In their famous work published in 1939, Oppenheimer and Volkoff [1] concluded that there is no solution for a stable configuration of dead star, without an internal source of energy, if the mass of the star exceeds a certain critical limit, which was later named by them. In the subsequent work, it was demonstrated that the star has to collapse below its event horizon [2] .

The conclusion about the non-existence of the stable configuration of very massive stars and the necessity of their collapse to become the black holes has been regarded as valid more than three quarters of century and is hard to believe that something could ever bring a doubt about this claim. In 2011, Chinese researcher Jun Ni [3] however published a solution, which forces us to deal again with the question on the stable configuration of very massive stars. Ni attempted to reproduce the creation of the Oppenheimer-Volkoff model of neutron star. He proceeded in the same way as the original authors of the model, except of the starting point of the numerical integration of the differential equations, which are relevant to the model. While Oppenheimer and Volkoff started the integration in the center of the object, Ni started it in a finite star-centric distance.

The inward proceeded part of the Ni’s integration always ended with zero pressure and energy density in a finite star-centric distance, i.e. he obtained a model with an inner physical surface of the object. In addition, he concluded that there is no upper-mass limit. And, the outer physical surface appears to be always situated above the corresponding event horizon.

In this paper, we attempt to analyze the reason why Ni was successful in the creation of the stable model for very massive compact object. Clearly, the Ni’s result evokes some fundamental questions on the principles and postulates within the general relativity, especially those, which were originally established within the Newtonian physics and appeared in general relativity after their formal generalization.

In this section, let us introduce the basic properties of the NS model constructed by using the Ni’s solution of Equations (11)-(13) in more detail. We numerically integrate these equations starting, for example, in the star-centric distance 10 km. We identify this distance to the distance of zero net gravity, therefore the input value of auxiliary quantity u is identical to, which is given by Equation (25). Let us consider the input value of the Fermi impulse equal to, e.g.,. It implies the corresponding input value according to relation (9). The integration is performed in two parts: in inward direction and outward direction. We arbitrarily terminate each part when decreases below to avoid a transgress to a negative value of in the last integration step.

The input value of auxiliary function is found via iteration to be equal to −3.488263. For this input value of, the outward processed integration of Equation (13) ends just with value and, thus, the metrics is continuous in the outer physical surface of NS.

The behaviors of density (calculated as the ratio of) and components and of metric tensor, as well as auxiliary quantity u in this example are shown in Figure 1. From the maximum in, the density (Figure 1(a)) decreases in as inward as outward processed integration. (The behavior of pressure is qualitatively the same.) In the interval of distances (), the gradient of the density is positive (negative). On contrary, the derivative of function (Figure 1(c)) with respect to r, which determines the orientation of the gravitational action (see the gravitational acceleration given by relation (40); it is proportional to), is negative (positive) in the same interval.

The metrics in the external vacuum is described by relations (15) and (18) with and in the outer space. Similarly, and given by relation (19) in the inner space. Specifically,

In Figure 1, one can see that this metrics is perfectly tailored with that valid for the NS’s interior. It means that the metrics is perfectly continuous in both outer and inner physical surfaces. Not only the values of, , and u are tailored, but their derivatives also equal each other correspondingly. Ni [3] assumed the Minkowski metrics in the internal void, , which was not however well tailored with the NS-body metrics.

The negative derivative of with respect to r in the interval (also in) implies an outward oriented net gravity, there. Such the behavior is reasoned in the following sections.

The stable configuration of matter in the interval from to occurs due to the change of the orientation of gas pressure gradient, which balances the outward acting gravity. The fact that the orientation of this gradient is always opposite to gravity can be seen in relation (4). Since there is always valid, the value of gradient has always the opposite sign than the value of and, therefore, the gradient is opposite. We remind that the gravity is proportional to and its orientation to the sign of this derivative.

In course to explain the outward acting gravitational attraction, let us, firstly, to deal with an important consequence of spherical symmetry in the Newtonian physics. In this theory, the space is Euclidean regardless the amount, compactness, and radial distribution of matter in the spherically symmetric object. Every such the object can then be divided into a large number of thin concentric material layers. The net gravitational force from every layer can be calculated separately and the total net force is simply the sum of the partial forces from every layer.

Let us consider an infinitesimally small material element of a thin, spherically symmetric layer in a distance from a test particle, which is situated inside the layer. In this distance, let the element is seen within a certain small space angle. Its volume is then proportional to. According to the Newton force law, the gravitational force between the element and the test particle is proportional to, therefore the product of and occurs in the formula for this force. The product equals unity, i.e. it is constant. The analogous force can be found between the test particle and small material element situated in exactly opposite direction and seen within the same space angle as the first element. If the second element is in distance from the test particle, its volume is proportional to and the corresponding force is proportional to the product of and (sign minus expresses the opposite orientation of force). This product equals −1, i.e. the same constant, but with the opposite sign. Therefore, if we add both partial forces, their sum is exactly zero.

In other words, the force from the element in the given direction is eliminated by the force from the element in the opposite direction. For every element, it is possible to find the element in the opposite direction, exactly eliminating the force of the first element, therefore the total net gravity of the thin spherical layer on the test particle wherever inside it is exactly zero. Consequently, there can be whatever number of outer concentric layers, with the radii larger than the object-centric distance of given test particle, their net gravitational force is the sum of zeros and, hence, zero.

However, it is important to realize that this calculation, with zero result, can be made only in the Euclidean space of Newtonian physics. Only in this case, it can be exactly and generally proved (analytically calculated) that the net gravity of the outer layers of spherically symmetric object is zero.

If we consider a curved spacetime or curved space in a static problem in the GR, there occur two differences, which yield problems. At first, the GR is not linear, therefore we cannot divide any spherical object into a number of thin concentric layers, calculate their partial gravitational actions separately, and obtain the net action by adding the partial results. At second, the distribution of matter, which is seen as spherically symmetric by the observer in its center, is not longer seen, in general, as spherically symmetric by an observer aside the center. The curved space changes the distance to a material element when seen by the observers in different places, in general.

In a calculation of gravitational acceleration of a test particle inside a spherical layer, the element volume is not longer proportional to the quadrate of distance in the curved space. And, the gravitational acceleration cannot be expected to be reciprocally proportional to this quadrate, either. Unless the gravity of outer layers is postulated to be zero, it cannot be expected to be zero, except of some special cases, perhaps. The Ni’s solution implies that this gravity is actually non-zero and oriented outward from the center of the object, which is seen as spherically symmetric by the observer in its center.

The EFEs are solved via integrations, which produce the integration constants. In an application of the solution of EFEs to a real object, we need to physically represent these constants and find their values. In this process, the fact that the GR must converge to its Newtonian approximation in the limit of weak field was utilized.

Sometimes, it seems that the representations of both these theories are not distinguished, properly. The rules being valid within the classical Newtonian physics are used in an argumentation within GR. Consequently, the GR is, in fact, constrained by these rules if one requires their validity in both theories. Below, we discuss this problem in more detail in the context of the Ni’s solution and demonstrate the consequences in the case when the GR is completely deliberated from the Newtonian physics, in this case. It appears that the most important quantity in this discussion is the gravitational potential, its understanding and definition.

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.

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,

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

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.

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 [16] . 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

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.

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 [1] , 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 [3] 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 [3] , 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 [2] 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 [17] 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.

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