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Neutron stars are known to have strong magnetic fields reaching as high as $10^{15}$ Gauss, besides having strongly curved interior spacetime. So for computing an equation of state for neutron-star matter, the effect of magnetic field as well as curved spacetime should be taken into account. In this article, we compute the equation of state for an ensemble of degenerate fermions in the curved spacetime of a neutron star in presence of a magnetic field. We show that the effect of curved spacetime on the equation of state is relatively stronger than the effect of observed strengths of magnetic field. Besides, a thin layer containing only spin-up neutrons is shown to form at the boundary of a degenerate neutron star., Comment: 8 pages, 4 figures, revtex4

We show that a primeval seed magnetic field arises due to spin-degeneracy breaking of fermions caused by the dragging of inertial frames in the curved spacetime of rotating astrophysical bodies. This seed magnetic field would arise even due to electrically neutral fermions such as neutrons. As an example, we show that an ideal neutron star rotating at $500$ revolutions per second, having mass $0.83$ M$_{\odot}$ and described by an ensemble of degenerate neutrons, would have $0.12$ Gauss seed magnetic field at its center arising through the breaking of spin-degeneracy., Comment: 5 pages, 2 figures, revtex4, improved discussion on the strength of seed field in rotating stars and galaxies

We compute the equation of state for an ensemble of degenerate fermions by using the curved spacetime of a slowly rotating axially symmetric star. We show that the equation of state computed in such curved spacetime depends on the gravitational time dilation as well as on the dragging of inertial frames, unlike an equation of state computed in a globally flat spacetime. The effect of gravitational time dilation leads to a significant enhancement of the maximum mass limit of a degenerate neutron star. However, such an enhancement due to the frame-dragging effect is extremely small. Nevertheless, in general relativity the frame-dragging effect is crucial for computing angular momentum of the star which is also shown to be enhanced significantly due to the usage of curved spacetime in computing the equation of state., Comment: v1 9 pages, 4 figures, revtex4; v2 10 pages, 6 figures, matches with published version

In order to solve the Tolman-Oppenheimer-Volkoff equations for neutron stars, one routinely uses the equation of states which are computed in the Minkowski spacetime. Using a first-principle approach, it is shown that the equation of states which are computed within the curved spacetime of the neutron stars include the effect of gravitational time dilation. It arises due to the radially varying interior metric over the length scale of the star and consequently it leads to a much higher mass limit. As an example, for a given set of parameters in a $\sigma-\omega$ model of nuclear matter, the maximum mass limit is shown to increase from $1.61 M_{\odot}$ to $2.24 M_{\odot}$ due to the inclusion of gravitational time dilation., Comment: v1 10 pages, 10 figures; v2 11 pages, matches with published version

In the study of spherical degenerate stars such as neutron stars, general relativistic effects are incorporated by using Tolman-Oppenheimer-Volkoff equations to describe their interior spacetime. However, the equation of states employed in such studies are invariably computed in flat spacetime. We show that the equation of states computed in the curved spacetime of these stars depend explicitly on the metric function. Further, we show that ignoring such metric-dependent gravitational time dilation effect leads one to grossly underestimate the mass limits of these compact stars., Comment: v1 5 pages, 2 figures; v2 6 pages, matches with published version

The present work investigates some exact solutions of the gravitational wave equation in some widely used cosmological spacetimes. The examples are taken from spatially flat and closed isotropic models as well as Kasner metric which is anisotropic. Various matter distributions are considered, from the standard dust or radiation distribution to exotic matters like that with an equation of state $P = - \frac{1}{3} \rho$. In almost all cases the frequency and amplitude of the wave are found to be decaying with evolution, except for a closed radiation universe where there is a resurgence of the waveform, consistent with the recollapse of the universe into the big crunch singularity., Comment: 17 pages, 6 figures

The electromagnetic interaction alters the Chandrasekhar mass limit by a factor which depends, as computed in the literature, on the atomic number of the positively charged nuclei present within the degenerate matter. Unfortunately, the methods employed for such computations break Lorentz invariance ab initio. By employing the methods of finite temperature relativistic quantum field theory, we show that in the leading order, the effect of electromagnetic interaction reduces the Chandrasekhar mass limit for non-general-relativistic, spherically symmetric white dwarfs by a universal factor of $(1-3\alpha/4\pi)$, $\alpha$ being the fine-structure constant., Comment: 5 pages, 1 figure, revtex

