Space-time singularity and Raychaudhuri equation

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Authors

S. Chaudhury
Department of Physics Lady Keane College
chaudhury.sumita120@gmail.com

Abstract

Prof A.K. Raychaudhuri’s deep involvement and observation with the singularity problem in General Relativity paved way to the formulation of famous Raychaudhuri Equation in General Relativity in general and Cosmology in particular. One of his early concerns was to construct a model of a collapsing homogenous dust ball and show that nothing prevented the ball from collapsing down to the centre r = 0 and thereby demystifying the so-called Schwarzschild singularity at r = 2M. Then, he addressed the most pertinent question of his time: is the cosmic singularity predicted by the FRW model an artifact of the homogeneity and isotropy of space or not? In fact, inspired by the famous Gödel solution, he was looking for a rotating non-singular solution without closed time-like lines. In the process, he discovered his celebrated equation which made the singularity analysis free and independent of these restrictions. The observation and consequent formulation ultimately led to the powerful Hawking-Penrose singularity theorems which established in a very general way, the inevitability of the occerence of singularity in Einstein’s gravity under reasonable energy and causality conditions.The present paper is a brief review of the celebrated Raychaudhuri equation.

Keywords Raychaudhuri Equation, singularity, General relativity, Cosmology.

Introduction

The field equations of general relativity, propounded by Einstein, in their most general form, are a set of interlinked non-linear partial differential equations, the solutions of which are practically beyond the range of techniques available in applied mathematics. It is apparent that in order to obtain any cosmological solution of the field equations, some assumptions are to be imposed regarding symmetry in order to make any progress towards a solution. The symmetry considerations assumed by Einstein were homogeneity and isotropy of space-time – known as the cosmological principle- and also the additional assumption that the space-time is static .This allowed him to choose a time coordinate t such that the line element of space-time could be described by

Further, additional imposition of Weyl’s postulate on the above line element, which states that the world lines of galaxies form a 3-bundle of non-intersecting geodesics orthogonal to a series of spacelike hypersurfaces, modified the line element (1) to the well known Robertson-Walker metric given by

where the 3-spaces t = constant are Euclidean for k = 0, closed with positive curvature for k = -1 and open with negative curvature for k = -1. The discrete parameter k characterize the Robertson- Walker metric. The term S(t) is often called the expansion factor.

Though the simplifying postulates of cosmology have reduced the number of unknowns in the metric tensor from 10 to the single function S(t) and the task remains in solving an ordinary differential equation in the independent variable t, the cause of concern to serious relativists including Einstein himself, was the singular nature of the exact solutions of General Relativistic equations. What really worried Einstein was: both Schwarzschild as well as cosmological solutions of general relativity turned out to be singular in nature. For the expansion factor S(t) = 0, the geometry of the universe represented by the Eq. 2 above becomes indeterminate. This exactly what really is the big bang. At the big bang, as S(t) = 0, the densities of both matter-dominated and radiation-dominated universe tend to have infinite values as is evident from the following equations which follow from analysis of the solution of Eq. 2:

Here tis the present instant of time. Thus the overall time dependence of matter as well as radiation energy density are determined by the scale factor a(t). The temperature may be expressed as

from where it is evident that the temperature varies inversely with the scale factor. If the universe began in a big bang, where a = 0, the temperature began at infinity and cooled as the universe expanded.

Relativists and cosmologists would not prefer having such indefinable singularities in their cosmological models. The known equations of cosmology would be incapable of explaining such singular states. It was immediately after Einstein in the year 1915 had deduced the equations of General relativity, that the dilemma of singularity started becoming apparent in exact solutions of cosmological models.

Both Schwarzschild and cosmological solutions were singular in nature. Is a singular solution admissible for representation of a real physical world? Under no circumstances, this is desirable to any serious relativist. Then is there a way out of this singularity?

Schwarzschild solution

Shortly after Einstein published his equations of General Relativity, Karl Schwarzschild solved them to find the spacetime geometry outside a spherical distribution of matter of mass M.

The problem is simplified by making use of symmetry arguments. If the spacetime outside such a spherical distribution is empty, then its geometry should be spherically symmetric about the center of distribution. So, he started with the most general form of the line element that fulfils this requirement of spherical symmetry.

It may be shown that such a line element would be of the form

where \upsilon and \lambda are functions of r and t. If \upsilon = \lambda = 0, we get back the Minkowski line element in spherical polar spatial coordinates. The non-Euclidean effects are therefore contained in the functions \upsilon and \lambda.

If the mass of the object under consideration is M, then using the boundary condition A = 2GM/c2, where \upsilon = e-\lambda = 1- A/r , [ A is a constant] we get the required solution as the line element

This is what is known as Schwarzschild line element and represents the metric of spacetime outside a spherical distribution of matter.

