Spacetime

The space-time or space-time continuum referred to in the theory of relativity the union of space and time in a single four-dimensional structure with special characteristics (such as " causality ", see below ) in which the spatial and temporal coordinates for transformations in other reference systems may be mixed together.

Historically, time and place are understood as separate terms. This is allowed at daily rates, but as approaching to the speed of light is no longer. It appears then that the time and place of an event always mutually dependent, regardless of the considered physical system. The coupling of space and time has to satisfy only the requirement that if event A caused event B, it must " causality " are valid in all coordinate systems.

A coordinate system change should therefore not alter the causality of events.

The causality is mathematically defined by a distance term, the dx of the three spatial coordinates differential, dy, dz and the events (see below) and its differential points in time dt depends. The requirement of " invariance " ( preservation of causality ) of the generalized distance between two events will cause physical models are described in mathematical spaces in which are coupled in a certain way time and space. It can be an absolute (absolute in the sense of invariance under change of coordinates ) a valid notion of distance (eg, the so-called proper time or the " generalized distance ", see below) for space-time points of the mentioned four-dimensional space -time continuum (so-called " defining events " ), but it is the state of motion of the observer, and the presence of mass or energy (such as in fields dependent ), which is measured as a spatial and as a temporal thereof distance. Mathematically, the space-time by using a pseudo-. described - Riemannian manifold, especially in the so-called Minkowski space example considered in this room for the "event" with the four coordinates cdt, dx, dy, dz - the speed of light c - that the corresponding " generalized distance " ds - or more precisely its square ( ds) 2 - not as usual by the Pythagoras ( ds) 2 = ( cdt ) 2 (dx ) 2 (dy ) 2 (dz ) 2, but by the indefinite expression ( ds) 2 = ( cdt ) 2 - ( dx) 2 - ( dy) 2 - ( dz) 2 is defined ( this is called a non-trivial "signature " of the four-dimensional space -time continuum, like so: ( , -, -, -) ).

• 2.1 General
• 2.2 Mathematical motivation of the Minkowski metric
• 2.3 Minkowski diagram

• 3.1 Non-Euclidean geometries
• 3.2 Space -time curvature
• 3.3 space curvature and centrifugal

Historic space-time concepts

Aristotle space-time

To construct the Aristotle space-time of the Euclidean space over the schema of Aristotle with the " Euclidean time " for so-called Aristotle space-time combined:

( The Euclidean time only differs from the space of real numbers that has no defined absolute zero. Both share an orientation ( time direction ) ).

( The three-dimensional Euclidean point space is an equivalence class of spaces which are obtained when one of the moves the zero point at different points in space In fact, the most important feature of the rooms of their homogeneity and isotropy. No point and no direction is in front of other excellent. Together is all a given scalar which gives the orthonormal bases. All rooms are oriented ( handedness ). )

Galileo space

In addition to Aristotle space-time and the Galileo space is defined. This is only slightly more general than the Aristotle space-time by the spaces are indeed viewed at different times as different, but time is defined globally.

This space is based on Galileo's observation that the Earth rotates around its axis and an observer on Earth yet so occurs when standing still, the earth. Even Galileo noticed that a liquid always drips in a ship that is moving uniformly straight down.

The problem is solved by independent inertial systems are defined, to which a relative movement is measured. Thus, the observer has the impression on the earth, that the earth is stationary and the sun revolves around the earth, while an observer near the sun is in a different inertial frame, and thus detects a movement of the earth relative to the sun.

The Galileo space is characterized by the fact that although different inertial frames exist, but only an absolute (that is, all points common ) time.

Newton - Cartan spacetime

Élie Cartan expanded the definition of space to the possibility of web-like movements (such as the orbit of the earth around the sun) to be considered as linear movements by each of the inertial system is moved. Extended Cartan space time is also referred to as space-time Newton.

Spacetime in special relativity theory

General

In the special relativity theory (SRT ), the three-dimensional space coordinates by a time component to a four-vector in the so-called Minkowski space ( " space-time " ) expands, so (or in another, less common convention, with the imaginary unit i).

A point in space-time has three space coordinates and one time coordinate and is called an event or world point. For events is an invariant space - time interval is defined. In the classical Euclidean space, a three-dimensional Cartesian coordinate system, the differential spatial square of the distance between two points

Since the n-dimensional Euclidean norm remains constant only under Galilean transformations, but not under Lorentz transformations, an invariant for all observers identical ( generalized ) distance is defined as:

This is the squared Minkowski norm that the improper metric ( distance function ) of the flat spacetime of special relativity theory (SRT ) is generated. It is induced by the ( indefinite ) invariant scalar product on the Minkowski space, which can be defined - metric tensor as the effect of ( pseudo):

The metric tensor in physical parlance simply as Minkowski " metric " or flat " metric " of space-time, in fact he is not to be confused in the proper sense of the metric itself. It is mathematically rather a scalar product on a pseudo- Riemannian manifold.

The line element is, except for the factor to the differential so-called proper time

This is simply measured by a comoving clock, ie in the " currently accompanying inertial system " in which the particle is located on the world-line resting.

