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Velocity of propagation of interaction
For the description of processes taking place in nature, one must have a system of
reference. By a system of reference we understand a system of coordinates serving to indicate
the position of a particle in space, as well as clocks fixed in this system serving to indicate
the time.
There exist systems of reference in which a freely moving body, i.e. a moving body which
is not acted upon by external forces, proceeds with constant velocity. Such reference systems
are said to be inertial.
If two reference systems move uniformly relative to each other, and if one of them is an
inertial system, then clearly the other is also inertial (in this system too every free motion will
be linear and uniform). In this way one can obtain arbitrarily many inertial systems of
reference, moving uniformly relative to one another.
Experiment shows that the so-called principle of relativity is valid. According to this
principle all the laws of nature are identical in all inertial systems of reference. In other
words, the equations expressing the laws of nature are invariant with respect to transforma-
tions of coordinates and time from one inertial system to another. This means that the
equation describing any law of nature, when written in terms of coordinates and time in
different inertial reference systems, has one and the same form.
The interaction of material particles is described in ordinary mechanics by means of a
potential energy of interaction, which appears as a function of the coordinates of the inter-
acting particles. It is easy to see that this manner of describing interactions contains the
assumption of instantaneous propagation of interactions. For the forces exerted on each
of the particles by the other particles at a particular instant of time depend, according to this
description, only on the positions of the particles at this one instant. A change in the position
of any of the interacting particles influences the other particles immediately.
However, experiment shows that instantaneous interactions do not exist in nature. Thus a
mechanics based on the assumption of instantaneous propagation of interactions contains
within itself a certain inaccuracy. In actuality, if any change takes place in one of the inter-
acting bodies, it will influence the other bodies only after the lapse of a certain interval of
time. It is only after this time interval that processes caused by the initial change begin to
take place in the second body. Dividing the distance between the two bodies by this time
interval, we obtain the velocity of propagation of the interaction.
We note that this velocity should, strictly speaking, be called the maximum velocity of
propagation of interaction. It determines only that interval of time after which a change
occurring in one body begins to manifest itself in another. It is clear that the existence of a maximum velocity of propagation of interactions implies, at the same time, that motions of
bodies with greater velocity than this are in general impossible in nature. For if such a motion
could occur, then by means of it one could realize an interaction with a velocity exceeding
the maximum possible velocity of propagation of interactions.
Interactions propagating from one particle to another are frequently called "signals",
sent out from the first particle and "informing" the second particle of changes which the
first has experienced. The velocity of propagation of interaction is then referred to as the
signal velocity.
From the principle of relativity it follows in particular that the velocity of propagation
of interactions is the same in all inertial systems of reference. Thus the velocity of propaga-
tion of interactions is a universal constant. This constant velocity (as we shall show later) is
also the velocity of light in empty space. The velocity of light is usually designated by the
letter c, and its numerical value is c = 2.998 × 10^10
cm/sec.
The large value of this velocity explains the fact that in practice classical mechanics
appears to be sufficiently accurate in most cases. The velocities with which we have occasion
to deal are usually so small compared with the velocity of light that the assumption that the
latter is infinite does not materially affect the accuracy of the results.
The combination of the principle of relativity with the finiteness of the velocity of propaga-
tion of interactions is called the principle of relativity of Einstein (it was formulated by
Einstein in 1905) in contrast to the principle of relativity of Galileo, which was based on an
infinite velocity of propagation of interactions.
The-mechanics based on the Einsteinian principle of relativity (we shall usually refer to it
simply as the principle of relativity) is called relativistic. In the limiting case when the
velocities of the moving bodies are small compared with the velocity of light we can neglect
the effect on the motion of the finiteness of the velocity of propagation. Then relativistic
mechanics goes over into the usual mechanics, based on the assumption of instantaneous
propagation of interactions; this mechanics is called Newtonian or classical. The limiting
transition from relativistic to classical mechanics can be produced formally by the transition
to the limit c -> oo in the formulas of relativistic mechanics.
In classical mechanics distance is already relative, i.e. the spatial relations between
different events depend on the system of reference in which they are described. The state-
ment that two nonsimultaneous events occur at one and the same point in space or, in
general, at a definite distance from each other, acquires a meaning only when we indicate the
system of reference which is used.
On the other hand, time is absolute in classical mechanics; in other words, the properties
of time are assumed to be independent of the system of reference; there is one time for all
reference frames. This means that if any two phenomena occur simultaneously for any one
observer, then they occur simultaneously also for all others. In general, the interval of time
between two given events must be identical for all systems of reference.
