Classical and Quantum Gravity and Its ApplicationsView this Special Issue
On the Jeans Theorem and the “Tolman–Oppenheimer–Volkoff Equation” in Gravity
Corda, Mosquera Cuesta, and Lorduy Gòmez have shown that spherically symmetric stationary states can be used as a model for galaxies in the framework of the linearized gravity. Those states could represent a partial solution to the Dark Matter Problem. Here, we discuss an improvement of this work. In fact, as the star density is a functional of the invariants of the associated Vlasov equation, we show that any of these invariants is in its turn a functional of the local energy and the angular momentum. As a consequence, the star density depends only on these two integrals of the Vlasov system. This result is known as the “Jeans theorem.” In addition, we find an analogy of the historical Tolman–Oppenheimer–Volkoff equation for the system considered in this paper. For the sake of completeness, in the final section of the paper, we consider two additional models which argue that Dark Matter could not be an essential element.
The Dark Matter problem started in the 30s of the last century . When one observes the Doppler shift of stars which move near the plane of our galaxy and calculates the velocities, one finds a large amount of matter inside the galaxy which prevents the stars from escaping. That (supposed and unknown) matter generates a very large gravitational force that the luminous mass in the galaxy cannot explain. In order to achieve the very large discrepancy, the sum of all the luminous components of the galaxy should be two or three times more massive. On the other hand, one can calculate the tangential velocity of stars in orbits around the galactic centre like a function of distance from the centre. The result is that stars which are far away from the galactic centre move with the same velocity independent of their distance from the centre.
These strange issues generate a portion of the Dark Matter problem. In fact, either the luminous matter is not able to correctly describe the radial profile of our galaxy or the Newtonian theory of gravitation cannot describe dynamics far from the galactic centre.
Other issues of the problem arise from the dynamical description of various self-gravitating astrophysical systems. Examples are stellar clusters, external galaxies, and clusters and groups of galaxies. In those cases, the problem is similar, as there is more matter arising from dynamical analyses with respect to the total luminous matter.
The more diffuse way to attempt to solve the problem is to assume that the Newtonian gravity holds at all scales that should exist and that they should exhibit nonluminous components which contribute to the missing mass. There are a lot of names which are used to define such nonluminous components. The MAssive Compact Halo Objects (MACHOs) are supposed to be bodies composed of normal baryon matter, which do not emit (or emit little) radiation and drift through interstellar space unassociated with planetary systems . They could be black holes and/or neutron stars populating the outer reaches of galaxies. The Weakly Interacting Massive Particles (WIMPs) are hypothetical particles which do not interact with standard matter (baryons, protons, and neutrons) . Hence, they should be particles outside the Standard Model of Particles Physics but they have not yet been directly detected. Dark Matter is usually divided into three different flavors: Hot Dark Matter (HDM) , Warm Dark Matter (WDM) , and Cold Dark Matter (CDM) . HDM should be composed of ultrarelativistic particles like neutrinos. CDM should consist of MACHOs, WIMPs, and axions, which are very light particles with a particular behavior of self-interaction . If we consider the standard model of cosmology, the most recent results from the Planck mission  show that the total mass energy of the known universe contains ordinary matter, 26.8% Dark Matter, and 68.3% Dark Energy.
An alternative approach is to explain large-scale structure without dark components in the framework of Extended Theories of Gravity [10–17] and the references within. In other words, we call “Dark Matter” a gravitational effect that we do not yet understand as a modification to both Newtonian and Einsteinian gravity could be needed (see [10–17]). The underlying idea in Extended Theories of Gravity is that General Relativity is a particular case of a more general effective theory which comes from basic principles [10–17]. The standard Einstein-Hilbert action of General Relativity [18, 19] is modified by adding new degrees of freedom, like high order curvature corrections (the so-called theories of gravity [10–17, 20]) and scalar fields (the generalization of the Nordström–Jordan–Fierz–Brans–Dicke theory of gravitation [21–25], which is known as scalar-tensor gravity [26–28]). In this different context, one assumes that gravity is not scale-invariant and takes into account only the “observed” ingredients, that is, curvature and baryon matter. Thus, it is not required to search for candidates for Dark Matter which have not yet been found [10–17, 20]. In this perspective, the gravitational wave astronomy should be the ultimate test for the physical consistency of General Relativity or of any other theory of gravitation .
