Special theory of relativity . The special theory of relativity (SRT), published by Einstein in 1905, describes relativistic processes and phenomena and manifests itself at a speed close to the speed of light. To create SRT, Einstein took two postulates: 1) the speed of light in all inertial frames of reference remains constant; 2) the laws of nature in all inertial frames of reference are invariant (the same). In addition, he applied the transformations of the Dutch theoretical physicist Hendrik Lorenz.
The interrelation of space and time, is shown in four-dimensional space-time. This relationship is clearly reflected in the formula for the distance (s) between two events in four-dimensional space:
where - time, ∆ℓ - distance between two points in three-dimensional space.
transformation Lorenz also contains the relationship between space and time in the form of a relationship between the coordinates of non-moving (K) and moving (K 1) reference systems х 1 = γּ(х─ ) and t 1 = γּ(t─ ), where γ = 1/- called relativistic coefficient. Lorentz found expressions for γ based on the linearity of the transformation and the constancy of the speed of light in moving (K 1) and non-moving (K) frames of reference.
Using the Lorentz transformation, Einstein created general relativity, according to which the length of a moving body shrinking according to law:
The mass of a body moving at a speed rise according to law:
The passage of time moving clock slows down according to law:
τ = τ 0 ּ ,
The following example more clearly shows the dilation of time when moving at high speeds. Suppose a spacecraft started at a speed of 0.99 km/s and returned after 50 years. According to SRT, according to the astronaut's watch, this flight lasted only one year. If an astronaut at the age of 20 left a newly born son on Earth, then a 50-year-old son will meet a 21-year-old father.
In SRT, the following formula has been obtained replacing speed summation law:
1 = (+u)/(1+u/c2) ,
if the body is moving at the speed of light =s. and the frame of reference moves at the speed of light u= c, then we get: 1 = with. Consequently, the speed of light remained constant, regardless of the speed of the frame of reference.
general theory of relativity . In frames of reference moving with acceleration, neither the principle of inertia nor the laws of mechanics are fulfilled. There was a need to create a theory that describes the motion of the body in non-inertial frames of reference. Einstein accomplished this task by creating general relativity(OTO).
IN GR Einstein extends the principle of relativity to non-inertial frames of reference. It proceeds from the fact that the gravitational and inertial masses of the body are equivalent. Back in 1890, a Hungarian physicist L. Eötvöshem the equivalence of the gravitational and inertial mass of the body up to 10 -9 was confirmed with high accuracy. This statement about the equivalence of gravitational and inertial mass was taken as the basis of GR.
General relativity showed that space near a concentration of masses, twisted and has the character of a Riemann space. General relativity replaces Newton's law of universal gravitation with Einstein's relativistic law of gravitation, from which, in a particular case, Newton's law follows. In 1919 and 1922 studied during a solar eclipse beam deflection coming from distant stars, from straightness in the gravitational field of the Sun. Experiments have shown curvature of space near the Sun and thus proved the correctness of general relativity.
General relativity describes the relativistic laws of gravity as the effect of matter on the properties of space and time. And the properties of space and time affect the physical processes occurring in them. Therefore, the movement of a material point in four-dimensional space occurs along the geodesic line of curved space. Consequently, the equation of motion of a material point describes the geodesic line of curved space. Einstein found this equation. It consists of 10 equations. In these equations, the gravitational field is described using 10 field potentials. The mathematical apparatus of general relativity is complex, almost all problems related to general relativity have not yet been solved, except for the simplest ones. Therefore, scientists are still trying to understand the meanings of general relativity.
Special Relativity, also known as Special Relativity, is an elaborate descriptive model for space-time relationships, motion, and the laws of mechanics, created in 1905 by Nobel laureate Albert Einstein.
Entering the Department of Theoretical Physics at the University of Munich, Max Planck sought advice from Professor Philipp von Jolly, who at that time was in charge of 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 close 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" subsequently emerged quantum theory, and from another - the theory of relativity, for which Max Planck and Albert Einstein received Nobel Prizes in physics.
Unlike many other theories that relied on physical experiments, Einstein's theory was based almost entirely on his thought experiments and only later was confirmed in practice. So back in 1895 (at the age of only 16) 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, the 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 where you are and the speed with 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 in 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 are ignited at the same time. For an outside observer passing by a train, both of these events occur simultaneously, however, for an observer located in the center of the train, these events will appear to have occurred at different times, since the flash of light from the beginning of the car will come earlier than from its end (due to the constant the speed of light).
From this he drew the 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”. In this case, for a moving observer, one of these pulses will have a greater energy than the other. In order for the law of conservation of momentum not to be violated in such a situation during the transition from one inertial frame of reference to another, it was necessary that the object, simultaneously with the loss of energy, should also lose mass. Thus, Einstein came up with 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
The constancy of the speed of light- by 1907, experiments were carried out to measure with an accuracy of ± 30 km / s (which was more than the orbital velocity of the Earth), which did not reveal its changes during the year. This was the first proof of the invariance 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 invariability 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 together with the Earth or in spaceship far beyond its limits - by setting up a physical experiment you will always come to the same results (if your ship is not accelerating or decelerating at that time). This principle was confirmed by all experiments on Earth, and Einstein reasonably considered this principle to be true for the rest of the Universe as well.
Consequences
By 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 himself, together with the ship, should shrink in size in the direction of movement (in order to compensate for the effects of movement and to observe principle of relativity). From the condition of finiteness of speed for a material body, it also followed that the rule for adding speeds (which had a simple arithmetic form in Newton's mechanics) should be replaced by more complex Lorentz transformations - in this case, even if we add two speeds 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 Einstein took only 11 years to form its general version from a private theory, no experiments were carried out directly to confirm SRT. However, in the same year that it was published, Einstein also published his calculations, which explained the shift of Mercury's perihelion to fractions of a percent, without the need for new constants and other assumptions that other theories required to explain this process. Since then, the correctness of general relativity has been confirmed experimentally with an accuracy of 10 -20 , and many discoveries have been made on its basis, which unambiguously proves the correctness of this theory.
Opening Championship
When Einstein published his first papers on special 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 so in honor of Hendrik Lorentz who received an approximate version of these transformations, although even in this role Voldemar Vogt was ahead of him.
The special theory of relativity was developed at the beginning of the 20th century by the efforts of G. A. Lorentz, A. Poincaré and A. Einstein.
Einstein's postulates
SRT is completely derived at the physical level of rigor from two postulates (assumptions):
Einstein's principle of relativity is valid, an extension of Galileo's principle of relativity.
The speed of light does not depend on the speed of the source in all inertial frames of reference.
Experimental verification of the SRT postulates is to a certain extent hampered by problems of a philosophical nature: the possibility of writing the equations of any theory in an invariant form, regardless of its physical content, and the complexity of interpreting the concepts of "length", "time" and "inertial frame of reference" in conditions of relativistic effects.
Essence of SRT
The consequences of the postulates of SRT are the Lorentz transformations, which replace the Galilean transformations for non-relativistic, “classical” motion. These transformations link the coordinates and times of the same events observed from different inertial frames of reference.
It is they who describe such famous effects as slowing down the passage of time and shortening the length of fast-moving bodies, the existence of a limiting speed of a body (which is the speed of light), the relativity of the concept of simultaneity (two events occur simultaneously according to clocks in one frame of reference, but at different points in time according to hours in another reference system).
The special theory of relativity has received numerous experimental confirmations and is undoubtedly the correct theory in its field of applicability. The special theory of relativity ceases to work on the scale of the entire Universe, as well as in cases of strong gravitational fields, where it is replaced by a more general theory - the general theory of relativity. The special theory of relativity is also applicable in the microcosm, its synthesis with quantum mechanics is quantum field theory.
Comments
Just as in the case of quantum mechanics, many predictions of the theory of relativity are counterintuitive, seem incredible and impossible. This, however, does not mean that the theory of relativity is wrong. In reality, how we see (or want to see) the world around us and how it actually is can be very different. For more than a century, scientists around the world have been trying to refute SRT. None of these attempts could find the slightest flaw in the theory. The fact that the theory is mathematically correct is evidenced by the strict mathematical form and clarity of all formulations. The fact that SRT really describes our world is evidenced by a huge experimental experience. Many consequences of this theory are used in practice. It is obvious that all attempts to "refute SRT" are doomed to failure because the theory itself is based on Galileo's three postulates (which are somewhat expanded), on the basis of which Newtonian mechanics is built, as well as on an additional postulate of the constancy of the speed of light in all frames of reference. All four do not raise any doubt within the maximum accuracy of modern measurements: better than 10 - 12, and in some aspects - up to 10 - 15. Moreover, the accuracy of their verification is so high that the constancy of the speed of light is put at the basis of the definition of the meter - units of length, as a result of which the speed of light becomes a constant automatically if measurements are carried out in accordance with metrological requirements.
