Spacetime

Space-time or space-time continuum is the common representation of the three-dimensional space and the one-dimensional time in a four-dimensional mathematical structure. This representation is used in the theory of relativity .

People experience time and place as two different givens, partly because of the causality associated with time (an effect cannot occur earlier than its cause). In classical physics and mostly in technology, time and place are treated as independent quantities . At speeds of the order of magnitude of the speed of light , however, it becomes apparent that the time and place of an event are mutually dependent. For example, the time interval between two events, as determined by a moving observer, also depends on their spatial distance. With the development of the special theory of relativity, it was recognized that it is advantageous to consider the two quantities as coordinates in a common four-dimensional space, the Minkowski space .

In the context of classical mechanics, the concept of space-time has been discussed by Penrose and Arnold .

Spacetime in the special theory of relativity

Causality and the Concept of Distance

Even with a coupling of space and time, if event A causes event B, this “ causality ” must apply in all coordinate systems ; a change of coordinate system must not change the causality of events. Causality is mathematically defined by a concept of distance . The distance between two events depends on the three location coordinates and the time coordinate . Because of the requirement for the maintenance of the causality of two events or, more generally, for Lorentz invariance , physical models must be described in mathematical spaces in which time and space are coupled in a certain way . ${\ displaystyle x, y, z}$${\ displaystyle t}$

An absolutely (absolutely in the sense of invariance towards change of coordinates) valid distance concept can be used, e.g. B. Define the so-called proper time or the “generalized distance” for spacetime points (“ events ”) of the four-dimensional space-time continuum, even for events that are as closely (“infinitesimally”) neighboring. What is measured as spatial and what is measured as temporal distance depends on the state of motion of the observer and (in the case of general relativity theory) on the presence of mass or energy (e.g. in fields ).

Mathematically, space-time is described using a pseudo-Riemannian manifold , especially in the so-called Minkowski space . In the Minkowski space, in addition to the location coordinates, the time coordinates of the events must also be taken into account to calculate distances , i.e. with the speed of light . The classic calculation of spatial distances in Cartesian coordinates - which is the squared distance - is therefore modified: The squared generalized distance of two events in Minkowski space is and is also called spacetime metric or spacetime interval . The signs used here are the signature of the metric and partly a question of convention. There are other, equivalent signatures, for example , or less common ones, such as where with is the imaginary unit of complex numbers . ${\ displaystyle \ mathrm {\ Delta} s}$${\ displaystyle c \ mathrm {\ Delta} t, \ mathrm {\ Delta} x, \ mathrm {\ Delta} y, \ mathrm {\ Delta} z}$${\ displaystyle c}$${\ displaystyle \ textstyle (\ mathrm {\ Delta} x) ^ {2} + (\ mathrm {\ Delta} y) ^ {2} + (\ mathrm {\ Delta} z) ^ {2}}$${\ displaystyle \ textstyle (\ mathrm {\ Delta} s) ^ {2} = (c \ mathrm {\ Delta} t) ^ {2} - (\ mathrm {\ Delta} x) ^ {2} - (\ mathrm {\ Delta} y) ^ {2} - (\ mathrm {\ Delta} z) ^ {2}}$${\ displaystyle \ textstyle (+, -, -, -)}$${\ displaystyle \ textstyle (-, +, +, +)}$${\ displaystyle \ textstyle (\ mathrm {i}, +, +, +)}$${\ displaystyle \ mathrm {i}}$${\ displaystyle \ mathrm {i} ^ {2} = - 1}$

Minkowski space, four-vectors

In the special theory of relativity (SRT), the three-dimensional space coordinates are extended by a time component to form a four-vector in Minkowski space (“space-time”), that is . ${\ displaystyle (x, y, z)}$${\ displaystyle ct}$ ${\ displaystyle \ mathbb {M} ^ {4} = \ mathbb {R} ^ {1,3}}$${\ displaystyle (ct, x, y, z)}$

A point in space-time has three space coordinates and one time coordinate and is called an event or world point .

