Raychaudhuri is the author of classical mechanics 4. His most significant contribution is the eponymous raychaudhuri equation, which demonstrates that singularities arise inevitably in general relativity and is a key ingredient in. Energy conditions and the raychaudhuri equation universe. One way to prove this is by using the raychaudhuri equation, discussions. This already has a number of interesting applications, both within the adm formalism and elsewhere. The gup and quantum raychaudhuri equation sciencedirect. Gowdy vcu deriving the raychaudhuri equation april 10, 20 1. A rare penetrant mutation in cfh confers high risk of agerelated macular degeneration. A selfsimilar distribution of a scalar field along with an imperfect fluid in a conformally flat spacetime is considered for the purpose. Expansion, rotation, shear what quantities characterise a. The derivation of the raychaudhuri equations, presented by raychaudhuri in his.
We compute the leading corrections to the raychaudhuri equation for the. The role of the raychaudhuri equation in studying gravitational collapse is discussed. Raychaudhuri equation in the finslerranders spacetime and generalized scalartensor theories. Pdf derivation of the raychaudhuri equation researchgate. These are physical examples of the raychaudhuri equation for a nonnormalized vector, unit timelike vector, and unit spacelike vector. The raychaudhuri equation for a spin fluid matter content is developed.
Cosmic censorship, einstein equation, gravitational focusing, raychaudhuri equation, singularities. The equation is applied to the behavior of an irrotational, unaccelerated fluid. The equation is important as a fundamental lemma for the penrosehawking singularity theorems and for the study of exact solutions in general relativity. Raychaudhuri equation in spacetimes with torsion repositorium. Now we present some new interesting applications of the generalized landauraychaudhuri equation. The raychaudhuri equation around 1955, raychaudhuri was interested mainly in two problems. Derivation raychaudhuri equation physics stack exchange. In general relativity, a test particle in free fall will trace out a geodesic. The raychaudhuri equations d finally, equating the two ways of writing r44 the equation for the evolution of the expansion rate is obtained. The raychaudhuri equation has also found applications in the recent developments in holography principle and quantum computation of black hole entropy. We obtain generalized raychaudhuri equations for spinning test particles corresponding to congruences of particles worldlines, momentum, and spin. It is the fundamental equation of gravitational attraction. The derived relation between the parameters appears to depend on the relative scale of the system black hole, which could be read as a beta function equation for the. Raychaudhuri equation plays a pioneer role in cosmology to describe the gravitational focusing and spacetime singularities.
Particles free fulltext effective null raychaudhuri. Write out the raychaudhuri equation for the geodesics of comoving matter in the flrw universe and show that it is reduced to the second friedman equation. Pdf as a homage to a k raychaudhuri, i derive in a straightforward. Raychaudhuri equation in the finslerranders spacetime and. Indeed, so do the theorems by hawking, penrose, hawking and penrose, geroch, gan. Thanks for contributing an answer to physics stack exchange. In particular, i attempted to derive, in a constructive way, the formula for the mukhanovsasaki scalar field. Raychaudhuri equation this equation has the special characteristic that it is true without reference to the einstein equations. The derivation of the raychaudhuri equation, as presented in the 1955 article. Self similar collapse and the raychaudhuri equation.
As a homage to a k raychaudhuri, i derive in a straightforward way his famous equation and also indicate the problems he was last engaged in. In general relativity, the raychaudhuri equation, or landauraychaudhuri equation, 1 is a fundamental result describing the motion of nearby bits of matter the equation is important as a fundamental lemma for the penrosehawking singularity theorems and for the study of exact solutions in general relativity, but has independent interest, since it offers a simple and general validation of. This course was given in heidelberg in the fall 2007. Raychaudhuri equation in spacetimes with torsion paulo luz1,2. A major difference in the nature of the raychaudhuri equation, when applied to a charged.
Raychaudhuri equation for shear thread starter julian. The raychaudhuri equation enables to examine the whole spacetime structure without specific solutions of einsteins equations, playing a central role for the understanding of the gravitational. Pdf as a homage to a k raychaudhuri, i derive in a straightforward way his famous equation and also indicate the problems he was last engaged in. Sergei winitzkis projects topics in general relativity. Gowdy virginia commonwealth university april 10, 20 r. But avoid asking for help, clarification, or responding to other answers. The standard raychaudhuri equation the raychaudhuri equation describes the proper time evolutionofthevolumescalar andfollowsbyapplying the ricci identity to the fundamental fourvelocity. The raychaudhuri equation is central to the understanding of gravitational attraction in astrophysics and cosmology, and in particular underlies the famous singularity theorems of general relativity theory. Raychaudhuri s, iartchouk o, chin k, tan pl, tai ak, ripke s, gowrisankar s, vemuri s, montgomery k, yu y, reynolds r, zack dj, campochiaro b, campochiaro p, katsanis n, daly mj, seddon jm. The derivation of the raychaudhuri equations, presented by.
