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Spacetime is equipped with an indefinite non-degenerate bilinear form, called the Minkowski metric, [2] the Minkowski norm squared or Minkowski inner product depending on the context. [nb 2] The Minkowski inner product is defined so as to yield the spacetime interval between two events when given their coordinate difference vector as an ...
e. In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive where and when ...
The interval imparts information about the causal structure of spacetime. When <, the interval is timelike and the square root of the absolute value of is an incremental proper time. Only timelike intervals can be physically traversed by a massive object.
The form of , being the difference of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances. [ note 7 ] The invariance of this interval is a property of the general Lorentz transform (also called the Poincaré transformation ), making it an isometry of ...
Certain types of world lines are called geodesics of the spacetime – straight lines in the case of flat Minkowski spacetime and their closest equivalent in the curved spacetime of general relativity. In the case of purely time-like paths, geodesics are (locally) the paths of greatest separation (spacetime interval) as measured along the path ...
Writing the coordinates in column vectors and the Minkowski metric η as a square matrix ′ = [′ ′ ′ ′], = [], = [] the spacetime interval takes the form (superscript T denotes transpose) = = ′ ′ and is invariant under a Lorentz transformation ′ = where Λ is a square matrix which can depend on parameters.
For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes η is replaced with −η, making the spatial terms produce negative contributions to the dot product or spacetime interval, while the time term makes a positive contribution. These differences can be used in any ...
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. The history of an object's location through time traces out a line or ...