What factors affect the speed of light

Astro-Lexicon L 6


The Lorentz or Γ factor (gamma factor) is one of the most important dimensionless quantities in the theory of relativity. This factor depends exclusively on the relative speed v from. The Lorentz factor becomes dimensionless because the speed v in the natural unit of the theory of relativity, the speed of light in a vacuum c is measured. In a vacuum, light moves at 2.99792458 × 108 m / s. The ratio of the relative speed to the speed of light is usually denoted by the Greek letter β.

Γ as a relativistic measure

The Γ factor increases very sharply as the speed increases. In the limit v = c, the Lorentz factor diverges and approaches infinity! The following links refer to diagrams in which the dependence of the Lorentz factor on the relative speed is shown:

The relative speed is a vector v and indicates the relative, unaccelerated movement of two reference systems to one another. For example, you can look at the speed of a fast particle. The two frames of reference are what we are familiar with Laboratory system. This is what physicists call the reference system in which one examines the moving particle. On the other hand, there is that Rest system the particle in which the particle does not move; the observer 'sits' on the particle, so to speak. The relative speed now indicates how the rest system moves in relation to the laboratory system.
When viewed in one spatial dimension, both systems move in the same direction, but at different speeds. Then it is enough to compare this difference with the relative speed v indicate, which in this special case has only one amount and one Scalar is. In general, however, the rest system and the laboratory system move in different directions and theVector character the relative speed must be taken into account. The amount goes into the Lorentz factor v this vector v one, which can be obtained according to the vector calculation according to the formula v2 = |v|2 = vx2+ vy2+ vz2 receives. (Here a Cartesian coordinate system {x, y, z} was chosen to indicate the components of the speed. The speed can, however, be represented in any other coordinate system.)

Why do you need the Lorentz factor?

On the one hand, it is used to assess how relativistic is a movement. So the question is whether the movement of the observed object is already so fast - namely, comparably fast with the movement of light - that effects of the special theory of relativity play a role and have to be taken into account. These effects are the time dilation and the Lorentz contraction (or length contraction): Length and the passage of time depend on how fast the object in question is moving!
The following notation has become established, which classifies how relativistic a movement is:

  • Lorentz factor equal to or comparable to 1: non-relativistic non-relativistic),
  • Lorentz factor greater than 2: relativistic relativistic),
  • Lorentz factor greater than 10: 'medium relativistic' mid-relativistic),
  • Lorentz factor greater than 100: ultra-relativistic ultra-relativistic).

Lorentz transformations

On the other hand, the Lorentz factor goes essentially into the Lorentz transformations, the so-called Boosts a. The Lorentz transformation mediates between the rest system and the laboratory system. If you want to compare the relativistic movement of a particle in one system with that in the other system, you carry out the Lorentz transformation.
In addition, the Γ factor weights the relativistic effects of time dilation and length contraction: the larger Γ, the more pronounced the relativistic phenomena. That goes relativistic mechanics steadily into the non-relativistic, the classical or Newtonian mechanics above. The laws of classical mechanics, which have long been known in physics, are contained as a limiting case in relativistic mechanics. This is precisely what the Lorentz factor quantifies, because it becomes very small in Newton's limit case and converges to one. This is illustrated in the first diagram above at low speeds v.

Particle physics: big Γs

In particle physics, relativistic effects are the order of the day: In particle accelerators, the accelerated particles (electrons, positrons, protons, atomic nuclei) reach relativistic speeds. The relativistic effects must be taken into account in the acceleration processes, e.g. the Lorentz contraction of the atomic nuclei during a collision.

great Γs in astronomy

In astronomy there are a number of very energetic processes that involve high Lorentz factors. In the vicinity of a black hole, the accretion flux moves relativistically, as does the jets that are accelerated from the central region by active galactic nuclei or compact objects. At the point of origin of the jets (the so-called base point), typical Lorentz factors are below 10. Extreme speeds have been observed with gamma ray bursts that suggest Γ factors of up to 1000.
Record holders in the universe are probably the pulsars: The magnetically driven pulsar winds achieve Lorentz factors of up to 10,000,000 through post-acceleration! A prominent example of this is the Crab Nebula in the constellation bull, is described in the lexicon entry SNR.

