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mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
, Minkowski space (or Minkowski spacetime) () is a combination of
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
and
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
into a
four-dimensional A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called '' dimensions'' ...
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
where the spacetime interval between any two events is independent of the
inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
in which they are recorded. Although initially developed by mathematician
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
for
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. Minkowski space is closely associated with Einstein's theories of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to
length contraction Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGera ...
and
time dilation In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events.This makes spacetime distance an invariant. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space. In 3-dimensional Euclidean space (e.g., simply ''space'' in Galilean relativity), the isometry group (the maps preserving the regular
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
) is the
Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations ...
. It is generated by rotations, reflections and translations. When time is appended as a fourth dimension, the further transformations of translations in time and
Galilean boost In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rota ...
s are added, and the group of all these transformations is called the
Galilean group In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rota ...
. All Galilean transformations preserve the 3-dimensional Euclidean distance. This distance is purely spatial. Time differences are ''separately'' preserved as well. This changes in the spacetime of special relativity, where space and time are interwoven. Spacetime is equipped with an indefinite
non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definit ...
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
, variously called the ''Minkowski metric'', the ''Minkowski norm squared'' or ''Minkowski inner product'' depending on the context.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. The Minkowski inner product is defined so as to yield the spacetime interval between two events when given their coordinate difference vector as argument. Equipped with this inner product, the mathematical model of spacetime is called Minkowski space. The analogue of the Galilean group for Minkowski space, preserving the spacetime interval (as opposed to the spatial Euclidean distance) is the Poincaré group. As manifolds, Galilean spacetime and Minkowski spacetime are ''the same''. They differ in what further structures are defined ''on'' them. The former has the Euclidean distance function and time interval (separately) together with inertial frames whose coordinates are related by Galilean transformations, while the latter has the Minkowski metric together with inertial frames whose coordinates are related by Poincaré transformations.


History


Complex Minkowski spacetime

In his second relativity paper in 1905–06
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
showed how, by taking time to be an imaginary fourth
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
coordinate , where is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
and is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, Lorentz transformations can be visualized as ordinary rotations of the four dimensional Euclidean sphere x^2 + y^2 + z^2 + (ict)^2 = \text. Poincaré set for convenience. 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 physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
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 hyperbolic rotation). This idea, which was mentioned only briefly by Poincaré, was elaborated by Minkowski in a paper in German published in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". Minkowski, using this formulation, restated the then-recent theory of relativity of Einstein. In particular, by restating the
Maxwell equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
as a symmetrical set of equations in the four variables combined with redefined vector variables for electromagnetic quantities, 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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
.


Real Minkowski spacetime

In a further development in his 1908 "Space and Time" lecture, Various English translations on Wikisource: " Space and Time." Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables of space and time in coordinate form in a four dimensional real
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. 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 of below appears unnamed when referring to orthogonality (which he calls ''normality'') of certain vectors, and the Minkowski norm squared is referred to (somewhat cryptically, perhaps this is 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 Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGera ...
) and to provide geometrical interpretation to the generalization of Newtonian mechanics to
relativistic mechanics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
. 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 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 Though Minkowski took an important step for physics,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
saw its limitation: For further historical information see references , and .