In the study of degenerate plasmas contained within compact astrophysical objects, both special relativity and general relativity play important roles. After reviewing the existing treatment in the literature, here we employ the methods of relativistic thermal quantum field theory to compute the equation of states of degenerate matter for compact astrophysical objects such as the white dwarfs and the neutron stars. In particular, we compute the equation of states that include leading order corrections due to the finite temperature, the fine-structure constant as well as the effect of gravitational time dilation. We show that the fine-structure constant correction remains well-defined even in the non-relativistic regime in contrast to the existing treatment in the literature., Comment: 9 pages, 3 figures, revtex; v2 a detailed comparison with existing treatment added, review is shortened, discussion improved, a new ref. added; v3 16 pages, 7 figures, significantly updated, title changed, to be appearing in Reviews of Modern Plasma Physics

In order to achieve a Hamiltonian-based canonical derivation of the Hawking effect, one usually faces multiple hurdles. Firstly, the spacetime foliation using Schwarzschild time does not lead to hyper-surfaces which are always spacelike. Secondly, the null coordinates which are frequently used in covariant approach, do not lead to a true matter Hamiltonian. Recently, an exact canonical derivation was presented using the so-called near-null coordinates. However, there too one faces the difficulty of having to deal with non-vanishing matter diffeomorphism generator as the spatial decomposition involves a non-zero shift vector. Here we introduce a new set of coordinates which allows one to perform an exact canonical derivation of Hawking effect without having to deal with matter diffeomorphism generator., Comment: 6 pages, 1 figure, revtex4

It is believed that extremal black holes do not emit Hawking radiation as understood by taking extremal limits of non-extremal black holes. However, it is debated whether one can make such conclusion reliably starting from an extremal black hole, as the associated Bogoliubov coefficients which relate ingoing and outgoing field modes do not satisfy the required consistency condition. We address this issue in a canonical approach firstly by presenting an exact canonical derivation of the Hawking effect for non-extremal Kerr black holes. Subsequently, for extremal Kerr black holes we show that the required consistency condition is satisfied in the canonical derivation and it produces zero number density for Hawking particles. We also point out the reason behind the reported failure of Bogoliubov coefficients to satisfy the required condition., Comment: 11 pages, 1 figure, revtex4

The Hawking effect is one of the most extensively studied topics in modern physics. Yet it remains relatively under-explored within the framework of canonical quantization. The key difficulty lies in the fact that the Hawking effect is principally understood using the relation between the ingoing modes which leave past null infinity and the outgoing modes which arrive at future null infinity. Naturally, these modes are described using advanced and retarded null coordinates instead of the usual Schwarzschild coordinates. However, null coordinates do not lead to a true Hamiltonian that describes the evolution of these modes. In order to overcome these hurdles in a canonical formulation, we introduce here a set of near-null coordinates which allows one to perform an exact Hamiltonian-based derivation of the Hawking effect. This derivation opens up an avenue to explore the Hawking effect using different canonical quantization methods such as polymer quantization., Comment: 11 pages, 3 figures, revtex4

It is widely believed that the Hawking effect might hold clues to the possible, yet unknown, trans-Planckian physics. On the other hand, one could ask whether the effect itself might be altered by such trans-Planckian physics. We seek an answer to this question within a framework where matter field is quantized using polymer quantization, a canonical quantization technique employed in loop quantum gravity. We provide an exact derivation of the Hawking effect using canonical formulation by introducing a set of near-null coordinates which allows one to overcome the challenges posed by a Hamiltonian-based derivation of the Hawking effect. Subsequently, we show that in polymer quantization the Hawking effect is short-lived and it eventually disappears for an asymptotic future observer. Such an observer finds the duration of the Hawking effect to be few milliseconds for a solar mass black hole whereas it is few years for an ultra-massive black hole. Consequently, it provides a new way to resolve the so-called information loss paradox., Comment: 5 pages, 1 figure, revtex4