The invariance under the Lorentz transformation is defined by the requirement that this four-dimensional distance (or the Minkowski metric ) constant ( invariant ) under a linear coordinate transformation, whereby the above- mentioned homogeneity of space-time is expressed.

An element ( "vector" ) of space-time is called time-like if the following applies. If applies, ie the vector space-like.

Light always moves exactly with the speed. Thus true for light in all reference systems. Hence the constancy of the speed of light, the output principle of special relativity. Moreover, it remains the classification of space-time vectors in space-like, light -like or time-like vectors in the admissible transformations ( Lorentz transformations ) unchanged ( invariance of the light cone ).

Two events for which is negative, space and time are so far apart that a light beam can not reach in time from one to another event. Since information is transmitted via light or matter either and matter in the theory of relativity can never reach the speed of light ( and therefore not faster than this may be ), such events can never stand in a cause -and-effect relationship. The space-time is thus divided into two parts: events with a real space-time distance an observer can see ( " time-like vectors" ). Events that are too far away and could only be perceived faster than light, because the space -time distance is imaginary, are basically invisible ( " spacelike vectors" ).

Mathematical motivation of the Minkowski metric

• Looking at the D' Alembert operator with

Can write, if the following two four-vectors are introduced:

• Since the four dimensions are linearly independent, can be diagonalized bring (main axis transformation).

• Because of the requirement that there is no excellent space-time coordinates, the diagonal elements can only have the value. For the spatial coordinates is usually selected. But this is a convention that is not used uniformly.

• The time component can not have the same sign as the space components. Therefore, we consider again the D' Alembert operator:

Minkowski diagram

In Minkowski diagram the relationships can be represented geometrically and analyzed. Because of the complex feature of the time component where the rotation of the time axis is shown with the opposite sign as the rotation of the coordinate axis.

Space-time in general relativity theory

Non-Euclidean geometries

Basis for the description of space-time ( ct, x, y, z) in the general theory of relativity is the pseudo- Riemannian geometry. The coordinate axes are here not linear, which can be interpreted as a space curve. For the four-dimensional space-time the same mathematical tools such as the description of a two-dimensional spherical surface, or for saddle surfaces are used. As irrefutable respected statements of Euclidean geometry, in particular the parallel axiom, must be abandoned in these theories and replaced by more general relationships. The shortest distance between two points is here, for example, no longer straight section. A straight line in Euclidean geometry corresponds to the geodesic in the non -Euclidean world; in the case of a spherical surface, the geodesics are the great circles. The sum of angles in - consisting of Geodätenabschnitten - triangle is no longer 180 degrees. In the case of the spherical surface is greater than 180 degrees, in the case of saddles, however, smaller.

Space-time curvature

The curvature of space and time is caused by any form of energy, such as mass, pressure, or radiation. These variables together form the energy -momentum tensor and go to the Einstein equations as a source of the gravitational field a. The resulting curvilinear motion of force-free bodies along geodesics of the gravitational acceleration is attributed to - in this model, something like a gravitational force does not exist anymore. In an infinitesimal area section ( local map ) the gravitational field generated always has the flat metric of special relativity. This is described by a constant curvature of space by a factor of g/c2. The curvature of the world-lines ( motion curves in space-time ) of all force-free bodies in that space section is the same.

In many popular representations of the general theory of relativity is often ignored, that not only the area but also the time must be curved in order to generate a gravitational field. That always space and time must be curved, is clearly easy to understand: If only the space is curved, the trajectory of a thrown stone would be always the same, no matter what initial velocity possessed the stone, as it would always just follow the curved space. Only by the additional curvature of the time can come about the different trajectories. As part of the ART, this can be shown mathematically.

In the normal three-dimensional space, only the projection of the world lines is visible on the motion plane. The body has a velocity v, the world line on the time axis is inclined, and that the angle. The projection of the web is longer by a factor of V increases, the radius of curvature is larger by the same factor, that is the angle change is smaller. The curvature ( change in angle per length portion ) is therefore smaller by a factor.

With

Then follows from the worldline curvature g/c2 for the observed curvature of the path in three-dimensional space

Space curvature and centrifugal

For small velocities v « c is the curvature of the path is g/v2 and thus corresponds to the value of a classical centrifugal acceleration. For light rays with v = c, the factor (1 v2/c2 ) has the value 2, the curvature corresponds to twice the value of the classical 2g/v2 consideration. The angular deviation of starlight of the stars near the sun should therefore be twice as large as in the classical case. This was verified for the first time by Arthur Eddington in the context of Africa expedition to observe the solar eclipse of 1919.

Because of this small deviation from the classical value of the planetary orbits are not exact ellipses more, but rosettes. At the perihelion of the planet Mercury it was first identified.

Symmetries

The space-time is characterized by a number of symmetries, which are very important for the types of force physics. These symmetries include not only the symmetries of the space ( translation, rotation ) and the symmetries under Lorentz transformations (change between reference systems of different speed). The latter ensures the principle of relativity.