# Reference system

A reference system is in physics an imaginary space - temporal entity that is required to change the behavior of location-dependent variables to describe clearly and completely. In particular, the position and movement of physical bodies can only be specified relative to a reference system. A reference system is defined by choosing a reference point and defining the spatial directions, as well as determining a physical process for measuring time. This first defines what is to be understood by "rest" and "movement". In addition, this enables a coordinate systemto introduce, with the help of which physical events can be described mathematically by specifying their space-time coordinates. If observers start from different reference systems, they can give different descriptions of a physical process, which nevertheless all apply if one takes into account their respective reference system. For example, a car driver could rightly claim that a tree is coming towards him, while an observer standing at the roadside, also rightly, sees the process in reverse. In physics, every frame of reference defined in this way may be chosen equally and that there is no fundamental process by which a particular frame of reference can be distinguished from all others.

## Reference systems in classical physics and in relativity theory

In classical physics, different reference systems correspond in the distances between two points, in the angles between two straight lines and in the time difference between two events. Therefore the time coordinate can be chosen uniformly for all reference systems, and the vectorial addition applies to the velocities. This means that the speed that a process has in a moving reference system K ' is added vectorially to the speed with which K' moves in a reference system K in order to obtain the speed of the same process in K : ${\ displaystyle t}$${\ displaystyle {\ vec {v}} '}$${\ displaystyle {\ vec {V}}}$${\ displaystyle {\ vec {v}}}$

${\ displaystyle {\ vec {v}} = {\ vec {V}} + {\ vec {v}} '}$ : Classical addition theorem of velocities.

In contrast, in reality the speed of light agrees in all reference systems, which cannot be reconciled with the classical addition theorem. The solution found in relativity theory assumes that distances, angles and time intervals can be different in different reference systems. One consequence is the relativistic addition theorem for velocities , according to which vectorial addition is a good approximation only for small velocities (compared to the speed of light). The deviations that occur noticeably at high speeds are confirmed by measurements.

## Reference points and coordinate systems

Sketch of a reference system

A point on a real body is often chosen as the reference point , e.g. B. "the left, front corner of the table", "the center of the platform" or "the center of the sun". But it can also be an imaginary point, e.g. B. "the common center of gravity of the earth and moon" or "a freely falling frame of reference".

In order to be able to determine the three spatial directions, at least two further points are required: A plane is spanned by these three points . The third dimension is then obtained e.g. B. as normal on this level. This means that you have all the requirements for the definition of a coordinate system that can be used to specify spatial points. Therefore the term reference system is sometimes used synonymously with the term coordinate system in the literature. However, a distinction is usually made between the terms because one and the same reference system (e.g. that of the earth) can be described by different coordinate systems (e.g. Cartesian coordinates and polar coordinates ). The space and time coordinates of any process can be converted from one coordinate system to the other by means of a coordinate transformation. Physical formulas that describe the same process in the same reference system can therefore take on completely different shapes when using different coordinate systems.

## Frequently chosen reference systems

### Rest system

A reference system in which the observed body rests is called its rest system. In this reference system it has no kinetic energy, neither through translation nor through rotation, and is in equilibrium of forces .

### Laboratory system

The rest system of the observer and the apparatus of the observed experiment is called the laboratory system. It is usually the closest reference system to describe an experiment, but not always the most suitable. The laboratory system is - if it is on earth - only approximately an inertial system.

### Absolute and relative system

In fluid mechanics , among other things , a distinction is made between two different reference systems: The reference system in which the outer casing of an object under consideration rests, for example the casing of a turbine , is defined as the "absolute system". A reference system that moves relative to this, for example a reference system that moves with the turbine blades, is therefore referred to as a “relative system”.

