Relativity Files - Cetin BAL - GSM:+90 05366063183

General Relativity and Special Relativity Files (Genel Rölativite ve Özel Rölativite Dosyası)

olarak bulmuştur.

Burada; M1, 2 cisimlerin kütlesi, R aralarındaki uzaklık, G ise $6,6710 - 11 Nm 2 kg - 2$ değerinde olan kütleçekim sabitidir.

Kütle, bir cisimdeki madde miktarının ölçüsüdür. Kütle her yerde aynı değere sahiptir.

Kütlenin SI birim sistemindeki birimi kilogram'dır. Bu kg. olarak kısaltılır. Kullanılan diğer birimler gram, ton ve pound'dur. Görelilik teorisine göre duran kütle m ile enerji E arasında E = mc2 bağlantısı olduğundan enerji birimi olan elektronVolt (eV) da kütle için kullanılabilir. Özellikle kütle ve enerjinin birbirine dönüşebildiği parçacık fiziğinde eV sık kullanılmaktadır. (yaklaşık 1 eV=1.783 × 10-36 kg).

Einstein teorisine göre uzay zaman eğridir.Uzay zamanın eğriliği kütle çekimi, yani gravitasyona eşittir. Bunu anlatacak bir örnek: Bir portakalın üstüne üç toplu iğne batıralım ve bu toplu iğnelere göre bir bıçakla portakalı keselim.Ortaya portakal kabuğundan yapılmış bir üçgen çıkacaktır.O üçgeni alıp masaya koyarsanız üçgenin kenarlarının düz olmadığını görürsününüz. Düz bıçakla kestiğimiz kenarlar eğridir. Şimdi aynı şekilde diyelim ki siz dünyadan bir uyduya bir sinyal gönderdiniz. O da bu sinyali başka bir uyduya gönderdi ve ikinci uydudan sinyal tekrar dünyaya aksettirildi. Işığın yörüngesi en kısa mesafelerden oluşan bir jeodezik üçgendir. Eğer güneş bu üçgenin içinde ise o zaman ortaya çıkan kenarları dışa doğru eğri bir üçgendir, tıpkı portakal kabuğu gibi. Çünkü güneşin kütlesinden dolayı ışık eğri bir yörünge takip ediyor. Bunu güneş tutulması esnasında arka plandaki yıldızlarının yerlerinin kaymasından görmüştük. Şimdi Einstein gibi şöyle düşünebilirsiniz : Ben güneşi ortadan kaldırayım ama uzay zamanı o üçgeni verecek şekilde eğri yapayım, tıpkı portakalın üstünde olduğu gibi.Bir bakış açısına göre güneşin kütle çekimi ışığın yörüngesini saptırıyor düz olmaktan. Öteki görüşte güneş hiç ortada yok, uzay zamanın eğriliği ışığın yörüngesinin düz olmamasını sağlıyor.

Einstein'in uzay zamanı bükük olarak nitelendirmesinde ilk olarak ivme ile çekim alanı arasında güçlü bir ilişki olduğunun farkına varması yatar. Asansör gibi, kapalı bir kutu içerisindeki bir kişi, kutunun dünyanın çekim alanında durağan mı olduğunu, yoksa serbest uzayda bir roketle ivme mi kazandığını ayırt edemez. Eğer dünya düz olsaydı, elmanın Newton'un başına yerçekimi nedeniyle veya Newton ve dünya yüzeyi yukarı doğru ivme kazandığı için düştüğüne eşit biçimde inanılabilirdi. Gene de, ivme ile yerçekimi arasındaki sözkonusu eşitlik, yuvarlak bir dünya için geçerliymiş gibi görünmüyordu. Çünkü, dünyanın zıt taraflarındaki kişilerin zıt yönlerde ivme kazanmaları ancak birbirlerinden sabit bir uzaklıkta kalmalarıyla mümkündü. İvme ve yerçekimi ancak, büyük bir kütle; uzay-zamanı bükerek, yakın çevresindeki nesnelerin yollarını eğerse denk olabilirdi!!!

Einstein kütle ve enerjinin, uzay-zamanı belirlenmesi gereken bir şekilde bükeceğini düşünüyordu. Gezegen gibi nesneler uzay-zaman boyunca düz doğrularla ilerlemeye çalışacak ancak uzay-zaman eğri olduğu için yolları, bir çekim alanı tarafından bükülmüş gibi görünecekti.

Evren madde ile doludur, ve madde uzay-zamanı kütleler birbiri üzerine düşecek şekilde büker. Einstein eşitliklerin zaman içerisinde değişmeyen, durağan bir evreni tanımlayan bir çözüme sahip olmadığını buldu. Kendisinin ve diğer birçok kişinin de inandığı sonsuz evrenden vazgeçmek yerine, kozmolojik sabit adı verilen bir terim ekleyerek eşitlikleri düzeltti. Bu sabit uzay-zamanı aksi anlamda büküyordu. Bunun sonucunda da kütleler birbirlerinden uzaklaşıyordu. Evrensel sabitin itici etkisi, maddenin çekici etkisini dengeleyerek evren için durağan bir çözüm sağlayabilecekti. Ancak bu 1920lerde yapılan gözlemlere kadar ciddiye alınmadı. Bu gözlemlerde diğer galaksilerin bizden ne kadar uzakta iseler o kadar hızla uzaklaşacakları ortaya çıkarıldı...

Genelde hep klasik bir örnek verilir, kocaman gerilmiş bir bez düşünün, üzerine de ağır küreler atın. Tabi kürenin durduğu yer çukur olacak. Büyük kütle büyük çukurluk yaratır, ve bez üzerinde bir top yuvarladığınız zaman yolları bu çukurlar tarafından bükülür. Yerçekiminin uzayı bükmesi bu benzetmeyle anlatılır.

Ancak işler bu örnekteki gibi de basit değil her zaman. Einstein'in teorisi de sonuçta matematiksel modellere dayanıyor ve bunlar tamamen soyut. Bu oluşan bükülmeler de bizim bildiğimiz anlamdaki bükülmelerden değil, yani üç boyutlu bir cismin yamulması şeklinde olmuyor. Einstein'in teorisinde evren 4 boyutlu bir manifold (Türkçe'si katman olabilir, emin değilim) şeklinde tasarlanıyor, tabi uzaydaki hareketlerin vs de karşılık gelecek şekilde modellemeleri var. Kütle çekimi denen şey de manifoldun yapısı üzerindeki bir takım özelliklere karşılık geliyor.

