Fontem paginae Relativitas generalis inspicere

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[[Fasciculus:Cassini-science-br.jpg|thumb|250px|Descriptio artistica de relativitate generale probatione quadam instrumento speculatorio spatiali [[Cassini-Huygens|Cassini]] facta. Monstrata sunt signalia [[radiophonum|radiophonica]] inter [[Tellus|Tellurem]] et instrumentum (unda viridis), quae prope solem tardantur ob flexionem spatii temporisque (lineae caerulae) ibi.]]

'''Relativitas generalis,''' seu '''theoria relativitatis generalis,''' est theoria [[physica]] de [[vis|vi]] [[gravitas (physica)|gravitatis]] ab [[Albertus Einstein|Alberto Einstein]] anno [[1915]] edita. Quae theoria [[relativitas specialis|theoriam relativitatis specialis Einsteinianam]] et [[gravitas (physica)|theoriam gravitatis Newtonianam]] coniungit, adamussim considerando [[corpus|corporis]] motum e perspectiva ubi corporis [[acceleratio]] gravitatis causa est effectus flexionis [[continuum spatio-temporale|spatio-temporalis]].

Einstein, novam theoriam petens, in [[gravitas (physica)|Principium Aequivalentiae]] et in [[geometria differentialis]] de superficiebus curvis maxime confisus est. [[Aequatio Einsteniana]], quae curvaturam spatio-temporalem ad tensorem <math>T_{\mu\nu}</math> energiae-motusque attribuit, patefacit clarissimum ''constantem  cosmologicum'' <math>\Lambda</math>, qui sicut [[vacuum|vacui]] energia interpretari potest.

Quamquam theoriae cum multis observationibus terrestialibus congruit, [[physicus|physici periti]] multum sollicitati sunt ob problemata de valore <math>\Lambda</math>, deque possibili incongruentia cum observationibus subtilibus [[Fragor Maximus|expansionis mundi universi]], deque incongruentia cum [[mechanica quantica|theoria quantica]].

== De limitibus theoriae Newtonianae ==
[[Fasciculus:Relativistic precession.svg|thumb|upright=0.8|left|Secundum theoriam Newtonianam, omnibus aliis planetis absentibus, circum corpus massivum planeta quaedam eandem [[ellipsem]] perfectam (gyrum rubrum) iterum interumque sequitur aeterniter. Alii planetae praesentes efficiunt hanc ellipsem paulatim precessare (gyrum caerulum). Relativitatis generalis causa, autem, precessio ab astronomis theoria Newtoniana praecenta discrepat cum precessione Mercuii anno [[1859]] observata.]] 

=== Quid Newtonus de sua lege sentiret ===
Anno [[1687]], [[Isaacus Newtonus]] in libro ''[[Leges motus Newtoni|Philosophiae Naturalis Principia Mathematica]]'' legem mathematicam de gravitatis vi inter corpora deduxit veram, omnia gyra planetarum circum solem adamussim explanans. Lex quam Newtonus deduxit ab legibus [[Iohannes Keplerus|Keplerianis]] fuit:<ref>Raymond A. Serway et John W. Jewett, ''Physics for Scientists and Engineers,'' 6a ed. (Brooks/Cole, 2004). ISBN 0-534-40842-7.</ref>

:::<math>\vec \mathbf{F}_{12} =  - G \frac{m_1 m_2}{r^2}\, \mathbf{\hat{r}}_{12}</math>

ubi 

* <math>\vec \mathbf{F}_{12}</math> est vis attractus in corpus 1 ob corpus 2,
* ''G'' est [[constans gravitatis]] quod valet vere ac 6.67 × 10<sup>-11</sup> Nm²/kg²,
* ''m''<sub>1</sub> et ''m''<sub>2</sub> sunt massae corporum,
* ''r'' est intervallum inter media corpora, et
* <math> \mathbf{\hat{r}}_{12} \ \stackrel{\mathrm{def}}{=}\  \frac{\vec \mathbf{r}_1 - \vec \mathbf{r}_2}{\vert \vec \mathbf{r}_1 - \vec \mathbf{r}_2\vert} </math> est directio ad corpus 1 a corpore 2.

