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Gravitational collapse in f(R) gravity

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Gravitational Perfect Fluid Collapse in f(R) Gravity PBIP Thesis by Soumya Chakrabarti Under the Supervision of Professor Narayan Banerjee Certificate This is to certify that the thesis entitled Gravitational Perfect Fluid Collapse in f(R) Gravity being submitted to the Indian Institute of Science Education and Research-Kolkata in partial fulfillment of the requirements for the award of the Integrated PhD degree, embodies the research work done by Soumya Chakrabarti under my supervision at IISER- Kolkata. The work presented here is original and has not been submitted so far, in part or full, for any degree or diploma of any other university/institute. Preface The work embodied in this thesis has been carried out in the Department of Physical Sciences, Indian Institute of Science Education and Research, Kolkata, during the period August 2011 to April 2012. In this work we have discussed aspects of gravitational collapse in metric f(R) gravity. Gravitational collapse is introduced in a very brief review. The recent relevance of f (R) gravity is also discussed. A collapse model in f(R) gravity with a constant curvature, which is already there in the literature, is discussed. The present work deals with a collapse model where the assumption of a constant curvature being relaxed. Date: May 2012 IISER-Kolkata Acknowledgements I bid my respect and thanks to my supervisor Professor Narayan Banerjee for helping me with his great vision and innovative ideas regarding the work. He s been my friend-philosopher-guide during my tough days and I m lucky to have a chance to work under him. I m very much thankful to my mother. Special thanks to Gopal-Da who helped me out with MATHEMATICA & Debjyoti, my old class-mate who helped me with some books. Ankan, Debmalya & Anandarup, my friends at IISER- Kolkata, talking to them, discussions, helped me quite a lot. It s always good for someone to have friends like them. A special thanks to Torsa; who was always there to motivate me. Contents Introduction: Concept of Gravitational Collapse. Chapter-I: Oppenheimer-Snyder Collapse. Chapter-II: f(R) Gravity. Chapter-III: Field equations of metric f(R) gravity. Chapter-IV: Perfect fluid collapse in metric f(R) gravity. Chapter-V: Assumption of constant scalar curvature. Chapter-VI: Apparent Horizons. Chapter-VII: An attempt to study collapse in exponential f(R) gravity Chapter-VIII: Indications. Introduction Gravitational collapse is the inward fall of the constituents of a body due to the influence of its own gravity. In any stable body, this inward gravitational force is counterbalanced by the internal pressure of the body, in the opposite direction to the f force orce of gravity. If the inwards pointing gravitational force, is stronger than the total combination of the outward pointing forces, the equilibrium becomes unbalanced and a collapse occurs until the internal pressure increases above that of the gravitational force and an equilibrium is once again attained. If the pressure is unable to halt the gravitational collapse, the whole matter is likely to crush to a central singularity. Because gravity is comparatively weak compared to other fundamental forces, gravitational collapse is usually associated with very massive bodies or collections of bodies, such as stars (including luding collapsed stars such as supernovae, neutron stars and black holes) and massive collections of stars such as globular clusters and galaxies. (A stellar collapse in process; courtesy: : burro.cwru.edu) In addition to the formation of singularity, gravitational collapse actually is also at the heart of structure formation in the universe. An initial smooth distribution of matter will eventually collapse and cause a hierarchy of structures, such as clusters of galaxies, stellar groups, stars and planets. For example, a star is born through the gradual gravitational collapse of a cloud of interstellar matter. The compression caused by the collapse raises the temperature until nuclear fuel reignites in the center of the star and the collapse comes to a halt. The thermal pressure gradient (leading to expansion) compensates the gravity (leading to compression) and a star is in dynamical equilibrium between these two forces. Gravitational collapse of a star occurs at the end of its lifetime, also called the death of the star. In this sense a star is in a "temporary" equilibrium state between a gravitational collapse at stellar birth and a further gravitational collapse at stellar death. The end states are called