External Gravitational Field Research Articles

Gravo-thermodynamics of the intracluster medium: Negative heat capacity and dilation of cooling time scales

The time scale for cooling of the gravitationally bound gaseous intracluster medium (ICM) is not determined by radiative processes alone. If the ICM is in quasi-hydrostatic equilibrium in the fixed gravitational field of the dark matter halo then energy losses incurred by the gravitational potential energy of the gas should also be taken into account. This “gravitational heating” has been known for a while using explicit solutions to the equations of motion. Here, we re-visit this effect by applying the virial theorem to gas in quasi-hydrostatic equilibrium in an external gravitational field, neglecting the gravity of the gas. For a standard NFW form of halo profiles and for a finite gas density, the response of the gas temperature to changes in the total energy is significantly delayed. The effective cooling time could be prolonged by more than an order of magnitude inside the scale radius (rs) of the halo. Gas lying at a distance twice the scale radius, has negative heat capacity so that the temperature increases as a result of energy losses. Although external heating (e.g. by AGN activity) is still required to explain the lack of cool ICM near the center, the analysis here may circumvent the need for heating in farther out regions where the effective cooling time could be prolonged to become larger than the cluster age and also explains the increase of temperature with radius in these regions.

Infinite slabs and other weird plane symmetric space–times with constant positive density

We present the exact solution of Einstein’s equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below z = 0. This solution depends essentially on two constants: the density ρ and a parameter κ. We show that these space–times finish down below at an inner singularity at finite depth. We show that for κ ≥ 0.3513 . . . the dominant energy condition is satisfied all over the space–time. We match this solution to the vacuum one and compute the external gravitational field in terms of slab’s parameters. Depending on the value of κ, these slabs can be attractive, repulsive or neutral. In the first case, the space–time also finishes up above at an empty repelling singular boundary. In the other cases, they turn out to be semi-infinite and asymptotically flat when z → ∞. We also find solutions consisting of joining an attractive slab and a repulsive one, and two neutral ones. We also discuss how to assemble a “gravitational capacitor” by inserting a slice of vacuum between two such slabs.

The Rayleigh–Taylor instability and internal waves in quantum plasmas

Influence of quantum effects on the internal waves and the Rayleigh–Taylor instability in plasma is investigated. It is shown that quantum pressure always stabilizes the RT instability. The problem is solved both in the limit of short-wavelength perturbations and exactly for density profiles with layers of exponential stratification. In the case of stable stratification, quantum pressure modifies the dispersion relation of the inertial waves. Because of the quantum effects, the internal waves may propagate in the transverse direction, which was impossible in the classical case. A specific form of pure quantum internal waves is obtained, which do not require any external gravitational field.

Spin–gravity coupling and gravity-induced quantum phases

External gravitational fields induce phase factors in the wave functions of particles. The phases are exact to first order in the background gravitational field, are manifestly covariant and gauge invariant and provide a useful tool for the study of spin–gravity coupling and of the optics of particles in gravitational or inertial fields. We discuss the role that spin–gravity coupling plays in particular problems.

Anomalies of the external and internal gravitational fields of the Earth’s isostatically balanced crust in the quadratic approximation

Formulas that include the contribution from dipole-distributed anomalous masses represented as layers distributed in height relative to the reference ellipsoid to the gravitational field in the quadratic approximation have been derived. The relationships between the expansion coefficients of some function and its square in terms of spherical functions have been established. The contribution from the relief masses and the density jump at the Mohorovicic discontinuity is used to illustrate the results.

Anomalies of an external and internal gravitational field of upper Earth layers in square law approximation

The explicit form for the quadratic contribution to the gravitational field from the dipole distributed anomalous masses is found. The anomalous masses are represented in the form of layers of variable height, arranged relative to the reference ellipsoid. The solution is reduced to the mathematical problem of finding an expression for the coefficients of expansion in terms of spherical harmonics of the square of any function that can be presented as a finite series of spherical harmonics, in terms of the coefficients of this initial series. The formulas have been calculated using mathematical modelling of symbol computation using computer algebra packages. The results obtained are illustrated using the example of the contribution from relief masses and density jumps on the Mohorovicic (Moho) discontinuity.

Energy-level shifts of a stationary hydrogen atom in a static external gravitational field with Schwarzschild geometry

The first order perturbations of the energy levels of a stationary hydrogen atom in a static external gravitational field, with Schwarzschild metric, are investigated. The energy shifts are calculated for the relativistic $1S$, $2S$, $2P$, $3S$, $3P$, $3D$, $4S$, $4P$, $4D$, and $4F$ levels. The results show that the energy-level shifts of the states with total angular momentum quantum number $1/2$ are all zero, and the ratio of absolute energy shifts with total angular momentum quantum number $5/2$ is $1\ensuremath{\mathbin:}4\ensuremath{\mathbin:}5$. This feature can be used to help us to distinguish the gravitational effect from other effects.