2. Theory of Gravity and Vlasov System
In the framework of theories of gravity, the theory is the simplest among the class of viable models with terms. Those models support the acceleration of the universe in terms of cosmological constant or quintessence as well as early time inflation [11, 15, 20]. Moreover, they should pass the Solar System tests, as they have an acceptable Newtonian limit, no instabilities, and no Brans–Dicke problem (decoupling of scalar) in scalar-tensor version. We recall that the theory was historically proposed in  with the aim of obtaining the cosmological inflation.
The theory arises from the action where represents the coupling constant of the term. In general, when the constant coupling of the term in the gravitational action (1) is much more minor than the linear term , the variation from standard General Relativity is very weak and the theory can pass the Solar System tests . In fact, as the effective scalar field arising from curvature is highly energetic, the constant coupling of the nonlinear term . In that case, the Ricci scalar, which represents an extra dynamical quantity in the metric formalism, should have a range longer than the size of the Solar System. This is correct when the effective length of the scalar field is much shorter than the value of mm . Hence, this effective scalar field results to be hidden from Solar System and terrestrial experiments. By analysing the deflection of light by the sun in the theory through a calculation of the Feynman amplitudes for photon scattering, one sees that, to linearized order, the result is the same as in standard General Relativity . By assuming that the dynamics of the matter (the stars making of the galaxy) can be described by the Vlasov system, a model of stationary, spherically symmetric galaxy can be obtained. This issue was described in detail in , in the framework of the theory. For the sake of completeness, in this section, we shortly review this issue.
In this paper, we consider Greek indices run from 0 to 3. By varying the action of (1) with respect to , one gets the field equations  By taking the trace of this equation, the associated Klein–Gordon equation for the Ricci curvature scalaris obtained , where is the d’Alembertian operator and the energy term, , has been introduced for dimensional motivations Hence, is positive . in (2) is the standard stress-energy tensor of the matter and General Relativity is easily reobtained for in (2).
As we study interactions between stars at galactic scales, we consider the linearized theory in vacuum (), which gives a better approximation than the Newtonian theory . Calling the linearized quantity which corresponds to , considering the plane wave with one can choose a gauge for a gravitational wave propagating in the direction in which a first-order solution of (2) with is given by the conformally flat line element We recall that the dispersion law for the modes of , that is, the second of (6), is that of a wave packet . The group velocity of a wave packet of centred in is From the second of (6) and (8), one gets which can be rewritten as If one assumes that the dynamics of the stars making of the galaxy are described by the Vlasov system, the gravitational forces between the stars will be mediated by metric (7). Thus, the key assumption is that, in a cosmological framework, the wave packet of centred in , which is given by the (linearized) space-time curvature, governs the motion of the stars . In this way, the “curvature” energy is identified as the Dark Matter content of a galaxy of typical mass energy g, in ordinary c.g.s. units . As g, from (4), one gets cm4 in natural units . Hence, the constant coupling of the term in action (1) is much more minor than the linear term and the variation from standard General Relativity is very weak. This implies that the theory can pass the Solar System tests as the effective length of the scalar field is mm .
Particles in space-time make up an ensemble with no collisions and are governed by a line element like (7) if the particle density satisfies the Vlasov equation [30, 32–34]where are the Christoffel coefficients, is the particle density, and is given by (), according to the relation [30, 32–34]
In general, the Vlasov-Poisson system is introduced through the system of equations [30, 32–34] In (16), is the time and and are the position and the velocity of the stars, respectively. is the average Newtonian potential generated by the stars. System (16) represents the nonrelativistic kinetic model for an ensemble of particles (stars in the galaxy) with no collisions. The stars interact only through the gravitational forces which they generate collectively and are considered pointlike particles, and we neglect the relativistic effects [30, 32–34]. The function in the Vlasov-Poisson system (16) is nonnegative and gives the density on phase space of the stars within the galaxy .
