Newtonian Gravitational Theory Research Articles

Topics in the foundations of general relativity and Newtonian gravitation theory

In in the Foundations of General Relativity and Newtonian Gravitation Theory, David B. Malament presents the basic logical-mathematical structure of general relativity and considers a number of special topics concerning the foundations of general relativity and its relation to Newtonian gravitation theory. These special topics include the geometrized formulation of Newtonian theory (also known as Newton-Cartan theory), the concept of rotation in general relativity, and Godel spacetime. One of the highlights of the book is a no-go theorem that can be understood to show that there is no criterion of orbital rotation in general relativity that fully answers to our classical intuitions. Topics is intended for both students and researchers in mathematical physics and philosophy of science.

Essay Review:Topics in the Foundations of General Relativity and Newtonian Gravitation Theory

The first portion of this book amounts to a formal introduction to differential geometry and the general theory of relativity (complete with problem sets and solutions). The second amounts to an investigation of special topics pursued by David Malament over the last 4 decades or so. In what follows, I intend to summarize the material. Along the way, I hope to offer some comments on the significance of the work. Before I begin the survey of chapters, it might be helpful to say something about what the book is not. Although there is quite a bit of overlap between the first portion of the book and a standard graduate-level physics text, the two do not fully intersect (more below). In addition, the book does not consider many common topics in the philosophy of space-time physics. For example, there is no discussion of whether space and time are absolute or relational in character (cf. Sklar 1974; Friedman 1983). Finally, one does not find any precisely formulated conjectures of physical or philosophical interest that gesture toward future work (cf. Wald 1984; Earman 1995). It simply is not a book of that kind. Nonetheless, a number of topics developed by Malament in later chapters have generated a significant number of interesting results. (Indeed, I think the previous sentence is wildly understated.) Because Malament does not always do so, I hope to draw attention to some of these lines of inquiry below (a portion of which is still currently active).

Gravitation in material media

When two gravitating bodies reside in a material medium, Newton's law of universal gravitation must be modified to account for the presence of the medium. A modified expression of Newton's law is known in the literature, but lacks a clear connection with existing gravitational theory. Newton's law in the presence of a homogeneous material medium is herein derived on the basis of classical, Newtonian gravitational theory and by a general relativistic use of Archimedes' principle. It is envisioned that the techniques presented herein will be most useful to graduate students and those undergraduate students having prior experience with vector analysis and potential theory.

Mathematical Aspects of General Relativity

Mathematical relativity, the subject of this conference, has recently become more and more devoted to the theory of nonlinear evolution equations, with global questions becoming ever more accessible. This is reflected by the fact that more than half of the talks given were concerned with the global dynamics of solutions of evolution equations related more or less directly to the Einstein equations of general relativity. Progress was reported in understanding subjects such as black holes, gravitational radiation, cosmology and the relation of general relativity to Newtonian gravitational theory.

Relativistic theory of tidal Love numbers

In Newtonian gravitational theory, a tidal Love number relates the mass multipole moment created by tidal forces on a spherical body to the applied tidal field. The Love number is dimensionless, and it encodes information about the body's internal structure. We present a relativistic theory of Love numbers, which applies to compact bodies with strong internal gravities; the theory extends and completes a recent work by Flanagan and Hinderer, which revealed that the tidal Love number of a neutron star can be measured by Earth-based gravitational-wave detectors. We consider a spherical body deformed by an external tidal field, and provide precise and meaningful definitions for electric-type and magnetic-type Love numbers; and these are computed for polytropic equations of state. The theory applies to black holes as well, and we find that the relativistic Love numbers of a nonrotating black hole are all zero.

Does wave function collapse cause gravity?

We give a twist to the assumption – discussed in various earlier works – that gravity plays a role in the collapse of the wave function. This time we discuss the contrary assumption that the collapse of the wave function plays a role in the emergence of the gravitational field. We start from the mathematical framework of a particular Newtonian gravitational collapse theory proposed by the author longtime ago, and we reconciliate it with the classical equivalence principle.

The non-peculiar velocity dispersion profile of the stellar system ω Centauri

We present the results of a survey of radial velocities over a wide region extending from r~10 arcmin out to r~80 arcmin (~1.5 tidal radii) within the massive star cluster omega Centauri. The survey was performed with FLAMES@VLT, to study the velocity dispersion profile in the outer regions of this stellar system. We derived accurate radial velocities for a sample of 2557 newly observed stars, identifying 318 bona-fide cluster red giants. Merging our data with those provided by Pancino et al. (2007), we assembled a final homogeneous sample of 946 cluster members that allowed us to trace the velocity dispersion profile from the center out to r~32 arcmin. The velocity dispersion appears to decrease monotonically over this range, from a central value of sigma_{v}~17.2 Km/s down to a minimum value of sigma_{v}~5.2 Km/s. The observed surface brightness profile, rotation curve, velocity dispersion profile and ellipticity profile are simultaneously well reproduced by a simple dynamical model in which mass follows light, within the classical Newtonian theory of gravitation. The comparison with an N-body model of the evolution of a system mimicking omega Cen during the last 10 orbits into the Galactic potential suggests that (a) the rotation of stars lying in the inner ~20 arcmin of the clusters is not due to the effects of the tidal field of the Milky Way, as hypothized by other authors, and (b) the overall observational scenario is still compatible with the possibility that the outer regions of the cluster are subject to some tidal stirring.