### Center of gravity

In the center of gravity of a physical system consisting of several bodies, their common center of gravity “rests” in the origin of the reference system. For some physical processes, e.g. B. the elastic impact , the center of gravity system allows a particularly simple description, because the impulses of the two bodies involved are here by definition opposite and equal. Even with several bodies involved, as they are, for. B. occur frequently in nuclear reactions, the center of gravity system makes sense: Here the vector sum of all impulses is to be considered. It is constantly equal to zero (see principle of centroid ).

### Inertial system

A reference system in which force-free particles rest or move through straight orbits at constant speed is called an inertial system . This is what the law of inertia says . The location coordinates of these orbits are linear inhomogeneous functions of time${\ displaystyle {\ vec {r}} (t)}$${\ displaystyle t}$

${\ displaystyle {\ vec {r}} (t) = {\ vec {r}} (0) + {\ vec {v}} \, t \ ,.}$

In it is the position of the particle at the time and its speed. Such reference systems are fixed except for the choice of the place and time origin, the choice of three directions (“up, front, right”) and the choice of a constant speed of the entire reference system (compared to another inertial system). This means: Every reference system that is at rest relative to an inertial system or that moves with constant speed in relation to it is also an inertial system. Currently (2017) the best known approximation to an inertial system is the inertial space defined in astronomy . ${\ displaystyle {\ vec {r}} (0)}$${\ displaystyle t = 0}$${\ displaystyle {\ vec {v}}}$

### Accelerated frame of reference

A frame of reference that is not an inertial frame is called an accelerated frame of reference. In relation to such a system, the bodies show movements that cannot always be explained with the forces known from the inertial system.

The following example should serve to explain the difference:

A train pulls up in the station, in it a man with a pram without the brakes on. The reference system in which the platform rests is (to a very good approximation) an inertial system. However, the reference system of the approaching train is an accelerated reference system. The stroller does not experience any force along the direction of travel when it starts moving and therefore remains at rest in the “platform” reference system. It therefore rolls backwards at an accelerated rate relative to the approaching train. To keep the stroller at rest relative to the train, the man has to exert a force on the stroller that accelerates it synchronously with the train. The stroller counteracts this accelerating force with its equally large inertial resistance, which acts back on the man as a real force.

In an accelerated reference system, bodies on which no forces act from the point of view of the inertial system move with an acceleration or on curved paths. A force seems to act on the body , which according to the basic equation of mechanics ${\ displaystyle {\ vec {a}}}$${\ displaystyle {\ vec {F}}}$

${\ displaystyle {\ vec {F}} = m {\ vec {a}}}$

caused this acceleration. Thus the observer in the accelerated frame of reference concludes that there is a force that does not exist in the inertial frame. Such forces are called pseudo or inertial forces . For the observer in the accelerated frame of reference, although no other cause can be found for them than his choice of frame of reference, they are just as real as all other forces. A body in the accelerated system only remains at rest if there is a force opposite to the inertial force that keeps the body at rest relative to the accelerated reference system. Apart from the acceleration observed in the accelerated reference system, all other consequences that are usually attributed to the inertial force, strictly speaking, result from the forces with which this acceleration is influenced (e.g. prevented). ${\ displaystyle {\ vec {a}}}$

### Rotating frame of reference

A rotating reference system is the special case in which an accelerated reference system does not perform a translation, but only a rotational movement. Although nothing appears to be accelerated in this situation (provided that the axis of rotation and angular velocity remain the same), the rotating reference system is counted among the accelerated reference systems. In the rotating reference system, bodies that are not on the axis of rotation experience an outwardly directed centrifugal force , and they only remain at rest if they are simultaneously acted on by an inwardly directed centripetal force . If you look at the same situation from an inertial system, the same body follows a circular path around the axis of rotation, and the centripetal force causes precisely the inward acceleration that keeps it on its circular path (see, for example, the chain carousel ).