compact stars. The types of compact stars are: White dwarfs, in which gravity is opposed by electron degeneracy pressure; Neutron stars, in which gravity is opposed by neutron degeneracy pressure and short-range repulsive neutron neutron interactions mediated by the strong force. Even more massive stars, above the Tolman Oppenheimer Volkoff limit cannot find a new dynamical equilibrium with any known force opposing gravity. Hence, the collapse continues with nothing to stop it. This unhindered collapse leads to the formation of a singularity where the curvature becomes infinity. Whether the singularity is visible to a distant observer or not depends on the formation of an event horizon. If an event horizon forms before the matter crushes to the singularity, one has black hole. If the horizon never forms, the singularity will be visible to a distant observer. This is called a naked singularity. If the formation of the horizon is after that of the singularity, the latter will be visible only for a finite amount of time. It might be thought that a sufficiently large neutron star could exist inside its Schwarzschild radius and appear like a black hole without having all the mass compressed to a singularity at the center; however, this is a misconception. Within the event horizon, matter would have to move outwards faster than the speed of light in order to remain stable and avoid collapsing to the center. No physical force can therefore prevent the star from collapsing to a singularity. Chapter-I On Continued Gravitational Contraction (J. R. OPPENHEIMER AND H. SNYDER) When all thermonuclear sources of energy are exhausted, a sufficiently heavy star will collapse. Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely. In their paper, they studied the solutions of the gravitational field equations. [1] General and qualitative arguments are given on the behavior of the metrical tensor as the contraction progresses: the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened, and can escape over a progressively narrower range of angles. An analytic solution of the field equations confirming these general arguments is obtained for the case that the pressure within the star can be neglected. The total time of collapse for an observer Co-moving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day; an external observer sees the star asymptotically shrinking to its gravitational radius. They considered stars which have large masses, >0.7Q, and which have used up their nuclear sources of energy. A star under these circumstances would collapse under the influence of its gravitational field and release energy. This energy could be divided into four parts: (1) Kinetic energy of motion of the particles in the star, (2) Radiation, (3) Potential and kinetic energy of the outer layers of the star which could be blown away by the radiation, (4) Rotational energy (does not apply for a spherical star) which could divide the star into two or more parts. If the mass of the original star were sufficiently small, or if enough of the star could be blown from the surface by radiation, or lost directly in radiation, or if the angular momentum of the star were great enough to split it into small fragments, then the remaining matter could form a stable static distribution, a white dwarf star. They considered the case where this cannot happen. Analytical Approach: The line element outside the boundary of the stellar matter must take the form (1) Here ro is the gravitational radius, connected with the gravitational mass m of the star by r0=2mg/c2, and constant. Here ro is the gravitational radius, connected with the gravitational mass m of the star by r0=2mg/c2, and constant. It is expected that since the pressure of the stellar matter is insufficient to support it against its own gravitational attraction, the star will contract, and its boundary r& will necessarily approach the gravitational radius ro. Near the surface of the star, where the pressure must in any case be low, we should expect to have a local observer see matter falling inward with a velocity very close to that of light; to a distant observer this motion will be slowed up by a factor (1-ro/rb). All energy emitted outward from the surface of the star will be reduced very much in escaping, by the Doppler effect from the receding source, by the large gravitational red-shift, (1 ro/rb)1/2, and by the gravitational deflection of light which will prevent the escape of radiation except through a cone about the outward normal of progressively shrinking aperture as the star contracts. The star thus tends to close itself off from any communication with a distant observer; only its