Galileo transformation

The Galilei transformation , named after Galileo Galilei , is the simplest coordinate transformation with which physical statements can be converted from one reference system into another. It can be used if the two reference systems differ in terms of rectilinear, uniform movement , rotation and / or displacement in space or time. All observations of distances, angles and time differences agree in both reference systems; all observed speeds differ by the constant relative speed of the two reference systems.

The Galileo transformation is fundamental to classical mechanics because it describes the transformation between two inertial systems . With regard to the sequential execution, the Galileo transformations form a group , the Galileo group. According to the relativity principle of classical mechanics , the laws of nature must be covariant with respect to this group .

In the field of electromagnetism , the Galileo transformation is not applicable, but has to be replaced by the Lorentz transformation . Historically, this formed the starting point for the special theory of relativity .

Galileo transformation

The Galileo transformation consists of the following individual transformations that can be combined with one another:

• Translation in time (1 parameter): ${\displaystyle t\rightarrow t+b}$
• Translation in space (3 parameters): ${\displaystyle {\vec {r}}\rightarrow {\vec {r}}+{\vec {a}}}$
• Rotation with the orthogonal rotation matrix (3 parameters): ${\displaystyle A}$ ${\displaystyle {\vec {r}}\rightarrow A{\vec {r}}}$
• Transformation to a reference system with uniform relative speed (3 parameters): ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {r}}\rightarrow {\vec {r}}+{\vec {v}}\cdot t}$

The vector notation was used: denotes the position vector and time. There are a total of 10 parameters for one time and three space dimensions. For , the spatial part of the Galileo transformation with 6 other free parameters represents the actual Euclidean group . The elements from are understood as spatial coordinate transformations ( passive or alias transformation ). Galileo transformations between stationary observers are a special case of the Euclidean transformation , which only requires the constancy of the distances between any two points during the transformation and which in classical mechanics serves to define invariant or objective quantities. ${\displaystyle {\vec {r}}}$ ${\displaystyle t}$
${\displaystyle b=0,{\vec {v}}={\vec {0}}}$ ${\displaystyle \mathrm {E} ^{+}(3)}$ ${\displaystyle \mathrm {E} ^{+}(3)}$

Limit of validity of the Galileo transformation

Classic mechanics

The independence of the laws of mechanics from the state of motion in the case of uniform motion was first recognized by Galileo Galilei and formulated by Isaac Newton in his book Principia . For Newton, forces are only dependent on the accelerations , and accelerations do not change under Galileo transformations. Velocities transform according to the usual vectorial addition law. The laws of classical mechanics are invariant or covariant under Galileo transformations (Galilean principle of relativity). For a long time this was considered a priori and unassailable.

Lorentz transformation

The electrodynamics continued until the end of the 19th century by a ether as carriers of electromagnetic waves, including the light from. However, Maxwell's equations and the resulting constant speed of light as the speed of propagation of the electromagnetic waves were not compatible with the Galilei transformation. ${\displaystyle c}$

Another example is a charged body that flies past a current-carrying conductor:

Cargo and ladder
Charge q and conductor with current j . This configuration is not Galileo transformable.

• A charge flies past a straight current-carrying but charge-neutral conductor at its initial speed (see picture). The current in the conductor creates a magnetic field which, by the Lorentz force, deflects the moving charge from its straight-line movement. If you now carry out a Galileo transformation into an inertial system in which the charge rests, no Lorentz force acts on the charge in this system. This apparent paradox can only be explained with the Lorentz transformation, in which the length of the conductor is shortened in the inertial system of the charge and the conductor thus receives a relative electric charge, which leads to an electric field. ${\displaystyle q}$ ${\displaystyle v}$ ${\displaystyle q}$

Hendrik Antoon Lorentz , Joseph Larmor and Henri Poincaré investigated the electrodynamics of moving bodies at the end of the 19th century and recognized that these problems could be solved by replacing the Galileo transformation with the Lorentz transformation . This ultimately led to Albert Einstein's special theory of relativity , which, however, required a modification of the ideas of time and space .

