Minkowski space

Complex Minkowski spacetime

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.

Real Minkowski spacetime

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).

Scalar product

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).

Norm and reversed Cauchy inequality

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 $${\displaystyle \eta (u_{1},u_{2})>\left\|u_{1}\right\|\left\|u_{2}\right\|}$$\left\|u_{1}\right\|\left\|u_{2}\right\|}"> or algebraically, $${\displaystyle c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}>{\sqrt {\left(c^{2}t_{1}^{2}-x_{1}^{2}-y_{1}^{2}-z_{1}^{2}\right)\left(c^{2}t_{2}^{2}-x_{2}^{2}-y_{2}^{2}-z_{2}^{2}\right)}}}$${\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.

Reversed triangle inequality

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.

Tangent vectors

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.

Metric signature

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

Terminology

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.

Pseudo-Euclidean metrics

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.

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 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.

Standard basis

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.

Raising and lowering of indices

Main articles: Raising and lowering indices and tensor contraction

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.

Chronological and causality relations

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.

Complexified Minkowski space

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.

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 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).

Curvature

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

Even in curved space, Minkowski space is still a good description in an 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.

Preliminaries

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.

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 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.

Pulling back the metric

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 $${\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}={\frac {4R^{2}\left[\left(du^{1}\right)^{2}+\cdots +\left(du^{n}\right)^{2}\right]}{\left(R^{2}-|u|^{2}\right)^{2}}}\equiv h_{R}^{2(n)}.}$$

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.