Carter-like Constants of the Motion in Newtonian Gravity and Electrodynamics

For a test body orbiting an axisymmetric body in Newtonian gravitational theory with mass m and multiple moments Q_{l} (and for a charge in orbit about a charge distribution with the same multipole moments) we show that there exists, in addition to the energy and angular momentum component along the symmetry axis, a conserved quantity analogous to the Carter constant of Kerr spacetimes for rotating black holes in general relativity, if the odd-l moments vanish, and the even-l moments satisfy Q_{2l}=m(Q_{2}/m);{l}. Strangely, this is precisely the relation among mass moments enforced by the no-hair theorems of rotating black holes. By contrast, if Newtonian gravity is supplemented by a multipolar gravitomagnetic field, whose leading term represents frame dragging, we are unable to find an analogous Carter-like constant. This further highlights the special nature of the Kerr geometry.

Circular orbits in Einstein-Gauss-Bonnet gravity

The stability under radial and vertical perturbations of circular orbits associated to particles orbiting a spherically symmetric center of attraction is studied in the context of the $n$-dimensional: the Newtonian theory of gravitation, Einstein's general relativity, and the Einstein-Gauss-Bonnet theory of gravitation. The presence of a cosmological constant is also considered. We find that this constant as well as the Gauss-Bonnet coupling constant are crucial to have stability for $ng4$.

On the orbital and physical parameters of the HDE 226868/Cygnus X-1 binary system

In this paper we explore the consequences of the recent determination of the mass m=(8.7 +/- 0.8)M_Sun of Cygnus X-1, obtained from the Quasi-Periodic Oscillation (QPO)-photon index correlation scaling, on the orbital and physical properties of the binary system HDE 226868/Cygnus X-1. By using such a result and the latest spectroscopic optical data of the HDE 226868 supergiant star we get M=(24 +/- 5)M_Sun for its mass. It turns out that deviations from the third Kepler law significant at more than 1-sigma level would occur if the inclination i of the system's orbital plane to the plane of the sky falls outside the range 41-56 deg: such deviations cannot be due to the first post-Newtonian (1PN) correction to the orbital period because of its smallness; interpreted in the framework of the Newtonian theory of gravitation as due to the stellar quadrupole mass moment Q, they are unphysical because Q would take unreasonably large values. By conservatively assuming that the third Kepler law is an adequate model for the orbital period we obtain i=(48 +/- 7) deg which yields for the relative semimajor axis a=(42 +/- 9)R_Sun. Our estimate for the Roche's lobe of HDE 226868 is r_M = (21 +/- 6)R_Sun.

Does ‘modified gravity’, not dark matter, explain galaxy motion?

New observations of the Bullet Cluster suggest that a revised version of Newton's gravitational theory can account for the patterns seen, just as well as the conventional explanation of dark matter affecting the mass distribution.

Presenting Newtonian gravitation

The basic principles of the Newtonian theory of gravitation are presented in a way which students may find more logically coherent, mathematically accessible and physically interesting than other approaches. After giving relatively simple derivations of the circular hodograph and the elliptical orbit from the inverse-square law, the concept of gravitational energy is developed from vector calculus. It is argued that the energy density of a gravitational field may reasonably be regarded as −g2/8πG, and that the inverse-square law may be replaced by a Schwarzschild-like force law without the need to invoke non-Euclidean geometry.

Gravitational wave signals from a chaotic system: A point mass with a disk

We study gravitational waves from a particle moving around a system of a point mass with a disk in Newtonian gravitational theory. A particle motion in this system can be chaotic when the gravitational contribution from a surface density of a disk is comparable with that from a point mass. In such an orbit, we sometimes find that there appears a phase in which particle motion becomes nearly regular (so-called ``stagnant motion'' or stickiness) for a finite time interval between more strongly chaotic phases. To study how these different chaotic behaviors affect observation of gravitational waves, we investigate a correlation of the particle motion and the waves. We find that such a difference in chaotic motions reflects on the wave forms and energy spectra. The character of the waves in the stagnant motion is quite different from that either in a regular motion or in a more strongly chaotic motion. This suggests that we may make a distinction between different chaotic behaviors of the orbit via the gravitational waves.

General theory of relativity - the power of speculative thought

As readers would be aware, Einstein presented his Special Theory of Relativity in 1905. It soon became clear that while this theory could encompass mechanics and electromagnetism, gravity lay beyond its reach. The effort to reconcile special relativity with Newtonian gravitation theory turned out to be exceptionally demanding, and it led to the General Theory which transcends both these starting points.

