25 pages, including 4 figures: change of title and other minor changes after peer review. Final version to be published in Foundations of Physics, Springer
This paper introduces a possible alternative model of gravity based on the theory of fractional-dimension spaces and its applications to Newtonian gravity. In particular, Gauss's law for gravity as well as other fundamental classical laws are extended to a $D$-dimensional metric space, where $D$ can be a non-integer dimension.
We show a possible connection between this Newtonian Fractional-Dimension Gravity (NFDG) and Modified Newtonian Dynamics (MOND), a leading alternative gravity model which accounts for the observed properties of galaxies and other astrophysical structures without requiring the dark matter hypothesis. The MOND acceleration constant $a_{0} \simeq 1.2 \times 10^{ -10}\mbox{m}\thinspace \mbox{s}^{ -2}$ can be related to a natural scale length $l_{0}$ in NFDG, i.e., $a_{0} \approx GM/l_{0}^{2}$, for astrophysical structures of mass $M$, and the deep-MOND regime is present in regions of space where the dimension is reduced to $D \approx 2$.
For several fundamental spherically-symmetric structures, we compare MOND results, such as the empirical Radial Acceleration Relation (RAR), circular speed plots, and logarithmic plots of the observed radial acceleration $g_{obs}$ vs. the baryonic radial acceleration $g_{bar}$, with NFDG results. We show that our model is capable of reproducing these results using a variable local dimension $D\left (w\right )$, where $w =r/l_{0}$ is a dimensionless radial coordinate. At the moment, we are unable to derive explicitly this dimension function $D\left (w\right )$ from first principles, but it can be obtained empirically in each case from the general RAR.
Additional work on the subject, including studies of axially-symmetric structures, detailed galactic rotation curves fitting, and a possible relativistic extension, will be needed to establish NFDG as a viable alternative model of gravity.
We study the post-Newtonian expansion of a class of Lorentz-violating gravity theories that reduce to khronometric theory (i.e. the infrared limit of Horava gravity) in high-acceleration regimes, and reproduce the phenomenology of the modified Newtonian dynamics (MOND) in the low-acceleration, non-relativistic limit. Like in khronometric theory, Lorentz symmetry is violated in these theories by introducing a dynamical scalar field (the "khronon") whose gradient is enforced to be timelike. As a result, hypersurfaces of constant khronon define a preferred foliation of the spacetime, and the khro…
Two incorrect arguments against MOND in elliptical galaxies could be that the equivalent circular velocity curves tend to become flat at much larger accelerations than in spiral galaxies, and that the Newtonian dark matter halos are more concentrated than in spirals. Here, we compare published scaling relations for the dark halos of elliptical galaxies to the scaling relations expected for MONDian phantom halos. We represent the baryonic content of galaxies by spherical profiles, and their corresponding MONDian phantom halos by logarithmic halos. We then derive the surface densities, central d…
Modified Newtonian dynamics (MOND) can be obtained by modifying the entropic formulation of gravity, this is achieved by considering the quantum statistical nature of the degrees of freedom on the holographic screen. Through this frame work, we find some constraints on a cosmological extension for MOND, with no additional auxiliary fields. The connections between MOND to conformal gravity and Rindler force gravity are examined. These two alternative gravity theories are subsequently considered as possible cosmological extensions of MOND.