SRT describes non-gravitational physical phenomena with very high accuracy. But this does not exclude the possibility of its clarification and addition. For example, the general theory of relativity is a refinement of SRT that takes into account gravitational phenomena. 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 give both SRT in combination with quantum field theory and general relativity within the limits. Most likely, SRT will face the same fate as Newton's mechanics - the limits of its applicability will be accurately outlined. At the same time, such a maximally general theory is still a very distant prospect, and not all scientists believe that its construction is even possible.
General theory of relativity
General theory of relativity(GR) is a geometric theory of gravity published by Albert Einstein in 1915-1916. Within the framework of this theory, which is a further development of the special theory of relativity, it is postulated that gravitational effects are caused not by the force interaction of bodies and fields located in space-time, but by the deformation of space-time itself, which is associated, in particular, with the presence of mass-energy.
General relativity is currently (2007) the most successful gravitational theory, well confirmed by observations. The first success of general relativity was to explain the anomalous precession of Mercury's perihelion. Then, in 1919, Arthur Eddington reported the observation of light deflection near the Sun at the moment total eclipse, which confirmed the predictions of general relativity In addition, numerous observations are interpreted as confirmation of one of the most mysterious and exotic predictions of general relativity - the existence of black holes.
Despite the overwhelming success of general relativity, there is discomfort in the scientific community that it cannot be reformulated as the classical limit of quantum theory due to the appearance of irremovable mathematical divergences when considering black holes and space-time singularities in general. To solve this problem, a number of alternative theories. Current experimental evidence indicates that any type of deviation from general relativity should be very small, if it exists at all.
Einstein began searching for a theory of gravity that would be compatible with the principle of the invariance of the laws of nature with respect to any frame of reference. The result of this search was the general theory of relativity, based on the principle of identity of gravitational and inertial mass.
The principle of equality of gravitational and inertial masses
In classical Newtonian mechanics, there are two concepts of mass: the first refers to Newton's second law, and the second to the law of universal gravitation. The first mass - inertial (or inertial) - is the ratio non-gravitational force acting on the body to accelerate it. The second mass is gravitational (or, as it is sometimes called, heavy) - determines the force of attraction of the body by other bodies and its own force of attraction. Generally speaking, these two masses are measured, as can be seen from the description, in different experiments, so they do not have to be proportional to each other at all. Their strict proportionality allows us to speak of a single body mass in both non-gravitational and gravitational interactions. By a suitable choice of units, these masses can be made equal to each other.
The principle of movement along geodesic lines
If the gravitational mass is exactly equal to the inertial mass, then in the expression for the acceleration of a body, on which only gravitational forces act, both masses are reduced. Therefore, the acceleration of the body, and consequently, its trajectory does not depend on the mass and internal structure body. If all bodies at the same point in space receive the same acceleration, then this acceleration can be associated not with the properties of the bodies, but with the properties of the space itself at this point.
Thus, the description of the gravitational interaction between bodies can be reduced to a description of the space-time in which the bodies move. It is natural to assume, as Einstein did, that bodies move by inertia, that is, in such a way that their acceleration in their own reference frame is zero. The trajectories of the bodies will then be geodesic lines, the theory of which was developed by mathematicians back in the 19th century.
Modern experiments confirm the motion of bodies along geodesic lines with the same accuracy as the equality of gravitational and inertial masses.
Curvature of space-time
If two bodies are launched from two close points parallel to each other, then in the gravitational field they will gradually either approach or move away from each other. This effect is called the deviation of geodesic lines. A similar effect can be observed directly if two balls are launched parallel to each other over a rubber membrane, on which a massive object is placed in the center. The balls will disperse: the one that was closer to the object pushing through the membrane will tend to the center more strongly than the more distant ball. This discrepancy (deviation) is due to the curvature of the membrane.
Main Consequences of General Relativity
According to the correspondence principle, in weak gravitational fields, the predictions of general relativity coincide with the results of applying Newton's law of universal gravitation with small corrections that increase as the field strength increases.
The first predicted and verified experimental consequences of general relativity were the three classical effects listed below in chronological order their first check:
- An additional shift in the perihelion of Mercury's orbit compared to the predictions of Newtonian mechanics.
- Deflection of a light beam in the gravitational field of the Sun.
- Gravitational redshift, or time dilation in a gravitational field.
Must be modified at speeds of bodies 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 velocity than in the nonrelativistic case.
The special theory of relativity has received numerous experimental confirmations and is a true theory in its field of applicability (see Experimental Foundations of Special Relativity). According to the apt remark of L. Page, “in our age of electricity, the rotating anchor 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 the physical theories built on its basis has now led to the fact that the term "special relativity" itself is practically not used in modern scientific articles usually speak only of the relativistic invariance of a single theory.
Basic concepts and postulates of SRT
The 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
SRT postulates the possibility of determining a single time within a given inertial frame of reference. To do this, a synchronization procedure is introduced for two clocks located at different points of the ISO. Let a signal (not necessarily light) be sent from the first clock to the second clock at a constant speed . Immediately upon reaching the second clock (according to their readings at time t), the signal is sent back at the same constant speed and reaches the first clock at time t. The clock is considered to be synchronized if the relation is satisfied.
It is assumed that such a procedure in a given inertial frame of reference can be carried out for any clocks that are stationary relative to each other, so the property of transitivity is valid: if the clocks A synchronized with clock B, and the clock B synchronized with clock C, then the clock A and C will also be synchronized.
Units of Measure Harmonization
To do this, it is necessary to consider three inertial frames S1, S2 and S3. Let the speed of the system S2 with respect to the system S1 be , the speed of the system S3 with respect to S2 is , and with respect to S1, respectively, . Writing down the sequence of transformations (S2, S1), (S3, S2) and (S3, S1), one 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 at in the first equation of the system and at in the second, then:
By dividing one equation by another, it is easy to obtain the desired ratio.
Since the relative velocities of frames of reference are both arbitrary and independent quantities, then this equality will be fulfilled only in the case when the ratio is equal to some constant , which is the same for all inertial frames of reference , and, therefore, .
The existence of an inverse transformation between IFRs, which differs from the direct one only by changing the sign of the relative velocity, makes it possible to find the function .
Proof
The postulate of the constancy of the speed of light
A historically 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 frames of reference. It was with the help of this postulate and the principle of relativity that Albert Einstein in 1905 obtained the Lorentz transformations with a fundamental constant that has the meaning of the speed of light. From the point of view of the axiomatic construction of SRT described above, the second postulate of Einstein turns out to be a theorem of the theory and follows directly from the Lorentz transformations (see relativistic addition of velocities). However, due to its historical importance, such a derivation of the Lorentz transformations is widely used in educational literature.
It should be noted that light signals, generally speaking, are not required when substantiating SRT. Although the non-invariance of Maxwell's equations with respect to Galilean transformations led to the construction of SRT, the latter is of a more general nature and is applicable to all types of interactions and physical processes. The fundamental constant arising in the Lorentz transformations has the meaning of the limiting speed of the movement 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 from this. Therefore, it makes sense to distinguish between fundamental speed and the speed of light. The first constant reflects general properties space and time, while the second is related to the properties of a particular interaction. To measure the fundamental velocity, there is no need to conduct electrodynamic experiments. It is enough, using, for example, the relativistic rule for adding velocities according to the values of the speed of an object relative to two ISOs, to obtain the value of the fundamental speed .
Consistency of the theory of relativity
The theory of relativity is a logically consistent theory. This means that it is impossible to logically deduce some assertion from its initial positions simultaneously with its negation. Therefore, many so-called paradoxes (like the twin paradox) are apparent. They arise as a result of the incorrect application of the theory to certain problems, and not due to the logical inconsistency of SRT.
The validity of the theory of relativity, like any other physical theory, is ultimately tested empirically. In addition, the logical consistency of SRT can be proved axiomatically. For example, within a group approach, it is shown that Lorentz transformations can be derived from 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 of a wider system of axioms are consistent, then they will be all the more consistent if only part of the axioms is used. From the point of view of logic, 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 SRT described above, this does not happen, so SRT is a consistent theory.
