Minkowski space: Difference between revisions - Wikipedia

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{{For|the ~~fictional concept~~use in ~~the ''Gundam'' franchise|Gundam Universal Century technology#Minovsky physics}}{{For|the use of~~ algebraic number theory|Minkowski space (number field)}}

[[File:De Raum zeit Minkowski Bild.jpg|250px|right|thumb|[[Hermann Minkowski]] (1864–1909) found that the theory of special relativity could be best understood as a four-dimensional space, since known as the Minkowski spacetime.]]

In [[~~mathematical~~ physics]], '''Minkowski space''' (or '''Minkowski spacetime''') ({{IPAc-en|m|ɪ|ŋ|ˈ|k|ɔː|f|s|k|i|,_|-|ˈ|k|ɒ|f|-}}<ref>[http://www.dictionary.com/browse/minkowski"Minkowski"] {{Webarchive|url=https://web.archive.org/web/20190622030121/https://www.dictionary.com/browse/minkowski |date=2019-06-22 }}. ''[[Random House Webster's Unabridged Dictionary]]''.</ref>) ~~combines~~is the main mathematical description of [[~~inertial~~spacetime]] in the absence of [[~~space~~general_relativity|gravitation]]. ~~and~~It combines [[~~time~~inertial]] [[~~manifolds~~space]] ~~with a~~and [[~~non-inertial reference frame~~time]] of [[~~Spacetime|space and time~~manifolds]] into a [[four-dimensional]] model ~~relating a position ([[inertial frame of reference]]) to the [[field (physics)|field]]~~.

The model helps show how a [[spacetime interval]] between any two [[Event (relativity)|events]] is independent of the [[inertial frame of reference]] in which they are recorded. Mathematician [[Hermann Minkowski]] developed it from the work of [[Hendrik Lorentz]], [[Henri Poincaré]], and others said it "was grown on experimental physical grounds".

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Spacetime is equipped with an indefinite [[Degenerate bilinear form|non-degenerate]] [[bilinear form]], called the ''Minkowski metric'',<ref>{{harvnb|Lee|1997|p=31}}</ref> the ''Minkowski norm squared'' or ''Minkowski inner product'' depending on the context.<ref group=nb>Consistent use of the terms "Minkowski inner product", "Minkowski norm" or "Minkowski metric" is intended for the bilinear form here, since it is in widespread use. It is by no means "standard" in the literature, but no standard terminology seems to exist.</ref> The Minkowski inner product is defined so as to yield the [[spacetime interval]] between two events when given their coordinate difference vector as an argument.<ref>{{cite book |title=Independent Axioms for Minkowski Space–Time |edition=illustrated |first1=John W. |last1=Schutz |publisher=CRC Press |year=1977 |isbn=978-0-582-31760-4 |pages=184–185 |url=https://books.google.com/books?id=eOO1SWD17GIC}} [https://books.google.com/books?id=eOO1SWD17GIC&pg=PA184 Extract of page 184]</ref> Equipped with this inner product, the mathematical model of spacetime is called Minkowski space. The group of transformations for Minkowski space that preserves the spacetime interval (as opposed to the spatial Euclidean distance) is the [[Poincaré group]] (as opposed to the [[Galilean transformation|Galilean group]]).

== History ==

=== Complex Minkowski spacetime ===

In his second relativity paper in 1905, [[Henri Poincaré]] showed<ref>{{harvnb|Poincaré|1905–1906|pp=129–176}} 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 {{~~mvar~~math|''c''}} is the [[speed of light]] and {{~~mvar~~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~~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=75–88}} 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".

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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}}.

== Causal structure ==

[[File:World line.svg|250px|thumb|right|Subdivision of Minkowski spacetime with respect to an event in four disjoint sets.: ~~The~~the [[light cone]], the '''causal future''' (also called the '''absolute future'''), the '''causal past''' (also called the '''absolute past'''), and '''elsewhere'''. The terminology is from {{harvtxt|Sard|1970}}, and from [[Causal structure]].]]

Where {{~~mvar~~math|''v''}} is velocity, {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are [[Cartesian coordinate system|Cartesian]] coordinates in 3-dimensional space, {{math|''c''}} is the constant representing the universal speed limit, and {{mvar|t}} is time, the four-dimensional vector {{math|1=''v'' = (''ct'', ''x'', ''y'', ''z'') = (''ct'', '''r''')}} is classified according to the sign of {{math|''c''{{i sup|2}}''t''{{i sup|2}} − ''r''{{i sup|2}}}}. A vector is '''timelike''' if {{math|''c''{{i sup|2}}''t''{{i sup|2}} > ''r''{{i sup|2}}}}, '''spacelike''' if {{math|''c''{{i sup|2}}''t''{{i sup|2}} < ''r''{{i sup|2}}}}, and '''null''' or '''lightlike''' if {{math|1=''c''{{i sup|2}}''t''{{i sup|2}} = ''r''{{i sup|2}}}}. This can be expressed in terms of the sign of {{math|''η''(''v'', ''v'')}}, also called [[#Scalar product|scalar product]], as well, which depends on the signature. The classification of any vector will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the spacetime interval under Lorentz transformation.

The set of all [[Null vector|null vectors]] at an event<ref group=nb>Translate the coordinate system so that the event is the new origin.</ref> of Minkowski space constitutes the [[light cone]] of that event. Given a timelike vector {{math|''v''}}, there is a [[worldline]] of constant velocity associated with it, represented by a straight line in a Minkowski diagram.

Once a direction of time is chosen,<ref group=nb>This corresponds to the time coordinate either increasing or decreasing when the proper time for any particle increases. An application of {{~~mvar~~math|''T''}} flips this direction.</ref> timelike and null vectors can be further decomposed into various classes. For timelike vectors, one has

# future-directed timelike vectors whose first component is positive (tip of vector located in causal future (also called the absolute future) in the figure) and

# past-directed timelike vectors whose first component is negative (causal past (also called the absolute past)).

Null vectors fall into three classes:

# the zero vector, whose components in any basis are {{math|(0, 0, 0, 0)}} (origin),

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[[Vector field]]s are called timelike, spacelike, or null if the associated vectors are timelike, spacelike, or null at each point where the field is defined.

== Properties of time-like vectors ==

Time-like vectors have special importance in the theory of relativity as they correspond to events that are accessible to the observer at (0, 0, 0, 0) with a speed less than that of light. Of most interest are time-like vectors that are ''similarly directed'', i.e. all either in the forward or in the backward cones. Such vectors have several properties not shared by space-like vectors. These arise because both forward and backward cones are convex, whereas the space-like region is not convex.

=== Scalar product ===

The [[scalar product]] of two time-like vectors {{math|1=''u''{{sub|1}} = (''t''{{sub|1}}, ''x''{{sub|1}}, ''y''{{sub|1}}, ''z''{{sub|1}})}} and {{math|1=''u''{{sub|2}} = (''t''{{sub|2}}, ''x''{{sub|2}}, ''y''{{sub|2}}, ''z''{{sub|2}})}} is

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Using the positivity property of time-like vectors, it is easy to verify that a linear sum with positive coefficients of similarly directed time-like vectors is also similarly directed time-like (the sum remains within the light cone because of convexity).

=== Norm and reversed Cauchy inequality ===

The norm of a time-like vector {{math|1=''u'' = (''ct'', ''x'', ''y'', ''z'')}} is defined as

<math display="block">\left\| u \right\| = \sqrt{ \eta(u, u) } = \sqrt{c^2 t^2 - x^2 - y^2 - z^2}</math>

''The reversed Cauchy inequality'' is another consequence of the convexity of either light cone.<ref>See Schutz's proof p 148, also Naber p. 48</ref> For two distinct similarly directed time-like vectors {{math|''u''{{sub|1}}}} and {{math|''u''{{sub|2}}}} this inequality is

<math display="block">\eta(u_1, u_2) > \left\| u_1 \right\| \left\| u_2 \right\|</math>

or algebraically,

<math display="block">c^2 t_1 t_2 - x_1 x_2 - y_1 y_2 - z_1 z_2 > \sqrt{\left(c^2 t_1^2 - x_1^2 - y_1^2 - z_1^2\right) \left(c^2 t_2^2 - x_2^2 - y_2^2 - z_2^2\right)}</math>

From this, the positive property of the scalar product can be seen.

=== Reversed triangle inequality ===

~~To prove this, you can start with one space coordinate x, and note \left(c t_1 x_2 - c x_1 t_2)\right ^2 >=0.~~

For two similarly directed time-like vectors {{mvar|u}} and {{mvar|w}}, the inequality is<ref>Schutz p. 148, Naber p. 49</ref>

~~===The reversed triangle inequality===~~

~~For two similarly directed time-like vectors {{mvar|u}} and {{mvar|w}}, the inequality is<ref>Schutz p.148, Naber p.49</ref>~~

where the equality holds when the vectors are [[linearly dependent]].

