Weak Field Expansion Research Articles

Second order Kerr deflection

We perform a full second order calculation of the deflection of light along the equatorial plane in the Kerr metric. Previous Kerr deflection calculations were interested in obtaining the correction due to rotation to the Einstein deflection. By expanding to first order in the rotational parameter a, they obtain the Einstein deflection of 4M/r o and the second order deflection due to rotation of $${4Ma/r_{\rm o}^2}$$ , (where r o is the point of closest approach). In this paper, we are interested in going beyond the rotational contribution for the purpose of astrophysical applications. We therefore keep all terms up to second order in our final weak field expansion. Besides the rotational contribution, we also obtain an extra second order term of $${7.78\,M^2/r_{\rm o}^2}$$ . Since M > a, this extra term is greater than the rotational contribution of $${4Ma/r_{\rm o}^2}$$ in astrophysical applications. When a/M is close to unity the terms are of the same order of magnitude.

Large-order perturbation theory and de Sitter/anti–de Sitter effective actions

We analyze the large-order behavior of the perturbative weak-field expansion of the effective Lagrangian density of a massive scalar in de Sitter and anti--de Sitter space, and show that this perturbative information is not sufficient to describe the nonperturbative behavior of these theories, in contrast to the analogous situation for the Euler-Heisenberg effective Lagrangian density for charged scalars in constant electric and magnetic background fields. For example, in even-dimensional de Sitter space there is particle production, but the effective Lagrangian density is nevertheless real, even though its weak-field expansion is a divergent nonalternating series whose formal imaginary part corresponds to the correct particle production rate. This apparent puzzle is resolved by considering the full nonperturbative structure of the relevant Feynman propagators, and cannot be resolved solely from the perturbative expansion.

Covariant long-distance modifications of Einstein theory and strong coupling problem

We present generic form of the covariant nonlocal action for infrared modifications of Einstein theory recently suggested within the weak-field curvature expansion. In the lowest order it is determined by two nonlocal operators -- kernels of Ricci tensor and scalar quadratic forms. In models with a low strong-coupling scale this action also incorporates the strongly coupled mode which cannot be perturbatively integrated out in terms of the metric field. This mode enters the action as a Lagrange multiplier for the constraint on metric variables which reduces to the curvature scalar and enforces the latter to vanish on shell. Generic structure of the action is demonstrated on the examples of the Fierz-Pauli and Dvali-Gabadadze-Porrati models and their extensions. The gauge-dependence status of the strong-coupling and VDVZ problems is briefly discussed along with the manifestly gauge-invariant formalism of handling the braneworld gravitational models at classical and quantum levels.

Weakly coupled metastable graviton

A graviton of a nonzero mass and decay width propagates five physical polarizations. The question of interactions of these polarizations is crucial for viability of models of massive/metastable gravity. This question is addressed in the context of the DGP model of a metastable graviton. First, we argue that the well-known breakdown of a naive perturbative expansion at a low scale is an artifact of the weak-field expansion itself. Then, we propose a different expansion---the constrained perturbation theory---in which the breakdown does not occur and the theory is perturbatively tractable all the way up to its natural ultraviolet cutoff. In this approach the couplings of the extra polarizations to matter and their self-couplings appear to be suppressed and should be neglected in measurements at subhorizon scales. The model reproduces results of General Relativity at observable distances with high accuracy, while predicting deviations from them at the present day horizon scale.

NONCOMMUTATIVE FIELD THEORY AND THE DYNAMICS OF QUANTUM HALL FLUIDS

We study the spectrum of density fluctuations of fractional Hall fluids in the context of the noncommutative hydrodynamical model of Susskind. We show that, within the weak-field expansion, the leading correction to the noncommutative Chern–Simons Lagrangian (a Maxwell term in the effective action), destroys the incompressibility of the Hall fluid due to strong UV/IR effects at one loop. We speculate on possible relations of this instability with the transition to the Wigner crystal, and conclude that calculations within the weak-field expansion must be carried out with an explicit ultraviolet cutoff at the noncommutativity scale. We point out that the noncommutative dipoles exactly match the spatial structure of the Halperin–Kallin quasiexcitons. Therefore, we propose that the noncommutative formalism must describe accurately the spectrum at very large momenta, provided no weak-field approximations are made. We further conjecture that the noncommutative open Wilson lines are "vertex operators" for the quasiexcitons.

Two-loop self-dual Euler-Heisenberg Lagrangians (II): Imaginary part and Borel analysis

We analyze the structure of the imaginary part of the two-loop Euler-Heisenberg QED effective Lagrangian for a constant self-dual background. The novel feature of the two-loop result, compared to one-loop, is that the prefactor of each exponential (instanton) term in the imaginary part has itself an asymptotic expansion. We also perform a high-precision test of Borel summation techniques applied to the weak-field expansion, and find that the Borel dispersion relations reproduce the full prefactor of the leading imaginary contribution.

