Trynka g, sandor c, han b, xu h, strange be, liu xs, raychaudhuri s. Kinematic quantities and raychaudhuri equations in a 5d. Raychaudhuri equation in the finslerranders spacetime and generalized scalartensor theories. We study the evolution of timelike geodesics for two dimensional black hole spacetimes arising in string theory and general theory of relativity by solving the raychaudhuri equation for expansion. The definition of singularity first appeared in the works of hawking and penrose after a decade of derivation of the. Raychaudhuri equation back in the past, it must drop below this straight line and reach arbitrarily small positive values of at a time less than 1h0 ago unless some other spacetime singularity intervenes before 0, which can happen only if the shear diverges. 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. Jun 28, 2011 friends, i am studying the raychaudhuri equation.
Pdf derivation of the raychaudhuri equation researchgate. Generalized raychaudhuri equations for strings and membranes sayan kar yz institute of physics, sachivalaya marg, bhubaneswar751005, india abstract a recent generalisation of the raychaudhuri equations for timelike geodesic. Userguided cytometric data analysis for largescale immunoprofiling studies. It is a covariant form of the time, time component, in a frame adapted to u, of the. Energy conditions and the raychaudhuri equation universe. 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. If the acceleration term is nonpositive and, together with the shear and matter terms, dominates the rotation and. Singularity theorems conjugate points due to raychaudhuri equation. Shortest distance along great circle longest distance along great circle conjugate points a b maximal geodesics predicted by global arguments, on the other hand. Derivation of the raychaudhuri equation internet archive. If the acceleration term is nonpositive and, together with the shear and matter terms, dominates the rotationand. Kinematic quantities and raychaudhuri equations in a 5d universe. Munich personal repec archive spacetime singularities and raychaudhuri equations mohajan, haradhan journal of natural sciences may 20 online at. Raychaudhuri equation and singularity theorems in finsler spacetimes e minguzzirecent citations on finsler spacetimes with a timelike killing vector field erasmo caponio and giuseppe stancaronethis content was downloaded from ip address 157.
One way to prove this is by using the raychaudhuri equation, discussions of which can be found in the textbooks by wald and by ciufolini and wheeler cited in the bibliography. See section iv for derivation of the general form of raychaudhuri equations for three kinematical quantities namely expansion scalar, shear and rotation. In particular, i attempted to derive, in a constructive way, the formula for the mukhanovsasaki scalar field. His most significant contribution is the eponymous raychaudhuri equation, which demonstrates that singularities arise inevitably in general relativity and is a key ingredient in. A generalised raychaudhuri equation for secondorder. This paper investigates the issue of the final fate of a gravitationally. In general relativity, the raychaudhuri equation, or landau raychaudhuri 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. Hence, we explicitly wrote down the friedmann and raychaudhuri equations in the framework of matter creation. Torsion and the second fundamental form for distributions in.
This course was given in heidelberg in the fall 2007. Chromatin marks identify critical celltypes for finemapping complex trait variants. Is the big bang singularity due to the strong symmetry of the friedmann lemaitrerobertsonwalker models due to the strong symmetry assumptions. The derived relation between the parameters appears to depend on the relative scale of the system black hole, which could be. Cambridge core economic history the cambridge economic history of india edited by tapan raychaudhuri. The gup and quantum raychaudhuri equation sciencedirect. The raychaudhuri equation for spinning test particles. Towards the raychaudhuri equation beyond general relativity daniel j burger. 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. Historically speaking, it is the equation for one of the quantitites the expansion, which is termed as the raychaudhuri equation. In the monetary system of the delhi sultanate a firm equation between gold and silver appears to have been established at 1.
Raychaudhuri equations for nulltimelike geodesic congruences 1,2,3. Particular attention is given to those stresses that resist contraction. Massachusetts institute of technology physics department physics 8. On the raychaudhuri equation george f r ellis mathematics department, university of cape town, rondebosch, cape town, south africa email. 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.