For the description of processes taking place in nature, one must have a system of
reference. By a system of reference we understand a system of coordinates serving to indicate
the position of a particle in space, as well as clocks fixed in this system serving to indicate
the time.
There exist systems of reference in which a freely moving body, i.e. a moving body which
is not acted upon by external forces, proceeds with constant velocity. Such reference systems
are said to be inertial.
If two reference systems move uniformly relative to each other, and if one of them is an
inertial system, then clearly the other is also inertial (in this system too every free motion will
be linear and uniform). In this way one can obtain arbitrarily many inertial systems of
reference, moving uniformly relative to one another.
Experiment shows that the so-called principle of relativity is valid. According to this
principle all the laws of nature are identical in all inertial systems of reference. In other
words, the equations expressing the laws of nature are invariant with respect to transforma-
tions of coordinates and time from one inertial system to another. This means that the
equation describing any law of nature, when written in terms of coordinates and time in
different inertial reference systems, has one and the same form.
The interaction of material particles is described in ordinary mechanics by means of a
potential energy of interaction, which appears as a function of the coordinates of the inter-
acting particles. It is easy to see that this manner of describing interactions contains the
assumption of instantaneous propagation of interactions. For the forces exerted on each
of the particles by the other particles at a particular instant of time depend, according to this
description, only on the positions of the particles at this one instant. A change in the position
of any of the interacting particles influences the other particles immediately.
However, experiment shows that instantaneous interactions do not exist in nature. Thus a
mechanics based on the assumption of instantaneous propagation of interactions contains
within itself a certain inaccuracy. In actuality, if any change takes place in one of the inter-
acting bodies, it will influence the other bodies only after the lapse of a certain interval of
time. It is only after this time interval that processes caused by the initial change begin to
take place in the second body. Dividing the distance between the two bodies by this time
interval, we obtain the velocity of propagation of the interaction.
We note that this velocity should, strictly speaking, be called the maximum velocity of
propagation of interaction. It determines only that interval of time after which a change
occurring in one body begins to manifest itself in another. It is clear that the existence of a maximum velocity of propagation of interactions implies, at the same time, that motions of
bodies with greater velocity than this are in general impossible in nature. For if such a motion
could occur, then by means of it one could realize an interaction with a velocity exceeding
the maximum possible velocity of propagation of interactions.
Interactions propagating from one particle to another are frequently called "signals",
sent out from the first particle and "informing" the second particle of changes which the
first has experienced. The velocity of propagation of interaction is then referred to as the
signal velocity.
From the principle of relativity it follows in particular that the velocity of propagation
of interactions is the same in all inertial systems of reference. Thus the velocity of propaga-
tion of interactions is a universal constant. This constant velocity (as we shall show later) is
also the velocity of light in empty space. The velocity of light is usually designated by the
letter c, and its numerical value is c = 2.998 × 10^10
cm/sec.
The large value of this velocity explains the fact that in practice classical mechanics
appears to be sufficiently accurate in most cases. The velocities with which we have occasion
to deal are usually so small compared with the velocity of light that the assumption that the
latter is infinite does not materially affect the accuracy of the results.
The combination of the principle of relativity with the finiteness of the velocity of propaga-
tion of interactions is called the principle of relativity of Einstein (it was formulated by
Einstein in 1905) in contrast to the principle of relativity of Galileo, which was based on an
infinite velocity of propagation of interactions.
The-mechanics based on the Einsteinian principle of relativity (we shall usually refer to it
simply as the principle of relativity) is called relativistic. In the limiting case when the
velocities of the moving bodies are small compared with the velocity of light we can neglect
the effect on the motion of the finiteness of the velocity of propagation. Then relativistic
mechanics goes over into the usual mechanics, based on the assumption of instantaneous
propagation of interactions; this mechanics is called Newtonian or classical. The limiting
transition from relativistic to classical mechanics can be produced formally by the transition
to the limit c -> oo in the formulas of relativistic mechanics.
In classical mechanics distance is already relative, i.e. the spatial relations between
different events depend on the system of reference in which they are described. The state-
ment that two nonsimultaneous events occur at one and the same point in space or, in
general, at a definite distance from each other, acquires a meaning only when we indicate the
system of reference which is used.
On the other hand, time is absolute in classical mechanics; in other words, the properties
of time are assumed to be independent of the system of reference; there is one time for all
reference frames. This means that if any two phenomena occur simultaneously for any one
observer, then they occur simultaneously also for all others. In general, the interval of time
between two given events must be identical for all systems of reference.
