Goedesics Completeness and Cauchy Hypersurfaces of Ricci Solitons on Pseudo-Riemannian Hypersurfaces at the Fictitious Singularity: Schwarzschild-Soliton Geometries and Generalized-Schwarzschild-Soliton Ones

Abstract

The methodology is developed here to write Ricci solitons on the newly found structure of the pseudo-spherical cylinder. The methodology is specified for Schwarzschild solitons and for Generalized-Schwarzschild solitons. Accordingly, a new classification is written for the Schwarzschild solitons and for the Generalized-Schwarzschild solitons. The rotational field is spelled out. The potential for a tangent vector field is used. The conditions are recalled to discriminate which submanifold of a Ricci manifold is a soliton or is an almost-Ricci soliton. It is my aim to prove that a concurrent vector field is uniquely determined after the 4-velocity vector of a Schwarzschild soliton. As a result, the analytically specified manifold, which is a spacelike submanifold of the Schwarzschild spacetime that admits Ricci solitons. The rotational killing fields are tangent to the event horizon. The conditions that are needed to match the new aspects are spelled out analytically. As a result, the two manifolds described in the work of Bardeen et al. about the requested mass of a stationary, axisymmetric solution of the Einstein Field Equations of the spacetime, which contains a blackhole surrounded with matter from the new results obtained after correcting the work of Hawking 1972 about would-be point ’beyond the conjugate point’ on the analytic continuation of the would-be geodesics: they are proven here to become the tangent manifold (which is expressed from the tangent bundle in General-Relativistic notation). The prescription here is based on one of the books of Landau et al., that the matter is not put into the metric tensor, not even in the ultra-Relativistic limit. This way, the pseudo-spherical cylinder is one implemented from the Minkowskian description and whose asymptotical limit is proven. The new methodology allows one to describe the outer region of the blackhole as one according to which the (union of the trapped) regions is one with null support. For the purpose of the present investigation, the definition of concurrent vector fields in General-Relativity is newly developed. As a further new result, the paradigm is implemented for the shrinking case, which admits as subcase the Schwarzschild manifolds and the Generalized-Schwarzschild manifolds. The Penrose 1965 Theorem is discussed for the framework outlined here; in particular, the presence of trapped hypersurfaces is discarded. The no-hair theorem can now be discussed.

1. Introduction

In the work of Israel [1], static, asymptotically flat spacetimes in vacuum are analyzed, which are endowed with simply-connected equipotential surfaces.

In the work of Hawking [2], the result is recalled—that a stationary blackhole has topologically spherical boundary.

From the work of Bardeen et al. [3], the Schwarzschild manifolds are studied according to the possibility of selecting Schwarzschild cylindrical manifolds from them, which are asymptotically flat at infinity.

These manifolds here are selected according their possibility to admit a Schwarzschild soliton. Such Schwarzschild soliton is characterized here. New insight is gained on the generalized-Schwarzschild solitons.

The geometrical analysis is organized after developing new results from [4,5,6,7,8,9,10,11].

In the work of Barros et al. [7], the immersion of an almost-Ricci soliton into a Riemannian manifold is studied. The relevant prescription from [10] is scrutinized.

In the work of Chen et al. [8], a Ricci soliton is studied, according to its concurrent vector fields.

In the work [11], some of the necessary and sufficient conditions are worked out, for a hypersurface of a Euclidean space to be a gradient Ricci soliton are given.

In the work of Bektas’ Demirci [4], the classification of Ricci solitons on pseudo-Riemannian hypersurfaces on a 4-dimensional Minkowski space is achieved. The suitable potential vector fields are taken, for which it is possible to prove the necessary and sufficient condition for the pseudo-Riemannian surface to admit a Ricci soliton. As a result, three kinds of manifolds are selected, i.e., the manifolds admitting shrinking Ricci solitons with the chosen potential vector filed are the totally umbilical isoperimetrical hypersurfaces, the hyperbolic hypersurfaces, and the pseudo-spherical cylinder in the Minkowski 4-dimensional Minkowski space.

