Minkowski space

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For spacetime graphics, see Minkowski diagram. For Minkowski space associated to a number field, see Minkowski space (number field). For geometry of the Minkowski plane, see Minkowski plane .

In mathematical physics, Minkowski space, Minkowski spacetime, named after the mathematician Hermann Minkowski, is the mathematical model of physical spacetime in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime.

The spacetime interval between any two events in Minkowski space is independent of the inertial system in which it recorded. This is an immediate consequence of the postulates of special relativity, and uniquely fixes the mathematical structure of Minkowski spacetime.[1]

Minkowski space is often contrasted with Euclidean space. While a Euclidean space has only spacelike dimensions, a Minkowski space also has one timelike dimension with different mathematical and physical properties. The isometry group, preserving Euclidean distances of a Euclidean space equipped with the regular inner product is the Euclidean group. The isometry group, preserving the spacetime interval of spacetime equipped with the non-positive definite bilinear form appropriate to Minkowski space, here called the Minkowski inner product,[nb 1] is the Poincaré group. The Minkowski inner product is defined as to yield the spacetime interval between two events when given their coordinate difference vector as argument.

History[edit]

Hermann Minkowski (1864 – 1909) was a German mathematician. He found that the theory of special relativity, introduced by his former student Albert Einstein, could best be understood in a four-dimensional space, since known as the Minkowski spacetime.

In 1905, with the publication in 1906, it was noted by Henri Poincaré that, by taking time to be the imaginary part of the fourth spacetime coordinate √−1 c t, a Lorentz transformation can be regarded as a rotation of coordinates in a four-dimensional Euclidean space with three real coordinates representing space, and one imaginary coordinate, representing time, as the fourth dimension. Since the space is then a pseudo-Euclidean space, the rotation is a representation of a hyperbolic rotation, although Poincaré did not give this interpretation, his purpose being only to explain the Lorentz transformation in terms of the familiar Euclidean rotation.[2]

This idea was elaborated by Hermann Minkowski,[3] who used it to restate the Maxwell equations in four dimensions, showing directly their invariance under the Lorentz transformation. He further reformulated in four dimensions the then-recent theory of special relativity of Einstein. From this 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,[4] he gave an alternative formulation of this idea that did not use the imaginary time coordinate, but represented the four variables (x, y, z, t) of space and time in coordinate form in a four dimensional affine space. Points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point (see diagram above), 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. 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, 1907[4]

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

Mathematical structure[edit]

For an overview, Minkowski space is a 4-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form on the tangent space at each point in spacetime, here simply called the Minkowski inner product, with signature either (−,+,+,+) or (+,−,−,−). In practice, one need not be concerned with the tangent spaces. The vector space nature of Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself.[5] 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.[6]

The signature refers to which sign the Minkowski inner product yields when given space and time basis vectors as arguments. In general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter. Arguments for the former (pure space vectors yield positive "norm-squared") include "continuity" from the Euclidean case corresponding to the non-relativistic limit c → ∞. Arguments for the latter (pure space vectors yield negative "norm-squared") include that otherwise ubiquitous minus signs in particle physics go away.

Mathematically associated to this 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 actually all the very same object. In coordinates, this is the 4×4 matrix representing the bilinear form. Keeping this in mind may facilitate reading what follows.

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 space is thus a comparatively simple special case of a Lorentzian manifold. Its metric tensor, called the Minkowski metric, is in coordinates the same symmetric matrix at every point of M, and its arguments can, per above, be taken as vectors in spacetime itself. More detail will be presented below.

Introducing more terminology (but not more structure), Minkowski space is thus a pseudo-Euclidean space with total dimension n = 4 and signature (3, 1) or (1, 3). Elements of Minkowski space are called events. Minkowski space is often denoted R3,1 or R1,3 to emphasize the chosen signature, or just M. It is perhaps the simplest example of a pseudo-Riemannian manifold.

Pseudo-Euclidean metric generalities[edit]

The Minkowski metric[nb 2] η is the metric tensor of Minkowski space. It is a Pseudo-Euclidean metric. As such it is a nondegenerate symmetric bilinear form, a type (0,2) tensor. It accepts two arguments up, vp, vectors in TpM, pM, 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 sans-serif italics, and not, as is common in the Eucliedean 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 \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 has the following properties.