The raychaudhuri type equations that we obtain in the paper, lead us to results on the evolution of both the 4d expansion and the 5d expansion in a 5d universe. We do this from an effective point of view, that is through a continuous. Even though the applications of the raychaudhuri equations are mostly con ned to the domain of gr, it is important to note that these equations contain some basic statements about the nature of geodesics in a riemannianpseudoriemannian geometry. In general relativity, the raychaudhuri equation, or landau raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter. Raychaudhuris equation and aspects of relativistic.
This is the generic form of the raychaudhuri equation eq. More than twenty years ago, the first of the two authors of this paper has deduced the generalized landauraychaudhuri equation and demonstrated its numerous applications. A recent generalisation of the raychaudhuri equations for timelike geodesic congruences to families. It is proved that the notable singularity theorems of lorentzian geometry extend to the finslerian domain. In particular a covariant bound on the black hole entropy can be obtained by studying the evolution of light sheets 6 as well as the flow of the renormalization group in the theory space could. However, in this article, we will refer to the full set of equations as raychaudhuri equations. It is an intrinsic property of the volume expansion say v is the timelike unit tangent vector of congruence of timelike geodesic is the acceleration of the flow line and is the tensor which project a vector x of the. Raychaudhuri was born on september 14, 1923 in barisal, in what is now bangladesh. Pdf in general relativity, gravity is universally attractive, a feature embodied.
The development of singularities in the expansion is studied for constant spin densities. That is to say a z decomposes into the sum of a multiple of the identity. Abstract as a homage to a k raychaudhuri, i derive in a straightforward way his famous equation and also indicate the problems he was last engaged in. Raychaudhuri oxford university press calcutta wikipedia citation please see wikipedias template documentation for further citation fields that may be required. Raychaudhuri equation wikipedia republished wiki 2.
Beginning with a summary of the essential features of the original article by raychaudhuri and subsequent work of numerous authors, we move on to a discussion of the equations in the context of alternate nonriemannian spacetimes as well as other theories of gravity, with a special mention on the equations in spacetimes with torsion. Lecture notes on general relativity matthiasblau albert einstein center for fundamental physics. Raychaudhuri equation in spacetimes with torsion core. Kinematic quantities and raychaudhuri equations in a 5d. In general relativity, the raychaudhuri equation, or landauraychaudhuri equation, is a. Spacetime singularities and raychaudhuri equations journal of. A course of lectures book online at best prices in india on. The raychaudhuri equation is sometimes referred to as the landauraychaudhuri equation. Shear, distortion in shape without change in volume, which is trace free and. The reason for this comparison is to connect the deformation parameters. Raychaudhuri equation and singularity theorems in finsler. This is done through the consideration of null congruences emerging from, or converging to, a generic point of spacetime, i. The connection between the raychaudhuri equation and the critical. The raychaudhuri equation for spinning test particles.
Gravity is attractive virginia commonwealth university. Akr was the president of the indian association of general relativity and gravitation from 1978 to 1980 and fellow of the indian academy of sciences since 1982. In particular a covariant bound on the black hole entropy can be obtained by studying the evolution of light sheets 6 as well as the. Landaus contribution appears in his treatise the classical theory of fields 10 and is also discussed in detail in 6,11.
The generalized landauraychaudhuri equation and its. After all, einsteins equation in its usual tensorial form is really a bunch of equations. In general relativity, the raychaudhuri equation, or landauraychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter the equation is important as a fundamental lemma for the penrosehawking singularity theorems and for the study of exact solutions in general relativity, but has independent interest, since it offers a simple and general validation of our. Kodama time, entropy bounds, the raychaudhuri equation. Pdf towards the raychaudhuri equation beyond general relativity. Second, a spacelike version of the raychaudhuri equation is brie.
The general focusing condition is found out and verified against the available exact solutions. We compute and compare the evolution of expansionlike parameters associated with these congruences. To see its implications, we rewrite it in the form 3. It may be worthwhile to point out precisely, the work of landau, in relation to this equation.
Our examples here will involve i rotationfree timelike. Historically speaking, it is the equation for one of the quantitites the expansion, which is termed as the raychaudhuri equation. The raychaudhuri equation and its consequences for chronality are studied in the context of finsler spacetimes. The key equation as regards gravitational attraction is the raychaudhuri equation 6, which shows that. See section iv for derivation of the general form of raychaudhuri equations for three kinematical quantities namely expansion scalar, shear and rotation. Raychaudhuri equations for nulltimelike geodesic congruences 1,2,3. Crucial in the derivation of these expressions is considering the. The raychaudhuri equation is sometimes referred to as the landau raychaudhuri equation. This course contains some of the material of the gr course as well as more modern topics, such as fr gravity, dynamics of inflation, and basics of inflationary perturbation theory.
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