Causal structure

Where is velocity, and , , and are Cartesian coordinates in 3-dimensional space, and is the constant representing the universal speed limit, and is time, the four-dimensional vector is classified according to the sign of . A vector is timelike if , spacelike if , and null or lightlike if . This can be expressed in terms of the sign of 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 interval. The set of all null vectors at an eventTranslate the coordinate system so that the event is the new origin. of Minkowski space constitutes the light cone of that event. Given a timelike vector , there is a
worldline The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from con ...
of constant velocity associated with it, represented by a straight line in a Minkowski diagram. Once a direction of time is chosen,This corresponds to the time coordinate either increasing or decreasing when proper time for any particle increases. An application of flips this direction. 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 absolute future in figure) and # past-directed timelike vectors whose first component is negative (absolute past). Null vectors fall into three classes: # the zero vector, whose components in any basis are (origin), # future-directed null vectors whose first component is positive (upper light cone), and # past-directed null vectors whose first component is negative (lower light cone). Together with spacelike vectors there are 6 classes in all. An
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases it is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors, called a null basis. Vector fields 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 which 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 which 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 and is \eta (u_1, u_2) = u_1 \cdot u_2 = c^2 t_1 t_2 - x_1 x_2 - y_1 y_2 - z_1 z_2 . ''Positivity of scalar product'': An important property is that the scalar product of two similarly directed time-like vectors is always positive. This can be seen from the reversed
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality f ...
below. It follows that if the scalar product of two vectors is zero then one of these at least, must be space-like. The scalar product of two space-like vectors can be positive or negative as can be seen by considering the product of two space-like vectors having orthogonal spatial components and times either of different or the same signs. 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 is defined as \left\, u \right\, = \sqrt = \sqrt ''The reversed Cauchy inequality'' is another consequence of the convexity of either light-cone. For two distinct similarly directed time-like vectors and this inequality is \eta(u_1, u_2) > \left\, u_1 \right\, \left\, u_2 \right\, or algebraically, c^2 t_1 t_2 - x_1 x_2 - y_1 y_2 - z_1 z_2 > \sqrt From this the positivity property of the scalar product can be seen.


The reversed triangle inequality

For two similarly directed time-like vectors and , the inequality is \left\, u + w \right\, \ge \left\, u \right\, + \left\, w \right\, , where the equality holds when the vectors are linearly dependent. The proof uses the algebraic definition with the reversed Cauchy inequality: \left\, u + w \right\, ^2 = \left\, u \right\, ^2 + 2 \left(u, w \right) + \left\, w \right\, ^2 \ge \left\, u \right\, ^2 +2 \left\, u \right\, \left\, w \right\, + \left\, w \right\, ^2 = \left( \left\, u \right\, + \left\, w \right\, \right)^2 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 In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
. This provides an ''origin'', which is necessary in order to be able to refer to spacetime as being modeled as a vector space. This is not really ''physically'' motivated in that a canonical origin ("central" event in spacetime) should exist. One can get away with less structure, that of an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
, but this would needlessly complicate the discussion and would not reflect how flat spacetime is normally treated mathematically in modern introductory literature. For an overview, Minkowski space is a -dimensional real
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
equipped with a nondegenerate, symmetric bilinear form on the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at each point in spacetime, here simply called the ''Minkowski inner product'', with metric signature either or . The tangent space at each event is a vector space of the same dimension as spacetime, .


Tangent vectors

In practice, one need not be concerned with the tangent spaces. The vector space nature 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. or These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as \begin \left(x^0,\, x^1,\, x^2,\, x^3\right) \ &\leftrightarrow\ \left. x^0 \mathbf e_0 \_p + \left. x^1 \mathbf e_1 \_p + \left. x^2 \mathbf e_2 \_p + \left. x^3 \mathbf e_3 \_p \\ &\leftrightarrow\ \left. x^0 \mathbf e_0 \_q + \left. x^1 \mathbf e_1 \_q + \left. x^2 \mathbf e_2 \_q + \left. x^3 \mathbf e_3 \_q \end with basis vectors in the tangent spaces defined by \left.\mathbf e_\mu\_p = \left.\frac\_p \text \mathbf e_0, _p = \left(\begin 1 \\ 0 \\ 0 \\ 0\end\right) \text. Here and 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 In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
operator on the set of smooth functions. This is promoted to a ''definition'' of tangent vectors in manifolds ''not'' necessarily being embedded in . This definition of tangent vectors is not the only possible one as ordinary ''n''-tuples can be used as well. A tangent vector at a point may be defined, here specialized to Cartesian coordinates in Lorentz frames, as column vectors associated to ''each'' Lorentz frame related by Lorentz transformation such that the vector in a frame related to some frame by transforms according to . This is the ''same'' way in which the coordinates transform. Explicitly, \begin x'^\mu &= _\nu x^\nu, \\ v'^\mu &= _\nu v^\nu. \end 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 with ''displacement vectors'' at , which is, of course, admissible by essentially the same canonical identification. The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in . They offer various degree 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. In general, but with several exceptions, mathematicians and general relativists prefer spacelike vectors to yield a positive sign, , while particle physicists tend to prefer timelike vectors to yield a positive sign, . Authors covering several areas of physics, e.g. Steven Weinberg and Landau and Lifshitz ( and respectively) stick to one choice regardless of topic. Arguments for the former convention include "continuity" from the Euclidean case corresponding to the non-relativistic limit . Arguments for the latter include that minus signs, otherwise ubiquitous in particle physics, go away. Yet other authors, especially of introductory texts, e.g. , do ''not'' choose a signature at all, but instead opt to coordinatize spacetime such that the time ''coordinate'' (but not time itself!) is imaginary. This removes the need of the ''explicit'' introduction of a
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
(which may seem as an extra burden in an introductory course), and one needs ''not'' be concerned with covariant vectors and contravariant vectors (or raising and lowering indices) to be described below. The inner product is instead effected by a straightforward extension of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
in to . This works in the flat spacetime of special relativity, but not in the curved spacetime of general relativity, see (who, by the way use ). MTW also argues that it hides the true ''indefinite'' nature of the metric and the true nature of Lorentz boosts, which aren't rotations. It also needlessly complicates the use of tools of
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
that are otherwise immediately available and useful for geometrical description and calculation – even in the flat spacetime of special relativity, e.g. of the electromagnetic field.


