Minkowski space

In physics, Minkowski space (or Minkowski spacetime) (/mɪŋˈkɔːfski, -ˈkɒf-/^[1]) is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.

The model helps show how a spacetime interval between any two 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".

Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure by which special relativity is formalized. While the individual components in Euclidean space and time might differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events.^{[nb 1]} Minkowski space differs from four-dimensional Euclidean space insofar as it treats time differently than the three spatial dimensions.

In 3-dimensional Euclidean space, the isometry group (maps preserving the regular Euclidean distance) is the Euclidean group. It is generated by rotations, reflections and translations. When time is appended as a fourth dimension, the further transformations of translations in time and Lorentz boosts are added, and the group of all these transformations is called the Poincaré group. Minkowski's model follows special relativity, where motion causes time dilation changing the scale applied to the frame in motion and shifts the phase of light.

Spacetime is equipped with an indefinite non-degenerate bilinear form, called the Minkowski metric,^[2] the Minkowski norm squared or Minkowski inner product depending on the context.^{[nb 2]} 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.^[3] 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 group).

Part of a series on
Spacetime
Special relativity General relativity
Spacetime concepts Spacetime manifold Equivalence principle Lorentz transformations Minkowski space
General relativity Introduction to general relativity Mathematics of general relativity Einstein field equations
Classical gravity Introduction to gravitation Newton's law of universal gravitation
Relevant mathematics Four-vector Derivations of relativity Spacetime diagrams Differential geometry Curved space Curved spacetime Mathematics of general relativity Spacetime topology
Physics portal  Category
v t e

In his second relativity paper in 1905, Henri Poincaré showed^[4] how, by taking time to be an imaginary fourth spacetime coordinate ict, where c is the speed of light and i is the imaginary unit, Lorentz transformations 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 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 (t, x, y, z). A Lorentz transformation is represented by a 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.$${\displaystyle x^{2}+y^{2}+z^{2}+(ict)^{2}={\text{constant}}.}$$

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 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".^[5] He reformulated Maxwell equations as a symmetrical set of equations in the four variables(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.

In a further development in his 1908 "Space and Time" lecture,^[6] Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables (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 (which he calls normality) of certain vectors, and the Minkowski norm squared is referred to (somewhat cryptically, perhaps this is a translation dependent) as "sum".

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

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

— Hermann Minkowski, 1908, 1909^[6]

Though Minkowski took an important step for physics, Albert Einstein saw its limitation:

At a time when Minkowski was giving the geometrical interpretation of special relativity by extending the Euclidean three-space to a 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.^[7]

For further historical information see references Galison (1979), Corry (1997) and Walter (1999).

Where v is velocity, x, y, and z are Cartesian coordinates in 3-dimensional space, c is the constant representing the universal speed limit, and t is time, the four-dimensional vector v = (ct, x, y, z) = (ct, r) is classified according to the sign of c²t² − r². A vector is timelike if c²t² > r², spacelike if c²t² < r², and null or lightlike if c²t² = r². This can be expressed in terms of the sign of η(v, v), also called 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 vectors at an event^{[nb 3]} of Minkowski space constitutes the light cone of that event. Given a timelike vector 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,^{[nb 4]} timelike and null vectors can be further decomposed into various classes. For timelike vectors, one has

1. 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
2. past-directed timelike vectors whose first component is negative (causal past (also called the absolute past)).

Null vectors fall into three classes:

1. the zero vector, whose components in any basis are (0, 0, 0, 0) (origin),
2. future-directed null vectors whose first component is positive (upper light cone), and
3. past-directed null vectors whose first component is negative (lower light cone).

Together with spacelike vectors, there are 6 classes in all.

An orthonormal 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.

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.

The scalar product of two time-like vectors u₁ = (t₁, x₁, y₁, z₁) and u₂ = (t₂, x₂, y₂, z₂) is $${\displaystyle \eta (u_{1},u_{2})=u_{1}\cdot u_{2}=c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}.}$$

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 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).