Newtonus autem ipse hanc legem credidit imperfectam, quod, secundum eam, vires gravitationales sine ulla mora per spatium translatae sunt.<ref>Scholium Generale: Philosophia Naturalis Principia Mathematica, Secunda Editio, anno [[1713]] </ref>

=== De quanto precessio Mercurii a theoria discrepet ===
Saecula duo haec theoria Newtoniana classica summa ratio fuit, usque ad annum [[1859]] cum astronomi patefecerunt discrepantiam inter precessionem perihelii [[Mercurius (planeta)|Mercurii]] observatam et precessionem aequationibus Newtonianis computatam.<ref>[http://adsabs.harvard.edu/abs/1859AnPar...5....1L Urbain Le Verrier "Theorie du Mouvement de Mercure" (195 pages, 17.6 MB)]</ref> Discrepabant, ut tabula infera monstrat, solo circiter 43 arcus secundis rotationis in quoque saeculo. <ref name="perhihelion">[http://www.nature.com/nature/journal/v320/n6057/abs/320039a0.html Anna M. Nobili & Clifford M. Will, The real value of Mercury's perihelion advance], ''Nature'' 320, 39 (1986); [http://math.ucr.edu/home/baez/physics/Relativity/GR/mercury_orbit.html situs interretialis.]</ref>

{| border="1" cellpadding="1" cellspacing="2" align="center" 
|- align="center" style="background:#A0A0ff;border: 2px solid #7070ff"
|+ '''Causae precessionis perihelii Mercurii observatae a Tellure.'''<ref name="perhihelion"/>
|- align="center" style="background:#A0A0ff;border: 2px solid #7070ff"
| Quantitas precessionis praecenta (arcus sec./saec.) || Causa  
|- align="center" style="background:#E0E0ff;"
| 5025.6 || Motus coordinatarum ob precessionem Telluris
|- align="center" style="background:#E0E0ff;" 
| 531.4 || Impulsus gravitationales aliorum planetarum
|- align="center" style="background:#E0E0ff;"
| 0.0254 || Protuberantia in figura Solis
|- align="center" style="background:#E0E0ff;"
| 42.98±0.04 || Relativitas generalis  
|- align="center" style="background:#A0A0ff;border: 2px solid #7070ff"
|  Precessio Universa ||   Praecenta aut Observata
|- align="center" style="background:#E0E0ff;"
| 5600.0±0.2 || Praecenta
|- align="center" style="background:#E0E0ff;"
| 5599.7±0.1 || Observata
|}<!--Although earlier measurements of planetary orbits were made using conventional telescopes, the most accurate measurements are now made with [[Radar astronomy|radar]]. The total observed precession of Mercury is 5600 [[arc-second]]s per century with respect to the position of the [[vernal equinox]] of the Sun. This precession is due to the following causes (the numbers quoted are the modern values):-->

Nemo poterat suspicari discrepantiam tam parvulam esse indicium novae theoriae eam Newtonianam surrogaturae!

=== De incongruentia cum principio relativitatis ===
Nova [[relativitas specialis|relativitatis specialis]] theoria, ab Alberto Einstein anno [[1905]] prolata, multa experimenta explicavit, sed contraria erat theoriae gravitatis Newtonianae. Secundum relativitatis specialis principia:<ref name="meaning">A. Einstein, The Meaning of Relativity, Including the Relativistic Theory of the Non-Symmetric Field, Princeton: Princeton University Press, 1922</ref>
* Omnes vires infra [[celeritas luminis|lucis celeritatem]] movendae sunt, sed secundum theoriam Newtonianam vires gravitationales trans distantias inter corpora fiunt sine mora, quae res Newtonum ipsum multum sollicitabat; 
* Corporis massa a celeritate sua dependet, sed haec dependentia efficit ut magnitudo gravitatis, celeritate tantum mutans, principium relativitatis plene violet.
Quam ob rem Einstein novam theoriam gravitatis mox petiturus erat. Nova theoria relativitatis generalis ambo problemata solvit: omnes vires gravitationales celeritate lucis movent et magnitudo gravitatis tantum corporis celeritate mutat, ut principium relativitatis conservetur. Nova theoria quoque adamussim precessionis perihelii Mercurii discrepantiam explanavit.

== Principia relativitatis generalis ==
=== Principium aequivalentiae ===
[[Fasciculus:Elevator gravity.svg|thumb|upright=1|Pila solum similiter cadit, et in [[rucheta]] accelerata (cui imago ad laevam) et in Tellure (ad dextram).]]
Einstein, novam theoriam petens, maxime confisus est in [[Gravitas (physica)|Principium Aequivalentiae]], quae ad systemata coordinatorum spectat, in forma: 
: ''Physicae leges in gravitatis campo esse identicas legibus in systemate coordinatorum accelerato'', 

quam postea Einstein dicit fuisse ''felicissimam cogitationem suae vitae.''

Percepit Einstein quod, secundum hoc principium, cum lucis traiectoria est curva in rucheta accelerata, tam curva necesse est prope corpus massivum. Et, quod in rucheta omnes effectus ob accelerationem per coordinatorum transformationes dantur, possumus expectare omnes effectus prope corpus massivum ob gravitatem per coordinatorum mutationes dari.