Geometric approach
Other approaches to the construction of the special theory of relativity are possible. Following Minkowski and earlier work by Poincaré, one can postulate the existence of a single metric four-dimensional space-time with 4-coordinates. In the simplest case of a 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. 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 a space corresponds to some inertial frame of reference, and the distance between two points corresponds to the modulus of the relative velocity between the ISO. By virtue of the principle of relativity, all points of such a space must be equal in rights, and, therefore, the space of velocities 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 for adding velocities. The space of constant negative curvature (Lobachevsky space) corresponds to the relativistic rule of addition of velocities and the special theory of relativity.
Different notation of the Lorentz transformation
Let the coordinate axes of two inertial reference frames S and S" be parallel to each other, (t, x, y, z) be the time and coordinates of some event observed relative to the frame 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 turn into Galilean transformations:
Such a passage to the limit is a reflection of the correspondence principle, according to which a more general theory (SRT) 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 reference systems is directed in an arbitrary direction (not necessarily along the axis):
where is the Lorentz factor, and are the radius vectors of the event with respect to the systems S and S".
Consequences of the Lorentz transformations
Addition of speeds
A direct 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 there is the following relationship between them:
In these relations, the relative velocity of the frames of reference v is directed along the x axis. The relativistic addition of velocities, like the Lorentz transformations, at low velocities () turns into classical law addition of speeds.
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 IFRs.
Time slowdown
If the clock is stationary in the system, then for two successive events takes place. Such clocks move relative to the system according to the law, so the time intervals are related as follows:
It is important to understand that in this formula, the time interval is measured alone moving clocks. It compares with the evidence several various, synchronously running clocks located in the system, past which the clock moves. As a result of this comparison, it turns out that a moving clock runs slower than a stationary clock. Related to this effect is the so-called twin paradox.
If the clock moves at a variable speed relative to the 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, by means of integration, the time intervals in locally inertial frames of reference (the so-called instantaneously accompanying IFRs) are summarized.
Relativity of Simultaneity
If two events spaced apart in space (for example, flashes of light) occur simultaneously in a moving frame of reference, then they will not be simultaneous with respect to the "fixed" frame. At , 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 leads to the impossibility of synchronizing clocks in different inertial frames of reference throughout space.
From the point of view of the system S
From the point of view of the system S"
Let in two reference systems along the x-axis there are clocks synchronized in each system, and at the moment of coincidence of the “central” clock (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 reference frame show different times. The clocks in the direction of the movement are behind, and those in the opposite direction of the 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 it follows from the Lorentz transformations that the linear dimensions of such a body relative to the “fixed” frame of reference are reduced:
,where is the length along the direction of motion relative to the fixed frame of reference, and is the length in the moving frame of reference associated with the body (the so-called proper length of the body). This reduces the longitudinal dimensions of the body (that is, measured along the direction of motion). The transverse dimensions do not change.
This reduction in size is also called Lorentz contraction. When visually observing moving bodies, in addition to the Lorentz contraction, it is necessary to take into account the propagation time of the light signal from the surface of the body. As a result, a rapidly moving body looks rotated, but not compressed in the direction of motion.
Doppler effect
Let a source moving at a speed v radiate a periodic signal at the speed of light with a 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 the natural frequency:
where is the angle between the direction to the source and its speed.
Distinguish between longitudinal and transverse Doppler effect. 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 (redshift), and if it approaches, then its frequency increases (blueshift):
The transverse effect occurs when , that is, the direction to 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 momentum, 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 the force to the acceleration if the force acts perpendicular to the velocity. In early work on the theory of relativity, it was called "transverse mass". It is her "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 velocity:
As noted above, these concepts are obsolete and are 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 the force and the speed of the body:
This leads to the fact that, as in classical mechanics, the force component perpendicular to the particle velocity 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, the relativistic energy and momentum measured relative to different inertial frames of reference are also related by certain relations:
where the components of the momentum vector are . The relative velocity and orientation of the inertial reference frames S, S" are defined in the same way as in the Lorentz transformations.
Covariant formulation
Four-dimensional space-time
Lorentz transformations leave invariant (unchanged) the following quantity, called the interval:
where , etc. - are the differences in times and coordinates of two events. If , then the events are said to be separated by a timelike interval; if , then spacelike. Finally, if , then such intervals are called lightlike. The light-like interval corresponds to events associated with a signal that propagates at the speed of light. The interval invariance means that it has same value with respect to two inertial frames of reference:
In its form, the 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 sets the distance in the pseudo-Euclidean four-dimensional space-time. It is also called the Minkowski spacetime. Lorentz transformations play the role of rotations in such a space. Basis rotations in four-dimensional space-time, mixing the time and space coordinates of 4-vectors, look like a transition to a moving frame of reference and are similar to rotations in ordinary three-dimensional space. In this case, the projections of four-dimensional intervals between certain events on the time and space axes of the reference system naturally change, which gives rise to relativistic effects of changing time and space intervals. It is the invariant structure of this space, given by the postulates of SRT, that does not change when moving from one inertial frame of reference to another. Using only two spatial coordinates (x, y), four-dimensional space can be represented in coordinates (t, x, y). The 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 repeated indices, summation from 0 to 3 is implied. In inertial reference systems with Cartesian coordinates, the metric tensor has the following form:
Briefly, this diagonal matrix is denoted as follows: .
Choosing a non-Cartesian coordinate system (for example, switching to spherical coordinates) or consideration of non-inertial frames of reference leads to a change in the values of the components of the metric tensor, but its signature remains unchanged. Within 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 frame of reference with Cartesian coordinates. In other words, the four-dimensional space-time of special relativity is flat (pseudo-Euclidean). In contrast, general relativity (GR) considers curved spaces in which the metric tensor cannot be reduced to a pseudo-Euclidean form in the entire space by any transformation of coordinates, 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 the component is its number, not the degree!). The zero component of the 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 the 4-vector is also denoted as follows: .
The components of the 4-vector, measured with respect to two inertial frames of reference 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 , the covector differs from the 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 value 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 "proving" SRT. The entire body of experimental data in high-energy physics, nuclear physics, spectroscopy, astrophysics, electrodynamics and other areas of physics is consistent with the theory of relativity within the accuracy of the experiment. For example, in quantum electrodynamics (combining SRT, 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 elementary particle accelerators. The processing of huge data arrays on the collision of particles moving at relativistic velocities in electromagnetic fields is based on the laws of relativistic dynamics, deviations from which have not been found. The corrections following from SRT and GRT are used in satellite navigation systems (GPS). SRT is at the heart of nuclear energy, and so on.
All this does not mean that SRT has no limits of applicability. On the contrary, as in any other theory, they exist, and their detection is an important task of experimental physics. For example, in Einstein's theory of gravity (GR), a generalization of the pseudo-Euclidean space of special relativity is considered for the case of space-time with curvature, which makes it possible to explain most of the astrophysical and cosmological observable data. There are attempts to detect space anisotropy and other effects that can change SRT relationships. However, it must be understood that if they are discovered, they will lead to more general theories, the limiting case of which will again be SRT. Similarly, at low speeds, classical mechanics, which is a special case of the theory of relativity, remains true. In general, by virtue of the correspondence principle, a theory that has received numerous experimental confirmations cannot turn out to be incorrect, although, of course, the area of its applicability can be limited.
Below are just some experiments illustrating the validity of SRT and its individual provisions.
Relativistic time dilation
The fact that the time of moving objects flows more slowly 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 accuracy according to the relativistic formula. In this experiment, the speed of muons was equal to 0.9994 of the speed of light, as a result of which their lifetime increased by 29 times. This experiment is also important because at a 7-meter radius of the ring, the muon acceleration reached values from the free fall acceleration . 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.
The measurement of time dilation was also carried out with macroscopic objects. For example, in the Hafele-Keating experiment, the readings of stationary atomic clocks were compared with those of atomic clocks flying on an airplane.
Independence of the speed of light from the motion of the source
At the dawn of the theory of relativity, the ideas of Walter Ritz gained some popularity that the negative result of Michelson's experiment could be explained using ballistic theory. In this theory, it was assumed that light was emitted at a speed relative to the source, and the speed of light and the speed of the source were added in accordance with the classical rule for adding velocities. Naturally, this theory contradicts SRT.