We determine the mean velocity dispersion of six Galactic outer halo globular clusters, AM 1, Eridanus, Pal 3, Pal 4, Pal 15, and Arp 2 in the weak acceleration regime to test classical vs. modified Newtonian dynamics (MOND). Owing to the non-linearity of MOND's Poisson equation, beyond tidal effects, the internal dynamics of clusters is affected by the external field in which they are immersed. For the studied clusters, particle accelerations are much lower than the critical acceleration a_0 of MOND, but the motion of stars is neither dominated by internal accelerations (a_i >> a_e) nor exter…
Non-spherical systems described by MOND theories of modified gravity arising from generalizations of the Poisson equations are affected by a MONDian extra-quadrupolar potential ϕ_M even if they are isolated and they are in deep Newtonian regime. In general MOND theories quickly approaching Newtonian dynamics for accelerations beyond A_0, ϕ_M is proportional to a multiplicative scaling coefficient α\sim 1, while in MOND models becoming Newtonian beyond κA_0, κ>> 1, it is enhanced by κ^2. We analytically work out some orbital effects due to ϕ_M in the framework of QUMOND, and compare them with t…
Recent experimental results for the gravitational constant G from Cavendish-type experiments were analysed in the framework of Modified Newtonian Dynamics (MOND). MOND corrections were applied to the equation of motion of a pendulum, under the assumption that the magnitude of the horizontal time dependent gravitational acceleration determines the amount of MOND corrections. The large vertical component of the local gravitational field of the earth is fully compensated by the alignment of the torsion pendulum in accordance with Newtons third law and therefore not considered for MOND corrections…
The Newtonian gravitational constant $G$ obeys the dimensional relation $[G] [M] [a] = [v]^4$, where $M$, $a$, and $v$ denote mass, acceleration, and speed, respectively. Since the baryonic Tully-Fisher (BTF) and Faber-Jackson (BFJ) relations are observed facts, this relation implies that $G\, a = {\rm constant}$. This result cannot be obtained in Newtonian dynamics which cannot explain the origin of the BTF and BFJ relations. An alternative, modified Newtonian dynamics (MOND) assumes that $G=G_0$ is constant in space and derives naturally a characteristic constant acceleration $a=a_0$, as wel…
No more salient issue exists in contemporary astrophysics and cosmology than that of the elusive "dark matter". For many years already Milgrom's paradigm of modified Newtonian dynamics (MOND) has provided an alternative way to interpret observations without appeal to invisible dark matter. MOND had been successful in elucidating economically the dynamics of disk galaxies of all scales, while doing less well for clusters of galaxies; in its original form it could not address gravitational lensing or cosmology. After reviewing some of the evidence in favor of MOND, I recollect the development of…
Employing a thermodynamic interpretation of gravity based on the holographic principle and assuming underlying particle statistics, fermionic or bosonic, for the excitations of the holographic screen leads to Modified Newtonian Dynamics (MOND). A connection between the acceleration scale $a_0$ appearing in MOND and the Fermi energy of the holographic fermionic degrees of freedom is obtained. In this formulation the physics of MOND results from the quantum-classical crossover in the fermionic specific heat. However, due to the dimensionality of the screen, the formalism is general and applies t…
I discuss a novel MOND effect that entails a correction to the dynamics of isolated mass systems even when they are deep in the Newtonian regime: systems whose extent R<<Rm, where Rm=sqrt(GM/a0) is the MOND radius, and M the total mass. Interestingly, even if the MOND equations approach Newtonian dynamics arbitrarily fast at high accelerations, this correction decreases only as a power of R/Rm. The effect appears in formulations of MOND as modified gravity, governed by generalizations of the Poisson equation. The MOND correction to the potential is a quadrupole field G*Pij*Ri*Rj, where R is th…
Based on Newtonian dynamics, observations show that the luminous masses of astrophysical objects that are the size of a galaxy or larger are not enough to generate the measured motions which they supposedly determine. This is typically attributed to the existence of dark matter, which possesses mass but does not radiate (or absorb radiation). Alternatively, the mismatch can be explained if the underlying dynamics is not Newtonian. Within this conceptual scheme, Modified Newtonian Dynamics (MOND) is a successful theoretical paradigm. MOND is usually expressed in terms of a nonlinear Poisson equ…
We show that the Modified Newtonian Dynamics (MOND) regime can be fully recovered as the weak-field limit of a particular theory of gravity formulated in the metric approach. This is possible when Milgrom's acceleration constant is taken as a fundamental quantity which couples to the theory in a very consistent manner. As a consequence, the scale invariance of the gravitational interaction is naturally broken. In this sense, Newtonian gravity is the weak-field limit of general relativity and MOND is the weak-field limit of that particular extended theory of gravity. We also prove that a Noethe…
MOdified Newtonian Dynamics (MOND) is an interesting alternative to dark matter in extragalactic systems. We here examine the possibility that mild or even strong MOND behavior may become evident well inside the solar system, in particular near saddle points of the total gravitational potential. Whereas in Newtonian theory tidal stresses are finite at saddle points, they are expected to diverge in MOND, and to remain distinctly large inside a sizeable oblate ellipsoid around the saddle point. We work out the MOND effects using the nonrelativistic limit of the T$e$V$e$S theory, both in the pert…