  • \eta(au + v, w) = a\eta(u, w) + \eta(v,  w), \quad \forall  u, v \in M, \forall a \in \mathbb R \qquad \text{(linearity in first slot)}
  • \eta( u,  v) = \eta( v,  u) \qquad \text{(symmetry)}
  • \eta( u,  v) = 0 \quad \forall v \in M \Rightarrow u = 0 \qquad \text{(non-degeneracy)}

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.

Two vectors v and w are said to be orthogonal if η(v, w) = 0.

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.

For a given inertial frame, an orthonormal basis in space, combined by 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[edit]

From the two postulates of special relativity follows that the spacetime interval between two events 1, 2,

\pm\left[c^2(t_1 - t_2)^2 - (x_1 - x_2)^2 - (y_1 - y_2)^2 - (z_1 - z_2)^2\right],

is independent of the inertial frame chosen. The factor ± simply means that the choice of signature is left open. The numerical values of η, viewed as a matrix representing the Minkowski inner product, follow from the theory of bilinear forms.

Just the signature of the metric is differently defined in the literature, this quantity is not consistently named. The interval (as defined here) is sometimes referred to as the interval squared.[7] Even the square root of the present interval occurs.[8] When signature and interval are fixed, ambiguity still remains as which coordinate is the time coordinate. It may be the fourth, or it may be the zeroth. This is not an exhaustive list of notational inconsistencies. It is a fact of life that one has to check out the definitions first thing when one consults the relativity literature.

The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of

\pm\left[c^2t^2 - x^2 - y^2 - z^2\right]

(with either sign ± preserved), provided the transformations are linear. This quadratic form can be used to define a bilinear form

 x \cdot  y = \pm\left[c^2t_1t_1 - x_1x_2 - y_1y_2 - z_1z_2\right].

via the polarization identity. This bilinear form can in turn be written as

 x \cdot  y =  x^{\mathrm T}[\eta] y,

where [η] is a 4×4 matrix associated with η. Possibly confusingly, denote [η] with just η as is common practice. The matrix is read off from the explicit bilinear form as

\eta = \pm \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix},

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. The 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.

Standard basis[edit]

A standard basis for Minkowski space is a set of four mutually orthogonal vectors { e0, e1, e2, e3 } such that

-\eta(e_0, e_0) = \eta(e_1, e_1) = \eta(e_2, e_2) = \eta(e_3, e_3) = 1 .

These conditions can be written compactly in the form

\eta(e_\mu, e_\nu) = \eta_{\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 = vμeμ. 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

\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. 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. 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 vμ are the components of a vector in a tangent space, then ημνvμ = vν 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 ημν. It happens that ημν = ημν. These maps between a vector space and its dual can be denoted η (eta-flat) and η (eta-sharp) by the musical analogy.[9]

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 is mathematically sound. Incorrect expressions tend to reveal themselves quickly.

Geometry[edit]

Generalizations and variations[edit]

The term "Minkowski space" is also used for analogues in any dimension. If n ≥ 2, n-dimensional Minkowski space is a vector space of real dimension n on which there is a constant Lorentz 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.

Lorentz transformations and symmetry[edit]

Standard configuration of coordinate systems for Lorentz transformations.

The Poincaré group is the group of all transformations preserving the interval. The interval is quite easily seen to be preserved by the translation group in 4 dimensions. The other transformations are those that preserve the interval and leave the origin fixed. Given the bilinear form associated with the Minkowski metric, the appropriate group follows directly from the theory (in particular the definition) of classical groups. In the linked article, one should identify η (in its a matrix representation) with the matrix Φ.

The appropriate group is O(3,1), in this context called the Lorentz group. Its elements are called (homogeneous) Lorentz transformations. For other methods of derivation, with a more physical twist, see derivations of the Lorentz transformations.

Among the simplest Lorentz transformations is a Lorentz boost. For reference, a boost in the x-direction is given by


\begin{bmatrix}
U'_0 \\ U'_1 \\ U'_2 \\ U'_3
\end{bmatrix}
=
\begin{bmatrix}
\gamma&-\beta \gamma&0&0\\
-\beta \gamma&\gamma&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{bmatrix}
\begin{bmatrix}
U_0 \\ U_1 \\ U_2 \\ U_3
\end{bmatrix},

where

\gamma = { 1 \over \sqrt{1 - {v^2 \over c^2}} }

is the Lorentz factor, and

\beta = { v \over c} \,.

