2002 (IPP)
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Item Why do we observe a small but non zero cosmological constant?(2002-03-03) Padmanabhan, T.The current observations seem to suggest that the universe has a positive cosmological constant of the order of H2 0 while the most natural value for the cosmological constant will be L−2 P where LP = (G¯ h/c3)1/2 is the Planck length. This reduction of the cosmological constant from L−2 P to L−2 P (LPH0)2 may be interpreted as due to the ability of quantum micro structure of spacetime to readjust itself and absorb bulk vacuum energy densities. Being a quantum mechanical process, such a cancellation cannot be exact and the residual quantum fluctuations appear as the “small” cosmological constant. I describe the features of a toy model for the spacetime micro structure which could allow for the bulk vacuum energy densities to be canceled leaving behind a small residual value of the the correct magnitude. Some other models (like the ones based on canonical ensemble for the four volume or quantum fluctuations of the horizon size) lead to an insignificantly small value of H2 0 (LPH0)n with n = 0.5 − 1 showing that obtaining the correct order of magnitude for the residual fluctuations in the cosmological constant is a nontrivial task, becaue of the existence of the small dimensionless number H0LP .Item THERMODYNAMICS AND/OF HORIZONS: A COMPARISION OF SCHWARZSCHILD, RINDER AND de SITTER SPACETIMES(2002-02-01) Padmanabhan, T.The notions of temperature, entropy and ‘evaporation’, usually associated with space- times with horizons, are analyzed using general approach and the following results, ap- plicable to different spacetimes, are obtained at one go. (i) The concept of temperature associated with the horizon is derived in a unified manner and is shown to arise from purely kinematic considerations. (ii) QFT near any horizon is mapped to a conformal field theory without introducing concepts from string theory. (iii) For spherically sym- metric spacetimes (in D = 1 + 3) with a horizon at r = l, the partition function has the generic form Z ∝ exp[S − βE], where S = (1/4)4πl 2 and |E| = (l/2). This analysis reproduces the conventional result for the blackhole spacetimes and provides a simple and consistent interpretation of entropy and energy for deSitter spacetime. (iv) For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. (v) In the case of a Schwarzschild black hole there exist quantum states (like Unruh vacuum) which are not invariant under time reversal and can describe blackhole evaporation. There also exist quantum states (like Hartle-Hawking vacuum) in which temperature is well-defined but there is no flow of radiation to infinity. In the case of deSitter universe or Rindler patch in flat spacetime, one usually uses quantum states analogous to Hartle-Hawking vacuum and obtains a temperature without the cor- responding notion of evaporation. It is, however, possible to construct the analogues of Unruh vacuum state in the other cases as well. Associating an entropy or a radiating vacuum state with a general horizon raises conceptual issues which are briefly discussed.Item Is gravity an intrinsically quantum phenomenon? Dynamics of gravity from the entropy of spacetime and the principle of equivalence(2002-05-01) Padmanabhan, T.The two surprising features of gravity are (a) the principle of equivalence and (b) the connection between gravity and thermodynamics. Using principle of equivalence and special relativity in the local inertial frame, one could obtain the insight that gravity must possess a geometrical description. I show that, using the same principle of equivalence, special relativity and quantum theory in the local Rindler frame one can obtain the Einstein-Hilbert action functional for gravity and thus the dynamics of the spacetime. This approach, which essentially involves postulating that the horizon area must be proportional to the entropy, uses the local Rindler frame as a natural extension of the local inertial frame and leads to the interpretation that the gravitational action represents the free energy of the spacetime geometry. As an aside, one also obtains a natural explanation as to: (i) why the covariant action for gravity contains second derivatives of the metric tensor and (ii) why the gravitational coupling constant is positive. The analysis suggests that gravity is intrinsically holographic and even intrinsically quantum mechanical.Item Statistical mechanics of gravitating systems in static and cosmological backgrounds(2002-03-01) Padmanabhan, T.This pedagogical review addresses several issues related to statistical de- scription of gravitating systems in both static and expanding backgrounds, focusing on the latter. After briefly reviewing the results for the static background, I describe the key issues and recent progress in the context of gravitational clustering of collision-less particles in an expanding universe. The questions addressed include: (a) How does the power injected into the system at a given wave number spread to smaller and larger scales? (b) How does the power spectrum of density fluctuations behave asymptotically at late times? (c) What are the universal characteristics of gravitational clustering that are independent of the initial conditions and manifest at the late time evolution of the system? The review is intended for non cosmologists and will be of interest to people working in fluid mechanics, non linear dynamics and condensed matter physics.Item Holography of gravity encoded in a relation between entropy, horizon area and action for gravity(2002-05-20) Padmanabhan, T.I provide a general proof of the conjecture that one