Editing Mathematical physics (section)

=== Relativistic ===

By the 1880s, there was a prominent paradox that an observer within Maxwell's electromagnetic field measured it at approximately constant speed, regardless of the observer's speed relative to other objects within the electromagnetic field. Thus, although the observer's speed was continually lost{{clarify|date=January 2018}} relative to the electromagnetic field, it was preserved relative to other objects ''in'' the electromagnetic field. And yet no violation of [[Galilean invariance]] within physical interactions among objects was detected. As Maxwell's electromagnetic field was modeled as oscillations of the [[luminiferous aether|aether]], physicists inferred that motion within the aether resulted in [[aether drift]], shifting the electromagnetic field, explaining the observer's missing speed relative to it. The [[Galilean transformation]] had been the mathematical process used to translate the positions in one reference frame to predictions of positions in another reference frame, all plotted on [[Cartesian coordinates]], but this process was replaced by [[Lorentz transformation]], modeled by the Dutch [[Hendrik Lorentz]] [1853–1928].

In 1887, experimentalists Michelson and Morley failed to detect aether drift, however. It was hypothesized that motion ''into'' the aether prompted aether's shortening, too, as modeled in the [[Lorentz contraction]]. It was hypothesized that the aether thus kept Maxwell's electromagnetic field aligned with the principle of Galilean invariance across all [[inertial frames of reference]], while Newton's theory of motion was spared.

Austrian theoretical physicist and philosopher [[Ernst Mach]] criticized Newton's postulated absolute space.  Mathematician [[Henri Poincaré|Jules-Henri Poincaré]] (1854–1912) questioned even absolute time. In 1905, [[Pierre Duhem]] published a devastating criticism of the foundation of Newton's theory of motion.<ref name=Lakatos1980/> Also in 1905, [[Albert Einstein]] (1879–1955) published his [[special theory of relativity]], newly explaining both the electromagnetic field's invariance and Galilean invariance by discarding all hypotheses concerning aether, including the existence of aether itself.  Refuting the framework of Newton's theory—[[absolute space and time|absolute space and absolute time]]—special relativity refers to ''relative space'' and ''relative time'', whereby ''length'' contracts and ''time'' dilates along the travel pathway of an object.

Cartesian coordinates arbitrarily used rectilinear coordinates. Gauss, inspired by Descartes' work, introduced the curved geometry, replacing rectilinear axis by curved ones. Gauss also introduced another key tool of modern physics, the curvature. Gauss's work was limited to two dimensions. Extending it to three or more dimensions introduced a lot of complexity, with the need of the (not yet invented) tensors. It was Riemman the one in charge to extend curved geometry to N dimensions. In 1908, Einstein's former mathematics professor [[Hermann Minkowski]], applied the curved geometry construction to model 3D space together with the 1D axis of time by treating the temporal axis like a fourth spatial dimension—altogether 4D spacetime—and declared the imminent demise of the separation of space and time. <ref>Minkowski, Hermann (1908–1909), "Raum und Zeit" [Space and Time], Physikalische Zeitschrift, 10: 75–88. Actually the union of space and time was implicit in Descartes's work first, with space and time being represented as axis of coordinates, and in Lorentz transformation later, but its physical interpretation was still hidden to common sense.
</ref> Einstein initially called this "superfluous learnedness", but later used [[Minkowski spacetime]] with great elegance in his [[general theory of relativity]],<ref>Salmon WC & Wolters G, eds, ''Logic, Language, and the Structure of Scientific Theories'' (Pittsburgh: University of Pittsburgh Press, 1994), p [https://books.google.com/books?id=Z9K8llQufcMC&pg=PA125&dq=superfluous+learnedness+Einstein+Minkowski+general+relativity 125]</ref> extending invariance to all reference frames—whether perceived as inertial or as accelerated—and credited this to Minkowski, by then deceased. General relativity replaces Cartesian coordinates with [[Gaussian coordinates]], and replaces Newton's claimed empty yet Euclidean space traversed instantly by Newton's [[Euclidean vector|vector]] of hypothetical gravitational force—an instant [[action at a distance]]—with a gravitational ''field''. The gravitational field is [[Minkowski spacetime]] itself, the 4D [[topology]] of Einstein aether modeled on a [[Lorentzian manifold]] that "curves" geometrically, according to the [[Riemann curvature tensor]]. The concept of Newton's gravity: "two masses attract each other" replaced by the geometrical argument: "mass transform curvatures of [[spacetime]] and free falling particles with mass move along a geodesic curve in the spacetime" ([[Riemannian geometry]] already existed before the 1850s, by mathematicians [[Carl Friedrich Gauss]] and [[Bernhard Riemann]] in search for intrinsic geometry and non-Euclidean geometry.), in the vicinity of either mass or energy. (Under special relativity—a special case of general relativity—even massless energy exerts gravitational effect by its [[mass–energy equivalence|mass equivalence]] locally "curving" the geometry of the four, unified dimensions of space and time.)