Deviation of the Earth’s crust and upper mantle from isostatic equilibrium

The isostatically unbalanced crust and upper mantle structures conforming to the external gravitational field and minimizing deviation of the internal gravitational field from the isostatic equilibrium field, are determined. A dipole-type distribution of these structures associated with the motion of anomalous masses toward the potential energy minimum is revealed. On the basis of seismic data, it is inferred that the compensation of isostatically unbalanced masses is possible at the core-mantle boundary.

Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions

Conformally invariant systems involving only dimensionless parameters are known to describe particle physics at very high energy. In the presence of an external gravitational field, the conformal symmetry may generalize to the Weyl invariance of classical massless field systems in interaction with gravity. In the quantum theory, the latter symmetry no longer survives: A Weyl anomaly appears. Anomalies are a cornerstone of quantum field theory, and, for the first time, a general, purely algebraic understanding of the universal structure of the Weyl anomalies is obtained, in arbitrary dimensions and independently of any regularization scheme.

Escaping from modified Newtonian dynamics

Abstract We present a new test of modified Newtonian dynamics (MOND) on galactic scales, based on the escape speed in the solar neighbourhood. This test is independent from other empirical successes of MOND at reproducing the phenomenology of galactic rotation curves. The galactic escape speed in MOND is entirely determined by the baryonic content of the Galaxy and the external field in which it is embedded. We estimate that the external field in which the Milky Way must be embedded to produce the observed local escape speed of ∼550 km s−1 is of the order of a0/100, where a0 is the dividing acceleration scale below which gravity is boosted in MOND. This is compatible with the external gravitational field actually acting on the Milky Way.

PHOENIX: MHD spectral code for rotating laboratory and gravitating astrophysical plasmas

The new PHOENIX code is discussed together with a sample of many new results that are obtained concerning magnetohydrodynamic (MHD) spectra of axisymmetric plasmas where flow and gravity are consistently taken into account. PHOENIX, developed from the CASTOR code [W. Kerner, J.P. Goedbloed, G.T.A. Huysmans, S. Poedts, E. Schwarz, J. Comput. Phys. 142 (1998) 271], incorporates purely toroidal, or both toroidal and poloidal flow and external gravitational fields to compute the entire ideal or resistive MHD spectrum for general tokamak or accretion disk configurations. These equilibria are computed by means of FINESSE [A.J.C. Beliën, M.A. Botchev, J.P. Goedbloed, B. van der Holst, R. Keppens, J. Comp. Physics 182 (2002) 91], which discriminates between the different elliptic flow regimes that may occur. PHOENIX makes use of a finite element method in combination with a spectral method for the discretization. This leads to a large generalized eigenvalue problem, which is solved by means of Jacobi–Davidson algorithm [G.L.G. Sleijpen, H.A. van der Vorst, SIAM J. Matrix Anal. Appl. 17 (1996) 401]. PHOENIX is compared with CASTOR, PEST-1 and ERATO for an internal mode of Soloviev equilibria. Furthermore, the resistive internal kink mode has been computed to demonstrate that the code can accurately handle small values for the resistivity. A new reference test case for a Soloviev-like equilibrium with toroidal flow shows that, on a particular unstable mode, the flow has a quantifiable stabilizing effect regardless of the direction of the flow. PHOENIX reproduces the Toroidal Flow induced Alfvén Eigenmode (TFAE, [B. van der Holst, A.J.C. Beliën, J.P. Goedbloed, Phys. Rev. Lett. 84 (2000) 2865]) where finite resistivity in combination with equilibrium flow effects causes resonant damping. Localized ideal gap modes are presented for tokamak plasmas with toroidal and poloidal flow. Finally, we demonstrate the ability to spectrally diagnose magnetized accretion disk equilibria where gravity acts together with either purely toroidal flow or both toroidal and poloidal flow. These cases show that the MHD continua can be unstable or overstable due to the presence of a gravitational field together with equilibrium flow-driven dynamics [J.P. Goedbloed, A.J.C. Beliën, B. van der Holst, R. Keppens, Phys. Plasmas 11 (2004) 28].

Effective dynamics for boson stars

We study solutions close to solitary waves of the pseudo-relativistic Hartree equation describing boson stars under the influence of an external gravitational field. In particular, we analyse the long-time effective dynamics of such solutions. In essence, we establish a (long-time) stability result for solutions describing boson stars that move under the influence of an external gravitational field. The proof of our main result tackles difficulties that are absent when deriving similar results on effective solitary wave motions for nonlinear Schrödinger equations or nonlinear wave equations. This is due to the fact that the pseudo-relativisitic Hartree equation does not exhibit Galilean or Lorentz covariance.

