Editing Curved spacetime (section)

==Introduction==
Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean [[reference frame]] that extends throughout all space and all time. Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space.<ref group=note>Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley: "That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it."</ref> In contrast, Einstein denied that there is any background Euclidean reference frame that extends throughout space. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself.<ref name="Taylor">{{cite book|url=https://archive.org/details/spacetime_physics/|title=Spacetime Physics: Introduction to Special Relativity|last1=Taylor|first1=Edwin F.|last2=Wheeler|first2=John Archibald|date=1992|publisher=Freeman|isbn=0-7167-0336-X|edition=2nd|location=San Francisco, California|access-date=14 April 2017}}</ref>{{rp|175–190}}

[[File:Principle of the tidal force.svg|thumb|Figure 5–1. Tidal effects.]]
In spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun. Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a [[Geodesics in general relativity|geodesic]]. No evidence of gravitation can be discovered following alongside the motions of a single particle.<ref name="Taylor" />{{rp|175–190}}

In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of ''two'' bodies or two separated particles. In Fig.&nbsp;5-1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The tidal accelerations that these particles exhibit with respect to each other do not require forces for their explanation. Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime. These tidal accelerations are strictly local. It is the cumulative total effect of many local manifestations of curvature that result in the ''appearance'' of a gravitational force acting at a long range from Earth.<ref name="Taylor" />{{rp|175–190}}

:<small>Different observers viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first observer, at the center of mass of particles 2 and 3 but unaware of the large mass 1, concludes that a force of repulsion exists between the particles in scenario A while a force of attraction exists between the particles in scenario B. (ii) A second observer, aware of the large mass 1, smiles at the first reporter's naiveté. This second observer knows that in reality, the apparent forces between particles 2 and 3 really represent tidal effects resulting from their differential attraction by mass 1. (iii) A third observer, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along geodesics in spacetime.</small>

Two central propositions underlie general relativity.
* The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion."<ref name="PrincipleOfRelativity">{{cite book |last1=Lorentz |first1=H. A. |last2=Einstein|first2=A. |last3=Minkowski |first3=H. |last4=Weyl |first4=H. |title=The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity |url=https://archive.org/details/principleofrelat00lore |url-access=registration |date=1952 |publisher=Dover Publications|isbn=0-486-60081-5}}</ref>{{rp|113}} This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.<ref name="Mook">{{cite book |last1=Mook |first1=Delo E. |last2=Vargish |first2=Thoma s |title=Inside Relativity |date=1987 |publisher=Princeton University Press |location=Princeton, New Jersey |isbn=0-691-08472-6 |url=https://archive.org/details/insiderelativity0000mook }}</ref>{{rp|137–149}}
[[File:Elevator gravity.svg|thumb|Figure 5–2. Equivalence principle]]
* The [[equivalence principle]] states that in any sufficiently small region of space, the effects of gravitation are the same as those from acceleration.{{br}} In Fig. 5-2, person A is in a spaceship, far from any massive objects, that undergoes a uniform acceleration of ''g''. Person B is in a box resting on Earth. Provided that the spaceship is sufficiently small so that tidal effects are non-measurable (given the sensitivity of current gravity measurement instrumentation, A and B presumably should be [[Lilliputian]]s), there are no experiments that A and B can perform which will enable them to tell which setting they are in.<ref name="Mook" />{{rp|141–149}}{{br}} An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation, {{nowrap|1=''F = GMm''<sub>g</sub>''/r''<sup>2</sup> = }} ''m''<sub>g</sub>''g'' and in Newton's second law, {{nowrap|1=''F = m''<sub>i</sub>''a'',}} there is no ''a&nbsp;priori'' reason why the [[gravitational mass]] ''m''<sub>g</sub> should be equal to the [[Mass#Inertial mass|inertial mass]] ''m''<sub>i</sub>. The equivalence principle states that these two masses are identical.<ref name="Mook" />{{rp|141–149}}

To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study. Without these mathematical tools, it is possible to write ''about'' general relativity, but it is not possible to demonstrate any non-trivial derivations.