An invariant space-time distance is defined for events. In classical Euclidean space , a three-dimensional Cartesian coordinate system , the differential spatial distance square ( Euclidean norm ) of two points remains constant only under Galileo transformations :

${\ displaystyle \ mathrm {d} s ^ {2} = \ mathrm {d} x ^ {2} + \ mathrm {d} y ^ {2} + \ mathrm {d} z ^ {2}}$

In the SRT, on the other hand, an identical (generalized) distance is defined for all observers, which remains constant (invariant) even under Lorentz transformations (this invariance is defined by the requirement that the four-dimensional distance or the Minkowski metric is constant (invariant) is under a linear coordinate transformation , which expresses the above-mentioned homogeneity of spacetime.):

${\ displaystyle \ mathrm {d} s ^ {2}: = \ eta _ {\ mu \ nu} \ mathrm {d} x ^ {\ mu} \ mathrm {d} x ^ {\ nu} = c ^ { 2} \ mathrm {d} t ^ {2} - \ mathrm {d} x ^ {2} - \ mathrm {d} y ^ {2} - \ mathrm {d} z ^ {2}.}$

This is the squared Minkowski norm that creates the improper metric (distance function) of flat spacetime. It is induced by the (indefinite) invariant scalar product on the Minkowski space, which can be defined as the effect of the (pseudo) -metric tensor : ${\ displaystyle \ eta _ {\ mu \ nu} = \ mathrm {diag} (+ 1, -1, -1, -1)}$

${\ displaystyle \ mathbf {x} \ cdot \ mathbf {y} = \ eta _ {\ mu \ nu} x ^ {\ mu} y ^ {\ nu}}$(note: Einstein's summation convention )

This metric tensor is also referred to in physical parlance as the “Minkowski metric” or “flat metric” of spacetime, although in the actual sense it should not be confused with the metric itself. Mathematically it is rather a scalar product on a pseudo-Siemens manifold .

With the exception of the factor, the line element is the differential proper time : ${\ displaystyle \ mathrm {d} s}$${\ displaystyle 1 / c}$

${\ displaystyle \ mathrm {d} \ tau = {\ frac {\ mathrm {d} s} {c}} = \ mathrm {d} t \ cdot {\ sqrt {1 - {\ frac {v ^ {2} } {c ^ {2}}}}}.}$

This is a co-moving measuring clock, ie in the "currently accompanying inertial system" in which the on the world line particles contained rests: . ${\ displaystyle (v (t) \ equiv 0)}$

An element ( vector ) of spacetime is called

• time- like , if (space-time distance real). Two events for which is positive are mutually visible, i.e. i.e., they lie within the light cone .${\ displaystyle \ mathrm {d} s ^ {2}> 0}$${\ displaystyle \ mathrm {d} s ^ {2}}$
• spacelike , if applies (spacetime-distance imaginary). Two events for which is negative are so far apart in space and time that a ray of light cannot get from one event to the other in time. Since information is transmitted either via light or matter and the speed of matter in the theory of relativity can never reach the speed of light (and therefore cannot exceed it), such events can never have a cause-effect relationship . They could only be perceived at faster than light speed, so they are in principle mutually invisible, i. that is, they lie outside the cone of light.${\ displaystyle \ mathrm {d} s ^ {2} <0}$${\ displaystyle \ mathrm {d} s ^ {2}}$
• light-like if applies. Light always moves exactly with the speed , so that it applies to all reference systems ( constancy of the speed of light , the starting principle of the special theory of relativity).${\ displaystyle \ mathrm {d} s ^ {2} = 0}$${\ displaystyle c}$${\ displaystyle \ mathrm {d} s ^ {2} \ equiv 0}$

The classification of the space-time vectors (space-like, light-like or time-like) remains unchanged for the permissible transformations (Lorentz transformations) ( invariance of the light cone ).

Calculating with space-time vectors finds practical application in the kinematics of fast particles .

