Minkowski space

The model helps show how a spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.

Mathematician Hermann Minkowski developed it from the work of Hendrik Lorentz, Henri Poincaré, and others said it "was grown on experimental physical grounds".

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".

From his reformulation, he concluded that time and space should be treated equally, and so arose his concept of events taking place in a unified four-dimensional spacetime continuum.

In 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.

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.

An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit 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.

This addition is not required, and more complex treatments analogous to an affine space can remove the extra structure.

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.

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.

Authors covering several areas of physics, e.g. Steven Weinberg and Landau and Lifshitz ((− + + +) and (+ − − −) respectively) stick to one choice regardless of topic.

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.

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.

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.

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).

The relation is preserved in a change of reference frames and consequently the computation of light speed yields a constant result.

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 = vμ eμ.

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.

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 present purpose is to show semi-rigorously how formally one may apply the Minkowski metric to two vectors and obtain a real number, i.e. to display the role of the differentials and how they disappear in a calculation.

The setting is that of smooth manifold theory, and concepts such as convector fields and exterior derivatives are introduced.

They provide a basis for the tangent space at p. The exterior derivative df of a function f is a covector field, i.e. an assignment of a cotangent vector to each point p, by definition such that

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.

However, the mathematics can easily be extended or simplified to create an analogous generalized Minkowski space in any number of dimensions.

Even in curved spacetime, Minkowski space is still a good description in an infinitesimal region surrounding any point (barring gravitational singularities).

[24] The 1 in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the n for its dimension.

meaning (with slight abuse of notation) the restriction of α to accept as input vectors tangent to some s ∈ S only.

Hermann Minkowski (1864–1909) found that the theory of special relativity could be best understood as a four-dimensional space, since known as the Minkowski spacetime.
Subdivision of Minkowski spacetime with respect to an event in four disjoint sets: the light cone , the causal future (also called the absolute future ), the causal past (also called the absolute past ), and elsewhere . The terminology is from Sard (1970) , and from Causal structure .
A pictorial representation of the tangent space at a point, x , on a sphere . This vector space can be thought of as a subspace of R 3 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.
The blue axes are hyperbolically-orthogonal; light speed is constant over all such pairs
Linear functionals (1-forms) α , β and their sum σ and vectors u , v , w , in 3d Euclidean space . The number of (1-form) hyperplanes intersected by a vector equals the inner product . [ 16 ]
Red circular arc is geodesic in Poincaré disk model ; it projects to the brown geodesic on the green hyperboloid.