l.scirp.org/file/2-7502479x249.png" /> is constant. So, we can re-write relation (41) to
If, we can analogously divide integral (41) into two parts. This time, the first integration runs from to and the second from to r. Let us denote the result of the first, again numerical integration by and use the OSS to express the components and of metric tensor in the second integral. Here, is again constant. The form of relation (41) changes to
In the last formula, term can be regarded as constant (it does not change, when we investigate a change of potential energy of a test particle in region) and we can see that the dependence of the potential on radial distance, r, is the same as in the effective potential introduced in the beginning of Section 5.2, i.e.
. The new potential, corresponding to the effective potential in region, differs from the later by its constant part. Such a difference is not important from the physical point
of view, because it disappears, when we calculate a difference of potential in two points. We know that there is also a difference in the constant term between the Newtonian potential, , and approximation of the effective potential for a weak field, which equals (assuming the gauging of in relation (34) as). The approximation of the effective potential contains constant term, which absents in its Newtonian counterpart. Anyway, both forms can be used in a practical calculation of energy gain or loss, because they give the same result.
The behavior of the new potential in the example of Ni’s hollow sphere presented is Section 3 is shown in Figure 2. We can observe that it acquires only positive values in the entire interval of radial distance (and zero in).
To establish the new potential, which is mathematically expressed by Formula (41), we propose to rule out the physical quantity “mass” from the GR and regard the quantity u, established originally by Oppenheimer and Volkoff  , as only an auxiliary metric quantity. In accord with its original definition (see relation (5)), this quantity replaces the auxiliary metric quantity in the EFEs and, therefore, it can be represented as the alternative parameter characterizing the metrics. In this new representation, the conversion of to mass M according to relation (22) is only formal, giving one aspect of metrics. We suggest to strictly distinguish between mass M, which is the quantity existing within the Newtonian physics, and GR parameter u (also). Although both parameters interact in the gauging, its physical meaning is different and should not be treated as identical (when unit G and c are used).
If we regard the negative values of u as physically acceptable, we accept, in fact, that the size of -component
of metric tensor, , can also be smaller than unity. Actually, there is no reason in the GR of why the in-
equality should always be valid.
Figure 2. The behavior of the newly defined gravitational potential in the GR in the example of the Ni’s hollow sphere presented in Section 3. The blue thick solid curve shows the potential inside the object’s body. This potential is calculated within the numerical integration of the EFEs by using relation (41). The dashed green (dashed violet) curves shows the potential in the region of radial distance (), which is calculated by relation (42) (relation (43)).
Within the GR, the concepts of energy (of all kinds) and potential remain. Since it has been empirically found that constant, appearing in the solutions of EFEs exclusively in the region, is always negative, arguments of the second square roots in relation (42) are always positive, therefore the potential is always real-valued.
When the quantity u is regarded as the auxiliary quantity characterizing the metrics, the new gravitational potential can be proved as always positive quantity, even in the region where. Thus, the requirement of the physically acceptable gravitational potential is obeyed in the whole space, from the center of hollow sphere up to an infinite distance.
5.4. Remarks Concerning the Mass Elimination
To support our suggestion from the previous subsection about the ruling out the mass as the physical quantity from the GR, we remind the suggestion, published in our earlier paper (arXiv: 1206.0405v1 [physics.gen-ph]), to eliminate not only the mass from the physics, but the electric charge and some fundamental physical constants as well.
We know, the fundamental laws, like the Newton force law, Coulomb law, or law of inertia, can exclusively be experimentally verified in a combination of one, e.g. Newton, force law and the law of inertia. In other words, we can verify the prediction made by the solution of the corresponding equation of motion. So, the real physical theory is never represented by a single law, but by an equation (or a set of equations).
Let us explain the principle of the application of fundamental physical laws in the example of two electrically charged, static particles. In this example, the laws are used to describe their dynamics. We determine the acceleration of the first, “test” particle (TP), when influenced by the second, “acting” particle (AP). The TP has mass and charge and AP has mass and charge. The AP acts on the TP by its gravity as well as electrically. Initially, both particles are assumed to be in rest in mutual distance r. The relevant equation of motion for the TP reads
where is the change of the velocity of the TP due to the action of the AP during a short time interval and is the permittivity of vacuum (the equation is given in the SI units).
