Geometry Minkowski

To treat the general situation, compatible with cartesian geometries, we will consider the (l+N)-dimensional Minkowski space. Then, taking... 

The received view, in which the phase factor of optics and electrodynamics is given by Eq. (554), can describe neither the Sagnac nor the Tomita-Chiao effects, which, as we have argued, are the same effects, differing only by geometry. Both are non-Abelian, and both depend on a round trip in Minkowski spacetime using 0(3) covariant derivatives. 

Frechet-Minkowski distance measure at the small scale, this distance approaches a constant value as the two lumps approach each other. This infinitesimal regime signals the devitation from the Euclidean geometry on small scales. In the asymptotic distance regime the Frechet-Minkowski distance becomes Euclidean distance. This phenomenon may play a significant role in the Planck regime. 

The Pauli and Dirac matrices may be related to the geometry of the E3 and the Minkowski M = R1 3 spaces, respectively. Despite their complex and complicated forms, they obey relations similar to the gij ones verified by the orthonormal frames of these real spaces. 

Theories like those of Lemaitre or Friedmann, which predict an expanding universe, are all based on forcing an affine metric, such as the Robertson-Walker metric, on the projective geometry of space-time. This has the effect of splitting local Minkowski space into separate space and time coordinates, without the natural complex relationship that ties space and time together. 

To specify the directions of two different vectors at nearby points it is necessary to define tangent vectors at these points. Stated in different terms, at each point of space-time, known as the contact point, there is an associated tangent Minkowski space. The theory of these spaces together with the underlying space becomes a Riemannian geometry if a Euclidean metric is introduced in each tangent space by means of a differential quadratic form. ... 

Locally, Minkowski and universal space are identical as causal manifolds, but the universal time is not equivalent to the time registered in the local Minkowski frame. Quantum mechanically temporal evolution and energy are defined by conjugate operators. The operator —i d/dt), which defines the energy (or frequency) depends on the geometry of the stationary state. 

The Minkowski space-time of special relativity differs from conventional Euclidean space only in the number of dimensions and gives the correct description of all forms of uniform relative motion. However, it fails when applied to accelerated motion, of which circular motion at constant orbital speed is the simplest example. Relativistic contraction only occurs in the direction of motion, but not in the perpendicular radial direction towards the centre of the orbit. The simple Euclidean formula that relates the circumference of the circle to its radius therefore no longer holds. The inevitable conclusion is that relativistic acceleration implies non-Euclidean geometry. 

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