The norm of a time-like vector u = (ct, x, y, z) is defined as $${\displaystyle \left\|u\right\|={\sqrt {\eta (u,u)}}={\sqrt {c^{2}t^{2}-x^{2}-y^{2}-z^{2}}}}$$

The reversed Cauchy inequality is another consequence of the convexity of either light cone.^[8] For two distinct similarly directed time-like vectors u₁ and u₂ this inequality is $${\displaystyle \eta (u_{1},u_{2})>\left\|u_{1}\right\|\left\|u_{2}\right\|}$$ or algebraically, $${\displaystyle c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}>{\sqrt {\left(c^{2}t_{1}^{2}-x_{1}^{2}-y_{1}^{2}-z_{1}^{2}\right)\left(c^{2}t_{2}^{2}-x_{2}^{2}-y_{2}^{2}-z_{2}^{2}\right)}}}$$

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

For two similarly directed time-like vectors u and w, the inequality is^[9] $${\displaystyle \left\|u+w\right\|\geq \left\|u\right\|+\left\|w\right\|,}$$ where the equality holds when the vectors are linearly dependent.

The proof uses the algebraic definition with the reversed Cauchy inequality:^[10] $${\displaystyle {\begin{aligned}\left\|u+w\right\|^{2}&=\left\|u\right\|^{2}+2\left(u,w\right)+\left\|w\right\|^{2}\\[5mu]&\geq \left\|u\right\|^{2}+2\left\|u\right\|\left\|w\right\|+\left\|w\right\|^{2}=\left(\left\|u\right\|+\left\|w\right\|\right)^{2}.\end{aligned}}}$$

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

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 4-dimensional 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 (+ − − −) or (− + + +). The tangent space at each event is a vector space of the same dimension as spacetime, 4.

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. Lee (2003, Proposition 3.8.) or Lee (2012, Proposition 3.13.) These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as^[11] $${\displaystyle {\begin{aligned}\left(x^{0},\,x^{1},\,x^{2},\,x^{3}\right)\ &\leftrightarrow \ \left.x^{0}\mathbf {e} _{0}\right|_{p}+\left.x^{1}\mathbf {e} _{1}\right|_{p}+\left.x^{2}\mathbf {e} _{2}\right|_{p}+\left.x^{3}\mathbf {e} _{3}\right|_{p}\\&\leftrightarrow \ \left.x^{0}\mathbf {e} _{0}\right|_{q}+\left.x^{1}\mathbf {e} _{1}\right|_{q}+\left.x^{2}\mathbf {e} _{2}\right|_{q}+\left.x^{3}\mathbf {e} _{3}\right|_{q}\end{aligned}}}$$ with basis vectors in the tangent spaces defined by $${\displaystyle \left.\mathbf {e} _{\mu }\right|_{p}=\left.{\frac {\partial }{\partial x^{\mu }}}\right|_{p}{\text{ or }}\mathbf {e} _{0}|_{p}=\left({\begin{matrix}1\\0\\0\\0\end{matrix}}\right){\text{, etc}}.}$$

Here, p and 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 Rⁿ. This definition of tangent vectors is not the only possible one, as ordinary n-tuples can be used as well.

For some purposes, it is desirable to identify tangent vectors at a point p with displacement vectors at p, which is, of course, admissible by essentially the same canonical identification.^[12] The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in Misner, Thorne & Wheeler (1973). They offer various degrees of sophistication (and rigor) depending on which part of the material one chooses to read.

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 in Relativity.

Mathematically associated with the bilinear form is a tensor of type (0,2) at each point in spacetime, called the Minkowski metric.^{[nb 5]} 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 4×4 matrix representing the bilinear form.

For comparison, in general relativity, a Lorentzian manifold L is likewise equipped with a metric tensor g, which is a nondegenerate symmetric bilinear form on the tangent space T_pL at each point p of L. In coordinates, it may be represented by a 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 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 n = 4 and signature (1, 3) or (3, 1). Elements of Minkowski space are called events. Minkowski space is often denoted R^1,3 or R^3,1 to emphasize the chosen signature, or just M. It is an example of a pseudo-Riemannian manifold.

Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space V, that is, $${\displaystyle \eta :V\times V\rightarrow \mathbf {R} }$$ where η has signature (+, -, -, -), and signature is a coordinate-invariant property of η. The space of bilinear maps forms a vector space which can be identified with ${\displaystyle M^{*}\otimes M^{*}}$, and η may be equivalently viewed as an element of this space. By making a choice of orthonormal basis ${\displaystyle \{e_{\mu }\}}$, ${\displaystyle M:=(V,\eta )}$ can be identified with the space ${\displaystyle \mathbf {R} ^{1,3}:=(\mathbf {R} ^{4},\eta _{\mu \nu })}$. The notation is meant to emphasize the fact that M and ${\displaystyle \mathbf {R} ^{1,3}}$ are not just vector spaces but have added structure. ${\displaystyle \eta _{\mu \nu }={\text{diag}}(+1,-1,-1,-1)}$.

