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Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions , to describe the sizes or locations of objects in the everyday world.
The spacetime underlying Albert Einstein's field equations, which mathematically describe gravitation, is a real 4 dimensional pseudo-Riemannian manifold. In quantum mechanics, wave functions describing particles are complex-valued functions of real space and time variables.
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.. In physics, Minkowski space (or Minkowski spacetime) (/ m ɪ ŋ ˈ k ɔː f s k i,-ˈ k ɒ f-/ [1]) is the main mathematical description of spacetime in the absence of gravitation.
Johan Ernest Mebius (2005). "A matrix-based proof of the quaternion representation theorem for four-dimensional rotations". arXiv: math/0501249. Johan Ernest Mebius (2007). "Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations". arXiv: math/0701759.
In Einstein's theory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions x μ (τ), where μ is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates.
Calculating the Minkowski norm squared of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass: = = = + | | = where = is the metric tensor of special relativity with metric signature for definiteness chosen to be (–1, 1, 1, 1).
This is a special case of a general fact about volumes in n-dimensional space: If K is a body (measurable set) in that space and RK is the body obtained by stretching in all directions by the factor R then the volume of RK equals R n times the volume of K. This is a direct consequence of the change of variables formula:
An N-dimensional unit sphere embedded in (N + 1)-dimensional Euclidean space may be defined as ∑ i = 0 N y i 2 = 1 {\displaystyle \sum _{i=0}^{N}y_{i}^{2}=1} This embedding induces a metric on the sphere, it is inherited directly from the Euclidean metric on the ambient space.