From the previous literature, new methods are developed in the present paper to be applied to the Schwarzschild solitons and to the Generalized-Schwarzschild solitons from [12] for the Penrose Theorem [13] to be applied.

In the present work, the procedures requested from [7] are used for Schwarzschild manifolds, for which the presented results are proven to hold under stronger conditions.

In the present paper, the pseudo-spherical cylinder in general relativity is newly developed.

After the definition of the new geometrical structures, the tasks in the present work are accomplished to study the geodesics completeness of the chosen geometries at the fictitious singularity $r=r_S$ following the new study of the Cauchy hypersurfaces. Furthermore, the new structures of concurrent vector fields are in the present work, being newly built and studied.

As a result, the outside region of the blackhole objects are studied as far as the presence of the union of trapped regions of null support is concerned.

The novel implications regarding the Penrose 1965 Theorem [13] are newly implemented.

The Penrose 1965 Theorem [13] is discussed in the framework outlined here. In particular, the presence of trapped hypersurfaces is discarded. The no-hair theorem can now be discussed.

As a result, the structures are studied in the cases of shrinking solitons, which are recently proven to be a subcase the Schwarzschild manifolds and the Generalized-Schwarzschild manifolds. The corresponding manifolds are therefore newly metrized here.

The paper is organized as follows.

In Section 1, the outline of the work is presented, and the concepts about the spacetime structure of the Schwarzschild solitons, which are developed in the present paper, are proven.

In Section 2, the introductory material is presented.

In Section 3, the spacetime structure of a blackhole is introduced.

In Section 5, the spacetime structure of a blackhole is newly refined.

In Section 7, Ricci solitons on Riemannian submanifolds are newly prepared.

In Section 8, pseudo-Riemannian hypersurfaces admitting Ricci solitons are newly discussed.

In Section 9, new theorems for Schwarzschild solitons are exposed.

In Section 10, a new classification of the Ricci solitons is provided; as a result, the Killing vector fields of Schwarzschild spacetimes are analyzed.

In Section 11, new theorems about the Schwarzschild solitons are introduced.

In Appendix C, the scalings of the metric are discussed.

The Outlook is presented in Section 13.

In Appendix A, the Christoffel symbols used to perform the calculations of Schwarzschild solitons are written.

In Appendix B, the Christoffel symbols used to perform the calculations of Generalized-Schwarzschild solitons are written.

In Appendix C, the geometrical objects to discuss the scaling invariances and the weight functions are written for the Schwarzschild solitons.

2. Introductory Material

From the work of Bektas’ Demirci [4], one recalls that, on a pseudo-Riemannian manifold $\mathcal{M}$, with $\mathtt{X}$ being the position vector, the position vector acts as a concurrent vector field as

$${\tilde{\nabla }}_X\mathtt{X}=X$$

(1)

The work [7,8,9] are outlined as follows.

From [7], the immersion of a Ricci soliton into a Riemannian manifold is considered. This paper admits a Ricci soliton with potential vector field.

From [7], the shrinking Ricci solitons are proven apt to be immersed into a space with constant means curvature as a Gauss soliton.

As an intermediate step, Ricci solitons on Euclidean manifolds are recalled here to be classified with potential vector field arising from the position vector field of a Euclidean hypersurface.

About the Killing Vector Field of the Schwarzschild Spacetime(s)

The velocity vector field of a Schwarzschild spacetime is characterized according to the work of Bardeen et al. [3].

In [3], the stationary, axisymmetric, asymptotically flat space of a blackhole spacetime is recalled to be the description of the asymptotical limit admitted after Schwarzschild-blackhole spacetime.

The vector filed is $K^{\mu }$, which admits the condition

$$k^a{k}_a=-1$$

(2)

i.e., it is timelike.

There exists a unique rotational Killing vector ${\tilde{K}}^{\mu }$ (i.e., one with non-vanishing rotor), which is normalized as

$${\tilde{K}}^{\mu }{\tilde{K}}_{\mu }=-1$$

(3)

The components of the velocity are characterized as

$$v_{[\rho ;\mu \nu ]}=-{\displaystyle \frac{1}{2}}{R}_{\sigma \rho \mu \nu }{v}^{\sigma }$$

(4)

The surface elements are defined as follows: the surface element of $\partial S$ is $d{\Sigma }_{\mu \nu }$, and the surface element of S is $d{\Sigma }_{\mu }$ as in

$${\int }_{\partial S}{k}^{[\mu ;\nu ]\sigma }d{\Sigma }_{\mu \nu }={\int }_S{R}^{\mu }{}_{\nu }k^{\nu }d{\Sigma }_{\mu }$$

(5)

For calculation, a spacelike surface is taken, which is asymptotically flat, tangent to ${\tilde{K}}^{\mu }$, and it intersects the Killing horizon as indicated in [2].

3. Selected Topics from the Previous Investigations

Selected investigation lines from [2] and from the pertinent items of bibliography are summarized here.

From ibidem, the result shows that a stationary blackhole has a topologically spherical boundary; furthermore, it is demonstrated that rotating blackholes are axisymmetric.

From ibidem, a blackhole on a spacelike surface is defined as the connected component of the surface bounded after the event horizon. The region of the surface bounded after the event horizon is taken.

Ibidem, the three requests are stated as follows:

(1)

The surface of a Relativistic star, which undergoes gravitational collapse, is described as ’passing’ the Schwarzschild fictitious singularity;

(2)

The spacetimes admits a singularity;

(3)

The observers from the outer region do not observe the singularity.

The outer region is described such that the singularity is not observed by the observers, which are outside the gravitational radius. In the outer region, the future is forecast from Cauchy data on a spacelike surface.

Ibidem, the study of the stability is performed after the perturbations. Small perturbations are studied in [14]. In [2], the conjecture that a blackhole establishes a Kerr solution of the EFEs is taken.

Ibidem, the surface of a blackhole is shown not to decrease with time.

4. New Introductory Results

The Structure of a Blackhole Spacetime

The spacetime structure of a blackhole is introduced as follows, and will be refined in Section 5.

The fictitious singularity is characterized from [1] as in the following outline.

The 3-dimensional part of the manifold of the fictitious singularity is prescribed in relation to the 4-dimensional metric as

$$d{s}^2=g_{ij}(x^1,x^2,x^3)d{x}^{i}d{x}^j-V^2\left(t\right)d{t}^2,$$

(6)

with

$$V^2=\mid {\zeta }_{\mu }{\zeta }^{\mu }\mid .$$

(7)

The surface $\Sigma$ is taken as the three-dimensional part of a four-dimensional manifold, which corresponds to the slides $x^0=const$.

This way, it is proven that

Theorem 1.

Σ is the only maximal extension of $V^2$ with respect to the vector field ${\zeta }^{\mu }$ when

$${\zeta }_{\mu }{\zeta }^{\mu }<0.$$

(8)

Proof of Theorem 1.

The surface $\Sigma$ is the largest domain of the integral curves of ${\zeta }^{\mu }$ when Equation (8) is requested. The vector field ${\zeta }^{\mu }$ is at least smooth, and there exists no strictly larger manifolds compatible with the construction of such maximal extension. □

The following description now holds and the definitions are therefore equivalent.

The gravitational potential V is taken from a Schwarzschild spacetime such that

$$V=\sqrt{g_{00}}.$$

(9)

The Killing vector field ${\zeta }^{\mu }$ characterizes a static spacetime manifold as it is static over a domain, in which it is hypersurface-orthogonal.

The surface $\Sigma$ is proven to be characterized as one with

$$g_{ij}={\delta }_{ij}+\mathcal{O}\left(r^{-1}\right),$$

(10)

with

$${\partial }_k{g}_{ij}=\mathcal{O}\left(r^{-2}\right).$$

(11)

The limit $r\to \infty$ is accomplished as the limit of $r\equiv \sqrt{\mid \overrightarrow{r}{\mid }^2}$ and defines V as

$$V=1-{\displaystyle \frac{m}{\mid \overrightarrow{r}\mid }}+\kappa$$

(12)

where the properties of $\kappa$ from [1] follow.