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.

The Minkowski inner product is not an inner product, since it is not positive-definite, i.e. the quadratic form η(v, v) need not be positive for nonzero v. 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 η 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 (0, 2) tensor. It accepts two arguments u_p, v_p, vectors in T_pM, p ∈ M, the tangent space at p in M. Due to the above-mentioned canonical identification of T_pM with M itself, it accepts arguments u, v with both u and v in M.

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

The definition ^[13] $${\displaystyle u\cdot v=\eta (u,\,v)}$$ yields an inner product-like structure on 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, $${\displaystyle u\cdot u=\eta (u,u)\equiv \|u\|^{2}\equiv u^{2},}$$ the resulting quantity will be called the Minkowski norm squared. The Minkowski inner product satisfies the following properties.

Linearity in the first argument

${\displaystyle \eta (au+v,\,w)=a\eta (u,\,w)+\eta (v,\,w),\quad \forall u,\,v\in M,\;\forall a\in \mathbb {R} }$

Symmetry

${\displaystyle \eta (u,\,v)=\eta (v,\,u)}$

Non-degeneracy

${\displaystyle \eta (u,\,v)=0,\;\forall v\in M\ \Rightarrow \ u=0}$

The first two conditions imply bilinearity. The defining difference between a pseudo-inner product and an inner product proper is that the former is not required to be positive definite, that is, η(u, u) < 0 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 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 Φ is identical in the case O(3, 1) (the Lorentz group) to the matrix η to be displayed below.

Two vectors v and w are said to be orthogonal if η(v, w) = 0. For a geometric interpretation of orthogonality in the special case, when η(v, v) ≤ 0 and η(w, w) ≥ 0 (or vice versa), see hyperbolic orthogonality.

A vector e is called a unit vector if η(e, e) = ±1. A basis for M consisting of mutually orthogonal unit vectors is called an orthonormal basis.^[14]

For a given inertial frame, 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 4-dimensional flat spacetime with the remaining ambiguity only being the signature convention.

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 1 and 2 is:^[15] $${\displaystyle c^{2}\left(t_{1}-t_{2}\right)^{2}-\left(x_{1}-x_{2}\right)^{2}-\left(y_{1}-y_{2}\right)^{2}-\left(z_{1}-z_{2}\right)^{2}.}$$ 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.^[16][17]

The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of $${\displaystyle c^{2}t^{2}-x^{2}-y^{2}-z^{2}}$$ provided the transformations are linear. This quadratic form can be used to define a bilinear form $${\displaystyle u\cdot v=c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}}$$ via the polarization identity. This bilinear form can in turn be written as $${\displaystyle u\cdot v=u^{\textsf {T}}\,[\eta ]\,v,}$$ where [η] is a ${\displaystyle 4\times 4}$ matrix associated with η. While possibly confusing, it is common practice to denote [η] with just η. The matrix is read off from the explicit bilinear form as $${\displaystyle \eta =\left({\begin{array}{r}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}}\right)\!,}$$ and the bilinear form $${\displaystyle u\cdot v=\eta (u,v),}$$ 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) 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 η has been used in a derivation, go back to the earliest point where it was used, substitute η for −η, and retrace forward to the desired formula with the desired metric signature.

A standard or orthonormal basis for Minkowski space is a set of four mutually orthogonal vectors {e₀, e₁, e₂, e₃} such that $${\displaystyle \eta (e_{0},e_{0})=-\eta (e_{1},e_{1})=-\eta (e_{2},e_{2})=-\eta (e_{3},e_{3})=1}$$ and for which $${\displaystyle \eta (e_{\mu },e_{\nu })=0}$$ when ${\textstyle \mu \neq \nu \,.}$

These conditions can be written compactly in the form $${\displaystyle \eta (e_{\mu },e_{\nu })=\eta _{\mu \nu }.}$$

Relative to a standard basis, the components of a vector v are written (v⁰, v¹, v², v³) where the Einstein notation is used to write v = v^μ e_μ. The component v⁰ is called the timelike component of v while the other three components are called the spatial components. The spatial components of a 4-vector v may be identified with a 3-vector v = (v₁, v₂, v₃).