Secundum principium relativitatis, non possumus per experimenta localia nostrum motum constantem detegere. Item, secundum principium aequivalentiae, non possumus distinguere inter motum constantem et motum libere cadendi. Ideo, quod libere cadendi status quodam transformationum coordinatorum effectu correspondet, necesse est quod omnes leges physicas sint ''covariantes'' sub coordinatorum transformationibus, id est ut legum forma sit immutabilis sub coordinatorum tranformationibus.

=== De tensore metrico gravitate absente ===
[[Fasciculus:Spacetime_curvature.png|upright=1.5|thumb|Secundum theoriam relativitatis generalis, materia ipsa natura flectit spatium et tempus, unde est vis gravitatis. Difficilis est animo fictu flexio dimensionum trium, difficilior quattuor, ergo haec imago modo monstrat flexionem duarum.]]

Einstein, ad theoriam relativitatis generalis conficiendam, usus est theoria geometriae differentialis [[Bernardus Riemann|Bernardi Riemann]] et [[Carolus Fridericus Gauss|Caroli Gauss]] de superficiebus curvis. 

Hac in theoria per tensorem <math>g_{\mu\nu}</math> metricum habemus intervallum differentiale quadratum
::<math>ds^2=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}</math>
inter duo puncta <math>x^\mu</math> et <math>x^\mu+dx^\mu</math>. Habemus quoque contextu spatio-temporali gravitate absente <math>dx^{\mu}=(dx,dy,dz,cdt)</math> et 
::<math>
g_{\mu\nu} = 
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&-1
\end{pmatrix}\,
</math>.

Cum tensor <math>g_{\mu\nu}</math> partes seu componentes habet quae a positione temporeque dependent, id est si <math>g_{\mu\nu}=g_{\mu\nu}(x^{\mu},t)</math>, hoc significat [[spatium quattuor dimensionum]], id est [[spatium|trium directionum]] ac [[tempus|temporis]], deformatum est, sicut in figura supera videri potest.

=== De tensore metrico prope corpus massivum irrotans ===
Aequationes Einstenianas circum massivum corpus ([[massa]] M) irrotans sequentem solutionem habent, quae a Schwarzschild anno [[1916]] patefacta est,<ref>K. Schwarzschild, "Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik, 189 (1916); K. Schwarzschild, "Uber das Gravitationsfeld einer Kugel aus inkompressibler Flussigkeit nach der Einsteinschen Theorie", Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technikm 424 (1916).</ref>
 
:::<math> ds^2 = - c^2 \left( 1 - {2Gm \over c^2 r} \right) dt^2 + \left( 1 - {2Gm \over c^2 r} \right)^{-1} dr^2 + r^2 d\Omega^2 </math>,

unde <math>d\Omega^2 = d\theta^2 + \sin^2\theta\; d\phi^2</math> est elementus anguli solidi infinitesimalis  et [[coordinata polaria|coordinatis polaribus]] <math>x^{\mu}=(r\,  \sin\theta\, \cos\phi, r\, \sin\theta\, \sin\phi, r\, \sin\theta, ct)\,</math>.

[[Fasciculus:BH LMC.png|thumb|upright=0.8|[[Foramen nigrum]] apud nubem Megallanicam.]]

Secundum hanc solutionem corpus quodlibet cuius radius est minor quam ''radius Schwarzschildianus'' 
:::<math> r_s = \frac{2GM}{c^2}</math>
in '''[[foramen nigrum]]''' transformatur.<ref>[[lingua Anglica|Anglice]]: black hole</ref><ref>[https://web.archive.org/web/20111120184631/http://prola.aps.org/abstract/PR/v55/i4/p374_1 "On Massive Neutron Cores"], J. R. Oppenheimer and G. M. Volkoff, Physical Review 55, 374 (1939).</ref> Intra regionem radii Schwarzschildiani, tam curvum est continuum spatiotemporale, ut ne lux quidem ex interiore parte exire possit (haec est ratio nominis ''atri'') et omnia quae in foramen nigrum cadant semper in eo loco perseverabunt.

=== De motu geodaetico ===
Secundum principium aequivalentiae, omnes leges physicae fundamentales oportet sic scribere, ut eae sub tranformationes coordinatorum immutabiles sint. Nobis non oportet leges fundamentales derivativo ordinario scribere, exempli gratia, quia, ob directiones cardinales mutantes, hic derivativus sub coordinatorum transformationes mutat.<ref name="meaning">A. Einstein, The Meaning of Relativity, Including the Relativistic Theory of the Non-Symmetric Field, Princeton: Princeton University Press, 1922</ref> Ne leges mutentur, idoneus ad leges scribendas est [[derivativus covarians]] <math>\nabla_\ell V^m</math> vectoris <math> V^m</math> cuiusdam in superficie quadam, qui definitur ac
:::<math>\nabla_\ell V^m = \frac{\partial V^m}{\partial x^\ell} + \Gamma^m {}_{k\ell} V^k\ </math>,
ubi quantitates Christofelae
:::<math> {\Gamma^\gamma}_{\alpha\beta} = \frac{1}{2} g^{\gamma\delta}(\partial_{\alpha}g_{\beta\delta} + \partial_{\beta}g_{\alpha\delta} -  \partial_{\delta}g_{\alpha\beta})</math>
describunt quo modo [[directiones cardinales]] mutant.