gravitational field persists. For this line element the field equations are: (2) (3) (4) (5) in which primes represent differentiation with respect to r and dots differentiation with respect to t. The energy-momentum tensor T is composed of two parts: (1) A material part due to electrons, protons, neutrons and other nuclei. (2) Radiation. The material part may be thought of as that of a fluid which is moving in a radial direction, and which in co-moving coordinates would have a definite relation between the pressure, density, and temperature. The radiation may be considered to be in equilibrium with the matter at this temperature, except for a flow of radiation due to a temperature gradient. From the first two field equations one can see that unless vanishes at least as rapidly as r2 when r goes to 0, T44 will become singular and that either or both T11 and ' will become singular. Physically such a singularity would mean that the expression used for the energy-momentum tensor does not take account of some essential physical fact which would really smooth the singularity out. Further, a star in its early stage of development would not possess a singular density or pressure; it is impossible for a singularity to develop in a finite time. Then, (6) Therefore 0, for all r since (T44 0). Thus, ' 0, since and (-T11) are equal to or greater than zero. Applying relevant boundary conditions, the field equations give: (7) When the pressure vanishes there are no static solutions to the field equations except when all components of T vanish. With p=0 we have the free gravitational collapse of the matter. The general features of the solution obtained this way give a valid indication even for the case that the pressure is not zero, provided that the mass is great enough to cause collapse. For the solution of this problem, it is convenient to follow the earlier work of Tolman [2] and use another system of coordinates, which are co-moving with the matter. Take a line element of the form: (8) Because the coordinates are co-moving with the matter and the pressure is zero, (9) & all other components of the energy momentum tensor vanish. (10) (11) (12) (13) with primes and dots here and in the following representing differentiation with respect to R and , respectively. The integral of the last equation is given by Tolman (14) with f2(R) being a positive but otherwise arbitrary function of R. We find a sufficiently wide class of solutions if we put f2(R) = 1. Substituting this in the first field equation we get: (15) The solution of this equation is: (16) in which F and G are arbitrary functions of R. For the density we obtain (17) Choose (18) At a particular time, say equal zero, we may assign the density as a function of R. The density equation then becomes a first-order differential equation (19) We now take, a particular special case: (20) A particular solution of this equation is: (21) in which the constant r0 is introduced for convenience, and is the gravitational radius of the star. They found a coordinate transformation which will change the line element into a form similar to the very first one. (22) Using the contra-variant form of the metric tensor: (23) (24) (25) The last equation is a first-order partial differential equation for t. Using the values of r, and the values of F and G: (26) The general solution of this is: (27) Outside the star, where R is greater than Rb, it is wished that the line element to be of the Schwarzschild form; again neglecting the gravitational effect of any escaping radiation; thus (28) At the surface of the star, R equal Rb, we must have L equal to M for all . The form of M is determined by this condition to be: (29) To find asymptotic behavior of e , e & for large t; the approximate relation looks: (30) From this relation we see that for a fixed value of R as t tends toward infinity, tends to a finite limit, which increases with R. After this time o an observer co-moving with the matter would not be able to send a light signal from the star; the cone within which a signal can escape has closed entirely. For a star which has an initial density of one gram per cubic centimeter and a mass of 1033 grams this time o is about a day. (31) For R equal to Rb, e tends to infinity like et/r as t approaches infinity. Where R is less than Rb, e tends to zero like e(-2t/r) and where R is equal to Rb, e tends to zero like e(-t/r). For tends to a finite limit, for r<ro as t approaches infinity, and for r=ro tends to infinity. Also for r<=ro, tends to minus infinity. We expect that this behavior will be realized by all collapsing stars which cannot end in a stable stationary state. Of course, actual stars