We consider a deviation of the physical length from the Riemann geometry toward the Randers'. We construct a consistent second-order relativistic theory of gravity that dynamically reduces to the Einstein-Hilbert theory for the strong and Newtonian gravity while its weak gravitational regime reproduces MOND and the gravitational lensing attributed to the dark matter halo. It also naturally accommodates the observed value of the cosmological constant. We show that it predicts a few percent deviation for the post Newtonian parameter $γ$ in a part of the regime that interpolates the Newtonian reg…
I provide a derivation of some characteristic effects of Milgrom's modified Newtonian dynamics (MOND) from a fractional version of Newton's theory based on the fractional Poisson equation. I employ the properties of the fractional Laplacian to investigate the features of the fundamental solution of the proposed model. The key difference between MOND and the fractional theory introduced here is that the latter is an inherently linear theory, featuring a characteristic length scale $\ell$, whilst the former is ultimately nonlinear in nature and it is characterized by an acceleration scale $a_0$.…
A speculative mathematical model is used to generate the modified Newtonian dynamics called MOND from fluctuations of the number of quanta of quantized acceleration. The one new parameter can be chosen either to make the transition to modification comparable to that obtained from the functions used to fit data with MOND, or to make the modification at larger accelerations comparable in magnitude to the unexplained accelerations of Pioneer 10 and 11.
A wealth of astronomical data indicate the presence of mass discrepancies in the Universe. The motions observed in a variety of classes of extragalactic systems exceed what can be explained by the mass visible in stars and gas. Either (i) there is a vast amount of unseen mass in some novel form - dark matter - or (ii) the data indicate a breakdown of our understanding of dynamics on the relevant scales, or (iii) both. Here, we first review a few outstanding challenges for the dark matter interpretation of mass discrepancies in galaxies, purely based on observations and independently of any alt…
This is a semi-popular account of the MOND paradigm and its comparison with the competing Newtonian-dynamics-plus-CDM paradigm. It was published recently in the online magazine of the Israel Physical Society in the IYA 2009 issue.
We numerically investigate the features of typical orbits occurring in the Oort cloud (r\approx 50-150 kAU) in the low-acceleration regime of the MOdified Newtonian Dynamics (MOND). We take into account the so-called External Field Effect (EFE) because the solar system is embedded in the Milky Way. In the framework of MOND this does matter since the gravitational acceleration of Galactic origin felt by the solar system is of the same order of magnitude of the characteristic MOND acceleration scale A_0\approx 10^-10 m s^-2. We use three different forms of the MOND interpolating function μ(x) an…
The cold dark matter paradigm has been extremely successful for explaining a wide range of cosmological phenomena. Nevertheless, since evidence for non-baryonic dark matter remains indirect, all reasonable alternatives should be explored. One proposed idea, involving a fundamental acceleration scale a_0 = 1-2e-10 m/s^2, is called MOdified Newtonian Dynamics or MOND. MOND was suggested to explain the flat rotation curves of galaxies without the need for dark matter. Whether or not it can adequately fit the available data has been debated for almost 20 years (and we summarise many of these studi…
The proximity of the MOND acceleration constant with cosmological accelerations -- for example, a0 cH0/2pi -- points to its possibly decreasing with cosmic time. I begin to consider the secular changes induced in galactic systems by such presumed variations, assumed adiabatic. It is important to understand these effects, in isolation from other evolutionary influences, in order to identify or constrain a0 variations by detection of induced effects, or lack thereof. I find that as long as the system is fully in the deep-MOND regime -- as applies to many galactic systems -- the adiabatic respons…
Modified Newtonian dynamics (MOND) is an empirically motivated modification of Newtonian gravity or inertia suggested by Milgrom as an alternative to cosmic dark matter. The basic idea is that at accelerations below a0 ~ 10^{-8} cm/s^2 ~ cH0/6 the effective gravitational attraction approaches sqrt{gN*a0} where gN is the usual Newtonian acceleration. This simple algorithm yields flat rotation curves for spiral galaxies and a mass-rotation velocity relation of the form M ~ V^4 that forms the basis for the observed luminosity-rotation velocity relation-- the Tully-Fisher law. We review the phenom…
We show that the Modified Newtonian Dynamics (MOND) regime can be fully recovered as the weak-field limit of a particular theory of gravity formulated in the metric approach. This is possible when Milgrom's acceleration constant is taken as a fundamental quantity which couples to the theory in a very consistent manner. As a consequence, the scale invariance of the gravitational interaction is naturally broken. In this sense, Newtonian gravity is the weak-field limit of general relativity and MOND is the weak-field limit of that particular extended theory of gravity.