Astrophysical observations are a convincing refutation of such an idea. For example, when observing binary stars rotating about a common center of mass, in accordance with Ritz's theory, effects would occur that are not actually observed (de Sitter's argument). Indeed, the speed of light ("images") from a star approaching the Earth would be higher than the speed of light from a star receding during rotation. At a large distance from the binary system, the faster "image" would significantly overtake the slower one. As a result, the apparent movement double stars it would look rather strange that it is not observed.
Sometimes there is an objection that the Ritz hypothesis is “actually” correct, but light, when moving through interstellar space, is re-emitted by hydrogen atoms, which have, on average, zero velocity relative to the Earth, and quickly acquires velocity .
However, if this were the case, there would be a significant difference in the image of binary stars in different ranges of the spectrum, since the effect of “entrainment” of light by the medium depends significantly on its frequency.
In the experiments of Tomaszek (1923), interference patterns from terrestrial and extraterrestrial sources (Sun, Moon, Jupiter, stars Sirius and Arcturus) were compared using an interferometer. All of these objects had different speeds relative to the Earth, however, the shift of the interference fringes expected in the Ritz model was not found. These experiments were subsequently repeated several times. For example, in the experiment of A. M. Bonch-Bruevich and V. A. Molchanov (1956), the speed of light was measured from different edges of the rotating Sun. The results of these experiments also contradict the Ritz hypothesis.
Historical outline
Relationship with other theories
gravity
classical mechanics
The theory of relativity comes into significant conflict with some aspects of classical mechanics. For example, Ehrenfest's paradox shows the incompatibility of SRT with the concept of an absolutely rigid body. It should be noted that even in classical physics it is assumed that the mechanical action on a solid body propagates at the speed of sound, and by no means with an infinite one (as it should be in an imaginary absolutely solid medium).
Quantum mechanics
Special relativity is (as opposed to general) fully compatible with quantum mechanics. Their synthesis is relativistic quantum field theory. However, both theories are quite independent of each other. It is possible to construct both quantum mechanics based on Galileo's nonrelativistic principle of relativity (see the Schrödinger equation) and theories based on SRT, completely ignoring 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 , successively cannot be described without involving the theory of relativity (see the 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 give both SRT in combination with quantum field theory and general relativity within the limits. Most likely, SRT will face the same fate as Newton's mechanics - the limits of its applicability will be accurately outlined. At the same time, such a maximally general theory is still a distant prospect.
see also
Notes
Sources
- Ginzburg V. L. Einstein's collection, 1966. - M .: Nauka, 1966. - S. 363. - 375 p. - 16,000 copies.
- Ginzburg V. L. How and who created the theory of relativity? in Einstein's collection, 1966. - M .: Nauka, 1966. - S. 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
- Mizner C., Thorne K., Wheeler J. Gravity. - M .: Mir, 1977. - T. 1. - S. 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. Meeting scientific papers. - M .: Nauka, 1965. - T. 1. - S. 7-35. - 700 s. - 32,000 copies.
- Matveev A. N. Mechanics and the theory of relativity. - 2nd edition, revised. - M .: Higher. school, 1986. - S. 78-80. - 320 s. - 28,000 copies.
- Pauly W. Theory of relativity. - M .: Science, 3rd edition, corrected. - 328 p. - 17,700 copies. - ISBN 5-02-014346-4
- von Philipp Frank und Hermann Rothe"Über die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme" Ann. der Physic, Ser. 4, Vol. 34, no. 5, 1911, pp. 825-855 (Russian translation)
- Fok V.A. Theory of space-time and gravity. - Edition 2, supplemented. - M .: State ed. Phys.-Math. lit., 1961. - S. 510-518. - 568 p. - 10,000 copies.
- "Lorentz Transformations" in The Relativistic World.
- Kittel Ch., Nait W., Ruderman M. Berkeley Physics Course. - 3rd edition, corrected. - M .: Nauka, 1986. - T. I. Mechanics. - S. 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. - S. 23-31. - 120 s. - 16,500 copies.
SPECIAL AND GENERAL RELATIVITY
One of the most important aspects of modern physics, which is of direct relevance to our analysis of theology, is the concept of time - its origin and the absence of a single, or constant and unchanging, measure of its flow. Because of the importance of chronology in interpreting the Bible, it is essential to try to understand how relativity treats our perception of the universe, its age, and everything that happens in it. time relativity quantum photon
It is difficult to name another theory that would have had such a profound impact on our understanding of the world and its creation as the theory of relativity (both special and general). Before the advent of this theory, time has always been regarded as an absolute category. The time elapsed from the beginning to the end of any process was considered independent of who measured its duration. As early as 300 years ago, Newton formulated this belief very eloquently: "Absolute, true and mathematical time, in itself and by virtue of its nature, flows uniformly and independently of any external factors." Moreover, time and space were considered as unrelated categories, not influencing each other in any way. Indeed, what other connection could exist between the distance separating two points of space and the passage of time, besides the fact that a greater distance required more time to overcome it; simple and pure logic.
Einstein's concepts in special relativity (1905) and later in general relativity (1916) changed our understanding of space and time in the same way that the light of a lamp on a lamp changes our perception of a previously darkened room.
The long road to Einstein's insight began in 1628, when Johannes Kepler discovered a curious phenomenon. He noticed that the tails of comets are always directed in the direction opposite to the Sun. Shooting stars tracing the night sky have a tail blazing, as it should be, behind. In the same way, the tail trails behind a comet when it approaches the Sun. But after the comet passes the Sun and begins its return flight to the far regions of the solar system, the situation changes in the most dramatic way. The comet's tail is in front of its main body. This picture strongly contradicts the very concept of a tail! Kepler suggested that the position of a comet's tail relative to its main body is determined by the pressure of sunlight. The tail has a lower density than the comet itself, and therefore it is more susceptible to solar radiation pressure than the comet's main body. The sun's radiation actually blows on the tail and pushes it away from the sun. If not for the gravitational pull of the comet's main body, the tiny particles that make up the tail would be blown away. Kepler's discovery was the first indication that radiation - for example, light - can have a mechanical (in this case, repulsive) force. This was a very important change in our conception of light, because it follows from this that light, which has always been considered something non-material, may have weight or mass. But only 273 years later, in 1901, the pressure exerted by the flow of light was measured. E.F. Nichols and J.F. Hull, by directing a powerful beam of light at a mirror suspended in a vacuum, measured the displacement of the mirror as a result of light pressure. It was a laboratory analogy for a comet's tail being repelled by sunlight.
In 1864, while exploring Michael Faraday's discoveries regarding electricity and magnetism, James Clark Maxwell proposed that light and all other types of electromagnetic radiation move through space like waves at the same fixed speed. Microwaves in the microwave oven in our kitchen, the light we use to read, X-rays that allow a doctor to see a broken bone, and gamma rays released from an atomic explosion are all electromagnetic waves that differ only in wavelength and frequency. The greater the radiation energy, the shorter the wavelength and the higher the frequency. In all other respects they are identical.
In 1900, Max Planck proposed a theory of electromagnetic radiation that was fundamentally different from all previous ones. Prior to this, it was believed that the energy emitted by a heated object, such as the red glow of hot metal, is emitted evenly and continuously. It was also assumed that the radiation process continues until the complete dissipation of all heat and the return of the object to its original state - and this was fully confirmed by cooling the heated metal to room temperature. But Planck showed that things were quite different. Energy is emitted not in a uniform and continuous flow, but in discrete portions, as if the hot metal gave up its heat, spewing out a stream of tiny hot particles.
Planck proposed a theory according to which these particles are single portions of radiation. He called them "quanta" and that's how quantum mechanics was born. Since all radiation travels at the same speed (the speed of light), the speed of the quanta must be the same. And although the speed of all quanta is the same, not all of them have the same energy. Planck suggested that the energy of an individual quantum is proportional to the frequency of its oscillations as it moves through space, like a tiny rubber ball that continually contracts and expands as it travels along its path. In the visible range, our eyes can measure the frequency of quantum pulsations, and we call this measure color. It is thanks to the quantized emission of energy that a slightly heated object begins to glow red, then, as the temperature rises, it begins to emit other colors of the spectrum corresponding to higher energies and frequencies. In the end, its radiation turns into a mixture of all frequencies, which we perceive as the white color of a hot body.