Manifold teorisi ve rölativitenin salt denklem takımları üzerinde formel anlamda derinlemesine çalışmadığım için ayrıntı veremeyeceğim, ama olaya bu model çerçevesinde bakılınca olayların karışıklığa yer vermeden anlaşılacağını düşünüyorum. Bazen üst düzey birşeyi ögrenciler yada meraklılar anlasın diye basit modellerle anlatmaya kalkınca olay kafa karıştırıcı bir hal alabiliyor. Matematik forumlarında bugüne kadar yazılanlardan edindiğim izlenim bu. Lise matematiğinde bazı şeyler ispatsız verilir, ya da muglak tanımlarla verilir, ögrenciye o tanımlarla (ki adamakıllı teorik bilgi ister bunlar) zaman kaybettirmemek için. Tabi biraz düşünen ögrenciler sonunda bazı terslikleri gözlemliyorlar ya da kendilerine sunulanı kesin doğru diye kabul edip bazı yanlış düşünceler geliştiriyorlar. Matematikte olduğu gibi fizikte de olayı tam manasıyla kavramak için, olayın içine bizzat girmek gerekiyor bence.

Genel rölativite teorisine ait yerçekimsel alanı niteleyen denklem: Einstein General Field Equation:

g_μ&nu = the Riemann tensor describing the scale factor (metric) of spacetime. This is essentially a determination of the distance between adjacent points in spacetime.
R_μ&nu = the Ricci tensor, and is essentially derivatives of the Riemann tensor.
R = the "trace" of the Ricci tensor describing the radius of spacetime's curvature
T_μ&nu = the stress-energy tensor describing the density of mass-energy in spacetime

Bükülme olarak nitelendirdiğimiz bir ipin bükülmesi gibi birşey değil. Uzay ve Zaman'i iki ayrı boyut olarak düşünürsek oluşturdukları bir alan olacaktır. O alanın bükülmesinden bahsediyoruz. Aslında bu teorik fiziği Einstein'dan beri gayet net ve anlaşılır açıklayan pek insan olmadı. Düz olarak hareket eden bir nesne düşünelim mesela bir uçak gökyüzünde aynı rotada ilerlesin. Yeryüzünden uçağa bakınca düz görürüz ama dağlar ve yüzeydeki engebeler üzerinde uçak sanki kıvrılarak gidiyormuş gibi görünür. Uzay zamanın bükülmesi de böyle birşey. Gezegenler uzay-zaman içinde düz ilerlemeye çalışacak ancak uzay-zamanın bükülmesiyle çekim alanıyla bükülmüş gibi görünecektir.

Soru: Kütle çekimini bir bez üzerindeki küre gibi düşündük.Ve tabi bu bezin bir esnekliği olacak.Sormak istediğim şey tıpkı bezin kendi türüne göre farklı bir şekilde esnemesi gibi bir olay gerçekte de iki kütlenin bulunduğu ortama bağlı mı?Yani boşlukta iki kütlenin birbirini çekmesiyle suda veya başka bir ortamda iki kütlenin birbirini çekmesi aynı şeymi.Toparlarsak Newton Evrensel Çekim yasasına göre; F=G.m.M/d2 de ki kütle ve uzaklık ortama göre değişmediğine göre G sabiti ortama göre değişir mi?

Çekim kuvveti kütle fazlalaştıkca artar. Güneşin kütlesi dünyadan 330000 kat fazladır. Bu durumda dünyada 60 kg olan bir kişi güneşte 20000 ton olacaktır. Çekim kuvvetini etkileyen ikinci özellik uzaklıktır. Cisimlerin merkezine doğru yaklaştıkca çekim kuvveti de artar. Yıldızların yakıtını tüketip çöküşü sırasında yarıçapı azaldığı için çekim kuvvetleri aşırı derecede artar. Bunlar pekçok insan için bilinen bilgiler ancak iyice oturması için tekrar vurgulamak istedim.

İki cisim arasındaki yerçekimi kuvveti, bu iki cismin merkezleri arasındaki uzaklık azaldıkca büyür. Yani uzaklığın karesi ile ters orantılı olarak artar. Yeryüzünden belli bir uzaklıkta olan birisi bu uzaklığın yarısına geldiğinde dünyanın o insan üzerindeki yerçekimi kuvveti dört kez artar. Yerçekimin en fazla olduğu yer yerkabuğunun üzerindedir. Zira dünyanın merkezine yaklaştıkca yerçekiminin etkisi de azalır! Yerçekiminin uzaklığa bağlı oluşu sadece çekiminde bulunan cismin dışında bulunma halinde geçerlidir. Ancak, o zaman cismin tüm kütlesinin merkezde toplanmış olduğu düşünülebilir. Dünyanın içine girildikçe, ancak dünyanın merkeze daha yakın olan kısmı çekecektir, üst tarafta kalan kısmın yerçekimine bir katkısı olmayacaktır ve sonunda dünyanın tam merkezine gelindiğinde ise hiçbir çekim kuvveti kalmayacaktır..

Ortamlara göre çekim kuvveti değişmez. Sadece ek kuvvetler girer araya. Suyun içinde ve aynı seviyedeki karada çekim kuvveti aynıdır ama sudaki kaldırma kuvveti çekim kuvvetine zıt yönde etkiler. Kütle ve çekim merkezine uzaklık aynı kaldığı sürece yerçekimi kuvveti değişmez. Örneğin dünyadan uzakta bir uzay gemisindeki insana göre dünya kütlesini kaybetmeden şimdiki yarıçapının yarısına büzüldüğü takdirde, o insan üzerinde etkiyen yerçekimi kuvveti değişmeyecektir. Çünkü insanın kütlesi dünyanın kütlesi ve insanın dünyanın merkezine uzaklığı hep aynı kalmaktadır. Ancak dünya üzerinde bulunan bir insan için bu farklı olacaktır. Dünyanın yarıçapının yarısına büzülmesi dünya yüzeyindeki insan üzerindeki çekimi 4 kat arttıracaktır..

Soru: Öğrendiğim kadarıyla ışık bir karadeliğin yakınından geçerken bir sapmaya uğruyor. Bize öğretilene göre ise ışık madde değil. Madde olmayan birşeyin kütle çekimden etkilenmesi mümkün mü? Eğer kütle çekimden dolayı sapmaya uğruyorsa zamanda yolculuk yapması da gerekmez miydi?

Cevap: Bu soruyu daha önce de yanıtlamıştık. Kütle uzay-zamanda bükülmeye yol açar. Karadelikler de çok büyük bir kütleyi çok küçük bir hacimde toplamış cisimler. Dolayısıyla uzay/zamanın bükülmesi sonsuz eğrilikte oluyor (dipsiz bir kuyu gibi). Böyle olunca da kendisi kütlesiz olmasına karşın ve dolayısıyla kütleçekimini duymayan fotonlar da (ışık parçacıkları) bu bükülmüş uzay/zaman dokusunu izleyerek yol aldıklarından hareketlerinde bir sapma görülüyor. Eğer karadeliğin yakınlarından geçmekte olan bir foton, karadeliğin olay ufku (yani uzayın dipsiz kuyu gibi büküldüğü alan) içine düşerse, bu eğriliği sonsuza kadar izlediği için bir daha dışarı çıkamıyor ve bu yüzden de karadelikler (adları üzerinde) gözlemlenemiyor, varlıkları ancak dolaylı yollardan belirleniyor.

Acaba kütle merkezinden dışarı doğru yayılan bu kuvvet engellenebilir mi?
Bunlar fotonlar gibi yansıtılabilir mi yada absorbe edilebilir mi ?

