Minkowski space


In physics, 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 space 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 space 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 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 space is a pseudo-Euclidean space equipped with an isotropic quadratic form called the spacetime interval or the Minkowski norm squared. An event in Minkowski space for which the spacetime interval is zero is on the null cone of the origin, called the light cone in Minkowski space. 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. Another misnomer is Minkowski metric, but Minkowski space is not a metric space.
The group of transformations for Minkowski space that preserves the spacetime interval is the Lorentz group.

History

Complex Minkowski spacetime

In his second relativity paper in 1905, Henri Poincaré showed how, by taking time to be an imaginary fourth spacetime coordinate, where is the speed of light and 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. 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.
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.
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". He reformulated Maxwell equations as a symmetrical set of equations in the four variables 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, Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables 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 of certain vectors, and the Minkowski norm squared is referred to as "sum".
Minkowski's principal tool is the Minkowski diagram, and he uses it to define concepts and demonstrate properties of Lorentz transformations 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 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
Though Minkowski took an important step for physics, Albert Einstein saw its limitation:
For further historical information see references, and.

Causal structure

Where is velocity,,, and are Cartesian coordinates in 3-dimensional space, is the constant representing the universal speed limit, and is time, the four-dimensional vector is classified according to the sign of. A vector is timelike if, spacelike if, and null or lightlike if. This can be expressed in terms of the sign of, 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 because of the invariance of the spacetime interval under Lorentz transformation.
The set of all null vectors at an event of Minkowski space constitutes the light cone of that event. Given a timelike vector, 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
  1. future-directed timelike vectors whose first component is positive and
  2. past-directed timelike vectors whose first component is negative.
Null vectors fall into three classes:
  1. the zero vector, whose components in any basis are ,
  2. future-directed null vectors whose first component is positive, and
  3. past-directed null vectors whose first component is negative.
Together with spacelike vectors, there are 6 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 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 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 and is
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.