Schwarzschild black hole

Refering to Eq. 6, is it possible to visualize an object that is so compact that its mass M and radius R are related by

A glance at the Schwarzschild solution will show that, for such an object, the spacetime geometry near the surface will be markedly non-Euclidean.

An object whose coordinate radius R satisfies the condition

is called a black hole. As its name implies, such an object is dark because its strong gravity traps light and prevents it from getting away. It may be shown that the gravitational red shift of a black hole is infinite. Since a redshift, Z, implies decrease in the energy of a photon of light by a factor (1+Z)-1, no photon with finite energy can come out of a black hole. The weird state of singularity thus continues to remain in the analysis of an exact solution of Einstein’s equation of General Relativity.

Cosmological solution

In view of equation (2) under the assumption of Weyl’s postulate and symmetry considerations imposed by the cosmological principle, it may be shown that the only non-trivial set of equations obtained using Einstein’s equations are of the form

Considering the universe at present to be dust-dominated, Friedmann models of the universe can be constructed for three special cases of k = 0 (Euclidean section), +1 (Closed sections) and -1( Open section).

Three types of Friedmann models are shown below as variation of cosmic time and Scale factor S(t).

In Figure 1 below, the three types of Friedmann models are shown together in this plot of S(t) versus t. Points P, Q and R indicate where the present epoch occurs on the various curves. As we go from open to closed models, this epoch falls closer to the initial singular point.

Cosmological black hole

Figure 1 shows that all the Friedmann models have the common feature of having S = 0 during a certain epoch (which we have chosen to be t = 0). As we approach S = 0, the Hubble constant which is related to S by the relation H(t) = S./S, increases rapidly becoming infinite at S = 0. This epoch, therefore corresponds to catastrophic activity and is given the name big bang. It was Sir Fred Hoyle, who in the late 1940s gave it this name, largely in a sarcastic vein, as he was critical of the big bang.

From a mathematical point of view, S = 0 describes a spacetime singularity. If we compute the components of Riklm and construct invariants out of these, such as

R, RikRik, RiklmRiklm, …….. ,

These invariants diverge. It is, therefore, meaningless to talk of a spacetime geometry at S = 0.

Thus, it is a matter of immense concern to the physicists as how to get rid of such an impasse that prevents further investigation into the epoch earlier than the state of singularity that has emerged in the process of mathematical analysis. FRW singularity, thus, must be interpreted as the catastrophic event from which the entire universe emerged, where all the known laws of physics and mathematics breakdown in such a way that we cannot know what state actually existed during and before the big bang singularity. The state S = 0 thus, presents insurmountable barrier to the physicists. Tolman and Eddington advanced the argument that this singularity has originated from the twin assumptioins of isotropy and homogeneity in the derivation of cosmological models.

Raychaudhuri Equation -The background scenerio

In the process of analysis of the singularity, certain very pertinent questions struck the mind of Prof Amol Kumar Raychaudhuri, who then, was a lecturer in the department of Physics, Presidency College, Calcutta : that what really is meant by a spacetime singularity? Under what conditions can they arise? Does the appearance of singularity signify an inherent defect in our description of spacetime? Both the Schwarzschild and the cosmological solutions were singular in nature. Was there a way out? Was it correct to believe in a theory which had singular solutions? Were singularities inevitable in General Relativity?

In the year 1955, Raychaudhuri obtained an equation which gave answers to all those questions and proved to be extremely useful in understanding how and why a singularity developes. The paper entitled ‘Relativistic Cosmology-l’ appeared in Physical Review 98,1123, (1955), which contains the derivation of the now-famous Raychaudhuri equation.

Fifty years hence, the Raychaudhuri equations have been discussed and analysed in a variety of contexts. Their rise to prominence was largely due to their use (through the notion of geodesic focusing) in the proofs of the seminal Hawking–Penrose Singularity Theorems of General Relativity. Today, the importance of this set of equations, as well as their applicability in diverse scenarios, is a well-known fact.

Insight into Raychaudhuri Equation

Prior to the formulation of Raychaudhuri equation, because of the appearance of the singularity, further progress in the study of cosmology was apparently considered to have come almost to a halt ! At this juncture, Raychaudhuri equation came to rescue and save Physics from the severe dilemma it was facing.

Prof Raychaudhuri proceeded with the remarkable idea that imposition of cosmological principle into the frame work of cosmological spacetime might gave given rise to the complexeity of singularity in cosmological models and absence of such a constraint in the formulation is likely to relieve Physics from the said deadlock created by the S = 0 singular state.