Çetin BAL: Einstein'ın özel görelilik (Special Relativity) kuramına göre hızlanan cisimlerde ( tren, roket, uçak, araba... vb gibi) zaman yavaşlar. Daha teknik bir ifade ile yerde duran bir gözlemci için zaman belli bir hızla akarken bu gözlemciye göre hareket eden bir uçak yada tren içindeki bir saatin tik takları arasındaki zaman dıştaki gözlemcinin saatine göre daha da genişler. Dolayısıyla zaman yerdeki gözlemciye göre daha hızlı geçerken gözlenen hareket halindeki araç içinde zaman daha yavaş geçmektedir.Dolayısıyla dıştaki gözlemciye göre farklı algılanan tek şey zamanın geçişi değildir. Hareket halindeki bir nesnenin boyutlarıda dıştaki gözlemciye göre hareket yönüne doğru kısalmaya uğrar. Bu kısalma ''hızla'' doğru orantılı bir şekilde artar. Hareket halindeki gözlenen araçlardaki zaman genişlemeside buna yani hareketin hızına paralel olarak artar.

En la imagen vemos una caja amarilla que se mueve en el interior de un tren.
El movimiento de este objeto es estudiado por dos observadores: Uno de ellos, O, está en reposo junto a la vía. Mientras que otro se mueve con el tren, aunque esté quieto en su interior.

Cada observador tiene su propio sistema de coordenadas y su reloj para estudiar el movimiento de la caja. Tratamos con dos sistemas de referencia en movimiento relativo.
Por otro lado, recordemos el primero de los principios de la Dinámica de Newton , el principio de inercia, que podemos enunciar así: Para que un cuerpo posea aceleración debe actuar una fuerza exterior sobre él.

Si desde un sistema de referencia se cumple este principio, el sistema se considera inercial. ¿Son todos los sistemas de referencia inerciales?. Para responder esta pregunta es mejor que nos introduzcamos en el siguiente apartado, el Principio de Relatividad.

Evrenin geometrisinin ( uzay/zamanın düz çizgilerinin) kapalı bir evreni tasvir edip etmemesine bağlı olarak eğrilen uzay-zamanın geçmişe yada gelecek zaman/uzay noktalarına geçit verecek şekilde bükülüp bitişmesi ve topolojik geometri kaynaşmaları (tüp geçitleri) söz konusu edilebilir. Matematik ve fiziğin alan geometrisi denklemleri içinde nokta ve çizgilerin bu şekilde bitiştirilip kaynaştırılması ile salt denklemler içinde zaman yolculuğuna olanak sağlayan solucan deliği tünelleri oluşturulabilir. Evren geometrisi içinde bu çeşit tünellere en yakın oluşumlar karadeliklerdir.

How the sun bends space,time and light

Although our sun is an ordinary star, and not a massive black hole, it distorts space-time into a "gravity well". As the sun moves against the background of fixed stars, their positions seem to move slightly. The effect is tiny - less than a thousandth of a degree - and is caused by gravity bending light. Light from the sun appears to be coming from a slightly different direction. It is bent because it follows a curving path across the sun's space-time "dimple", instead of the straight line it would follow in gravitationally undistorted space.

Extreme conditions at the Large Hadron Collider may produce wormholes in space-time. An advanced civilisation might be able to manipulate one of these to create a tunnel back to the point in time when the wormhole was first created. Colliding gravitational wavea from accelerated proton rip a wormhole in space-time ( 14 TeV concentrated into the space of 10 ^ -15 m ) The wormhole helps form a closed timelike curve, which allows particles to flow into the past, or from the future to the present. Dark energy might keep wormhole open, and could even make it wide enough for a person.

wormhole

Physics. A theoretical distortion of space-time in a region of the universe that would link one location or time with another, through a path that is shorter in distance or duration than would otherwise be expected.

Analogy to a wormhole in a curved 2D space

Artist's impression of a wormhole as seen by an observer crossing the event horizon of a Schwarzschild wormhole, which is similar to a Schwarzschild black hole but with the singularity replaced by an unstable path to a white hole in another universe. The observer originates from the right, and another universe becomes visible in the center of the wormhole shadow once the horizon is crossed. This new region is, however, unreachable in the case of a Schwarzschild wormhole, as the bridge between the black hole and white hole will always collapse before the observer has time to cross it. See White Holes and Wormholes for a more technical discussion and an animation of what an observer sees when falling into a Schwarzschild wormhole.

In physics, a wormhole is a hypothetical topological feature of spacetime that is essentially a 'shortcut' through space and time. A wormhole has at least two mouths which are connected to a single throat. If the wormhole is traversable, matter can 'travel' from one mouth to the other by passing through the throat. While there is no observational evidence for wormholes, spacetimes containing wormholes are known to be valid solutions in general relativity.

The term wormhole was coined by the American theoretical physicist John Wheeler in 1957. However, the idea of wormholes was invented already in 1921 by the German mathematician Hermann Weyl in connection with his analysis of mass in terms of electromagnetic field energy.[1]

This analysis forces one to consider situations...where there is a net flux of lines of force through what topologists would call a handle of the multiply-connected space and what physicists might perhaps be excused for more vividly terming a ‘wormhole’.

John Wheeler in Annals of Physics

The name "wormhole" comes from an analogy used to explain the phenomenon. If a worm is travelling over the skin of an apple, then the worm could take a shortcut to the opposite side of the apple's skin by burrowing through its center, rather than travelling the entire distance around, just as a wormhole traveler could take a shortcut to the opposite side of the universe through a hole in higher-dimensional space.

Definition
The basic notion of an intra-universe wormhole is that it is a compact region of spacetime whose boundary is topologically trivial but whose interior is not simply connected. Formalizing this idea leads to definitions such as the following, taken from Matt Visser's

Lorentzian Wormholes:
If a Lorentzian spacetime contains a compact region Ω, and if the topology of Ω is of the form Ω ~ R x Σ, where Σ is a three-manifold of nontrivial topology, whose boundary has topology of the form dΣ ~ S², and if, furthermore, the hypersurfaces Σ are all spacelike, then the region Ω contains a quasipermanent intra-universe wormhole.

Characterizing inter-universe wormholes is more difficult. For example, one can imagine a 'baby' universe connected to its 'parent' by a narrow 'umbilicus'. One might like to regard the umbilicus as the throat of a wormhole, but the spacetime is simply connected.

