In physics, Minkowski space (or Minkowski spacetime) (/mɪŋˈkɔːfski, -ˈkɒf-/) 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 (maps preserving the regular Euclidean distance) 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 (as opposed to the spatial Euclidean distance) is the Lorentz group (as opposed to the Galilean group).

In his second relativity paper in 1905, Henri Poincaré showed how, by taking time to be an imaginary fourth spacetime coordinate ict, where c is the speed of light and i 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 (t, x, y, z). 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.$${\displaystyle x^{2}+y^{2}+z^{2}+(ict)^{2}={\text{constant}}.}$$