One of the tasks of physics is to identify the basic laws according to which the movement of matter occurs, that is, any changes in its state over time. To describe the motion in physics, the basic concepts of an inertial and non-inertial reference frame are introduced. This is a kind of material basis, on the basis of which, it is possible to measure the characteristics of the state of the physical system.
What is a frame of reference
Any movement can be considered only in relation to any object, from the parameters of which state changes are counted. With regard to mechanical motion, as such an object, a complex of bodies with a strictly fixed mutual spatial arrangement is chosen, called the reference body, to which the spatial coordinate system is attached. Together with the clock measuring the time for this complex, it is a reference system. It does not have to include a real body - an indispensable condition is the presence of a coordinate system and a clock.
In order to understand what reference frames are called non-inertial, it is necessary to turn to one of the fundamental properties inherent in physical bodies.
Reference systems and body inertia
The property of bodies to resist the action of forces that cause acceleration is called inertia. A sign is associated with it, according to which two types of reference systems are distinguished . An inertial system is considered to be if the bodies associated with it are at rest or move by inertia (rectilinearly and evenly) under conditions when they are not affected by external forces.
If, in the absence of external forces, the body’s movement in any reference frame changes, then this reference frame is non-inertial. In relation to a certain inertial spatial system, it moves accelerated (including that it can rotate), and the body associated with it, due to its inertia, tends to maintain peace or uniform direct movement.
Inertia forces
The observer will record the change in movement as a fact of acceleration of the body (“material point”). Therefore, the question of which reference frames are called non-inertial can give the following answer: these are systems in which inertia externally manifests itself as a source of force that informs the body of acceleration.
The concept of inertia is rather vague and ambiguous. Sometimes they are called fictitious, or pseudo-forces, due to the fact that their origin is not associated with the interaction of bodies. These forces are real for the observer within the framework of a non-inertial reference frame: here they are applied to the body mass and do the work. But an outside observer in an inertial system, in general, will not record their occurrence - for him only the forces of interaction of the observed material points will be real. The work of these forces will cause a change in the kinetic energy of the body.
Non-inertial reference systems in classical mechanics
Newtonian laws of motion were formulated for inertial reference systems. However, using them, one can describe the motion of the body in a non-inertial system as well, only in this case it is necessary to introduce inertia forces into the equation as actually acting. The general form of the equation of motion of a material point in a non-inertial reference frame will be: ma '= F + F in . Here m is the mass of the body, a 'is the acceleration in a non-inertial system, F is the result of the real forces of interaction, F in are the forces of inertia.
For clarity, it is necessary to give several examples of non-inertial reference systems and the forces acting in them. In all cases, in order to simplify, friction and gravity are not taken into account.
Translational reference frame acceleration
Let an unsecured barrel lie on the last open platform of the train. When the train starts and accelerates, the barrel will roll along the platform and fall onto the rails. What will the observer in the inertial reference frame, standing on the platform (except the fact of mismanagement), note? He will see that the train went with some acceleration and simply “left” from under the barrel, which in the reference system of this observer was at rest.
An observer leaving with the train will note that, for no apparent reason, the loose object gained acceleration –a in in the opposite direction to the composition acceleration. Since it is not equal to zero, for observing the laws of mechanics, the observer on the platform will have to assume that a certain special force F in = –ma in acted on the barrel. This will be the force of inertia. Note that it manifests itself to this observer as an external force of attraction. He himself will feel how, when accelerating a train, something pulls him “backward”.
Centrifugal force
Now imagine the same composition moving uniformly and entering a turn. In this example of a non-inertial reference system, a person riding on a platform will immediately feel that a certain force (obviously, again inertial) shifts it to the outer edge of the platform. The value of this force F in = mw 2 R = mv 2 / R (w is the angular velocity, R is the turning radius) is proportional, as we see, to the passenger mass and the squared speed of the train. It is directed radially from the center of rotation.
But the observer, standing near the embankment, will see that the person on the platform seeks to maintain inertial motion, and the train, again, "leave" from under it. Let, in order not to fly off the platform, the passenger rests on some fixed load. Then this load presses on it with a force F = mv 2 / R, which imparts a centripetal acceleration, which allows it to remain on the train. So in this case, the observer associated with the inertial reference system will not record any inertia forces.