Fine vertical structure of the galactic gaseous disk

The three-dimensional evolution of an ensemble of N particles (N = 8 × 105) in the external gravitational field of a galaxy perturbed by a spiral density wave is considered. The particles simulate clouds of interstellar gas, and inelastic two-body collisions between them are taken into account. The three-dimensional structure of the gaseous galactic layer and the vertical profile of the spiral arms are computed. It is shown that: (1) the local thickness of the gaseous galactic disk has a minimum where the volume gas density has a maximum (the maximum density of the interstellar medium is shifted downstream relative to the galactic shock front), (2) the configuration of the vertical profile of the spiral arms changes radically when the corotation region is crossed. Our first result explains the negative correlation between the thickness of the gas layer and the density derived from neutral-hydrogen observations. The second result can be used in the next generation of neutral-hydrogen observations to localize the corotation radius in the Galaxy.

Binary black hole initial data from matched asymptotic expansions

We present an approximate metric for a binary black-hole spacetime to construct initial data for numerical relativity. This metric is obtained by asymptotically matching a post-Newtonian metric for a binary system to a perturbed Schwarzschild metric for each hole. In the inner zone near each hole, the metric is given by the Schwarzschild solution plus a quadrupolar perturbation corresponding to an external tidal gravitational field. In the near zone, well outside each black hole but less than a reduced wavelength from the center of mass of the binary, the metric is given by a post-Newtonian expansion including the lowest-order deviations from flat spacetime. When the near zone overlaps each inner zone in a buffer zone, the post-Newtonian and perturbed Schwarzschild metrics can be asymptotically matched to each other. By demanding matching (over a 4-volume in the buffer zone) rather than patching (choosing a particular 2-surface in the buffer zone), we guarantee that the errors are small in all zones. The resulting piecewise metric is made formally C{sup {infinity}} with smooth transition functions so as to obtain the finite extrinsic curvature of a 3-slice. In addition to the metric and extrinsic curvature, we present explicit results for the lapse and themore » shift, which can be used as initial data for numerical simulations. This initial data is not accurate all the way to the asymptotically flat ends inside each hole, and therefore must be used with evolution codes which employ black hole excision rather than puncture methods. This paper lays the foundations of a method that can be straightforwardly iterated to obtain initial data to higher perturbative order.« less

On the phenomenological three-graviton vertex

In General Relativity, the graviton interacts in three-graviton vertex with a tensor that is not the energy-momentum tensor of the gravitational field. We consider the possibility that the graviton interacts with the definite gravitational energy-momentum tensor that we previously found in the G 2 approximation. This tensor in a gauge, where nonphysical degrees of freedom do not contribute, is remarkable, because it gives positive gravitational energy density for the Newtonian center in the same manner as the electromagnetic energy-momentum tensor does for the Coulomb center. We show that the assumed three-graviton vertex does not lead to contradiction with the precession of Mercury’s perihelion. In the S-matrix approach used here, the external gravitational field has only a subsidiary role, similar to the external field in quantum electrodynamics. This approach with the assumed vertex leads to the gravitational field that cannot be obtained from a consistent gravity equation.

Overlap Lattice Fermion in a Gravitational Field

We construct a lattice Dirac operator of overlap type that describes the propagation of a Dirac fermion in an external gravitational field. The local Lorentz symmetry is manifestly realized as a lattice gauge symmetry, while it is believed that the general coordinate invariance is restored only in the continuum limit. Our doubler-free Dirac operator satisfies the conventional Ginsparg-Wilson relation and possesses gamma_5 hermiticity with respect to the inner product, which is suggested by the general coordinate invariance. The lattice index theorem in the presence of a gravitational field holds, and the classical continuum limit of the index density reproduces the Dirac genus. Reduction to a single Majorana fermion is possible for 8k+2 and 8k+4 dimensions, but not for 8k dimensions, which is consistent with the existence of the global gravitational/gauge anomalies in 8k dimensions. Other Lorentz representations, such as the spinor-vector and the bi-spinor representations, can also be treated. Matter fields with a definite chirality (with respect to the lattice-modified chiral matrix) are briefly considered.