In terms of components, the Minkowski inner product between two vectors v and w is given by

$${\displaystyle \eta (v,w)=\eta _{\mu \nu }v^{\mu }w^{\nu }=v^{0}w_{0}+v^{1}w_{1}+v^{2}w_{2}+v^{3}w_{3}=v^{\mu }w_{\mu }=v_{\mu }w^{\mu },}$$ and $${\displaystyle \eta (v,v)=\eta _{\mu \nu }v^{\mu }v^{\nu }=v^{0}v_{0}+v^{1}v_{1}+v^{2}v_{2}+v^{3}v_{3}=v^{\mu }v_{\mu }.}$$

Here lowering of an index with the metric was used.

There are many possible choices of standard basis obeying the condition ${\displaystyle \eta (e_{\mu },e_{\nu })=\eta _{\mu \nu }.}$ Any two such bases are related in some sense by a Lorentz transformation, either by a change-of-basis matrix ${\displaystyle \Lambda _{\nu }^{\mu }}$, a real 4 × 4 matrix satisfying $${\displaystyle \Lambda _{\rho }^{\mu }\eta _{\mu \nu }\Lambda _{\sigma }^{\nu }=\eta _{\rho \sigma }.}$$ or Λ, a linear map on the abstract vector space satisfying, for any pair of vectors u, v, $${\displaystyle \eta (\Lambda u,\Lambda v)=\eta (u,v).}$$

Then if two different bases exist, {e₀, e₁, e₂, e₃} and {e′₀, e′₁, e′₂, e′₃}, ${\displaystyle e_{\mu }'=e_{\nu }\Lambda _{\mu }^{\nu }}$ can be represented as ${\displaystyle e_{\mu }'=e_{\nu }\Lambda _{\mu }^{\nu }}$ or ${\displaystyle e_{\mu }'=\Lambda e_{\mu }}$. While it might be tempting to think of ${\displaystyle \Lambda _{\nu }^{\mu }}$ and Λ as the same thing, mathematically, they are elements of different spaces, and act on the space of standard bases from different sides.

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 M and the cotangent spaces of M. At a point in M, the tangent and cotangent spaces are dual vector spaces (so the dimension of the cotangent space at an event is also 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.^[19]

Thus if v^μ are the components of a vector in tangent space, then 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 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 η in matrix representation, can be used to define raising of an index. The components of this inverse are denoted η^μν. It happens that η^μν = η_μν. These maps between a vector space and its dual can be denoted η^♭ (eta-flat) and η^♯ (eta-sharp) by the musical analogy.^[20]

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, 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).

One quantum mechanical analogy explored in the literature is that of a de Broglie wave (scaled by a factor of Planck's reduced constant) associated with a momentum four-vector 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 times the arrow pierces the planes.^[18] The mathematical reference, Lee (2003), offers the same geometrical view of these objects (but mentions no piercing).

The electromagnetic field tensor is a differential 2-form, which geometrical description can as well be found in MTW.

One may, of course, ignore geometrical views altogether (as is the style in e.g. Weinberg (2002) and 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.

Given a bilinear form ${\displaystyle \eta :M\times M\rightarrow \mathbf {R} }$, the lowered version of a vector can be thought of as the partial evaluation of ${\displaystyle \eta }$, that is, there is an associated partial evaluation map $${\displaystyle \eta (\cdot ,-):M\rightarrow M^{*};v\mapsto \eta (v,\cdot ).}$$

The lowered vector ${\displaystyle \eta (v,\cdot )\in M^{*}}$ is then the dual map ${\displaystyle u\mapsto \eta (v,u)}$. Note it does not matter which argument is partially evaluated due to the symmetry of ${\displaystyle \eta }$.

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 ${\displaystyle M}$ to ${\displaystyle M^{*}}$. This then allows the definition of the inverse partial evaluation map, $${\displaystyle \eta ^{-1}:M^{*}\rightarrow M,}$$ which allows the inverse metric to be defined as $${\displaystyle \eta ^{-1}:M^{*}\times M^{*}\rightarrow \mathbf {R} ,\eta ^{-1}(\alpha ,\beta )=\eta (\eta ^{-1}(\alpha ),\eta ^{-1}(\beta ))}$$ where the two different usages of ${\displaystyle \eta ^{-1}}$ 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 η⁻¹ is indeed the matrix inverse to η.