Prof Raychaudhuri, in his formulation, considered an inhomogenous, stationery space time of infinite curvature having no constraint of cosmological principle asserted on it. The spacetime was such that it was considered to have undergone every possible deformations , like rotation, acceleration, twist and shear. Prof Raychaudhuri’s primary investigation would be to examine now the path of light rays and material particles through such a deformed spacetime – whether they are geodetic in the spacetime. What result emerged out of this study is what goes now by the name of Raychaudhuri equation- supposedly one of the marvels of cosmology today! While Prof Raychaudhuri wanted to achieve a singularity-free cosmological solution by not imposing the constraints of cosmological principle of homogeneity and isotropy, the end result of the analysis came out with a remarkable idea that the congruence of geodesics in such a generally deformed spacetime, would be intersecting- thereby resulting in ultimate conclusion that singularity is inevitable in the cosmological solution. Hence, the occurrence of big bang and presence of black hole – both of which signify singularity, are true and inevitable in the universe we live in! Big bang is as much a reality as is black hole!

Almost a decade later, Stephen Hawking, Roger Penrose and Robert Geroch, used this idea laid down by Prof Raychaudhuri in deriving some path-breaking singularity theorems, the gist of which may considered  as follows: based on the ideas of general relativity, it may be proved that sometimes in the past or future, it is inevitable to have singularities in the universe we live in. These theorems were proved by them more rigorously using the advanced branch of differential topology. While, Prof Raychaudhuri’s analysis gave solution pertaining to local distribution of matter, the theorems given by Hawking and others was true for the whole universe considered as an entity. It was at that time, along with the proofs of the singularity theorems, the term Raychaudhuri equations came into existence in the physics literature and henceforth came to be recognized as one of the milestones in the study of relativistic cosmology. In their famous book The Large Scale Structure of Space and Time (1973), Stephen Hawking and George Ellis have referred to Raychaudhuri equation as many as thirteen times.

Some mathematical groundwork required for the derivation of Raychaudhuri equation

Here, we shall follow the treatment by A.K.Raychaudhuri,  S. Banerji and A. Banerjee *.

We shall do some ground work before deriving the Raychaudhuri equation and its implication in cosmology.

The Groundwork

Let us consider the elastic deformation of a material medium in classical physics. Supposing, u(x,y,z) is the displacement of a point (x,y,z,). Then the change in volume of a region bounded by S is given by

where the integral is over the entire surface S. Applying Gauss’ theorem’

Where V is the volume bounded by the surface S. Since the LHS of E\(qn. (11) represents the change in volume of a certain region, \Delta.u must be of the form \frac{d(dV)}{dV} which is just the dilation denoted by \Theta. Therefore,

Now, writing \frac{\partial u_{i}}{\partial x^{k}}

in the expanded form

Now, the RHS of Eq. 13 consists of three parts which have the following meanings:

a. The first part represents a change of shape without a change in volume. It is trace-free, i.e., if it is denoted by Aik, then Ai= 0. This part is called shear.

b. \frac{\partial u_{i}}{\partial x^{k}} -  \frac{\partial u_{k}}{\partial x^{i}} is a component of \bigtriangledown × which represents a rotation. So, the second part of Eq. 13 is a rotation.

c. The third part represents dilation, i.e., volume expansion.

The above ideas may be carried over to general relativity in the following manner:

a. The displacement vector u is replaced by a unit velocity vector \vartheta ^{\mu }

b. The volume expansion is a scalar and is taken over as such.

c. Classically, the shear and rotation are Cartesian tensors. They now have the general tensorial form, but these tensors should be in the 3-space orthogonal to the velocity vector  \vartheta ^{\mu }. This may be clarified as follows:

A vector or a tensor can be projected into the 3-space orthogonal \vartheta ^{\mu } to  by contracting with the projection tensor g_{\mu \upsilon } - \partial _{\mu }\partial _{\upsilon } We may check it by projecting \vartheta ^{\mu } into the 3-space orthogonal to itself. Thus,

Now, if we project \vartheta _{\mu };_{\upsilon } into the 3-space, then we have

where \dot{\vartheta }\mu \equiv \vartheta _{\mu };_{\upsilon }\vartheta ^{\upsilon } is called acceleration , for, in the local Lorentz frame in which the medium is at rest,

which represents acceleration.

We have thus completed construction of the mathematical ground work required for the derivation of Raychaudhuri equation. We shall derive the expressions for shear and vorticity by replacing \frac{\partial ui}{\partial xk} occurring in the classical expressions for shear and vorticity in Eq. 13.

