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A fuller explanation of the concept of coordinate time arises from its relations with proper time and with clock synchronization. Synchronization, along with the related concept of simultaneity, has to receive careful definition in the framework of general relativity theory, because many of the assumptions inherent in classical mechanics and classical accounts of space and time had to be removed.
In mathematical physics, when we complexify a real coordinate space we create a complex coordinate space, referred to in differential geometry as a "complex manifold". The space C n {\displaystyle \mathbb {C} ^{n}} can be related to R 2 n {\displaystyle \mathbb {R} ^{2n}} , since every complex number constitutes two real numbers.
In physics, as well as mathematics, a vector is often identified with a tuple, or list of numbers, which depend on a coordinate system or reference frame. If the coordinates are transformed, such as by rotation or stretching the coordinate system, the components of the vector also transform.
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all position vectors r in Euclidean space, and has dimensions of length; a position vector defines a point in space.
In physics, particularly special relativity, light-cone coordinates, introduced by Paul Dirac [1] and also known as Dirac coordinates, are a special coordinate system where two coordinate axes combine both space and time, while all the others are spatial.
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.
In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple (r, θ, z). [8] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates ( r , z ) to polar coordinates ( ρ , φ ) giving a triple ( ρ , θ ...
The relationship between general covariance and general relativity may be summarized by quoting a standard textbook: [3] Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics.