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The metrics of the compact objects should be the continuous function of coordinates. The metrics inside every object is set by its internal structure. The metrics in the adjacent empty space is described by the outer Schwarzschild or Kerr solution of the Einstein field equations. It appears that the linkup of both object-interior and empty-space metrics is not continuous at the physical surfaces of the objects for the common, generally (by convention) accepted set of assumptions. We suggest the new way of how to achieve the success in the linkup, which does not assume the higher value of the relativistic speed limit in the empty space governed by the object, in contrast to our previous suggestion. We also give a more detailed explanation of the existence of inner physical surface of compact objects and suggest the way of the linkup of metrics in this surface. To achieve the continuous linkup, we assume a lower value of the speed limit in the object’s interior as well as a new gauging of the outer Schwarzschild solution for the inner empty space of the object. Newly established gauging constants are calculated and the success of the linkup is shown in several examples. The new gauging implies a lower gravitational attraction (lower gravitational constant) in the inner empty space in comparison with that in the outer space, which is measured in the common, observed, gravitational interactions of material objects.

The first description of the internal structure of dead stars, which spent all storage of their nuclear fuel, was published by Oppenheimer and Volkoff [

In 2011, Chinese researcher Ni [^{1}. In our earlier paper [

In Paper I, we pointed out the second feature that every realistic concept of NS must possess: the spacetime metrics must be the continuous function of radial distance from the object’s interior up to the adjacent outer space. It appears, the continuity is not obvious at the outer physical surface of the object. The metrics inside the object's body can be calculated from the model of the object’s internal structure. In the case of non-rotating object, the metrics of the outer space is described by the outer Schwarzschild (OSCH, hereinafter) solution [

To solve the problem of the linkup, we suggested the alternative way of gauging of one integration constant in the OSCH solution. Namely, there should be a specific relativistic speed limit,

The fact that there is assumed the speed limit larger than

Meanwhile we found that the assumption of the limit

We note, we create a “global” model, which describes the overall internal structure of compact neutron object, here. It means, we do not give any description of local phenomena or fields (e.g., magnetic field), which can oc- cur at the surface of the object or in its close vicinity. The ignorance of these phenomena or fields does not re- present, however, any principal problem, since the outer physical surface in all presented models is situated above the corresponding event horizon. Thus, the modeling is analogous to that of the internal structure of common stars. The local phenomena at the stars often used to be described in addition to the stellar model as a whole.

Concerning the structure of our paper, it is following. In the second section of our paper, we explain, once more, the mechanism of the formation of inner physical surface of compact object. In Section 3, we recall the basic equations determining the internal structure of non-rotating neutron object and also explain how the alternative gauging of the OSCH solution can be done. In the beginning of Section 4, the first way of the alternative gauging of integration constant in the OSCH solution is reminded. Then, the new assumption is presented in detail and the appropriate modified constants are given. In Section 5, we present a series of solutions for the internal structure of neutron objects with the description of whole metrics as well as some interesting dependencies concerning the relationship between the mass and outer radius or that between both outer radius and Schwarzschild gravitational radius, etc. In this paper, we also deal with the linkup in the inner physical surface, in Section 6. Some conclusion remarks are written in Section 7.

In the Newtonian physics with the flat, Euclidean space, the area of a spherical surface delineated by a spherical angle is proportional to the quadrate of the distance from the area. The Newtonian gravitational force and, hence, the corresponding acceleration is proportional to the reciprocal quadrate of the distance.

In general relativity, the metrics of the vacuum in a vicinity of spherically symmetric distribution of matter is OSCH, according to the Birkhoff theorem. It means that the time component,

In the following, let us nevertheless to discuss the case when we abandon the postulate and consider the non- degenerated OSCH metrics also inside the shell (or, the postulate

Taking into account this assumption, let us consider a spherically symmetric, thin material shell (

Let us further consider the analogous shell, but so compact that its gravity significantly curves the spacetime, therefore the relativistic description of gravity is relevant. Inside the shell, there is a point-like test particle in the position characterized with a radius vector. The plane containing the particle and perpendicular to the particle’s radius vector (indicated with the violet dashed straight-line in the bottom schemes in

The solution of the Einstein field equations found by Ni implies that

The object with the inner surface is further referred to as the “hollow sphere”. Inside such the object, there is always a distance

Distance

The schemes helping to explain the formation of the inner physical surface of compact object. The green (upper scheme) and blue-red (bottom schemes) circles represent a thin, spherically symmetric material shell in the case of Newtonian (upper scheme) and general-relativistic (bottom schemes) description of gravity. The violet dashed straight-line indicates the plane passing through a test particle, situated inside the shell, and perpendicular to the particle’s radius vector. The acceleration of a particle being outside of shell due to its gravity is denoted by a_{Nout} or a_{GRout}. The acceleration of test particle being inside the shell due to its outer (inner) part is denoted by a_{GRin}_{1}(a_{GRin}_{2}) and the acceleration of this particle due to the net force is a_{Nin} or a_{GRin}

The behavior of density inside four compact objects with the extremely small distance r_{max} and inner radius R_{in}. The density is given in the logarithmic scale and the radial distance in both linear (a) and logarithmic (b) scales. The thick, solid curves illustrate the behavior in the distances from r_{max} to the outer radius R_{out} and thin, dashed curves that from r_{max} down to the inner radius R_{in}

below this particle and it is obviously more attracted by these layers than by few layers above it. If we consider the particle in a shorter and shorter object-centric distances, there are less and less layers below and more and more layers above it. Therefore, the particle is in a lesser and lesser degree attracted by the lower layers and more and more attracted by the upper layers. Considering the solution by Ni, we get to the distance, where the attraction of the lower layers (toward the center) is balanced by the attraction of the upper layers (outward the center).

The gradient of pressure, balancing the gravity, also changes its orientation in distance

can be given with the help of

where

The pressure reaches its maximum value when

We note that, according to some solutions of the Einstein field equations, the matter can still be distributed from the true center of NS in such a way that the vector of the gravitational attraction is oriented inward from this center to infinity and gradient of pressure always acts against this attraction inside the NS body.

The equations to describe the non-rotating object, without any internal source of energy, consisting of a cold, de- generated, Fermi-Dirac gas, has been presented many times, starting with the famous paper by Oppenheimer and Volkoff [

Components

(

therefore the product

where

A model of internal structure of a neutron object as the hollow sphere can be calculated assuming a set of input values

in this distance. The latter is derived from the condition for the local maximum of function

The sum of the rest masses of all neutrons constituting the object, i.e. the object’s rest mass, can be calculated as

and the object’s total or gravitational mass as

where

In the case of the spherical symmetry, only the components

In the process of the derivation of OSCH solution, we obtain the differential equation

in the spherical coordinate frame

Although the line element (1.13) with constant

Using

inside the NS. Let us denote function

where

In Paper I, we constructed the dependencies of the ratio

The second requirement of acceptable linkup of metrics is equality of

When the input values of

The continuous linkup however requires that also the derivatives of

Because of the above-mentioned displacement of

In Paper I, we assumed that the active agent that shapes the local spacetime and also determines its intrinsic properties is the mass accumulation in the object. The metrics of the adjacent empty space should thus be adapted to become a smooth continuation of that, which was found for the object alone. Consequently, we suggested the alternative gauging of the OSCH solution postulating that the requirement (1.15) must be satisfied and replacing the value of speed limit,

Product

After the new, alternative gauging, components

When this OSCH solution is considered, the displacement in

The dependence of

We empirically found, in Paper I, that

Meanwhile, we noticed another possibility of the successful linkup of the metrics under our study, with velocity

Considering the new assumption, we retain the original, historical gauging of the integration constants in the OSCH solution and change constant

We remind, constant

Ratio

The decrease of the velocity of light in a material environment is the effect well described in optics. It is known that the relative permittivity in an optically thick environment is larger than that in vacuum (unity) and, thus, the velocity of light in this environment is smaller than the velocity of light in vacuum. Our new gauging of the speed limit is, however, related to the properties of spacetime resulting in the effects that discriminate between the relativistic physics and its Newtonian approximation. It is disputable if the optical effects can be identified with the latter. There are also further questions like, e.g., what is the mass of a particle if the value of speed limit decreases from

In gases, the relative permittivity depends on their density. Analogously, a similar dependence can be expected for the value of gauged speed limit,

Let us now to describe two examples of the metrics, when the new gauging of the velocity of light in the equations of internal structure is done. The examples are for the hollow spheres, but the same principles are also valid for the full-sphere solutions. The behaviors of gas density, component of metric tensor

In both discussed types of gauging, the numerical integration in the first example is started in

In the first example in

In

To gain a more complex information of how some quantities are mutually related, we construct a set of models for all combinations of

In

The relative excess of mass

Two examples of behaviors of the density (a), (b) and components g_{11} (c), (d) and g_{44} (e), (f) of metric tensor inside a compact object described by the concept of hollow sphere (thick curves) and corresponding OSCH solutions (thin, dotted curves). The blue, dashed, thick curves, labeled by “1” (red, solid, thick curves labeled by “2”), give the behaviors if the OSCH solution is gauged alternatively as in Paper I (if the constants of the internal structure are gauged as suggested in this paper). While the outer physical radius of the object in the first example (left-hand plots (a), (c), (e)) is larger than the gravitational radius, it is smaller than R_{g} in the second example (right-hand plots (b), (d), (f)), when no alternative gauging is done

increases for the low values of

The occupied radial extent, from the inner radius,

The comparison of the dependence of the object’s mass, M, on the maximum Fermi impulse, , in the case. The de- pendence is shown for both kinds of gauging, (blue, full squares) as well as (red, full circles). Impulse is given in the unit of for the first (second) kind of the gauging

The dependence of neutron-object mass, M, on the maximum Fermi impulse,. The given curve corresponds to the specific distance. Its value, in, is indicated in the legend

The relative excess of the total mass of object, M, over its rest mass, M_{o}, for various and

The relation between mass of object, M, and its inner and outer physical radius. The positions of the radii in the graphs are shown with solid circles and the extent from to with the lines linking the corresponding pair of the solid circles. The empty circles show the position of the distance (given in legend in). The black, dashed, straight line shows the behavior of the gravitational radius

It appears that the OSCH solution can be linked also to the NS-body metrics at the non-rotating-NS inner surface, when the hollow-sphere model of NS is considered (Our attempt with a linking up the inner Schwar- zschild solution did not result in any continuous metrics, with the equal derivatives in

We found that constant

for

Let us denote the speed limit in the internal empty space by symbol

Constant

Since it always appears that

Let us denote

Or, the equality can be divided by analogous equality which is set by the requirement of the linkup in

We empirically found that there is valid, at least in our toy model with the constant

An example of the successful linkup of metrics in both physical surfaces of a hollow-sphere NS is shown in

In

When we consider a particle inside the compact, relativistic object, the material layers situated above the particle attract it outward from the center if the flat metrics is not postulated in the interior of a spherically symmetric distribution of matter. Such the attraction is a qualitative difference from the Newtonian physics with the zero net gravity of the upper layers. In the compact object, there can, consequently, be such an object-centric distance,

If the numerical integration of the equations to describe the internal structure of compact pure-neutron object starts in an object-centric distance larger than zero, then we can demonstrate that it never provides any model in the form of exact full sphere. At the present, all models of the NSs are constrained to be the full spheres by a postulate. In the general relativity deliberated from such the postulate, there are also possible the NS models as the hollow spheres. In the model of this kind, the energy density and pressure reach the maximum values at the non-zero object-centric radial distance

The example of behaviors of the components g_{11} (a) and g_{44} (b) of metric tensor inside the hollow-sphere compact object (thick red curves) and corresponding OSCH solutions (thin, dotted curves) in both inner and outer empty space (blue and green thin dotted curves)

The relation between the velocity of light, c_{m}, inside the neutron object (a) as well as the velocity of light, c_{in}, in the vacuum in region (b) and the maximum Fermi impulse, , for a series of values of the distance, in which this impulse occurs. Each curve is for a single value of r_{max} which is indicated in the legend in plot (b)

Distance

In this paper, we demonstrated that the physically acceptable, continuous linkup of metrics is possible also in the concept of unique relativistic speed limit in the vacuum outside the material objects, in the interstellar and intergalactic space. If this value of the speed limit is the maximum limit of velocity of any entity motion, then the velocity of light inside the compact neutron objects has to be reduced to achieve the continuous linkup. The reduction has, however, an impact on the total mass of the objects, which subsequently appears to be relatively lower.

If we consider the neutron object as the hollow sphere and allow the distances

If the spacetime of vacuum below the inner surface of compact hollow sphere is again described with the OSCH solution of the Einstein field equations, as demanded due to the Birkhoff theorem, then this metrics can be successfully linked up to the NS-body metrics also in the inner physical surface of radius

We believe that the concept of hollow sphere, with the perfectly continuous behavior of spacetime metrics, will be further developed and used to construct the models of real compact objects. Because the outer surfaces of the compact objects with the continuous behavior of metrics are always situated above the event horizon (re- gardless the objects are hollow or full spheres), the advanced realistic models will have to deal with an atmos- phere, magnetic field, and a variety of local phenomena, which can be potentially observed at the outer surface. So, there is a lot of work for a number of another researchers who will, let us hope, also join this interesting, new stream in the astrophysics of relativistic compact objects.

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.