Editing Minkowski space (section)

== History ==
{{Spacetime|cTopic=Types}}

=== Complex Minkowski spacetime ===
{{See also|Four-dimensional space}}
In his second relativity paper in 1905, [[Henri Poincaré]] showed<ref>{{harvnb|Poincaré|1905–1906|pp=129&ndash;176}} Wikisource translation: [[s:Translation:On the Dynamics of the Electron (July)|On the Dynamics of the Electron]]</ref> how, by taking time to be an imaginary fourth [[spacetime]] coordinate {{math|''ict''}}, where {{math|''c''}} is the [[speed of light]] and {{math|''i''}} is the [[imaginary unit]], [[Lorentz transformation]]s can be visualized as ordinary rotations of the four-dimensional Euclidean sphere. The four-dimensional spacetime can be visualized as a four-dimensional space, with each point representing an event in spacetime. The [[Lorentz transformation|Lorentz transformations]] can then be thought of as rotations in this four-dimensional space, where the rotation axis corresponds to the direction of relative motion between the two observers and the rotation angle is related to their relative velocity.

To understand this concept, one should consider the coordinates of an event in spacetime represented as a four-vector {{math|(''t'', ''x'', ''y'', ''z'')}}. A Lorentz transformation is represented by a [[Matrix (mathematics)|matrix]] that acts on the four-vector, changing its components. This matrix can be thought of as a rotation matrix in four-dimensional space, which rotates the four-vector around a particular axis.<math display="block">x^2 + y^2 + z^2 + (ict)^2 = \text{constant}. </math>

Rotations in planes spanned by two space unit vectors appear in coordinate space as well as in physical spacetime as Euclidean rotations and are interpreted in the ordinary sense. The "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a [[Lorentz boost]] in physical spacetime with ''real'' inertial coordinates. The analogy with Euclidean rotations is only partial since the radius of the sphere is actually imaginary, which turns rotations into rotations in hyperbolic space (see [[Lorentz transformation#Hyperbolic rotation of coordinates|hyperbolic rotation]]).

This idea, which was mentioned only briefly by Poincaré, was elaborated by Minkowski in a paper in [[German language|German]] published in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies".<ref>{{harvnb|Minkowski|1907–1908|pp=53–111}} *Wikisource translation: [[s:Translation:The Fundamental Equations for Electromagnetic Processes in Moving Bodies]]</ref> He reformulated [[Maxwell equations]] as a symmetrical set of equations in the four variables {{math|(''x'', ''y'', ''z'', ''ict'')}} combined with redefined vector variables for electromagnetic quantities, and he was able to show directly and very simply their invariance under Lorentz transformation. He also made other important contributions and used matrix notation for the first time in this context.
From his reformulation, he concluded that time and space should be treated equally, and so arose his concept of events taking place in a unified four-dimensional [[spacetime continuum]].

=== Real Minkowski spacetime ===
In a further development in his 1908 "Space and Time" lecture,<ref name="raumzeit">{{harvnb|Minkowski|1908–1909|pp=75&ndash;88}} Various English translations on Wikisource: "[[s:Translation:Space and Time|Space and Time]]"</ref> Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables {{math|(''x'', ''y'', ''z'', ''t'')}} of space and time in the coordinate form in a four-dimensional real [[vector space]]. Points in this space correspond to events in spacetime. In this space, there is a defined [[light-cone]] associated with each point, and events not on the light cone are classified by their relation to the apex as ''spacelike'' or ''timelike''. It is principally this view of spacetime that is current nowadays, although the older view involving imaginary time has also influenced special relativity.

In the English translation of Minkowski's paper, the Minkowski metric, as defined below, is referred to as the ''line element''. The Minkowski inner product below appears unnamed when referring to [[Orthogonality (mathematics)|orthogonality]] (which he calls ''normality'') of certain vectors, and the Minkowski norm squared is referred to (somewhat cryptically, perhaps this is a translation dependent) as "sum".

Minkowski's principal tool is the [[Minkowski diagram]], and he uses it to define concepts and demonstrate properties of Lorentz transformations (e.g., [[proper time]] and [[length contraction]]) and to provide geometrical interpretation to the generalization of Newtonian mechanics to [[relativistic mechanics]]. For these special topics, see the referenced articles, as the presentation below will be principally confined to the mathematical structure (Minkowski metric and from it derived quantities and the [[Poincaré group]] as symmetry group of spacetime) ''following'' from the invariance of the spacetime interval on the spacetime manifold as consequences of the postulates of special relativity, not to specific application or ''derivation'' of the invariance of the spacetime interval. This structure provides the background setting of all present relativistic theories, barring general relativity for which [[flat (geometry)|flat]] Minkowski spacetime still provides a springboard as curved spacetime is locally Lorentzian.

Minkowski, aware of the fundamental restatement of the theory which he had made, said
{{blockquote|The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.| Hermann Minkowski, 1908, 1909<ref name="raumzeit"/>}}

Though Minkowski took an important step for physics, [[Albert Einstein]] saw its limitation:
{{blockquote|At a time when Minkowski was giving the geometrical interpretation of special relativity by extending the Euclidean three-space to a [[pseudo-Euclidean space|quasi-Euclidean]] four-space that included time, Einstein was already aware that this is not valid, because it excludes the phenomenon of [[gravitation]]. He was still far from the study of [[curvilinear coordinates]] and [[Riemannian geometry]], and the heavy mathematical apparatus entailed.<ref>[[Cornelius Lanczos]] (1972) "Einstein's Path from Special to General Relativity", pages 5–19 of ''General Relativity: Papers in Honour of J. L. Synge'', L. O'Raifeartaigh editor, [[Clarendon Press]], see page 11</ref>}}

For further historical information see references {{harvtxt|Galison|1979}}, {{harvtxt|Corry|1997}} and {{harvtxt|Walter|1999}}.