Cursus geodaeticus, in physica relativistica, dicitur ille cursus quem particula libere cadens sequitur. Est etiam cursum secundum superficiem quod inter duo puncta intervallum minimum accumulat. Quia principium aequivalentiae dicit liberi cadendi statum esse motui gravitate absente indistinguibilem, speramus 4-velocitas <math>V^m = \frac {dx^m}{d\tau}</math> sit constans secundum cursum et acceleratio covarians <math>\nabla_\ell V^m = 0</math>. Aequivalenter, supponiendo in definitione derivativi covariantis supera, obtinemus

:::<math>\frac{d^2x^a}{d \tau^2} + \Gamma^a_{bc}\frac{dx^b}{d \tau} \frac{dx^c}{d \tau} = 0</math>,

quae formula dat corporis accelerationem, etiam integrando cursum geodaeticum, ob curvaturam spatio-temporalem.

=== De curvatura spatio-temporali ===
In geometria differentiali, curvatura spatio-temporalis per tensores derivatos a tensore <math>g_{\mu\nu}</math> metrico quantificatur. Habemus igitur tensorem Riemannianum<ref name = "baez">[http://math.ucr.edu/home/baez/einstein/ ``The Meaning of Einstein’s Equation''], John C. Baez and Emory F. Bunn†, (2006).</ref> 

:::<math>{R^\delta}_{\alpha\beta\gamma} =
 \partial_{\alpha} {\Gamma^\delta}_{\beta\gamma} - 
 \partial_{\beta}  {\Gamma^\delta}_{\alpha\gamma} +
 {\Gamma^\delta}_{\alpha\varepsilon} {\Gamma^\varepsilon}_{\beta\gamma} -
 {\Gamma^\delta}_{\beta\varepsilon} {\Gamma^\varepsilon}_{\alpha\gamma}
</math>,

[[Fasciculus:Parallel transport.png|thumb|Vector, circum limitem clausum transportatus, angulo torquetur <math>\alpha</math> in proportione areae inclusae.]]
[[Fasciculus:Tensor de Ricci.PNG|thumb|180px|Effectus tensori Ricciani in volumine tridimensionali sphaerico. <!--En la ilustración se reproducen los efectos del tensor de Ricci (concretamente su componente '''<math>R^{00}</math>''' sobre un volumen tridimensional esférico: conforme aumenta el tiempo, dicho volumen se reduce. Él autor de la imagen se ha permitido la siguiente licencia: Aunque los ejes de coordenadas representan dos dimensiones espaciales y una temporal, el volumen de la esfera está definido por tres dimensiones espaciales.-->]]

tensorem Riccianum
  
:::<math>R_{\alpha\gamma}={R^\beta}_{\alpha\beta\gamma}</math>,

et curvaturam scalarem

:::<math>R=g^{\alpha\gamma}R_{\alpha\gamma}\,</math>.

Tensor Riemannianus fundamentalis dat quanto vector <math>V^\mu</math> quidam mutat vel torquetur postquam is circum rectangulum infinitesimale <math>dx^\nu dx^\sigma</math> transportatur, ut vector novus 
:::<math> V'^\mu = V^\mu + R^\mu_{\nu\sigma\tau} dx^\nu dx^\sigma V^\tau</math>.
Idem tensor dat accelerationem <math>A^\mu</math> corporis 1 relativam positioni corporis 2 in statu libere cadendi
:::<math> A^\mu = -R^\mu_{\nu\sigma\tau} V_1^\nu dx^\sigma V_2^\tau</math>
ubi <math>V_i</math> denotat 4-velocitas corporis ''i'' et <math>dx^\sigma</math> positionem corporis 2 corpori 1 relativam.

Ex hac accelerationis formula, significationem tensoris Ricciani respectu theoriae relativitatis possumus extrahere. Ad propositum, imaginamus multas particulas statu libere cadendi vel quiescentes, cuius velocitates <math>V^\nu</math> tempore <math>t=0</math> sunt identicae, in sphaera voluminis <math>\mathbb{V}</math> infinitesimalis quadam circum corpus 1 sitas. Summando super omnibus positionibus <math>dx^\sigma</math> in sphaera et omnibus directionibus <math>\mu</math>, obtinemus<ref name = "baez"/>

:::<math> \frac{\ddot{\mathbb{V}}}{\mathbb{V}} = -R^\sigma_{\nu\sigma\tau} V^\nu V^\tau=-R_{\nu\tau} V^\nu V^\tau = -R_{44} c^2</math>

ubi ultimo passo adsumimus particulas quiescentes esse, i.e. <math>V^\nu=(0,0,0,c)</math>.  Possumus igitur intepretare tensorem Riccianum ut voluminis fluxionem, curvaturae spatio-temporalis causa, quae in casu particularum quiescentum formam simplissimam adsumit.

