Spacetime combines spatial position and time into one geometric setting for physical events. In special relativity it is modeled by Minkowski spacetime; in general relativity its geometry is dynamical.
A spacetime event is an idealized occurrence at one point of spacetime, such as a light pulse reaching a detector. An inertial frame assigns spatial coordinates and a time coordinate to it. Different frames can assign different coordinates to the same spacetime event; the event itself is not changed by a coordinate transformation.
A spacetime extension is an isometric embedding of a spacetime as a proper open subset of a larger connected spacetime with a nondegenerate Lorentzian metric. The allowed differentiability class must be specified: analytic, smooth and low-regularity extensions are different questions. Divergent curvature can obstruct a regular extension even when an incomplete geodesic has finite affine length.
A maximal analytic extension of an analytic spacetime is an analytic spacetime extension admitting no further proper analytic extension as a regular spacetime of the same dimension. It is different from a maximal Cauchy development, which can end at a smooth Cauchy horizon. Additional global choices or quotients can affect uniqueness. The simply connected extension of positive-mass Schwarzschild spacetime through both horizons is the Kruskal spacetime; its curvature singularities remain boundaries.
An extendible spacetime admits a spacetime extension in the specified regularity class. A coordinate horizon need not obstruct an extension: the exterior Schwarzschild chart extends through its horizon in regular null coordinates. Maximality refers to extensions of the given spacetime, not to the mere absence of a singularity in one coordinate chart.
Proper time is the time measured by a clock following a timelike worldline. With metric signature and units ,along that worldline.
Restoring the speed of light, the same metric signature gives for a future-directed timelike curve. Integrating along a clock's world line gives its elapsed proper time, even when its velocity varies. For uniform motion this becomes the time dilation formula; it is not generally the square root of the interval between endpoints of a curved world line.
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Spacetime is a fundamental concept in physics that combines the three dimensions of space with the dimension of time into a single four-dimensional continuum. This framework is essential for understanding the behavior of objects in the universe, particularly in the context of Einstein's theory of relativity. In classical physics, space and time were treated as separate entities; however, Einstein's Special Theory of Relativity (published in 1905) demonstrated that space and time are interwoven.