For speeds that are much smaller than the speed of light of approx. 300,000 km / s, the Galileo transformation is often a good approximation of the Lorentz transformation in practice. For the Lorentz transformation goes exactly into the Galilean transformation. But for small velocities the Galileo transformation is not a borderline case of the Lorentz transformation, as is often wrongly claimed. For example, the time dilation does not disappear if one considers two events with increasingly larger spatial distances. Galileo and Lorentz transformations are essentially different transformations, both of which converge to the identity transformation for small speeds. ${\displaystyle c\to \infty }$

Practical use

In everyday life, the Galileo transformation can almost always be used for mechanical problems, as the correction in the Lorentz transformation is very small at earthly speeds. The correction factor is often below the measurable limit; even in the celestial mechanics of our planetary system it is z. B. below 10 ⁻⁸ for the already quite high orbital speed of the earth around the sun (about 30 km / s).

Therefore, the Galileo transformation applies, for example, when calculating the drift of a ship or airplane. Even with the collision processes considered in nuclear physics , it is usually sufficient to convert between the laboratory and center of gravity system (see kinematics (particle processes) ). However, it is not applicable to electrodynamic phenomena.

Use of the Galileo transformation in the derivation of the laws of collision by Huygens

A historically important application of the Galilean relativity theory , i.e. the use of the fact that the physical description is the same in different reference systems connected by Galileo transformation, is the correct derivation of the laws of elastic shock by Christian Huygens (1650s, published 1669 and 1703 in his De Motu Corporum ). In doing so, he corrected the mostly wrong representation by René Descartes , who at least had the right idea to use conserved quantities for the analysis (with Descartes wrongly with the values ​​of the speeds). Descartes was correct only in the case of the collision of the same masses with the same but opposite velocities of the particles 1,2 before ( ) and after the collision ( ), whereby the motion is considered in one dimension: ${\displaystyle \sum _{i=1}^{2}m_{i}\cdot |v_{i}|}$ ${\displaystyle v_{1},v_{2}}$ ${\displaystyle u_{1},u_{2}}$

${\displaystyle (v_{1},v_{2})=(v,-v)\rightarrow (u_{1},u_{2})=(-v,v)}$

His other results were wrong. As an essential new element, Huygens introduced the consideration of another reference system moving at constant speed , a boat or a man on the bank who observes impact experiments in the boat (sketched in a picture in Huygens' book as a collision of two pendulum balls on the outstretched arm of two men, of which one is in the boat and the other on the bank, but there exactly the movement of the balls perceived by him in the boat): ${\displaystyle w}$

Discussion of the Huygens law of collision

${\displaystyle (v_{1},v_{2})=(v+w,-v+w)\rightarrow (u_{1},u_{2})=(-v+w,v+w)}$

If you choose z. B. one obtains: ${\displaystyle v=w}$

${\displaystyle (v_{1},v_{2})=(2v,0)\rightarrow (u_{1},u_{2})=(0,2v)}$

for which Descartes got the wrong result . Huygens, on the other hand, obtained the correct result with the help of the Galilean principle of relativity, that one ball stops and transfers its momentum completely to the other, previously stationary ball. Huygens could handle other cases by appropriate choice of . In general it can be shown in today's conceptualization that he proved the law of the conservation of momentum in elastic collisions, whereby, in contrast to Descartes, he treated the momentum correctly with algebraic signs and used the conservation of kinetic energy (in Huygens formulated indirectly as a condition of the elastic Shock). If one uses today's terms, this can be shown simply by considering the conservation of the kinetic energy in a reference system moving with constant speed (the pre-factors are omitted): ${\displaystyle (2v,0)\rightarrow (-{\frac {3v}{2}},{\frac {v}{2}})}$ ${\displaystyle w}$ ${\displaystyle {\frac {1}{2}}}$

${\displaystyle \sum _{i=1}^{2}m_{i}\cdot {(v_{i}+w)}^{2}=\sum _{i=1}^{2}m_{i}\cdot {(u_{i}+w)}^{2}}$

One multiplies and uses the energy law in the resting system

${\displaystyle \sum _{i=1}^{2}m_{i}\cdot {(v_{i})}^{2}=\sum _{i=1}^{2}m_{i}\cdot {(u_{i})}^{2}}$

the momentum conservation law follows:

${\displaystyle \sum _{i=1}^{2}m_{i}\cdot v_{i}=\sum _{i=1}^{2}m_{i}\cdot u_{i}}$

Huygen's use of the principle of relativity is highlighted in Ernst Mach's book on the development of mechanics, which has been shown to have a strong influence on Albert Einstein and thus possibly stimulated his use of reference systems.