=== De tensore energiae motusque ===
In physica relativistica tensor energiae motusque <math>T_{\alpha \beta}</math> sic definitur ut 4-motus fluxio per aream A in directionem <math>n^\beta</math> correspondat
:::<math>\frac{dP_{\alpha}}{dt}= T_{\alpha \beta}\; n^\beta A</math> 
Hac definitione habemus
:::<math>T_{\alpha \beta} 
=\begin{pmatrix}  
 G_{xx} & G_{xy} & G_{xz} & \frac{S_x}{c}\\ 
 G_{yx} & G_{yy} & G_{yz} & \frac{S_y}{c}\\ 
 G_{zx} & G_{zy} & G_{zz} & \frac{S_z}{c}\\
\frac{S_x}{c} & \frac{S_y}{c} & \frac{S_z}{c} & W \\
\end{pmatrix}</math>

ubi
* <math>W</math> est [[densitas]] [[energia]]e,
* <math>(S_x,S_y,S_z)</math> est  densitas [[motus]], et
* <math>G_{ii}</math> est pressio mechanica in directionem ''i'',
* <math>G_{ik}</math> est contentio virium in superficie ''i'' in directionem ''k''.

Pro liquore ideali inviscido qui calorem non conducit, possumus scribere
:::<math>T_{\alpha \beta} \, = (\rho + {p\over c^2})u_{\alpha}u_{\beta} + pg_{\alpha \beta}</math>.

ut gravitate paene absente
:::<math>g_{\alpha \beta} \, = \mathrm{diag}(1,1,1, -1)</math>,
et celeritate paene zeri
:::<math>u_{\alpha}u_{\beta} \, = \mathrm{diag}(0,0,0,c^2)</math>, 
habeamus
:::<math>
T_{\alpha \beta} = 
\left( \begin{matrix}
p & 0 & 0 & 0 \\
0 & p & 0 & 0 \\
0 & 0 & p & 0 \\    
0 & 0 & 0 & \rho c^2 \\
\end{matrix} \right).
</math>

=== Lex gravitatis Einsteiniana ===
[[Fasciculus:Universe expansion-en.svg|thumb|200px|Secundum aequationem Einsteiniam, si constans cosmologicus esset zerum, continuum spatio-temporale expandere necesse est. Et si expandit, necesse est continuum fuisse olim minus quam hodie--punctum singulare tantum! Mundus noster igitur potest olim [[Fragor Maximus|fragore maximo]] incipere ex puncto singulari.]]

Secundum theoriam Einsteinianam, [[massa]] (sive [[energia]]), tensore <math>T_{\mu\nu}</math> energiae motusque data, curvaturam spatio-temporalem efficit. ''Aequatio Einsteniana'' 
:::<math> R_{\mu\nu} -\frac{1}{2}g_{\mu\nu}R - \Lambda g_{\mu\nu}= \frac{8\pi G}{c^4}T_{\mu\nu} \,</math>,
igitur curvaturam spatio-temporalem ad tensorem <math>T_{\mu\nu}</math>  attribuit, unde <math>R_{\mu\nu}</math> est tensor Riccianus, <math>R</math> curvatura scalaris, et <math>\Lambda</math> clarissimus ''constans cosmologicus'', qui sicut [[vacuum|vacui]] energia interpretari potest. 

Ad hanc aequationem, Einstein modo per multa conata pervenit. Sine constante <math>\Lambda</math>, Einstein mox repperit concentrationem<ref>Ernestus Gotthold Struve D., ''Paradoxum chymicum sine igne'', Ienae, 1715, apud Ernestum Claudium Bailliar, p. 55. [http://books.google.com.br/books?id=s8MPAAAAQAAJ&pg=PT62&lpg=PT62&dq=concentrationibus&source=bl&ots=g6-lXUDh65&sig=_LFxoUn2xECZDsD0_jbEtnhzhIk&hl=pt-BR&sa=X&ei=hfdfT8_FCOni0QHy2tynBw&sqi=2&ved=0CEYQ6AEwBQ#v=onepage&q=concentrationibus&f=false] [[Libri Googles]] {{Ling|Latine}}</ref> massae energiaeque in mundo universo esse tam parvam, ut secundum suam aequationem continuum spatio-temporale assidue expanderet. Haec expansio autem ei apparuit tam absurda, ut in prima editione suae theoriae Einstein constantem <math>\Lambda</math> proprosuerit magnitudinem tantum habere ut expansionem adusque consistat. Et postea, cum [[Edwinus Hubble]] ex [[stella]]rum observationibus legem deduxit,<ref>[http://antwrp.gsfc.nasa.gov/diamond_jubilee/1996/hub_1929.html ''A relation between distance and radial velocity among extra-galactic nebulae''], E. Hubble, Proceedings of the National Academy of Sciences 15 (1929).</ref> quae continuum universum sic expandere sicut <math>\Lambda \approx 0</math> dicit, Einstein funditus constantem reliquit, eam ''maximum erratum suae vitae'' dicens.<ref>G. Gamow, My World Line—An Informal Autobiography, Viking Press, New York (1970), p. 44. I thank Lawrence Krauss for this reference.; Vide etiam: https://web.archive.org/web/20071203030652/http://www.physicstoday.org/vol-58/iss-11/p31.html</ref>