In this approach, the first-order solutions of the Klein–Gordon equation (3) for the Ricci curvature scalar are considered like galactic high energy scalarons, which are expressed in terms of wave packets having stationary solutions within the Vlasov system . The energy of the wave packet is interpreted like the Dark Matter component which guarantees the galaxy’s equilibrium . This approximation is not as precise as one would aspect , but here we consider it as the starting point of our analysis.
As one is interested in stationary states, one calls the wavelength of the “galactic” gravitational wave (7), that is, the characteristic length of the gravitational perturbation . One further assumes that being the galactic scale of order light-years [35, 36]. In other words, the gravitational perturbation can be considered “frozen-in” with respect to the galactic scale .
In that way, one can write down the system of equations defining the stationary solutions of (17) :Therefore, the idea in  is that the spin-zero degree of freedom arising from the term in the gravitational Lagrangian, that is, the scalaron, is a potential candidate for the Dark Matter. In this approach, the dominant contribution to the curvature within a galaxy comes from the scalaron field equation (3). That equation has a proper baryon source term. This enables the baryons themselves to evolve obeying a collisionless Boltzmann equation. Then, the baryons can propagate on the curvature generated by the scalaron.
3. The Jeans Theorem and the “Tolman–Oppenheimer–Volkoff Equation”
Let us consider the stationary solution of our stellar dynamics model, that is, (18), (19), and (20). The particle density is a functional of the invariants of the Vlasov equation (14). The Jeans theorem states that any of these invariants must be a functional of the local energy and the angular momentum. In that way, the particle density depends only on these two integrals of the system under consideration. In order to prove these statements, one introduces the new coordinatesWe note that, with being spherically symmetric, one can write it as a function of , , ; that is, . Thus, (20) in the new coordinates readsThis equation has the same form of equation in  with . Thus, by applying the result in , one obtains that must have the formwhereReturning to the system of (18), (19), and (20), one gets immediatelywhereis the local energy of the particles andis the modulus squared of the local angular momentum. Using (27) and a transformation of variables, system (21) becomesA direct computation permits obtainingwhich is analogous to the historical Tolman–Oppenheimer–Volkoff equation [38, 39] for the system under analysis in this paper.
For the sake of completeness, we stress that the Jeans instability (gravitational stability) has been analysed in the context of modified theories of gravity for rotating/nonrotating configurations in [40–42]. Related questions on the Tolman–Oppenheimer–Volkoff equation and the Jeans instability (but not for plane waves) were also studied in [43–45]. The Tolman–Oppenheimer–Volkoff equation has been also analysed in the dimensional gravity’s rainbow in the presence of cosmological constant in  and in the framework of dilaton gravity in .
4. Two Additional Models
For the sake of completeness, we play the devil’s advocate and consider two models which argue that Dark Matter is not an essential element, even though popular models postulate that it comprises roughly a fourth of a universe. Our starting point is the relation [48, 49]where is the present value of , is the present age of the universe, and is the time elapsed from the present epoch. Similarly, one could deduce that In this scheme, the gravitational constant varies slowly with time. This is suggested by Sidharth’s 1997 cosmology , which correctly predicted a Dark Energy driven accelerating universe at a time when the accepted paradigm was the Standard Big Bang cosmology in which the universe would decelerate under the influence of Dark Matter.
We reiterate the following: the problem of galactic rotational curves [49, 50]. We would expect, on the basis of straightforward dynamics, that the rotational velocities at the edges of galaxies would fall off according toHowever, it is found that the velocities tend towards a constant value:This, as known, had led to the postulation of the as yet undetected additional matter, the so-called Dark Matter. We observe that from (33) it can be easily deduced that as we are considering infinitesimal intervals and nearly circular orbits. Equation (36) shows that there is anomalous inward acceleration, as if there is an extra attractive force, or an additional central mass .
Thus, From (37), it follows thatEquation (38) shows that, at distances within the edge of a typical galaxy, that is, cm, (34) holds but as we reach the edge and beyond, that is, for cm, we have cm/sec, in agreement with (35).
Then, the time variation of explains observation without invoking Dark Matter. It may also be mentioned that other effects like the Pioneer anomaly and shortening of the period of binary pulsars can be deduced , while new effects also are predicted.