Raychaudhuri equation and its implication

The Riemann curvature tensor may be defined by the commutator of the second covariant derivative of any vector \vartheta ^{\mu } :

Contracting Eq. 15 with \vartheta ^{\alpha } and also over the indices, \beta and \mu, we obtain

Now, we take \vartheta \mu as a unit vector and present some definitions as follows:

(a) \vartheta \mu ;\mu is a generalization of Eq. 12. It is, therefore, called expansion scalar and denoted by \Theta, i.e.,

(b) The equation of geodesicity of \vartheta ^{\mu } is given by

Thus, if \vartheta ^{\alpha } vanishes, it represents the geodesicity of \vartheta \alpha . But, if \vartheta ^{\alpha }\neq 0, then it represents the departure of \vartheta ^{\mu } from geodesicity and is therefore called the acceleration.

We will now show that

Now,

Therefore

(c) We now construct the vorticity ( rotation ) tensor in the following form [refer to Eq. 13]

One can see that \omega _{\alpha \beta } is antisymmetric. Now, we show that \omega _{\alpha \beta } is orthogonal to the vector \vartheta ^{\alpha }. Thus,

From Eqn (18) and since \vartheta ^{\alpha } is a unit vector.

Hence,

Similarly,

(d) We now proceed to construct the shear in the following form [ refer to Eq.13]:

Evidently, \sigma _{\alpha \beta } is symmetric. Further, following Eqn(20), it may be shown that,

Now, using expressions of \omega _{\alpha \beta } and \sigma _{\alpha \beta } from Eqs. 19 and 22, we find

Now we look back to Eq. 16. The L.H.S. of the equation can be written in the following manner

Using Eqs. 15 and 25,

We shall now evaluate the first and second terms on the L.H.S. of Eq. 26 making use of Eq. 24 and the properties of \dot{\vartheta ^{\mu }}\omega _{\alpha \beta } and \sigma _{\alpha \beta } .

Let us take the first term first :

Now, the second term:

Now, using Eqs. 27 and 28 in Eq. 25, we obtain the well-known Raychaudhuri equation

R_{\gamma \alpha }\vartheta ^{\gamma }\vartheta ^{\alpha } here represents the gravity effect. The magnitude of R_{\gamma \alpha }\vartheta ^{\gamma }\vartheta ^{\alpha } can be evaluated from Einstein’s field equation

where

Since \vartheta ^{\alpha } is a unit velocity vector, Eqs. 30 and 31 give for the magnitude of  R_{\gamma \alpha }\vartheta ^{\gamma }\vartheta ^{\alpha },

Using Eq. 32 in Eq. 29, we obtain the Raychaudhuri equation in an alternative form

For convenience, we rewrite Eq. 33 in the form ( with \Theta _{,\alpha }\vartheta ^{\alpha }= \frac{d\Theta }{ds} )

where s parameterizes a geodesic.

Some implications of Raychaudhuri equation will be discussed briefly now :

(a). The cosmological field equations with FRW metric

are given by

Adding Eqs. A and B ( with  \Lambda = 0 ) we obtain

Now, FRW metric [Eq. A] give for \Theta

Now, defining, for a general metric, a new scale function R such that \Theta =3\tfrac{R}{Rc} , we have,

Taking note of Eq. 35 and using Eq. 37 in Eq. 34, we can write the raychaudhuri equation in the form

Now, let us have a look at Eq. 38. The last term represents the gravity effect and causes deceleration of the expansion. This is abetted by \sigma ^{2} in bringing about a collapse. The vorticity (rotation) term , on the other hand, has the effect of repulsion and may cause a bounce instead of a collapse. However, the study of Gödel’s rotating universe shows that vorticity may involve some awkward situation. Lastly, the acceleration term may be positive or negative. We may, therefore, arrive at the following singularity theorem from Eq. 38 under certain conditions:

In the absence of \omega ^{2} and , spatial volumes inevitably collapse to singularity.

(b) We may arrive at the same conclusion in an alternative way following R.M.Wald. Assuming \dot{\vartheta }^{\mu }_{;\mu } and \omega ^{2} to be absent, we can write Eq. 34
as an inequality

On integration,

Where \Theta _{0} is the value of \Theta at s = 0. Supposing \Theta _{0} to be negative, which means that the congruence (i.e., the bundle) of geodesics is initially converging. Then, at s\leq \frac{3}{\left | \grave{e_{0}} \right |},\Theta ^{-1} must pass through zero, i.e., through \Theta \rightarrow -\infty, which implies obviously a singularity.

Thus, according to the Raychaudhuri equation, singularity appears to be inevitable in classical relativity developed by Einstein in 1916.

This literally means that the universe we live in, had its origin in a state which cannot, till date, be apprehended by the known laws of physics we so far have at our disposal. The origin of the universe has, thus, remained mathematically undefinable till today. This is exactly what Raychaudhuri equation describes.

Acknowledgement

The author expresses her profound gratitude to her research guide Prof K. D. Krori, UGC Emeritus Professor, for explaining more than twenty years back, the essence and physical significance of RayChaudhuri equation and encouraging her recently in writing the present paper.

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