The Wightman function i.e. vacuum two-point function, for a massless free scalar field in Fock quantization is inversely proportional to the invariant distance squared between the corresponding spacetime points. Naturally it diverges when these two points are taken to be infinitesimally close to each other. Using a combination of analytical and numerical methods, we show that the Wightman function is bounded from above in polymer quantization of scalar field. The bounded value of the Wightman function is governed by the polymer scale and its bounded nature can be viewed as if the spacetime has a zero-point length. This emergence of effective discreteness in spacetime appears from polymer quantization of matter field alone as the geometry is treated classically. It also demonstrates that the polymer quantization of matter field itself can capture certain aspects of the effective discrete geometry. We discuss the implications of the result on the response function of a Unruh-DeWitt detector that depends on the properties of the Wightman function., Comment: 10 pages, 3 figures, revtex4

If an Unruh-DeWitt detector operates for an infinite proper time along the trajectory of a uniformly accelerated observer in Fock space then induced transition rate of the detector is proportional to Planck distribution. For a realistic detector which operates only for a finite period, the instantaneous transition rate contains both transient and non-transient terms. In particular, the non-transient term contains a residue evaluated at the pole of the two-point function. We show here by considering a massless scalar field that unlike in Fock quantization, the short-distance two-point function contains no pole in polymer quantization, the quantization techniques used in loop quantum gravity. Consequently, corresponding transition rate of the Unruh-DeWitt detector contains only transient terms. Thus, the result presented here provides an alternative evidence for absence of Unruh effect in polymer quantization which was shown earlier by the authors using methods of Bogoliubov transformation and Kubo-Martin-Schwinger condition., Comment: 9 pages, 1 figure, revtex4; v2 new reference added, few typos fixed; a constant term which was missed earlier incorporated; results unchanged

Existence of Unruh effect is often understood from the property of two-point function along Rindler trajectory where it satisfies KMS condition. In particular, it exhibits the so-called KMS periodicity along imaginary time direction. Corresponding period is then identified with reciprocal of Unruh temperature times Boltzmann constant. We show here that the two-point function including leading order perturbative corrections due to polymer quantization, the quantization method used in loop quantum gravity, violates KMS condition in low-energy regime. This violation is caused by correction terms which are not Lorentz invariants. Consequently, polymer corrected two-point function along Rindler trajectory looses its thermal interpretation. We discuss its implications on existence of Unruh effect in the context of polymer quantization., Comment: 6 pages, revtex4; v2 few clarifications on domain of perturbation added; v3 initials in title expanded, few discussions and a reference are added, to match with the published version

Unruh effect is a landmark prediction of standard quantum field theory in which Fock vacuum state appears as a thermal state with respect to an uniformly accelerating observer. Given its dependence on trans-Planckian modes, Unruh effect is often considered as an arena for exploring a candidate theory of quantum gravity. Here we show that Unruh effect disappears if, instead of using Fock quantization, one uses polymer quantization or loop quantization, the quantization method used in loop quantum gravity. Secondly, the polymer vacuum state remains a vacuum state even for the accelerating observer in the sense that expectation value of number density operator in it remains zero. Finally, if experimental measurement of Unruh effect is ever possible then it may be used either to verify or rule out a theory of quantum gravity., Comment: 6 pages, 1 figure; v2 with improved discussions and notation; several new references added; published in CQG with a slightly different title but the same results

Quantum matter in quantum space-time is discussed using general properties of energy-conservation laws. As a rather radical conclusion, it is found that standard methods of differential geometry and quantum field theory on curved space-time are inapplicable in canonical quantum gravity, even at the level of effective equations., Comment: 17 pages, v2: further references and more detailed conclusions

We study the generation of primordial fluctuations in pure de Sitter inflation where the quantum scalar field dynamics are governed by polymer (not Schrodinger) quantization. This quantization scheme is related to, but distinct from, the structures employed in Loop Quantum Gravity; and it modifies standard results above a polymer energy scale $M_{\star}$. We recover the scale invariant Harrison Zel'dovich spectrum for modes that have wavelengths bigger than $M_{\star}^{-1}$ at the start of inflation. The primordial spectrum for modes with initial wavelengths smaller than $M_{\star}^{-1}$ exhibits oscillations superimposed on the standard result. The amplitude of these oscillations is proportional to the ratio of the inflationary Hubble parameter $H$ to the polymer energy scale. For reasonable choices of $M_{\star}$, we find that polymer effects are likely unobservable in CMB angular power spectra due to cosmic variance uncertainty, but future probes of baryon acoustic oscillations may be able to directly constrain the ratio $H/M_{\star}$., Comment: Additional comments about polymer momentum operator (see footnote 4). Version to appear in JCAP. 18 pages, 10 figures

Loop quantum gravity, a non-perturbative and manifestly background free, quantum theory of gravity implies that at the kinematical level the spatial geometry is discrete in a specific sense. The spirit of background independence also requires a non-standard quantum representation of matter. While loop quantization of standard model fields has been proposed, detail study of its implications is not yet available. This review aims to survey the various efforts and results.