Milgrom  approached the problem by modifying Newtonian dynamics at large distances. This approach is purely phenomenological. The idea was that perhaps standard Newtonian dynamics work at the scale of the Solar System but, at galactic scales involving much larger distances, the situation might be different. However, although it produces the asymptotically at rotation curves of galaxies, a simple modification of the distance dependence in the gravitation law, as pointed out by Milgrom, should be not sufficient. Such a law would predict the wrong form of the mass velocity relation. So, Milgrom suggested the following modification to Newtonian dynamics: a test particle at a distance from a large mass is subject to the acceleration given bywhere is acceleration such that standard Newtonian dynamics are a good approximation only for acceleration values much larger than . The above equation however would be true when is much less than . Both statements can be combined in the heuristic relationIn (40), when , and when . It is worthwhile to note that (39) and (40) are not deduced from any theory but rather are an ad hoc fit to explain observations. Interestingly, it must be mentioned that most of the implications of the Modified Newton Dynamics (MOND) do not depend strongly on the exact form of .
It is interesting to note that there is an interesting relationship between the varying approach, which has a theoretical base, and the purely phenomenological MOND approach. Let us writeHence,So, At this stage, we can see a similarity with MOND. If , we are with the usual Newtonian dynamics and if then we get back to the varying case exactly as with MOND.
5. Concluding Remarks
The results in  have shown that spherically symmetric stationary states can be used as a model for galaxies in the framework of the linearized gravity. Those states could, in principle, be a partial solution to the Dark Matter problem. In this paper, an improvement of this work has been discussed. As the star density is a functional of the invariants of the associated Vlasov equation, it has been shown that any of these invariants is in turn a functional of the local energy and the angular momentum. Then, the star density depends only on these two integrals of the Vlasov system. This result represents the so-called “Jeans theorem.” In addition, an analogy of the historical Tolman–Oppenheimer–Volkoff equation [38, 39] for the system considered in this paper has been discussed. We tried this extension of previous work in  because, on the one hand, the Jeans theorem is important in galaxy dynamics and in the framework of molecular clouds . On the other hand, the historical Tolman–Oppenheimer–Volkoff equation constrains the structure of a spherically symmetric body of isotropic material which is in static gravitational equilibrium, as modelled by metric theories of gravity, starting from the general theory of relativity [38, 39]. Thus, a viable extended theory of gravity, like the gravity, must show consistence with these two important issues.
For the sake of completeness, in Section 4 of this paper, two additional models which argue that Dark Matter could not be an essential element have been discussed. In fact, Dark Matter is considered a mysterious and controversial issue. There are indeed a minority of researchers who think that the dynamics of galaxies could not be determined by massive, invisible Dark Matter halos (see [10, 11, 51–59]). We think that, at the present time, there is not a final answer to the Dark Matter issue. In other words, it is undoubtedly true that the universe exhibits a plethora of mysterious phenomena for which many unanswered questions still exist. Dark Matter is an important part of this intriguing puzzle. Thus, when one works in classical, modern, and developing astrophysical and cosmological theories, it is imperative to repeatedly question their capabilities, identify possible shortcomings, and propose corrections and alternative theories for experimental submission. In the procedures and practice of scientific professionals, no such clues, evidence, or data may be overlooked.
Finally, we take the chance to stress that important future impacts which could help in a better understanding of the important Dark Matter issue could arise from the nascent gravitational wave astronomy . The first direct detection of gravitational waves by the LIGO Collaboration, the so-called event GW150914 , represented a cornerstone for science and for astrophysics in particular. We hope that the gravitational wave astronomy will become an important branch of observational astronomy which will aim to use gravitational waves to collect observational data not only about astrophysical objects such as neutron stars and black holes, but also about the mysterious issues of Dark Matter and Dark Energy. In order to achieve this prestigious goal, a network including interferometers with different orientations is required and we hope that future advancements in ground-based projects and space-based projects will have a sufficiently high sensitivity [61–63]. For the benefits of the reader, we also signal two important works on self-gravitating systems  and on Jeans mass for anisotropic matter .
The authors declare that there are no competing interests regarding the publication of this paper.
Christian Corda has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Research Project no. 1/4717-113.
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