Wormhole types

Intra-universe wormholes connect one location of a universe to another location of the same universe (in the same present time or unpresent). A wormhole should be able to connect distant locations in the universe by creating a shortcut through spacetime, allowing travel between them that is faster than it would take light to make the journey through normal space. See the image above. Inter-universe wormholes connect one universe with another [1], [2]. This gives rise to the speculation that such wormholes could be used to travel from one parallel universe to another. A wormhole which connects (usually closed) universes is often called a Schwarzschild wormhole. Another application of a wormhole might be time travel. In that case, it is a shortcut from one point in space and time to another. In string theory, a wormhole has been envisioned to connect two D-branes, where the mouths are attached to the branes and are connected by a flux tube [3]. Finally, wormholes are believed to be a part of spacetime foam [4]. There are two main types of wormholes: Lorentzian wormholes and Euclidean wormholes. Lorentzian wormholes are mainly studied in general relativity and semiclassical gravity, while Euclidean wormholes are studied in particle physics. Traversable wormholes are a special kind of Lorentzian wormholes which would allow a human to travel from one side of the wormhole to the other. Serguei Krasnikov suggested the term spacetime shortcut as a more general term for (traversable) wormholes and propulsion systems like the Alcubierre drive and the Krasnikov tube to indicate hyperfast interstellar travel.

Theoretical basis
It is known that (Lorentzian) wormholes are not excluded within the framework of general relativity, but the physical plausibility of these solutions is uncertain. It is also unknown whether a theory of quantum gravity, merging general relativity with quantum mechanics, would still allow them. Most known solutions of general relativity which allow for traversable wormholes require the existence of exotic matter, a theoretical substance which has negative energy density. However, it has not been mathematically proven that this is an absolute requirement for traversable wormholes, nor has it been established that exotic matter cannot exist.

In March 2005, Amos Ori envisioned a wormhole which allows time travel, does not require any exotic matter and satisfies the weak, dominant, and strong energy conditions [5]. The stability of this solution is uncertain, so it is unclear whether infinite precision would be required for it to form in a way that allows time travel and also whether quantum effects would uphold chronology protection in this case, as analyses using semiclassical gravity have suggested they might do in the case of traversable wormholes.

Schwarzschild wormholes
Lorentzian wormholes known as Schwarzschild wormholes or Einstein-Rosen bridges are bridges between areas of space that can be modeled as vacuum solutions to the Einstein field equations by sticking a model of a black hole and a model of a white hole together. This solution was discovered by Albert Einstein and his colleague Nathan Rosen, who first published the result in 1935. However, in 1962 John A. Wheeler and Robert W. Fuller published a paper showing that this type of wormhole is unstable, and that it will pinch off instantly as soon as it forms, preventing even light from making it through.Before the stability problems of Schwarzschild wormholes were apparent, it was proposed that quasars were white holes forming the ends of wormholes of this type.

While Schwarzschild wormholes are not traversable, their existence inspired Kip Thorne to imagine traversable wormholes created by holding the 'throat' of a Schwarzschild wormhole open with exotic matter (material that has negative mass/energy).

Traversable wormholes
Lorentzian traversable wormholes would allow travel from one part of the universe to another part of that same universe very quickly or would allow travel from one universe to another. The possibility of traversable wormholes in general relativity was first demonstrated by Kip Thorne and his graduate student Mike Morris in a 1988 paper; for this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, is referred to as a Morris-Thorne wormhole. Later, other types of traversable wormholes were discovered as allowable solutions to the equations of general relativity, including a variety analyzed in a 1989 paper by Matt Visser, in which a path through the wormhole can be made in which the traversing path does not pass through a region of exotic matter. A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al., [2], in which it was proposed that such wormholes could have been naturally created in the early universe.

Wormholes connect two points in spacetime, which means that they would in principle allow travel in time as well as in space. In a 1988 paper, Morris, Thorne and Yurtsever[3] worked out explicitly how to convert a wormhole traversing space into one traversing time.

Wormholes and faster-than-light travel
Special relativity only applies locally. Wormholes allow superluminal (faster-than-light) travel by ensuring that the speed of light is not exceeded locally at any time. While traveling through a wormhole, subluminal (slower-than-light) speeds are used. If two points are connected by a wormhole, the time taken to traverse it would be less than the time it would take a light beam to make the journey if it took a path through the space outside the wormhole. However, a light beam traveling through the wormhole would always beat the traveler. As an analogy, running around to the opposite side of a mountain at maximum speed may take longer than walking through a tunnel crossing it. You can walk slowly while reaching your destination more quickly because the length of your path is shorter.

Wormholes and time travel

A wormhole could allow time travel. This could be accomplished by accelerating one end of the wormhole to a high velocity relative to the other, and then sometime later bringing it back; relativistic time dilation would result in the accelerated wormhole mouth aging less than the stationary one as seen by an external observer, similar to what is seen in the twin paradox. However, time connects differently through the wormhole than outside it, so that synchronized clocks at each mouth will remain synchronized to someone traveling through the wormhole itself, no matter how the mouths move around. This means that anything which entered the accelerated wormhole mouth would exit the stationary one at a point in time prior to its entry. For example, if clocks at both mouths both showed the date as 2000 before one mouth was accelerated, and after being taken on a trip at relativistic velocities the accelerated mouth was brought back to the same region as the stationary mouth with the accelerated mouth's clock reading 2005 while the stationary mouth's clock read 2010, then a traveler who entered the accelerated mouth at this moment would exit the stationary mouth when its clock also read 2005, in the same region but now five years in the past. Such a configuration of wormholes would allow for a particle's world line to form a closed loop in spacetime, known as a closed timelike curve.

It is thought that it may not be possible to convert a wormhole into a time machine in this manner: some analyses using the semiclassical approach to incorporating quantum effects into general relativity indicate that a feedback loop of virtual particles would circulate through the wormhole with ever-increasing intensity, destroying it before any information could be passed through it, in keeping with the chronology protection conjecture. This has been called into question by the suggestion that radiation would disperse after traveling through the wormhole, therefore preventing infinite accumulation. The debate on this matter is described by Kip S. Thorne in the book Black Holes and Time Warps. There is also the Roman ring, which is a configuration of more than one wormhole. This ring seems to allow a closed time loop with stable wormholes when analyzed using semiclassical gravity, although without a full theory of quantum gravity it is uncertain whether the semiclassical approach is reliable in this case.

Wormhole metrics
Theories of wormhole metrics describe the spacetime geometry of a wormhole and serve as theoretical models for time travel. An example of a (traversable) wormhole metric is the following:

In Einsteins Universum haben Raum und Zeit eine übersichtliche Ordnung. Massereiche Körper krümmen die Raumzeit - in schöner Regelmäßigkeit

TIME DILATION
At this stage one may probably be thinking "Oh man, so length contracts...what happens to time?" So, let's do another thought experiment, a rather classical one at that... Consider the following situation, you are sitting in a train carriage which is travelling near the speed of light to some unknown destination (yes it is the train again - not very imaginative, but saves me time when making the graphics...). In front of you is a light hanging from the roof and there's a mirror on the floor which reflects the light back up. You have a large clock with which to time how long it takes for the light to reach the mirror and be reflected back to the top. Your good old buddy Einstein is standing outside, watching you and the train ride past. He too has a large clock, and is timing how long it takes for the light to travel down to the mirror and back.

As you can see in diagram 1, within your reference frame, the light only has to travel up and down. But in diagram 2, from Einstein's perspective, the light has to travel not only up and down but horizontally as well. Since the light has a longer distance to travel, more time is taken (remember that the speed of light is always constant!). Hence, the dilation of time.