We study free scalar field theory on flat spacetime using a background independent (polymer) quantization procedure. Specifically we compute the propagator using a method that takes the energy spectrum and position matrix elements of the harmonic oscillator as inputs. We obtain closed form results in the infrared and ultraviolet regimes that give Lorentz invariance violating dispersion relations, and show suppression of propagation at sufficiently high energy., Comment: 5 pages, 1 figure, clarified discussion of boundary conditions, improved figure, matches version published in PRD

It is shown that polymer quantization leads to a modified uncertainty principle similar to that obtained from string theory and non-commutative geometry. When applied to quantum field theory on general background spacetimes, corrections to the uncertainty principle acquire a metric dependence. For Friedmann-Robertson-Walker cosmology this translates to a scale factor dependence which gives a large effect in the early universe., Comment: 6 pages

We apply a type of background independent "polymer" quantization to a free scalar field in a flat spacetime. Using semi-classical states, we find an effective wave equation that is both nonlinear and Lorentz invariance violating. We solve this equation perturbatively for several cases of physical interest, and show that polymer corrections to solutions of the Klein-Gordon equation depend on the amplitude of the field. This leads to an effective dispersion relation that depends on the amplitude, frequency and shape of the wave-packet, and is hence distinct from other modified dispersion relations found in the literature. We also demonstrate that polymer effects tend to accumulate with time for plane-symmetric waveforms. We conclude by discussing the possibility of measuring deviations from the Klein-Gordon equation in particle accelerators or astrophysical observations., Comment: 15 pages, minor revision to match PRD version

We consider a polymer quantization of a free massless scalar field in a homogeneous and isotropic cosmological spacetime. This quantization method assumes that field translations are fundamentally discrete, and is related to but distinct from that used in loop quantum gravity. The semi-classical Friedman equation yields a universe that is non-singular and non-bouncing, without quantum gravity. The model has an early de Sitter-like inflationary phase with sufficient expansion to resolve the horizon and entropy problems, and a built in mechanism for a graceful exit from inflation., Comment: 4 pages, 1 figure; v2 clarifications added, reference updated

A consistent implementation of quantum gravity is expected to change the familiar notions of space, time and the propagation of matter in drastic ways. This will have consequences on very small scales, but also gives rise to correction terms in evolution equations of modes relevant for observations. In particular, the evolution of inhomogeneities in the very early universe should be affected. In this paper consistent evolution equations for gauge-invariant perturbations in the presence of inverse triad corrections of loop quantum gravity are derived. Some immediate effects are pointed out, for instance concerning conservation of power on large scales and non-adiabaticity. It is also emphasized that several critical corrections can only be seen to arise in a fully consistent treatment where the gauge freedom of canonical gravity is not fixed before implementing quantum corrections. In particular, metric modes must be allowed to be inhomogeneous: it is not consistent to assume only matter inhomogeneities on a quantum corrected homogeneous background geometry. In this way, stringent consistency conditions arise for possible quantization ambiguities which will eventually be further constrained observationally., Comment: 40 pages

We study the tensor modes of linear metric perturbations within an effective framework of loop quantum cosmology. After a review of inverse-volume and holonomy corrections in the background equations of motion, we solve the linearized tensor modes equations and extract their spectrum. Ignoring holonomy corrections, the tensor spectrum is blue tilted in the near-Planckian superinflationary regime and may be observationally disfavoured. However, in this case background dynamics is highly nonperturbative, hence the use of standard perturbative techniques may not be very reliable. On the other hand, in the quasi-classical regime the tensor index receives a small negative quantum correction, slightly enhancing the standard red tilt in slow-roll inflation. We discuss possible interpretations of this correction, which depends on the choice of semiclassical state., Comment: 12 pages; v3: typos corrected. Invited contribution to the special issue of Advanced Science Letters on "Quantum gravity, Cosmology and Black Holes"