We assume that the AP and TP are composed of matter consisting only of elementary particles of k kinds, whereby the mass of j-th kind is. While the AP consists of particles of j-th kind, the TP consists of particles of j-th kind. Taking into account such the composition, the masses and can be given as
Further, if the AP consists of () carriers of positive (negative) elementary charge and the TP consists of () carriers of positive (negative) elementary charge, then the charges and can be given as and. The elementary electric charge (charge of proton) is denoted by. Now, Equation (44) can be re-written to
The dimensional analysis enables combining the fundamental physical constants to obtain special length, time, and mass, which are known as the Planck length, , Planck time, , and Planck mass,. Specifically, these quantities are defined by
where is the Planck constant, h, divided by. The Planck time is supposed to be the shortest time interval in the universe. So, there should not be problem to identify the time interval in Equation (44) to the Planck time, i.e. to put.
In quantum physics, a wave is associated to every elementary particle. The angular frequency, , of this wave is related to the particle’s mass, m, according to the de Broglie’s relation   . Since the angular frequency is related to the wavelength, b, as, mass m can also be given as
If we use this relation to convert masses to the corresponding wavelengths, and assume the identity, equation of motion (45) changes to
In the last term, the fraction can be re-written with the help of the dimensionless fine-structure constant, , as. Taking this possibility into account, using the relation giving the Planck length, and multiplying (48) by, this equation can be written in the form
We can see that the last equation is dimensionless and the fundamental constants G, , and disappeared. Only the dimensionless fine-structure constant and the ratios of various lengths and velocities figures there, except of the numbers of particles and charge carriers. Because the universe is not static, the occurrence of the velocities, which represent the change (the first derivative of length) and change of change (the second derivative of length), is obvious in the description.
The fundamental equations yielding the TOV model (11)-(13) can also be re-written to the dimensionless form. If we denote the wavelength of the wave associated with neutron by, then the product of constants and, which equals, can be converted, using Formula (47), to. With this new form of and after multiplication of Equations (11) and (13) by, the three fundamental equations acquire form
Because this is the static problem, the equations contain only the ratios of lengths and derivatives with respect to length.
The GR was originally intended to be formulated as a geometric theory and Equation (49) as well as Equations (50)-(52) indicate the way toward the GR as the pure geometric theory. Mass does not seem to be any component of such a theory. Hence, our proposal to rule out this quantity from any GR description and argumentation appears to be reasonable.
We note that such the units as meter, second, or kilogram were defined by man and, thus, cannot be regarded as the “natural” physical units. As well, the quantities as mass or electric charge, which were also established by man, can be, in fact, only the artificial quantities not really existing in the nature. Instead, the quantities as length, frequency, their change, and change of their change seem to be natural. And, concerning the physical units, the above mentioned Planck length, Planck time, and speed of light are, likely, the “natural” units.
In the two schematic examples presented above, we could see that if the man-established quantities are replaced by the natural quantities, expressed in the natural units (with the help of Planck length and speed of light) then the fundamental constants, as the gravitational constant, permittivity of vacuum, and Planck constant, simply disappear from the equations describing a physical problem. This circumstance implies that these constants are, most probably, only the transformation constants between the artificial, man-established and natural quantities.
6. Full-Sphere versus Hollow-Sphere Models
In practice, every inward-proceeded part of the numerical integration of Equations (11)-(13), when it is started in a distance larger than zero, ends with the implication of the inner physical surface. (In principle, the numerical integration from a finite distance can end in the center with finite energy density and pressure. This is, however, only a single of infinite variety of possibilities. If there is no special attention to the choice of initial conditions, the probability of its occurrence approaches zero.) Such the property can be also expected for some other equations of state, not only the Chandrasekhar’s Equation (7) and Equation (8). (Actually, the same qualitative behavior occurs using, e.g., the polytrope. And, the energy-density and pressure maximum in a finite distance also occurs for the equation state of radiation,.) On the other-hand side, there is possible to construct a model of NS in the form of full sphere, if the integration starts in the center. (Then, we force the single of the infinite variety of possibilities to be the case.) In this section, let us reveal the conditions implying the full-sphere model of NS.
The realistic solution of the Equations (11)-(13) in the TOV problem has to satisfy the following demands. The pressure and energy density inside the object must acquire only finite positive values and have to decrease to zero at the object’s physical surface. In the full-sphere model, these quantities must be finite in the center. The last requirement implies the asymptotic behavior and for., , , and are the constant coefficients. Because the energy density approaches the constant for, Equation (7) and Equation (12) yield in this limit, where is an integration constant. If, then (see Equation (6)) for. Therefore, it is necessary to demand. Consequently, for.
