# Inertial system

A reference system in physics is called an inertial system (from the Latin inertia for “inertia”), when every force-free body remains at rest relative to this reference system or moves uniformly (unaccelerated) and in a straight line. Force-free means that the body does not experience any forces from other objects or that they cancel each other out, so that the resulting force is zero .

If a body, although it is free of forces in this sense, accelerates or moves in a curvilinear manner relative to a certain reference system, the accelerations that occur are explained by inertial forces . These are due to the fact that the reference system is in rotation or otherwise accelerated movement compared to an inertial system. Inertial forces do not originate from other bodies and are not included in the assessment of the freedom of forces. There are no inertial forces in an inertial system.

For example, because of the rotation of the earth, a reference system connected to the earth's surface is not an inertial system. The inertial forces caused by the rotation, however, are usually not noticeable, which is why such a system can be regarded as an inertial system in a very good approximation. In a real inertial system the sky of the fixed star would not rotate.

A three-dimensional space , for which a (strictly or approximately valid) inertial system can be used reproducibly as a reference system, is called an inertial space in some specialist fields .

In the modern works on theoretical mechanics , the inertial system is often defined solely with the aid of the inertia law, which corresponds to the first of Newton's three axioms . For a complete definition, however, all three Newtonian axioms are required: The first calls the rectilinear, uniform motion of force-free bodies as an essential property of an inertial system. The second generally defines the forces resulting from the accelerations they cause. Finally, the third requires that there must be a counterforce for every force, so that only forces are meant here that are based on interactions between bodies, which is precisely not the case with inertial forces.

The term “inertial system” was first worked out in 1885 by Ludwig Lange , who (according to Ernst Mach ) specified the term “force-free body” as follows: The force-free body can be thought of as being “infinitely” far removed from other matter. Equivalent is (according to James Maxwell ) to express the principle of inertia negatively: Whenever a body observed in an inertial system does not move in a straight line, it is caused by forces that emanate from other bodies. (P. 271)

## background

The same physical process is generally described differently by different observers . An example: for an observer on earth the sun rotates around the earth and the other planets move on sometimes loop-shaped orbits , while an observer on the sun sees that the earth and all other planets move around the sun. The movement can therefore only be described relative to a reference system , i.e. to the point of view of an observer. If the movements appear different, observers who do not take into account the influence of the choice of the frame of reference would have to explain the same process by different physical causes .

This is especially true for movements of bodies that do not run in a straight line. Inertial systems are the reference systems in which any deviation from the rectilinear, uniform motion of a body can be traced back to the influence of a force emanating from another body. So in them the principle of inertia applies. Different inertial systems can differ by a linear, uniform translational movement. Every rotation or other acceleration of the reference system leads to the fact that force-free bodies do not always move in a straight line. This is described by the action of inertial forces, which are not generated by other bodies, but for the observer concerned only by the acceleration of his reference system. Since there is no inertial force in an inertial system, the equations of motion of mechanics can in principle have the simplest form. Nevertheless, in many areas it is advantageous to consider the processes in an accelerated reference system if this is more favorable for practical reasons (e.g. in the geosciences ).

An indeterminacy arises from the fact that the inertial force, which is caused by a uniform acceleration, does not differ in any way from a gravitational force in a constant, homogeneous gravity field with a correspondingly selected strength ( principle of equivalence ). Therefore, one can also see a uniformly accelerated reference system as an inertial system in which only a changed gravity prevails. If the gravitational field is homogeneous and the acceleration of the reference system just corresponds to the free fall , the gravitation is even exactly compensated by the inertial force. The state of weightlessness in space stations is a local approximation, insofar as the earth's gravitational field can be viewed as homogeneous. (There is no such thing as an exactly homogeneous gravitational field.) In this sense, reference systems that are accelerated against each other can also be called inertial systems. The basic idea of general relativity goes even further : only reference systems that are in free fall are inertial systems, and the whole phenomenon of gravitation is explained by the inertial force observed in a reference system that is accelerated against it.

## Newtonian mechanics

The easiest way to imagine an inertial system as a reference system at a distant place in space in complete weightlessness, i.e. far away from larger masses that could disturb the movement of bodies through their gravitation. The spatial coordinates can then be specified relative to any force-free reference body that is considered to be “at rest”. Which of these reference bodies is selected is completely arbitrary. This is what the Galilean principle of relativity says. A second body that moves uniformly and in a straight line in this frame of reference is also free of forces. So it could itself be the point of reference for a second inertial system. In other words: every frame of reference that moves uniformly and in a straight line relative to an inertial frame is also an inertial frame. Hence there are infinitely many inertial systems in Newtonian mechanics. The spatial and temporal coordinates of two inertial systems depend on a transform Galilei together.

Conversely, every frame of reference that moves accelerated relative to an inertial frame is not itself an inertial frame. In such an accelerated frame of reference, the law of inertia cannot easily be applied. In order to be able to justify the accelerated or curvilinear movements of bodies in accelerated reference systems correctly, the assumption of so-called inertial forces is required , for which no real cause can be found and no reaction can be given.

Galileo transformations form a group with regard to the sequential execution . Simple shifts in time or space belong to it. Since an inertial system changes into an inertial system in the event of a spatial or temporal shift, inertial systems do not distinguish any place or point in time. Space and time are homogeneous .

The Galileo group also includes the finite rotation, which maps the reference directions (front, left, top) of one system to the temporally unchangeable directions of the other system. Since an inertial system changes into an inertial system during a rotation, inertial systems do not indicate any direction. The space is isotropic .

