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 -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, .
Tangent vectors . This vector space can be thought of as a subspace of itself. Then vectors in it would be called
geometrical tangent vectors. By the same principle, the tangent space at a point in flat spacetime can be thought of as a subspace of spacetime, which happens to be
all of spacetime. In practice, one need not be concerned with the tangent spaces. The vector space structure of Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself. See e.g. or These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as \begin{align} \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{align} with basis vectors in the tangent spaces defined by \left.\mathbf e_\mu\right|_p = \left.\frac{\partial}{\partial x^\mu}\right|_p \text{ or } \mathbf e_0|_p = \left(\begin{matrix} 1 \\ 0 \\ 0 \\ 0\end{matrix}\right) \text{, etc}. Here, and are any two events, and the second basis vector identification is referred to as
parallel transport. The first identification is the canonical identification of vectors in the tangent space at any point with vectors in the space itself. The appearance of basis vectors in tangent spaces as first-order differential operators is due to this identification. It is motivated by the observation that a geometrical tangent vector can be associated in a one-to-one manner with a
directional derivative operator on the set of smooth functions. This is promoted to a
definition of tangent vectors in manifolds
not necessarily being embedded in . This definition of tangent vectors is not the only possible one, as ordinary
n-tuples can be used as well. A tangent vector at a point may be defined, here specialized to Cartesian coordinates in Lorentz frames, as column vectors associated to
each Lorentz frame related by Lorentz transformation such that the vector in a frame related to some frame by transforms according to . This is the
same way in which the coordinates transform. Explicitly, \begin{align} x'^\mu &= {\Lambda^\mu}_\nu x^\nu, \\ v'^\mu &= {\Lambda^\mu}_\nu v^\nu. \end{align} This definition is equivalent to the definition given above under a canonical isomorphism. For some purposes, it is desirable to identify tangent vectors at a point with
displacement vectors at , which is, of course, admissible by essentially the same canonical identification. The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in . They offer various 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. In general, but with several exceptions, mathematicians and general relativists prefer spacelike vectors to yield a positive sign, , while particle physicists tend to prefer timelike vectors to yield a positive sign, . Authors covering several areas of physics, e.g.
Steven Weinberg and
Landau and Lifshitz and , respectively stick to one choice regardless of topic. Arguments for the former convention include "continuity" from the Euclidean case corresponding to the non-relativistic limit . Arguments for the latter include that minus signs, otherwise ubiquitous in particle physics, go away. Yet other authors, especially of introductory texts, e.g. , do
not choose a signature at all, but instead, opt to coordinatize spacetime such that the time
coordinate (but not time itself!) is imaginary. This removes the need for the
explicit introduction of a
metric tensor (which may seem like an extra burden in an introductory course), and one need
not be concerned with
covariant vectors and
contravariant vectors (or raising and lowering indices) to be described below. The inner product is instead effected by a straightforward extension of the
dot product from \mathbb{R}^3 over to \mathbb{C} \times \mathbb{R}^3. This works in the flat spacetime of special relativity, but not in the curved spacetime of general relativity, see who, by the way use MTW also argues that it hides the true
indefinite nature of the metric and the true nature of Lorentz boosts, which are not rotations. It also needlessly complicates the use of tools of
differential geometry that are otherwise immediately available and useful for geometrical description and calculation – even in the flat spacetime of special relativity, e.g. of the electromagnetic field.