The Physical Content of the Leading Orders of Principal Component Analysis of Spectral Profiles

We consider the principal component analysis (PCA) method of expanding Stokes intensity and net polarization profiles in terms of eigenfunctions (or principal components) of the spectral covariance matrix. The expansion is ordered by the magnitude of the relevant eigenvalue from largest to smallest. We find that the ordering represents a perturbation expansion. This allows us to examine the physical content of the first few orders of the basis set for 40,000 profiles for each Stokes parameter for a solar active region. For the intensity profile, we find that the expansion represents a Taylor series with the highest ranked, or first, eigenfunction being the zeroth order term, the second as the (scaled) first derivative of the zeroth term, and the third as the (scaled) second derivative term. Thus, we can derive a velocity from the coefficients of the first derivative term and a magnetic splitting parameter from those of the second using the standard velocity perturbation and weak-field expansion. For the net polarization profiles, we find that the zeroth order terms yield, using the weak-field expansion, the vector magnetic field. A comparison with a Stokes profile fitting inversion shows that the thus-estimated velocity and magnetic parameters are in good agreement with the more time-consuming profile fitting values, but do show a roll-off, or saturation, for sufficiently large values. We also find that the bright quiet-Sun points have an upflow signature, while the dark regions have a downflow—one in good agreement with that derived by traditional analysis.

Weak-field expansion for processes in a homogeneous background magnetic field

The weak-field expansion of the charged fermion propagator under a uniform magnetic field is studied. Starting from Schwinger's proper-time representation, we express the charged fermion propagator as an infinite series corresponding to different Landau levels. This infinite series is then reorganized according to the powers of the external field strength $B$. For illustration, we apply this expansion to $\gamma\to \nu\bar{\nu}$ and $\nu\to \nu\gamma$ decays, which involve charged fermions in the internal loop. The leading and subleading magnetic-field effects to the above processes are computed.

A Classical Theory of Coronal Emission Line Polarization

We present a classical theory of formation for polarized, magnetic dipole emission lines in the magnetized solar corona. Because of the small Einstein A-coefficients of forbidden lines and the expected magnetic field strengths in the corona, coherences between different magnetic substates can be neglected, so the observed Stokes vector for these lines is the result of the incoherent superposition of the Stokes vectors emitted in the de-excitation of the individual magnetic substates of the transition's upper level. Using classical electrodynamics and a weak-field expansion formalism, we could derive the main polarization properties of the transition J = 1 to J = 0, in the collisionless regime. In particular, we derived the correct amount of atomic alignment in the upper level, induced by the anisotropic, unpolarized illumination from the photosphere, and the dependence of Stokes Q and U linear polarization on the magnetic field direction in the plane of the sky. The influence of atomic alignment on the V profile is also correctly reproduced. This work provides a classical interpretation of the physical process that generates atomic alignment in the radiating ion and how the associated Van Vleck effect in resonance-scattering linear polarization and the alignment contribution to Zeeman effect circular polarization come about.

LETTER: Some Remarks on the Neutrino Oscillation Phase in a Gravitational Field

The weak gravitational field expansion method to account for the gravitationally induced neutrino oscillation effect is critically examined. It is shown that the splitting of the neutrino phase into a ``kinematic'' and a ``gravitational'' phase is not always possible because the relativistic factor modifies the particle interference phase splitting condition in a gravitational field.

WEAK FIELD EXPANSION OF GRAVITY: GRAPHS, MATRICES AND TOPOLOGY

We present some approaches to the perturbative analysis of the classical and quantum gravity. First we introduce a graphical representation for a global SO (n) tensor (∂)dhαβ, which generally appears in the weak field expansion around the flat space: gμν=δμν+hμν. Making use of this representation, we explain (1) Generating function of graphs (Feynman diagram approach), (2) Adjacency matrix (Matrix approach), (3) Graphical classification in terms of "topology indices" (Topology approach), (4) The Young tableau (Symmetric group approach). We systematically construct the global SO (n) invariants. How to show the independence and completeness of those invariants is the main theme. We explain it taking simple examples of ∂∂h-, and (∂∂h)2-invariants in the text. The results are applied to the analysis of the independence of general invariants and (the leading order of) the Weyl anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10 dimensions).

On the measure in simplicial gravity

Functional measures for lattice quantum gravity should agree with their continuum counterparts in the weak field, low momentum limit. After showing that the standard simplicial measure satisfies the above requirement, we prove that a class of recently proposed non-local measures for lattice gravity do not satisfy such a criterion, already to lowest order in the weak field expansion. We argue therefore that the latter cannot represent acceptable discrete functional measures for simplicial geometries.

Graphical classification of global SO(n) invariants and independent general invariants

This paper treats some basic points in general relativity and in its perturbative analysis. Firstly a systematic classification of global SO(n) invariants, which appear in the weak-field expansion of n-dimensional gravitational theories, is presented. Through the analysis, we explain the following points: (a) a graphical representation is introduced to express invariants clearly; (b) every graph of invariants is specified by a set of indices; (c) a number, called weight, is assigned to each invariant. It expresses the symmetry with respect to the suffix-permutation within an invariant. Interesting relations among the weights of invariants are given. Those relations show the consistency and the completeness of the present classification; (d) some reduction procedures are introduced in graphs for the purpose of classifying them. Secondly the above result is applied to the proof of the independence of general invariants with the mass-dimension M6 for the general geometry in a general space dimension. We take a graphical representation for general invariants too. Finally all relations depending on each space-dimension are systematically obtained for 2, 4, and 6 dimensions.

