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In physics and mathematics,
Minkowski space (or Minkowski spacetime) is the mathematical setting 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 spacetime.
Note that this article describes only the mathematical structure of Minkowski space. For descriptions of the physics see
the special relativity page.
History
Minkowski space is named for the German mathematician Hermann
Minkowski, who around 1907 realized that the theory of special relativity previously
worked out by Einstein and Lorentz could be elegantly described using a four-dimensional spacetime, which combines the dimension of
time with the three dimensions of space.
“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, 1908
Structure
Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature
(+,-,-,-). The overall sign is a matter of convention and
many prefer to use the signature (-,+,+,+). Minkowski space is often denoted R1,3 to
emphasize the signature, although it is also denoted M 4 or simply M.
The Minkowski inner product
The inner product between two vectors v, w in
Minkowski space is a map M × M → R, denoted <v, w>, that
satisfies four properties. Three of which are that it be
- bilinear: <au + v, w> = a<u, w> + <v,
w>, for all a, u, v, and w
- symmetric: <v, w> = <w, v> for all v and w,
and
- nondegenerate: if <v, w> = 0 for all w then v = 0,
where a is in R and u, v, w are vectors in M.
Note that this is not an inner product in the usual sense of the word since it is not positive-definite, i.e. the norm-squared of a vector v, defined as
||v||2 = <v, v>, need not be positive. The positive-definite condition has been
replaced by the weaker condition of nondegeneracy (every positive-definite form is nondegenerate but not vice-versa).
Just as in Euclidean space, two vectors are said to be
orthogonal if <v, w> = 0. A vector v
is called a unit vector if ||v||2 = ±1. A basis
for M consisting of mutually orthogonal unit vectors is called an orthonormal basis.
There is a theorem stating that any inner product space satisfying conditions 1-3 above always has an orthonormal basis.
Furthermore, the theorem states that the number of positive and negative unit vectors in any such basis is fixed. This pair of
numbers is called the signature of the inner product.
We can then state then fourth condition on the Minkowski inner product:
Four-vectors
Vectors in Minkowski space are also called four-vectors
in order to distinguish them from three-dimensional spatial vectors. In this article, however, we use the two terms
interchangeably. We shall refer instead to the spatial component of a four-vector (which, of course, depends on the choice of
basis).
Lorentz transformations
To be completed.
See: Lorentz transformations, Lorentz group, Poincaré
group
Causal structure
Four-vectors are classified according to the sign of their norm squared. Vectors are said to be timelike or
spacelike if their norms squared are positive or negative respectively. Vectors with zero norm are called
null or lightlike. This terminology comes from the use of Minkowski space in the theory of relativity. The set of all lightlike vectors constitutes
what is called the light cone. Note that all of these notions are independent
of one's frame of reference.
Once a direction of time is chosen, timelike and null vectors can be further decomposed into various classes. For timelike
vectors we have
- future timelike vectors whose first component is positive, and
- past timelike vectors whose first component is negative.
Null vectors fall into three class:
- the zero vector, whose components in any basis are (0,0,0,0),
- future null vectors whose first component is positive, and
- past 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
(non-orthonormal) basis consisting entirely of null vectors, called a null basis.
Locally flat spacetime
Strictly speaking, the use of the Minkowski space to describe physical systems over finite distances applies only in the
Newtonian limit of 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 a infinitesimally small 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 limit of weak gravity, spacetime becomes flat and looks globally, not just locally, like Minkowski space. For this
reason Minkowski space is often referred to a flat spacetime.
Related Topics
Euclidean space | spacetime | speed of light | world line | metric tensor | pseudo-Riemannian manifold | Erlangen program
References
- Naber, Gregory L., The Geometry of Minkowski Spacetime, Springer-Verlag, New York, 1992. ISBN 0-387-97848-8 (hardcover), ISBN 0-486-43235-1 (Dover paperback
edition).
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