respects both systems are completely equal.

The interval in the new notation takes the form:

By direct substitution, you can check that this expression does not change its form under Lorentz transformations, i.e. sў 2 = s 2.

Clocks and rulers.

The most surprising (from the point of view of classical physics) consequences of the Lorentz transformations are the statements that observers in two different inertial frames of reference will receive different results when measuring the length of a rod or the time interval between two events that occurred in the same place.

Reducing the length of the rod.

Let the rod be located along the axis xў reference systems Sў and rests in this system. Its length Lў = xў 2 – xў 1 is recorded by an observer in this system. Moving to an arbitrary system S, we can write expressions for the coordinates of the end and beginning of the rod, measured at the same moment in time according to the observer’s clock in this system:

xў 1 = g ( x 1 – b x 0), xў 2 = g ( x 2 – b x 0).

Lў = xў 2 – xў 1 = g ( x 2 – x 1) = g L.

This formula is usually written as:

L = Lў /g .

Since g > 1, this means that the length of the rod L in the reference system S turns out to be less than the length of the same rod Lў in the system Sў , in which the rod is at rest (Lorentzian contraction of length).

Slowing down the pace of time.

Let two events occur at the same place in the system Sў , and the time interval between these events according to the clock of an observer at rest in this system is equal to

Dt = tў 2 – tў 1.

Proper time is usually called time t, measured by the clock of an observer at rest in a given frame of reference. proper time and the time measured by the clock of a moving observer are related. Because

Where xў is the spatial coordinate of the event, then subtracting one equality from the other, we find:

D t = g Dt .

From this formula it follows that the clock in the system S shows a longer time interval between two events than the clock in the system Sў , moving relative to S. In other words, the interval of proper time between two events, which is shown by a clock moving with the observer, is always less than the time interval between the same events, which is shown by the clock of a stationary observer.

The effect of time dilation is directly observed in experiments with elementary particles. Most of these particles are unstable and decay after a certain time interval t (more precisely, the half-life or average lifetime of the particle is known). It is clear that this time is measured by a clock at rest relative to the particle, i.e. this is the particle's own lifetime. But the particle flies past the observer at high speed, sometimes close to the speed of light. Therefore, its clockwise life time in the laboratory becomes equal to t= gt , and for g >> 1 time t>> t . For the first time, researchers encountered this effect when studying muons produced in the upper layers of the Earth's atmosphere as a result of the interaction of cosmic radiation particles with atomic nuclei in the atmosphere. The following facts were established:

muons are born at an altitude of about 100 km above the Earth's surface;

the muon's own lifetime t @ 2H 10 –6 s;

a stream of muons generated in the upper layers of the atmosphere reaches the Earth's surface.

But this seems impossible. After all, even if muons moved at a speed equal to the speed of light, they could still fly a distance equal to only c t » 3H 10 8 H 2H 10 –6 m = 600 m. Thus, the fact that muons, without decaying, fly 100 km, i.e., a distance 200 times greater, and are recorded near the Earth’s surface, may be explained by only one thing: from the point of view of an earthly observer, the lifetime of the muon has increased. Calculations completely confirm the relativistic formula. The same effect is observed experimentally in particle accelerators.

It should be emphasized that the main essence of SRT is not the conclusions about length reduction and time dilation. The most significant thing in the special theory of relativity is not the relativity of the concepts of spatial coordinates and time, but the immutability (invariance) of some combinations of these quantities (for example, an interval) in a single space-time, therefore, in a certain sense, SRT should be called not the theory of relativity, but the theory of absoluteness (invariance) of the laws of nature and physical quantities in relation to transformations of transition from one inertial reference system to another.

Addition of speeds.

Let the reference systems S And Sў move relative to each other with a speed directed along the axis x (xў ). Lorentz transformations for changing the coordinates of a body D x,D y V has only one component along the axis x, so the scalar product Vvў = Vvў x):

In the limiting case, when all speeds are much less than the speed of light, V c and vў c (non-relativistic case), we can neglect the second term in the denominator and this leads to the law of addition of velocities of classical mechanics

v = vў + V.

In the opposite, relativistic case (velocities close to the speed of light), it is easy to see that, contrary to the naive idea, when adding velocities it is impossible to obtain a speed exceeding the speed of light in vacuum. Let, for example, all velocities be directed along the axis x And vў = c, then it is clear that v = c.

One should not think that when adding velocities within the framework of SRT, velocities greater than the speed of light can never be obtained. Here's a simple example: two starships approaching each other at a speed of 0.8 With each relative to an earthly observer. Then the approach speed of starships relative to the same observer will be equal to 1.6 With. And this in no way contradicts the principles of SRT, since we are not talking about the speed of signal (information) transmission. However, if you ask the question, what is the speed of approach of one starship to another from the point of view of an observer in a starship, then the correct answer is obtained by applying the relativistic formula for adding speeds: the speed of the starship relative to the Earth (0.8 With) is added to the speed of the Earth relative to the second spaceship (also 0.8 With), and as a result v = 1,6/(1+0,64)c = 1,6/1,64c = 0,96c.

Einstein's relation.

The main applied formula of SRT is the Einstein relation between energy E, impulse p and mass m freely moving particle:

This formula replaces the Newtonian formula relating kinetic energy to momentum:

E kin = p 2/(2m).