Hodie observationes astronomicae monstrant <math>\Lambda</math> non posse excedere 10<sup>-46</sup> km<sup>-2</sup>.<ref>Christopher S. Kochanek (August 1996). ""Is There a Cosmological Constant?"". The Astrophysical Journal 466 (2): 638-659.</ref>

=== De interpretatione legis Einsteinianae ===
Lex Einsteinana, 16 aequationes differentiales gradus secundi in se continens, est instricatissima legi Newtonianae composita. Quid lex significat?

Memoramus, cum particulae quiescentes quaedam tempore <math>t=0</math> volumen <math>\mathbb{V}</math> complent, aequationem Riccianam
:::<math> \frac{\ddot{\mathbb{V}}}{\mathbb{V}} = -R_{44} c^2</math>
eorum voluminis fluxionem dare curvaturae causa. Aequationem Einsteinianam per <math>g^{\mu\nu}</math> multiplicando, et <math>\Lambda = 0</math>, <math>g^{\mu\nu}g_{\mu\nu}=4</math>, et <math>g^{\mu\nu}R_{\mu\nu}=R</math> supponendo, statim obtinemus
:::<math> R = -\frac{8\pi G}{c^4}T^{\mu}_{\mu} \,</math>.
Aequationem igitur Einteinianam possumus rescribere
:::<math> R_{\mu\nu}= \frac{8\pi G}{c^4}\left( T_{\mu\nu} -\frac{1}{2}g_{\mu\nu} T^{\alpha}_{\alpha}\right)\,</math>
unde, supponendo etiam <math>T^{\alpha}_{\alpha} = P_x+P_y+P_z-W</math> et <math>g_{44}=-1\,</math> et adsumendo omnem energiae densitatem <math>W=\rho c^2\,</math> esse ob materiae densitatem <math>\rho</math>, habemus
:::<math> \frac{\ddot{\mathbb{V}}}{\mathbb{V}}= -R_{44} c^2 = -4\pi G \left(\rho+ \frac{P_x + P_y + P_z}{c^2}\right) \approx -4\pi G\rho</math>,
ultimo passo quoque adsumendo condicionem <math>\frac{P_x + P_y + P_z}{c^2} \ll \rho </math>, quae fere semper tenet.

Lex Einsteiniana igitur possumus sic dicere: cum manus particularum quiescentum incipit libere cadere, eius volumen incipit decrescere secundum productum voluminis initialis et summae densitatis energiae in centro atque pressionum directiones ''x'', ''y'', et ''z'' versus.<ref name = "baez"/>

=== Quomodo legem Newtonianam ex Einsteiniana derivare ===
Quod lex Einsteinana haud modo apparet legi Newtonianae similis, nobis oportet etiam investigare quomodo legem Newtoniana e lege Einsteiniana derivare. 

Ex aequatione, quae volumen perditum propter massam terrestrem <math>\rho \mathbb{V}_0 = M</math> dat,
:::<math> \frac{\ddot{\mathbb{V}}}{\mathbb{V}}= -4\pi G\rho</math>,
possumus aequationem Newtonianam continuo potiri,<ref name = "baez"/> quia, quoniam <math>\dot{\mathbb{V}} =0</math> adsumimus, bis integrando obtinemus, post tempus <math>\delta t</math>,
:::<math> \delta \mathbb{V} = -4\pi G\rho\mathbb{V}_0\left( \frac{1}{2}\;\delta t^2 \right)</math>.

Etiam, quoniam distantia <math>r</math> radium terrestrem <math>R</math> excedens volumen <math>\mathbb{V}=\frac{4}{3}\pi r^3</math> complectitur, simul habemus
:::<math> \delta \mathbb{V} =  4\pi r^2 \delta r</math>,
ut, quoniam <math> \delta r = \frac{1}{2} a\,\delta t^2</math>, comparando obtinamus 
:::<math>a = 2 \frac{\delta r}{\delta t^2} = -\frac{G M}{r^2}</math>, 
eam quae erat demonstanda: formulam accelerationis Newtonianam.