would collapse more slowly than the example studied analytically because of the effect of the pressure of matter, of radiation, & of rotation. Chapter-II f(R) gravity: The acceleration of the universe discovered ten years ago with type Ia supernovae still has no truly satisfactory explanation. This discovery has important implications not only for cosmology, but also for fundamental physics. According to WMAP and the other cosmic microwave background experiments, if General Relativity (GR) is the correct description of our universe, then approximately 76% of its energy content is a mysterious form of dark energy, exotic, invisible, and unclustered. Three main classes of models for this cosmic acceleration have been proposed: [3] 1) A cosmological constant ; 2) Dark energy, and 3) Modified gravity. Naively, a cosmological constant propelling the cosmic acceleration and eventually coming to dominate the universe, causing it to enter a de Sitter phase without return, seems the most obvious explanation. However, brings with it the notorious cosmological constant and coincidence problems. To admit that has the tiny value required to explain the current acceleration amounts to an extreme fine-tuning. It is not surprising, therefore, that most cosmologists reject this explanation, postulating that is exactly zero for unknown reasons, and that a different explanation is to be found for the cosmic acceleration. The second class of models, (mostly) within the context of GR, postulates the existence of a dark energy fluid with equation of state p= (- ) (where and p are the energy density and pressure, respectively), which comes to dominate late in the matter era. Dark energy could even be phantom energy with equation of state p< (- ). Many dark energy models have been studied, none of which is totally convincing or free of fine-tuning problems. A third possibility consists in dispensing entirely with the mysterious dark energy and modifying gravity at the largest scales. [4-a, 4-b, 4-c] Here we focus on modified gravity and, specifically, on the so-called f(R) gravity theories. f(R) or modified gravity consists of infrared modifications of GR that become important only at low curvatures, late in the matter era. The Einstein-Hilbert action is modified to: (32) where f(R) is a non-linear function of its argument. In principle, the metric tensor contains several degrees of freedom: tensor, vector, and scalar, mass-less or massive. In GR only the familiar mass-less spin 2 graviton propagates. When the Einstein-Hilbert action is modified, other degrees of freedom appear. The change in the action brings to life, in addition to the mass-less graviton, a massive scalar mode which can drive the cosmic acceleration and is analogous to the inflaton field driving the accelerated expansion of the early universe, although at a much lower energy scale. If terms quadratic in the Ricci and Riemann tensor, and possibly other curvature invariants, are included in the gravitational Lagrangian, massive gravitons and vector degrees of freedom appear. f(R) gravity has a long history: its origins can be loosely traced to Weyl's 1919 theory. Later, f(R) gravity received the attention of many authors, including Eddington, Bach, Lanczos, Schrodinger, and Buchdahl. In the 1960's and 1970's, it was found that quadratic corrections to SEH were necessary to improve the renormalizability of GR, and in 1980 they were found to fuel inflation without scalar fields. Non-linear corrections are also motivated by string theories and play a role in studies of the cosmological constant with the Wheeler-De Witt equation. The prototype of f(R) gravity is the model f(R) = R-( 4/R), where is a mass scale of the order of the present value of the Hubble parameter. Although ruled out by its weak-field limit and by a violent instability, this model gives the basic idea: the 1/R correction is negligible in comparison with R at high curvatures, and kicks in only late as R tends to 0. The three versions of f(R) gravity Modified gravity comes in three versions: 1) metric (or second order) formalism; 2) Palatini (or first order) formalism; and 3) metric-affine gravity. Chapter-III Field Equations in metric f(R) Gravity: The action of f(R) gravity in the presence of matter described by the matter Lagrangian LM is given by: (33) Where the matter Lagrangian depends on the metric tensor g and the matter fields. Varying this action with respect to metric tensor, we obtain the following field equations [5] (34) (= 8 ) is the coupling constant in gravitational units. These are the fourth order partial differential equations in the metric tensor. The fourth order is due to the