The proof uses the algebraic definition with the reversed Cauchy inequality:<ref>Schutz p. 148</ref>

<math display="block">\begin{align}

\left\| u + w \right\| ^2

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The result now follows by taking the square root on both sides.

== Mathematical structure ==

It is assumed below that spacetime is endowed with a coordinate system corresponding to an [[inertial frame]]. This provides an ''origin'', which is necessary for spacetime to be modeled as a vector space. This addition is not required, and more complex treatments analogous to an [[affine space]] can remove the extra structure. However, this is not the introductory convention and is not covered here.

For an overview, Minkowski space is a {{math|4}}-dimensional [[real number|real]] [[vector space]] equipped with a non-degenerate, [[symmetric bilinear form]] on the [[tangent space]] at each point in spacetime, here simply called the ''Minkowski inner product'', with [[metric signature]] either {{math|(+ − − −)}} or {{math|(− + + +)}}. The tangent space at each event is a vector space of the same dimension as spacetime, {{math|4}}.

=== Tangent vectors ===

[[Image:Image Tangent-plane.svg|thumb|A pictorial representation of the tangent space at a point, {{mvar|x}}, on a [[sphere]]. This vector space can be thought of as a subspace of {{math|'''R'''3}} itself. Then vectors in it would be called ''geometrical tangent vectors''. By the same principle, the tangent space at a point in flat spacetime can be thought of as a subspace of spacetime, which happens to be ''all'' of spacetime.]]

In practice, one need not be concerned with the tangent spaces. The vector space structure of Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself. See e.g. {{harvtxt|Lee|2003|loc=Proposition 3.8.}} or {{harvtxt|Lee|2012|loc=Proposition 3.13.}} These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as<ref>{{harvnb|Lee|1997|p=15}}</ref>

<math display="block">\begin{align}

\left(x^0,\, x^1,\, x^2,\, x^3\right)

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&\leftrightarrow\ \left. x^0 \mathbf e_0 \right|_q + \left. x^1 \mathbf e_1 \right|_q + \left. x^2 \mathbf e_2 \right|_q + \left. x^3 \mathbf e_3 \right|_q

\end{align}</math>

with basis vectors in the tangent spaces defined by

\left.\mathbf e_\mu\right|_p = \left.\frac{\partial}{\partial x^\mu}\right|_p \text{ or }

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</math>

Here, {{math|''p''}} and {{math|''q''}} are any two events, and the second basis vector identification is referred to as [[parallel transport]]. The first identification is the canonical identification of vectors in the tangent space at any point with vectors in the space itself. The appearance of basis vectors in tangent spaces as first-order differential operators is due to this identification. It is motivated by the observation that a geometrical tangent vector can be associated in a one-to-one manner with a [[directional derivative]] operator on the set of smooth functions. This is promoted to a ''definition'' of tangent vectors in manifolds ''not'' necessarily being embedded in {{math|'''R'''''n''}}. This definition of tangent vectors is not the only possible one, as ordinary ''n''-tuples can be used as well.

~~{{Hidden begin| titlestyle = color:green; background:lightgrey;| title=Definitions of tangent vectors as ordinary vectors}}~~

A tangent vector at a point {{mvar|p}} may be defined, here specialized to Cartesian coordinates in Lorentz frames, as {{math|4 × 1}} column vectors {{mvar|v}} associated to ''each'' Lorentz frame related by Lorentz transformation {{math|Λ}} such that the vector {{mvar|v}} in a frame related to some frame by {{math|Λ}} transforms according to {{math|''v'' → Λ''v''}}. This is the ''same'' way in which the coordinates {{math|''x''''μ''}} transform. Explicitly,

{{hidden begin |titlestyle = color:green; background:lightgrey;| title=Definitions of tangent vectors as ordinary vectors}}

A tangent vector at a point {{math|''p''}} may be defined, here specialized to Cartesian coordinates in Lorentz frames, as {{math|4 × 1}} column vectors {{math|''v''}} associated to ''each'' Lorentz frame related by Lorentz transformation {{math|Λ}} such that the vector {{math|''v''}} in a frame related to some frame by {{math|Λ}} transforms according to {{math|''v'' → Λ''v''}}. This is the ''same'' way in which the coordinates {{math|''x''''μ''}} transform. Explicitly,

<math display="block">\begin{align}

x'^\mu &= {\Lambda^\mu}_\nu x^\nu, \\

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This definition is equivalent to the definition given above under a canonical isomorphism.

For some purposes, it is desirable to identify tangent vectors at a point {{mvar|p}} with ''displacement vectors'' at {{mvar|p}}, which is, of course, admissible by essentially the same canonical identification.<ref>{{harvnb|Lee|2003|loc=See Lee's discussion on geometric tangent vectors early in chapter 3.}}</ref> The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in {{harvtxt|Misner|Thorne|Wheeler|1973}}. They offer various degrees of sophistication (and rigor) depending on which part of the material one chooses to read.

=== Metric signature ===

The metric signature refers to which sign the Minkowski inner product yields when given space (''spacelike'' to be specific, defined further down) and time basis vectors (''timelike'') as arguments. Further discussion about this theoretically inconsequential but practically necessary choice for purposes of internal consistency and convenience is deferred to the hide box below. See also the page treating [[Sign_convention#Relativity| sign convention]] in Relativity.

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=== Terminology ===

Mathematically associated with the bilinear form is a [[tensor]] of type {{math|(0,2)}} at each point in spacetime, called the ''Minkowski metric''.<ref group=nb>For comparison and motivation of terminology, take a [[Riemannian metric]], which provides a positive definite symmetric bilinear form, i. e. an [[inner product]] proper at each point on a manifold.</ref> The Minkowski metric, the bilinear form, and the Minkowski inner product are all the same object; it is a bilinear function that accepts two (contravariant) vectors and returns a real number. In coordinates, this is the {{math|4×4}} matrix representing the bilinear form.

For comparison, in [[general relativity]], a [[Lorentzian manifold]] {{math|''L''}} is likewise equipped with a [[Metric tensor (general relativity)|metric tensor]] {{math|''g''}}, which is a nondegenerate symmetric bilinear form on the tangent space {{math|''T''''p''''L''}} at each point {{mvar|p}} of {{math|''L''}}. In coordinates, it may be represented by a {{math|4×4}} matrix ''depending on spacetime position''. Minkowski space is thus a comparatively simple special case of a [[Lorentzian manifold]]. Its metric tensor is in coordinates with the same symmetric matrix at every point of {{math|''M''}}, and its arguments can, per above, be taken as vectors in spacetime itself.

Introducing more terminology (but not more structure), Minkowski space is thus a [[pseudo-Euclidean space]] with total dimension {{math|1=''n'' = 4}} and [[signature (quadratic form)|signature]] {{math|(3, 1)}} or {{math|(1, 3)}}. Elements of Minkowski space are called [[Event (relativity)|events]]. Minkowski space is often denoted {{math|'''R'''3,1}} or {{math|'''R'''1,3}} to emphasize the chosen signature, or just {{~~mvar~~math|''M''}}. It is ~~perhaps the simplest~~an example of a [[pseudo-Riemannian manifold]].

~~Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space <math>V</math>, that is,~~

~~<math display=block>\eta:V\times V\rightarrow \mathbb{R}</math>~~

Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space {{math|''V''}}, that is,

where <math>\eta</math> has signature <math>(-,+,+,+)</math>, and signature is a coordinate-invariant property of <math>\eta</math>. The space of bilinear maps forms a vector space which can be identified with <math>M^*\otimes M^*</math>, and <math>\eta</math> may be equivalently viewed as an element of this space. By making a choice of orthonormal basis <math>\{e_\mu\}</math>, <math>M:=(V,\eta)</math> can be identified with the space <math>\mathbb{R}^{1,3}:=(\mathbb{R}^{4},\eta_{\mu\nu})</math>. The notation is meant to emphasize the fact that <math>M</math> and <math>\mathbb{R}^{1,3}</math> are not just vector spaces but have added structure. <math>\eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)</math>.

<math display="block">\eta:V\times V\rightarrow \mathbf{R}</math>

where {{math|''η''}} has signature {{math|(−, +, +, +)}}, and signature is a coordinate-invariant property of {{math|''η''}}. The space of bilinear maps forms a vector space which can be identified with <math>M^*\otimes M^*</math>, and {{math|''η''}} may be equivalently viewed as an element of this space. By making a choice of orthonormal basis <math>\{e_\mu\}</math>, <math>M:=(V,\eta)</math> can be identified with the space <math>\mathbf{R}^{1,3}:=(\mathbf{R}^{4},\eta_{\mu\nu})</math>. The notation is meant to emphasize the fact that {{math|''M''}} and <math>\mathbf{R}^{1,3}</math> are not just vector spaces but have added structure. <math>\eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)</math>.

An interesting example of non-inertial coordinates for (part of) Minkowski spacetime is the [[Born coordinates]]. Another useful set of coordinates is the [[light-cone coordinates]].