== Status theoriae hodie respectu observationibus ==

=== Quid stellae pulsantes patefaciant ===

=== Quid galaxiae patefaciant ===

=== Quid expansio mundi universi patefaciat ===

== Ad theoriam gravitatis quanticam ==
Quamvis theoria relativitatis generalis accurate phaenomena ad gravitatis vim pertinentia describit, [[physicus|physici]] relativitatem generalem cum mechanica quantica coniungere maxime volunt, quia notum est circa [[longitudo Planckiana|longitudinem Planckianam]] effectus quanticos ob gravitatis vim non ignorandos esse. [[Theoria chordarum]] est nostri theoria temporis quae has duas claras theorias optime coniungat, sed non plane confecta est, atque nondum experimentorum confirmationem habet. In hac descriptione vis gravitatis ex particularum minimarum ''gravitonorum'' permutationibus nascitur, quemadmodum [[theoria camporum quanticorum]] corporum attractiones describere solet, e.g., attractio electrica corporum [[onus electricum]] habentium in se mutuo ex [[photon]]orum permutationibus fiat.

Quamquam ab experimentis multis constantem cosmologicam esse minimam comprobatur, [[theoria camporum quanticorum]] illam esse permagnam praevidet, plus quam <math>10^{100}</math> magnitudinis ab experimentis mensuratae. Nunquam aliqua theoria tam longe ab experimentis veritatibus absit et usque ad hodie hoc ''constantis cosmologicae problema'' non solutum est. Hoc autem problema [[supersymmetria]] minuit (discrepantia est circa <math>10^{50}</math> post supersymmetriae fracturam), sed id non solvit.

== Hodiernus theoriae status ==
Relativitas generalis facta est mirifice prosperum [[gravitas (physica)|gravitationis]] [[cosmologia]]eque exemplar, quod hactenus omne observationis experimentique examen certum probavit; nihilominus, sunt clarae indicationes theoriae imperfectae.<ref>Cf. {{Harvnb|Maddox|1998|loc=pp. 52–59 and 98–122}}; {{Harvnb|Penrose|2004|loc=section 34.1 et capitulum 30}}.</ref> Quaestiones gravitatis quanti et realitatis [[singtularitas spatiotemporis|singularitatum spatiotemporis]] apertae manent.<ref>Vide sectionem [[Relativitas generalis#Gravitas quanti]] supra.</ref> <!--

Observational data taken as evidence for [[dark energy]] and [[dark matter]] could indicate the need for new physics,<ref>Cf. the section [[General relativity#Cosmology|Cosmology]], above.</ref> and while the so-called [[Pioneer anomaly|''Pioneer'' anomaly]] might yet admit of a conventional explanation, it, too, could be a harbinger of new physics.<ref>See {{Harvnb|Nieto|2006}}.</ref> Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,<ref>See {{Harvnb|Friedrich|2005}}.</ref> and increasingly powerful computer simulations (such as those describing merging black holes) are run.<ref>A review of the various problems and the techniques being developed to overcome them, see {{Harvnb|Lehner|2002}}.</ref> The race for the first direct detection of gravitational waves continues apace,<ref>See {{Harvnb|Bartusiak|2000}} for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as [http://geo600.aei.mpg.de GEO 600] and [http://www.ligo.caltech.edu/ LIGO].</ref> in the hope of creating opportunities to test the theory's validity for much stronger gravitational fields than has been possible to date.<ref>For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see {{Harvnb|Blanchet|Faye|Iyer|Sinha|2008}}, and {{Harvnb|Arun|Blanchet|Iyer|Qusailah|2007}}; for a review of work on compact binaries, see {{Harvnb|Blanchet|2006}} and {{Harvnb|Futamase|Itoh|2006}}; for a general review of experimental tests of general relativity, see {{Harvnb|Will|2006}}.</ref>--> Abhinc annos plusquam nonaginta divulgata, relativitas generalis exstat fertilissimus investigationis campus.<ref>A good starting point for a snapshot of present-day research in relativity is the electronic review journal [https://web.archive.org/web/20161227064809/http://relativity.livingreviews.org/ ''Living Reviews in Relativity''].</ref>

{{NexInt}}
{{div col|2}}
* [[Albertus Einstein]]
* [[E=mc²]]
* [[Fragor Maximus]]
* [[Gravitas (physica)]]
* [[Foramen nigrum]]
* [[Iter transtemporale]]
* [[Relativitas specialis]]
* [[Singularitas nuda]]
* [[Spatium Calabi-Yau]]
* [[Theoria Chordarum]]
{{div col end}}

