Dynamical Friction in Nonlocal Gravity

Imagine the scattering state of a Newtonian two-body system. Under the Newtonian inverse-square law of gravity, the bodies deviate from their original paths during the scattering process, but the mechanical system is conservative and the overall positive energy is conserved. Let us choose one of the bodies as the reference and focus attention on its initial momentum vector. In the scattering process, the component of the final deflected momentum of the reference body along the direction of its initial momentum does not change to first order in the gravitational coupling constant $G$, but decreases beyond the linear order. The net loss of momentum of the reference particle along its initial direction of motion is the source of the dynamical friction force. Normally, dynamical friction becomes physically significant when an astronomical body of mass $M$ moves relatively slowly through a population of stars of average mass $m\ll M$. It was first calculated in a collisionless gravitational system by Chandrasekhar Chandra . For the applications of dynamical friction in galactic dynamics, see Binney . The main purpose of this paper is to study dynamical friction within the framework of nonlocal gravity theory and derive the analogue of Chandrasekhar’s formula in the Newtonian regime of NLG. We now turn to a brief introduction of NLG and its Newtonian regime.

In the special and general theories of relativity, physics is local Einstein . For instance, the standard approach for extending relativity theory to an accelerated system in Minkowski spacetime involves the pointwise application of Lorentz transformations, which makes physical sense if the velocity of the accelerated system is in effect locally uniform during an elementary act of measurement. The locality assumption is certainly valid for pointlike coincidences of classical point particles and rays of radiation. However, wave phenomena are intrinsically nonlocal by the Huygens principle; moreover, such nonlocality extends to the measurement of the electromagnetic field. To determine the frequency content of an incident wave packet, for example, an accelerated observer must in general employ instantaneous Lorentz transformations over an extended period of time, during which the state of the observer varies continuously. At the same time, there are invariant local acceleration scales of length and time associated with the accelerated system. Therefore, accelerated systems are in general nonlocal and the past history of accelerated motion must be taken into account Mashhoon:1993zz , thus leading to nonlocal special relativity Mashhoon:2008vr . The deep connection between inertia and gravitation, as revealed in Einstein’s development of general relativity Einstein , suggests that gravity should be nonlocal as well. The most natural approach to a nonlocal gravity theory would be to introduce history dependence via a nonlocal constitutive relation in close analogy with the nonlocal electrodynamics of media Jackson ; L+L ; HeOb . Thus, nonlocal gravitational field equations would reflect an influence (“memory”) from the past that endures. History dependence could in fact be a natural feature of the universal gravitational interaction.

To implement this idea, the first step would involve expressing Einstein’s general relativity (GR) in a form that resembles Maxwell’s electrodynamics. This can be simply accomplished by the introduction of a preferred frame field in spacetime and employing an extended geometric framework known as teleparallelism. Indeed, there is a well-known teleparallel equivalent of general relativity (TEGR), which is the gauge theory of the group of spacetime translations Cho . Therefore, TEGR, though nonlinear, is formally analogous to electrodynamics and can be rendered nonlocal via history-dependent constitutive relations as in the nonlocal electrodynamics of media Hehl:2008eu ; Hehl:2009es . In the resulting theory of nonlocal gravity (NLG), the gravitational field is locally defined but satisfies partial integro-differential field equations. The only known exact solution of NLG is the trivial solution; that is, we simply recover Minkowski spacetime in the absence of gravity. Thus far, the nonlinearity of NLG has prevented finding exact solutions for strong-field regimes such as those involving black holes or cosmological models Bini:2016phe . However, linearized NLG and its Newtonian limit have been the subject of extensive investigations Blome:2010xn ; Rahvar:2014yta ; Mashhoon:2014jna ; Chicone:2015coa ; MaHe . A comprehensive account of NLG is contained in BMB .