A weakly continuous, asymptotically simple space is taken from [14] of a spacetime manifold $\mathcal{M}$ with metric $g_{\mu \nu }$. $\mathcal{M}$ is then embedded in a larger manifold $\tilde{\mathcal{M}}$ with metric ${\tilde{g}}_{\mu \nu }$ as ${\tilde{g}}_{\mu \nu }{\mathsf{\Omega }}^2{g}_{\mu \nu }$ (the metric ${\tilde{g}}_{\mu \nu }$ is further discussed in Section 11 and in Section 13).

5. New Results

Concurrent Vector Fields in General Relativity

The concurrent vector fields are characterized as follows from [15].

Let ${\mathcal{R}}^m$ be an m-dimensional Riemannian manifold, with non-vanishing Christoffel symbols that define the covariant derivative $\mathcal{D}$, And let ${\mathcal{R}}^m$ an immersed manifold of an n-dimensional n-dimensional manifold ${\mathcal{M}}^n$ into ${\mathcal{R}}^m$: the following definition is given.

Definition 1.

$X^{\mu }$ is a concurrent vector field if it is defined as

$$dx+\mathcal{D}X=0$$

(13)

with x being the differential of the immersion.

The concurrent vector fields are further studied in [16]. Ibidem, complete linear connection on the manifold which is called standard connection on the manifold is recalled. Concurrency with respect to the standard connection is ibidem discussed.

From [16], the following theorem is worked out.

Theorem 2.

A Riemannian connection is uniquely determined after the Riemannian metric.

Proof of Theorem 2.

The Christoffel symbols are unique. Furthermore, there is no torsion, no non-metric object, and no metric-asymmetricity object. □

Furthermore,

Definition 2.

A vector is concurrent with respect to the metric tensor if it is concurrent with respect to the corresponding Riemannian connection.

Some of the properties of the submanifold of a Riemannian manifold admitting a Ricci soliton are studied in section 3 from [10] Section 3 ibidem. In the present paper, the concept is developed into pseudo-Riemannian manifolds.

The Riemann immersions will be discussed in future work.

6. New Construction of the Spacetime

The space part of the manifold $\mathcal{M}$ is taken from [1], from which a thin-sandwich conjecture can be implemented in a straightforward manner as with [17].

The boundaries of $\mathcal{M}$ are two null hypersurfaces ${\mathcal{J}}^{+}$ and ${\mathcal{J}}^{-}$. $\mathcal{J}\ast +$ and ${\mathcal{J}}^{-}$, which have the topology of ${\mathbb{S}}^2$; ${\mathcal{J}}^{-}$ is the past infinity, ${\mathcal{J}}^{+}$ is the future infinity. The possibility of forecasting events in the neighborhood of ${\mathcal{J}}^{+}$ is now investigated.

The following definition is taken:

Definition 3.

A weakly asymptotically simple space is ’future-asymptotically predictable’ if there exists a spacelike Cauchy surface $\mathcal{S}$ which does not intersect any non-spacelike curve more than once.

Let $D^{+}\mathcal{S}$ be the set of all points q such that, as requested in the guidelines from [2], ’every past-directed non-spacelike curve from the set $\left\{q\right\}$ intersects $\mathcal{S}$ if $D^{+}\left(\mathcal{S}\right)$ is extended far enough’. An asymptotically predictable space $J^{\pm }$ is defined as follows:

Definition 4.

In an asymptotically predictable space $J^{\pm }$, there are no naked singularities if the future infinity is $J^{+}\left(\mathcal{S}\right)$, and there exists at least one closed trapped surface $\mathcal{T}$ in $D^{+}\mathcal{S}$.

The following aspect of blackhole spacetimes is now fixed:

Theorem 3.

The union of the closed trapped surfaces $\mathcal{T}$ of $D^{+}\mathcal{S}$ is a null support.

From [2], the case of non-rotating blackholes is chosen here.

Different from [2], the blackhole spacetime regions from a connected component of the region of the boundary after the event horizon is investigated. The singularities that happen inside the fictitious singularity are not observed by the observer.