When and, therefore, , function, calculated from relation (5), approaches for. For, , and, Equation (6) implies and, further, for. This constant is unique for the object of given mass (implied by the given value of the maximum Fermi impulse, , in the object’s center), because the metrics is continuous in the outer surface only for a single specific behavior of. Hence, the component of metric tensor converges to its value for the flat spacetime and converges to a unique constant. So, the metrics of the full-sphere NS must obey and in the limit.
We note that a difference between the full-sphere and hollow-sphere models can be practically negligible. In Figure 3, there is an example of such two similar models. In the hollow-sphere model, the distance of zero net gravity, , is chosen in the distance of 1 meter from the center. In this distance, the input value of Fermi impulse is chosen to be the same as in the center of full-sphere model. We put it to equal to, e.g.,. In Figure 3, we can see that the models are practically identical from the outer radius of both spheres down to the star-centric distance. Of course, there can be created, in principle, a hollow-sphere model with an arbitrarily small. The interval of similarity can be prolonged to even a shorter distance.
Since can be even smaller than a step of numerical integration, , we can sometimes construct a model in form of a quasi full sphere. It is a hollow sphere, in fact, but one cannot recognize its true character because.
We can conclude that the border between the traditionally unacceptable and acceptable models seems to be only formal: the model with exactly zero is acceptable, but almost identical model with a negligibly small, but finite is not longer acceptable in the traditional concept of NSs. One can, however, seriously doubt whether the hollow sphere model of NS is actually unacceptable and postulate P2 is an actually reasonable requirement for a realistic model of NS.
7. A Wider Variety of the Ni’s Models of Neutron Stars
In this section, we introduce several sequences of the NS models constructed by using the Ni’s solution of the TOV equations in course to map some properties of them. To obtain the models, we again start the numerical integration of Equations (11)-(13) in the distance of zero net gravity, , where the initial value of function u is given by relation (25). The values of are chosen to be 1 m, 1, 5, 10, 15, 20, and 20 km. For each value of,
Figure 3. The comparison of almost identical TOV full-sphere and Ni’s hollow-sphere models of NS. The top, middle, and bottom plots show the behavior of density, component, and components of metric tensor, respectively. The left (right) plots show the behavior of the quantities in the linear (decadic-logarithm) scale of distance r. While the thick, solid, blue line shows the behavior of the quantities inside the NS body for the hollow-sphere model, the thick, dashed, red line does so for the full-sphere model. The thin, dotted lines touching the ends of the thick curves show the behavior of the components of metric tensor in the corresponding OSS, the violet (green) curve for NS-exterior vacuum in the outer (inner) space.
we integrate the sequence of models considering the series of input values of Fermi impulse from 0.2 to, with the step of. For each model, we perform an iteration to find the appropriate input value of, which yields the continuous metrics at the outer physical surface of NS.
The result for some considered values of is illustrated in Figure 4. This figure shows the relation between the size of the object in given model and its gravitational mass. The pair of circles linked together by a solid line show the extent of the neutron object in given model. The square between these circles shows the position of the
Figure 4. The size of NSs of various mass. Each pair of circles linked with a solid horizontal line shows the position of inner (left circle), , and outer (right circle), , radius in the model of given mass, M. The square between both circles shows the radial distance with zero net gravity and maximum energy density and pressure. The dotted straight-line shows the behavior of the Schwarzschild gravitational radius. The plots (a) and (c) show the same models as plots (b) and (d), respectively, but the distance scale in plots (b) and (d) is logarithmic.
distance, where the net gravity is zero and energy density and pressure are largest. The dotted straight-line shows the behavior of the Schwarzschild gravitational radius in the dependence on the mass.
For a relatively small value of (Figures 4(a)-(d) for and), the outer radius, , monotonously decreases with increasing maximum Fermi impulse,. On the other-hand side, the mass is not any monotonous function of, but reaches a maximum and again decreases with increasing.
For a relatively large value of (Figure 4(e) for and 20 km; but it is found that the same is qualitatively also valid at least for, 15, and 25 km), the outer (inner) radius decreases (increases) with increasing at small value of. For -values above a critical value, the outer (inner) radius on contrary increases (decreases) with increasing. In Figure 4(e), one can see that the outer radius asymptotically approaches, from outside, to the Schwarzschild gravitational radius. We can expect that both radii become identical in the limit of and, consequently, in the limit of infinite internal energy of object.