From Einstein's formula it follows that when p = 0

E 0 = mc 2.

The meaning of this famous formula is that a massive particle in a comoving frame of reference (that is, in an inertial frame of reference moving along with the particle, so that the particle is at rest relative to it) has a certain rest energy E 0, which is uniquely related to the mass of this particle. Einstein postulated that this energy is quite real and when the mass of a particle changes, it can transform into other types of energy and this is the basis of nuclear reactions.

It can be shown that from the point of view of an observer, relative to whom the particle moves with speed v , the energy and momentum of the particle change:

Thus, the values ​​of energy and momentum of a particle depend on the frame of reference in which these quantities are measured. Einstein's relation expresses the universal law of equivalence and interconvertibility of mass and energy. Einstein's discovery became the basis not only for many technical achievements of the 20th century, but also for understanding the birth and evolution of the Universe.

Alexander Berkov

SRT, also known as the special theory of relativity, is a sophisticated descriptive model for the relationships of space-time, motion, and the laws of mechanics, created in 1905 by Nobel Prize winner Albert Einstein.

Entering the department of theoretical physics at the University of Munich, Max Planck turned for advice to Professor Philipp von Jolly, who at that time headed the department of mathematics at this university. To which he received advice: “in this area almost everything is already open, and all that remains is to patch up some not very important problems.” Young Planck replied that he did not want to discover new things, but only wanted to understand and systematize already known knowledge. As a result, from one such “not very important problem” quantum theory subsequently emerged, and from another, the theory of relativity, for which Max Planck and Albert Einstein received the Nobel Prize in physics.

Unlike many other theories that relied on physical experiments, Einstein's theory was based almost entirely on his thought experiments and was only later confirmed in practice. So back in 1895 (at the age of only 16 years) he thought about what would happen if he moved parallel to a beam of light at its speed? In such a situation, it turned out that for an outside observer, particles of light should have oscillated around one point, which contradicted Maxwell’s equations and the principle of relativity (which stated that physical laws do not depend on the place where you are and the speed at which you move). Thus, young Einstein came to the conclusion that the speed of light should be unattainable for a material body, and the first brick was laid into the basis of the future theory.

The next experiment was carried out by him in 1905 and consisted in the fact that at the ends of a moving train there are two pulsed light sources that light up at the same time. For an outside observer passing by a train, both of these events occur simultaneously, but for an observer located in the center of the train, these events will seem to have occurred at different times, since the flash of light from the beginning of the car will arrive earlier than from its end (due to constant speed of light).

From this he made a very bold and far-reaching conclusion that the simultaneity of events is relative. He published the calculations obtained on the basis of these experiments in the work “On the Electrodynamics of Moving Bodies.” Moreover, for a moving observer, one of these pulses will have greater energy than the other. In order for the law of conservation of momentum to not be violated in such a situation when moving from one inertial reference system to another, it was necessary that the object simultaneously with the loss of energy should also lose mass. Thus, Einstein came to a formula characterizing the relationship between mass and energy E=mc 2 - which is perhaps the most famous physical formula at the moment. The results of this experiment were published by him later that year.

Basic postulates

Constancy of the speed of light– by 1907, experiments were carried out to measure with an accuracy of ±30 km/s (which was greater than the Earth’s orbital speed) and did not detect its changes during the year. This was the first proof of the invariability of the speed of light, which was subsequently confirmed by many other experiments, both by experimenters on earth and by automatic devices in space.

The principle of relativity– this principle determines the immutability of physical laws at any point in space and in any inertial frame of reference. That is, regardless of whether you are moving at a speed of about 30 km/s in the orbit of the Sun along with the Earth or in a spaceship far beyond its borders - when you perform a physical experiment, you will always come to the same results (if your ship is in this time does not speed up or slow down). This principle was confirmed by all experiments on Earth, and Einstein wisely considered this principle to be true for the rest of the Universe.

Consequences

Through calculations based on these two postulates, Einstein came to the conclusion that time for an observer moving in a ship should slow down with increasing speed, and he, along with the ship, should shrink in size in the direction of movement (in order to thereby compensate for the effects of movement and maintain principle of relativity). From the condition of finite velocity for a material body, it also followed that the rule for adding velocities (which had a simple arithmetic form in Newtonian mechanics) should be replaced by more complex Lorentz transformations - in this case, even if we add two velocities to 99% of the speed of light, we will get 99.995% of this speed, but we will not exceed it.

Status of the theory

Since it took Einstein only 11 years to form a general version from a particular theory, no experiments were carried out to directly confirm STR. However, in the same year as it was published, Einstein also published his calculations that explained the shift in the perihelion of Mercury to within a fraction of a percent, without the need to introduce new constants and other assumptions that were required by other theories that explained this process. Since then, the correctness of general relativity has been confirmed experimentally with an accuracy of 10 -20, and on its basis many discoveries have been made, which clearly proves the correctness of this theory.

Championship in opening

When Einstein published his first works on the special theory of relativity and began to write its general version, other scientists had already discovered a significant part of the formulas and ideas underlying this theory. So let's say the Lorentz transformations in general form were first obtained by Poincaré in 1900 (5 years before Einstein) and were named after Hendrik Lorentz, who received an approximate version of these transformations, although even in this role he was ahead of Waldemar Vogt.