It is believed that the modification of Newtonian dynamics (MOND) is possible alternate for dark matter hypothesis. Although Bekenstein's TeVeS supplies a relativistic version of MOND, one may still wish a more concise covariant formulism of MOND. In this paper, within covariant geometrical framwork, we present another version of MOND. We show the spacetime structure of MOND with properties of Tully-Fisher relation and Lorentz invariance violation.
The recently proposed Skordis-Złośnik theory is the first relativistic MOND theory that can recover the success of the standard $Λ$CDM model at matching observations of the cosmic microwave background. This paper aims to revisit the Newtonian and MOND approximations and the gravitational wave analysis of the theory. For the local gravitational parameters, we show that one could obtain both time-varying effective Newtonian gravitational \textit{constant} $G_\textrm{N}$ and time-varying characteristic MOND acceleration scale $a_\textrm{MOND}$, by relaxing the static assumption extensively adopte…
The Modified Newtonian Dynamics (MOND) has been formulated as a modification of the Poisson equation for the Newtonian gravitational field. This theory generically predicts a violation of the strong version of the equivalence principle, and as a result the gravitational dynamics of a system depends on the external gravitational field in which the system is embedded. This so-called external field effect has been recently shown to imply the existence of an anomalous quadrupolar correction, along the direction of the external Galactic field, in the gravitational potential felt by planets in the S…
Verlinde's heuristic argument for the interpretation of the standard Newtonian gravitational force as an entropic force is generalized by the introduction of a minimum temperature (or maximum wave length) for the microscopic degrees of freedom on the holographic screen. With the simplest possible setup, the resulting gravitational acceleration felt by a test mass m from a point mass M at a distance R is found to be of the form of the modified Newtonian dynamics (MOND) as suggested by Milgrom. The corresponding MOND-type acceleration constant is proportional to the minimum temperature, which ca…
The modified Newtonian dynamics (MOND) paradigm is discussed in the context of asymptotic safe gravity. We estimate quantum correction to the logarithmic potential which is well known to account for the constancy of the circular velocity $v$ of the spiral galaxies. We determine plausible bounds on $v$.
Intermediate-richness galaxy groups are an important test ground for MOND. First, they constitute a distinct type of galactic systems, with their own evolution histories and underlying physical processes; secondly, they probe little-chartered regions of parameter space, as they have baryonic masses similar to massive galaxies, and similar velocity dispersions, but much larger sizes -- similar to cluster cores (or even to clusters), but much lower dispersions. Importantly in the context of MOND, they have the lowest internal accelerations reachable inside galactic systems. I analyze a sample of…
The modified Newtonian dynamics (MOND) has been proposed as an alternative to the dark matter paradigm; the philosophy behind is that there is no dark matter and we witness a violation of the Newtonian law of dynamics. In this article, we interpret differently the phenomenology sustaining MOND, as resulting from an effect of "gravitational polarization", of some cosmic fluid made of dipole moments, aligned in the gravitational field, and representing a new form of dark matter. We invoke an internal force, of non-gravitational origin, in order to hold together the microscopic constituents of th…
After reviewing the modified Newtonian dynamics (MOND) proposal, we advocate that the associated phenomenology may actually not result from a modification of Newtonian gravity, but from a mechanism of "gravitational polarization" of some dipolar medium playing the role of dark matter. We then build a relativistic model within standard general relativity to describe (at some phenomenological level) the dipolar dark matter polarizable in a gravitational field. The model naturally involves a cosmological constant, and is shown to reduce to the concordance cosmological scenario (Lambda-CDM) at ear…
MOND predicts that the asymptotic gravitational potential of an isolated, bounded (baryonic) mass, M, is phi(r)=(MGa0)^{1/2}ln(r); a0 is the MOND constant. Relativistic MOND theories predict that the lensing effects of M are dictated by phi(r) as general-relativity lensing is dictated by the Newtonian potential. Thus, MOND predicts that the asymptotic Newtonian potential deduced from galaxy-galaxy gravitational lensing will have: (1) a logarithmic r dependence, and (2) a normalization (parametrized standardly as 2s^2) that depends only on M: s=(MGa0/4)^{1/4}. I compare these predictions with r…
We introduce the Gauge Vector-Tensor (GVT) theory by extending the AQUAL's approach to the GravitoElectroMagnetism (GEM) approximation of gravity. GVT is a generally covariant theory of gravity composed of a pseudo Riemannian metric and two U(1) gauge connections that reproduces MOND in the limit of very weak gravitational fields while remains consistent with the Einstein-Hilbert gravity in the limit of strong and Newtonian gravitational fields. GVT also provides a simple framework to study the GEM approximation to gravity. We illustrate that the gravitomagnetic force at the edge of a galaxy c…
A heuristic entropic-gravity derivation has previously been given of the gravitational two-body force of modified Newtonian dynamics (MOND). Here, it is shown that also another characteristic of MOND can be recovered, namely, the external field effect (implying a violation of the Strong Equivalence Principle). In fact, the derivation gives precisely the modified Poisson equation which Bekenstein and Milgrom proposed as a consistent nonrelativistic theory of MOND.