When nonlocal TEGR is expressed as modified GR, the source of gravity turns out to include besides the standard symmetric energy-momentum tensor $T_{\mu\nu}$ of matter, certain purely nonlocal gravity terms which we interpret in terms of nonlocally induced effective dark matter. That is, the nonlocal aspect of gravity appears to simulate dark matter. What is now considered dark matter in astrophysics and cosmology may indeed be the manifestation of the nonlocal component of the gravitational interaction. This circumstance constitutes a significant observational consequence of NLG and has been explored in connection with the gravitational physics of nearby spiral galaxies and clusters of galaxies Rahvar:2014yta . Though much remains to be done, thus far NLG appears to be consistent with observational data regarding the solar system as well as nearby galaxies and clusters of galaxies. However, there are still many issues regarding the astrophysical and cosmological implications of nonlocal gravity that must be resolved Chicone:2015sda ; Chicone:2017oqt ; Roshan:2019xda ; Ghafourian:2020uae ; Roshan:2021mfc .

It follows from Eq. (4) that in the Newtonian regime of NLG, Newtonian gravitation theory can be employed provided we take due account of the corresponding effective dark matter as well. Indeed, every point particle of mass $m$ is surrounded by a spherically symmetric distribution of effective dark matter of density $m\,q(r)$, where $r$ is the radial coordinate in a spherical polar coordinate system centered on the point particle $m$. The net amount of effective dark matter is $m_{D}=m\,\alpha_{0}\,w$, where $\alpha_{0}\approx 11$. If we choose $q=q_{0}$, then $w=1$. The result is slightly less if we choose $q_{1}$ or $q_{2}$, see Fig. 7.2 of BMB . In any case, because of spherical symmetry, the center of mass of the total effective dark matter associated with $m$ is exactly at the location of $m$.

If we now have a Newtonian system of $N$ baryons, say, each of inertial mass $m_{i}$, $i=1,2,\cdots,N$, then each baryon brings with it the accompanying spherically symmetric effective dark matter of mass $m_{i}\,\alpha_{0}\,w$ centered on the baryon. Since $\alpha_{0}\,w$ is a constant and the same for each baryon, the Newtonian center of mass coincides with the center of mass of the total effective dark matter. This will not be true, however, if we ask for the effective dark matter inside a galaxy, for example; then, the range of integration of the effective dark matter will be restricted by the boundary of the galaxy and instead of $\alpha_{0}\,w$ for each baryon, we will get a fraction of this quantity depending on the location of the baryon $m_{i}$ in the galactic system. Therefore, for a galaxy the baryonic center of mass is in general different from the center of mass of the corresponding effective dark matter.

For any isolated system of $N$ baryons, we have the total baryonic mass $M_{B}$ and the corresponding total effective dark matter $\alpha_{0}\,wM_{B}$ over all space, where $w<1$. However, the amount of dark matter, $M_{D}$, confined within the system depends on the shape and volume of the system such that

The dynamic mass of the system is thus given by $M_{B}+M_{D}=M_{B}(1+\phi)$, where $\phi$ for the system is defined by

In NLG, we have the basic result that for any finite system the fraction $\phi$ must be smaller than $\alpha_{0}\approx 11$. If the system is so huge that most of the baryonic matter is at distances larger than $\mu_{0}^{-1}\approx 17$ kpc, then we expect that the effective dark matter fraction approaches the limiting value of $\alpha_{0}\,w$. Moreover, very far away from a baryonic system of mass $M_{B}$, it exhibits an effective gravitational mass that approaches $M_{B}\,(1+\alpha_{0}\,w)$. Therefore, for a cluster of galaxies, NLG predicts that we should have $\phi\approx\alpha_{0}\,w$. This appears to be consistent with observation of nearby clusters of galaxies Rahvar:2014yta . The standard definition of dark matter fraction in astrophysics is

Let us note that $f_{DM}$ as a function of $\phi$ increases monotonically from zero and approaches unity as $\phi\to\infty$. In NLG, $\phi<\alpha_{0}$; hence, $f_{DM}$ is less than $\alpha_{0}/(1+\alpha_{0})$. Therefore, for $\alpha_{0}\approx 11$, the effective dark matter ratio in NLG is such that $f_{DM}<0.917$.