Now for the mathematical part. The time interval between two events as seen by an observer (in this case, you) in the same reference frame where the events are taking place, is called proper time (t_o).The proper length (l_o) of an object is the length of the object measured in the frame where it is at rest. Let's simplify the above diagrams to the following two below. Diagram one is from your point of view in the carriage, and diagram 2 is from Einstein's perspective outside the train.

From diagram 1, t_o = 2d / c
From diagram 2 using Pythagoras' Theorem:
(c Δ t / 2)² = (v Δ t / 2)² + d ²
Solving for Δ t gives:
Δ t = 2d / (c² - v²)^1/2
Δ t = 2d/c x [1/ (1 - v²/c²)^1/2 ]
Since Δ t₀ = 2d/c, sub this into the equation:
Δ t = t_o / (1 - v²/c²)^1/2

Einstein's Theory of Gravitation

-Spacetime curving around a star-

Einstein realized that those observers in the presence of a gravitational field, and those in accelerating reference frames would not be able to determine their situation. He in turn, found that acceleration and gravity were equivalent. But in order to prove his idea, he needed to come up with a theoretical framework that would account for this. Einstein knew that a framework explaining gravity could also resolve the conflict in his theory of special relativity, and so he began working on a new theory of gravitation.

Einstein's theory of gravitation says that space will curve in the presence of matter, and this curvature will cause effects that we know as gravity. While Einstein was developing his equations, he quickly realized that standard mathematics (including calculus) would not be able to handle the situations described by his theory. He began researching the branch of mathematics called tensor analysis, which would allow him to develop equations to describe curved four-dimensional spaces.

Gravitational Time Dilation

A clock in a gravitational field runs more slowly according to the gravitational time dilation relationship from general relativity

where T is the time interval measured by a clock far away from the mass. For a clock on the surface of the Earth, this expression becomes

The violent deaths of large stars (supernovas) and the collisions of extremely dense stars such as neutron stars with each other can cause spacetime disturbances to happen and to spread out in a wavelike manner. These waves are called gravity waves or gravitational radiation. Within the next five years gravitational radiation should be seen for the first time.

When we talk about black hole, you may think of concepts like "wormhole" or "time tunnel" which give you a heavy flavour of sci-fi. In science fictions or movies, spaceship may enter a black hole, travel through spacetime via "wormhole" and finally emerge from somewhere in the Universe. However, most of the present astronomers will not take the concepts like spacetime travel seriously for reasons that they can neither find any mechanism leading to the formation of "wormhole", nor can they prove the existence of "wormhole" by way of astronomical observations. Moreover, recent studies on the topics demonstrate that even if "wormhole" does exist, it will be very unstable. An extremely small amount of matter passing through it will suffice to make it collapse. Up to now, the concept of spacetime travel by way of black hole can only be regarded as subjects of sci-fi and not a serious science.

Other than some improvable guesses, interests in black hole studies in recent years are on the rise. What new discoveries have pushed astronomers into such relentless researches in everything connected with the black hole? How does the study of black hole help to the understanding of the evolution of the Universe?

Back in the 18th century, scientists like Laplace has already pointed out that highly compact objects might prevent their nearby light from escaping. Soon after Einstein published his General Theory of Relativity in the beginning of 20th century, Karl Schwarzschild found a mathematical solution of the theory to describe the spacetime structure of such object with spherical symmetry. That was the prelude to the study of black holes. Later on, Oppenheimer and others through calculation proved that supermassive stars under gravitational force could really collapse to form black holes. By the 70's, astronomers started to carry out systematic observations to look for evidence of black holes in binary system. The flush of observational evidence from the Hubble Space Telescope launched at the end of the 20th century further convinces us that black holes really exist. To the surprise of astronomers, black holes come in various sizes and origins, and are far more complicated than we can think of. For example, the sizes of black holes can vary immensely from a few to a few billions of solar masses! What is more important, the existence of these different kinds of black holes and the respective astronomical phenomena associated with them always brings far-reaching revelation to our understanding of the evolution of stars, galaxies and at last the whole Universe.

In the early days, man turned their eyes to the binary systems to look for black holes. From the spectral analysis of the orbit, if the invisible companion of a star in a binary system is 3.5 times heavier than that of the Sun, this dark celestial object is most probably a black hole. We take 3.5 solar masses as the benchmark for judging whether something is a black hole because we know we know that theoretically the mass of other compact objects (like neutron star) cannot exceed that maximum threshold. Otherwise those bodies will collapse under their own gravitational force into black holes. However, it is never an easy task to determine the mass of companion stars in binary systems just by way of spectral analysis. Miscalculations did always happen, as it is difficult to accurately measure factors like the luminosity of the visible stars in the pairs and the tilting of the orbits. Accretion disks formed when compact objects suck in matters of companion stars may also betray the existence of black holes. For neutron stars and black holes, accretion disk will emit high-energy X-rays when matters are spiraling in the compact objects, since immense gravitational force can cause substantial heat up of the matters. Searching for X-ray sources in the sky becomes the most important ways to locate neutron stars and black holes among binary system.

In recent years, with the help of X-ray satellites, astronomers make remarkable progress in the search of the Holy Grail. The main difference between neutron stars and black holes lies in the fact that neutron star has a solid surface whereas the black hole does not. Studies show that, in a binary system, huge energy is released when matter of companion star fall onto the surface of neutron star. On the contrary, when matter falls into black hole, it together with the energy generated will disappear behind the event horizon. For that reason, X rays emitted by neutron star binaries are stronger and their spectra exhibit special characteristics. For the total energy generated during the accretion process, the part confiscated by a black hole could be 100 times higher than the radiation that can narrowly escape from the formidable gravitational force. Astronomers are almost sure that the dark companions of many X-ray binaries are black holes and not neutron stars. V404 Cygni is one of the most well-known examples.

For a long time, black hole's event horizon and its association bizarre behaviours are only mathematical game. But observation by the Hubble Space Telescope in recent years provided convincing evidence for the existence of the event horizon. Researchers analysed a huge amount of ultra-violet radiation data coming from a compact object called Cygnus XR-1. They found two events showing the shortening of pulsating cycle and decaying of radiations intensity. The signatures matched theories of what scientists would predict to see. When matter is falling so close to the event horizon, it will be circling the black hole with increasing speed and its light will rapidly dim as it is stretched by gravity to ever-longer wavelengths. However, it is impossible for astronomers to see the even horizon directly due to the current technical limitation. Therefore what scientists discovered so far is only an indirect evidence of the gravitational redshift or similar phenomenon caused by black holes. But those results undoubtedly become an important bridge for linking black hole theory with actual observation.

se si considera la traiettoria della luce in un sistema accelerato, questa non risulta più rettilinea, ma curva :

Gli studi di Einstein lo portarono a ritenere che la luce possa essere deviata anche dalla curvatura dello spazio-tempo, quindi da una massa.