Terminology

Mathematically associated to the bilinear form is a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
of type at each point in spacetime, called the ''Minkowski metric''.For comparison and motivation of terminology, take a Riemannian metric, which provides a positive definite symmetric bilinear form, i. e. an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
proper at each point on a manifold.
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 matrix representing the bilinear form. For comparison, in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, a Lorentzian manifold is likewise equipped with a
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
, which is a nondegenerate symmetric bilinear form on the tangent space at each point of . In coordinates, it may be represented by a matrix ''depending on spacetime position''. Minkowski space is thus a comparatively simple special case of a Lorentzian manifold. Its metric tensor is in coordinates the same symmetric matrix at every point of , 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 and signature or . Elements of Minkowski space are called events. Minkowski space is often denoted or to emphasize the chosen signature, or just . It is perhaps the simplest example of a
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
. Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space V, that is, :\eta:V\times V\rightarrow \mathbb where \eta has signature (-,+,+,+), and signature is a coordinate-invariant property of \eta. The space of bilinear maps forms a vector space which can be identified with M^*\otimes M^*, and \eta may be equivalently viewed as an element of this space. By making a choice of orthonormal basis \, we can identify M:=(V,\eta) with the space \mathbb^:=(\mathbb^,\eta_). The notation is meant to emphasise the fact that M and \mathbb^ are not just vector spaces but have added structure. \eta_ = \text(-1, +1, +1, +1). An interesting example of non-inertial coordinates for (part of) Minkowski spacetime are the
Born coordinates In relativistic physics, the Born coordinate chart is a coordinate chart for (part of) Minkowski spacetime, the flat spacetime of special relativity. It is often used to analyze the physical experience of observers who ride on a ring or disk r ...
. Another useful set of coordinates are the light-cone coordinates.


