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. Crucial in the derivation of these expressions is considering the. Pdf as a homage to a k raychaudhuri, i derive in a straightforward. Raychaudhuri was born on september 14, 1923 in barisal, in what is now bangladesh. Gowdy virginia commonwealth university april 10, 20 r. Raychaudhuri is the author of classical mechanics 4. Gravity is attractive virginia commonwealth university. Indeed, so do the theorems by hawking, penrose, hawking and penrose, geroch, gan. The key equation as regards gravitational attraction is the raychaudhuri equation 6, which shows that. The derivation of the raychaudhuri equations, presented by. The raychaudhuri equation for a spin fluid matter content is developed. 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. 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. Pdf towards the raychaudhuri equation beyond general relativity.
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. It may be worthwhile to point out precisely, the work of landau, in relation to this equation. The equation is important as a fundamental lemma for the penrosehawking singularity theorems and for the study of exact solutions in general relativity. 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.
Thanks for contributing an answer to physics stack exchange. The derivation of the raychaudhuri equations, presented by raychaudhuri in his. Energy conditions and the raychaudhuri equation universe. This is done through the consideration of null congruences emerging from, or converging to, a generic point of spacetime, i. 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. Cosmic censorship, einstein equation, gravitational focusing, raychaudhuri equation, singularities. The development of singularities in the expansion is studied for constant spin densities. 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. Kodama time, entropy bounds, the raychaudhuri equation. Expansion, rotation, shear what quantities characterise a. The gup and quantum raychaudhuri equation sciencedirect.
Raychaudhuri equation plays a pioneer role in cosmology to describe the gravitational focusing and spacetime singularities. Landaus contribution appears in his treatise the classical theory of fields 10 and is also discussed in detail in 6,11. A recent generalisation of the raychaudhuri equations for timelike geodesic congruences to families. The raychaudhuri equation and its consequences for chronality are studied in the context of finsler spacetimes. A rare penetrant mutation in cfh confers high risk of agerelated macular degeneration. Raychaudhuri equation for shear thread starter julian. 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. The role of the raychaudhuri equation in studying gravitational collapse is discussed. Raychaudhuri equation and singularity theorems in finsler. The raychaudhuri equation has also found applications in the recent developments in holography principle and quantum computation of black hole entropy. The equation is applied to the behavior of an irrotational, unaccelerated fluid.
This is the generic form of the raychaudhuri equation eq. We compute the leading corrections to the raychaudhuri equation for the. But avoid asking for help, clarification, or responding to other answers. It is proved that the notable singularity theorems of lorentzian geometry extend to the finslerian domain. We compute and compare the evolution of expansionlike parameters associated with these congruences. 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. In particular, i attempted to derive, in a constructive way, the formula for the mukhanovsasaki scalar field. It is the fundamental equation of gravitational attraction. 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.
Our examples here will involve i rotationfree timelike. The generalized landauraychaudhuri equation and its. 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. Raychaudhuri oxford university press calcutta wikipedia citation please see wikipedias template documentation for further citation fields that may be required. Historically speaking, it is the equation for one of the quantitites the expansion, which is termed as the raychaudhuri equation. Self similar collapse and the raychaudhuri equation.
The reason for this comparison is to connect the deformation parameters. Particles free fulltext effective null raychaudhuri. 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. We obtain generalized raychaudhuri equations for spinning test particles corresponding to congruences of particles worldlines, momentum, and spin. Second, a spacelike version of the raychaudhuri equation is brie. Gowdy vcu deriving the raychaudhuri equation april 10, 20 1. Sergei winitzkis projects topics in general relativity. A major difference in the nature of the raychaudhuri equation, when applied to a charged. A selfsimilar distribution of a scalar field along with an imperfect fluid in a conformally flat spacetime is considered for the purpose. 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. The connection between the raychaudhuri equation and the critical. The raychaudhuri equations d finally, equating the two ways of writing r44 the equation for the evolution of the expansion rate is obtained. Spacetime singularities and raychaudhuri equations journal of. Kinematic quantities and raychaudhuri equations in a 5d.
Derivation raychaudhuri equation physics stack exchange. Pdf derivation of the raychaudhuri equation researchgate. Raychaudhuri equation this equation has the special characteristic that it is true without reference to the einstein equations. Pdf in general relativity, gravity is universally attractive, a feature embodied. Raychaudhuri equations for nulltimelike geodesic congruences 1,2,3. One way to prove this is by using the raychaudhuri equation, discussions. A course of lectures book online at best prices in india on.
Raychaudhuri equation in the finslerranders spacetime and. Raychaudhuri equation in spacetimes with torsion repositorium. That is to say a z decomposes into the sum of a multiple of the identity. To see its implications, we rewrite it in the form 3. See section iv for derivation of the general form of raychaudhuri equations for three kinematical quantities namely expansion scalar, shear and rotation. The standard raychaudhuri equation the raychaudhuri equation describes the proper time evolutionofthevolumescalar andfollowsbyapplying the ricci identity to the fundamental fourvelocity. Shear, distortion in shape without change in volume, which is trace free and. In general relativity, a test particle in free fall will trace out a geodesic. This course was given in heidelberg in the fall 2007.
Raychaudhuri equation in spacetimes with torsion paulo luz1,2. 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 equation wikipedia republished wiki 2. This already has a number of interesting applications, both within the adm formalism and elsewhere. Spacetime singularities and raychaudhuri equations munich. In general relativity, the raychaudhuri equation, or landau raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter. Raychaudhuri equation in spacetimes with torsion core. 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. These are physical examples of the raychaudhuri equation for a nonnormalized vector, unit timelike vector, and unit spacelike vector. The general focusing condition is found out and verified against the available exact solutions. 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. Now we present some new interesting applications of the generalized landauraychaudhuri equation. The raychaudhuri equation is sometimes referred to as the landauraychaudhuri equation. Raychaudhuri equation in the finslerranders spacetime and generalized scalartensor theories.
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. The derivation of the raychaudhuri equation, as presented in the 1955 article. Lecture notes on general relativity matthiasblau albert einstein center for fundamental physics. The raychaudhuri equation around 1955, raychaudhuri was interested mainly in two problems. The raychaudhuri equation is sometimes referred to as the landau raychaudhuri equation. 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. 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 landauraychaudhuri equation, is a. We do this from an effective point of view, that is through a continuous. His most significant contribution is the eponymous raychaudhuri equation, which demonstrates that singularities arise inevitably in general relativity and is a key ingredient in. However, in this article, we will refer to the full set of equations as raychaudhuri equations.
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