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.

Let x, y ∈ M. Here,

1. x chronologically precedes y if y − x is future-directed timelike. This relation has the transitive property and so can be written x < y.
2. x causally precedes y if y − x is future-directed null or future-directed timelike. It gives a partial ordering of spacetime and so can be written x ≤ y.

Suppose x ∈ M is timelike. Then the simultaneous hyperplane for x is {y : η(x, y) = 0}. Since this hyperplane varies as x varies, there is a relativity of simultaneity in Minkowski space.

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

Complexified Minkowski space is defined as M_c = M ⊕ iM.^[22] 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 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 i is introduced, which also changes sign with a change of orientation. Thus, elements of M_c are independent of the choice of the orientation.

The inner product-like structure on M_c is defined as u ⋅ v = η(u, v) for any u,v ∈ M_c. A relativistic pure spin of an electron or any half spin particle is described by ρ ∈ M_c as ρ = u + is, where u is the four-velocity of the particle, satisfying u² = 1 and s is the 4D spin vector,^[23] which is also the Pauli–Lubanski pseudovector satisfying s² = −1 and u ⋅ s = 0.

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 n ≥ 2, n-dimensional Minkowski space is a vector space of real dimension n on which there is a constant Minkowski metric of signature (n − 1, 1) or (1, n − 1). These generalizations are used in theories where spacetime is assumed to have more or less than 4 dimensions. String theory and M-theory are two examples where n > 4. In string theory, there appears conformal field theories with 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).

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 infinitesimal region surrounding any point (barring gravitational singularities).^{[nb 6]} 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.

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 submanifolds endowed with a Riemannian metric yielding hyperbolic geometry.

Model spaces of hyperbolic geometry of low dimension, say 2 or 3, cannot be isometrically embedded in Euclidean space with one more dimension, i.e. ${\displaystyle \mathbf {R} ^{3}}$ or ${\displaystyle \mathbf {R} ^{4}}$ respectively, with the Euclidean metric ${\displaystyle {\overline {g}}}$, preventing easy visualization.^{[nb 7][24]} By comparison, model spaces with positive curvature are just spheres in Euclidean space of one higher dimension.^[25] Hyperbolic spaces can be isometrically embedded in spaces of one more dimension when the embedding space is endowed with the Minkowski metric ${\displaystyle \eta }$.

Define ${\displaystyle \mathbf {H} _{R}^{1(n)}\subset \mathbf {M} ^{n+1}}$to be the upper sheet (${\displaystyle ct>0}$) of the hyperboloid $${\displaystyle \mathbf {H} _{R}^{1(n)}=\left\{\left(ct,x^{1},\ldots ,x^{n}\right)\in \mathbf {M} ^{n}:c^{2}t^{2}-\left(x^{1}\right)^{2}-\cdots -\left(x^{n}\right)^{2}=R^{2},ct>0\right\}}$$ in generalized Minkowski space ${\displaystyle \mathbf {M} ^{n+1}}$ of spacetime dimension ${\displaystyle n+1.}$ This is one of the surfaces of transitivity of the generalized Lorentz group. The induced metric on this submanifold, $${\displaystyle h_{R}^{1(n)}=\iota ^{*}\eta ,}$$ the pullback of the Minkowski metric ${\displaystyle \eta }$ under inclusion, is a Riemannian metric. With this metric ${\displaystyle \mathbf {H} _{R}^{1(n)}}$ 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 ${\displaystyle -1/R^{2}}$.^[26] The 1 in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the n for its dimension. A ${\displaystyle 2(2)}$ corresponds to the Poincaré disk model, while ${\displaystyle 3(n)}$ corresponds to the Poincaré half-space model of dimension ${\displaystyle n.}$

In the definition above ${\displaystyle \iota :\mathbf {H} _{R}^{1(n)}\rightarrow \mathbf {M} ^{n+1}}$ is the inclusion map and the superscript star denotes the pullback. The present purpose is to describe this and similar operations as a preparation for the actual demonstration that ${\displaystyle \mathbf {H} _{R}^{1(n)}}$ actually is a hyperbolic space.

