In physics, Minkowski spacetime (or Minkowski space; ) 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, and said it "was grown on experimental physical grounds". Minkowski spacetime 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. Minkowski spacetime differs from four-dimensional Euclidean space insofar as it treats time differently from 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. Minkowski spacetime is a pseudo-Euclidean space equipped with an isotropic quadratic form called the spacetime interval or the Minkowski norm squared. In Euclidean space, the distance between any two distinct points is greater than zero, but in Minkowski spacetime, the interval between two distinct events can be zero. This occurs when one event is on the light cone of the other. Using the polarization identity the quadratic form is converted to a symmetric bilinear form called the Minkowski inner product, though it is not a geometric inner product. The group of transformations for Minkowski spacetime that preserves the spacetime interval (as opposed to the spatial Euclidean distance) is the Lorentz group (as opposed to the Galilean group).
History
Geometry of Lorentz transformations
In the 1890s, Hendrik Lorentz began to develop theories of electrodynamics based on a pervasive luminiferous aether uncoupled from matter. In a series of papers from 1892 to 1904 he used transformations which would come to bear his name as mathematical aids in analyzing these theories. Many other physicists were also involved in the development and understanding of these transformation. Minkowski was especially familiar with work of Henri Poincaré and especially his second relativity paper in 1905. In that paper Poincaré showed 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 sphere, 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. As part of his analysis Poincaré described invariants under the rotation, like the quantity
x 2 + y 2 + z 2 − t 2 , {\displaystyle x^{2}+y^{2}+z^{2}-t^{2},}
where the x , y , z , t {\displaystyle x,y,z,t} are differences between points in the four dimensional space, as distances. (The sum of the first three differences would be the square of the Euclidean distance in 3 space; such distances are invariant under rotation in 3 space). The rotation and concept of invariants as distances was elaborated by Minkowski, in a paper in German published in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". The paper introduced tensors to the mathematical representation of the physics and 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. He was able to show their invariance under Lorentz transformation. 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.
"Space and Time" lecture In 1908 Minkowski presented his ideas at a lecture in Cologne, Germany before an international audience including both physicists and mathematicians. The lecture came to be known as his 1908 "Space and Time" lecture and was later published in several forms. Minkowski started his lecture with what would become his most famous comment:
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. Despite his starting point in experimental physics, the remainder of Minkowski's lecture emphasized mathematics. He emphasized the advantages of four-vector representation of coordinates and velocities. He introduced the Minkowski diagram and used it to define concepts and demonstrate properties of Lorentz transformations (e.g., proper time and length contraction) as well as to provide geometrical interpretation to the generalization of Newtonian mechanics to relativistic mechanics. The effect was to position the mathematics as revealing important concepts of physical reality, rather than simply a convenient tool.
Impact In Minkowski's own point of view, Einstein's 1905 paper established a new understanding of time as a local phenomenon, a consequence of the relativity of simultaneity. Minkowski saw his contribution as altering the fundamental idea of space. The Minkowski concept of spacetime, developed from studies of electrodynamics, became the basis for the development of relativistic mechanics. The four-vector approach to the geometry of spacetime became the standard language for understanding relativity. Minkowski himself showed that the 3-space current density, j, combines with the charge density, ρ, to form a four-vector. The 3-space electric field, E, combines with the 3-space magnetic field, B, to create a tensor in the four-vector formalism. This approach makes it clear that the E and B fields transform into one another depending on the inertial frame of the observer. Though Minkowski took an important step for physics, Albert Einstein was not initially enthusiastic. Minkowski's approach excluded gravitation, a problem Einstein knew needed to be included.
Causal structure
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 of Minkowski spacetime 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, timelike and null vectors can be further decomposed into various classes. For timelike vectors, one has
future-directed timelike vectors whose first component is positive (tip of vector located in causal future (also called the absolute future) in the figure) and past-directed timelike vectors whose first component is negative (causal past (also called the absolute past)). Null vectors fall into three classes:
the zero vector, whose components in any basis are (0, 0, 0, 0) (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 basis for Minkowski spacetime 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 that are accessible to the observer at (0, 0, 0, 0) with a speed less than that of light. Of most interest are time-like vectors that are similarly directed, i.e. all either in the forward or in the backward cones. Such vectors have several properties not shared by space-like vectors. These arise because both forward and backward cones are convex, whereas the space-like region is not convex.
Scalar product The scalar product of two time-like vectors u1 = (t1, x1, y1, z1) and u2 = (t2, x2, y2, z2) is
η ( u 1 , u 2 ) = u 1 ⋅ u 2 = c 2 t 1 t 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 . {\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
‖ u ‖ = η ( u , u ) = c 2 t 2 − x 2 − y 2 − z 2 {\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. For two distinct similarly directed time-like vectors u1 and u2 this inequality is
η ( u 1 , u 2 ) > ‖ u 1 ‖ ‖ u 2 ‖ {\displaystyle \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 > ( c 2 t 1 2 − x 1 2 − y 1 2 − z 1 2 ) ( c 2 t 2 2 − x 2 2 − y 2 2 − z 2 2 ) {\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.