last two terms on the left side of the equation. Capozziello, et al. have discussed the general solutions of these fourth order equation in the context of spherically symmetry. If we take f(R) = R, these equations reduce to the field equations in general relativity. Chapter- IV Perfect Fluid Collapse in f(R) Gravity The energy-momentum tensor for perfect fluid is: (35) where is the energy density, p is the pressure and u = 0 is the four-vector velocity in co-moving coordinates. Putting this equation in the field equation, we get: (36) Let the space-time be denoted by the metric defined by: (37) where X and Y are functions of t and r. So the field equation in the context of this space-time metric comes to be: (38) (39) (40) (41) (42) To solve this set of the field equations, we need to integrate the last equation to get the explicit value of X. It follows that: (43) We see that this cannot be integrated unless we have some value of F such that derivative of the denominator should be the numerator. Chapter- V The condition of constant scalar curvature: This analytical approach has already been studied by Sharif & Kausar [6]. Here the work in details has been reviewed. We wish to apply this approach to some other particular models, e.g. exponential f(R) gravity, and look into certain collapse models. The condition of constant scalar curvature (R = R0), according to which F(R0) = constant, enables us to solve this integral. This means that the Ricci scalar must be constant which is possible only if we take pressure and density constant, i.e., p = p0, & = 0. Consequently, the field equations become: (44) (45) (46) (47) Integration of the last equation with respect to t yields: (48) Where W = W(r) is an arbitrary function of r. Using this equation in the modified set of field equations: (49) Integrating this equation with respect to t, it follows that: (50) where m = m(r) is an arbitrary function of r and is related to the mass of the collapsing system. Substituting these into the first field equation: (51) Integrating this equation with respect to r, we obtain (52) where c(t) is a function of integration. Since mass cannot be negative due to physical reasons, we must take the function m(r) positive. The total energy M(r, t) up to a radius r at time t inside the hyper-surface can be calculated by using the definition of mass function: [7] (53) For the interior metric, it becomes: (54) Using this: (55) Here we assume: [6] (56) & the condition: [8] (57) Thus solving the field equation give: (58) (59) Where, (60) Here ts(r) is an arbitrary function of r which denotes the time formation of singularity for a particular shell at distance r. When (61) The above solution corresponds to the famous Tolman-Bondi solution: [8] (62) (63) Chapter- VI Apparent Horizons The apparent horizon is formed when the boundary of trapped two spheres is formed. The formula to find such a boundary with null outward normals is given as follows: [6] (64) For this particular case the equation gives us: (65) The values of Y yield the apparent horizons. For f(R0) = 8 (p0 0), we have Y = 2m, i.e., Schwarzschild horizon. The last equation can have the following positive roots. Case (i): For (66) We obtain two horizons, namely cosmological horizon, Yc, and black hole horizon, Ybh, i.e. (67) (68) Where, (69) If we take m = 0, it follows: (70) Case (ii): For (71) there is only one positive root which corresponds to a single horizon i.e., (72) This shows that both horizons coincide. The ranges for these horizons can be written as follows: (73) The largest proper area of the black hole horizon is given by: (74) and the cosmological horizon has its area between (75) Case (iii): For (76) There are no positive roots and consequently there are no apparent horizons. Now we calculate the time of formation for the apparent horizon (77) When f(R0)= 8 (p0 0), this corresponds to Tolman-Bondi solution given as: (78) From the equation of the time of formation for the apparent horizon: (79) Where (80) These equations imply that Yc>Ybh and t2>t1 respectively. Here t1 denotes the time formation of cosmological horizon and t2 denotes the time formation of black hole horizon. The inequality t2>=t1 indicates that the cosmological horizon forms earlier than the black hole horizon. The time difference between the formation of cosmological horizon and singularity and the formation of black hole horizon and singularity respectively can be found as follows. (81) (82) The time difference between the formation of singularity and apparent horizons is: (83) It follows: (84) (85) It shows that T1 is a decreasing function of mass m. This means that time interval between the formation