These considerations are naturally reflected in the gravitational force (16) on a point mass $m$ due to a point mass $m^{\prime}$ located at $\mathbf{r}$. For $r\gg 1/\mu_{0}$, Eq. (16) reduces to

That is, as the distance $r$ between $m$ and $m^{\prime}$ increases, the magnitude of $\mathbf{F}_{\rm NLG}$ in Eq. (16) slowly increases from the Newtonian $1/r^{2}$ and for $r\gg\frac{1}{\mu_{0}}\approx 17$ kpc becomes $1/r^{2}$ again, but with $m^{\prime}\to m^{\prime}\,(1+\alpha_{0}\,w)$. The physical interpretation of this result is that $m$ is now subject to the attractive Newtonian gravitational force of $m$ as well as the total spherically symmetric effective dark matter distribution associated with $m^{\prime}$. For $r\to\infty$, this net amount of dark matter becomes $m^{\prime}\,\alpha_{0}\,w$, as expected, and hence the effective gravitational mass of $m^{\prime}$ becomes $m^{\prime}\,(1+\alpha_{0}w)$. As is evident in the force formula above, this situation is symmetric between $m$ and $m^{\prime}$ and hence our discussion could be repeated for the gravitational force experienced by $m^{\prime}$.

It is interesting to mention an alternative explanation of the behavior of the two-body gravitational force that is important for the consideration of dynamical friction that is the focus of the present work. We can imagine that the strength of the gravitational coupling increases with distance in NLG; that is,

so that the gravitational coupling “constant” ranges from $G$ to $G(1+\alpha_{0}\,w)$ as $r$ goes from $0$ to $\infty$.

Let us briefly digress here and mention the connection of these results with Fourier space. From

we note that $\hat{\rho}(0)=M_{B}$ and similarly $\hat{\rho}_{D}(0)$ is the total mass of dark matter over all space, namely, $\alpha_{0}\,w\,M_{B}$. The convolution theorem implies $\hat{\rho}_{D}(0)=\hat{q}(0)\,\hat{\rho}(0)$, which is consistent with the fact that $\hat{q}(0)=\alpha_{0}\,w$. Moreover, $\hat{q}_{0}(0)>\hat{q}_{1}(0)>\hat{q}_{2}(0)$, a relation that actually holds for any $\mathbf{k}$ BMB .

Finally, let us imagine an isolated astrophysical system (“galaxy”) consisting of $N$ particles and let $D$ be the baryonic diameter of the system, namely, the diameter of the smallest sphere that completely surrounds the baryonic system at the present time. For each baryon $m_{i}$ in the system, $i=1,2,\cdots,N$, the corresponding effective dark matter that is within the confines of the system and contributes to the dynamic mass of the whole system is evidently less than

This means that the total effective dark matter in the galaxy, $M_{D}$, is such that

On the other hand, $q<q_{0}$ implies

Moreover, Eq. (21) can be written as

where

is a function that starts from $H(0)=0$ at $D=0$, monotonically increases with increasing $D$ and asymptotically approaches unity as $D\to\infty$. It follows that in the Newtonian regime of NLG, we have

For systems with $D\gg 1/\mu_{0}\approx 17\,$kpc, such as for nearby clusters of galaxies, $H(D)\approx 1$ and we find

in agreement with the relation $\phi\approx\alpha_{0}\,w$ discussed earlier for such systems. On the other hand, for systems in the intermediate regime where $D$ is less than or comparable to $1/\mu_{0}\approx 17\,$kpc, we note that $H(D)$ always stays below the line $\mu_{0}\,D/2$ for $D>0$. Hence,

where $\lambda_{0}\approx 3$ kpc. Thus for a globular star cluster, the effective amount of dark matter is at most a few percent of the baryonic content of the globular cluster. For a dwarf galaxy, the effective amount of dark matter could be at most comparable to its baryonic content. For giant galaxies and clusters of galaxies, $\phi$ can at most approach $\alpha_{0}\,w$. Further discussion of these issues is contained in BMB . For recent observational advances in connection with dwarf galaxies, see vanDokkum:2018vup ; Guo:2019wgb ; Pina:2019rer ; Hammer:2020qcd ; Shen:2021zka and the references cited therein; the prediction of NLG that there is less dark matter in dwarf galaxies seems to be consistent with these studies.

The purpose of this section is to discuss the gravitational two-body system in accordance with the Newtonian limit of NLG in order to derive the analogue of Chandrasekhar’s formula for dynamical friction. For the sake of simplicity and convenience, we follow closely the standard treatment presented in Binney . An astronomical body of mass $M$ moves without direct collision through a population of stars, each with a typical mass $m$. The result of two-body interaction of $M$ with a star of mass $m$ will then be extended to include all of the stars.