The property is newly defined as 3bis. The outer region is requested to be endowed with at least one observer (i.e., the outsider) who is not ’observing’ the singularity.

The statement 3bis is now proven to be equivalent to the choice of the surface on the boundary as one single point $\pi$.

The limit is now studied, in which a ’small portion’ of the surface area sets the Cauchy data on the outsider spacelike surface.

The ’small portion of the surface area’ considered here therefore consists of one single point $\pi$ (whose area will not nevertheless decrease): the outsider spacelike region considered here is therefore one accessed after the pertinent Cauchy data.

This way, the closure of $\mathcal{M}$ (rather that the closure of $\tilde{\mathcal{M}}$) of $D^{+}\left(\mathcal{S}\right)$ is therefore taken into account.

Accordingly, the new theorem holds as follows:

Theorem 4.

Points of the geodesics p of the closure of $\mathcal{M}$ of $D^{+}\left(\mathcal{S}\right)$, which are located ’beyond’ the conjugate point, are unpredictable.

Proof of Theorem 4.

There should be one unique ’would-be’ geodesics line passing through one chosen point from the future, through the point $\pi$ on the fictitious singularity and through the would-be point ’beyond the conjugate point’ on the analytic continuation of the would-be geodesics; differently, the would-be point $\sigma$ would be accessed starting from the future in the would-be geodesics. □

Intermediate study of a ’thin’ manifold $\mathcal{M}$ is thus analogous to that of a thin-sandwich conjecture.

The would-be point $\sigma$ not being accessible implies that there no accessed geodesics lines passing through it.

The points beyond which the analytical continuation is in Einstein–Hilbert GR are therefore defined: they consist of the bases of the tangent cylinder.

The tagent cylinder here is newly proven to consist of exactly all the geodesics that access (attain at) the point from the future and on the point $\pi$ on the fictitious singularity. Therefore,

Theorem 5.

The tangent cylinder is geodesically complete.

Therefore,

Theorem 6.

An unperturbed Schwarzschild solution of the EFE’s establishes a Schwarzschild blackhole.

Proof of Theorem 6.

All the cylinders tangent to the Schwarzschild blackhole are geodesically complete. □

7. About Ricci Solitons on Riemannian Submanifolds

From [4], the results of [8,9] are recalled, according to which it is possible to write the equations of the Ricci tensor of a submanifold in a Riemannian manifold. It admits a Ricci soliton with potential vector field as the tangential part of the concurrent vector field.

As a result, Ricci solitons on Euclidean surfaces are classified with a potential vector field arising from the position vector field of a Euclidean hypersurface.

The hypersurface in the Euclidean space is made to be a gradient Ricci soliton, for which the necessary and sufficient conditions are spelled in [11].

The conditions for a submanifold of a Ricci soliton to be a Ricci soliton are studied in [10].

The use of concurrent vector fields for Ricci solitons is discussed in [18].

The methodologies connected to the concurrent vector fields will subsequently be specified to Schwarszchild solitons and to Generalized-Schwarszchild solitons from [19]. Ibidem, the uniqueness of the Schwarzschild solitons is studied. The Ricci flow is reconciled with the EFE’s. The scaling functions are used to investigate the rotational properties of the geometrical objects and are utilized to spell out the determinant of the metric tensor.

8. About Pseudo-Riemannian Hypersurfaces Admitting Ricci Soliton

From the work of Bektas’ Demirci [4], the classification theorems of Ricci solitons on pseudo-Riemannian hypersurfaces are used on on a 4-dimensional Minkowski space ${E_1}^4$.

The new methodology developed here is summarized as follows.

The potential vector field is taken as the tangent component of the position vector of the pseudo-Riemannian hypersurfaces.

The theorems needed for the constructions are now enunciated.

Theorem 7.

The potential vector fields are unique.

Proof of Theorem 7.

The 4-velocity is unique. □

Theorem 8.

The tangent component of the position vector of the pseudo-Riemannian hypersurfaces is unique.

Proof of Theorem 8.

The connection is unique, and the concurrent vector field is unique. □

Theorem 9.