The actual, orthochronous Lorentz transformations form a mathematical group.

Group properties

Generally there are in the group structure elements a certain quantityconnected to each other through a mathematical operation linked. For the groups, this operation again results in an element that belongs to the initial set.
Groups as a mathematical structure meet certain mathematical criteria, such as

  • the existence of one neutral element;
  • the existence of a inverse element and
  • the Associativity.

Is also the Commutativity given, i.e. swapping the order of operations leads to the same result, this is what the group is called abelian.
This catalog of criteria can now be applied to the Lorentz transformations and it can be proven that they are the so-called Lorentz group form:

  • the identity forms the transformation matrix with vanishing relative speed, v = 0. The Lorentz transformation matrix then becomes just thatIdentity matrix and converts four-vectors into themselves.
  • The inverse element is the inverse transformation matrix that is obtained when replacing β with -β: The velocity is inverted.
  • The associativity of the Lorentz transformations can also be traced back to the associative law for matrix multiplication.

four types of Lorentz transformations

    One distinguishes four types of Lorentz transformations (see picture):
  • the actual or orientation preserving orientation-preserving) Lorentz transformations,
  • the improper Lorentz transformations,
  • the orthochronous or time orientation preserving (eng. orthochronous or time-preserving) Lorentz transformations
  • and the non-orthochronous Lorentz transformations.

further Lorentz groups

Just that actual, orthochronous Lorentz transformations (engl. proper orthochronous Lorentz transformations) form a Subgroup the Lorentz group. It is asix-parameter, continuous transformation group.
In addition, executing two Lorentz transformations one after the other with different relative speeds corresponds to the multiplication of matrices or rotations executed one after the other with different angles in the Minkowski space (Distributive law).

Relation to rotations

The Lorentz group is divided into special and general Lorentz groups and assigns them according to the special theory of relativity and the general theory of relativity. In the Group theory it turns out that the Lorentz group is closely related to the Rotation group, SO (3). The full Lorentz transformation then sets any two inertial systems in relation to one another and can be broken down into a chain of normal space rotation, boost transformation and further space rotation.
The physical and epistemological content of the physical group theories is very profound and carries further than this mathematical apparatus appears: While Newton's law invariant under Galileo transformations is and with the structure of the Galileo group related, a new group structure was found with the theory of relativity: the Lorentz group. The laws of the theory of relativity are invariant under Lorentz transformations, one also says shortening Lorentz invariant. This is a consequence of theRelativity Principle: All inertial observers are equivalent. (see also equivalence principle). Next to it is that Covariance principle: Physical laws are form invariant under Lorentz transformations. This results in the necessary mathematical description with tensors, i.e. structures that have the same shape in all coordinate systems.

Poincaré group

However, there is still a group structure that is superordinate to the Lorentz group, the Poincaré group.

The principle of Lorentz invariance is an essential property of the theory of relativity. Lorentz invariance means that the observers or physical quantities can be converted into one another by Lorentz transformations without changing the physical relationships. This non-change is known in mathematical physics with the term Invariance. Ultimately, this is a symmetry property. The corresponding symmetry group of this transformation is called the Lorentz group. A variable of Lorentz is identical in all reference systems.

Lorentz invariance in Einstein's theories

Lorentz invariance applies to both theories, the Special Theory of Relativity (SRT), where systems that move in a straight line relative to one another or systems that are relatively at rest are considered; but also in the general theory of relativity (GTR), where the relative movements were generalized to uniformly accelerated or free falling systems. There is one important difference, however: The SRT is global Lorentz invariant, the ART is only locally Lorentz invariant. This means that the Lorentz invariance in the Minkowski metric, the spacetime of the SRT, applies everywhere. One can change from any world point on the manifold to another by means of the Lorentz transformation; the size remains the same. In ART, this only applies locally, i.e. in a world point with immediate surroundings, because space-time is global in general curved is. In other words: In an arbitrarily small environment around a world point in globally curved spacetime applies local flatness and Lorentz invariance.