Galileo transformation and conservation laws

The laws of nature do not change under Galileo transformation. The outcome of an experiment remains the same if one subjects its location to a Galileo transformation. A shift in place, or in time, or even the alignment does not change anything. Such an invariance is also called symmetry . According to Noether's theorem , every such symmetry is associated with a conservation law. From the invariance of the laws of nature under Galileo transformation, the conservation laws of classical mechanics follow . In detail:

• From the invariance under displacement of the location which follows momentum conservation .
• Conservation of energy follows from the invariance with a shift in time .
• Conservation of angular momentum follows from the invariance under rotation .

Galileo group and quantum mechanics

If one considers a quantum mechanical system that is realized in a representation of the Galilei group, there is, in contrast to the usual treatment as a representation of the Poincaré group of the special theory of relativity, an exact conservation of the mass (so-called super-selection rule), i.e. there is none unstable particles.

In quantum mechanics, unitary , projective representations in Hilbert space are considered. In the case of the Poincaré, Lorentz or rotation group commonly used in elementary particle physics, according to Valentine Bargmann, faithful representations are obtained by considering the universal superposition group. This is not the case with the Galileo group. You only get faithful representations except for a prefactor in which the mass is included as a parameter. There is a one-dimensional-infinite set of non-equivalent classes of projective representations (parameterized by mass), all non-equivalent to faithful representations, and they are just the physically relevant representations.

It can also be deduced that the internal energy of a particle can also be chosen arbitrarily. In 3 space and one time dimension there are three Casimir invariants of the Lie algebra belonging to the Galileo group , mass , the mass shell invariant ( is the energy, the momentum) and with (where the boost operator corresponds to the transition to a system with another Speed) and the angular momentum. The third invariant can be given for as with the spin . ${\displaystyle E_{0}}$ ${\displaystyle M}$ ${\displaystyle ME-{\frac {P^{2}}{2}}=mE_{0}}$ ${\displaystyle E}$ ${\displaystyle P}$ ${\displaystyle {\vec {W}}^{2}}$ ${\displaystyle {\vec {W}}=M{\vec {L}}+{\vec {P}}\times {\vec {C}}}$ ${\displaystyle {\vec {C}}}$ ${\displaystyle {\vec {L}}}$ ${\displaystyle m>0}$ ${\displaystyle sm}$ ${\displaystyle s}$

An example of the application is the light front formalism (Infinite Momentum Frame) in elementary particle physics, in which one changes to a reference system with, in the limit case, infinitely high speed (as in typical high-energy scattering experiments). Since one moves approximately to a system with Galileo symmetry, there are considerable simplifications such as similarities with the non-relativistic perturbation theory, elimination of Feynman diagrams with pair creation and annihilation and new conservation quantities.

Individual evidence

1. ↑ z. BAP French: Special Theory of Relativity . Vieweg 1971, chapter 8.
2. ↑ R. Baierlein. Two myths about special relativity. American Journal of Physics, 74 (3): 193-195, 2006
3. ^ Julian Barbour : The Discovery of Dynamics . Oxford UP, 2001, pp. 458ff.
4. ↑ Falk, Ruppel: Mechanik, Relativität, Gravitation , Springer 1973, p. 27ff. The condition is that if the magnitude of the velocity of one of the particles is the same before and after the collision, this also applies to the other particle. A derivation of the momentum theorem with this condition using Galilei transformations can be found in the book by Falk and Ruppel.
5. ↑ So u. a. Martin J. Klein in the preface to Mach's English translation; Principles of Thermodynamics , 1986, quoted from Julian Barbour : The Discovery of Dynamics . Oxford UP, 2001, p. 470.
6. ↑ Jean-Marc Lévy-Leblond , Galilei group and non relativistic quantum mechanics, Journal of Mathematical Physics, Volume 4, 1963, p. 776, doi : 10.1063 / 1.1724319
7. ↑ Valentine Bargmann, On unitary ray representations of continuous groups, Annals of Mathematics, Volume 59, 1954, pp. 1-46, JSTOR 1969831
8. ↑ Steven Weinberg , Dynamics at Infinite Momentum, Physical Review, Volume 150, 1966, 1313, doi : 10.1103 / PhysRev.150.1313 . Applications, for example, extensively in Stanley Brodsky's school .