è plausibile ritenere che la curvatura dello spazio-tempo abbia effetti
anche sulla propagazione della luce.

Una verifica sperimentale della teoria era impossibile sulla Terra, che ha massa troppo piccola.

Einstein cercò tale conferma mediante l'osservazione della deviazione dello spazio-tempo causata dalla massa del Sole.

La relativité générale et les trous noirs

Pour bien comprendre comment fonctionne un trou noir, il nous faut introduire la théorie de la Relativité Générale. Einstein a élaboré celle ci car la théorie de Newton posait deux principaux problèmes à ses yeux:

Pour résoudre ces problèmes, Einstein part du principe d' équivalence: on ne peut distinguer un champs de forces d' inertie dans un mouvement uniformément accéléré d' un champs de gravitation. Par exemple, une personne dans un vaisseau en accélération constante de 9,8 m.s-2 ressentira exactement la même chose que si elle était sur terre. Maintenant, faisons une petite expérience de pensée: Soit une personne A immobile et une une personne B dans un vaisseau transparent en très forte accélération. Un photon est émis perpendiculairement au vaisseau depuis l' extérieur et traverse celui ci. Prenons d' abord le point de vue de A, voici ce qu' il observe:

On voit que le trajet de la lumière est courbé par l' accélération. Or d' après le principe d' équivalence, la gravité est identique à l' accélération, on peut donc en déduire que la gravité courbe les rayons lumineux. De plus, la lumière emprunte toujours le chemin le plus court entre deux points (géodésique), l' espace est donc courbé par la gravité. Grâce à des expériences de pensée similaire, nous pouvons montrer que le temps est ralenti à proximité d' une masse. En fait, la relativité générale montre que la gravitation n' est pas une force, mais une propriété géométrique de l' espace-temps. En effet, celui ci est courbé par la masse (et l' énergie, via E=mc^2). Ceci peut être résumé par la formule suivante: "L 'espace-temps agit sur la matière et lui indique comment elle doit se déplacer. Réciproquement, la matière agit sur l' espace-temps et lui indique comment il doit se courber". Dans un espace-temps à quatre dimensions, les planètes ne ressentent aucune force au voisinage du soleil, cependant, nous ne pouvons percevoir que la projection en trois dimension de cet espace-temps, et dans celle ci, les planètes sembles être "attirées" par le soleil par une force. La conséquence de tout ceci est que le temps et l' espace perdent totalement leur caractère absolu, puisque ces deux grandeurs dépendent maintenant de l' endroit où on se trouve (d' ou le nom de relativité).

Bien sur, il est impossible de s' imaginer un espace-temps courbé à quatre dimensions, mais on peut par exemple le représenter en deux dimensions de la manière suivante:

En l' absence de masse (schémas a), l' espace temps est plat (espace-temps de Minkowski), et la trajectoire de la lumière est rectiligne. Lorsqu' une masse est présente (schémas b), l' espace se courbe, et il en va de même pour la trajectoire des rayons lumineux.

Pour élaborer sa théorie quantitativement, Einstein a dû utiliser des outils mathématiques plus compliqués qu' auparavant. En effet, lorsque l' on parle de d' espace courbe, la géométrie Euclidienne ne s' applique plus et il faut utiliser une nouvelle géométrie développée par Riemann au XIXème siècle. Dans celle ci, la courbure peut être négative (comme la surface d' une selle de cheval), dans ce cas la, la somme des angle d' un triangle est inférieure à 180°. Elle peut être aussi positive (surface d' une sphère), la somme valant cette fois plus que 180°. Nous ne pouvons malheureusement pas approfondir la partie mathématique de la relativité générale car ceci demande demande un niveau bien trop élevé, cependant, des liens on été ajoutés dans la section bibliographie.

II] Application aux trous noirs:
Nous avons vus que les trous noirs pouvaient être abordés grâce à la mécanique classique dans la première partie. Après cette première approche de la relativité, nous allons voir à quoi cela correspond avec cette nouvelle théorie. Considérons une étoile massive en effondrement gravitationnel représentée par le schémas suivant:

Avant que l' étoile ne commence à s' effondrer, les rayons ne sont que très peu courbés (1), et ils s' échappent en ligne droite. Au fur et a mesure que l' étoile s' effondre, les rayons émis ont de plus en plus de mal à s' échapper de l' étoile (2). Lorsque le rayon de l' étoile atteint Rs, si un photon est émis a ce moment, il restera à tout jamais au niveau de l' horizon du trou noir (3). Maintenant, si photon est émis après cette limite, il tombera irrémédiablement vers la singularité (4).

Si l' on veut représenter l' espace-temps aux alentours d' un trou noir, cela donne ceci:

Les cônes rouge représentent les trajets possibles pour un photon venant du passé, les cônes vert ceux du futur. On peut remarquer que plus on s' approche de l' horizon, plus les cônes sont déformés vers l' intérieur du trou noir. Cela signifie que la seul trajectoire possible pour un photon (et pour les autres particules) est de plonger vers la singularité.

Lorsqu' une masse se déplace, elle produit des sortes de rides dans l' espace temps, celles ci sont appelées ondes gravitationnelles. On peut faire l' analogie entre ces ondes dû a un mouvement de matière et un champs magnétique, qui lui est dû a un déplacement de charges. Pour produire ces ondes, la masse doit perdre de l' énergie cinétique (principe de la conservation d' énergie). Cela signifie que si deux trous noirs tournent l' un autour de l' autre, ils vont spiraler pour enfin fusionner entre eux. Pour produire ces ondes, il faut des quantités phénoménale d' énergie, c' est pourquoi on ne peut les rencontrer que lors d'évenements violents comme les supernovae et le mouvement de trou noir, lorsqu' il est extrêmement rapide. La terre crée bien des ondes gravitationnelles, mais en tellement faible quantité, qu 'elle ne se rapproche du soleil que de moins d' un atome par ans.

Il faut ajouter que lorsqu' un trou noir est en rotation (trou noir de Kerr), il entraîne avec lui l' espace-temps. Ceci a pour effet de réduire le rayon de l' horizon (en rouge), comme le montre l' animation suivante:

De plus, les rayons rayons lumineux sont déviés d' une manière différente suivant la rotation, en effet, celle les entraîne avec elle:

Forte rotation: on peut observer que cette rotation va jusqu'a faire "revenir en arrière" les photon: leur sens de rotation est inversé lorsqu 'ils s' approchent de l' horizon. Nous verrons en détail plus tard pourquoi les rayons sont de plus en plus décalés vert le violet.

General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. The conceptual core of general relativity, from which its other consequences largely follow, is the Principle of Equivalence, which describes gravitation and acceleration as different perspectives of the same thing, and which was originally stated by Einstein in 1907 as:

We shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame.

In other words, he postulated that no experiment can locally distinguish between a uniform gravitational field and a uniform acceleration.