Pseudo-Euclidean metrics

The Minkowski inner product is not an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, since it is not positive-definite, i.e. the
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
need not be positive for nonzero . The positive-definite condition has been replaced by the weaker condition of non-degeneracy. The bilinear form is said to be ''indefinite''. The Minkowski metric is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally a ''constant'' pseudo-Riemannian metric in Cartesian coordinates. As such it is a nondegenerate symmetric bilinear form, a type tensor. It accepts two arguments , vectors in , the tangent space at in . Due to the above-mentioned canonical identification of with itself, it accepts arguments with both and in . As a notational convention, vectors in , called 4-vectors, are denoted in italics, and not, as is common in the Euclidean setting, with boldface . The latter is generally reserved for the -vector part (to be introduced below) of a -vector. The definition u \cdot v = \eta(u,\, v) yields an inner product-like structure on , previously and also henceforth, called the ''Minkowski inner product'', similar to the Euclidean
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, but it describes a different geometry. It is also called the ''relativistic dot product''. If the two arguments are the same, u \cdot u = \eta(u, u) \equiv \, u\, ^2 \equiv u^2, the resulting quantity will be called the ''Minkowski norm squared''. The Minkowski inner product satisfies the following properties. ; Linearity in first argument : \eta(au + v,\, w) = a\eta(u,\, w) + \eta(v,\, w),\quad \forall u,\, v \in M,\; \forall a \in \R ; Symmetry : \eta(u,\, v) = \eta(v,\, u) ; Non-degeneracy : \eta(u,\, v) = 0,\; \forall v \in M\ \Rightarrow\ u = 0 The first two conditions imply bilinearity. The defining ''difference'' between a pseudo-inner product and an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
proper is that the former is ''not'' required to be positive definite, that is, is allowed. 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 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 is identical in the case (the Lorentz group) to the matrix to be displayed below. Two vectors and are said to be
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
if . For a geometric interpretation of orthogonality in the special case when and (or vice versa), see
hyperbolic orthogonality In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyp ...
. A vector is called a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
if . A basis for consisting of mutually orthogonal unit vectors is called an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
. For a given
inertial frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
, an orthonormal basis in space, combined with the unit time vector, forms an orthonormal basis in Minkowski space. The number of positive and negative unit vectors in any such basis is a fixed pair of numbers, equal to the signature of the bilinear form associated with the inner product. This is Sylvester's law of inertia. 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 -dimensional flat spacetime with the remaining ambiguity only being the signature convention.


Minkowski metric

From the 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 and is: 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. 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. The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of c^2 t^2 - x^2 - y^2 - z^2 provided the transformations are linear. This
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
can be used to define a bilinear form u \cdot v = c^2 t_1 t_2 - x_1 x_2 - y_1 y_2 - z_1 z_2. via the
polarization identity In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product the ...
. This bilinear form can in turn be written as u \cdot v = u^\textsf
eta Eta (uppercase , lowercase ; grc, ἦτα ''ē̂ta'' or ell, ήτα ''ita'' ) is the seventh letter of the Greek alphabet, representing the close front unrounded vowel . Originally denoting the voiceless glottal fricative in most dialects, ...
v\,. Where is a 4\times 4 matrix associated with . While possibly confusing, it is common practice to denote with just . The matrix is read off from the explicit bilinear form as \eta = \begin -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end, and the bilinear form u \cdot v = \eta(u, v), with which this section started by assuming its existence, is now identified. For definiteness and shorter presentation, the signature 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
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...
) 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 has been used in a derivation, go back to the earliest point where it was used, substitute for , 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 such that -\eta(e_0, e_0) = \eta(e_1, e_1) = \eta(e_2, e_2) = \eta(e_3, e_3) = 1 . These conditions can be written compactly in the form \eta(e_\mu, e_\nu) = \eta_. Relative to a standard basis, the components of a vector are written where the
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
is used to write . The component is called the timelike component of while the other three components are called the spatial components. The spatial components of a -vector may be identified with a -vector . In terms of components, the Minkowski inner product between two vectors and is given by \eta(v, w) = \eta_ v^\mu w^\nu = v^0 w_0 + v^1 w_1 + v^2 w_2 + v^3 w_3 = v^\mu w_\mu = v_\mu w^\mu, and \eta(v, v) = \eta_ v^\mu v^\nu = v^0v_0 + v^1 v_1 + v^2 v_2 + v^3 v_3 = v^\mu v_\mu. Here lowering of an index with the metric was used. There are many possible choices of standard basis obeying the condition \eta(e_\mu, e_\nu) = \eta_. Any two such bases are related in some sense by a Lorentz transformation, either by a change-of-basis matrix \Lambda^\mu_\nu, a real 4\times 4 matrix satisfying \Lambda^\mu_\rho\eta_\Lambda^\nu_\sigma = \eta_. or \Lambda, a linear map on the abstract vector space satisfying, for any pair of vectors u,v, \eta(\Lambda u, \Lambda v) = \eta(u, v). Then if we have two different bases \ and \, we can write e_\mu' = e_\nu\Lambda^\nu_\mu or e_\mu' = \Lambda e_\mu. While it might be tempting to think of \Lambda^\mu_\nu and \Lambda 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