Behavior of tensors under inclusion, pullback of covariant tensors under general maps and pushforward of vectors under general maps

Behavior of tensors under inclusion: For inclusion maps from a submanifold S into M and a covariant tensor α of order k on M it holds that $${\displaystyle \iota ^{*}\alpha \left(X_{1},\,X_{2},\,\ldots ,\,X_{k}\right)=\alpha \left(\iota _{*}X_{1},\,\iota _{*}X_{2},\,\ldots ,\,\iota _{*}X_{k}\right)=\alpha \left(X_{1},\,X_{2},\,\ldots ,\,X_{k}\right),}$$ where X1, X1, …, Xk are vector fields on 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 $${\displaystyle \iota ^{*}\alpha =\alpha |_{S},}$$ meaning (with slight abuse of notation) the restriction of α to accept as input vectors tangent to some s ∈ S only. Pullback of tensors under general maps: The pullback of a covariant k-tensor α (one taking only contravariant vectors as arguments) under a map F: M → N is a linear map $${\displaystyle F^{*}\colon T_{F(p)}^{k}N\rightarrow T_{p}^{k}M,}$$ where for any vector space V, $${\displaystyle T^{k}V=\underbrace {V^{*}\otimes V^{*}\otimes \cdots \otimes V^{*}} _{k{\text{ times}}}.}$$ It is defined by $${\displaystyle F^{*}(\alpha )\left(X_{1},\,X_{2},\,\ldots ,\,X_{k}\right)=\alpha \left(F_{*}X_{1},\,F_{*}X_{2},\,\ldots ,\,F_{*}X_{k}\right),}$$ where the subscript star denotes the pushforward of the map F, and X1, X2, …, Xk are vectors in TpM. (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 F∗X1 ≠ X1 in general.) The pushforward of vectors under general maps: Heuristically, pulling back a tensor to p ∈ M from F(p) ∈ N feeding it vectors residing at p ∈ M is by definition the same as pushing forward the vectors from p ∈ M to F(p) ∈ N feeding them to the tensor residing at F(p) ∈ N. Further unwinding the definitions, the pushforward F∗: TMp → TNF(p) of a vector field under a map F: M → N between manifolds is defined by $${\displaystyle F_{*}(X)f=X(f\circ F),}$$ where f is a function on N. When M = Rm, N= Rn the pushforward of F reduces to DF: Rm → Rn, the ordinary 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 F from Rm to Rn. 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 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, $${\displaystyle F^{*}\colon T_{F(p)}^{*}N\rightarrow T_{p}^{*}M.}$$

In order to exhibit the metric, it is necessary to pull it back via a suitable parametrization. A parametrization of a submanifold S of a manifold M is a map U ⊂ R^m → M whose range is an open subset of S. If S has the same dimension as M, a parametrization is just the inverse of a coordinate map 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 n = 2. It is instructive to compare to stereographic projection for spheres.

Stereographic projection σ: Hⁿ
_R → Rⁿ and its inverse σ⁻¹: Rⁿ → Hⁿ
_R are given by $${\displaystyle {\begin{aligned}\sigma (\tau ,\mathbf {x} )=\mathbf {u} &={\frac {R\mathbf {x} }{R+\tau }},\\\sigma ^{-1}(\mathbf {u} )=(\tau ,\mathbf {x} )&=\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}},{\frac {2R^{2}\mathbf {u} }{R^{2}-|u|^{2}}}\right),\end{aligned}}}$$ where, for simplicity, τ ≡ ct. The (τ, x) are coordinates on Mⁿ⁺¹ and the u are coordinates on Rⁿ.

One has $${\displaystyle h_{R}^{1(n)}=\eta |_{\mathbf {H} _{R}^{1(n)}}=\left(dx^{1}\right)^{2}+\cdots +\left(dx^{n}\right)^{2}-d\tau ^{2}}$$ and the map $${\displaystyle \sigma ^{-1}:\mathbf {R} ^{n}\rightarrow \mathbf {H} _{R}^{1(n)};\quad \sigma ^{-1}(\mathbf {u} )=(\tau (\mathbf {u} ),\,\mathbf {x} (\mathbf {u} ))=\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}},\,{\frac {2R^{2}\mathbf {u} }{R^{2}-|u|^{2}}}\right).}$$