Terminology Mathematically associated with the bilinear form is a
tensor of type 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 matrix representing the bilinear form. For comparison, in
general relativity, a
Lorentzian manifold is likewise equipped with a
metric tensor , which is a nondegenerate symmetric bilinear form on the tangent space at each point of . In coordinates, it may be represented by a matrix
depending on spacetime position. Minkowski space is thus a comparatively simple special case of a
Lorentzian manifold. Its metric tensor is in coordinates with the same
symmetric matrix at every point of , and its arguments can, per above, be taken as vectors in spacetime itself. Introducing more terminology (but not more structure), Minkowski space is thus a
pseudo-Euclidean space with total dimension and
signature or . Elements of Minkowski space are called
events. Minkowski space is often denoted or to emphasize the chosen signature, or just . It is an example of a
pseudo-Riemannian manifold. Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space , that is, \eta:V\times V\rightarrow \mathbf{R} where has signature , and signature is a coordinate-invariant property of . The space of bilinear maps forms a vector space which can be identified with M^*\otimes M^*, and may be equivalently viewed as an element of this space. By making a choice of orthonormal basis \{e_\mu\}, M:=(V,\eta) can be identified with the space \mathbf{R}^{1,3}:=(\mathbf{R}^{4},\eta_{\mu\nu}). The notation is meant to emphasize the fact that and \mathbf{R}^{1,3} are not just vector spaces but have added structure. \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 is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally, a
constant pseudo-Riemannian metric in Cartesian coordinates. As such, it is a nondegenerate symmetric bilinear form, a type tensor. It accepts two arguments , vectors in , the tangent space at in . Due to the above-mentioned canonical identification of with itself, it accepts arguments with both and in . As a notational convention, vectors in , called
4-vectors, are denoted in italics, and not, as is common in the Euclidean setting, with boldface . The latter is generally reserved for the -vector part (to be introduced below) of a -vector. The definition u \cdot v = \eta(u,\, v) yields an inner product-like structure on , previously and also henceforth, called the
Minkowski inner product, similar to the Euclidean
inner product, but it describes a different geometry. It is also called the
relativistic dot product. If the two arguments are the same, u \cdot u = \eta(u, u) \equiv \|u\|^2 \equiv u^2, the resulting quantity will be called the
Minkowski norm squared. The Minkowski inner product satisfies the following properties. ; Linearity in the first argument : \eta(au + v,\, w) = a\eta(u,\, w) + \eta(v,\, w),\quad \forall u,\, v \in M,\; \forall a \in \R ; Symmetry : \eta(u,\, v) = \eta(v,\, u) ; Non-degeneracy : \eta(u,\, v) = 0,\; \forall v \in M\ \Rightarrow\ u = 0 The first two conditions imply bilinearity. The 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 is identical in the case (the Lorentz group) to the matrix to be displayed below.
Orthogonality Minkowski space 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: . 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 space. 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 and is: c^2\left(t_1 - t_2\right)^2 - \left(x_1 - x_2\right)^2 - \left(y_1 - y_2\right)^2 - \left(z_1 - z_2\right)^2. This quantity is not consistently named in the literature. The interval is sometimes referred to as the square root of the interval as defined here. The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of c^2 t^2 - x^2 - y^2 - z^2 provided the transformations are linear. This
quadratic form can be used to define a bilinear form u \cdot v = c^2 t_1 t_2 - x_1 x_2 - y_1 y_2 - z_1 z_2 via the
polarization identity. This bilinear form can in turn be written as u \cdot v = u^\textsf{T} \, [\eta] \, v, where is a 4\times 4 matrix associated with . While possibly confusing, it is common practice to denote with just . The matrix is read off from the explicit bilinear form as \eta = \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 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 has been used in a derivation, go back to the earliest point where it was used, substitute for , and retrace forward to the desired formula with the desired metric signature.
Standard basis A standard or orthonormal basis for Minkowski space is a set of four mutually orthogonal vectors such that \eta(e_0, e_0) = -\eta(e_1, e_1) = -\eta(e_2, e_2) = -\eta(e_3, e_3) = 1 and for which \eta(e_\mu, e_\nu) = 0 when \mu \neq \nu\,. These conditions can be written compactly in the form \eta(e_\mu, e_\nu) = \eta_{\mu \nu}. Relative to a standard basis, the components of a vector are written where the
Einstein notation is used to write . The component is called the
timelike component of while the other three components are called the
spatial components. The spatial components of a -vector may be identified with a -vector . In terms of components, the Minkowski inner product between two vectors and is given by \eta(v, w) = \eta_{\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 \eta(v, v) = \eta_{\mu \nu} v^\mu v^\nu = v^0v_0 + v^1 v_1 + v^2 v_2 + v^3 v_3 = v^\mu v_\mu. Here