== Notae ==
<references/>

== Bibliographia ==
[[Fasciculus:Calabi-Yau.png|thumb|200px|Proiectio [[Multiplex Calabi-Yau|Multiplicis Calabi-Yau]], unius ex modis quibus additae [[dimensio]]nes quas [[theoria chordarum]] poscit [[compactificatio (physica)|compactificantur]]]]
[[Fasciculus:Spin network.svg|thumb|200px|Simplex [[rete versans]]<!--spin network--> generis quod in [[gravitas quantum loop|gravitate quantum loop]]{{dubsig}} adhibetur]]
* Carroll, Sean M. [[2004]]. ''Spacetime and Geometry: An Introduction to General Relativity.'' San Francisco: Addison-Wesley. ISBN 0-8053-8732-3. http://spacetimeandgeometry.net/.
* Dirac, P. A. M. [[1975]]. ''General Theory of Relativity.'' Princeton: Princeton University Press.
* [[Albertus Einstein|Einstein, Albert]], H. A. Lorentz, H. Weyl, et H. Minkowski. [[1952]]. ''The Principle of Relativity.'' A W. Perrett et G. G. Jeffery conversus. Toronti: Dover Publications.
* [[Albertus Einstein|Einstein, Albert]]. [[1922]]. ''The Meaning of Relativity, Including the Relativistic Theory of the Non-Symmetric Field.'' Princeton: Princeton University Press.
* Grøn, Øyvind, et Sigbjørn Hervik. [[2007]]. ''Einstein's General Theory of Relativity.'' Novi Eboraci: Springer. ISBN 978-0-387-69199-2.
* Landau, Lev D., et Evgeny F. Lifshitz. [[1980]]. ''The Classical Theory of Fields.'' 4a ed. Londini: Butterworth-Heinemann. ISBN 0-7506-2768-9.
* Misner, Charles W., Kip S. Thorne, Kip. S., et John A. Wheeler. [[1973]]. ''Gravitation.'' W. H. Freeman. ISBN 0-7167-0344-0.
* [http://math.ucr.edu/home/baez/gr/ Praecepta de relativitate generali comparata a Ioanne Baez, Ph.D., professore physicae.]
* Stephani, Hans. [[1990]]. ''General Relativity: An Introduction to the Theory of the Gravitational Field.'' Cantabrigiae: Cambridge University Press. ISBN 0-521-37941-5.
* Wald, Robert M. [[1984]]. ''General Relativity''. Sicagi: University of Chicago Press. ISBN 0-226-87033-2.
* Weinberg, Steven. [[1972]]. ''Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity''. Novi Eboraci: John Wiley & Sons. ISBN 0-471-92567-5.

;Libri populares
* Geroch, Robert. [[1981]]. ''General Relativity from A to B.'' Sicagi: University of Chicago Press. ISBN 0-226-28864-1.
* Lieber, Lillian. [[2008]]. ''The Einstein Theory of Relativity: A Trip to the Fourth Dimension.'' Philadelphia: Paul Dry Books, Inc. ISBN 978-1-58988-044-3.
* Wald, Robert M. [[1992]]. ''Space, Time, and Gravity: the Theory of the Big Bang and Black Holes.'' Sicagi: University of Chicago Press. ISBN 0-226-87029-4.

;Libri pro scholis scripti
* Callahan, James J. [[2000]]. ''The Geometry of Spacetime: an Introduction to Special and General Relativity.'' Novi Eboraci: Springer, ISBN 0-387-98641-3.
* Taylor, Edwin F., et John Archibald Wheeler. [[2000]]. ''Exploring Black Holes: Introduction to General Relativity.'' Addison Wesley. ISBN 0-201-38423-X.

;Libri pro collegiis et universitatibus scripti 
* Cheng, Ta-Pei. [[2005]]. ''Relativity, Gravitation and Cosmology: A Basic Introduction.'' Oxoniae et Novi Eboraci: Oxford University Press. ISBN 0-19-852957-0.
* Gron, O., et S. Hervik. [[2007]]. ''Einstein's General theory of Relativity.'' Springer. ISBN 978-0-387-69199-2.
* Hartle, James B. [[2003]]. ''Gravity: an Introduction to Einstein's General Relativity.'' San Francisco: Addison-Wesley. ISBN 0-8053-8662-9.
* Hughston, L. P., et K. P. Tod. [[1991]]. ''Introduction to General Relativity.'' Cantabrigiae: Cambridge University Press. ISBN 0-521-33943-X.
* d'Inverno, Ray. [[1992]]. ''Introducing Einstein's Relativity.'' Oxoniae: Oxford University Press. ISBN 0-19-859686-3.
* Schutz, B. F. [[2009]]. ''A First Course in General Relativity.'' Ed. 2a. Cantabrigiae: Cambridge University Press. ISBN 978-0-521-88705-2.

[[Categoria:Physica]]
{{1000 paginae}}

{{Myrias|Physica}}