Here, we give a different derivation that avoids the raychaudhuri equation but instead makes use of index form methods. Raychaudhuri equation in the finslerranders spacetime and. This was the precise and profound question which raychaudhuri addressed in mid 50s and obtained his celebrated equation 4. It may be worthwhile to point out precisely, the work of landau, in relation to this equation. Raychaudhuri equation plays a pioneer role in cosmology to describe the gravitational focusing and spacetime singularities. Focusing of timelike geodesic congruences in gravitational.
We shall now consider the derivation of this beautiful equation. The reason for this comparison is to connect the deformation parameters. The relationshipbetween dimensional stability derivatives and dimensionless aerodynamic. Pdf comments on derivation of the raychaudhuri equation. Is the big bang singularity due to the strong symmetry of the friedmann lemaitrerobertsonwalker models due to the strong symmetry assumptions. Raychaudhuris equation and aspects of relativistic. Venturi meter is a device or component which is used for measuring the rate of flow of liquid through pipes. Now we use the roots to solve equation 1 in this case. A recent generalisation of the raychaudhuri equations for timelike geodesic congruences to families of d. Raychaudhuri equation and singularity theorems in finsler. Robert oppenheimer and george volkoff in their 1939 paper, on massive neutron cores. Our philosophy is to avoid excessive use of linear finsler connections.
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. In general relativity, the raychaudhuri equation, or landauraychaudhuri equation, is a. First, we shall deriveall the equations of stellar structure in a general, non spherical case, but very quickly we shall restrict. The raychaudhuri equation for a spin fluid matter content is developed. If the acceleration term is nonpositive and, together with the shear and matter terms, dominates the rotation and. At present the raychaudhuri equations have been discussed and analyzed in a variety of frameworks of general relativity, quantum field theory, string. In general relativity, gravity is universally attractive, a feature embodied by the raychaudhuri equation which requires that the expansion of a congruence of geodesics is always nonincreasing. A new derivation of singularity theorems with weakened. 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. 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. Spacetime singularities and raychaudhuri equations munich. Pdf derivation of the raychaudhuri equation researchgate.
A new derivation of singularity theorems with weakened energy. If you want to learn differential equations, have a look at differential equations for engineers if your interests are matrices and elementary linear algebra, try matrix algebra for engineers if you want to learn vector calculus also known as multivariable calculus, or calculus three, you can sign up for vector calculus for engineers. Kinematic quantities and raychaudhuri equations in a 5d. Equations of stellar structure general equations we shall consider a spherically symmetric, selfgravitating star. The derivation of the raychaudhuri equation, presented in the 1955 article, is somewhat di. It is proved that the notable singularity theorems of lorentzian geometry extend to the finslerian domain. Jun 28, 2011 friends, i am studying the raychaudhuri equation.
Use the einstein equation only to evaluate r abuaub in terms of. Institute of physics, sachivalaya marg, bhubaneswar751005, india. The statistical weight of an ion in the lower ionization state to be used in the equation i. See section iv for derivation of the general form of raychaudhuri equations for three kinematical quantities namely expansion scalar, shear and rotation. Spacetime singularities and raychaudhuri equations journal of. The system of these equations is obtained by analyzing the linearresponses to the conformal perturbations in the system of the raychaudhuri type equations. The raychaudhuri equation and its consequences for chronality are studied in the context of finsler spacetimes.
Raychaudhuri equation in spacetimes with torsion core. Pdf scope of raychaudhuri equation in cosmological. It is an intrinsic property of the volume expansion. The equations in contexts outside of general relativity are still referred to as the einstein field equations. Historically speaking, it is the equation for one of the quantitites the expansion, which is termed as the raychaudhuri equation. The raychaudhuri equation around 1955, raychaudhuri was interested mainly in two problems. We have only one exponential solution, so we need to multiply it by t. The definition of singularity first appeared in the works of hawking and penrose after a decade of derivation of the. A major difference in the nature of the raychaudhuri equation, when applied to a charged. Bibtex entry for this abstract preferred format for this abstract see preferences. In that year, he published three articles, on light quanta, on brownian motion, and on the foundations of the theory of special relativity and, almost as an afterthought, a short note containing a. Although the einstein field equations were initially formulated in the context of a fourdimensional theory, some theorists have explored their consequences in n dimensions.
We recall that in finsler geometry one has both a nonlinear connection on the bundle tm m and many. Raychaudhuri equation this equation has the special characteristic that it is true without reference to the einstein equations. Raychaudhuri was born on september 14, 1923 in barisal, in what is now bangladesh. Raychaudhuri equation with zero point length sciencedirect. But landau also deduced this equation, in independent way. Gravitational focusing, raychaudhuri equation s, ingularities.