Mathematical motivation of the Minkowski metric

• If one considers the D'Alembert operator with${\ displaystyle \ Box}$
${\ displaystyle \ Box = {\ frac {1} {c ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} - {\ vec {\ nabla}} ^ {2},}$
so it can be seen that one can also abbreviate
${\ displaystyle \ Box = \ partial _ {\ mu} \ partial ^ {\ mu}}$
can write if the following two four-vectors are introduced:
${\ displaystyle \ partial _ {\ mu} = \ left ({\ frac {1} {c}} {\ frac {\ partial} {\ partial t}}, {\ vec {\ nabla}} \ right)}$
${\ displaystyle \ partial ^ {\ mu} = \ left ({\ frac {1} {c}} {\ frac {\ partial} {\ partial t}}, - {\ vec {\ nabla}} \ right) }$
In this case time appears as the fourth dimension, so the metric must be induced by a matrix.${\ displaystyle \ eta _ {\ mu \ nu}}$${\ displaystyle 4 \ times 4}$
${\ displaystyle (\ eta _ {\ mu \ nu}) = \ left ({\ begin {array} {cccc} \ alpha _ {0} & 0 & 0 & 0 \\ 0 & \ alpha _ {1} & 0 & 0 \\ 0 & 0 & \ alpha _ {2} & 0 \\ 0 & 0 & 0 & \ alpha _ {3} \ end {array}} \ right)}$
• Due to the requirement that there are no excellent space-time coordinates, the diagonal elements can only have the value . The space coordinates are selected here . However, this is a convention that is not used uniformly.${\ displaystyle \ pm 1}$${\ displaystyle -1}$
${\ displaystyle (\ eta _ {\ mu \ nu}) = \ left ({\ begin {array} {cccc} \ pm 1 & 0 & 0 & 0 \\ 0 & \ mp 1 & 0 & 0 \\ 0 & 0 & \ mp 1 & 0 \\ 0 & 0 & 0 & \ mp 1 \ end {array}} \ right)}$
• The time component cannot have the same sign as the space components. To do this, consider the D'Alembert operator again :${\ displaystyle \ Box}$
${\ displaystyle \ Box = \ partial _ {\ mu} \ partial ^ {\ mu} = \ eta _ {\ mu \ nu} \ partial ^ {\ mu} \ partial ^ {\ nu}}$
This would result in a homogeneous wave equation for a wave${\ displaystyle \ psi}$
${\ displaystyle \ left ({\ vec {\ nabla}} ^ {2} + {\ frac {1} {c ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial t ^ { 2}}} \ right) \ psi = 0}$
If one now applies for a plane wave , i. H. , this would result in a complex frequency and thus exponentially attenuated. In this case there would be no permanent plane waves, which is contrary to observation. Consequently, the time component must have a different sign:${\ displaystyle \ psi}$${\ displaystyle \ psi ({\ vec {r}}, t) = A \, e ^ {\ mathrm {i} ({\ vec {k}} \ cdot {\ vec {r}} - \ omega t) }}$${\ displaystyle \ psi}$
${\ displaystyle (\ eta _ {\ mu \ nu}) = \ left ({\ begin {array} {cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \ end {array}} \ right)}$
This results in the correct homogeneous wave equation
${\ displaystyle \ left ({\ vec {\ nabla}} ^ {2} - {\ frac {1} {c ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial t ^ { 2}}} \ right) \ psi = 0}$

Minkowski diagram

The relationships can be geometrically represented and analyzed in the Minkowski diagram . Because of the complex property of the time component, the rotation of the time axis is shown there with the opposite sign than the rotation of the coordinate axis.

Spacetime in general relativity

Non-Euclidean geometries

The basis for the description of spacetime in general relativity is pseudo-Riemannian geometry . The coordinate axes are non-linear here, which can be interpreted as the curvature of space. The same mathematical tools are used for four-dimensional space-time as for describing a two-dimensional spherical surface or for saddle surfaces. Statements of Euclidean geometry that are regarded as irrefutable, especially the axiom of parallels , must be given up in these theories and replaced by more general relationships. For example, the shortest connection between two points is no longer a straight line segment . The geodesic in the non-Euclidean world corresponds to a straight line in Euclidean geometry ; in the case of a spherical surface, the geodesics are the great circles . The angle sum in the triangle - consisting of geodesic sections - is no longer 180 degrees. In the case of the spherical surface it is greater than 180 degrees, in the case of saddle surfaces it is smaller.

