Editing Curved spacetime (section)

== Curvature of space ==
The <math>(1 - 2GM/(c^2 r) )</math> coefficient in front of <math>(c \Delta t)^2</math> describes the curvature of time in Newtonian gravitation, and this curvature completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to <math>G</math> and <math>M</math>, and because of the <math>r</math> in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is curved.

But general relativity is a theory of curved space ''and'' curved time, so if there are terms modifying the spatial components of the spacetime interval presented above, should not their effects be seen on, say, planetary and satellite orbits due to curvature correction factors applied to the spatial terms?

The answer is that they ''are'' seen, but the effects are tiny. The reason is that planetary velocities are extremely small compared to the speed of light, so that for planets and satellites of the [[Solar System]], the <math>(c \Delta t)^2</math> term dwarfs the spatial terms.<ref name="Schutz" />{{rp|234–238}}

Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In 1859, [[Urbain Le Verrier]], in an analysis of available timed observations of transits of [[Mercury (planet)|Mercury]] over the Sun's disk from 1697 to 1848, reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury. The perihelion of Mercury's orbit exhibited an [[Tests of general relativity#Perihelion precession of Mercury|excess rate of precession]] over that which could be explained by the tugs of the other planets.<ref>{{cite journal|last1=Le Verrier|first1=Urbain|title=Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète|journal=Comptes rendus hebdomadaires des séances de l'Académie des Sciences |date=1859 |volume=49 |pages=379–383 |url=https://archive.org/stream/comptesrendusheb49acad#page/378/mode/2up}}</ref> The ability to detect and accurately measure the minute value of this anomalous precession (only 43 [[arc seconds]] per [[tropical year|tropical century]]) is testimony to the sophistication of 19th century [[astrometry]].

[[File:General relativity time and space distortion frame 1.png|thumb|Figure 5–4. General relativity is a theory of curved time ''and'' curved space. [[:File:General relativity time and space distortion extract.gif|'''Click here to animate.''']] ]]
As the astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing irregularities in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed.<ref>{{cite web|last1=Worrall |first1=Simon |title=The Hunt for Vulcan, the Planet That Wasn't There |url=http://news.nationalgeographic.com/2015/11/151104-newton-einstein-gravity-vulcan-planets-mercury-astronomy-theory-of-relativity-ngbooktalk/ |website=National Geographic|date = 4 November 2015|archive-url= https://web.archive.org/web/20170524004444/http://news.nationalgeographic.com/2015/11/151104-newton-einstein-gravity-vulcan-planets-mercury-astronomy-theory-of-relativity-ngbooktalk/ |url-status=dead |archive-date=24 May 2017}}</ref>

In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct.

The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of&nbsp;±1 on a spacetime diagram. Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components.<ref name="Schutz" />{{rp|234–238}}
: <math>\Delta s^2 = \left( 1 - \frac{2GM}{c^2 r} \right) (c \Delta t)^2</math><math>- \, \left( 1 + \frac{2GM}{c^2 r} \right) \left[  (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2  \right] </math>

In Newton's gravitation, the <math>(1 - 2GM/(c^2 r) )</math> coefficient in front of <math>(c \Delta t)^2</math> predicts bending of light around a star. In general relativity, the <math>(1 + 2GM/(c^2 r) )</math> coefficient in front of <math>\left[  (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2  \right] </math> predicts a ''doubling'' of the total bending.<ref name="Schutz" />{{rp|234–238}}

The story of the [[Eddington experiment|1919 Eddington eclipse expedition]] and Einstein's rise to fame is well told elsewhere.<ref>{{cite web|last1=Levine|first1=Alaina G.|title=May 29, 1919: Eddington Observes Solar Eclipse to Test General Relativity|url=https://www.aps.org/publications/apsnews/201605/physicshistory.cfm |website=APS News|department= This Month in Physics History|publisher=American Physical Society|date =May 2016|volume = 25|number = 5|archive-url= https://web.archive.org/web/20170602134913/http://www.aps.org/publications/apsnews/201605/physicshistory.cfm|url-status=live|archive-date=2 June 2017}}</ref>