For a pseudo-Riemannian hypersurface, the necessary and sufficient condition given for it to admit a Ricci soliton is the uniqueness of the connections.

Proof of Theorem 9.

The Christoffel symbols that define the connection are unique. □

From [9], the position vector field is selected as a concurrent vector field.

Accordingly, the 4-velocity corresponds to the time-derivative of the 4-position.

9. Schwarzschild Solitons: New Theorems

The new result is here presented, according to which

Theorem 10.

The position concurrent vector field of a Schwarzschild soliton is uniquely determined after the 4-velocity.

Proof of Theorem 10.

The 4-velocity vector uniquely determines a manifold, which is tangent to the Killing horizon. □

Accordingly,

Theorem 11.

From the hypotheses of [11], a unique hypersurface exists, which uniquely defines a Schwarzschild manifold.

The gradient Ricci solitons can therefore be newly analyzed.

In particular, the following Theorem holds.

Theorem 12.

A non-compact gradient Ricci soliton exists and is unique, whose curvature operator is pseudo-Riemannian (i.e., it is defined after a pseudo-Riemannian metric).

Furthermore, from the hypotheses ibidem, the following Theorem is given as follows:

Theorem 13.

A Schwarzschild manifold is a gradient Ricci soliton.

10. Spelling of the New Construction of the Solitons

The vector field ${\xi }_{\mu }$ is here chosen.

The Lie derivative of the metric tensor with respect to ${\xi }^{\mu }$ defines the manifolds:

$${\pounds }_{\xi }{g}_{\mu \nu }+R_{\mu \nu }=\lambda g_{\mu \nu }.$$

(14)

Equation (14) is studied here to produce the following solitons.

(1)

The solitons with vanishing derivative of the metric tensor with respect to the vector field ${\xi }^{\mu }$ is as follows:

$${\pounds }_{\xi }{g}_{\mu \nu }\equiv 0$$

(15)

i.e., according to which the Steady Ricci Schwarzschild solitons are described as

$$\lambda =0.$$

(16)

(2)

The steady Generalized-Schwarzschild solitons such that

$$R_{\mu \nu }\equiv 0$$

(17)

(when they are found).

(3)

The soliton

$$R_{\mu \nu }\ne 0,$$

(18)

with

$$R_{\mu \nu }=\lambda g_{\mu \nu }.$$

(19)

Now, the solitons (1)–(3) are constructed by imposing the most general sets of conditions on the weight functions f of

$$\nabla f=\xi ,$$

(20)

i.e., which is spelled as

$${\nabla }_{\mu }f={\xi }_{\mu },$$

(21)

with ${\xi }^{\mu }$ being a Killing vector field. The Killing vector field ${\xi }^{\mu }$ is proven to correspond with the 4-velocity field. The velocity v is a concurrent vector field; it is unique because the Ricci scalar can be constructed from the unique Christoffel symbols. More specifically, the uniqueness of the (radial part of the) 4-velocity is taken as an example in [16].

The conditions from Equation (21) that are compatible with the Schwarzschild manifolds are

$$\begin{array}{cc}\hfill & {\nabla }_{t}f={\xi }_t,\hfill \end{array}$$

(22a)

$$\begin{array}{cc}\hfill & {\nabla }_{r}f={\xi }_r,\hfill \end{array}$$

(22b)

$$\begin{array}{cc}\hfill & {\nabla }_{\varphi }f=0,\hfill \end{array}$$

(22c)

$$\begin{array}{cc}\hfill & {\nabla }_{\theta }f=0.\hfill \end{array}$$

(22d)

10.1. Equations of the Spacelike Submanifold of Pseudo-Spherical Cylinders

The spacelike submanifold of pseudo-spherical cylinders is described after integration of Equation (22) as

$$\begin{array}{cc}\hfill & {\nabla }_0{f}_{\alpha }={\xi }_0={\Gamma }_{0\alpha }^{\sigma }{f}_{\sigma },\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\hfill & {\nabla }_r{f}_{\alpha }={\xi }_r={\partial }_r{f}_{\alpha }+{\Gamma }_{r\alpha }^{\sigma }{f}_{\sigma }.\hfill \end{array}$$

