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The Minkowski distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance. It is named after the Polish mathematician Hermann Minkowski .
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.
The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in space, to arbitrary measurable sets.
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime , being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
A closed string is a generalization of a particle. The spatial coordinate of a point on the string is conveniently described by a parameter which runs from to .Time is appropriately described by a parameter .
A Riemannian metric is a metric with a positive definite signature (v, 0). A Lorentzian metric is a metric with signature ( p , 1) , or (1, p ) . There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as ( v − p ) , where v and p are as above, which is equivalent to the above definition when ...
An equivalence class of such metrics is known as a conformal metric or conformal class. Thus, a conformal metric may be regarded as a metric that is only defined "up to scale". Often conformal metrics are treated by selecting a metric in the conformal class, and applying only "conformally invariant" constructions to the chosen metric.
This allows us to define a new metric on any of these hyperslices which is conformally related to the original metric inherited from the spacetime, but with the property that geodesics in the new metric (note this is a Riemannian metric on a Riemannian three-manifold) are precisely the projections of the null geodesics of spacetime.