While the centrifugal force acts on every body in a rotating frame of reference, a second inertial force, the Coriolis force , only acts on bodies that move relative to the rotating system. As long as you z. B. only stands on a rotating disk, you only feel the centrifugal force. If you now try to walk on the disk, the Coriolis force comes into play. It is always directed sideways to the direction of movement and lets you describe a curve. If you try z. B. to go straight towards the axis of rotation (or away from it), you will be deflected in the direction of rotation (or opposite to it). If, on the other hand, you walk around the axis of rotation of the disk at a constant distance, the Coriolis force is directed radially, i.e. parallel or antiparallel to the centrifugal force. If you then run against the direction of rotation just so fast that you stay in the same place when viewed from the outside, then you are at rest in the inertial system, i.e. free from the action of force. In the reference system of the disk, however, a circular movement around the axis of rotation is described, which in turn requires an inwardly directed relative acceleration in this reference system. This is generated by the Coriolis force, which in this case is directed inwards and not only neutralizes the omnipresent centrifugal force in the rotating reference system, but also provides the force that is required for the relative acceleration towards the axis of rotation.

The earth as a reference body defines a rotating reference system. However, due to the slow rotation of the earth, the differences to an inertial system can often be neglected. B. in many physical processes in everyday life. In the laboratory, the differences can only be demonstrated with special experiments such as the Foucault pendulum . On a large scale, however, they have an unmistakable influence, e.g. B. on ocean currents and weather.

### Free falling frame of reference

If the reference system is linked to a point that is in free fall, then gravitational and inertial forces cancel each other out. The term " free fall " is to be taken here in the physical sense; that is, it is not limited to bodies falling straight down. Rather, it means all bodies that are not prevented by any external supporting, holding or frictional forces from freely following the force of gravity. Even a space station that orbits the earth in orbit outside of the atmosphere is therefore in free fall. Here the gravitational pull cannot be felt because gravity accelerates all masses, including the astronauts, evenly and no other forces act. So-called weightlessness prevails .

The “disappearance” of gravitational and inertial forces in freely falling frames of reference can be explained in two ways: One can choose the rest system of the earth as the frame of reference and then find out that a falling body is accelerated by its weight . His own rest system is therefore an accelerated reference system in which inertial forces occur in addition to the force of weight. The magnitude and direction of these inertial forces are such that they precisely compensate for the weight. So a freely falling body behaves in its rest system as if no external forces were acting on it. The other approach is based on the fact that not the rest system of the earth, but the freely falling reference system is an inertial system. From this perspective, a body resting on the earth is constantly accelerating upwards and its weight is nothing but the inertial force caused by this acceleration. In this way the gravitational force itself becomes an "apparent force". Both approaches are mathematically equivalent.

Einstein placed the equivalence of gravitational forces with inertial forces in the form of the equivalence principle at the beginning of his general theory of relativity .

## Change of the reference system

The exact description of a physical phenomenon generally depends on the chosen reference system, for example the observed values ​​for spatial coordinates and times and thus all of the values ​​derived from them such as speed, acceleration, etc. Depending on the reference system, the observations of the same specific process appear different, so that different Formulas can be read and, under certain circumstances, various conclusions can be drawn regarding the course of the process or the physical laws on which it is based.

Quantities and mathematical relationships that remain unchanged when the reference system is changed are called invariant .

### Simple examples

#### Which ball hits the other?

During a game of billiards, an observer standing at the billiard table, i. H. in the laboratory system, like a white billiard ball colliding with a resting red one and then lying there. In another frame of reference, which moves at constant speed so that the white ball is initially resting in it, the red ball approaches the stationary white ball at the opposite speed, bumps it and then stops, while the white ball now moves with it the initial speed of the red moves away. In a third reference system, the center of gravity system of both balls, both balls first move towards each other, collide and move away from each other, always at the same speed, which is just half the initial speed of the red ball in the first reference system. The question of which ball hits the other is not a physically meaningful question insofar as it can be answered differently, depending on the reference system in which an observer interprets the process.

