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Spacetime
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=== Curved manifolds === {{Main|Manifold|Lorentzian manifold|Differentiable manifold}} For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected [[Lorentzian manifold]] <math>(M, g)</math>. This means the smooth [[Lorentz metric]] <math>g</math> has [[metric signature|signature]] <math>(3,1)</math>. The metric determines the ''{{vanchor|geometry of spacetime|SPACETIME_GEOMETRY}}'', as well as determining the [[geodesic]]s of particles and light beams. About each point (event) on this manifold, [[coordinate charts]] are used to represent observers in reference frames. Usually, Cartesian coordinates <math>(x, y, z, t)</math> are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light <math>c</math> is equal to 1.<ref name="Pfaffle">{{cite book|last1=Bär|first1=Christian|last2=Fredenhagen|first2=Klaus|title=Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations|date=2009|publisher=Springer|location=Dordrecht|isbn=978-3-642-02779-6|pages=39–58|chapter-url=https://www.springer.com/cda/content/document/cda_downloaddocument/9783642027796-c1.pdf?SGWID=0-0-45-800045-p173910618|access-date=14 April 2017|archive-url=https://web.archive.org/web/20170415201236/http://www.springer.com/cda/content/document/cda_downloaddocument/9783642027796-c1.pdf?SGWID=0-0-45-800045-p173910618|archive-date=15 April 2017|chapter=Lorentzian Manifolds|url-status=dead}}</ref> A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event <math>p</math>. Another reference frame may be identified by a second coordinate chart about <math>p</math>. Two observers (one in each reference frame) may describe the same event <math>p</math> but obtain different descriptions.<ref name="Pfaffle" /> Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing <math>p</math> (representing an observer) and another containing <math>q</math> (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a [[Singularity (mathematics)|non-singular]] coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.<ref name="Pfaffle" /> For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event <math>p</math>). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples <math>(x, y, z, t)</math> (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces [[tensors]] into relativity, by which all physical quantities are represented. Geodesics are said to be timelike, null, or spacelike if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by timelike and null (lightlike) geodesics, respectively.<ref name="Pfaffle" /> {{anchor|Privileged character of 3+1 spacetime}}
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