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 and the cotangent spaces of . At a point in , the tangent and cotangent spaces are dual vector spaces (so the dimension of the cotangent space at an event is also ). Just as an authentic inner product on a vector space with one argument fixed, by
Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, ...
, may be expressed as the action of a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
on the vector space, the same holds for the Minkowski inner product of Minkowski space. Thus if are the components of a vector in a tangent space, then 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 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 in matrix representation, can be used to define raising of an index. The components of this inverse are denoted . It happens that . These maps between a vector space and its dual can be denoted (eta-flat) and (eta-sharp) by the musical analogy. Contravariant and covariant vectors are geometrically very different objects. The first can and should be thought of as arrows. A linear functional can be characterized by two objects: its kernel, which is a
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
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 In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
(though the latter is usually reserved for covector ''fields''). uses a vivid analogy with wave fronts of a
de Broglie wave Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
(scaled by a factor of Planck's reduced constant) quantum mechanically associated to a
momentum four-vector In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum ...
to illustrate how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought of as the action of the covariant version of one of them on the contravariant version of the other. The inner product is then how many time the arrow pierces the planes. The mathematical reference, , offers the same geometrical view of these objects (but mentions no piercing). The
electromagnetic field tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
is a differential 2-form, which geometrical description can as well be found in MTW. One may, of course, ignore geometrical views all together (as is the style in e.g. and ) and proceed algebraically in a purely formal fashion. The time-proven robustness of the formalism itself, sometimes referred to as
index gymnastics In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions. Vectors, covectors and the metric M ...
, 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 \eta:M\times M\rightarrow \mathbb, the lowered version of a vector can be thought of as the partial evaluation of \eta, that is, there is an associated partial evaluation map :\eta(\cdot, -):M\rightarrow M^*; v \mapsto \eta(v,\cdot). The lowered vector \eta(v,\cdot)\in M^* is then the dual map u\mapsto\eta(v,u). Note it does not matter which argument is partially evaluated due to symmetry of \eta. Non-degeneracy is then equivalent to injectivity of the partial evaluation map, or equivalently non-degeneracy tells us the kernel of the map is trivial. In finite dimension, as we have 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 M to M^*. This then allows definition of the inverse partial evaluation map, :\eta^:M^*\rightarrow M, which allows us to define the inverse metric :\eta^:M^*\times M^* \rightarrow \mathbb, \eta^(\alpha,\beta) = \eta(\eta^(\alpha),\eta^(\beta)) where the two different usages of \eta^ can be told apart by the argument each is evaluated on. This can then be used to raise indices. If we work in a coordinate basis, we find that the metric \eta^ is indeed the matrix inverse to \eta.


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. A full-blown version of the Minkowski metric in coordinates as a tensor field on spacetime has the appearance \eta_ dx^\mu \otimes dx^\nu = \eta_ dx^\mu \odot dx^\nu = \eta_ dx^\mu dx^\nu. Explanation: The coordinate differentials are 1-form fields. They are defined as the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
of the coordinate functions . These quantities evaluated at a point provide a basis for the cotangent space at . The
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
(denoted by the symbol ) yields a tensor field of type , i.e. the type that expects two contravariant vectors as arguments. On the right hand side, the symmetric product (denoted by the symbol or by juxtaposition) has been taken. The equality holds since, by definition, the Minkowski metric is symmetric. The notation on the far right is also sometimes used for the related, but different, line element. It is ''not'' a tensor. For elaboration on the differences and similarities, see ''Tangent'' vectors are, in this formalism, given in terms of a basis of differential operators of the first order, \left.\frac\_p, where is an event. This operator applied to a function gives the
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
of at in the direction of increasing with fixed. They provide a basis for the tangent space at . The exterior derivative of a function is a covector field, i.e. an assignment of a cotangent vector to each point , by definition such that df(X) = Xf, for each vector field . A vector field is an assignment of a tangent vector to each point . In coordinates can be expanded at each point in the basis given by the . Applying this with , the coordinate function itself, and , called a ''coordinate vector field'', one obtains dx^\mu\left(\frac\right) = \frac = \delta_\nu^\mu. Since this relation holds at each point , the provide a basis for the cotangent space at each and the bases and are dual to each other, \left. dx^\mu \_p \left(\left.\frac\_p\right) = \delta^\mu_\nu. at each . Furthermore, one has \alpha \otimes \beta(a, b) = \alpha(a)\beta(b) for general one-forms on a tangent space and general tangent vectors . (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 , both expanded in terms of the basis coordinate vector fields, the result is \eta_ dx^\mu \otimes dx^\nu(a, b) = \eta_ a^\mu b^\nu, where , are the ''component functions'' of the vector fields. The above equation holds at each point , and the relation may as well be interpreted as the Minkowski metric at applied to two tangent vectors at . As mentioned, in a vector space, such as that modelling the spacetime of special relativity, tangent vectors can be canonically identified with vectors in the space itself, and vice versa. This means that the tangent spaces at each point are canonically identified with each other and with the vector space itself. This explains how the right hand side of the above equation can be employed directly, without regard to spacetime point the metric is to be evaluated and from where (which tangent space) the vectors come from. This situation changes in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. There one has g(p)_ \left. dx^\mu \_p \left. dx^\nu \_p(a, b) = g(p)_ a^\mu b^\nu, where now , i.e., is still a metric tensor but now depending on spacetime and is a solution of
Einstein's field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the for ...
s. Moreover, ''must'' be tangent vectors at spacetime point and can no longer be moved around freely.