And here we run into a paradox - the very theory that describes light as a stream of particles, called quanta, simultaneously describes the energy of light using frequency (see Fig. 1). But frequency is associated with waves, not particles. In addition, we know that the speed of light is always constant. But what happens if the object emitting light, or the observer registering this light, moves itself? Will the speed of their movement be added to or subtracted from the speed of light? Logic tells us that yes, it must be added or subtracted, but then the speed of light will not be constant! The pressure exerted by light on the tail of a comet or on the mirror in the Nichols-Hull experiment means that there is a change in the momentum (also called momentum) of the light as it hits the surface. It is for this reason that any moving object exerts pressure on an obstacle. The jet of water from the hose drives the ball along the ground, because the water has mass and this mass has a velocity that turns to zero at the moment the jet hits the ball. In this case, the momentum of the water is transferred to the ball and the ball rolls back. The very definition of momentum (momentum) as the product of an object's mass (m) or weight and its velocity (v), or mv, requires moving light to have mass. Somehow, these undulating particles of light have mass, even though no material traces are left on the surface upon which the light falls. After the light has "shed" on the surface, there is no "dirt" left on it from which it could be cleaned. So far we're still trying to create unified theory, which would fully explain this phenomenon of light and any other radiation.
Simultaneously with the study of the nature of radiant energy, studies were carried out relating to the propagation of light. It seemed quite logical that since light and other types of electromagnetic radiation are, in a certain sense, waves, they need some kind of medium in which these waves could propagate. It was believed that waves could not propagate in a vacuum. Just as sound needs a certain material substance, such as air, to carry its wave-like energy, so light seemed to need some special substance to propagate it. At one time, it was suggested that the Universe should be filled with an invisible and intangible medium, which ensures the transfer of radiation energy through outer space - for example, light and heat from the Sun to the Earth. This medium was called the ether, which was supposed to fill even the vacuum of space.
The postulate of the propagation of light through the ether made it possible to explain the paradox of the constancy of its speed. According to this explanation, light must propagate at a constant speed, not relative to the light source or observer, but relative to this omnipresent ether. For an observer moving through the ether, the light could propagate faster or slower, depending on the direction of its movement relative to the direction of the light, but relative to the motionless ether, the speed of light should remain constant.
Rice. one.
The same is true for the propagation of sound. Sound travels through still air at sea level at a constant speed of about 300 meters per second whether the sound source is moving or not. The explosion-like sound an aircraft makes when it crosses the sound barrier is actually the result of the aircraft hitting its own sound wave as it overtakes it, moving faster than 300 meters per second. In this case, the source of the sound, the aircraft, is moving faster than the sound it produces. The dual nature of light is such that if we put a hole of small diameter in its path, the light behaves exactly like an ocean wave passing through a narrow harbor entrance. Both the light and the ocean wave, having passed through the hole, propagate on the other side of the hole in circles. On the other hand, if light illuminates the surface of some metal, it behaves like a stream of tiny particles bombarding this surface. Light knocks electrons out of the metal one at a time, just as small pellets, hitting a paper target, will pull out pieces of paper from it, one piece per pellet. The energy of a light wave is determined by its length. The energy of light particles is determined not by their speed, but by the frequency with which light particles - photons - pulsate during their movement at the speed of light.
When scientists discussed the alleged properties of the aether, which had yet to be discovered, no one suspected that the passage of time was due to the movement of light. But this discovery was not far off.
In 1887, Albert Michelson and Edward Morley published the results of their attempt to experimentally observe what followed from the theory of the ether. They compared the total time required for light to travel the same distance back and forth in two directions - parallel and perpendicular to the Earth's motion in its orbit around the Sun. Since the Earth moves in its orbit around the Sun at a speed of about 30 kilometers per second, it was assumed that it was moving with the same speed relative to the ether. If light radiation obeys the same laws that govern all other waves, the Earth's motion relative to the ether must have affected the light travel time measured in their experiments. This effect should not have been any different from the effect of a strong wind blowing sound away.
To everyone's surprise, Michelson and Morley did not record the slightest trace of the impact of this speed of 30 kilometers per second. The initial experiment, as well as subsequent, technically more advanced versions of the same experiment, led to a completely unexpected conclusion - the movement of the Earth does not have any effect on the speed of light.
This caused confusion. The speed of light (c) is always 299,792.5 kilometers per second whether the light source or the observer is moving or stationary. In addition to this, the same beam of light behaves both as a wave and as a particle, depending on the method of observation. It was as if we were standing on a dock and watching the waves rolling in from the ocean, and suddenly, in the twinkling of an eye, the usual crests of the waves and the troughs between them would turn into a stream of individual water balls moving, pulsing, in the air above the very sea level. And the next moment the balls would disappear and the waves would reappear.
In 1905, in the midst of this confusion, Albert Einstein appeared on the scientific scene with his theory of relativity. During that year, Einstein published a series of papers that literally changed humanity's understanding of our universe. Planck had proposed the quantum theory of light five years earlier. Using Planck's theory, Einstein was able to explain an interesting phenomenon. Light hitting the surface of some metals releases electrons, resulting in an electric current. Einstein postulated that this "photoelectric" effect is due to light quanta (photons) literally knocking electrons out of their orbits around the atomic nucleus. It turns out that photons have mass when they are moving (recall that they are moving at the speed of light c), but their "rest mass" is zero. A moving photon has the properties of a particle - at every moment it is at a certain point in space and, moreover, it has mass, and therefore, as Kepler suggested at one time, it can act on material objects, such as the tail of a comet; at the same time, it has the properties of a wave - it is characterized by an oscillation frequency that is proportional to its energy. It turned out that matter and energy are closely connected in a photon. Einstein discovered this relationship and formulated it into a well-known equation. Einstein came to the conclusion that this equation applies to all kinds of mass and forms of energy. These provisions became the basis of the special theory of relativity.
The perception of these ideas is not so simple and requires considerable mental effort. For example, let's take an object. The mass (what we usually call "weight") of a stationary object is called, in scientific terms, rest mass. Now let's give this object a strong push. It will begin to move at a certain speed and, as a result, will acquire kinetic energy, the greater, the higher its speed. But since the e in E=mc2 refers to all forms of energy, the total energy of an object will be the sum of its rest energy (related to its rest mass) and its kinetic energy (the energy of its motion). In other words, Einstein's equation requires that the mass of an object actually increase with its speed.
So, according to the theory of relativity, the mass of an object changes with a change in its speed. At low velocities, the mass of an object practically does not differ from its rest mass. That is why in our daily activities the Newtonian description of the laws of nature turns out to be quite accurate. But for galaxies soaring through space, or for subatomic particles in an accelerator, things are quite different. In both cases, the speed of these objects can be a large fraction of the speed of light, and therefore the change in their masses can be very, very significant.
This interchange between mass and energy is discussed quite eloquently both by Steven Weinberg in his book The First Three Minutes and by Nachmanides in his commentaries on Genesis. They both talk about mass-energy duality, describing the first minutes of the life of the universe.
The special theory of relativity is based on two postulates: the principle of relativity and the constancy of the speed of light. The principle of relativity, postulated by Galileo Galilei 300 years ago, was refined by Einstein. This principle states that all the laws of physics (which are nothing but the laws of nature) operate in the same way in all systems moving without acceleration, that is, uniformly and rectilinearly. Such systems are called in the language of physicists inertial frames of reference.
The frame of reference determines the relationship of the observer with the outside world. The principle of relativity tells us that, being in an inertial frame of reference, we cannot, using the laws of physics, determine whether the system itself is moving, since its movement does not affect the results of measurements made inside the system. That is why we do not feel movement when we fly at a constant speed in calm weather. But, rocking in a rocking chair, we find ourselves in a non-inertial frame of reference; since the speed and direction of the rocking chair is constantly changing, we can feel our movement.
We all had to deal with examples of the impossibility of measuring absolute motion. For example, we are standing in front of a traffic light, and the car in front of us begins to slowly roll back. Or are we moving forward? At the first moment it is difficult to understand who exactly is moving. Our train slowly and smoothly begins to move along the platform. Waking up from a slumber, we notice that a train standing on a nearby track begins to slowly move back. Or at least it seems to us that this is the case. Until our frame of reference - our car or train - begins to move with acceleration (ceasing to be an inertial frame), it is not clear what is moving and what is at rest.
This may seem like a contradiction: Einstein convinced us that the mass of an object is a function of its velocity, and now we claim that we cannot determine the motion by measuring how the mass changes under its influence. But there is a very subtle difference here. Within the inertial frame of reference, all quantities remain unchanged. When they are measured from another frame of reference, which moves relative to the first, then the values of dimensions and mass will change. If all parts of the Universe moved equally and uniformly, the theory of relativity would have nothing to do with the topic of our study. But things are different. It is the possibility of observing the same events from different frames of reference that plays an essential role in our biblical analysis of cosmology.