A fully consistent linear perturbation theory for cosmology is derived in the presence of quantum corrections as they are suggested by properties of inverse volume operators in loop quantum gravity. The underlying constraints present a consistent deformation of the classical system, which shows that the discreteness in loop quantum gravity can be implemented in effective equations without spoiling space-time covariance. Nevertheless, non-trivial quantum corrections do arise in the constraint algebra. Since correction terms must appear in tightly controlled forms to avoid anomalies, detailed insights for the correct implementation of constraint operators can be gained. The procedures of this article thus provide a clear link between fundamental quantum gravity and phenomenology., Comment: 54 pages, no figures

Cosmological constant a.k.a. dark energy problem is considered to be one major challenge in modern cosmology. Here we present a model where large scale structure formation causes spatially-flat FRW universe to fragment into numerous `FRW islands' surrounded by vacuum. We show that this mechanism can explain the origin of dark energy as well as the late time cosmic acceleration. This explanation of dark energy does not require any exotic matter source nor an extremely fine-tuned cosmological constant. This explanation is given within classical general relativity and relies on the fact that our universe has been undergoing structure formation since its recent past., Comment: 4 pages, 2 figs

Cosmological tensor perturbations equations are derived for Hamiltonian cosmology based on Ashtekar's formulation of general relativity, including typical quantum gravity effects in the Hamiltonian constraint as they are expected from loop quantum gravity. This translates to corrections of the dispersion relation for gravitational waves. The main application here is the preservation of causality which is shown to be realized due to the absence of anomalies in the effective constraint algebra used., Comment: 27 pages

In contrast to scalar and tensor modes, vector modes of linear perturbations around an expanding Friedmann--Robertson--Walker universe decay. This makes them largely irrelevant for late time cosmology, assuming that all modes started out at a similar magnitude at some early stage. By now, however, bouncing models are frequently considered which exhibit a collapsing phase. Before this phase reaches a minimum size and re-expands, vector modes grow. Such modes are thus relevant for the bounce and may even signal the breakdown of perturbation theory if the growth is too strong. Here, a gauge invariant formulation of vector mode perturbations in Hamiltonian cosmology is presented. This lays out a framework for studying possible canonical quantum gravity effects, such as those of loop quantum gravity, at an effective level. As an explicit example, typical quantum corrections, namely those coming from inverse densitized triad components and holonomies, are shown to increase the growth rate of vector perturbations in the contracting phase, but only slightly. Effects at the bounce of the background geometry can, however, be much stronger., Comment: 20 pages

The leading quantum correction to Einstein-Hilbert Hamiltonian coming from large volume vacuum isotropic loop quantum cosmology, is independent of quantization ambiguity parameters. It is shown here that this correction can be viewed as finite volume gravitational Casimir energy due to one-loop `graviton' contributions. In vacuum case sub-leading quantum corrections and in non-vacuum case even leading quantum correction depend on ambiguity parameters. It may be recalled that these are in fact analogous features of perturbative quantum gravity where it is well-known that pure gravity (on-shell) is one-loop finite whereas higher-loops contributions are not even renormalizable. These features of the quantum corrections coming from non-perturbative quantization, sheds a new light on a major open issue; how to communicate between non-perturbative and perturbative approaches of quantum gravity., Comment: revtex4, 4 pages

In isotropic loop quantum cosmology, non-perturbatively modified dynamics of a minimally coupled scalar field violates weak, strong and dominant energy conditions when they are stated in terms of equation of state parameter. The violation of strong energy condition helps to have non-singular evolution by evading singularity theorems thus leading to a generic inflationary phase. However, the violation of weak and dominant energy conditions raises concern, as in general relativity these conditions ensure causality of the system and stability of vacuum via Hawking-Ellis conservation theorem. It is shown here that the non-perturbatively modified kinetic term contributes negative pressure but positive energy density. This crucial feature leads to violation of energy conditions but ensures positivity of energy density, as scalar matter Hamiltonian remains bounded from below. It is also shown that the modified dynamics restricts group velocity for inhomogeneous modes to remain sub-luminal thus ensuring causal propagation across spatial distances., Comment: 29 pages, revtex4; few clarifications, references added, to appear in CQG