Lorentz invariance leads to Einstein's principles

As I said, Lorentz invariant observers measure the same quantities in a physical experiment and will arrive at the same test result. These Equality of the observers flows into the principle of relativity and further generalized into the principle of equivalence.

Does Lorentz invariance always apply?

Some variants of the quantum gravity theories say one Violation of the Lorentz invariance ahead, e.g. the string theories. So far this could not be confirmed with astronomical measurements, for example the electronic synchrotron radiation in the Crab Nebula. Further Tests of Lorentz invariance are nevertheless urgently required in order to sound out the scope of the theory of relativity.

The Lorentz contraction or actually Fitzgerald-Lorentz contraction goes back to the physicists George Francis Fitzgerald (1851-1901) and Hendrik Antoon Lorentz (1853-1928). Together with the mathematician Jules Henri Poincare (1854 - 1912) they interpreted in 1895 with this contraction of lengths and time dilation the zero result of the Michelson-Morley experiment (1881/87). In this experiment the world ether, which was postulated as the carrier substance for light waves, was to be demonstrated. The zero result consisted of the fact that the light spreads at the same constant speed on all running routes. Fitzgerald, Lorentz and Poincaré nevertheless stuck to the world ether and postulated ad hoc the length contraction and the time dilation, which the measuring devices may influence accordingly, so that the zero result comes about. The Lorentz transformation provided the mathematics to describe these mechanisms.

Away with the ether!

Albert Einstein suggested doing without the world ether entirely. His special theory of relativity (SRT) only postulates the constancy of the speed of light and the principle of relativity. In this way, the zero result can be explained very elegantly. The Lorentz transformation turned out to be correct, but not the associated original interpretation by Fitzgerald, Lorentz and Poincaré. Only Einstein succeeded in establishing a new, revolutionary view of space and time.

A relativistic effect

In the theory of relativity, the Lorentz or length contraction describes a relativistic effect, where a relativistically moving body experiences a length reduction in the direction of movement, specifically by the Lorentz factor or Γ factor (1- (v / c)2)-1/2. Mathematically, this goes back to the properties of the special Lorentz transformation, which mediates between two different inertial systems. Velocities in the theory of relativity are usually measured in units of the speed of light c. Therefore the dimensionless size is available v / c which is usually abbreviated as β.

No science fiction!

The Lorentz contraction has also been proven experimentally. In particle accelerators in the rest system, heavy ions that appear spherical and consist of many nucleons are compressed in the direction of movement (see figure above right: comparison of moving and static observer systems). The Lorentz-contracted, heavy ions therefore have a flattened shape in the laboratory system and resemble a pancake rather than a ball. It is very impressive that Einstein wrote this fact back in his miracle year 1905 in the legendary paper On the electrodynamics of moving bodies calculated! Of course, there was no talk of Lorentz-contracted atomic nuclei, but he explicitly presented the calculation (§4) of how a sphere is deformed into an ellipsoid of revolution via Lorentz boost.

2nd relativistic effect: time dilation

The relativistic effect related to the Lorentz contraction is called time dilation and, however, affects time intervals.

The Lorentz transformation is a mathematical operation that mediates between reference systems that are uniformly moved in a straight line against each other (special theory of relativity, SRT) or freely falling (accelerated) reference systems (general theory of relativity). She has the classic Galileo transformation replaced, in which the time transformation was an identity, t = t ', and thus the concept of a absoluteTime allowed. In the theory of relativity, time has one relative character, which can be seen very well from the equations of the Lorentz transformation.

In mathematical terms

Mathematically speaking, the Lorentz transformation is one linear, homogeneous transformation (see first figure above right), which can be represented by a 4 × 4 matrix (usually denoted by Λ, see equation) in relation to the four-vector vectors of relativity theory to be transformed. The Lorentz transformation is a special case of the Poincaré transformation. The latter also relates Translations with a and is therefore a linear, inhomogeneous transformation. Both types of transformation form mathematical groups: the Lorentz group and the Poincaré group.