The second element of the foundation of special relativity is even more difficult to understand. One might even say that it is incomprehensible to the extreme. He claims that the speed of light, c, is constant (c = 2.997925 x 108 meters per second in vacuum - always) and is the same in all frames of reference. This fact came to light from the results of the Michelson-Morley experiment. If you think about the meaning of this statement, you will be able to appreciate all its audacity. Einstein took the liberty of declaring that regardless of the speed of the observer's movement towards the light source or away from it, the speed of light remains equal to the same c. No other form of motion (such as a sound wave) has this property. This looks highly illogical.
If a pitcher throws a ball at 90 miles per hour to a catcher, the catcher sees the ball approaching him at 90 miles per hour. Now, if, in violation of any rules, the catcher runs towards the pitcher at 20 miles per hour, the speed of the ball in relation to the catcher will be 110 miles per hour (90 + 20). The speed of the ball relative to the pitcher will be, as before, 90 miles per hour. The next time the pitcher, instead of throwing the ball, shows the catcher a picture of the ball. It is moving towards the catcher at the speed of light (c), which is approximately 300 million meters per second. The fast-footed catcher, in turn, rushes towards the pitcher at a speed equal to one tenth of the speed of light, that is, 30 million meters per second. And what will this catcher of ours see? An image of a ball approaching him at 330 million meters per second? Not! This is precisely the paradox of light - confusing, annoying, sometimes even infuriating, but at the same time liberating us.
The catcher sees the image of the ball approaching him at exactly the speed of light, 300 million meters per second, even if he runs towards him and thereby adds his speed to the speed of light. Light, regardless of the speed of the observer in relation to the light source, always moves at a speed c. Always. And what is the speed of movement of the image of the ball fixes standing motionless pitcher? That's right, also. How, then, do two observers, one moving and the other standing still, fix the same speed of light? Logic and common sense say that this is impossible. But relativity says this is reality. And this reality was confirmed in the Michelson-Morley experiment.
Both observers register the same speed of light, because the fact of changing mass, space and time - no matter how incomprehensible it may seem - is a fundamental law of relativistic mechanics and the Universe in which we live. The laws governing these changes are such that nothing happens within the system that seems absurd. The one who is inside it does not notice any changes. But when we observe another system moving past us, we see that the dimensions of the object along the direction of motion are reduced relative to the same dimensions of the object when it is at rest. Moreover, clocks that showed the exact time when they were at rest, moving, begin to lag behind clocks "resting" in our frame of reference.
The combination of the constancy of the speed of light and the principle of relativity inevitably entails the dilation of time. Time dilation can be demonstrated using a thought experiment similar to one used by Einstein when he developed the basic principles of the theory of relativity. An example of such a thought experiment is given by Taylor and Wheeler in their classic book The Physics of Space and Time.
Consider two frames of reference, one of which is stationary and the other is moving. The fixed system is an ordinary physical laboratory. The second system is a rocket moving at high speed, completely transparent and permeable, inside which there is a crew consisting of absolutely transparent and permeable scientists. The rocket, due to its complete transparency and permeability, can pass through our laboratory without any interaction with it and its contents. In the laboratory, from point A (Fig. 2) there is a flash of light that moves diagonally to the mirror located at point M. The light reflected from the mirror also travels diagonally to point B. The time of arrival of the rocket to the laboratory is determined in such a way that at the moment of the flash point A of the rocket is the same as point A of the laboratory. Let the speed of the rocket be such that point A of the rocket coincides with point B in the laboratory at exactly the moment when the flash of light reaches point B. To observers in the rocket, it will appear that the light sent from point A of the rocket goes directly to the point of the rocket M and returns back to point A of the rocket. Because the rocket's speed is constant (it is an inertial frame), the people in the rocket don't know it's moving.
The distance traveled by the light, as perceived by the passengers of the rocket, is 2y (from point A to point M and back). The same path of light seen by those in the laboratory is the sum of the two sides of the triangle - from point A to point M and from point M to point B. Obviously, this path must be greater than the path seen by the passengers of the rocket. We can accurately calculate the difference between them using the Pythagorean theorem. Thus, we conclude that the path of light observed from a rocket is shorter than the path of light observed from a laboratory.
Rice. 2.
Recall that the speed of light in both systems is the same. This is one of the firmly established fundamental principles theory of relativity. It is also known that in all cases the time taken to move is equal to the distance traveled divided by the speed of movement. The time it takes to travel 100 miles at 50 miles per hour is two hours. Since the speed of light for both scientists in the laboratory and for scientists traveling in a rocket is the same c, and the distance traveled by light in the laboratory is greater than the distance traveled by it in the rocket, the time interval between the flash of light at point A and the arrival of light at point B should be more in the laboratory than in the rocket.
Only one event happened. There was only one flash of light, and the light observed in two frames of reference made its way once. Nevertheless, the duration of this event was different when measured in two different reference systems.
This difference in measured time is called relativistic time dilation, and it is this dilation that convincingly reconciles the six days of Creation with the 15 billion years of cosmology.
The concepts underlying general relativity are a development of the ideas of special relativity, but are more complex. While special relativity deals with inertial systems, general relativity deals with both inertial and non-inertial (accelerated) systems. In non-inertial systems, external forces - such as gravitational forces - affect the movement of objects. A special relativistic property of gravity, which is directly related to the problem we are studying, is that gravity - as well as speed - causes time dilation. The same clock on the Moon runs faster than on Earth because the Moon's gravity is weaker. As we shall see, gravity plays a crucial role in reconciling Creation and the Big Bang.
The forces of gravitational attraction are felt in exactly the same way as the forces that cause acceleration. For example, in an elevator going up, we feel the force with which the floor presses down on our feet; it actually pushes us up along with the elevator. This is perceived as the force that we would feel while standing in a stationary elevator if, somehow, the gravitational pull of the Earth suddenly increased. Einstein reasoned that since gravity is treated in exactly the same way as any other force that causes a change in motion, it should lead to the same results. Since accelerating forces cause a change in motion and time dilation, changes in gravity must also cause time dilation.
Since the time dilation aspect of the theory of relativity is essential to the problem of unifying the cosmological and biblical calendars, it is very important to show that time dilation does exist. After all, relativistic changes become noticeable only in those cases when the relative velocities of motion approach the speed of light. Even at 30 million meters per second, which is one-tenth the speed of light, the time dilation is less than one percent.
Speeds close to the speed of light are rare in everyday life, but common in cosmology and high energy physics. True, it should be noted that the real possibility of measuring the dilation of time does not make the idea itself more accessible to understanding. Nevertheless, this allows us to move it from the category of a purely theoretical concept to the realm of empirical facts. Quite a wide range of species human activity- from experiments in laboratories of high energy physics to regular flights of commercial airliners - allows you to demonstrate time dilation.
One of the many elementary particles that arise in the course of experiments in physical laboratories is the muon. It has a half-life of one and a half microseconds. Mu-mesons, however, appear not only in high-energy physics laboratories, but also in the upper layers earth's atmosphere when cosmic rays collide with the atomic nuclei of atmospheric gases. Since the energy of cosmic radiation is very high, muons at the moment of their formation acquire a speed almost equal to the speed Sveta. At such a high speed, there is a time dilation that can be measured. Even when moving at close to the speed of light, it takes 200 microseconds for a muon to travel the distance of 60 kilometers from the atmospheric layer in which they originate to the Earth's surface. Since the muon has a half-life of one and a half microseconds, the 200 microsecond transit time covers 133 of its half-lives. Recall that for each such half-cycle, half of the remaining particles decay. After 133 half-cycles, the proportion of muons that must survive and reach the Earth's surface will be equal to "/2 x 1/2 x" / 2, and so on 133 times, which is one millionth millionth billionth billionth of the number of muons that began their travel to the surface of the earth. This number is so small that practically almost no muon should reach the Earth. The vast majority of them will fall apart along the way. Nevertheless, if we compare the number of muons produced in the upper atmosphere with the number of muons that reach the Earth's surface, we find, to our surprise, that "/8 of their initial number successfully arrive at their destination." Survival" of 1/8 muons means that only three half-cycles are completed during their 60-kilometer journey. Thus, for a mu-meson moving at a speed close to the speed of light, the elapsed (relativistic) time is only three half-cycle is 4.5 microseconds (3 x 1.5 microseconds) For an observer on the Earth's surface, at least 200 microseconds will elapse - the minimum time required to cover 60 kilometers from upper layers atmosphere to the surface. The same single event occurs during two different periods of time - 4.5 microseconds in the reference frame of a rapidly moving muon and 200 microseconds in the reference frame of an observer standing on the surface. Recall again that we are talking about one event. But due to the fact that the observer and the observed object are moving relative to each other, there are two different time intervals for this one event. And both of them are absolutely true!