MOND dynamics consists of a modification of the acceleration with respect to the one provided by Newtonian mechanics. In this paper we investigate whether it can be derived from a velocity-dependent deformation of the coordinates of the systems. The conclusion is that it cannot be derived this way because of the intrinsic non-local character in time of the MOND procedure. This is a feature pointed out some time ago already by Milgrom himself.
The Local Group (LG) is {an appropriate} test system for Modified Newtonian Dynamics, since the acceleration of M31 galaxy is fully in the deep MOND regime $a \ll a_0$. We model the LG as a two body problem of $M31$ and the Milky Way (MW) galaxies. {Extending previous studies, we also include the Cosmological Constant.} The assumption that in the big bang the galaxies emerged from the same place and approach to the measured distance and velocity today (the Timing Argument), predicts the total mass for the LG: $(0.447 \pm 0.005)\cdot 10^{12} M_{\odot}$. The corresponding motion of the LG predic…
We looked at the orbital motions of test particles according to the External Field Effect (EFE) predicted by the MOdified Newtonian Dynamics (MOND) in the Oort cloud which falls in the deep MONDian regime (r\approx 50-150 kAU). Concerning the interpolating function μ(x), we extensively used the forms μ_1=1/(1+x),μ_2=x/(1+x^2)^1/2,μ_3/2=x/(1+x^3/2)^2/3. We integrated both the MOND and the Newtonian equations of motion in Cartesian coordinates sharing the same initial conditions. We considered both ecliptic and nearly polar trajectories, all with high eccentricities (e>0.1). In order to evaluate…
The modified newtonian dynamics (MOND) paradigm of Milgrom can boast of a number of successful predictions regarding galactic dynamics; these are made without the assumption that dark matter plays a significant role. MOND requires gravitation to depart from Newtonian theory in the extragalactic regime where dynamical accelerations are small. So far relativistic gravitation theories proposed to underpin MOND have either clashed with the post-Newtonian tests of general relativity, or failed to provide significant gravitational lensing, or violated hallowed principles by exhibiting superluminal s…
The proposed Satellite Test of the Equivalence Principle (STEP) will detect possible violations of the Weak Equivalence Principle by measuring relative accelerations between test masses of different composition with a precision of one part in $10^{18}$. A serendipitous byproduct of the experimental design is that the absolute (common-mode) acceleration of the test masses is also measured to high precision as they oscillate along a common axis under the influence of restoring forces produced by the position sensor currents, which in drag-free mode lead to Newtonian accelerations as small as $10…
We quantify the potential for testing MOdified Newtonian Dynamics (MOND) with LISA Pathfinder (LPF), should a saddle point flyby be incorporated into the mission. We forecast the expected signal to noise ratio (SNR) for a variety of instrument noise models and trajectories past the saddle. For standard theoretical parameters the SNR reaches middle to high double figures even with modest assumptions about instrument performance and saddle approach. Obvious concerns, like systematics arising from LPF self-gravity, or the Newtonian background, are examined and shown not to be a problem. We also i…
Galaxy groups, which have hardly been looked at in MOND, afford probing the acceleration discrepancies in regions of system-parameter space that are not accessible in well-studied galactic systems, such as galaxies, galaxy clusters, and dwarf-spheroidal satellites of galaxies. Groups are typically the size of galaxy-cluster cores, but have masses typically only a few times that of a single galaxy. Accelerations in groups get far below those in galaxies, and far below the MOND acceleration. So much so, that many groups might be affected by the external-field effect, which is unique to MOND, due…