There is an alternative way to view Chandrasekhar’s dynamical friction formula. As the reference body of mass $M$ moves through an infinite homogeneous medium consisting of stars of average mass $m$, the ensuing disturbance leads to a density enhancement in the reference body’s wake, thereby slowing it down via gravitational attraction. This approach has been developed by a number of authors; see TW and the references cited therein. To determine the role of NLG in this process, we follow the treatment presented in TW .

In the background medium, we ignore the gravitational interaction between the stars. Therefore, the stars in their unperturbed states move according to $\mathbf{x}=\mathbf{x}_{0}+\mathbf{v}_{m}t$ with constant momenta $\mathbf{p}=m\mathbf{v}_{m}$ starting from $t=-\infty$ with $\mathbf{x}_{0}$ ranging over all space. The motion of the reference body generates a density perturbation

where $\beta>0$ is a constant auxiliary parameter that we need in the course of our calculations; however, we will eventually assume that $\beta\to 0$. In the Newtonian regime of NLG, Poisson’s Eq. (4) takes the form

where $q$ is the reciprocal kernel. Working in Fourier space, we find

where we have used the fact that the reciprocal kernel is spherically symmetric by assumption and its Fourier integral transform $\hat{q}(k)$ is thus only a function of $k=|\mathbf{k}|$. Neglecting any interaction between the stars, the motion of a star is perturbed by the gravitational influence of the reference body alone. In general,

We can express the resulting perturbation in $\mathbf{x}$ and $\mathbf{p}$ of a star in powers series in terms of the gravitational coupling constant $G$. That is,

where, for instance, $\Delta_{n}\mathbf{p}$ is the momentum perturbation of order $n$. Substituting Eq. (57) in Eq. (56), the first-order perturbation in $\mathbf{p}$ can be obtained from Eq. (56) with unperturbed $\mathbf{x}=\mathbf{x}_{0}+\mathbf{v}_{m}t$. Therefore,

where

It is important to note that the net force of gravity on the stars vanishes to first order in $G$. For instance, the $j$-component of the net force can be calculated by integrating Eq. (58) over all the stars. To this end, let us multiply the right-hand side of Eq. (58) by $N\,d^{3}x_{0}$, where $N$ is the constant density of stars. The integration over the $j$-component of $\mathbf{x}_{0}$ results in $\xi\,\delta(\xi)=0$ in the integrand, where $\xi=k^{j}$. Thus, the net rate of momentum transfer is zero to linear order in $G$. We therefore proceed to the evaluation of this quantity to second order in $G$.

The second-order perturbation in the momentum $\Delta_{2}\mathbf{p}$ can be determined from

where $K_{jl}=\partial^{2}\Phi/\partial x^{j}\partial x^{l}$ is the tidal matrix evaluated along the unperturbed path of a star. Moreover, the first-order change in the position of a star can be calculated by integrating the force Eq. (58) twice over time starting from $t=-\infty$. The result is

To evaluate this second-order rate of momentum transfer, let us first note that

where $\omega^{\prime}:=\mathbf{k^{\prime}}\cdot(\mathbf{v}_{M}-\mathbf{v}_{m})+i\beta$. It proves convenient at this point to replace $\mathbf{k^{\prime}}$ by $-\mathbf{k^{\prime}}$ and hence $\omega^{\prime}$ by $-\omega^{{}^{\prime}*}$ in Eq. (62); then, combining Eqs. (61) and (62) we find

To find the net rate of momentum transfer to second order, we must sum over all the stars; therefore, let us integrate Eq. (63) over $N\,d^{3}x_{0}$. The result is

where we have used the fact that $i(\omega^{{}^{\prime}*}-\omega)=2\beta$ when $\mathbf{k^{\prime}}=\mathbf{k}$. In Appendix A, we prove a useful identity, namely,

Substituting this relation in Eq. (64) and taking advantage of the following representation of Dirac’s delta function:

we find by letting $\beta\to 0^{+}$ that $\exp(2\beta t)\to 1$ and