of cosmological horizon and singularity is decreased with the increase of mass. Similarly, (86) This indicates that T2 is an increasing function of mass m showing that the time difference between the formation of black hole horizon and singularity is increased with the increase of mass. Chapter-VII An attempt to study collapse in Exponential f(R) gravity So far, the study has been more or less a review of a previously published work by M. Sharif & R. Kausar. Following the same procedure, an attempt to study collapse in a model where: f(R)= exp(- R) Then, from equation (43) X= & + R & Y 2 RR & Y 2Y dt 2Y + R Y (87) This can only be integrated if & Y 2 RR & Y (88) ( R Y & ) = R Or Y = k ( r ) e R (89) Where k(r)= some arbitrary function of r, which comes as a constant of integration (over time). Then, X = R + ln{3k2 (r )k (r ) R 2k2 (r )k (r )} (90) We have been trying to study this typical model over the past month, & the detailed analytical study is on progress. We ll study formation of naked singularity/apparent horizon in this particular model. As we ve found out the metric after some messy attempts, we re hopeful this will be a good analytical study. Chapter-VIII: Indications In this study we have discussed different aspects of gravitational collapse in metric f(R) gravity. For this purpose, the field equations are solved using the assumption of constant scalar curvature. To meet the requirement of gravitational collapse, it is assumed that scalar curvature is very large constant quantity. We obtain two physical apparent horizons named as black hole horizon and cosmological horizon. It is concluded that black hole horizon requires more time for its formation as compared to cosmological horizon. However, both the horizons form earlier than singularity which indicates that singularity is covered, i.e., black hole. In this way, f(R) gravity seems to support CCC. The analysis of positive and negative acceleration provides the same results on f(R0) as one can obtain from the Newtonian force. We would like to mention here why to explore the constant curvature solution in typical modified f (R) gravity. To generate the accelerating expansion in the present universe, f(R) could be a small constant. The universe starts from the inflation driven by the effective cosmological constant at the early stage, where curvature is very large. When the curvature becomes smaller, the effective cosmological constant also becomes smaller. [9] There appears the small effective cosmological constant when the density of the radiation and the matter becomes small and the curvature goes to the value R0. Thus expansion could start and cosmological constant can be identified as f (R0) in the present accelerating era. [6] It is well-known that only those f(R) models are cosmologically viable whose solution correspond to general relativity. The class of constant scalar curvature solutions with spherically symmetry contains black hole solutions in the presence of cosmological constant. For example, the Schwarzschild anti de-Sitter solutions and all the topological solutions associated with a negative . Exact solutions of cylindrically symmetric spacetimes with assumptions of constant scalar curvature are applicable to the exterior of a string. [10] Finally, we conclude that the term f(R0) plays the same role as that of the cosmological constant in the Einstein field equations. In general relativity, the cosmological constant acts as a repulsive force (hence slows down the collapse of matter) and same does f(R0). It would be worthwhile to investigate these issues in f(R) theory by dropping the assumption of scalar curvature for complete understanding of gravitational collapse. . References: [1] Oppenheimer, J.R. and Synder, H.: Phys. Rev. 56(1939)455 [2] R. C. Tolman, Proc. Nat. Acad. Sci. 20, 3 (1934) [3] P. S. Joshi, Gravitational Collapse and Space-time Singularities, (Cambridge University Press, Cambridge, 2008). [4-a] V. Faraoni, arXiv:gr-qc/0607116; arXiv:gr-qc/0511094; arXiv:0710.1291[gr-qc]; [4-b] S. Capozziello, S. Carloni and A. Troisi, arXiv:astro-ph/0303041; [4-c] Alejandro Guarnizo, 1 Leonardo Casta~neda 2 & Juan M. Tejeiro, arXiv:1010.5279v3 [gr-qc] 3 May 2011. [5] Thomas P. Sotiriou, Valerio Faraoni; arXiv:0805.1726v4 [gr-qc] 4 Jun 2010 [6] M. Sharif _and H. Rizwana Kausar, arXiv:1007.2852v1 [gr-qc] 18 Jun 2010 [7] Misner, C.W. and Sharp, D.: Phys. Rev. 136(1964) b571 [8] Eardley, D.M. and Smarr, L.: Phys. Rev. D19(1979)2239 [9] Nojiri, S., Odintsov, S.D.: Phys. Lett. B657(2007)238. [10] Sharif, M. and Ahmad, Z.: Mod. Phys. Lett. A22(2007)1493; ibid. A22(2007)2947.

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