Non-linear conformally invariant generalization of the Poisson equation to D>2 dimensions

I propound a non-linear generalization of the Poisson equation describing a "medium" in D dimensions with a "dielectric constant" proportional to the field strength to the power D-2. It is the only conformally invariant scalar theory that is second order, and in which the scalar $phi$ couples to the sources $\rho$ via a $\phi\rho$ contact term. The symmetry is used to generate solutions for the field for some non-trivial configurations (e.g. for two oppositely charged points). Systems comprising N point charges afford further application of the symmetry. For these I derive e.g. exact expressions for the following quantities: the general two-point-charge force; the energy function and the forces in any three-body configuration with zero total charge; the few-body force for some special configurations; the virial theorem for an arbitrary, bound, many-particle system relating the time-average kinetic energy to the particle charges. Possible connections with an underlying conformal quantum field theory are mentioned.Comment: Revtex, 16 pages. To be published in Phys. Rev.

Dwarf satellite galaxies in the modified dynamics

In the modified dynamics (MOND) the inner workings of dwarf satellites can be greatly affected by their mother galaxy-over and beyond its tidal effects. Because of MOND's nonlinearity a system's internal dynamics can be altered by an external field in which it is immersed (even when this field, by itself, is constant in space). As a result, the size and velocity dispersion of the satellite vary as the external field varies along its orbit. A notable outcome of this is a substantial increase in the dwarf's vulnerability to eventual tidal disruption-rather higher than Newtonian dynamics (with a dark-matter halo) would lead us to expect for a satellite with given observed parameters.Comment: 15 pages with 5 embedded figures; Astrophysical J in press Corrected error in names of author

Practically linear analogs of the Born-Infeld and other nonlinear theories

I discuss theories that describe fully nonlinear physics, while being practically linear (PL), in that they require solving only linear differential equations. These theories may be interesting in themselves as manageable nonlinear theories. But, they can also be chosen to emulate genuinely nonlinear theories of special interest, for which they can serve as approximations. The idea can be applied to a large class of nonlinear theories, exemplified here with a PL analogs of scalar theories, and of Born-Infeld (BI) electrodynamics. The general class of such PL theories of electromagnetism are governed by a Lagrangian L=-(1/2)F_mnQ^mn+ S(Q_mn), where the electromagnetic field couples to currents in the standard way, while Qmn is an auxiliary field, derived from a vector potential that does not couple directly to currents. By picking a special form of S(Q_mn), we can make such a theory similar in some regards to a given fully nonlinear theory, governed by the Lagrangian -U(F_mn). A particularly felicitous choice is to take S as the Legendre transform of U. For the BI theory, this Legendre transform has the same form as the BI Lagrangian itself. Various matter-of-principle questions remain to be answered regarding such theories. As a specific example, I discuss BI electrostatics in more detail. As an aside, for BI, I derive an exact expression for the short-distance force between two arbitrary point charges of the same sign, in any dimension.Comment: 20 pages, Version published in Phys. Rev.

MOND and the lensing Fundamental Plane: No need for dark matter on galaxy scales

Bolton et al. (2007) have derived a mass-based fundamental plane using photometric and spectroscopic observations of 36 strong gravitational lenses. The lensing allows a direct determination of the mass-surface density and so avoids the usual dependence on mass-to-light ratio. We consider this same sample in the context of modified Newtonian dynamics (MOND) and demonstrate that the observed mass-based fundamental plane coincides with the MOND fundamental plane determined previously for a set of high-order polytropic spheres chosen to match the observed range of effective radii and velocity dispersions in elliptical galaxies. Moreover, the observed projected mass within one-half an effective radius is consistent with the mass in visible stars plus a small additional component of ``phantom dark matter'' resulting from the MOND contribution to photon deflection.Comment: Minor revisions in response to referee. Revised title. Accepted in MNRA

MOND mass-to-light ratios for galaxy groups

I estimate MOND M/L values for nine galaxy groups that were recently studied by Tully et al.. Instead of the large M/L values that they find with Newtonian dynamics (up to 1200 solar units) the MOND estimates cluster around 1 solar unit. Tully et al. find a systematic and significant difference between the M/L values of groups that do not contain luminous galaxies and those that do: Dwarfs-only groups have larger M/L values (by a factor of about 5). The MOND M/L values do not show this trend; the Newtonian disparity is traced back to the dwarfs-only groups having systematically smaller intrinsic accelerations (similar sizes, but rather smaller velocity dispersions).Comment: 7 pages, Astrophys. J. Lett., in pres