As a homage to a k raychaudhuri, i derive in a straightforward way his famous equation and also indicate the. View academics in raychaudhuri equation derivation on academia. 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 derived relation between the parameters appears to depend on the relative scale of the system black hole, which could be. In this work, we obtain the raychaudhuri equations for various types of finsler spaces as the finslerranders fr spacetime and in a generalized geometrical structure of the spacetime manifold w. Indian scientist amal kumar raychaudhuri established raychaudhuri equation in 1955 to describe gravitational focusing properties in cosmology. The equation is applied to the behavior of an irrotational, unaccelerated fluid. 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.
Crucial in the derivation of these expressions is considering the. 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. We describe the generalisation to congruences of solutions of arbitrary secondorder ordinary differential equations on a manifold. 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. In this paper, the equation of state for a degenerate fermi gas of neutrons was used to calculate an upper limit of 0. Their rise to prominence was largely due to their use through the notion. Is the bigbang singularity predicted by the frw model generic or special to homogeneity and isotropy of matter distribution. The form of the equation given here was derived by j. Fifty years hence, the raychaudhuri equations have been discussed and analysed in a variety of contexts. In introducing the raychaudhuri equations and its gen.
The raychaudhuri equation for a geodesic congruence in the presence of a zeropoint length has been investigated. 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 for nulltimelike geodesic congruences 1,2,3. Pdf comment on derivation of the raychaudhuri equation by. Lecture notes on general relativity matthiasblau albert einstein center for fundamental physics institut fu. Derivation of the raychaudhuri equation internet archive. The original derivation for the most general case of d dimensional timelike, extremal, nambugoto surfaces embedded in an n dimensional lorentzian background is due to capovilla and guven 7. 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. An attempt has been taken here to describe gravitational focusing and spacetime singularities in some detail with easier mathematical calculations. Case iii critical damping repeated real roots if b2 4mk then the term under the square root is 0 and the characteristic polynomial has repeated roots. The form of the equation for string world sheets given.
We then apply it to the simplest model of the gravitational collapse, the timereversed friedmann robertsonwalker model, and explicitly analyze the property of these equations. Raychaudhuris equation is fundamental for the analysis of behaviour of geodesic congruences. Raychaudhuri equation wikipedia republished wiki 2. Various authors have proved versions of these results under weakened hypotheses, by considering the riccati inequality obtained from raychaudhuri s equation. Laplace equation in rectangular coordinates uniqueness theorems bibliography table of contents 1 derivation from maxwells equations 2 example. We show how our results improve over existing methods and how they can be applied to hypotheses inspired by quantum energy inequalities. 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.
Even though the applications of the raychaudhuri equations are mostly con. 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. Note that a positive pressure adds to the attraction of energy density. In a recent preprint, grqc0511123, dadhich has given a brief yet beautiful exposition on some of the research works by prof. The venturi meter is invented by an american engineer clemans herchel 18411932 and named him after italian giovanni venturi 18421932. Pdf geodesic deviation, raychaudhuri equation, and tidal. In doing so, we introduce two families of observers. 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. 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. Derivation of the raychaudhuri equation inspire inspire hep. 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 particular, we illustrate the problems associated with the technical derivation of the so called quantum raychaudhuri equation, as well as its invalid physical implications. The development of singularities in the expansion is studied for constant spin densities. The basic equation of general relativity is called einsteins equation. For an emission line, that initial level is the upper level of the transition. Unfortunately, the beautiful geometrical meaningof this equation is a bit hardto nd in most treatments of relativity. It is a covariant form of the time, time component, in a frame adapted to u, of the. Chapter 4 dynamical equations for flight vehicles these notes provide a systematic background of the derivation of the equations of motion fora.
The raychaudhuri equation is sometimes referred to as the landau raychaudhuri equation. Laplace equation in rectangular coordinates 3 uniqueness theorems 4 bibliography yes e j. Here dadhich highlights the fact that the apparently selfevident assumption of occurrence of trapped. Although this topic has 23 been investigated before in ec theory, the major problems that we see involve a welldefined relationship between torsion and spin angular momentum and a general lack of awareness of the difference. Landaus contribution appears in his treatise the classical theory of fields 10 and is also discussed in detail in 6,11. Abstract 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. Energy conditions and the raychaudhuri equation universe in. It must however be mentioned, that in a subsequent paper in 1957 4, raychaudhuri presented further results which bear a similarity with the modern approach to the derivation. 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. Academics in raychaudhuri equation derivation academia. The derivation of the raychaudhuri equations, presented by raychaudhuri in his seminal paper in 1955, was different from which we derived here. Pdf towards the raychaudhuri equation beyond general. All the physical quantities will depend on two independent variables.
Pdf comment on derivation of the raychaudhuri equation. 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. Their rise to prominence was largely due to their use through the notion of geodesic focusing in the proofs of the seminal hawking. See section iv for derivation of the general form of raychaudhuri equations for three kinematical. This is directly related to the smallscale structure of spacetime and possibly captures some quantum gravity effects. This equation is extensively used in general relativity, quantum field theory, string theory and the. Cosmic censorship, einstein equation, gravitational focusing, raychaudhuri equation, singularities. Say v is the timelike unit tangent vector of congruence of timelike geodesic. Energy conditions and the raychaudhuri equation universe.
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