Simulation of the growth of nanoparticle aggregates reproducing their natural structure in disperse systems

A three-dimensional continuum model of the generation of aggregates of spherical particles is developed that allows us to reproduce natural conditions of structure formation in real disperse systems with an arbitrary viscosity of a dispersion medium. The exchange between translational and rotational degrees of freedom for aggregates, subaggregates, and single particles at an arbitrary interparticle interaction potential, as well as at the preset functions of the distribution over particle sizes and thicknesses of the adsorption layer, is taken into account in the model. The interaction potential of the particles includes electrostatic (among them, screened), elastic, and van der Waals interactions and allows for the effect of external electric, magnetic, and gravitational fields. Structural and statistical characteristics of aggregates formed at different stages of particle coagulation are studied, and the quasi-ordering of single subaggregates in the elastic adsorption layer of particles is revealed. The model can be applied to the study of optical absorption by large aggregates of nanoparticles with natural structure and the kinetics of sol aggregation as functions of the properties of the adsorption layer of particles and the action of external physical factors.

Weakly perturbed Schwarzschild lens in the strong deflection limit

We investigate the strong deflection limit of gravitational lensing by a Schwarzschild black hole embedded in an external gravitational field. The study of this model, analogous to the Chang & Refsdal lens in the weak deflection limit, is important to evaluate the gravitational perturbations on the relativistic images that appear in proximity of supermassive black holes hosted in galactic centers. By a simple dimensional argument, we prove that the tidal effect on the light ray propagation mainly occurs in the weak field region far away from the black hole and that the external perturbation can be treated as a weak field quadrupole term. We provide a description of relativistic critical curves and caustics and discuss the inversion of the lens mapping. Relativistic caustics are shifted and acquire a finite diamond shape. Sources inside the caustics produce four sequences of relativistic images. On the other hand, retro-lensing caustics are only shifted while remaining point-like to the lowest order.

Tidal coupling of a Schwarzschild black hole and circularly orbiting moon

We describe the possibility of using the laser interferometer space antenna (LISA) 's gravitational-wave observations to study, with high precision, the response of a massive central body (e.g. a black hole or a soliton star) to the tidal gravitational pull of an orbiting, compact, small-mass object (a white dwarf, neutron star, or small-mass black hole). Motivated by this LISA application, we use first-order perturbation theory to study tidal coupling for a special, idealized case: a Schwarzschild black hole of mass $M$, tidally perturbed by a ``moon'' with mass $\ensuremath{\mu}\ensuremath{\ll}M$ in a circular orbit at a radius $b\ensuremath{\gg}M$ with orbital angular velocity $\ensuremath{\Omega}$. We investigate the details of how the tidal deformation of the hole gives rise to an induced quadrupole moment ${\mathcal{I}}_{ij}$ in the hole's external gravitational field at large radii, including the vicinity of the moon. In the limit that the moon is static, we find, in Schwarzschild coordinates and Regge-Wheeler gauge, the surprising result that there is no induced quadrupole moment. We show that this conclusion is gauge dependent and that the static, induced quadrupole moment for a black hole is inherently ambiguous, and we contrast this with an earlier result of Suen, which gave, in a very different gauge, a nonzero static induced quadrupole moment with a sign opposite to what one would get for a fluid central body. For the orbiting moon and the central Schwarzschild hole, we find (in agreement with a recent result of Poisson) a time-varying induced quadrupole moment that is proportional to the time derivative of the moon's tidal field, ${\mathcal{I}}_{ij}=(32/45){M}^{6}{\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\mathcal{E}}}_{ij}$ and that therefore is out of phase with the tidal field by a spatial angle $\ensuremath{\pi}/4$ and by a temporal phase shift $\ensuremath{\pi}/2$. This induced quadrupole moment produces a gravitational force on the moon that reduces its orbital energy and angular momentum at the same rate as the moon's tidal field sends energy and angular momentum into the hole's horizon. As a partial analog of a result derived long ago by Hartle for a spinning hole and a static distant companion, we show that the orbiting moon's tidal field induces a tidal bulge on the hole's horizon, and that the rate of change of the horizon shape (i.e. the horizon shear) leads the perturbing tidal field at the horizon by an angle $4M\ensuremath{\Omega}$. We prefer to avoid introducing an ingoing null geodesic, as Hartle did in his definition of the phase shift, because the moon is in the central body's near zone ($b\ensuremath{\ll}1/\ensuremath{\Omega}$) and thus should interact with the horizon instantaneously, not causally. We discuss the implications of these results for LISA's future observations of tidal coupling, including the inappropriateness of using the concepts of tidal polarizability and tidal lag or lead angle, for the massive central body, when discussing LISA's observations.

External Gravitational Field of a Nonstatic Spherically Symmetric Body in the Inertial Frame

We show that the external gravitational field of a nonstatic spherically symmetric source described by a diagonal metric tensor can only be static in the field theory of gravity.