Using the Ni’s solution, it appears that there is a variety of NSs constituted by the same number of neutrons. In other words, we can construct a variety of models for a given rest mass, but with the different other characteristics. In Table 1, we present several such the models for the NS with the same rest mass as in the example shown in Figure 3. This rest mass is. For lower-mass part of these models, there exist two models with the same distance (the second models are listed in the second part of Table 1) since two input values, , yielding the demanded rest mass can be found.
8. On the Central Singularity
In the past astrophysical applications of GR, there was well-known the existence of the true singularity in the center of black hole. Because this singularity was situated below the event horizon, it was regarded as having no concern to any observer residing in our universe, i.e. outside of the black hole. Because of this reason, there is the convention that this kind of singularity is not problematic and the metrics containing it can be accepted for a description of real objects. (The singularity is not “naked” singularity.)
On contrary, some solutions of the EFEs yield a true singularity that is not situated inside the event horizon and that can be, in principle, “experienced” by an observer. This kind of singularity is known as the “naked singularity” and its acceptability in a realistic description was ruled out by so-called “cosmic censorship” theorem  . According to the latter, no naked singularity, other than the Big-Bang singularity, can exist in the universe and, hence, in the theory, when all the considered conditions are realistic. We note, the naked singularity, other
Table 1. Some characteristics of the models of NSs consisting of the same number of neutrons, i.e. with the same rest mass,. The models are created by using the Ni’s solution of the EFEs. The explanation of symbols can be found in the text.
than that of Big-Bang type, would cause a collapse of matter into a point in the center, in which the density would increase above all limits.
In the model of NS based on the Ni’s solution, the metrics inside the internal cavity, which is bordered by the sphere of radius, is characterized with the metric tensor, the components and of which are given by relations (27) and (29). In the last relation, we see that when. In this section, we discuss the basic properties of this central singularity, especially its concern to the naked singularity and cosmic censorship.
We note, there is not, either, the local Lorentz frame in the central point, with, in the cavity.
It was empirically found that always. Hence, the denominator in relation (19) for is positive. As well, it was empirically found that and, thus, the nominator in relation (19), , is also positive. Consequently, constant given by relation (19) is positive. Since the gravity is proportional to, which coincidentally equals in (as well as in), it must be oriented outward within the sphere of radius as seen after performing the derivative of given by relation (29). (Its sign is opposite than that of the derivative of for the outer empty space given by relation (28).) Thus, the central singularity is repulsive in the sense that every material object is attracted away from it. Maybe, this claim seems to be a paradox. We should however realize that a test particle in the region is not repelled by the abstract singularity (an empty-vacuum-point in the NS center), but attracted by the circumambient real matter of NS, in fact. This attraction diverges in the limit.
Since it is found that the outer radius is always larger than the Schwarzschild gravitational radius and all components of metric tensor are continuous (and everywhere finite) functions of radial distance in every distance, the NS body including, there is no other than the central singularity. Because of this fact, the central singularity seems to be the “naked singularity”. However, its repulsive character discriminates between it and the true, problematic, naked singularity. This singularity is the naked singularity, but that of Big-Bang type because of its repulsive character. It does not lead to the collapse of matter into any infinitesimally small volume and infinite increase of density. And, no material object can enter it.
Since the Ni’s solution implies the models of compact objects without any singularity other than the above-mentioned Big-Bang type singularity, it enables us to accept a more strict than the original cosmic censorship. Specifically, we can demand that “no singularity other than the Big-Bang type singularity can exist in the universe and, hence, in the theory, when all the considered conditions are realistic”. It means that adjective “naked” can be omitted in the theorem.
In conclusion, the main reason to postulate the Minkowski metrics in the vacuum inside a spherical shell is abolished in the Ni’s concept of hollow sphere.
9. Conclusion Remarks
The GR was originally created as the geometric theory. Since the geometry of spacetime is determined by stress-energy tensor, the quantities as energy and (effective) gravitational potential are the integral part of the theory, except of the geometrical aspects.