Spacetime curvature

The curvature of space and time is caused by any form of energy, such as mass, radiation or pressure. These quantities together form the energy-momentum tensor and are included in the Einstein equations as the source of the gravitational field. The resulting curvilinear movement of force-free bodies along geodesics is attributed to gravitational acceleration - in this model, something like a gravitational force no longer exists. In an infinitesimal space segment (local map), the generated gravitational field always has the flat metric of the special theory of relativity . This is described by a constant curvature of space with the factor . The curvature of the world lines (movement curves in space-time) of all force-free bodies in this space is the same. ${\ displaystyle g / c ^ {2}}$

In many popular representations of the general theory of relativity, it is often ignored that not only space but also time must be curved in order to create a gravitational field. The fact that space and time always have to be curved is clearly easy to understand: If only the space were curved, the trajectory of a thrown stone would always be the same, regardless of the initial speed of the stone, since it would always only follow the curved space. The different trajectories can only come about through the additional curvature of time. This can also be shown mathematically within the framework of the ART.

In normal, three-dimensional space, only the projection of the world lines onto the plane of movement is visible. If the body has the speed , the world line is inclined with respect to the time axis, namely by the angle α with . The projection of the path increases by a factor of longer, the radius of curvature increases by the same factor , so the change in angle is smaller. The curvature (angle change per length segment) is therefore smaller by a factor . ${\ displaystyle v}$${\ displaystyle \ tan \ alpha = v / c}$${\ displaystyle v}$${\ displaystyle 1 / \ sin \ alpha}$${\ displaystyle 1 / \ sin \ alpha}$${\ displaystyle \ sin ^ {2} \ alpha}$

With

${\ displaystyle \ sin \ alpha = {\ frac {v} {c}} {\ frac {1} {\ sqrt {1 + {\ frac {v ^ {2}} {c ^ {2}}}}} }}$

then follows from the curvature of the world line for the observed orbital curvature in three-dimensional space ${\ displaystyle g / c ^ {2}}$${\ displaystyle \ kappa = 1 / R}$

${\ displaystyle \ kappa = {\ frac {g} {v ^ {2}}} \ cdot \ left (1 + {\ frac {v ^ {2}} {c ^ {2}}} \ right)}$.

Space curvature and centrifugal acceleration

For small speeds vc the curvature of the path is g / v 2 and thus corresponds to the value for a classic centrifugal acceleration. For light rays with v = c , the factor (1 + v 2 / c 2 ) has the value 2 , the curvature 2 g / c 2 therefore corresponds to twice the value of the classic consideration g / c 2 . The angular deviation of starlight from the fixed stars in the vicinity of the sun should therefore be twice as large as in the classic case. This was verified for the first time by Arthur Eddington as part of an Africa expedition to observe the solar eclipse of 1919, which attracted a great deal of attention and contributed significantly to the implementation of the general theory of relativity. His observations turned out to be imprecise in later analyzes, but subsequent observations of solar eclipses confirmed the predictions of the general theory of relativity.

Because of this small deviation from the classic value, the planetary orbits are no longer exact ellipses, but are subject to a rotation of the apse . Such a rotation of the apse, which was previously inexplicable in celestial mechanics, had previously been observed on the planet Mercury and was explained by the general theory of relativity.

Symmetries

Spacetime is characterized by a number of symmetries that are very important for the physics that apply in it. In addition to the symmetries of space ( translation , rotation ), these symmetries also include the symmetries under Lorentz transformations (change between reference systems of different speeds). The latter ensures the principle of relativity .

literature

Philosophical Books:

• Paul Davies : The Immortality of Time. Modern physics between rationality and God. Scherz, Munich 1995, ISBN 3502131430 (Original: About Time - Einstein's unfinished revolution . Simon and Schuster 1995).
• Robert DiSalle: Understanding space-time: the philosophical development of physics from Newton to Einstein. Cambridge Univ. Press, Cambridge 2007, ISBN 978-0-521-85790-1 .
• Moritz Schlick : Space and Time in Contemporary Physics. Springer, Berlin 1922, doi: 10.1007 / BF02448303 .
• Lawrence Sklar : Space, Time, and Spacetime , University of California Press 1977.