But muons are rather exotic particles, and the skeptic may well chuckle and shake his head in disbelief. After all, no observer can travel in the company of muons. We rely only on their half-life as clocks moving with them.
What about real clocks and a person moving with them and measuring the dilation of time in the most direct way? It would definitely look more convincing. And that is exactly what was reported in the prestigious journal Science by Hafele and Keating12 from the University of Washington and the US Naval Laboratory. They sent four sets of cesium clocks around the world in Boeing 707s and Concordes operated by TWA and Pan Am and operating regular commercial flights. This watch was chosen because it has exceptionally high accuracy.
The earth rotates from west to east. If we look at the Earth from space, while being above its north pole, we will see that when flying to the east, the speed of the aircraft is added to the speed of the Earth. As predicted by relativity, the clocks on board the aircraft lagged behind those in the US Naval Laboratory in Washington, DC (all clocks used in this experiment were provided by the lab). When flying west, the speed of the plane is subtracted from the speed of the Earth's rotation and, in full agreement with the theory of relativity, the clock on board this plane has gone ahead. In the words of Häfele and Keating, “in science, relevant empirical facts are more powerful than theoretical arguments. These results provide an unambiguous empirical solution to the famous clock paradox.
Not only the perception of time, but also the actual course of time changes depending on the relative motion of the observers. Within each given frame of reference, everything looks quite normal. But when the two systems are first separated and then reconnected and the clocks are compared, the course of time in them turns out to be different (actual "aging").
A particularly interesting aspect of the Häfele-Keating time dilation experiments was that they tested both special and general relativity. According to the general theory of relativity, the difference in the strength of gravity affects the duration in the same way as the difference in relative speed, which is postulated by the special theory of relativity. The effect of a gravitational field on any object is inversely proportional to the square of the distance between objects. When the distance doubles, the gravitational attraction decreases by a factor of four. The farther an object is from the Earth, the weaker the Earth pulls it. Since airplanes in flight are high above the Earth's surface (the typical flight altitude of a Boeing 707 is 10 km, and a Concorde is 20 km), the Earth's gravitational effect on the clock on board the aircraft was different from the effect on the clock that was on the Earth's surface in the Naval laboratories. The changes in the course of the clock, recorded in the experiment, corresponded to the predictions of the general theory of relativity (which takes into account the influence of both movement and gravity).
This experiment, like all others like it, proved that Einstein's special and general theories of relativity correctly describe the real characteristics of our universe. The theory of relativity is no longer a pure theory. Relativity is a proven, empirically confirmed fact. In other words, the theory of relativity has become the law of relativity.
And now, based on this law, based on one of the natural sciences that describe the universe, we can continue to discuss the first six days of Creation - that period in which natural science and theology, at first glance, contradict each other.
Let's look at the changes in the relationship between the Creator, the Universe and man that have taken place since that moment, which we call "beginning". At the same time, we should not for a moment lose sight of the fact that the difference in the course of time can be fixed only if we compare the observation of the same events from two different frames of reference. But this is not enough - it is also necessary that either the gravitational forces in these two frames of reference differ significantly from each other, or that the relative speed of their movement approaches 300 million meters per second, that is, the speed of light. Inside each system, regardless of its relative speed or the gravitational force acting in it, everything happens in full accordance with Newton's laws, that is, everything looks normal and logical, just like we do on Earth, although we are rushing at high speed through space.
The Creator had and still has a certain interest in the creation of the Universe. We can assume this based on the fact that the universe exists. However, we do not know what this interest is. However, we can find some hints of this by analyzing the interaction between the Creator and the Universe during the entire time of its creation and existence. Traditional theology maintains that if the Creator had wished to create the universe in one fell swoop, he would have done so. But it is clear from the biblical account that it was not his plan to create a fully formed universe through a single act. For some reason, the method of gradual development was chosen. And the first two chapters of the book "Genesis" are devoted to the description of the gradual formation of the Universe.
If we play by the rules according to which the universe operates today - and these rules are the physical laws we know - then the gradual development of the universe from the primary substance that existed at the time of the Big Bang was absolutely necessary for the emergence of man. But the Earth itself and everything that exists on it are not direct products of the Big Bang. We are told quite unequivocally that in the very beginning the Earth was formless and empty, or in Hebrew gohu and bohu. Leading nuclear particle physicists now refer to T and B (toxu and bohu) as the two original "bricks" from which all matter is built. The force of the Big Bang literally compressed these Gi Bs into hydrogen and helium - almost no other elements were formed at that moment. And only cosmic alchemy subsequently created all other elements from these primordial hydrogen and helium.
The earth and the entire solar system are a hodgepodge of matter that has come down to us after countless cycles of supercontraction in the depths of stars. This pressure compressed hydrogen and helium so tightly that their nuclei joined and separated again, forming such heavier elements as carbon (truly the substance of life), iron, uranium and other 89 elements that make up the universe. The stars then exploded and spewed out the newly formed elements into the Universe, which greedily devoured them, using them to create other stars. The birth of stars and their death were necessary in order to eventually turn the hydrogen and helium formed in the first moments after the Big Bang into the elements necessary to create life in the form that we know. In their interpretations of the Bible, commentators such as Maimonides and Rashi explained that God created and destroyed many worlds in the process of creating life on Earth. But here I am not relying on Maimonides; I got the above information from the astrophysicists Woosley and Phillips.
So, if we have six days to go before Adam appears, how can we fit all the cycles of world formation and destruction into this time span? The biblical commentators we rely on clearly state that the first six days of Creation are six days of 24 hours each. This means that someone who kept track of time then had to record the passage of these same 24 hours a day. But who could be present at that time to measure the passage of time? Until the moment when, after six days, Adam appeared, only the Lord God could watch the clock. And that's the whole point.
When our Universe was created - until the very moment of the appearance of man - God was not closely connected with the Earth. For the first one or two days out of the six days of Creation, the Earth didn't even exist yet! Although Genesis 1:1 says, “In the beginning God created the heavens and the earth,” the next verse states that the earth was empty and formless. The first verse of the book of Genesis is, in fact, a statement of the most general plan, meaning that at the very beginning the primary substance was created, from which, during the next six days, the heavens and the earth were to be formed. Later, in verse 31:17 of the book "Exodus", this is stated more clearly: "... in six days the Lord created the heavens and the earth ...". What were heaven and earth "made" of during those six days? From the substance created "at the beginning" of those six days. Since there was no Earth in the early Universe, and since there was no possibility of establishing a close connection or interpenetration of frames of reference, there was no common calendar for God and for the Earth.
The law of relativity taught us that even for God there is no way to choose a calendar that would be valid for all parts of the universe, or at least for a limited number of them that played a role in the emergence of mankind. The law of relativity, one of the basic laws of the universe, established at its creation, makes existence impossible common system reference for the Creator and for each part of that totality of matter, which eventually turned into humanity and the planet Earth on which it lives.
We know that according to the law of relativity, in an expanding Universe it is impossible to describe the time spanning a certain sequence of events in one part of the Universe in such a way that it is equal to the time of the same events, but observed from another part of the Universe. Differences in the motion and gravitational forces of different galaxies or even stars in the same galaxy turn absolute time into a purely local phenomenon. Time flows differently in different parts of the universe.
The Bible is a guidebook describing the journey of mankind through life and time. To instill in man respect for the physical wonder of the universe, this guide includes a description of the process that led from an empty, formless universe to a home in which humanity can exist. But it is almost impossible to choose some unified time reference system to describe this process, since too many factors most directly affect the speed of the passage of time. These factors include gravitational forces in many stars, in the depths of which primary hydrogen and helium were converted into the elements underlying life, and the movement of intergalactic gas, which condenses in the process in a nebula, and then into stars, and explosions. supernovae, marking the death and subsequent rebirth of the stars from which the Milky Way is formed, and the mass of the Earth. The passage of time was that aspect of life that we, prior to Einstein's insight, mistakenly believed to be unchanging. It is unrealistic, no, it is simply impossible for the same clock in all ages to measure the age of all that cosmic substance of which we are composed.