This principle explains the experimental observation that inertial and gravitational mass are equivalent. Moreover, the principle implies that some frames of reference must obey a non-Euclidean geometry: that spacetime is curved (by matter and energy), and gravity can be seen purely as a result of this geometry. This then yields many predictions such as gravitational redshifts and light bent around stars, black holes, time slowed by gravitational fields, and slightly modified laws of gravitation even in weak gravitational fields. However, it should be noted that the equivalence principle does not uniquely determine the field equations of curved spacetime, and there is a parameter known as the cosmological constant which can be adjusted.

The modifications to Isaac Newton's law of universal gravitation produced the first great theoretical success of general relativity: the correct prediction of the precession of the perihelion of Mercury's orbit. Many other quantitative predictions of general relativity have since been confirmed by astronomical observations. However because of the difficulty in making these observations, theories which are similar but not identical to general relativity, such as the Brans-Dicke theory and the Rosen bi-metric theory cannot be ruled out completely, and current experimental tests can be viewed at reducing the deviation from GR which is allowable. There are no known experimental results that suggest that a theory of gravity radically different from general relativity is necessary. (For example, the Allais effect was initially speculated to demonstrate "gravitational shielding," but was subsequently explained by conventional phenomena.)

However, there are good theoretical reasons for considering general relativity to be incomplete. General relativity does not include quantum mechanics, and this causes the theory to break down at sufficiently high energies. A continuing unsolved challenge of modern physics is the question of how to correctly combine general relativity with quantum mechanics, thus applying it also to the smallest scales of time and space.

Mathematicians use the term "curved" to refer to any space whose geometry is non-Euclidean. Frequently, this curvature is illustrated by an image something like the following:

This graphic shows spacetime as a higher-dimensional flat space, with the "weight" of a massive object "stretching" the trampoline-like spacetime "fabric", which would result in trajectories around this "dent" being curved due to the "slope" and the pull of gravity in some higher dimension . This image, however, is only suggestive of the reality. It is important to remember that spacetime is curved, not merely space, and that space is three-dimensional, not two-dimensional as shown.

Another approach used to understand spacetime as a curved surface in three-dimensional space is to instead begin by imagining a universe of one-dimensional beings living in one dimension of space and one dimension of time. Each bit of matter is not a point on whatever curved surface you imagine, but a line showing where that point moves as it goes from the past to the future. These lines are called world lines.

While it can be helpful for visualization to imagine a curved surface sitting in space of a higher dimension, that model is not thought to be true in any meaningful sense for the real universe. Curvature can be measured entirely within a surface, and similarly within a higher-dimensional manifold such as space or spacetime. On earth, if you start at the North Pole, walk south for about 10,000 km (to the Equator), turn left by 90 degrees, walk for 10,000 more km, and then do the same again, you will be back where you started. Such a triangle with three right angles is only possible because the surface of the earth is curved. The curvature of spacetime can be evaluated, and indeed given meaning, in essentially the same way.

Relationship to special relativity
The special theory of relativity (1905) modified the equations used in comparing the measurements made by differently moving bodies, in view of the constant value of the speed of light, i.e. its observed invariance in reference frames moving uniformly relative to each other. This had the consequence that physics could no longer treat space and time separately, but only as a single four-dimensional system, "space-time," which was divided into "time-like" and "space-like" directions differently depending on the observer's motion. The general theory added to this that the presence of matter "warped" the local space-time environment, so that apparently "straight" lines through space and time have the properties we think of "curved" lines as having.
On May 29, 1919, observations by Arthur Eddington of shifted star positions during a solar eclipse confirmed the theory.

Foundations
General relativity's mathematical foundations go back to the axioms of Euclidean geometry and the many attempts over the centuries to prove Euclid's fifth postulate, that parallel lines remain always equidistant, culminating with the realisation by Lobachevsky, Bolyai and Gauss that this axiom need not be true. The general mathematics of non-Euclidean geometries was developed by Gauss' student, Riemann, but these were thought to be wholly inapplicable to the real world until Einstein developed his theory of relativity.

Gauss had realised that there is no a priori reason that the geometry of space should be Euclidean. What this means is that if a physicist holds up a stick, and a cartographer stands some distance away and measures its length by a triangulation technique based on Euclidean geometry, then he is not guaranteed to get the same answer as if the physicist brings the stick to him and he measures its length directly. Of course for a stick he could not in practice measure the difference between the two measurements, but there are equivalent measurements which do detect the non-Euclidean geometry of space-time directly; for example the Pound-Rebka experiment (1959) detected the change in wavelength of light from a cobalt source rising 22.5 meters against gravity in a shaft in the Jefferson Physical Laboratory at Harvard, and the rate of atomic clocks in GPS satellites orbiting the Earth has to be corrected for the effect of gravity.

Newton's theory of gravity had assumed that objects did in fact have absolute velocities: that some things really were at rest while others really were in motion. He realized, and made clear, that there was no way these absolutes could be measured. All the measurements one can make provide only velocities relative to one's own velocity (positions relative to one's own position, and so forth), and all the laws of mechanics would appear to operate identically no matter how one was moving. Newton believed, however, that the theory could not be made sense of without presupposing that there are absolute values, even if they cannot be determined. In fact, Newtonian mechanics can be made to work without this assumption: the outcome is rather innocuous, and should not be confused with Einstein's relativity which further requires the constancy of the speed of light.

In the nineteenth century Maxwell formulated a set of equations—Maxwell's field equations—that demonstrated that light should behave as a wave emitted by electromagnetic fields which would travel at a fixed velocity through space. This appeared to provide a way around Newton's relativity: by comparing one's own speed with the speed of light in one's vicinity, one should be able to measure one's absolute speed--or, what is practically the same, one's speed relative to a frame of reference that would be the same for all observers.

The assumption was whatever medium light was travelling through—whatever it was waves of—could be treated as a background against which to make other measurements. This inspired a search to determine the earth's velocity through this cosmic backdrop or "aether"—the "aether drift." The speed of light measured from the surface of the earth should appear to be greater when the earth was moving against the aether, slower when they were moving in the same direction. (Since the earth was hurtling through space and spinning, there should be at least some regularly changing measurements here.) A test made by Michelson and Morley toward the end of the century had the astonishing result that the speed of light appeared to be the same in every direction.

In his 1905 paper "On the Electrodynamics of Moving Bodies", Einstein explained these results in his theory of special relativity.

Outline of the theory
The fundamental idea in relativity is that we cannot talk of the physical quantities of velocity or acceleration without first defining a reference frame, and that a reference frame is defined by choosing particular matter as the basis for its definition. Thus all motion is defined and quantified relative to other matter. In the special theory of relativity it is assumed that reference frames can be extended indefinitely in all directions in space and time. The theory of special relativity concerns itself with inertial (non-accelerating) frames while general relativity deals with all frames of reference. In the general theory it is recognised that we can only define local frames to given accuracy for finite time periods and finite regions of space (similarly we can draw flat maps of regions of the surface of the earth but we cannot extend them to cover the whole surface without distortion). In general relativity Newton's laws are assumed to hold in local reference frames. In particular free particles travel in straight lines in local inertial (Lorentz) frames. When these lines are extended they do not appear straight, and are known as geodesics. Thus Newton's first law is replaced by the law of geodesic motion.