Coriolis Force
If the body moves in a rotating spatial system, another inertial force arises - the Coriolis force, which deflects its trajectory in a plane oriented perpendicular to the axis of rotation of the system. When approaching this axis, the Coriolis force deflects the body in the direction of rotation, while moving away from the axis, the direction of action of the force is opposite to rotation. In the general case, it is equal to the vector product of the linear and angular velocities of the body multiplied by its doubled mass: F in = F K = 2m [v × w].
So, in the frame of reference of the Earth due to its rotation, the bodies associated with it experience the action of the Coriolis force. In the northern hemisphere, it deflects to the right the trajectory of the flight of shells, the movement of rivers, the oscillation of pendulums, currents in the oceans and the movement of air masses. In the southern hemisphere, displacement occurs, respectively, to the left. The manifestation of the Coriolis force is minimal near the equator and grows closer to the poles, since there the surface of the planet approaches a plane perpendicular to the axis of its rotation.
In an inertial reference frame, this force is again not observed. For example, the Foucault pendulum for such an observer preserves the plane of oscillations, and the Earth, on the contrary, “turns” under it.
Mass, inertia and gravity
Newtonian laws of motion operate with mass as a measure characterizing the inertia of the body, while gravitational mass appears in the law of universal gravitation. The joint application of these laws is possible only due to the equality of the two types of masses, which in the framework of classical physics cannot be explained, but is only an observational fact.
It was noted somewhat above that the behavior of material points in a non-inertial reference system is a process that is locally indistinguishable from the manifestation of a gravitational field superimposed externally on an inertial system. This fact was formulated by A. Einstein in 1913 as the principle of equivalence of inertia and gravity forces, meaning the unity of their physical nature, which made it possible to generalize the concept of mass. A few years later, Einstein used it when creating a holistic theory.
Non-inertial reference systems in the general theory of relativity
The Einstein theory of gravity makes it possible to explain the nature of the action of gravitational forces through changes in the geometry of the space-time continuum in the presence of mass. Any body in its own frame of reference moves inertially along the so-called geodesic, or straight (but not straight!) Lines passing through points of the same curvature of space-time.
For example, a stone in a fall by inertia follows a geodesic line in space-time deformed by the mass of the Earth. The lying object is subjected to the action of force from the side of the planet’s surface, and its spatiotemporal trajectory - the world line - is deviated from the straightest.
Due to this uneven and inconsistent curvature in the four-dimensional continuum, one cannot distinguish any privileged inertial reference frame. In reality, all of them are non-inertial and equal: the laws of physics expressed by the mathematical apparatus of this theory do not change shape during any transitions between reference systems, that is, they are invariant with respect to all systems.
Reference systems in the quantum world
The energy state of quantum fields is related by coordinates or time, and theoretically, the motion associated with a non-inertial reference frame should also have features in quantum theory. So, one of the consequences of quantum field theory, developed in 1976 by the Canadian physicist B. Unru, suggests a change in the observed properties of vacuum during motion with acceleration.
Vacuum is the lowest, zero state of a quantum field. In accordance with the principle of uncertainty, it is subject to fluctuations - zero oscillations, which cannot be seen with inertial motion. However, in an accelerating reference system, the observer will not detect a zero state of vacuum, but a background of particles having the form of radiation with a thermal spectrum.
The increase in vacuum temperature predicted in the Unruh effect depends on acceleration and is very small with a coefficient of the order of 10 -21 K, which makes the possibility of experimental verification extremely difficult. And it is very desirable for physicists, since it would give them a tool for determining absolute acceleration.
On the importance of the concept of reference system
There is no doubt that both inertial and non-inertial reference systems are the very basis for understanding the essence of physical motion. One cannot do without them either in classical mechanics, which perfectly copes with the description of a world close to us in scale, nor in the theory of relativity, which reveals the nature of space-time and motion in it, nor even in the physics of quantum systems. Even a superficial acquaintance with them convincingly indicates that the world is built much more complicated than intuitive ideas sometimes limited to us by direct experience suggest us.