=== Pseudo-Euclidean metrics ===

The Minkowski inner product is not an [[inner product]], since it is not [[Definite bilinear form|positive-definite]], i.e. the [[quadratic form]] {{math|''η''(''v'', ''v'')}} need not be positive for nonzero {{mvar|v}}. The positive-definite condition has been replaced by the weaker condition of non-degeneracy. The bilinear form is said to be ''indefinite''.

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As a notational convention, vectors {{mvar|v}} in {{mvar|M}}, called [[4-vector]]s, are denoted in italics, and not, as is common in the Euclidean setting, with boldface {{math|'''v'''}}. The latter is generally reserved for the {{math|3}}-vector part (to be introduced below) of a {{math|4}}-vector.

The definition <ref>Giulini 2008 pp. 5, 6</ref>

yields an inner product-like structure on {{math|''M''}}, previously and also henceforth, called the ''Minkowski inner product'', similar to the Euclidean [[inner product]], but it describes a different geometry. It is also called the ''relativistic dot product''. If the two arguments are the same,

yields an inner product-like structure on {{mvar|M}}, previously and also henceforth, called the ''Minkowski inner product'', similar to the Euclidean [[inner product]], but it describes a different geometry. It is also called the ''relativistic dot product''. If the two arguments are the same,

<math display="block">u \cdot u = \eta(u, u) \equiv \|u\|^2 \equiv u^2,</math>

the resulting quantity will be called the ''Minkowski norm squared''. The Minkowski inner product satisfies the following properties.

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The most important feature of the inner product and norm squared is that ''these are quantities unaffected by Lorentz transformations''. In fact, it can be taken as the defining property of a Lorentz transformation in that it preserves the inner product (i.e. the value of the corresponding bilinear form on two vectors). This approach is taken more generally for ''all'' classical groups definable this way in [[classical group]]. There, the matrix {{math|Φ}} is identical in the case {{math|O(3, 1)}} (the Lorentz group) to the matrix {{math|''η''}} to be displayed below.

Two vectors {{~~mvar~~math|''v''}} and {{~~mvar~~math|''w''}} are said to be [[orthogonal]] if {{math|1=''η''(''v'', ''w'') = 0}}. For a geometric interpretation of orthogonality in the special case, when {{math|''η''(''v'', ''v'') ≤ 0}} and {{math|''η''(''w'', ''w'') ≥ 0}} (or vice versa), see [[hyperbolic orthogonality]].

A vector {{mvar|e}} is called a [[unit vector]] if {{math|1=''η''(''e'', ''e'') = ±1}}. A [[Basis (linear algebra)|basis]] for {{mvar|M}} consisting of mutually orthogonal unit vectors is called an [[orthonormal basis]].<ref>{{cite book |title=The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity |author1=Gregory L. Naber |edition=illustrated |publisher=Courier Corporation |year=2003 |isbn=978-0-486-43235-9 |page=8 |url=https://books.google.com/books?id=pNfRHzwdVZ0C |access-date=2022-12-26 |archive-date=2022-12-26 |archive-url=https://web.archive.org/web/20221226231018/https://books.google.com/books?id=pNfRHzwdVZ0C |url-status=live }} [https://books.google.com/books?id=pNfRHzwdVZ0C&pg=PA8 Extract of page 8] {{Webarchive|url=https://web.archive.org/web/20221226231020/https://books.google.com/books?id=pNfRHzwdVZ0C&pg=PA8 |date=2022-12-26 }}</ref>

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More terminology (but not more structure): The Minkowski metric is a [[pseudo-Riemannian metric]], more specifically, a [[Lorentzian metric]], even more specifically, ''the'' Lorentz metric, reserved for {{math|4}}-dimensional flat spacetime with the remaining ambiguity only being the signature convention.

=== Minkowski metric ===

{{Distinguish|text=[[Minkowski distance]] which is also called Minkowski metric}}

From the [[Postulates of special relativity|second postulate of special relativity]], together with homogeneity of spacetime and isotropy of space, it follows that the [[spacetime interval]] between two arbitrary events called {{math|1}} and {{math|2}} is:<ref>{{cite book |title=Spacetime and Geometry |author1=Sean M. Carroll |edition=illustrated, herdruk |publisher=Cambridge University Press |year=2019 |isbn=978-1-108-48839-6 |page=7 |url=https://books.google.com/books?id=PTGdDwAAQBAJ&pg=PA7}}</ref>

<math display="block">c^2\left(t_1 - t_2\right)^2 - \left(x_1 - x_2\right)^2 - \left(y_1 - y_2\right)^2 - \left(z_1 - z_2\right)^2.</math>

This quantity is not consistently named in the literature. The interval is sometimes referred to as the square root of the interval as defined here.<ref>{{harvnb|Sard|1970|p=71}}</ref><ref>Minkowski, {{harvnb|Landau|Lifshitz|2002|p=4}}</ref>

The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of

provided the transformations are linear. This [[quadratic form]] can be used to define a bilinear form

~~<math display="block">u \cdot v = c^2 t_1 t_2 - x_1 x_2 - y_1 y_2 - z_1 z_2.</math>~~

via the [[polarization identity]]. This bilinear form can in turn be written as

<math display="block">u \cdot v = u^\textsf{T} \, [\eta] \, v,</math>

where {{math|[''η'']}} is a <math>4\times 4</math> matrix associated with {{mvar|η}}. While possibly confusing, it is common practice to denote {{math|[''η'']}} with just {{mvar|η}}. The matrix is read off from the explicit bilinear form as

~~<math display="block">u \cdot v = u^\textsf{T} [\eta] v\,.~~

<math display="block">\eta = \left(\begin{array}{r}

~~</math>~~

1 & 0 & 0 & 0 \\

0 & -1 & 0 & 0 \\

Where {{math|[''η'']}} is a <math>4\times 4</math> matrix associated with {{mvar|η}}. While possibly confusing, it is common practice to denote {{math|[''η'']}} with just {{mvar|η}}. The matrix is read off from the explicit bilinear form as

0 & 0 & -1 & 0 \\

0 & 0 & 0 & -1

~~<math display="block">\eta = \begin{pmatrix}~~

\end{array}\right)\!,</math>

~~-1 & 0 & 0 & 0 \\~~

~~0 & 1 & 0 & 0\\~~

~~0 & 0 & 1 & 0\\~~

~~0 & 0 & 0 & 1~~

~~\end{pmatrix},</math>~~

and the bilinear form

with which this section started by assuming its existence, is now identified.

For definiteness and shorter presentation, the signature {{math|(− + + +)}} is adopted below. This choice (or the other possible choice) has no (known) physical implications. The symmetry group preserving the bilinear form with one choice of signature is isomorphic (under the map given [[Classical group#O(p, q) and O(n) – the orthogonal groups|here]]) with the symmetry group preserving the other choice of signature. This means that both choices are in accord with the two postulates of relativity. Switching between the two conventions is straightforward. If the metric tensor {{~~mvar~~math|''η''}} has been used in a derivation, go back to the earliest point where it was used, substitute {{~~mvar~~math|''η''}} for {{math|−''η''}}, and retrace forward to the desired formula with the desired metric signature.

~~===Standard basis===~~

A standard or orthonormal basis for Minkowski space is a set of four mutually orthogonal vectors {{math|{''e''0, ''e''1, ''e''2, ''e''3}<nowiki/>}} such that

=== Standard basis ===

A standard or orthonormal basis for Minkowski space is a set of four mutually orthogonal vectors {{math|{{mset|''e''0, ''e''1, ''e''2, ''e''3}}}} such that

and for which <math display="block">\eta(e_\mu, e_\nu) = 0</math> when <math display="inline">\mu \neq \nu\,.</math>

These conditions can be written compactly in the form

Line 259 ⟶ 222:

and

Here '''lowering of an index''' with the metric was used.

There are many possible choices of standard basis obeying the condition <math>\eta(e_\mu, e_\nu) = \eta_{\mu \nu}.</math> Any two such bases are related in some sense by a Lorentz transformation, either by a change-of-basis matrix <math>\Lambda^\mu_\nu</math>, a real <{{math>|4~~\times~~ × 4~~</math>~~}} matrix satisfying

<math display="block">\Lambda^\mu_\rho\eta_{\mu \nu}\Lambda^\nu_\sigma = \eta_{\rho \sigma}.</math>

or {{math|Λ}}, a linear map on the abstract vector space satisfying, for any pair of vectors {{math|''u''}}, {{math|''v''}},

~~or <math>\Lambda,</math> a linear map on the abstract vector space satisfying, for any pair of vectors <math>u,v,</math>~~

<math display="block">\eta(\Lambda u, \Lambda v) = \eta(u, v).</math>

Then if two different bases exist, <{{math>\|{~~e_0~~{mset|''e''{{sub|0}},~~e_1~~ ''e''{{sub|1}},~~e_2~~ ''e''{{sub|2}},~~e_3\~~ ''e''{{sub|3}}}}}}~~</math>~~ and <{{math>\|{~~e_0~~{mset|''e''′{{sub|0}},~~e_1~~ ''e''′{{sub|1}},~~e_2~~ ''e''′{{sub|2}},~~e_3~~ '\'e''′{{sub|3}}}}}}~~</math>~~, <math>e_\mu' = e_\nu\Lambda^\nu_\mu</math> can be represented as <math>e_\mu' = e_\nu\Lambda^\nu_\mu</math> or <math>e_\mu' = \Lambda e_\mu</math>. While it might be tempting to think of <math>\Lambda^\mu_\nu</math> and <{{math~~>\Lambda</math>~~|Λ}} as the same thing, mathematically, they are elements of different spaces, and act on the space of standard bases from different sides.