The raychaudhuri equation for spinning test particles the raychaudhuri equation for spinning test particles mohseni, morteza 20150217 00. This was the precise and profound question which raychaudhuri addressed in mid 50s and obtained his celebrated equation 4. 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. 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. We use the raychaudhuri equation to probe certain aspects related to the gravitational collapse of a charged medium. If the inline pdf is not rendering correctly, you can download the pdf file here. Raychaudhuri equation this equation has the special characteristic that it is true without reference to the einstein equations. Testing some fr,t gravity models from energy conditions. But landau also deduced this equation, in independent way. Say v is the timelike unit tangent vector of congruence of timelike geodesic. Raychaudhuris equation is fundamental for the analysis of behaviour of geodesic congruences. Akr was very delighted about this recognition, which helped his standing in india a lot. Energy conditions and the raychaudhuri equation universe in. A k raychaudhuri and his equation max planck society.
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. Unfortunately, the beautiful geometrical meaningof this equation is a bit hardto nd in most treatments of relativity. As winlabs director, he is responsible for an internationally recognized industryuniversity research center specializing in wireless technology. Amal kumar raychaudhuri established it in 1955 to describe gravitational focusing properties in cosmology. As a homage to a k raychaudhuri, i derive in a straightforward way his famous equation and also indicate the problems he was last. Raychaudhuri equations play important roles to describe the gravitational focusing and spacetime singularities. Spacetime singularities and raychaudhuri equations journal of. 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. Kathryns summer project with the raychaudhuri lab juneaugust 2019 r 1 0 0 0 updated jul 11, 2019. We shall now consider the derivation of this beautiful equation. Expansion, rotation, shear what quantities characterise a. 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. His most significant contribution is the eponymous raychaudhuri equation, which demonstrates that singularities arise.
Rqwkhlqvohupdqlirog finslerkawaguchi lagrangian formulation. 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 equation in the finslerranders spacetime and. The qre is derived from considering the pilot wave theory of quantum mechanics to assign to each particle moving in the congruence a semi. In the monetary system of the delhi sultanate a firm equation between gold and silver appears to have been.
How an effective cosmological constant may affect a. When the star is heavier than a few solar masses, it could undergo an endless gravitational collapse without achieving any equilibrium state. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a pdf plugin installed and enabled in your browser. Is the bigbang singularity predicted by the frw model generic or special to homogeneity and isotropy of matter distribution. Einstein equations as equations of state for a non equi librium spacetime thermodynamics 79. The most recent of which is the quantum raychaudhuri equation qre 3. 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. Sergei winitzkis projects topics in general relativity. The remonetarization of the economy might have occurred by the middle of the thirteenth century, for at this period the suhrawardi shaykhs of multan left assets of lakhs of tankas. This equation is extensively used in general relativity, quantum field theory, string theory and the theory of relativistic membranes. Pdf towards the raychaudhuri equation beyond general. The cambridge economic history of india edited by tapan.
It is an intrinsic property of the volume expansion. The geodesic spray, the vertical projection, and raychaudhuris equation. Raychaudhuri equation in spacetimes with torsion core. Scope of raychaudhuri equation in cosmological gravitational. The development of singularities in the expansion is studied for constant spin densities. However, in this article, we will refer to the full set of equations as raychaudhuri equations. The field equations form an autonomous system of differential equations, where the friedmann equation constrains the dynamics of the universe and the raychaudhuri equation essentially describes its evolution. Google drive or other file sharing services please confirm that you accept the.
The derivation of the raychaudhuri equation, as presented in the 1955 article, is some what different from the way it is arrived at in standard. 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. In general relativity, the raychaudhuri equation, or landau raychaudhuri 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. 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. Raychaudhuri is the author of classical mechanics 4. Jul 31, 2007 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. The raychaudhuri equation around 1955, raychaudhuri was interested mainly in two problems.
The aim is to identify the stresses the maxwell field exerts on the fluid and discuss their potential implications. The reason for this comparison is to connect the deformation parameters. 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. 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. The raychaudhuri equation for a spin fluid matter content is developed. A recent generalisation of the raychaudhuri equations for timelike geodesic congruences to families of d. The equation is applied to the behavior of an irrotational, unaccelerated fluid.
These are physical examples of the raychaudhuri equation. Crucial in the derivation of these expressions is considering the. We describe the generalisation to congruences of solutions of arbitrary secondorder ordinary differential equations on a manifold. Indian scientist amal kumar raychaudhuri established raychaudhuri equation in 1955 to describe gravitational focusing properties in cosmology.
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