The Bright Side of Dark Matter

We show that it is not possible in the absence of dark matter to construct a four-dimensional metric that explains galactic observations. In particular, by working with an effective potential it is shown that a metric which is constructed to fit flat rotation curves in spiral galaxies leads to the wrong sign for the bending of light i.e. repulsion instead of attraction. Hence, without dark matter the motion of particles on galactic scales cannot be explained in terms of geodesic motion on a four- dimensional metric. This reveals a new bright side to dark matter: it is indispensable if we wish to retain the cherished equivalence principle.Comment: 7 pages, latex, no figures. Received an honorable mention in the 1999 Gravity research Foundation Essay Competition. Submitted to Phys. Rev. Let

Dark matter and non-Newtonian gravity from General Relativity coupled to a fluid of strings

An exact solution of Einstein's field equations for a point mass surrounded by a static, spherically symmetric fluid of strings is presented. The solution is singular at the origin. Near the string cloud limit there is a $1/r$ correction to Newton's force law. It is noted that at large distances and small accelerations, this law coincides with the phenomenological force law invented by Milgrom in order to explain the flat rotation curves of galaxies without introducing dark matter. When interpreted in the context of a cosmological model with a string fluid, the new solution naturally explains why the critical acceleration of Milgrom is of the same order of magnitude as the Hubble parameter.Comment: 12 pages, REVTeX, no figure

X-ray Group and cluster mass profiles in MOND: Unexplained mass on the group scale

Although very successful in explaining the observed conspiracy between the baryonic distribution and the gravitational field in spiral galaxies without resorting to dark matter (DM), the modified Newtonian dynamics (MOND) paradigm still requires DM in X-ray bright systems. Here, to get a handle on the distribution and importance of this DM, and thus on its possible form, we deconstruct the mass profiles of 26 X-ray emitting systems in MOND, with temperatures ranging from 0.5 to 9 keV. Initially we compute the MOND dynamical mass as a function of radius, then subtract the known gas mass along with a component of galaxies which includes the cD galaxy with $M/L_K=1$. Next we test the compatibility of the required DM with ordinary massive neutrinos at the experimental limit of detection ($m_{\nu}=2$ eV), with density given by the Tremaine-Gunn limit. Even by considering that the neutrino density stays constant and maximal within the central 100 or 150 kpc (which is the absolute upper limit of a possible neutrino contribution there), we show that these neutrinos can never account for the required DM within this region. The natural corollary of this finding is that, whereas clusters (T \ga 3 keV) might have most of their mass accounted for if ordinary neutrinos have a 2 eV mass, groups (T \lsim 2 keV) cannot be explained by a 2 eV neutrino contribution. This means that, for instance, cluster baryonic dark matter (CBDM, Milgrom 2007) or even sterile neutrinos would present a more satisfactory solution to the problem of missing mass in MOND X-ray emitting systems.Comment: 13 pages, 8 figures, 1 table, accepted in MNRA

Born reciprocity and the 1/r potential

Many structures in nature are invariant under the transformation (p,r)->(br,-p/b), where b is some scale factor. Born's reciprocity hypothesis affirms that this invariance extends to the entire Hamiltonian and equations of motion. We investigate this idea for atomic physics and galactic motion, where one is basically dealing with a 1/r potential and the observations are very accurate, so as to determine the scale $b = m\Omega$. We find that an $\Omega \sim 1.5\times 10^{-15}$ Hz has essentially no effect on atomic physics but might possibly offer an explanation for galactic rotation, without invoking dark matter.Comment: 14 pages, with 4 figures, Latex, requires epsf.tex and iop style file

Does MOND follow from the CDM paradigm?

In a recent paper, Kaplinghat and Turner (2001) (KT) advertise that MOND can be derived naturally in the CDM paradigm. They actually proceed to produce a more limited result: Every galaxy should have a transition radius, $r_t$, below which baryons dominate, and above which dark matter (DM) takes over; the acceleration at $r_t$ is nearly the same for all galaxies; and due to a coincidences this is of order $a_0\sim cH_0$. This follows from their tacit, intermediate result, whereby CDM halos of galaxies have a very nearly universal acceleration profile $a(r)\approx v^2(r)/r\approx A\hat a(r/\ell)$, where A is universal, and only the scale $\ell$ varies from halo to halo. (This remains so when baryons are added because they assume a universal baryon-collapse factor.) The KT scenario is phenomenologically wrong--observed galaxies are simply not like that. For example, it precludes altogether the existence of LSB galaxies, in which the acceleration is everywhere smaller than $a_0$. The phenomenologically sound outcome--i.e., the role of $a_0$ as a transition acceleration in high-surface-brightness galaxies--pertains to only a small part of the statement of MOND. There are several other, independent roles that $a_0\sim cH_0$ plays in MOND phenomenology, and other predictions of MOND, not related to the value of $a_0$, that are not explainable in the KT scenario. The results of KT also disagree with those of CDM simulations, which, as they now stand, do not reproduce any aspect of MOND phenomenology.Comment: 6 page