Minkowski room

The Minkowski space , named after Hermann Minkowski , is a four-dimensional space in which the theory of relativity can be elegantly formulated. Around 1907, Minkowski recognized that the work of Hendrik Antoon Lorentz (1904) and Albert Einstein (1905) on the theory of relativity can be understood in a non-Euclidean space . He assumed that space and time are connected to one another in a four-dimensional space-time continuum . This is also known as the Minkowski world .

Three of its coordinates are those of Euclidean space ; there is also a fourth coordinate for time. The Minkowski space thus has four dimensions. Nevertheless, the Minkowski space differs significantly from a four-dimensional Euclidean space due to the different structure of space and time coordinates (see below).

In mathematics one also considers Minkowski spaces of any dimension .

Real definition

The Minkowski space is a four-dimensional real vector space on which the scalar product is not given by the usual expression, but by a non-degenerate bilinear form with index 1. So this is not definitely positive . The Minkowski four-vectors (so-called "events") are assigned four-component elements and are usually set ${\ displaystyle \ mathbf {x}}$ ${\ displaystyle \ mathbf {y}}$

{\ displaystyle \ mathbf {x \ cdot y}: = - x_ {0} y_ {0} + x_ {1} y_ {1} + x_ {2} y_ {2} + x_ {3} y_ {3}, }

where the coordinate is also real: it is derived from the time coordinate with the aid of the speed of light . ${\ displaystyle x_ {0} = ct}$ ${\ displaystyle c}$ ${\ displaystyle t}$

Instead of the signature chosen here, which is most frequently used in general relativity today (it is the convention in the influential textbook by Charles Misner , Kip Thorne and John Archibald Wheeler from 1973), the physically equivalent is often used, especially in more recent literature reverse signature chosen. The latter is also widespread in particle physics and is used, for example, in the well-known series of textbooks by Landau and Lifschitz . is therefore also called the Particle Physics Convention in English (also known as the West Coast Convention), and the Relativity Theory Convention (also known as the East Coast Convention). The time is sometimes recorded as a fourth instead of a zeroth coordinate. ${\ displaystyle {(-, +, +, +)},}$ ${\ displaystyle {(+, -, -, -)}}$ ${\ displaystyle {(+, -, -, -)}}$ ${\ displaystyle {(-, +, +, +)},}$

Alternatively, the inner product of two elements of the Minkowski space can also be understood as the action of the metric tensor : ${\ displaystyle \ eta _ {\ mu \ nu}}$

{\ displaystyle \ mathbf {x \ cdot y}: = \ eta _ {\ mu \ nu} x ^ {\ mu} y ^ {\ nu} \ ,,}

by distinguishing contravariant and covariant vector components (upper and lower indices, e.g. but ). ${\ displaystyle x ^ {0} = + ct \ ,,}$ ${\ displaystyle x_ {0} = \ eta _ {0 \ nu} \, x ^ {\ nu} = - ct \ ,,}$ ${\ displaystyle \ eta _ {\ mu \ nu} = {\ rm {diag}} (- 1, + 1, + 1, + 1) \,}$

Definition with imaginary time

Some older textbooks use equivalent notation that avoids the mixed signature of the inner product by using an imaginary timeline. By setting you can use those with positive definite, Euclidean metrics and you still get the correct Minkowski signature ${\ displaystyle x_ {0} = \ mathrm {i} ct, x_ {1} = x, x_ {2} = y, x_ {3} = z}$ ${\ displaystyle x_ {i}}$

{\ displaystyle x_ {0} ^ {2} + x_ {1} ^ {2} + x_ {2} ^ {2} + x_ {3} ^ {2} = - c ^ {2} t ^ {2} + x ^ {2} + y ^ {2} + z ^ {2} \.}

A characteristic of this convention is that no distinction is made between contravariant and covariant components. The change from the Minkowski signature to the Euclidean signature of the metric is called the Wick rotation . This convention is not used in modern textbooks and its use is discouraged.