#### At what angle do the balls fly apart?

In the reference system “billiard table” the general rule applies that after a non-central impact of the white against the stationary red billiard ball, both of them move at exactly 90 ° apart. In the center of gravity system, however, their directions of movement always form an angle of 180 ° after the collision (just like before the collision, only along a different direction). Neither of these two rules is a general law of nature.

### Coordinate transformation from one reference system to another

If a second reference system is defined on the basis of a reference system, the coordinates valid in one reference system can be expressed by means of a coordinate transformation for each point and each point in time using the coordinates from the other reference system. In the case of a constant speed of the reference systems against each other, the Galilei transformation is to be used for Cartesian coordinates in classical mechanics . This means that during the transition from one reference system to the other for all speeds, the relative speed of the reference system is added vectorially and the translation for all location coordinates . Although mathematically very simple and immediately clear, this type of coordinate transformation is only correct at relative speeds that are very small compared to the speed of light . If this cannot be assumed, the Galileo transformation is replaced by the Lorentz transformation of relativistic physics. While temporal intervals and spatial distances are invariant in relation to the Galilei transformation, this does not apply to the Lorentz transformation. In particular, speeds cannot simply be added here. (see relativistic addition theorem for speeds ) ${\ displaystyle {\ vec {v}}}$${\ displaystyle {\ vec {v}} t}$

### Relativity Principle

According to the principle of relativity, any two reference systems that move in a straight and uniform manner relative to one another are equivalent. In other words, there is no physical process in which, besides the fact that the two frames of reference move relative to one another (and the necessary consequences such as the Doppler effect), another distinguishing feature between them could be observed. Therefore, the fundamental laws of physics must be invariant to the change between these frames of reference. If the law has the form of a formula in which the coordinates of the respective reference system appear, then the formulas for both coordinate systems must look exactly the same, and one must result from the other if the coordinates in it are expressed by those of the other reference system. In mathematical terms, the laws of nature must be invariant to the coordinate transformation. As a result, terms such as "absolute rest" or "absolute movement" are physically meaningless because they cannot be detected.

## history

### mechanics

Huygens: Two observers (one in the boat, one on land) describe the collision between two spheres differently.

For Aristotle , the natural state of a body is absolute calm. When the body moves, it is only through an internal drive or an external compulsion. For him, rest and movement are objectively distinguishable things, so there is only one objective reference system in Aristotle's physics: the earth.

With the beginning of modern times, Galileo Galilei and Isaac Newton recognized in the 17th century that force-free bodies do not go into a state of rest by themselves, but continue to move in a straight line at their current speed and thus remain in their state of motion. This “perseverance” is called inertia and applies equally to bodies at rest and in motion. Whether a body moves in a straight line or rests therefore depends only on the point of view of the observer, i.e. H. from its frame of reference. The transition between the inertial systems is described in classical mechanics by the Galilei transformation.

Also in the 17th century, Christiaan Huygens examined the differences in the descriptions of a simple mechanical process in different frames of reference. He described, for example, an elastic collision between two objects seen from the bank and from a passing ship (see Galileo transformation ). That served him u. a. to specify the term "movement quantity" or momentum .

Until the beginning of the 20th century, various elementary quantities were tacitly viewed as invariant when the frame of reference changed. B. Spatial and temporal distances. Einstein postulated in the special theory of relativity in 1905 that all inertial systems are physically equivalent (see principle of relativity ) and that the speed of light does not depend on the state of motion of the light source. The invariance of the speed of light follows directly from this. However, if the speed of light, in contrast to all other speeds, is the same in all frames of reference, then times and lengths cannot be invariant.

### Electrodynamics

Up until the beginning of the 20th century, a medium was sought that would allow light to propagate. The impossibility effects of the movement against this hypothetical ether prove led to the preparation of the above-mentioned principle of relativity and the resulting theory of relativity . Accordingly, the idea of ​​the ether had to be dropped. In his special theory of relativity, Einstein was still able to explain the relationship between electric and magnetic fields, which had already been shown in Maxwell's equations. Accordingly, magnetic fields emerge from the electric fields when one changes the frame of reference, and vice versa.