lowering of an index with the metric was used. There are many possible choices of standard basis obeying the condition \eta(e_\mu, e_\nu) = \eta_{\mu \nu}. Any two such bases are related in some sense by a Lorentz transformation, either by a change-of-basis matrix \Lambda^\mu_\nu, a real matrix satisfying \Lambda^\mu_\rho\eta_{\mu \nu}\Lambda^\nu_\sigma = \eta_{\rho \sigma}. or , a linear map on the abstract vector space satisfying, for any pair of vectors , , \eta(\Lambda u, \Lambda v) = \eta(u, v). Then if two different bases exist, and , e_\mu' = e_\nu\Lambda^\nu_\mu can be represented as e_\mu' = e_\nu\Lambda^\nu_\mu or e_\mu' = \Lambda e_\mu. While it might be tempting to think of \Lambda^\mu_\nu and 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 Euclidean space. The number of (1-form)
hyperplanes intersected by a vector equals the
inner product. Technically, a non-degenerate bilinear form provides a map between a vector space and its dual; in this context, the map is between the tangent spaces of and the
cotangent spaces of . At a point in , the tangent and cotangent spaces are
dual vector spaces (so the dimension of the cotangent space at an event is also ). Just as an authentic inner product on a vector space with one argument fixed, by
Riesz representation theorem, 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. Thus if are the components of a vector in tangent space, then are the components of a vector in the cotangent space (a linear functional). Due to the identification of vectors in tangent spaces with vectors in itself, this is mostly ignored, and vectors with lower indices are referred to as
covariant vectors. In this latter interpretation, the covariant vectors are (almost always implicitly) identified with vectors (linear functionals) in the dual of Minkowski space. The ones with upper indices are
contravariant vectors. In the same fashion, the inverse of the map from tangent to cotangent spaces, explicitly given by the inverse of in matrix representation, can be used to define
raising of an index. The components of this inverse are denoted . It happens that . These maps between a vector space and its dual can be denoted (eta-flat) and (eta-sharp) by the musical analogy. Contravariant and covariant vectors are geometrically very different objects. The first can and should be thought of as arrows. A linear 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. The notation on the far right is also sometimes used for the related, but different,
line element. It is
not a tensor. For elaboration on the differences and similarities, see
Tangent vectors are, in this formalism, given in terms of a basis of differential operators of the first order, \left.\frac{\partial}{\partial x^\mu}\right|_p\ , where is an event. This operator applied to a function gives the
directional derivative of at in the direction of increasing with fixed. They provide a basis for the tangent space at . The exterior derivative of a function is a
covector field, i.e. an assignment of a cotangent vector to each point , by definition such that \operatorname{d}f(X) = X\ f, for each
vector field . A vector field is an assignment of a tangent vector to each point . In coordinates can be expanded at each point in the basis given by the Applying this with , the coordinate function itself, and called a
coordinate vector field, one obtains \operatorname{d}x^\mu\left(\frac{\partial}{\partial x^\nu}\right) = \frac{\partial x^\mu}{\partial x^\nu} = \delta_\nu^\mu ~. Since this relation holds at each point , the provide a basis for the cotangent space at each and the bases and are
dual to each other, \Bigl.\operatorname{d}x^\mu \Bigr|_p \left(\left.\frac{\partial}{\partial x^\nu}\right|_p\right) = \delta^\mu_\nu ~. at each . Furthermore, one has \alpha\ \otimes\ \beta(a, b)\ =\ \alpha(a)\ \beta(b)\ for general one-forms on a tangent space and general tangent vectors . (This can be taken as a definition, but may also be proved in a more general setting.) Thus when the metric tensor is fed two vectors fields , , both expanded in terms of the basis coordinate vector fields, the result is \eta_{\mu\nu}\ \operatorname{d}x^\mu \otimes \operatorname{d}x^\nu(a, b)\ =\ \eta_{\mu\nu}\ a^\mu\ b^\nu\ , where , are the
component functions of the vector fields. The above equation holds at each point , and the relation may as well be interpreted as the Minkowski metric at applied to two tangent vectors at . As mentioned, in a vector space, such as modeling the spacetime of special relativity, tangent vectors can be canonically identified with vectors in the space itself, and vice versa. This means that the tangent spaces at each point are canonically identified with each other and with the vector space itself. This explains how the right-hand side of the above equation can be employed directly, without regard to the spacetime point the metric is to be evaluated and from where (which tangent space) the vectors come from. This situation changes in
general relativity. There one has g_{\mu\nu}\!(p)\ \Bigl.\operatorname{d}x^\mu \Bigr|_p\ \left. \operatorname{d}x^\nu \right|_p(a, b)\ =\ g_{\mu\nu}\!(p)\ a^\mu\ b^\nu\ , where now , i.e., is still a metric tensor but now depending on spacetime and is a solution of
Einstein's field equations. Moreover,
must be tangent vectors at spacetime point and can no longer be moved around freely.
Chronological and causality relations Let . Here, •
chronologically precedes if is future-directed timelike. This relation has the
transitive property and so can be written . •
causally precedes if is future-directed null or future-directed timelike. It gives a
partial ordering of spacetime and so can be written . Suppose is timelike. Then the
simultaneous hyperplane for is . Since this
hyperplane varies as varies, there is a
relativity of simultaneity in Minkowski space. == Generalizations ==