can attribute an en- tropy to the area of any horizon. This is done by constructing a canonical ensemble of a subclass of spacetimes with a fixed value for the temperature T = β−1 and evaluating the exact partition function Z(β). For spherically symmetric spacetimes with a horizon at r = a, the partition function has the generic form Z ∝ exp[S −βE], where S = (1/4)4πa2 and |E| = (a/2). Both S and E are determined entirely by the properties of the metric near the horizon. This analysis reproduces the conventional result for the black-hole spacetimes and provides a simple and consistent interpretation of entropy and energy for De Sitter spacetime. For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. Further, I show that the relationship between entropy and area allows one to construct the action for the gravitational field on the bulk and thus the full theory. In this sense, gravity is intrinsically holographic.Item Hawking radiation in different coordinate settings : Complex paths approach(2002-06-06) Shankaranarayanan, S.; Padmanabhan, T.; Srinivasan, K.We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild space-time. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semiclassical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation as far as these coordinates are concerned.Item Evolution of the correlation function for a class of processes involving non local self-Replication(2002-07-06) Padmanabhan, T.A large class of evolutionary processes can be modeled by a rule that involves self-replication of some physical quantity with a non local rescaling. We show that a class of such models are exactly solvable — in the discrete as well as continuum limit — and can represent several physical situations as varied from the formation of galaxies in some cosmological models to growth of bacterial cultures. This class of models, in general, has no steady state solution and evolve unstably as t → ∞ for generic initial conditions. They can however exhibit (unstable) power law correlation function in the continuum limit, for an intermediate range of times and length scales.Item Entropy and energy of a class of spacetimes with horizon : a general derivation(2002-03-01) Padmanabhan, T.Euclidean continuation of several Lorentzian spacetimes with horizons requires treating the Eu- clidean time coordinate to be periodic with some period β. Such spacetimes (Schwarzschild, de- Sitter,Rindler .....) allow a temperature T = β−1 to be associated with the horizon. I construct a canonical ensemble of a subclass of such spacetimes with a fixed value for β and evaluate the par- tition function Z(β). For spherically symmetric spacetimes with a horizon at r = a, the partition function has the generic form Z ∝ exp[S −βE], where S = (1/4)4πa2 and |E| = (a/2). Both S and E are determined entirely by the properties of the metric near the horizon. This analysis reproduces the conventional result for the blackhole spacetimes and provides a simple and consistent interpre- tation of entropy and energy for deSitter spacetime. For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. The implications are discussed.Item Cosmology with tachyon field as dark energy(2011-07-06) Bagla, J. S.; Jassal, H. K.; Padmanabhan, T.We present a detailed study of cosmological effects of homogeneous tachyon matter coexisting with non-relativistic matter and radiation, concentrating on the inverse square potential and the expo- nential potential for the tachyonic scalar field. A distinguishing feature of these models (compared to other cosmological models) is that the matter density parameter and the density parameter for tachyons remain comparable even in the matter dominated phase. For the exponential potential, the solutions have an accelerating phase, followed by a phase with a(t) ∝ t 2/3 as t → ∞. This elimi- nates the future event horizon present in ΛCDM models and is an attractive feature from the string theory perspective. A comparison with supernova Ia data shows that for both the potentials there exists a range of models in which the universe undergoes an accelerated expansion at low redshifts and are also consistent with requirements of structure formation. They do require fine tuning of parameters but not any more than in the case of ΛCDM or quintessence models.Item Cosmological constant - The weight of the vacuum(2011-07-06) Padmanabhan, T.Recent cosmological observations suggest the existence of a positive cosmological constant Λ with the magnitude Λ(G~/c3) ≈ 10−123. This review discusses several aspects of the cosmological constant both from the cosmological (sections 1–6) and field theoretical (sections 7–11) perspectives. After a brief introduction to the key issues related to cosmological constant and a historical overview, a summary of the kinematics and dynamics of the standard Friedmann model of the universe is provided. The observational evidence for cosmological constant, especially from the supernova results, and the constraints from the age of the universe, structure for- mation, Cosmic Microwave Background Radiation (CMBR) anisotropies and a few others are described in detail, followed by a discussion of the theoretical models (quintessence, tachyonic scalar field, ...) from different perspectives. The latter part of the review (sections 7–11) concentrates on more conceptual and fundamental aspects of the cosmological constant like some alternative interpretations of the cosmological constant, relaxation mechanisms to reduce the cosmological constant to the currently observed value, the geometrical structure of the de Sitter space- time, thermodynamics of the de Sitter universe and the role of string theory in the cosmological constant problem.