We distinguish inertial reference frames, in which bodies maintain a uniform state of motion unless acted upon by another body, from non-inertial frames in which freely moving bodies have an acceleration deriving from the reference frame itself. In non-inertial frames there is a perceived force which is accounted for by the acceleration of the frame, not by the direct influence of other matter. Thus we feel acceleration when cornering on the roads when we use a car as the physical base of our reference frame. Similarly there are coriolis and centrifugal forces when we define reference frames based on rotating matter (such as the Earth or a child's roundabout). The principle of equivalence in general relativity states that there is no local experiment to distinguish non-rotating free fall in a gravitational field from uniform motion in the absence of a gravitational field. In short there is no gravity in a reference frame in free fall. From this perspective the observed gravity at the surface of the Earth is the force observed in a reference frame defined from matter at the surface which is not free, but is acted on from below by the matter within the Earth, and is analogous to the acceleration felt in a car.

Mathematically, Einstein models space-time by a four-dimensional pseudo-Riemannian manifold, and his field equation states that the manifold's curvature at a point is directly related to the stress energy tensor at that point; the latter tensor being a measure of the density of matter and energy. Curvature tells matter how to move, and matter tells space how to curve.

The field equation is not uniquely proven, and there is room for other models, provided that they do not contradict observation. General relativity is distinguished from other theories of gravity by the simplicity of the coupling between matter and curvature, although we still await the unification of general relativity and quantum mechanics and the replacement of the field equation with a deeper quantum law. Few physicists doubt that such a theory of everything will give general relativity in the appropriate limit, just as general relativity predicts Newton's law of gravity in the non-relativistic limit.
Einstein's field equation contains a parameter called the "cosmological constant" Λ which was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations by Hubble a decade later confirmed that our universe is in fact not static but expanding. So Λ was abandoned, with Einstein calling it the "biggest blunder [I] ever made". However, quite recently, improved astronomical techniques have found that a non-zero value of Λ is needed to explain some observations.

where

R ik

is the Ricci curvature tensor, R is the scalar curvature,

g ik

is the metric tensor,

Λ

is the cosmological constant,

T ik

is the stress-energy tensor describing the non-gravitational matter, energy and forces at any given point in space-time,

π

is pi, c is the speed of light in a vacuum and

G

is the gravitational constant which also occurs in Newton's law of gravity.

The Ricci tensor and scalar curvature are themselves derivable from the

g ik

g ik

describes the metric of the manifold and is a symmetric 4 x 4 tensor, so it has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations that make up the Einstein field equation reduce to 6.

Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side, written as part of the stress-energy tensor, and then interpreted as a form of dark energy whose density is constant in space-time.

The study of the solutions of this equation is one of the activities of a branch of astronomy named cosmology. It leads to the prediction of black holes and to the different models of evolution of the universe.

The vierbein formulation of general relativity
This is an alternative equivalent formulation of general relativity using four reference vector fields, called a vierbein or tetrad. We have e_a, a = 0, 1, 2, 3 such that g(e_a, e_b) = η_ab where

.
See sign convention. One thing to note is that we can perform an independent proper, orthochronous Lorentz transformation at each point (subject to smoothness, of course) and still get a valid tetrad. So, the tetrad formulation of GR is a gauge theory, but with a noncompact gauge group SO(3,1). It is also invariant under diffeomorphisms.

See vierbein and Palatini action for more details. See Einstein-Cartan theory for an extension of general relativity to include torsion. See teleparallelism for another theory which predicts the same results as general relativity but with FLAT spacetime with no curvature.

Relativity is a catch-all phrase for both the theory of special relativity and the theory of general relativity. Albert Einstein is the father of both theories, even though special relativity has it's roots in earlier work.

Contrary to popular belief, the theory of relativity does not say that everything is relative. It does say that the speed of light is constant. Since light is constant, special relativity reasons, things that we once thought were constant, namely length, mass, and time, are not constant.

What do we mean by constant? Think of it this way: in the newtonian model, no velocity was constant. For example, if you see someone riding their bike, you might say that they are going 15 miles per hour in relation to you. However, if you then started jogging along side of the bike, you might say that the bikes velocity is only 5 miles per hour, because you are running 10 miles per hour. Velocities are relative. In relativity theory, this is still for the most part true, with the exception of light. If you are not moving and light is riding a bicycle (for discussion's sake) toward you at the speed of light, you would measure it's speed as the speed of light. However, if you then started jogging along side of the bike, you would still measure it's velocity as the speed of light, even though you are now going 10 miles per hour. In fact, you could be going a million miles per hour in relation to your original position, but you would still measure the speed of light the same.

That's not all relativity tells us. Relativity also tells us that

and that gravity is the net effect of the curvature of space-time as a result of mass.

Special relativity is the theory published by Einstein in 1905. Specifically, special relativity says that light is constant, and as velocity increases length decreases, mass increases, and time slows down.

What is General Relativity?

In this illustration, space-time is like a rubber sheet on which a massive ball is placed. The mass of the ball "warps" space-time.

General Relativity is a "generalized" and enhanced version of special relativity. General relativity describes the same odd behavior at high velocities as special relativity, but adds a twist. General relativity throws in gravity.

Einstein realized that there was no difference in the force of gravity and the force of acceleration. For example, someone in a rocket ship without windows cannot tell whether the ship is a rest on Earth or is accelerating through outer-space. The net effect (the person's feet pushing against the floor of the spaceship) is identical. If gravity has characteristics of motion, then strong gravitational feilds must make matter behave similar to high velocities.

The "general" comes from the way the theory explains the phenomena. General relativity can theoretically explain any scenario in space-time.

What is meant by "the curvature of space-time"? To know that, we must first learn what space-time is.

Just about everything you can see is made out of atoms. Those atoms combine to form molecules, etc. Air is made out of atoms, water is made out of atoms, just about everything around us is made out of atoms. Imagine if we could see down to the atomic level. We could see all the atoms that make up everything. But what is in between the atoms? Nothing?

The answer is space-time. Time is an important part of it, because the theories of relativity tell us that as you move through space you also move through time. Einstein thought of space-time having four dimensions: up-down, east-west, north-south, and a fourth dimension that is time multiplied by i ( i is defined as the square root of -1). Thus, you have all the dimensions of space and one time-like dimension that makes up space-time. How does it curve?

On the right you can see an illustration of this concept. Imagine space-time is the grid, and the blue sphere is something that is massive, like a star. The mass of the star causes space-time to curve. The greater the curve, the greater the attractive force of gravity. Notice that space-time is more curved closer to the object.

In the lower illustration, imagine that you are walking on the top straight line of the grid. You can measure the strength of the force pulling you closer to the ball by measuring the distance from the top straight line you are standing on to the bottom curved line. As you walk toward the ball, the distance between the line and the curve increases, thus the strength of the force increases.

This, of course, is not a perfect illustration, as it is a 3 dimensional representation of a 4 dimensional concept.

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