Gauge invariance in simplicial gravity

The issue of local gauge invariance in the simplicial lattice formulation of gravity is examined. We exhibit explicitly, both in the weak-field expansion about flat space, and subsequently for arbitrarily triangulated background manifolds, the exact local gauge invariance of the gravitational action, which includes in general both cosmological constant and curvature-squared terms. We show that the local invariance of the discrete action and the ensuing zero-modes correspond precisely to the diffeomorphism invariance in the continuum, by carefully relating the fundamental variables in the discrete theory (the edge lengths) to the induced metric components in the continuum. We discuss mostly the two-dimensional case, but argue that our results have general validity. The previous analysis is then extended to the coupling with a scalar field, and the invariance properties of the scalar field action under lattice diffeomorphisms are exhibited. The construction of the lattice conformal gauge is then described, as well as the separation of lattice metric perturbations into orthogonal conformal and diffeomorphism part. The local gauge invariance properties of the lattice action show that no Faddeev-Popov determinant is required in the gravitational measure, unless lattice perturbation theory is performed with a gauge-fixed action, such as the one arising in the lattice analog of the conformal or harmonic gauges.

The Post-Newtonian Limit of Dilaton Gravity

We study the post-Newtonian limit of a generalized dilaton gravity in which gravity is coupled to dilaton and eletromagnetic fields. The field equations are derived using the post-Newtonian scheme, and the approximate solution is presented for a point mass with electric and dilaton charges. The result indicates that the dilaton effect can be detected, in post-Newtonian level, using a charged test particle but not a neutral one. We have also checked that the approximate solution is indeed consistent with the weak field expansion of charged dilaton black hole solution in the harmonic coordinate. E-mail address: rrhsu@mail.ncku.edu.tw E-mail address: phycrl@ccunix.ccu.edu.tw 1

Galaxy formation just after the recombination period — a supplement

The basic equation in an approach of galaxy formation proposed by the author in a recent paper is derived from a consistent weak-field expansion of the field equations and the equation of motion.

Electromagnetic fields in a thermal background

The one-loop effective action for a slowly varying electromagnetic field is computed at finite temperature and density using a real-time formalism. We discuss the gauge invariance of the result. Corrections to the Debye mass from an electric field are computed at high temperature and high density. The effective coupling constant, defined from a purely electric weak-field expansion, behaves at high temperature very differently from the case of a magnetic field, and does not satisfy the renormalization group equation. The issue of pair production in the real-time formalism is discussed and also its relevance for heavy-ion collisions.

Newtonian potential in quantum Regge gravity

We show how the Newtonian potential between two heavy masses can be computed in simplicial quantum gravity. On the lattice we compute correlations between Wilson lines associated with the heavy particles and which are closed by the lattice periodicity. We check that the continuum analog of this quantity reproduces the Newtonian potential in the weak field expansion. In the smooth anti-de Sitter-like phase, which is the only phase where a sensible lattice continuum limit can be constructed in this model, we attempt to determine the shape and mass dependence of the attractive potential close to the critical point in G. It is found that non-linear graviton interactions give rise to a potential which is Yukawa-like, with a mass parameter that decreases towards the critical point where the average curvature vanishes. In the vicinity of the critical point we give an estimate for the effective Newton constant.

GRAVITY AND YANG-MILLS THEORY: TWO FACES OF THE SAME THEORY?

We introduce a gauge and diffeomorphism invariant theory on the Yang-Mills phase space. The theory is well defined for an arbitrary gauge group with an invariant bilinear form, it contains only first class constraints, and the spacetime metric has a simple form in terms of the phase space variables. With gauge group SO (3, C), the theory equals the Ashtekar formulation of gravity with a cosmological constant. For Lorentzian signature, the theory is complex, and we have not found any good reality conditions. In the Euclidean signature case, everything is real. In a weak field expansion around de Sitter spacetime, the theory is shown to give the conventional Yang-Mills theory to the lowest order in the fields. We show that the coupling to a Higgs scalar is straightforward, while the naive spinor coupling does not work. We have not found any way of including spinors that gives a closed constraint algebra. For gauge group U(2), we find a static and spherically symmetric solution.

Towards a unification of gravity and Yang-Mills theory.

We introduce a gauge and diffeomorphism invariant theory on the Yang-Mills phase space. The theory is well defined for an arbitrary gauge group whose Lie algebra admits a nondegenerate invariant bilinear form, and it contains only first class constraints. With gauge group $\mathrm{SO}(3,C)$, the theory equals the Ashtekar formulation of gravity with a cosmological constant. For Lorentzian signature, the theory is complex, and we have not found any good reality conditions. In the Euclidean signature case, everything is real. By using gauge group $\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}{G}^{\mathrm{YM}}$ and doing a weak field expansion around de Sitter spacetime, the theory is shown to give the conventional Yang-Mills theory to the lowest order in the fields.