Lorentz transformations

The Lorentz transformations play a role analogous to the rotations around the coordinate origin in Euclidean spaces: They are those homogeneous- linear transformations that leave the object and thus the inner product of the Minkowski space invariant , which explains the importance of the Minkowski space in the special theory of relativity . This formalism is also suitable for generalization in the general theory of relativity . In contrast to the rotating groups, the Lorentz transformations also have the causal structure of the systems as a consequence. ${\ displaystyle \ eta _ {\ mu \ nu}}$

Causal structure (space-like, time-like and light-like vectors)

The elements of the Minkowski space can be divided into three classes according to the sign of : ${\ displaystyle y ^ {2}}$

time-like Minkowski vectors (this corresponds to " pairs of events" that can be causally influenced by " massive bodies"),

space-like Minkowski vectors ( event pairs that cannot be influenced causally )

- as a borderline case - light-like Minkowski vectors (causally only pairs of events that can be influenced by light signals).

The invariance of this division in all Lorentz transformations follows from the invariance of the light cone . The time-like interior of the light cone describes the causal structure : possible causes of an event lie in the “past” (backward area of the light cone inside), possible effects in the “future” (forward area of the light cone inside); there is also the space-like outer area of the light cone, which is not "causally related" to the observed event in the center, because this would require the transmission of information at faster than light speeds.

Minkowski spaces in mathematics

In mathematics , especially differential geometry , one also considers Minkowski spaces of any dimension. These are -dimensional vector spaces with a symmetrical bilinear form of the signature . In a suitable basis, it can be called ${\ displaystyle \ mathbb {R} ^ {1, n}}$ ${\ displaystyle (n + 1)}$ ${\ displaystyle b}$ ${\ displaystyle (1, n)}$ ${\ displaystyle b}$

{\ displaystyle b (x, y) = - x_ {0} y_ {0} + x_ {1} y_ {1} + \ ldots + x_ {n} y_ {n}}

,

represent, this form is called Lorentz form .

literature

Francesco Catoni: The mathematics of Minkowski space-time. Birkhäuser, Basel 2008, ISBN 978-3-7643-8613-9 .
John W. Schutz: Independent axioms for Minkowski space-time. Longman, Harlow 1997, ISBN 0-582-31760-6 .

Web links

Wikibooks: Special Theory of Relativity - Learning and Teaching Materials

References and footnotes

↑ For example, in the well-known textbooks by Michael Peskin and Daniel Schroeder, An introduction to quantum field theory, 1995, and almost all particle textbooks since the classic textbooks by James Bjorken and Sidney Drell Relativistic Quantum Mechanics , 1964.
↑ It was used, among others, by Wolfgang Pauli in his influential article on the theory of relativity in the Encyclopedia of Mathematical Sciences. Einstein used different conventions in his essay on general relativity from 1916 the convention (+, -, -, -) and also Hermann Minkowski in 1908 in his lecture Raum und Zeit .
↑ See for example the textbook on theoretical physics by Friedrich Hund , Volume II.
^ Charles W. Misner , Kip S. Thorne and John A. Wheeler : Gravitation . Freeman, San Francisco 1973, ISBN 0-7167-0334-3 .
↑ That we are dealing with pairs of events becomes clear when we use the infinitesimal differences . ${\ displaystyle \, \ mathbf {y} ^ {2}}$ ${\ displaystyle \, \ mathrm {d} \ mathbf {y} ^ {2}}$

[1] For example, in the well-known textbooks by Michael Peskin and Daniel Schroeder, An introduction to quantum field theory, 1995, and almost all particle textbooks since the classic textbooks by James Bjorken and Sidney Drell Relativistic Quantum Mechanics , 1964.

[2] It was used, among others, by Wolfgang Pauli in his influential article on the theory of relativity in the Encyclopedia of Mathematical Sciences. Einstein used different conventions in his essay on general relativity from 1916 the convention (+, -, -, -) and also Hermann Minkowski in 1908 in his lecture Raum und Zeit .

[3] See for example the textbook on theoretical physics by Friedrich Hund , Volume II.

[4] Charles W. Misner , Kip S. Thorne and John A. Wheeler : Gravitation . Freeman, San Francisco 1973, ISBN 0-7167-0334-3 .

[5] That we are dealing with pairs of events becomes clear when we use the infinitesimal differences . ${\ displaystyle \, \ mathbf {y} ^ {2}}$ ${\ displaystyle \, \ mathrm {d} \ mathbf {y} ^ {2}}$