The pulled back metric can be obtained by straightforward methods of calculus; $${\displaystyle \left.\left(\sigma ^{-1}\right)^{*}\eta \right|_{\mathbf {H} _{R}^{1(n)}}=\left(dx^{1}(\mathbf {u} )\right)^{2}+\cdots +\left(dx^{n}(\mathbf {u} )\right)^{2}-\left(d\tau (\mathbf {u} )\right)^{2}.}$$

One computes according to the standard rules for computing differentials (though one is really computing the rigorously defined exterior derivatives), $${\displaystyle {\begin{aligned}dx^{1}(\mathbf {u} )&=d\left({\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}\right)={\frac {\partial }{\partial u^{1}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{1}+\cdots +{\frac {\partial }{\partial u^{n}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{n}+{\frac {\partial }{\partial \tau }}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}d\tau ,\\&\ \ \vdots \\dx^{n}(\mathbf {u} )&=d\left({\frac {2R^{2}u^{n}}{R^{2}-|u|^{2}}}\right)=\cdots ,\\d\tau (\mathbf {u} )&=d\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}}\right)=\cdots ,\end{aligned}}}$$ and substitutes the results into the right hand side. This yields $${\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}={\frac {4R^{2}\left[\left(du^{1}\right)^{2}+\cdots +\left(du^{n}\right)^{2}\right]}{\left(R^{2}-|u|^{2}\right)^{2}}}\equiv h_{R}^{2(n)}.}$$

Detailed outline of computation

One has $${\displaystyle {\begin{aligned}{\frac {\partial }{\partial u^{1}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{1}&={\frac {2\left(R^{2}-|u|^{2}\right)+4R^{2}\left(u^{1}\right)^{2}}{\left(R^{2}-|u|^{2}\right)^{2}}}du^{1},\\{\frac {\partial }{\partial u^{2}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{2}&={\frac {4R^{2}u^{1}u^{2}}{\left(R^{2}-|u|^{2}\right)^{2}}}du^{2},\end{aligned}}}$$ and $${\displaystyle {\frac {\partial }{\partial \tau }}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}d\tau ^{2}=0.}$$ With this one may write $${\displaystyle dx^{1}(\mathbf {u} )={\frac {2R^{2}\left(R^{2}-|u|^{2}\right)du^{1}+4R^{2}u^{1}(\mathbf {u} \cdot d\mathbf {u} )}{\left(R^{2}-|u|^{2}\right)^{2}}},}$$ from which $${\displaystyle \left(dx^{1}(\mathbf {u} )\right)^{2}={\frac {4R^{2}\left(r^{2}-|u|^{2}\right)^{2}\left(du^{1}\right)^{2}+16R^{4}\left(R^{2}-|u|^{2}\right)\left(\mathbf {u} \cdot d\mathbf {u} \right)u^{1}du^{1}+16R^{4}\left(u^{1}\right)^{2}\left(\mathbf {u} \cdot d\mathbf {u} \right)^{2}}{\left(R^{2}-|u|^{2}\right)^{4}}}.}$$ Summing this formula one obtains $${\displaystyle {\begin{aligned}&\left(dx^{1}(\mathbf {u} )\right)^{2}+\cdots +\left(dx^{n}(\mathbf {u} )\right)^{2}\\={}&{\frac {4R^{2}\left(R^{2}-|u|^{2}\right)^{2}\left[\left(du^{1}\right)^{2}+\cdots +\left(du^{n}\right)^{2}\right]+16R^{4}\left(R^{2}-|u|^{2}\right)(\mathbf {u} \cdot d\mathbf {u} )(\mathbf {u} \cdot d\mathbf {u} )+16R^{4}|u|^{2}(\mathbf {u} \cdot d\mathbf {u} )^{2}}{\left(R^{2}-|u|^{2}\right)^{4}}}\\={}&{\frac {4R^{2}\left(R^{2}-|u|^{2}\right)^{2}\left[\left(du^{1}\right)^{2}+\cdots +\left(du^{n}\right)^{2}\right]}{\left(R^{2}-|u|^{2}\right)^{4}}}+R^{2}{\frac {16R^{4}(\mathbf {u} \cdot d\mathbf {u} )}{\left(R^{2}-|u|^{2}\right)^{4}}}.\end{aligned}}}$$ Similarly, for τ one gets $${\displaystyle d\tau =\sum _{i=1}^{n}{\frac {\partial }{\partial u^{i}}}R{\frac {R^{2}+|u|^{2}}{R^{2}+|u|^{2}}}du^{i}+{\frac {\partial }{\partial \tau }}R{\frac {R^{2}+|u|^{2}}{R^{2}+|u|^{2}}}d\tau =\sum _{i=1}^{n}R^{4}{\frac {4R^{2}u^{i}du^{i}}{\left(R^{2}-|u|^{2}\right)}},}$$ yielding $${\displaystyle -d\tau ^{2}=-\left(R{\frac {4R^{4}\left(\mathbf {u} \cdot d\mathbf {u} \right)}{\left(R^{2}-|u|^{2}\right)^{2}}}\right)^{2}=-R^{2}{\frac {16R^{4}(\mathbf {u} \cdot d\mathbf {u} )^{2}}{\left(R^{2}-|u|^{2}\right)^{4}}}.}$$ Now add this contribution to finally get $${\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}={\frac {4R^{2}\left[\left(du^{1}\right)^{2}+\cdots +\left(du^{n}\right)^{2}\right]}{\left(R^{2}-|u|^{2}\right)^{2}}}\equiv h_{R}^{2(n)}.}$$