The pioneer

The designation Lorentz transformation is with the Dutch physicistHendrik Antoon Lorentz (1853 - 1928), who did fundamental work in the field of relativity theory and - how Albert Einstein - the negative outcome of theMichelson-Morley experiment to measure a World ethers seizure. The terms Lorentz contraction, Lorentz factor and Lorentz group are still associated with it today.

absolute speed of light

The natural and only unit of special relativity is that Vacuum speed of lightc. she is unchangeable in every frame of reference and in that sense absolutely as postulated in the SRT (Not everything is relative in the theory of relativity!). It is therefore advisable to measure speeds in units of the speed of light. This defines the dimensionless quantityv / c (usually referred to as β), like the second figure shows. The so-called Lorentz factor (γ, see second figure, bottom line) appears in the derivation of the Lorentz transformation law. This factor that depends on the relative speed v depends on two reference systems moving against each other, must always be taken into account when changing from one reference system to the other. It is of immense importance for the entire theory of relativity and is the factor that determines the length or Lorentz contraction and the time dilation.
Mathematically, the Lorentz transformation can be written as a matrix-vector product (first equation above). The vectors x are Four-vectors with a temporal component and three spatial components. The is accordingly Transformation matrix, which just mediates the Lorentz transformation, a 4 × 4 matrix, thus has 16 entries. The transformation matrix for a Lorentz transformation in the x direction, a so-called one, is shown on the left x boost.
It is clear Special Lorentz transformation a rotation in the Minkowski room. The Lorentz group is related to the rotation group and contains the rotations in space. A Boost is in principle also a rotation, in which, however, space and time are converted into one another (this becomes clear when looking at the explicit transformation laws below). That's why they are called Boosts also Pseudo-rotations. The General Lorentz Transformation on the other hand corresponds to a special Lorentz transformation concatenated with a space rotation.
This clearly means that the Relative speedv between uncoated (Rest system) and the deleted system (relatively moving system) is parallel to the x-direction. As the next figure on the right shows, the y and z components of the four-vector position remain unchanged (invariant), while the room component moves in the boost direction (namely x)and the time component t change if you change to another reference system! This property proves the close connection of space and time to space-time or the Space-time continuum. You can quickly recalculate the four transformation laws noted down by component by calculating the matrix-vector product from the transformation matrix above and a position four-vector x = (x0, x1, x2, x3)T = (ct, x, y, z)T (Comment: T stands for the transposed vector, because: matrix × column vector = column vector.).

Another note

There are also notations of the Lorentz transformation in which the imaginary uniti = (-1)1/2 is used. These approaches are mathematically equivalent, but out of date. A use of i is not recommended for educational reasons, because it confuses beginners in the theory of relativity in particular.

Properties of the Lorentz Transformation

  • It is one linear transformation. Through this property, physically speaking, the type of movement (uniform or freely falling) is retained.
  • The Determinant the transformation matrix is ​​1 (see brief calculation below).
  • For the Limes of small speeds versus the speed of light in a vacuum c the Lorentz transformation merges into the classical Galilei transformation.
  • The inverse Lorentz transformation is obtained by replacing v/c (beta) in the transformation matrix by -v/c. From a physical point of view, the direction of movement is simply reversed.

Lorentz invariants

Lorentz invariants do not change with a Lorentz transformation, i.e. they are the same in all reference systems! That's how it is Length of a world vector a Lorentz invariant, because world vectors are rotated under Lorentz transformations only in Minkowski space. This reveals the close relationship between Rotation groups and the Special Lorentz group.

Addition theorem for velocities

The Addition theorem for (relative) speeds can easily be demonstrated by a concatenation of Lorentz transformations. With this law (see equation on the right) one can easily see that the light from a moving light source is Not moves at about the speed of light plus the speed of the light source, but - as required in Einstein's postulate - the Speed ​​of light constant remains. This addition theorem works for small velocities (v much smaller than c) into the well-known law for speeds, according to which a projectile that originates from a moving source is also added to the speed of the source.


A generalized mathematics of the Lorentz transformation in general relativity is offered by the Spinor Algebra.



© Andreas Müller, August 2007