A unique signature of the modified Newtonian dynamics (MOND) paradigm is its peculiar behavior in the vicinity of the points where the total Newtonian acceleration exactly cancels. In the Solar System, these are the saddle points of the gravitational potential near the planets. Typically, such points are embedded into low-acceleration bubbles where modified gravity theories a` la MOND predict significant deviations from Newton's laws. As has been pointed out recently, the Earth-Sun bubble may be visited by the LISA Pathfinder spacecraft in the near future, providing a unique occasion to put th…
General Relativity is able to describe the dynamics of galaxies and larger cosmic structures only if most of the matter in the Universe is dark, namely it does not emit any electromagnetic radiation. Intriguingly, on the scale of galaxies, there is strong observational evidence that the presence of dark matter appears to be necessary only when the gravitational field inferred from the distribution of the luminous matter falls below an acceleration of the order of 10^(-10) m/s^2. In the standard model, which combines Newtonian gravity with dark matter, the origin of this acceleration scale is c…
We reconsider the recently proposed mimetic gravity, focusing in particular on whether the theory is able to reproduce the inferred flat rotation curves of galaxies. We extend the theory by adding a non-minimal coupling between matter and mimetic field. Such coupling leads to the appearance of an extra force which renders the motion of test particles non-geodesic. By studying the weak field limit of the resulting equations of motion, we demonstrate that in the Newtonian limit the acceleration law induced by the non-minimal coupling reduces to a Modified Newtonian Dynamics (MOND)-like one. In t…
In this article we construct a relativistic extended metric theory of gravity, for which its weak field limit reduces to the non-relativistic MOdified Newtonian Dynamics regime of gravity. The theory is fully covariant and the way to achieve this is to introduce torsion in the description of gravity as well as with the addition of a particular function of the matter lagrangian into the gravitational action.
A new formulation of MOND as a modified-potential theory of gravity is propounded. In effect, the theory dictates that the MOND potential phi produced by a mass distribution rho is a solution of the Poisson equation for the modified source density rho*=-(1/4 pi G)divergence(g), where g=nu(|gN|/a0)gN, and gN is the Newtonian acceleration field of rho. This makes phi simply the scalar potential of the algebraic acceleration field g. The theory thus involves solving only linear differential equations, with one nonlinear, algebraic step. It is derivable from an action, satisfies all the usual cons…
Verlinde has suggested that the gravity has an entropic origin, and a gravitational system could be regarded as a thermodynamical system. It is well-known that the equipartition law of energy is invalid at very low temperature. Therefore, entropic force should be modified while the temperature of the holographic screen is very low. It is shown that the modified entropic force is proportional to the square of the acceleration, while the temperature of the holographic screen is much lower than the Debye temperature $T_D$. The modified entropic force returns to the Newton's law of gravitation whi…
We propose a simple field theory reproducing the MOND phenomenology at galaxy scale, while predicting negligible deviations from general relativity at small scales thanks to an extended Vainshtein ("k-mouflage") mechanism induced by a covariant Galileon-type Lagrangian. The model passes solar-system tests at the post-Newtonian order, including those of local Lorentz invariance, and its anomalous forces in binary-pulsar systems are orders of magnitude smaller than the tightest experimental constraints. The large-distance behavior is obtained as in Bekenstein's tensor-vector-scalar (TeVeS) model…
We consider structure formation in a nonlocal, metric-based realization of Milgrom's MOdified Newtonian Dynamics (MOND). We derive the general equations for linearized scalar perturbations about the $Λ$CDM expansion history. These equations are considerably simplified for sub-horizon modes, and it becomes obvious (in this model) that the MOND enhancement is not sufficient to allow ordinary matter to drive structure formation. We discuss ways in which the model might be changed to correct the problem.