==== Raising and lowering of indices ====

[[File:1-form linear functional.svg|thumb|400px|Linear functionals (1-forms) {{math|'''α'''}}, {{math|'''β'''}} and their sum {{math|'''σ'''}} and vectors {{math|'''u'''}}, {{math|'''v'''}}, {{math|'''w'''}}, in [[three-dimensional space|3d]] [[Euclidean space]]. The number of (1-form) [[hyperplane]]s intersected by a vector equals the [[inner product]].<ref name=":0">{{harvnb|Misner|Thorne|Wheeler|1973}}</ref>]]

Technically, a non-degenerate bilinear form provides a map between a vector space and its dual; in this context, the map is between the tangent spaces of {{mvar|M}} and the [[cotangent space]]s of {{mvar|M}}. At a point in {{mvar|M}}, the tangent and cotangent spaces are [[dual vector space]]s (so the dimension of the cotangent space at an event is also {{math|4}}). Just as an authentic inner product on a vector space with one argument fixed, by [[Riesz representation theorem]], may be expressed as the action of a [[linear functional]] on the vector space, the same holds for the Minkowski inner product of Minkowski space.<ref>{{harvnb|Lee|2003}}. One point in Lee's proof of the existence of this map needs modification (Lee deals with [[Riemannian metric]]s.). Where Lee refers to positive definiteness to show the injectivity of the map, one needs instead appeal to non-degeneracy.</ref>

Thus if {{math|''v''''μ''}} are the components of a vector in tangent space, then {{math|1=''η''''μν'' ''v''''μ'' = ''v''''ν''}} are the components of a vector in the cotangent space (a linear functional). Due to the identification of vectors in tangent spaces with vectors in {{math|''M''}} itself, this is mostly ignored, and vectors with lower indices are referred to as '''covariant vectors'''. In this latter interpretation, the covariant vectors are (almost always implicitly) identified with vectors (linear functionals) in the dual of Minkowski space. The ones with upper indices are '''contravariant vectors'''. In the same fashion, the inverse of the map from tangent to cotangent spaces, explicitly given by the inverse of {{math|''η''}} in matrix representation, can be used to define '''raising of an index'''. The components of this inverse are denoted {{math|''η''''μν''}}. It happens that {{math|1=''η''''μν'' = ''η''''μν''}}. These maps between a vector space and its dual can be denoted {{math|''η''♭}} (eta-flat) and {{math|''η''♯}} (eta-sharp) by the musical analogy.<ref>{{harvnb|Lee|2003|loc=The tangent-cotangent isomorphism p.  282.}}</ref>

Contravariant and covariant vectors are geometrically very different objects. The first can and should be thought of as arrows. A linear function can be characterized by two objects: its [[kernel (linear algebra)|kernel]], which is a [[hyperplane]] passing through the origin, and its norm. Geometrically thus, covariant vectors should be viewed as a set of hyperplanes, with spacing depending on the norm (bigger = smaller spacing), with one of them (the kernel) passing through the origin. The mathematical term for a covariant vector is 1-covector or [[1-form]] (though the latter is usually reserved for covector ''fields'').

Line 291 ⟶ 249:

One may, of course, ignore geometrical views altogether (as is the style in e.g. {{harvtxt|Weinberg|2002}} and {{harvnb|Landau|Lifshitz|2002}}) and proceed algebraically in a purely formal fashion. The time-proven robustness of the formalism itself, sometimes referred to as [[index gymnastics]], ensures that moving vectors around and changing from contravariant to covariant vectors and vice versa (as well as higher order tensors) is mathematically sound. Incorrect expressions tend to reveal themselves quickly.

==== Coordinate free raising and lowering ====

Given a bilinear form <math>\eta:M\times M\rightarrow \~~mathbb~~mathbf{R}</math>, the lowered version of a vector can be thought of as the partial evaluation of <math>\eta</math>, that is, there is an associated partial evaluation map

<math display=block>\eta(\cdot, -):M\rightarrow M^*; v \mapsto \eta(v,\cdot).</math>

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Non-degeneracy is then equivalent to injectivity of the partial evaluation map, or equivalently non-degeneracy indicates that the kernel of the map is trivial. In finite dimension, as is the case here, and noting that the dimension of a finite-dimensional space is equal to the dimension of the dual, this is enough to conclude the partial evaluation map is a linear isomorphism from <math>M</math> to <math>M^*</math>. This then allows the definition of the inverse partial evaluation map,

<math display=block>\eta^{-1}:M^*\rightarrow M,</math>

which allows the inverse metric to be defined as

<math display="block">\eta^{-1}:M^*\times M^* \rightarrow \mathbf{R}, \eta^{-1}(\alpha,\beta) = \eta(\eta^{-1}(\alpha),\eta^{-1}(\beta))</math>

where the two different usages of <math>\eta^{-1}</math> can be told apart by the argument each is evaluated on. This can then be used to raise indices. If a coordinate basis is used, the metric {{math|''η''{{sup|−1}}}} is indeed the matrix inverse to {{math|''η''}}.

==== Formalism of the Minkowski metric ====

~~<math display="block">\eta^{-1}:M^*\times M^* \rightarrow \mathbb{R}, \eta^{-1}(\alpha,\beta) = \eta(\eta^{-1}(\alpha),\eta^{-1}(\beta))</math>~~

where the two different usages of <math>\eta^{-1}</math> can be told apart by the argument each is evaluated on. This can then be used to raise indices. If a coordinate basis is used, the metric <math>\eta^{-1}</math> is indeed the matrix inverse to <math>\eta.</math>

~~====The formalism of the Minkowski metric====~~

The present purpose is to show semi-rigorously how ''formally'' one may apply the Minkowski metric to two vectors and obtain a real number, i.e. to display the role of the differentials and how they disappear in a calculation. The setting is that of smooth manifold theory, and concepts such as convector fields and exterior derivatives are introduced.

{{~~Hidden~~hidden begin

|titlestyle=color:green;background:lightgrey;

|title=A formal approach to the Minkowski metric

}}

A full-blown version of the Minkowski metric in coordinates as a tensor field on spacetime has the appearance

<math display="block">\eta_{\mu\nu} dx^\mu \otimes dx^\nu = \eta_{\mu\nu} dx^\mu \odot dx^\nu = \eta_{\mu\nu} dx^\mu dx^\nu.</math>

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''Tangent'' vectors are, in this formalism, given in terms of a basis of differential operators of the first order,

<math display="block">\left.\frac{\partial}{\partial x^\mu}\right|_p,</math>

where {{mvar|p}} is an event. This operator applied to a function {{mvar|f}} gives the [[directional derivative]] of {{mvar|f}} at {{mvar|p}} in the direction of increasing {{math|''x''''μ''}} with {{math|''x''''ν'', ''ν'' ≠ ''μ''}} fixed. They provide a basis for the tangent space at {{mvar|p}}.

The exterior derivative {{math|''df''}} of a function {{mvar|f}} is a '''covector field''', i.e. an assignment of a cotangent vector to each point {{math|p}}, by definition such that

for each [[vector field]] {{mvar|X}}. A vector field is an assignment of a tangent vector to each point {{math|p}}. In coordinates {{mvar|X}} can be expanded at each point {{mvar|p}} in the basis given by the {{math|∂/∂''x''''ν''{{!}}''p''}}. Applying this with {{math|1=''f'' = ''x''''μ''}}, the coordinate function itself, and {{math|1=''X'' = ∂/∂''x''''ν''}}, called a ''coordinate vector field'', one obtains

<math display="block">dx^\mu\left(\frac{\partial}{\partial x^\nu}\right) = \frac{\partial x^\mu}{\partial x^\nu} = \delta_\nu^\mu.</math>

Since this relation holds at each point {{mvar|p}}, the {{math|''dx''''μ''{{!}}''p''}} provide a basis for the cotangent space at each {{mvar|p}} and the bases {{math|''dx''''μ''{{!}}''p''}} and {{math|∂/∂''x''''ν''{{!}}''p''}} are [[dual basis|dual]] to each other,

<math display="block">\left. dx^\mu \right|_p \left(\left.\frac{\partial}{\partial x^\nu}\right|_p\right) = \delta^\mu_\nu.</math>

at each {{mvar|p}}. Furthermore, one has

<math display="block">\alpha \otimes \beta(a, b) = \alpha(a)\beta(b)</math>

for general one-forms on a tangent space {{math|''α'', ''β''}} and general tangent vectors {{math|''a'', ''b''}}. (This can be taken as a definition, but may also be proved in a more general setting.)