Other Lorentz transformations are pure rotations, and hence elements of the SO(3) subgroup of O(3,1). A general homogeneous Lorentz transformation is a product of a pure boost and a pure rotation. An inhomogeneous Lorentz transformation is a homogeneous transformation followed by a translation in space and time. Special transformations are those that invert the space coordinates (P) and time coordinate (T) respectively, or both (PT).

All four-vectors in Minkowski space transform, by definition, according to the same formula under Lorentz transformations. Minkowski diagrams illustrate Lorentz transformations.

Causal structure[edit]

Subdivision of Minkowski spacetime with respect to an event in four disjoint sets. The light cone, the absolute future, the absolute past, and elsewhere. The terminology is from Sard (1970).
Main article: Causal structure

Vectors v = (ct, x, y, z) = (ct, r) are 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) as well, but depends on the signature.

These notions are independent of the frame of reference due to the invariance of the interval.

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 absolute future in figure) and
  2. past-directed timelike vectors whose first component is negative (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).

Spacelike vectors are in elsewhere. The terminology stems from the fact that spacelike separated events cannot possibly influence each other. Together with spacelike and lightlike vectors there are 7 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. Over the reals, if two null vectors are orthogonal (zero Minkowski tensor value), then they must be proportional. However, allowing complex numbers, one can obtain a null tetrad, which is a basis consisting of null vectors, some of which are orthogonal to each other.

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.

Causality relations[edit]

Let x, yM. We say that

  1. x chronologically precedes y if yx is future-directed timelike.
  2. x causally precedes y if yx is future-directed null or future-directed timelike

Reversed triangle inequality[edit]

If v and w are both future-directed timelike four-vectors, then

 \left\| v+w \right\| \ge \left\| v \right\| + \left\| w \right\| .

Locally flat spacetime[edit]

Strictly speaking, the use of the Minkowski space to describe physical systems over finite distances applies only in systems without significant gravitation. In the case of significant gravitation, spacetime becomes curved and one must abandon special relativity in favor of the full theory of general relativity.

Nevertheless, even in such cases, Minkowski space is still a good description in an infinitesimal region surrounding any point (barring gravitational singularities). More abstractly, we say 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.

In the realm of weak gravity, spacetime becomes flat and looks globally, not just locally, like Minkowski space. For this reason Minkowski space is often referred to as flat spacetime.

See also[edit]

Remarks[edit]

  1. ^ Consistent use of the term "Minkowski inner product" is intended for the bilinear form here, since it is in spread use. It is by no means "standard" in the literature, but no such standard seems to exist.
  2. ^ 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.
  3. ^ Translate the coordinate system so that the event is the new origin.
  4. ^ This corresponds to the time coordinate either increasing or decreasing when proper time for any particle increases. An application of T flips this direction.

Notes[edit]

  1. ^ Landau & Lifshitz 2002, p. 5
  2. ^ Poincaré 1905–1906, pp. 129–176 Wikisource translation: On the Dynamics of the Electron
  3. ^ Minkowski 1907–1908, pp. 53–111 *Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies.
  4. ^ a b Minkowski 1907–1909, pp. 75–88 Various English translations on Wikisource: Space and Time.
  5. ^ Lee 2003, Proposition 3.8. The identification is routinely done in mathematics.
  6. ^ Lee 2003, See Lee's discussion on geometric tangent vectors early in chapter 3.
  7. ^ Sard 1970, p. 71
  8. ^ Landau & Lifshitz 2002, p. 4
  9. ^ Lee 2003, The tangent-cotangent isomorphism p. 282.

References[edit]

  • Lee, J. M. (2003). Introduction to Smooth manifolds. Springer Graduate Texts in Mathematics 218. ISBN 0-387-95448-1. 
  • Minkowski, Hermann (1907–1909), "Raum und Zeit" [Space and Time], Physikalische Zeitschrift 10: 75–88  Various English translations on Wikisource: Space and Time
  • Sard, R. D. (1970). Relativistic Mechanics - Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 978-0805384918. 
  • Shaw, R. (1982). "§ 6.6 Minkowski space, § 6.7,8 Canonical forms pp 221–242". Linear Algebra and Group Representations. Academic Press. ISBN 0-12-639201-3. 

External links[edit]

Media related to Minkowski diagrams at Wikimedia Commons