Chronological and causality relations

Let . We say that # chronologically precedes if is future-directed timelike. This relation has the transitive property and so can be written . # causally precedes if is future-directed null or future-directed timelike. It gives a partial ordering of spacetime and so can be written . Suppose ''x'' ∈ ''M'' is timelike. Then the simultaneous hyperplane for x is \. Since this
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
varies as ''x'' varies, there is a
relativity of simultaneity In physics, the relativity of simultaneity is the concept that ''distant simultaneity'' – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame. This possib ...
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 ( 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 . Its real part is the Minkowski space of
four-vectors In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
, such as the four-velocity and the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
, which are independent of the choice of orientation of the space. The imaginary part, on the other hand, may consist of four-pseudovectors, such as
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an object ...
and
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
, which change their direction with a change of orientation. We introduce a pseudoscalar which also changes sign with a change of orientation. Thus, elements of are independent of the choice of the orientation. The
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
-like structure on is defined as for any . A relativistic pure
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
of an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
or any half spin particle is described by as , where is the four-velocity of the particle, satisfying and is the 4D spin vector, which is also the Pauli–Lubanski pseudovector satisfying and .


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 , -dimensional Minkowski space is a vector space of real dimension on which there is a constant Minkowski metric of signature or . These generalizations are used in theories where spacetime is assumed to have more or less than dimensions.
String theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
and
M-theory M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witt ...
are two examples where . In string theory, there appears conformal field theories with spacetime dimensions.
de Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canoni ...
can be formulated as a submanifold of generalized Minkowski space as can the model spaces of
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
(see below).


Curvature

As a ''flat spacetime'', the three spatial components of Minkowski spacetime always obey the
Pythagorean Theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
. Minkowski space is a suitable basis for
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, a good description of physical systems over finite distances in systems without significant
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
. However, in order to take gravity into account, physicists use the theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, 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 Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
. Even in curved space, Minkowski space is still a good description in an infinitesimal region surrounding any point (barring gravitational singularities).This similarity between flat and curved space at infinitesimally small distance scales is foundational to the definition of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
in general.
More abstractly, we say that in the presence of gravity spacetime is described by a curved 4-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
for which the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
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 a Euclidean geometry, and not with any of the generalized Riemannian geometries with intrinsic curvature, those exposed by the ''model spaces'' in
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
(negative curvature) and the geometry modeled by the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
(positive curvature). The reason is the indefiniteness of the Minkowski metric. Minkowski space is, in particular, not a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
and not a Riemannian manifold with a Riemannian metric. However, Minkowski space contains submanifolds endowed with a Riemannian metric yielding hyperbolic geometry. Model spaces of hyperbolic geometry of low dimension, say or , ''cannot'' be isometrically embedded in Euclidean space with one more dimension, i.e. or respectively, with the Euclidean metric , disallowing easy visualization.There ''is'' an isometric embedding into according to the
Nash embedding theorem The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instan ...
(), but the embedding dimension is much higher, for a Riemannian manifold of dimension .
By comparison, model spaces with positive curvature are just spheres in Euclidean space of one higher dimension. Hyperbolic spaces ''can'' be isometrically embedded in spaces of one more dimension when the embedding space is endowed with the Minkowski metric . Define to be the upper sheet () of the
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
:\mathbf H_R^ = \left\ in generalized Minkowski space of spacetime dimension . This is one of the surfaces of transitivity of the generalized Lorentz group. The
induced metric In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback. It may be determined using t ...
on this submanifold, :h_R^ = \iota^* \eta, the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
of the Minkowski metric under inclusion, is a Riemannian metric. With this metric is a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
. It is one of the model spaces of Riemannian geometry, the
hyperboloid model In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbo ...
of hyperbolic space. It is a space of constant negative curvature . The in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the for its dimension. A corresponds to the
Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk ...
, while corresponds to the
Poincaré half-space model Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Luci ...
of dimension .