Thus when the metric tensor is fed two vectors fields {{math|''a''}}, {{math|''b''}}, both expanded in terms of the basis coordinate vector fields, the result is

<math display="block">\eta_{\mu\nu} dx^\mu \otimes dx^\nu(a, b) = \eta_{\mu\nu} a^\mu b^\nu,</math>

where {{math|''a''''μ''}}, {{math|''b''''ν''}} are the ''component functions'' of the vector fields. The above equation holds at each point {{mvar|p}}, and the relation may as well be interpreted as the Minkowski metric at {{mvar|p}} applied to two tangent vectors at {{mvar|p}}.

Line 353 ⟶ 295:

This situation changes in [[general relativity]]. There one has

<math display="block">g(p)_{\mu\nu} \left. dx^\mu \right|_p \left. dx^\nu \right|_p(a, b) = g(p)_{\mu\nu} a^\mu b^\nu,</math>

where now {{math|''η'' → ''g''(''p'')}}, i.e., {{mvar|g}} is still a metric tensor but now depending on spacetime and is a solution of [[Einstein's field equation]]s. Moreover, {{math|''a'', ''b''}} ''must'' be tangent vectors at spacetime point {{mvar|p}} and can no longer be moved around freely.

=== Chronological and causality relations ===

Let {{math|''x'', ''y'' ∈ ''M''}}. Here,

# {{math|''x''}} '''chronologically precedes''' {{math|''y''}} if {{math|''y'' − ''x''}} is future-directed timelike. This relation has the [[transitive property]] and so can be written {{math|''x'' < ''y''}}.

# {{math|''x''}} '''causally precedes''' {{math|''y''}} if {{math|''y'' − ''x''}} is future-directed null or future-directed timelike. It gives a [[partial ordering]] of spacetime and so can be written {{math|''x'' ≤ ''y''}}.

Suppose {{math|''x'' ∈ ''M''}} is timelike. Then the '''simultaneous hyperplane''' for {{math|''x''}} is <{{math>\|{{mset|1=''y'' : ~~\eta~~''η''(''x'', ''y'') = 0\}}}}.~~</math>~~ Since this [[hyperplane]] varies as {{math|''x''}} varies, there is a [[relativity of simultaneity]] in Minkowski space.

== Generalizations ==

A Lorentzian manifold is a generalization of Minkowski space in two ways. The total number of spacetime dimensions is not restricted to be {{math|4}} ({{math|2}} or more) and a Lorentzian manifold need not be flat, i.e. it allows for curvature.

=== Complexified Minkowski space ===

Complexified Minkowski space is defined as {{math|1=''M''c'' = ''M'' ⊕ ''iM'' }}.<ref>Y. Friedman, A Physically Meaningful Relativistic Description of the Spin State of an Electron, Symmetry 2021, 13(10), 1853; https://doi.org/10.3390/sym13101853 {{Webarchive|url=https://web.archive.org/web/20230813114023/https://www.mdpi.com/2073-8994/13/10/1853 |date=2023-08-13 }}</ref> Its real part is the Minkowski space of [[four-vectors]], such as the [[four-velocity]] and the [[four-momentum]], which are independent of the choice of [[orientation (geometry)|orientation]] of the space. The imaginary part, on the other hand, may consist of four pseudovectors, such as [[angular velocity]] and [[magnetic moment]], which change their direction with a change of orientation. A [[pseudoscalar]] {{math|''i''}} is introduced, which also changes sign with a change of orientation. Thus, elements of {{math|''M''c''}} are independent of the choice of the orientation.

The [[inner product]]-like structure on {{math|''M''c''}} is defined as {{math|1=''u'' ⋅ ''v'' = ''η''(''u'', ''v'')}} for any {{math|''u'',''v'' ∈ ''M''c''}}. A relativistic pure [[Spin (physics)|spin]] of an [[electron]] or any half spin particle is described by {{math|''ρ'' ∈ '' M''c''}} as {{math|1=''ρ'' = ''u'' + ''is''}}, where {{math|''u''}} is the four-velocity of the particle, satisfying {{math|1=''u''2 = 1}} and {{mvar|s}} is the 4D spin vector,<ref>Jackson, J.D., Classical Electrodynamics, 3rd ed.; John Wiley \& Sons: Hoboken, NJ, US, 1998</ref> which is also the [[Pauli–Lubanski pseudovector]] satisfying {{math|1=''s''2 = ~~−''1''~~−1}} and {{math|1=''u'' ⋅ ''s'' = ''0''}}.

=== Generalized Minkowski space ===

Minkowski space refers to a mathematical formulation in four dimensions. However, the mathematics can easily be extended or simplified to create an analogous generalized Minkowski space in any number of dimensions. If {{math|''n'' ≥ 2}}, {{math|''n''}}-dimensional Minkowski space is a vector space of real dimension {{math|''n''}} on which there is a constant Minkowski metric of signature {{math|(''n'' − 1, 1)}} or {{math|(1, ''n'' − 1)}}. These generalizations are used in theories where spacetime is assumed to have more or less than {{math|4}} dimensions. [[String theory]] and [[M-theory]] are two examples where {{math|''n'' > 4}}. In string theory, there appears [[conformal field theory|conformal field theories]] with {{math|1 + 1}} spacetime dimensions.

[[de Sitter space]] can be formulated as a submanifold of generalized Minkowski space as can the model spaces of [[hyperbolic geometry]] (see below).

=== Curvature ===

As a '''flat spacetime''', the three spatial components of Minkowski spacetime always obey the [[Pythagorean Theorem]]. Minkowski space is a suitable basis for special relativity, a good description of physical systems over finite distances in systems without significant [[gravitation]]. However, in order to take gravity into account, physicists use the theory of [[general relativity]], which is formulated in the mathematics of a [[non-Euclidean geometry]]. When this geometry is used as a model of physical space, it is known as ''[[curved space]]''.

Even in curved space, Minkowski space is still a good description in an [[Local reference frame|infinitesimal region]] surrounding any point (barring gravitational singularities).<ref group=nb>This similarity between [[flat space]] and curved space at infinitesimally small distance scales is foundational to the definition of a [[manifold]] in general.</ref> More abstractly, it can be said that in the presence of gravity spacetime is described by a curved 4-dimensional [[manifold]] for which the [[tangent space]] to any point is a 4-dimensional Minkowski space. Thus, the structure of Minkowski space is still essential in the description of general relativity.

== Geometry ==

The meaning of the term ''geometry'' for the Minkowski space depends heavily on the context. Minkowski space is not endowed with Euclidean geometry, and not with any of the generalized Riemannian geometries with intrinsic curvature, those exposed by the ''model spaces'' in [[hyperbolic geometry]] (negative curvature) and the geometry modeled by the [[sphere]] (positive curvature). The reason is the indefiniteness of the Minkowski metric. Minkowski space is, in particular, not a [[metric space]] and not a Riemannian manifold with a Riemannian metric. However, Minkowski space contains [[submanifold]]s endowed with a Riemannian metric yielding hyperbolic geometry.

Model spaces of hyperbolic geometry of low dimension, say ~~{{math|~~2}} or ~~{{math|~~3}}, ''cannot'' be isometrically embedded in Euclidean space with one more dimension, i.e. {{<math~~|ℝ<sup~~>\mathbf{R}^3</~~sup~~math>}} or {{<math~~|ℝ<sup~~>\mathbf{R}^4</~~sup~~math>}} respectively, with the Euclidean metric {{<math|>\overline{~~{overline|''~~g~~''}}}~~}</math>, ~~disallowing~~preventing easy visualization.<ref group=nb>There ''is'' an isometric embedding into {{math|ℝ''n''}} according to the [[Nash embedding theorem]] ({{harvtxt|Nash|1956}}), but the embedding dimension is much higher, {{math|''n'' {{=}} (''m''/2)(''m'' + 1)(3''m'' + 11)}} for a Riemannian manifold of dimension {{mvar|m}}.</ref><ref>{{harvnb|Lee|1997|p=66}}</ref> By comparison, model spaces with positive curvature are just spheres in Euclidean space of one higher dimension.<ref>{{harvnb|Lee|1997|p=33}}</ref> Hyperbolic spaces ''can'' be isometrically embedded in spaces of one more dimension when the embedding space is endowed with the Minkowski metric {{<math~~|''η''}}~~>\eta</math>.