An inertial system can therefore be defined as a reference system with respect to which space is homogeneous and isotropic and time is homogeneous.

Finally, transformation belongs to the Galileo group

${\ displaystyle t ^ {\ prime} = t}$
${\ displaystyle \ mathbf {x} ^ {\ prime} = \ mathbf {x} - \ mathbf {v} \, t,}$

through which one coordinate system is moved against another with constant speed . ${\ displaystyle \ mathbf {v}}$

Since the laws of Newtonian mechanics apply in the same form in all inertial systems, there is no preferred reference system and no possibility to measure a speed absolutely. This is the relativity principle of Newtonian mechanics.

## Special theory of relativity

Instead of the Galileo transformation between inertial systems of Newtonian physics, Lorentz transformations and spatiotemporal shifts mediate in relativistic physics how the coordinates are connected with which uniformly moving observers indicate when and where events take place. Together with the spatial and temporal shifts, Lorentz transformations form the Poincaré group .

According to the following idealized procedure, a uniformly moving observer assigns its inertial coordinates to each event, as with radar: He sends a light beam to the event and uses his watch to measure the start time and the time at which the light beam reflected during the event arrives at him again. He uses the mean value as the time at which the event occurred ${\ displaystyle t _ {-}}$${\ displaystyle t _ {+}}$

${\ displaystyle t = {\ frac {1} {2}} {\ bigl (} t _ {+} + t _ {-} {\ bigr)},}$

the distance is half the transit time of the light moving back and forth times the speed of light : ${\ displaystyle c}$

${\ displaystyle r = {\ frac {c} {2}} {\ bigl (} t _ {+} - t _ {-} {\ bigr)}}$

In addition, he determines angles and between reference directions he has chosen and the outgoing light beam. With this he assigns the following coordinates to the event : ${\ displaystyle \ theta}$${\ displaystyle \ varphi}$

${\ displaystyle x = {\ begin {pmatrix} t \\ r \, \ sin \ theta \ cos \ varphi \\ r \, \ sin \ theta \ sin \ varphi \\ r \, \ cos \ theta \ end { pmatrix}}}$

The reflected light beam only comes back from the direction of the outgoing light beam for each event if the observer is not turning. In this way the observer can distinguish whether he is turning or whether he is being orbited by other objects.

## general theory of relativity

The general theory of relativity is formulated so that their equations apply in any coordinate system. The world lines of freely falling particles are the straight lines (more precisely geodesics ) of curved spacetime . In free fall, gravitation is shown by the tidal effect, that neighboring geodesics strive towards or away from each other and can repeatedly intersect. For example, if two space stations orbit the earth with the same constant distance in different planes, their trajectories intersect where the orbital planes intersect, then their distance increases until they have passed through a quarter circle, then decreases again until their orbits after one Semicircle crosses again. This effect of uneven gravitation (it acts in different places in different directions or with different strengths) is called the tidal effect. With small distances it increases with the distance. If the tidal effect can be neglected, then the special theory of relativity applies in free fall .

Wiktionary: Inertial system  - explanations of meanings, word origins, synonyms, translations

## Individual evidence

1. ^ Rudolf Brockhaus: flight control. 2nd edition, Springer 2001, ISBN 978-3-662-07265-3
2. ^ Günter Seeber: Satellite Geodesy: Basics, Methods and Applications. De Gruyter, Berlin, 1989, ISBN 3-11-010082-7 , p. 396
3. Manuela Seitz, Detlef Angermann, Mathis Bloßfeld: Geometric Reference Systems . In: Earth measurement and satellite geodesy . Springer Berlin Heidelberg, Berlin, Heidelberg 2017, ISBN 978-3-662-47099-2 , pp. 325-348 , doi : 10.1007 / 978-3-662-47100-5_17 ( springer.com [accessed July 7, 2020]).
4. Fließbach: Textbook on Theoretical Physics I - Mechanics. Springer, 7th edition, 2015, p. 9 .: “There are reference systems in which the force-free movement takes place at constant speed. These are inertial systems. "
5. Henz, long Hanke: pathways through theoretical mechanics 1. Springer, 2016, p 42. "There are coordinate systems in which each force-free mass point moves rectilinearly uniform or suspended. These particularly important coordinate systems are called inertial systems. ”
Almost the same also in Nolting: Basic Course Theoretical Physics 1 - Classical Mechanics , Springer, 10th edition, 2013, p. 173.
6. Nayaran Rana, Pramod Joag: Classical Mechanics . 24th edition. Tata McGraw-Hill Education, New Delhi 2001, ISBN 0-07-460315-9 , pp. 9 .
7. Ludwig Lange: About the scientific version of Galileo's law of inertia . In: W. Wundt (Ed.): Philosophical Studies . tape 2 , 1885, p. 266 ff . ( online [accessed June 12, 2017]).
8. ^ LD Landau, EM Lifshitz: Mechanics . Pergamon Press, 1960, pp. 4-6 .

## literature

• Ernst Schmutzer: Basics of Theoretical Physics . 3. Edition. tape 1 . Wiley-VCH, 2005, ISBN 978-3-527-40555-8 .
• Walter Greiner: Theoretical Physics - 1. Classical Mechanics 1 . 8th edition. tape 1 . Europa-Lehrmittel, 2007, ISBN 978-3-8085-5564-4 .
• Martin Mayr: Technical mechanics: statics, kinematics, kinetics, vibrations, strength theory . 7th edition. Carl Hanser Verlag, 2012, ISBN 978-3-446-43400-4 .