Preliminaries

In the definition above is the inclusion map and the superscript star denotes the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
. The present purpose is to describe this and similar operations as a preparation for the actual demonstration that actually is a hyperbolic space.


Hyperbolic stereographic projection

In order to exhibit the metric it is necessary to pull it back via a suitable ''parametrization''. A parametrization of a submanifold of is a map whose range is an open subset of . If has the same dimension as , a parametrization is just the inverse of a coordinate map . The parametrization to be used is the inverse of ''hyperbolic stereographic projection''. This is illustrated in the figure to the left for . It is instructive to compare to
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
for spheres. Stereographic projection and its inverse are given by \begin \sigma(\tau, \mathbf x) = \mathbf u &= \frac,\\ \sigma^(\mathbf u) = (\tau, \mathbf x) &= \left(R\frac, \frac\right), \end where, for simplicity, . The are coordinates on and the are coordinates on .


Pulling back the metric

One has :h_R^ = \eta, _ = \left(dx^1\right)^2 + \ldots + \left(dx^n\right)^2 - d\tau^2 and the map : \sigma^:\mathbb^n \rightarrow \mathbf_R^;\quad \sigma^(\mathbf) = (\tau(\mathbf),\, \mathbf(\mathbf)) = \left(R\frac,\, \frac\right). The pulled back metric can be obtained by straightforward methods of calculus; : \left.\left(\sigma^\right)^* \eta\_ = \left(dx^1(\mathbf u)\right)^2 + \ldots + \left(dx^n(\mathbf u)\right)^2 - \left(d\tau(\mathbf u)\right)^2. One computes according to the standard rules for computing differentials (though one is really computing the rigorously defined exterior derivatives), :\begin dx^1(\mathbf u) &= d\left(\frac\right) = \frac\fracdu^1 + \ldots + \frac\fracdu^n + \frac\fracd\tau,\\ &\ \ \vdots\\ dx^n(\mathbf u) &= d\left(\frac\right) = \cdots,\\ d\tau(\mathbf u) &= d\left(R\frac\right) = \cdots, \end and substitutes the results into the right hand side. This yields : \left(\sigma^\right)^* h_R^ = \frac \equiv h_R^. This last equation shows that the metric on the ball is identical to the Riemannian metric in the Poincaré ball model, another standard model of hyperbolic geometry.


See also

* Hyperspace * Introduction to the mathematics of general relativity * Minkowski plane


Remarks


Notes


References

* * * * Giulini D The rich structure of Minkowski space, https://arxiv.org/abs/0802.4345v1 * * * * * * **Published translation: **Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies * Various English translations on Wikisource: Space and Time * * * * * Wikisource translation: 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 * * * *


External links

* visualizing Minkowski space in the context of special relativity.
The Geometry of Special Relativity: The Minkowski Space - Time Light Cone

Minkowski space
at PhilPapers {{DEFAULTSORT:Minkowski Space Equations of physics Geometry Lorentzian manifolds Special relativity Exact solutions in general relativity Hermann Minkowski