~~Define {{math|'''H'''{{su|p=1(''n'')|b=''R''}} ⊂ '''M'''''n''+1}} to be the upper sheet ({{math|''ct'' > 0}}) of the [[hyperboloid]]~~

Define <math>\mathbf{H}^{1(n)}_R \subset \mathbf{M}^{n+1}</math>to be the upper sheet (<math>ct > 0</math>) of the [[hyperboloid]]

<math display="block">\mathbf H_R^{1(n)} = \left\{\left(ct, x^1, \ldots, x^n\right) \in \mathbf M^n: c^2 t^2 - \left(x^1\right)^2 - \cdots - \left(x^n\right)^2 = R^2, ct > 0\right\}</math>

in generalized Minkowski space <math>\mathbf{M}^{n+1}</math> of spacetime dimension <math>n + 1.</math> This is one of the [[Lorentz group#Restricted Lorentz group|surfaces of transitivity]] of the generalized Lorentz group. The [[induced metric]] on this submanifold,

in generalized Minkowski space {{math|'''M'''''n''+1}} of spacetime dimension {{math|''n'' + 1}}. This is one of the [[Lorentz group#Restricted Lorentz group|surfaces of transitivity]] of the generalized Lorentz group. The [[induced metric]] on this submanifold,

the [[Pullback (differential geometry)|pullback]] of the Minkowski metric <math>\eta</math> under inclusion, is a [[Riemannian metric]]. With this metric <math>\mathbf{H}^{1(n)}_R</math> is a [[Riemannian manifold]]. It is one of the model spaces of Riemannian geometry, the [[hyperboloid model]] of [[hyperbolic space]]. It is a space of constant negative curvature <math>-1/R^2</math>.<ref>{{harvnb|Lee|1997}}</ref> The 1 in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the {{math|''n''}} for its dimension. A <math>2(2)</math> corresponds to the [[Poincaré disk model]], while <math>3(n)</math> corresponds to the [[Poincaré half-plane model|Poincaré half-space model]] of dimension <math>n.</math>

=== Preliminaries ===

the [[Pullback (differential geometry)|pullback]] of the Minkowski metric {{math|''η''}} under inclusion, is a [[Riemannian metric]]. With this metric {{math|'''H'''{{su|p=1(''n'')|b=''R''}}}} is a [[Riemannian manifold]]. It is one of the model spaces of Riemannian geometry, the [[hyperboloid model]] of [[hyperbolic space]]. It is a space of constant negative curvature {{math|−1/''R''2}}.<ref>{{harvnb|Lee|1997}}</ref> The {{math|1}} in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the {{math|''n''}} for its dimension. A {{math|2(2)}} corresponds to the [[Poincaré disk model]], while {{math|3(''n'')}} corresponds to the [[Poincaré half-plane model|Poincaré half-space model]] of dimension {{math|''n''}}.

In the definition above <math>\iota: \mathbf{H}^{1(n)}_R \rightarrow \mathbf{M}^{n+1}</math> is the [[inclusion map]] and the superscript star denotes the [[Pullback (differential geometry)|pullback]]. The present purpose is to describe this and similar operations as a preparation for the actual demonstration that <math>\mathbf{H}^{1(n)}_R</math> actually is a hyperbolic space.

~~===Preliminaries===~~

In the definition above {{math|''ι'': '''H'''{{su|p=1(''n'')|b=''R''}} → '''M'''''n''+1}} is the [[inclusion map]] and the superscript star denotes the [[Pullback (differential geometry)|pullback]]. The present purpose is to describe this and similar operations as a preparation for the actual demonstration that {{math|'''H'''{{su|p=1(''n'')|b=''R''}}}} actually is a hyperbolic space.

{| class="wikitable collapsible collapsed"

Line 411 ⟶ 346:

'''Behavior of tensors under inclusion:'''

For inclusion maps from a submanifold {{mvar|S}} into {{mvar|M}} and a covariant tensor {{mvar|α}} of order {{mvar|k}} on {{mvar|M}} it holds that

<math display=block>\iota^*\alpha\left(X_1,\, X_2,\, \ldots,\, X_k\right) = \alpha\left(\iota_* X_1,\, \iota_*X_2,\, \ldots,\, \iota_* X_k\right) = \alpha\left(X_1,\, X_2,\, \ldots,\, X_k\right),</math>

where {{math|''X''1, ''X''1, …, ''X''k}} are vector fields on {{mvar|S}}. The subscript star denotes the pushforward (to be introduced later), and it is in this special case simply the identity map (as is the inclusion map). The latter equality holds because a tangent space to a submanifold at a point is in a canonical way a subspace of the tangent space of the manifold itself at the point in question. One may simply write

<math display=block>\iota^*\alpha = \alpha|_S,</math>

meaning (with slight [[abuse of notation]]) the restriction of {{mvar|α}} to accept as input vectors tangent to some {{math|''s'' ∈ ''S''}} only.

'''Pullback of tensors under general maps:'''

The pullback of a covariant {{mvar|k}}-tensor {{mvar|α}} (one taking only contravariant vectors as arguments) under a map {{math|''F'': ''M'' → ''N''}} is a linear map

<math display=block>F^*\colon T_{F(p)}^k N \rightarrow T_p^k M,</math>

where for any vector space {{mvar|V}},

<math display=block>T^k V = \underbrace{V^* \otimes V^* \otimes \cdots \otimes V^*}_{k\text{ times}}.</math>

It is defined by

<math display=block>F^*(\alpha)\left(X_1,\, X_2,\, \ldots,\, X_k\right) = \alpha\left(F_* X_1,\, F_*X_2,\, \ldots,\, F_* X_k\right),</math>

where the subscript star denotes the [[pushforward (differential)|pushforward]] of the map {{mvar|F}}, and {{math|''X''1, ''X''2, …, X''k''}} are vectors in {{math|''T''''p''''M''}}. (This is in accord with what was detailed about the pullback of the inclusion map. In the general case here, one cannot proceed as simply because {{math|''F''∗''X''1 ≠ ''X''1}} in general.)

Line 439 ⟶ 365:

Further unwinding the definitions, the pushforward {{math|F∗: ''TM''''p'' → ''TN''''F''(''p'')}} of a vector field under a map {{math|''F'': ''M'' → ''N''}} between manifolds is defined by

where {{mvar|f}} is a function on {{mvar|N}}. When {{math|''M'' {{=}} '''R'''''m'', ''N''{{=}} '''R'''''n''}} the pushforward of {{mvar|F}} reduces to {{math|''DF'': '''R'''''m'' → '''R'''''n''}}, the ordinary [[Total derivative#The total derivative as a linear map|differential]], which is given by the [[Jacobian matrix]] of partial derivatives of the component functions. The differential is the best linear approximation of a function {{mvar|F}} from {{math|'''R'''''m''}} to {{math|'''R'''''n''}}. The pushforward is the smooth manifold version of this. It acts between tangent spaces, and is in coordinates represented by the Jacobian matrix of the ''coordinate representation'' of the function.

where {{mvar|f}} is a function on {{mvar|N}}. When {{math|''M'' {{=}} ℝ''m'', ''N''{{=}} ℝ''n''}} the pushforward of {{mvar|F}} reduces to {{math|''DF'': ℝ''m'' → ℝ''n''}}, the ordinary [[Total derivative#The total derivative as a linear map|differential]], which is given by the [[Jacobian matrix]] of partial derivatives of the component functions. The differential is the best linear approximation of a function {{mvar|F}} from {{math|ℝ''m''}} to {{math|ℝ''n''}}. The pushforward is the smooth manifold version of this. It acts between tangent spaces, and is in coordinates represented by the Jacobian matrix of the ''coordinate representation'' of the function.

The corresponding pullback is the [[Transpose of a linear map|dual map]] from the dual of the range tangent space to the dual of the domain tangent space, i.e. it is a linear map,

<math display=block>F^*\colon T^*_{F(p)}N \rightarrow T^*_p M.</math>

=== Hyperbolic stereographic projection ===

[[File:HyperboloidProjection.png|thumb|right|Red circular arc is geodesic in [[Poincaré disk model]]; it projects to the brown geodesic on the green hyperboloid.]]

In order to exhibit the metric, it is necessary to pull it back via a suitable ''parametrization''. A parametrization of a submanifold {{mvar|S}} of a manifold {{mvar|M}} is a map {{math|''U'' ⊂ '''R'''''m'' → ''M''}} whose range is an open subset of {{math|''S''}}. If {{mvar|S}} has the same dimension as {{math|''M''}}, a parametrization is just the inverse of a coordinate map {{math|''φ'': ''M'' → ''U'' ⊂ '''R'''''m''}}. The parametrization to be used is the inverse of ''hyperbolic stereographic projection''. This is illustrated in the figure to the right for {{math|1=''n'' = 2}}. It is instructive to compare to [[stereographic projection]] for spheres.