The odyssey of matter from the substance of the Big Bang to its current state was too complex, too diverse, for the passage of time in it to be measured by the same clock. Who can say now how many galaxies or what kind of supernova ultimately gave rise to the elements that make up our physical bodies? We humans, and everything else in the solar system, including the Sun and the planets, are fragments of long-gone stars. We are literally made of stardust. To which carbon, nitrogen, or oxygen atoms does this time refer? To yours or to the atoms of your neighbor? To those that are part of a particle of your skin, or to those that are in a drop of your blood? It is likely that each of them began in the depths of different stars, and therefore each of them has its own unique age. The transformations of cosmic matter that took place before the formation of the Earth took place in myriads of stars, simultaneously and sequentially. Each star, each supernova, had its own gravity and its own speed, and therefore its own space-time frame of reference.
Billions of cosmic clocks ticked (and still tick), each at its own locally correct pace. They all started ticking at the same moment - the moment of the Big Bang - and they all simultaneously reached the time period when Adam appeared. But the absolute, local time that passed from the "beginning" to the moment when each of these particles of matter contributed to the creation of mankind was very different for each star and for each particle. Although the transformations of matter began and ended at the same time, it follows from Einstein's theory that the age of each given particle of matter differs very significantly from the age of other particles of matter with which it eventually joined, forming solar system and then humanity. Our reasoning is no more and no less sophisticated than, say, finding 200 microseconds in those 4.5 microseconds that elapse until muons, formed in the upper atmosphere under the impact of cosmic radiation, reach the Earth's surface. In 4.5 microseconds, 200 microseconds pass. This proven fact can be better understood with the help of Einstein's thought experiment, in which scientists on board a fast-moving rocket and scientists in a stationary laboratory record two different times for the same event. This situation has nothing to do with the statement of the late U.K. Fields, who said that during one long evening in Philadelphia he lived for a whole week. His statement belongs to the realm of emotional sensation; in our case we are dealing with a physical fact. When we talk about a billion years, we don't mean that we experience them as a billion years. A billion years have indeed passed! If during those same six days there were some clocks in that part of the Universe that is now occupied by the Earth, they would not necessarily fix 15 billion years. In the early universe, the curvature of space and time in this place was probably quite different from what it is now.
In order to describe consistent development The universe needed to find some kind of compromise. As such a compromise, the Creator chose for the time before the appearance of Adam, his own frame of reference, in which the entire universe was perceived as a single whole.
The creation of Adam was qualitatively different from all other events that accompanied the creation of the universe. It indicated a fundamental change in God's relationship to the universe. We know that all objects in the universe, organic and inorganic, living and inanimate, are composed of matter, the origin of which can be traced back to the primary creation. In this sense, humanity is no exception. We were unequivocally explained that the material source of our origin is the "dust of the earth." All living beings (Genesis 1:30), including man (Genesis 2:7), were given a living soul (Hebrew nefesh). However, only Adam was given something new, unique to the entire universe - the living breath of God ("Genesis" 2:7).
And it was at that moment, when God breathed into Adam his breath of life (neshama in Hebrew), both - the Creator and his creation - became inextricably linked with each other. It was at this moment that, out of billions of possible hours, the one and only one was irrevocably chosen, by which from now on the course of all future events was to be measured.
In the jargon of relativistic physicists, at the moment of the appearance of Adam, that part of the Universe, which became the habitat of man, began to function in the same space-time reference frame as its Creator. Starting from this point, the chronology of the Bible and the course of time on Earth became one - the general spatio-temporal relationship between God and man was henceforth fixed.
The results of this new connection are evident from a glance at the biblical text. There is a parallelism between the dates to which the Bible refers events that occurred after the creation of Adam, and the corresponding archaeological estimates of the chronology of the same events. The Bronze Age of the biblical calendar and the Bronze Age of archeology do coincide. According to the Bible, Hatzor was destroyed by Joshua 3,300 years ago; archeology, as it turned out after detailed research, refers this event to the same period. The part of the biblical calendar that begins with the creation of Adam looks quite logical to our eyes, and the discovery of the Dead Sea Scrolls proves that the Bible correctly describes events thousands of years before they are confirmed by modern archaeological finds. If we did not know the law of relativity and if we tried to date events that took place on Earth in the time after Adam, from another point in the Universe, we would now be surprised why in our perception the elapsed time differs from what is fixed by clocks on Earth.
In the first six days of the existence of our Universe, the Eternal Clock measured 144 hours. We now know that this span of time does not necessarily coincide with the same span of time measured elsewhere in the universe. As inhabitants of this universe, we evaluate the passage of time with the help of clocks in our local frame of reference; such clocks include radioactive dating, geological data, and measurements of speeds and distances in the expanding universe. It is with this watch that humanity travels through time and space.
When the Bible describes how our universe evolved day by day during the first six days following Creation, it is indeed talking about six days of 24 hours each. But the frame of reference in which these days were counted included the entire universe. This first week of Creation is by no means a fairy tale, designed to satisfy the curiosity of a child so that later, with the advent of adult wisdom, to be discarded as unnecessary. On the contrary, it contains allusions to events that humanity is only now beginning to understand.
The sages, the interpreters of the Bible, have long warned that our understanding of the events of the first six days of Creation will not match our understanding of nature in the times following the appearance of Adam. They understood this from the description of the Sabbath rest contained in the Ten Commandments. If we compare the text in Exodus 20:11 with the text in Zechariah 5:11 and 2 Samuel 21:10, we see that the same word for rest is used in these texts, but with different shades. From the way the word is used there, it can be deduced that God did not actually "rest" on the first Sabbath. Rather, the Creator paused in his work to survey the universe that had been created in the first six days. Our perception of this break, according to Maimonides, is that at all times from this first Sabbath onwards, the laws of nature, including the passage of time, will function in a "normal" way. By contrast, the course of events that took place during the first six days could look illogical, as if there had been a violation of the laws of nature and time. As you can see, the prediction of the sages that we would perceive the biblical and scientific pictures of the early universe as contradictory, actually came true.
The first Sabbath marks the beginning of the calendar, counting the time since the creation of Adam. And it is this part of the calendar that corresponds to our logical perception of reality. Thanks to the extraordinary fact of time relativity, Einstein's law of relativity, the biblical calendar is correct on those six days as well. It became unnecessary to explain the discovery of fossil finds by saying that the Creator deliberately placed them where they were found, to test our faith in the act of Creation or to satisfy our curiosity. The rate of radioactive decay in rocks, meteorites and fossil finds correctly reflects the passage of time, but this passage of time has been measured and continues to be measured by clocks in our earth system reference. The time fixed by these clocks was and continues to be only relatively, that is, only locally correct. Other clocks, located in other reference systems, attribute other, but no less correct, moments of time to the events taking place on Earth. And it will always be so, as long as the Universe obeys the laws of nature.
LITERATURE
- 1. Rashi. "Comments on the Book of Genesis". 1:1.
- 2. Nachmanides. "Comments on the Torah". "Genesis" 5:4.
- 3. "Archaeology and studies of the Old Testament." Ed. Thomas. (Thomas, ed., Archeology and Old Testament Study).
- 4. Newton. "Mathematical principles of natural philosophy". (Newton, Mathematical Principles of Natural Philosophy).
- 5. Einstein. "Relativity: special and general theories". (Einstein, Relativity: The Special and General Theories).
- 6. Cohen. "The Birth of a New Physics". (Cohen, The Birth of a New Physics).
- 7. Pagels. "Perfect Symmetry". (Pagels, Perfect Symmetry).
- 8. Shankland. "The Michelson-Morley Experiment". (Shankland, "The Michelson-Morley experiment", American Journal of Physics, 32 (1964):16).
- 9. German. "The Origin of Quantum Theory" (1899-1913). (Hermann, The Genesis of the Quantum Theory (1899-1913)).
- 10. Taylor and Wheeler. "Physics of space-time". (Taylor and Wheeler, Spacetime Physics).
- 11. Häfele and Keating, "Circumnavigating Atomic Clocks: Observations of Relativistic Time Shift". (Hafele and Keating, "Around-the-world atomic clocks: observed relativistic time gains." Science, 117 (1972): 168).
- 12. Woosley and Phillips, Supernova 1987A1. (Woosley and Phillips, "Supernova 1987A!" Science, 240 (1988): 750).
- 13. Maimonides. "The Mentor of the Hesitating", part 1, ch. 67.