Meanwhile, an alternative to the original intent of truly geometrical theory started to be used, especially in the theory of NSs: the concept of mass within the Newtonian concept of potential was regarded as the integral part of the GR. This concept of the potential differs from the approximation of the GR effective potential for a weak field by an absence of constant term. The demand of the validity of the Newtonian concept implies that the size of -component of metric tensor in the TOV problem must always be larger than unity. As a consequence of this constraint, there is no stable solution of the EFEs for an object with mass above the Oppenheimer-Volkoff limit. And, we can obtain only the solution of the EFEs with the monotonous, everywhere inward oriented gravity.
Consequently, the traditional model of spherically symmetric NS, as firstly found by Oppenheimer and Volkoff  , is only the single of infinite variety of realistic NS models in GR. A further, infinite set of models appears when we permit that the size of the -component of metric tensor is also lower than unity. The essential effect corresponding to this permission is the outward oriented net gravitational attraction of upper stellar layers (material being in a larger distance from the center than a given test particle).
In the Euclidean space of Newtonian physics, the net gravity of upper layers is proved to be exactly zero in the case of spherical symmetry. However, the conception of spherical symmetry in the curved spacetime of GR is more complicated. A concentric layer, which is observed as spherically symmetric by the observer in the stellar center, is not longer observed, in general, as spherically symmetric by an observer aside the center. Hence, its gravitational action on a particle situated inside the layer but aside its center is generally non-zero. The solution of the EFEs found by Ni  implies that it is actually finite and oriented outward from the center.
Inside the NS, the outward oriented net gravity of upper layers increases with the decreasing radial distance, since the mass of upper layers increases. The behavior of net gravity of lower layers is opposite, of course. In the NS’s interior, there is a critical distance in which the partial gravitational actions of both upper and lower layers equal each other. Below this distance, the net gravity of upper layers becomes dominant and, thus, the total net gravity is oriented outward. In the stable configuration, this gravity is again balanced by the gradient of pressure, which is always oriented against the gravity, according to the equation for the gradient derived from the EFEs. The inner physical surface is formed by the same mechanism as the outer surface.
So, the NS model constructed by using the Ni’s solution of the EFEs is the hollow sphere with a cavity in its interior. The existence of this cavity is enabled by the fact that the metrics inside it is again described by the OSS, but with sign plus in front of the fraction figuring in the formulas for the components of metric tensor. It means that the corresponding quantity u is negative. Thus, the gravitational attraction in the cavity is oriented outward. Because of this orientation, there is no problem with the naked, but of Big-Bang type central singularity. And, there is no reason to reject the application of the Ni solution of EFEs in the astrophysics of real for-central-observer spherically symmetric objects.
The negative values of u become physically acceptable, when this quantity is no longer represented in the term of Newtonian-type gravitational potential, but is simply regarded as the metric quantity (alternative form of). This requires ruling out “mass” as the regular physical quantity from the GR. In gauging of the integration constants yielded by integrations of the EFEs with the help of Newtonian physics, it is then necessary to strictly keep the different meanings of u and mass. In a weak field and region of radial distances, where always, the correspondence between this parameter and can be supposed, but we should not inversely consider mass M within the GR, which is intended to be the purely geometrical theory. (We demonstrated that the EFEs in the TOV problem can be re-written to the dimensionless form. Most likely, this is possible generally. The concept of mass then becomes useless.) And, we should not further generalize the concept of mass and use it in any argumentation within the GR.
The concept of hollow sphere is able, as already claimed by Ni  , to create a model of stable compact object of whatever large mass. Such that the conclusion can seem to be in a contradiction with that by Oppenheimer and Snyder  who described the collapse of every compact object with a mass larger than a certain upper limit. As well, it contradicts to the conclusion published by Rhoades and Ruffini  about the maximum mass of NSs. These works were, however, implicitly based on postulate that the total net gravity must be oriented inward inside the whole NS. Thus, they are valid only for the full-sphere concept of NS.
With the concept of hollow sphere, we can moreover avoid any singularity, other than the central Big-Bang type singularity, in the astrophysics of real objects. Thus, it seems that we can eliminate several serious problems in the current astrophysics with the help of the Ni’s solution of EFEs.
The work was supported, in part, by the VEGA―the Slovak Grant Agency for Science, grant No. 2/0031/14, and by the Slovak Research and Development Agency under the contract No. APVV-0158-11.
Cite this paper
LubošNeslušan, (2015) The Ni’s Solution for Neutron Star and Outward Oriented Gravitational Attraction in Its Interior. Journal of Modern Physics,06,2164-2183. doi: 10.4236/jmp.2015.615220