This last equation shows that the metric on the ball is identical to the Riemannian metric h²⁽ⁿ⁾
_R in the Poincaré ball model, another standard model of hyperbolic geometry.

Alternative calculation using the pushforward

The pullback can be computed in a different fashion. By definition, $${\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}(V,\,V)=h_{R}^{1(n)}\left(\left(\sigma ^{-1}\right)_{*}V,\,\left(\sigma ^{-1}\right)_{*}V\right)=\eta |_{\mathbf {H} _{R}^{1(n)}}\left(\left(\sigma ^{-1}\right)_{*}V,\,\left(\sigma ^{-1}\right)_{*}V\right).}$$ In coordinates, $${\displaystyle \left(\sigma ^{-1}\right)_{*}V=\left(\sigma ^{-1}\right)_{*}V^{i}{\frac {\partial }{\partial u^{i}}}=V^{i}{\frac {\partial x^{j}}{\partial u^{i}}}{\frac {\partial }{\partial x^{j}}}+V^{i}{\frac {\partial \tau }{\partial u^{i}}}{\frac {\partial }{\partial \tau }}=V^{i}{\frac {\partial }{x}}^{j}{\partial u^{i}}{\frac {\partial }{\partial x^{j}}}+V^{i}{\frac {\partial }{\tau }}{\partial u^{i}}{\frac {\partial }{\partial \tau }}=Vx^{j}{\frac {\partial }{\partial x^{j}}}+V\tau {\frac {\partial }{\partial \tau }}.}$$ One has from the formula for σ–1 $${\displaystyle {\begin{aligned}Vx^{j}&=V^{i}{\frac {\partial }{\partial u^{i}}}\left({\frac {2R^{2}u^{j}}{R^{2}-|u|^{2}}}\right)={\frac {2R^{2}V^{j}}{R^{2}-|u|^{2}}}-{\frac {4R^{2}u^{j}\langle \mathbf {V} ,\,\mathbf {u} \rangle }{\left(R^{2}-|u|^{2}\right)^{2}}},\quad \left({\text{here }}V|u|^{2}=2\sum _{k=1}^{n}V^{k}u^{k}\equiv 2\langle \mathbf {V} ,\,\mathbf {u} \rangle \right)\\V\tau &=V\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}}\right)={\frac {4R^{3}\langle \mathbf {V} ,\,\mathbf {u} \rangle }{\left(R^{2}-|u|^{2}\right)^{2}}}.\end{aligned}}}$$ Lastly, $${\displaystyle \eta \left(\sigma _{*}^{-1}V,\,\sigma _{*}^{-1}V\right)=\sum _{j=1}^{n}\left(Vx^{j}\right)^{2}-(V\tau )^{2}={\frac {4R^{4}|V|^{2}}{\left(R^{2}-|u|^{2}\right)^{2}}}=h_{R}^{2(n)}(V,z,V),}$$ and the same conclusion is reached.

• Hyperbolic quaternion
• Hyperspace
• Introduction to the mathematics of general relativity
• Minkowski plane

Media related to Minkowski diagrams at Wikimedia Commons

• Animation clip on YouTube visualizing Minkowski space in the context of special relativity.
• The Geometry of Special Relativity: The Minkowski Space – Time Light Cone
• Minkowski space at PhilPapers