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A bubble chart is a type of chart that displays three dimensions of data. Each entity with its triplet ( v 1 , v 2 , v 3 ) of associated data is plotted as a disk that expresses two of the v i values through the disk's xy location and the third through its size.
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.
A spacetime diagram is a graphical illustration of locations in space at various times, especially in the special theory of relativity.Spacetime diagrams can show the geometry underlying phenomena like time dilation and length contraction without mathematical equations.
causal (or non-spacelike) if the tangent vector is timelike or null at all points in the curve. The requirements of regularity and nondegeneracy of Σ {\displaystyle \Sigma } ensure that closed causal curves (such as those consisting of a single point) are not automatically admitted by all spacetimes.
The geometry of space-time within the bubble is that of anti-de Sitter space, a space much like conventional de Sitter space except that its group of symmetries is O(3, 2) rather than O(4, 1). Although this space-time is free of singularities, it is unstable under small perturbations, and inevitably suffers gravitational collapse of the same ...
The need for exotic matter raises questions about whether one can distribute the matter in an initial spacetime that lacks a warp bubble in such a way that the bubble is created at a later time, although some physicists have proposed models of dynamical warp-drive spacetimes in which a warp bubble is formed in a previously flat space. [4]
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Mathematically, the frame fields {} define an isomorphism at each point where they are defined from the tangent space to ,. Then abstract indices label the tangent space, while greek indices label ,. If the frame fields are position dependent then greek indices do not necessarily transform tensorially under a change of coordinates.