### Astronomy and cosmology

Aristotle used only the geocentric frame of reference and only formulated his laws of motion in relation to this. Ptolemy followed him and created the geocentric worldview that dominated until the 17th century. a. was strongly defended by the Catholic Church (cf. Galileo Trial ). Copernicus described the planetary system in the heliocentric frame of reference in the middle of the 16th century. In it the observer moves with the earth, whereby the loop movements of the outer planets, which seem complicated in his reference system, find a simple explanation. With the apparatus of Newtonian mechanics, the planetary movements could be predicted very precisely if the center of gravity of the solar system was used as the reference point. However, since this is not too far from the center of the sun, the heliocentric worldview is a usable approximate model.

If you mentally move away from the earth, a different reference system makes sense depending on the size scale: In the earth-moon reference system, both celestial bodies move around the common center of gravity. In the solar system reference system, the earth moves on an ellipse around the sun. In the Milky Way reference system, the solar system moves around the center of the Milky Way. Etc.

According to the theory of relativity, there should be no universal reference system in and of itself. However, there is only one frame of reference in which the cosmic background radiation is isotropic . This could theoretically be seen as the "rest system of the universe". However, this does not change the principle of relativity.

## Individual evidence

1. ^ Arnold Sommerfeld: Lectures on theoretical physics, Volume 1: Mechanics . Leipzig 1943, Harri Deutsch 1994, ISBN 978-3-87144-374-9 . On page 9, Sommerfeld writes: “What requirements do we have to make of the ideal reference system for mechanics? We understand it to be a spatiotemporal structure according to which we can determine the position of the mass points and the passage of time, for example a right-angled coordinate system x, y, z and a time scale. "
2. ^ Klaus Lüders, Robert Otto Pohl: Pohl's introduction to physics: Volume 1: Mechanics, acoustics and thermodynamics . Springer DE, 2008, ISBN 978-3-540-76337-6 , pp. 11 ( limited preview in Google Book search).
3. ^ Dieter Meschede: Gerthsen Physics . Springer DE, ISBN 978-3-642-12893-6 , p. 643 ( limited preview in Google Book search).
4. Willy JG Bräunling: Aircraft engines: Fundamentals, aero-thermodynamics, cycle processes, thermal turbo machines, component and emissions . Springer-Verlag, 2013, ISBN 3-662-07268-8 , pp. 527 ( limited preview in Google Book search).
5. From: C. Huygens, Oeuvres Complètes , Vol. 16, The Hague: Martinus Nijhoff, 1940
6. Aristotle: Physics ; Aristotle, Physics , trans. by RP Hardie and RK Gaye .
7. ^ CD Andriesse: Huygens: The Man Behind the Principle . Cambridge University Press, 2005, ISBN 0-521-85090-8 ( limited preview in Google Book Search).
8. ^ Helmar Schramm, Ludger Schwarte, Jan Lazardzig: Collection - Laboratory - Theater: Scenes of Knowledge in the 17th Century . Walter de Gruyter, 2005, ISBN 3-11-020155-0 , p. 47 ( limited preview in Google Book search).
9. ^ A b Albert Einstein: On the electrodynamics of moving bodies. In: Annals of Physics and Chemistry. 17, 1905, pp. 891–921 (as facsimile ; PDF; 2.0 MB).
10. Marcelo Alonso, Edward J. Finn: Physics . Oldenbourg Verlag, 2000, ISBN 3-486-25327-1 , p. 304 ( limited preview in Google Book Search).
11. Bergmann, Schaefer: Textbook of Experimental Physics, Volume 2: Electrodynamics , Author: Wilhelm Raith, 8th edition. 1999, ISBN 3-11-016097-8 , p. 363.