Reversed triangle inequality For two similarly directed time-like vectors u and w, the inequality is
‖ u + w ‖ ≥ ‖ u ‖ + ‖ w ‖ , {\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:
‖ u + w ‖ 2 = ‖ u ‖ 2 + 2 ( u , w ) + ‖ w ‖ 2 ≥ ‖ u ‖ 2 + 2 ‖ u ‖ ‖ w ‖ + ‖ w ‖ 2 = ( ‖ u ‖ + ‖ w ‖ ) 2 . {\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.
Mathematical structure It is assumed below that spacetime is endowed with a coordinate system corresponding to an inertial frame. This provides an origin, which is necessary for spacetime to be modeled as a vector space. This addition is not required, and more complex treatments analogous to an affine space can remove the extra structure. However, this is not the introductory convention and is not covered here. For an overview, Minkowski spacetime 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.
Tangent vectors
In practice, one need not be concerned with the tangent spaces. The vector space structure of Minkowski spacetime allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski spacetime 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
( x 0 , x 1 , x 2 , x 3 ) ↔ x 0 e 0 | p + x 1 e 1 | p + x 2 e 2 | p + x 3 e 3 | p ↔ x 0 e 0 | q + x 1 e 1 | q + x 2 e 2 | q + x 3 e 3 | q {\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
e μ | p = ∂ ∂ x μ | p or e 0 | p = ( 1 0 0 0 ) , etc . {\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. 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. 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 TpL at each point p of L. In coordinates, it may be represented by a 4×4 matrix depending on spacetime position. Minkowski spacetime 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 spacetime is thus a pseudo-Euclidean space with total dimension n = 4 and signature (1, 3) or (3, 1). Elements of Minkowski spacetime are called events. Minkowski spacetime is often denoted R1,3 or R3,1 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,
η : V × V → R {\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 M ∗ ⊗ M ∗ {\displaystyle M^{*}\otimes M^{*}} , and η may be equivalently viewed as an element of this space. By making a choice of orthonormal basis { e μ } {\displaystyle \{e_{\mu }\}} , M := ( V , η ) {\displaystyle M:=(V,\eta )} can be identified with the space R 1 , 3 := ( R 4 , η μ ν ) {\displaystyle \mathbf {R} ^{1,3}:=(\mathbf {R} ^{4},\eta _{\mu \nu })} . The notation is meant to emphasize the fact that M and R 1 , 3 {\displaystyle \mathbf {R} ^{1,3}} are not just vector spaces but have added structure. η μ ν = diag ( + 1 , − 1 , − 1 , − 1 ) {\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 has non-zero null vectors. Since it is not a definite bilinear form it is called indefinite. The Minkowski metric η is the metric tensor of Minkowski spacetime. 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 up, vp, vectors in TpM, p ∈ M, the tangent space at p in M. Due to the above-mentioned canonical identification of TpM 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
u ⋅ v = η ( u , v ) {\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,
u ⋅ u = η ( u , u ) ≡ ‖ u ‖ 2 ≡ u 2 , {\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
η ( a u + v , w ) = a η ( u , w ) + η ( v , w ) , ∀ u , v ∈ M , ∀ a ∈ R {\displaystyle \eta (au+v,\,w)=a\eta (u,\,w)+\eta (v,\,w),\quad \forall u,\,v\in M,\;\forall a\in \mathbb {R} }
Symmetry
η ( u , v ) = η ( v , u ) {\displaystyle \eta (u,\,v)=\eta (v,\,u)}
Non-degeneracy
η ( u , v ) = 0 , ∀ v ∈ M ⇒ u = 0 {\displaystyle \eta (u,\,v)=0,\;\forall v\in M\ \Rightarrow \ u=0}
The first two conditions imply bilinearity. 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.
Orthogonality
Minkowski spacetime is constructed so that the speed of light will be the same constant regardless of the reference frame in which it is measured. This property results from the relation of the time axis to a space axis. Two events u and v are orthogonal when the bilinear form is zero for them: η(v, w) = 0. When both u and v are both space-like, then they are perpendicular, but if one is time-like and the other space-like, then the relation is hyperbolic orthogonality. The relation is preserved in a change of reference frames and consequently the computation of light speed yields a constant result. The change of reference frame is called a Lorentz boost and in mathematics it is a hyperbolic rotation. Each reference frame is associated with a hyperbolic angle, which is zero for the rest frame in Minkowski spacetime. Such a hyperbolic angle has been labelled rapidity since it is associated with the speed of the frame.
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:
c 2 ( t 1 − t 2 ) 2 − ( x 1 − x 2