It is widely believed that quantum field fluctuation in an inflating background creates the primeval seed perturbation which through subsequent evolution leads to the observed large scale structure of the universe. The standard inflationary scenario produces scale invariant power spectrum quite generically but it produces, unless fine tuned, too large amplitude for the primordial density perturbation than observed. Using similar techniques it is shown that loop quantum cosmology induced inflationary scenario can produce scale invariant power spectrum as well as small amplitude for the primordial density perturbation without fine tuning. Further its power spectrum has a qualitatively distinct feature which is in principle falsifiable by observation and can distinguish it from the standard inflationary scenario., Comment: 33 pages, revtex4, few typos and a confusing notation changed; several clarifications, references added, to appear in CQG

The absence of isotropic singularity in loop quantum cosmology can be understood in an effective classical description as the universe exhibiting a Big Bounce. We show that with scalar matter field, the big bounce is generic in the sense that it is independent of quantization ambiguities and details of scalar field dynamics. The volume of the universe at the bounce point is parametrized by a single parameter. It provides a minimum length scale which serves as a cut-off for computations of density perturbations thereby influencing their amplitudes., Comment: 4 pages, revtex4, two figures. In version 2, only e-print archive numbers are corrected. In version 3 a Reference is added. Final version to appear in Phys. Rev. Lett. Some explanations improved

For a class of solutions of the fundamental difference equation of isotropic loop quantum cosmology, the difference equation can be replaced by a differential equation valid for {\em all} values of the triad variable. The differential equation admits a `unique' non-singular continuation through vanishing triad. A WKB approximation for the solutions leads to an effective continuum Hamiltonian. The effective dynamics is also non-singular (no big bang singularity) and approximates the classical dynamics for large volumes. The effective evolution is thus a more reliable model for further phenomenological implications of the small volume effects., Comment: 18 pages, revtex4, one figure. In version 2, only e-print archive numbers are corrected. In this final version minor typos are corrected and an extra para of explanation added in the discussion. To appear in Class. Quantum Grav

Non-perturbative corrections from loop quantum cosmology (LQC) to the scalar matter sector is already known to imply inflation. We prove that the LQC modified scalar field generates exponential inflation in the small scale factor regime, for all positive definite potentials, independent of initial conditions and independent of ambiguity parameters. For positive semi-definite potentials it is always possible to choose, without fine tuning, a value of one of the ambiguity parameters such that exponential inflation results, provided zeros of the potential are approached at most as a power law in the scale factor. In conjunction with generic occurrence of bounce at small volumes, particle horizon is absent thus eliminating the horizon problem of the standard Big Bang model., Comment: 4 pages, revtex4, one figure. Only e-print archive numbers correctedi in the second version. Reference added in the 3rd version. Final version to appear in Phys. Rev. Lett. Explanations improved

The Bianchi IX model has been used often to investigate the structure close to singularities of general relativity. Its classical chaos is expected to have, via the BKL scenario, implications even for the approach to general inhomogeneous singularities. Thus, it is a popular model to test consequences of modifications to general relativity suggested by quantum theories of gravity. This paper presents a detailed proof that modifications coming from loop quantum gravity lead to a non-chaotic effective behavior. The way this is realized, independently of quantization ambiguities, suggests a new look at initial and final singularities., Comment: 41 pages, 3 figures, revtex4

We present a construction of the Hubble operator for the spatially flat isotropic loop quantum cosmology. This operator is a Dirac observable on a subspace of the space of physical solutions. This subspace gets selected dynamically, requiring that its action be invariant on the physical solution space. As a simple illustrative application of the expectation value of the operator, we do find a generic phase of (super)inflation, a feature shown by Bojowald from the analysis of effective Friedmann equation of loop quantum cosmology., Comment: 20 pages, 3 eps figures, few comments and clarifications added to match with the published version

Assuming a flat Friedmann-Robertson-Walker cosmology with a single perfect fluid, we propose a pressure-density ratio that evolves as a specific universal function of the scale parameter. We show that such a ratio can indeed be consistent with several observational constraints including those pertaining to late-time accelerated expansion. Generic dynamical scalar field models of Dark energy (with quadratic kinetic terms in their Lagrange density) are shown to be in accord with the proposed equation-of-state ratio, provided the current matter density parameter $\Omega_{m0} < 0.23$ - a value {\it not} in agreement with recent measurements., Comment: 5 pages Revtex, 2 eps figures, a reference and a comment added