Stereographic projection {{math|''σ'': '''H'''{{su|p=''n''|b=''R''}} → '''R'''''n''}} and its inverse {{math|''σ''−1: '''R'''''n'' → '''H'''{{su|p=''n''|b=''R''}}}} are given by

<math display="block">\begin{align}

\sigma(\tau, \mathbf x) = \mathbf u &= \frac{R\mathbf x}{R + \tau},\\

\sigma^{-1}(\mathbf u) = (\tau, \mathbf x) &= \left(R\frac{R^2 + |u|^2}{R^2 - |u|^2}, \frac{2R^2\mathbf u}{R^2 - |u|^2}\right),

\end{align}</math>

where, for simplicity, {{math|''τ'' ≡ ''ct''}}. The {{math|(''τ'', '''x''')}} are coordinates on {{math|'''M'''''n''+1}} and the {{math|'''u'''}} are coordinates on {{math|'''R'''n}}.

Line 466 ⟶ 387:

| title = Detailed derivation

| proof = Let

<math display="block">\mathbf H_R^n = \left\{\left(\tau, x^1, \ldots, x^n\right) \subset \mathbf M: -\tau^2 + \left(x^1\right)^2 + \cdots + \left(x^n\right)^2 = -R^2, \tau > 0\right\}</math>

and let

<math display=block>S = (-R, 0, \ldots, 0) .</math>

~~<math display=block>S = (-R, 0, \ldots, 0)</math>~~

<math display=block>P = \left(\tau, x^1, \ldots, x^n\right) \in \mathbf H_R^n,</math>

then it is geometrically clear that the vector

<math display=block>\overrightarrow{PS}</math>

intersects the hyperplane

<math display=block>\left\{\left(\tau, x^1, \ldots, x^n\right) \in M: \tau = 0\right\}</math>

once in point denoted

<math display="block">U = \left(0, u^1(P), \ldots, u^n(P)\right) \equiv (0, \mathbf u).</math>

One has

<math display="block">\begin{align}

S + \overrightarrow{SU} &= U \Rightarrow \overrightarrow{SU} = U - S,\\

S + \overrightarrow{SP} &= P \Rightarrow \overrightarrow{SP} = P - S

\end{align}.</math>

<math display="block">\begin{align}

\overrightarrow{SU} &= (0, \mathbf u) - (-R,\mathbf 0) = (R, \mathbf u),\\

\overrightarrow{SP} &= (\tau, \mathbf x) - (-R,\mathbf 0) = (\tau + R, \mathbf x).

\end{align}.</math>

By construction of stereographic projection one has

<math display=block>\overrightarrow{SU} = \lambda(\tau)\overrightarrow{SP}.</math>

This leads to the system of equations

<math display="block">\begin{align}

R &= \lambda(\tau + R),\\

Line 513 ⟶ 420:

\end{align}</math>

The first of these is solved for <{{math~~>\lambda</math>~~|''λ''}} and one obtains for stereographic projection

<math display=block>\sigma(\tau, \mathbf x) = \mathbf u = \frac{R\mathbf x}{R + \tau}.</math>

Next, the inverse ~~<math>\sigma^~~{-{math|1=''σ''{{sup|−1}}(''u'') = (~~\tau~~''τ'', ~~\mathbf~~ '''x''')~~</math>~~}} must be calculated. Use the same considerations as before, but now with

<math display=block>\begin{align}

U &= (0, \mathbf u)\\

P &= (\tau(\mathbf u), \mathbf x(\mathbf u)).

\end{align} ,</math>

one gets

~~One gets~~

<math display=block>\begin{align}

\tau &= \frac{R(1 - \lambda)}{\lambda},\\

\mathbf x &= \frac{\mathbf u}{\lambda},

\end{align}</math>

but now with {{math|''λ''}} depending on {{math|'''u'''}}. The condition for {{math|''P''}} lying in the hyperboloid is

~~but now with <math>\lambda</math> depending on <math>\mathbf u.</math> The condition for {{mvar|P}} lying in the hyperboloid is~~

<math display=block>-\tau^2 + |\mathbf x|^2 = -R^2,</math>

<math display=block>-\frac{R^2(1 - \lambda)^2}{\lambda^2}+\frac{|\mathbf u|^2}{\lambda^2}=-R^2,</math>

leading to

<math display=block>\lambda = \frac{R^2 - |u|^2}{2R^2}.</math>

With this <{{math~~>\lambda</math>~~|''λ''}}, one obtains

<math display="block">\sigma^{-1}(\mathbf u) = (\tau, \mathbf x) = \left(R\frac{R^2 + |u|^2}{R^2 - |u|^2}, \frac{2R^2\mathbf u}{R^2 - |u|^2}\right).</math>

}}

=== Pulling back the metric ===

One has

<math display="block">h_R^{1(n)} = \eta|_{\mathbf H_R^{1(n)}} = \left(dx^1\right)^2 + \cdots + \left(dx^n\right)^2 - d\tau^2</math>

and the map

\sigma^{-1}:\~~mathbb~~mathbf{R}^n \rightarrow \mathbf{H}_R^{1(n)};\quad

\sigma^{-1}(\mathbf{u}) = (\tau(\mathbf{u}),\, \mathbf{x}(\mathbf{u})) = \left(R\frac{R^2 + |u|^2}{R^2 - |u|^2},\, \frac{2R^2\mathbf{u}}{R^2 - |u|^2}\right).</math>

The pulled back metric can be obtained by straightforward methods of calculus;

\left.\left(\sigma^{-1}\right)^* \eta\right|_{\mathbf H_R^{1(n)}} =

Line 567 ⟶ 459:

One computes according to the standard rules for computing differentials (though one is really computing the rigorously defined exterior derivatives),

<math display="block">\begin{align}

dx^1(\mathbf u) &= d\left(\frac{2R^2 u^1}{R^2 - |u|^2}\right)

Line 577 ⟶ 468:

d\tau(\mathbf u) &= d\left(R\frac{R^2 + |u|^2}{R^2 - |u|^2}\right) = \cdots,

\end{align}</math>

and substitutes the results into the right hand side. This yields

\left(\sigma^{-1}\right)^* h_R^{1(n)} =

Line 591 ⟶ 480:

One has

<math display=block>\begin{align}

\frac{\partial}{\partial u^1}\frac{2R^2 u^1}{R^2 - |u|^2}du^1 &= \frac{2\left(R^2 -|u|^2\right) + 4R^2 \left(u^1\right)^2}{\left(R^2 - |u|^2\right)^2}du^1, \\

\frac{\partial}{\partial u^2}\frac{2R^2 u^1}{R^2 - |u|^2}du^2 &= \frac{4R^2 u^1 u^2}{\left(R^2 - |u|^2\right)^2}du^2,

\end{align}</math>

and

<math display=block>\frac{\partial}{\partial \tau}\frac{2R^2 u^1}{R^2 - |u|^2}d\tau^2 = 0.</math>

With this one may write

<math display=block>dx^1(\mathbf u) = \frac{2R^2 \left(R^2 - |u|^2\right)du^1 + 4R^2 u^1(\mathbf u \cdot d\mathbf u)}{\left(R^2 - |u|^2\right)^2},</math>

from which

\left(dx^1(\mathbf{u})\right)^2 =

Line 616 ⟶ 499:

Summing this formula one obtains

<math display=block>\begin{align}

&\left(dx^1(\mathbf u)\right)^2 + \cdots + \left(dx^n(\mathbf u)\right)^2 \\

Line 628 ⟶ 510:

Similarly, for {{mvar|τ}} one gets

d\tau = \sum_{i=1}^n \frac{\partial}{\partial u^i} R\frac{R^2 + |u|^2}{R^2 + |u|^2}du^i + \frac{\partial}{\partial\tau}R\frac{R^2 + |u|^2}{R^2 + |u|^2}d\tau

= \sum_{i=1}^n R^4\frac{4R^2 u^idu^i}{\left(R^2 - |u|^2\right)},

</math>

yielding

-d\tau^2 = -\left(R\frac{4R^4\left(\mathbf u \cdot d\mathbf u\right)}{\left(R^2 - |u|^2\right)^2}\right)^2

Line 642 ⟶ 521:

Now add this contribution to finally get

<math display=block>\left(\sigma^{-1}\right)^* h_R^{1(n)} = \frac{4R^2\left[\left(du^1\right)^2 + \cdots + \left(du^n\right)^2\right]}{\left(R^2 - |u|^2\right)^2} \equiv h_R^{2(n)}.</math>

Line 652 ⟶ 530:

The pullback can be computed in a different fashion. By definition,

\left(\sigma^{-1}\right)^* h_R^{1(n)}(V,\, V) =

Line 660 ⟶ 537:

In coordinates,

\left(\sigma^{-1}\right)_* V

Line 672 ⟶ 548:

One has from the formula for {{math|''σ''–1}}

<math display=block>\begin{align}

Vx^j &= V^i\frac{\partial}{\partial u^i}\left(\frac{2R^2 u^j}{R^2 - |u|^2}\right)

Line 682 ⟶ 557:

Lastly,

\eta\left(\sigma_*^{-1}V,\, \sigma_*^{-1}V\right) =

Line 689 ⟶ 563:

h_R^{2(n)}(V,z, V),

</math>

and the same conclusion is reached.