Class of exact solutions of Einstein’s field equations in higher dimensional spacetimes,d≥4: Majumdar-Papapetrou solutions

The Newtonian theory of gravitation and electrostatics admits equilibrium configurations of charged fluids where the charge density can be equal to the mass density, in appropriate units. The general relativistic analog for charged dust stars was discovered by Majumdar and by Papapetrou. In the present work we consider Einstein-Maxwell solutions in $d$-dimensional spacetimes and show that there are Majumdar-Papapetrou type solutions for all $d\ensuremath{\ge}4$. It is verified that the equilibrium is independent of the shape of the distribution of the charged matter. It is also showed that for perfect fluid solutions satisfying the Majumdar-Papapetrou condition with a boundary where the pressure is zero, the pressure vanishes everywhere, and that the $(d\ensuremath{-}1)$-dimensional spatial section of the spacetime is conformal to a Ricci-flat space. The Weyl $d$-dimensional axisymmetric solutions are generalized to include electric field and charged matter.

Is It Possible to Nominalize Quantum Mechanics?

Hartry Field (1980) has developed an interesting nominalization strategy for Newtonian gravitation theory—a strategy that reformulates the theory without quantification over abstract entities. According to David Malament (1982), Field's strategy cannot be extended to quantum mechanics (QM), and so it only has a limited scope. In a recent work, Mark Balaguer has responded to Malament's challenge by indicating how QM can be nominalized, and by “doing much of the work needed to provide the nominalization” (Balaguer 1998, 114). In this paper, I critically assess Balaguer's proposal, and argue that it ultimately fails. Balaguer's strategy is incompatible with a number of interpretations of QM, in particular with Bas van Fraassen's version of the modal interpretation. And given that Balaguer's strategy invokes physically real propensities, it is unclear whether it is even compatible with nominalism. I conclude that the nominalization of QM remains a major problem for the nominalist.

Calculation of So‐Called General Relativistic Phenomena by Advancing Newton's Theory of Gravitation, Maintaining Classical Conceptions of Space and Relativity

With the example of the motion of Mercury around the Sun it is shown how Newton’s theory of gravitation should be advanced by taking into consideration the finite velocity of gravitational expansion and the present concept of transference of forces by particles to be able to calculate so-called general relativistic phenomena such as the additional motion of Mercury’s perihelion, the curvature of a light beam at the surface of the Sun, and the phenomena observed at the binary pulsar PSR 1913+16, maintaining classical conceptions of a Euclidean space and the Galilean principle of relativity.

The Reception of Newton's Gravitational Theory by Huygens, Varignon, and Maupertuis: How Normal Science may be Revolutionary

This paper first discusses the current historical and philosophical framework forged during the last century to account for both the history and the epistemic status of Newton's theory of general gravitation. It then examines the conflict surrounding this theory at the close of the seventeenth century and the first steps towards the revolutionary shift in rational mechanics in the eighteenth century. From a historical point of view, it shows the crucial contribution of the Cartesian mechanistic philosophy and Leibnizian analytic methods to the emergence of so-called Newtonian mechanics which can also be fairly characterized as a synthetic theory of attraction. From a philosophical standpoint, the paper suggests that the reworking of Newton's theory in the 18th century is better understood in a theoretical framework that reconciles Kuhn's notion of “invisible revolution” rather than his notion of “normal science” with Whewell's ascription of the completion of dynamical studies to the post Newtonian period.

Indeterminism, classical gravitation and non-collision singularities

Until the present, the Newtonian theory of gravitation has only been studied in any detail through the usual, presupposed ontology of point particles. This paper shows that changing our ontology into one which makes use of continuous bodies (non-point particles) allows us to obtain in a simple way two important results relevant to the theory: (a) The Newtonian theory of gravitation is indeterministic in a way apparently unparalleled when non-point particle models of it are used. (b) In the Newtonian theory of gravitation it is possible to find non-collision singularities qualitatively different from the ones presented in point particle models.

The 1995-99measurements of the Lense-Thirring effect using laser-rangedsatellites

In general relativity a current of mass-energy, such as a spinning body,gives rise to peculiar phenomena on bodies, particles and clocks in itsvicinity, which are not predicted by the Newtonian gravitational theory;one of these phenomena is the Lense-Thirring effect on particles orbitinga spinning central body. In this paper we first review the methodused tomeasure the Lense-Thirring effect, by analysing the orbits of the twolaser-ranged satellites LAGEOS and LAGEOS II, that has provided adirect measurements of this effect; we then report on these detections ofthe Lense-Thirring effect, obtained by analysing the nodes of LAGEOS andLAGEOS II and the perigee of LAGEOS II with the orbital programsGEODYN-SOLVE, using the Earth gravitational models JGM-3 and EGM-96and this new method. The first detection was obtained in 1995, the mostaccurate measurements were obtained in 1998 using EGM-96, with about20-30% accuracy. Finally, we briefly review the proposed LARESexperiment to measure the Lense-Thirring effect withan accuracy of about2-3% and to provide other basic tests of general relativity andgravitation.