== See also ==

* [[Hyperbolic quaternion]]

* [[Hyperspace]]

Line 699 ⟶ 572:

* [[Minkowski plane]]

== Remarks ==

== Notes ==

== References ==

*{{cite journal|last=Corry|first=L.|title=Hermann Minkowski and the postulate of relativity|journal=Arch. Hist. Exact Sci.|volume=51|issue=4|year=1997|doi=10.1007/BF00518231|pages=273–314|s2cid=27016039|issn=0003-9519}}

* {{cite journal|last=Corry|first=L.|title=Hermann Minkowski and the postulate of relativity|journal=Arch. Hist. Exact Sci.|volume=51|issue=4|year=1997|doi=10.1007/BF00518231|pages=273–314|s2cid=27016039|issn=0003-9519}}

*{{cite book|last1=Catoni|first1=F.|title=The Mathematics of Minkowski Space-Time |year=2008|series=Frontiers in Mathematics|publisher=[[Birkhäuser Verlag]]|issn=1660-8046|location=Basel|isbn=978-3-7643-8613-9|doi=10.1007/978-3-7643-8614-6|display-authors=etal}}

* {{cite book|~~last~~last1=~~Galison~~Catoni|~~first~~first1=~~P. L~~F.|title=~~Minkowski's~~The ~~Space–Time:~~Mathematics ~~from~~of ~~visual~~Minkowski ~~thinking~~Space-Time ~~to the absolute world~~|year=2008|series=~~Historical Studies~~Frontiers in ~~the Physical Sciences|volume=10|doi=10.2307/27757388|editor=R McCormach~~Mathematics|publisher=[[~~Johns Hopkins University~~Birkhäuser ~~Press~~Verlag]]|~~year~~issn=~~1979~~1660-8046|~~pages~~location=~~85–121~~Basel|~~jstor~~isbn=~~27757388~~978-3-7643-8613-9|doi=10.1007/978-3-7643-8614-6|display-~~editors~~authors=etal}}

* {{cite book|last=Galison|first=P. L.|title=Minkowski's Space–Time: from visual thinking to the absolute world|series=Historical Studies in the Physical Sciences|volume=10|doi=10.2307/27757388|editor=R McCormach|publisher=[[Johns Hopkins University Press]]|year=1979|pages=85–121|jstor=27757388|display-editors=etal}}

* Giulini D The rich structure of Minkowski space, https://arxiv.org/abs/0802.4345v1.

* {{cite book|first1=D.|last1=Kleppner|author-link1=Daniel Kleppner|first2=R. J.|last2=Kolenkow|author-link2=Robert J. Kolenkow|title=An Introduction to Mechanics|year=1978|orig-year=1973|isbn=978-0-07-035048-9|publisher=[[McGraw-Hill]]|location=London|url=https://archive.org/details/introductiontome00dani}}

* {{cite book|last1=Landau|first1=L.D.|author-link1=Lev Landau|last2=Lifshitz|first2=E.M.|author-link2=Evgeny Lifshitz|title=The Classical Theory of Fields|series=Course of Theoretical Physics|volume=2|edition=4th|publisher=[[Butterworth–Heinemann]]|isbn=0-7506-2768-9|year=2002|orig-year=1939}}

* {{cite book|last=Lee|first= J. M.|title=Introduction to Smooth manifolds|year=2003|series=Springer Graduate Texts in Mathematics|isbn=978-0-387-95448-6|volume=218}}

* {{cite book|last=Lee|first= J. M.|title=Introduction to Smooth manifolds|year=2012|series=Springer Graduate Texts in Mathematics|isbn=978-1-4419-9981-8}}

* {{cite book|last=Lee|first= J. M.|title=Riemannian Manifolds – An Introduction to Curvature|year=1997|publisher=Springer Verlag|location=New York · Berlin · Heidelberg|series=Springer Graduate Texts in Mathematics|volume=176|isbn=978-0-387-98322-6}}

* {{~~Citation~~citation|last=Minkowski|first=Hermann|author-link=Hermann Minkowski|year=1907–1908|title=Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern|trans-title=The Fundamental Equations for Electromagnetic Processes in Moving Bodies|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=53–111|title-link=s:de:Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern}}

** Published translation: {{Cite journal|author-last=Carus|author-first=Edward H.|title=Space and Time|journal=[[The Monist]]|year=1918|volume=28|issue=288|pages=288–302|doi=10.5840/monist19182826 |url=https://archive.org/details/monistquart28hegeuoft/page/288/mode/2up?view=theater}}

** Wikisource translation: [[s:Translation:The Fundamental Equations for Electromagnetic Processes in Moving Bodies|The Fundamental Equations for Electromagnetic Processes in Moving Bodies]].

* {{~~Citation~~citation|last=Minkowski|first=Hermann|year=1908–1909|title=Raum und Zeit|trans-title=Space and Time|journal=Physikalische Zeitschrift|volume=10|pages=75–88|title-link=s:de:Raum und Zeit (Minkowski)}} Various English translations on Wikisource: [[s:Translation:Space and Time|Space and Time]].

* {{~~Citation~~citation|first1=Charles W.|last1=Misner|author-link=Charles W. Misner|first2=Kip. S.|last2=Thorne|author2-link=Kip Thorne|first3=John A.|last3=Wheeler|author3-link=John A. Wheeler|title=Gravitation|publisher= W. H. Freeman|date=1973|isbn=978-0-7167-0344-0|title-link=Gravitation (book)}}.

* {{~~Cite~~cite book|last=Naber|first=G. L.|title=The Geometry of Minkowski Spacetime|year=1992|publisher=[[Springer-Verlag]]|location=New York|isbn=978-0-387-97848-2}}

* {{cite journal|first=J.|last=Nash|author-link=John Forbes Nash|title=The Imbedding Problem for Riemannian Manifolds|journal=[[Annals of Mathematics]]|year=1956|volume=63|issue=1|pages=20–63|doi=10.2307/1969989|mr=0075639|jstor=1969989}}

* {{cite book|last=Penrose|first=Roger|author-link=Roger Penrose|year=2005|title=Road to Reality : A Complete Guide to the Laws of the Universe|chapter=18 Minkowskian geometry|publisher=[[Alfred A. Knopf]]|isbn=9780679454434|url=https://archive.org/details/roadtorealitycom00penr_0}}

* {{~~Citation~~citation|last=Poincaré|first=Henri|author-link=Henri Poincaré|year=1905–1906|title=Sur la dynamique de l'électron|trans-title=On the Dynamics of the Electron|journal=Rendiconti del Circolo Matematico di Palermo|volume=21|pages=129–176|doi=10.1007/BF03013466|bibcode=1906RCMP...21..129P |hdl=2027/uiug.30112063899089|s2cid=120211823|url=https://zenodo.org/record/1428444}} Wikisource translation: [[s:Translation:On the Dynamics of the Electron (July)|On the Dynamics of the Electron.]]

* Robb A A: Optical Geometry of Motion; a New View of the Theory of Relativity Cambridge 1911, (Heffers). http://www.archive.org/details/opticalgeometryoOOrobbrich.

* Robb A A: Geometry of Time and Space, 1936 Cambridge Univ Press http://www.archive.org/details/geometryoftimean032218mbp.

* {{cite book|last=Sard|first=R. D.|title=Relativistic Mechanics - Special Relativity and Classical Particle Dynamics|url=https://archive.org/details/relativisticmech0000sard|url-access=registration|year=1970|publisher=W. A. Benjamin|location=New York|isbn=978-0805384918}}

* {{cite book|last=Shaw|first=R.|year=1982|title=Linear Algebra and Group Representations|url=https://archive.org/details/linearalgebragro0000shaw|url-access=registration|chapter=§ 6.6 Minkowski space, § 6.7,8 Canonical forms pp 221–242|publisher=[[Academic Press]]|isbn=978-0-12-639201-2}}

* {{~~Cite~~cite book|last=Walter|first=Scott A.|title=The Expanding Worlds of General Relativity |chapter=Minkowski, Mathematicians, and the Mathematical Theory of Relativity |editor=Goenner, Hubert|year=1999 |publisher=Birkhäuser |location=Boston |isbn=978-0-8176-4060-6 |pages=45–86 |chapter-url=http://scottwalter.free.fr/papers/1999-mmm-walter.html |display-editors=etal}}

* {{citation|last=Weinberg|first=S.|year=2002|title=The Quantum Theory of Fields|volume=1|isbn=978-0-521-55001-7|author-link=Steven Weinberg|publisher=[[Cambridge University Press]]|url=https://archive.org/details/quantumtheoryoff00stev}}.

== External links ==

* {{YouTube|C2VMO7pcWhg|Animation clip}} visualizing Minkowski space in the context of special relativity.

* [https://web.archive.org/web/20150402140655/http://www.relativityscience.com/Minkowski_special_relativity_geometry.shtml The Geometry of Special Relativity: The Minkowski Space -– Time Light Cone]

* [https://philpapers.org/s/Minkowski%20space Minkowski space] at [[PhilPapers]]