(23b)

Furthermore, one of them has

$${\partial }_{\theta }{f}_{\theta }=0$$

(24)

and

$${\partial }_{\theta }{f}_{\varphi }=0.$$

(25)

10.2. The Killing Vector Fields for Schwarzschild Spacetime

From Equation (2), the Killing vector field is taken as $k^{\mu }$ such that

$$k^{\mu }{k}_{\mu }=-1$$

(26)

where the killing vector field comes from the r component of the 4-velocity; it is defined here as

$$k_{\mu }\equiv {\nabla }_{\mu }{v}_r$$

(27)

There exists a further degrees of freedom associated with the manifolds, which is defined after the further vector field as

$$K_{\mu }\equiv {\nabla }_{\mu }{v}_0$$

(28)

i.e., from the t-component of the 4-velocity.

The corresponding unique rotational vectors are therefore studied.

11. Schwarzschild Solitons: Further New Theorems

The results from [10] are here newly studied in the case of a pseudo-Riemannian manifold.

Theorem 14.

The condition of a gradient Ricci soliton

$$V_{\mu }=\nabla f_{\mu }$$

(29)

are studied here based on the framework of [2]; in particular, a 1-dimensional submanifold tangent to the event horizon exists, and it is written after the potential of a tangent vector field ${\mathsf{\Phi }}^{\mu }$

$$V^{\mu }=v^{\nu }{\displaystyle \frac{\partial {\mathsf{\Phi }}^{\mu }}{\partial x^{\nu }}}.$$

(30)

Accordingly, the following theorem holds.

Theorem 15.

For a Schwarzschild soliton, the event horizon corresponds to the Killing horizon.

According to the paradigm presented here, the following Theorem holds.

Theorem 16.

The immersion of a shrinking Ricci soliton $({\mathcal{M}}^m,g,\lambda )$ into a space form with constant mean curvature is for Schwarzschild manifolds of a shrinking Ricci soliton $({\mathcal{M}}^n,g,\lambda )$, from which the construction of a Gaussian solitons always holds.

Furthermore,

Corollary 1.

The transformations that leave the soliton $({\mathcal{M}}^m,g,\lambda )$ invariant [20] are the transformations which leave the metric g invariant in General Relativity.

Corollary 2.

The transformations that leave the metric of $({\mathcal{M}}^m,g,\lambda )$ invariant are the diffeomorphisms, i.e., $({\mathcal{M}}^m,g,\lambda )$ is always upgraded to $({\mathcal{M}}^n,g,\lambda )$.

Corollary 3.

The diffeomorphism that leave the metric of a Schwarzschild soliton invariant are the coordinate transformations.

Remark 1.

The diffeomorphisms discussed for a Schwarzschild solitons are the General-Relativistic transformations.

This present analysis allows one to work out that

Corollary 4.

The scalings of the metric are always trivial for Schwarzschild solitons.

The local scalings of the metric are studied in [19], from which the consiitons from [2] are now discussed in the Table in Appendix C. Indeed, from [2], the scalings of the metric are written as ${\tilde{g}}_{\mu \nu }$ as ${\tilde{g}}_{\mu \nu }={\mathsf{\Omega }}^2{g}_{\mu \nu }$, where, ibidem, the function $\mathsf{\Omega }$ is described to be a smooth function equal to zero with a non-vanishing gradient on the boundaries of $\mathcal{M}$. The geometrical objects are summarized from [19] and in Appendix C.

12. Generalized-Schwarzschild Solitons: Further New Theorems

The following relevant results are provided as far as the Generalized-Schwarzschild solitons are concerned.

The definition of concurrent vector fields and the consequent analysis of the 4-velocity vector allows one to establish the characterization of the fictitious singularity.

Theorem 17.

For a Generalized-Schwarzschild soliton, the event horizon corresponds to the Killing horizon.

Moreover, the local conformal invariance is analyzed.

Theorem 18.

The scaling of the metric are always trivial for Generalized-Schwarzschild solutions, which admit $R=0$ even with $R_{\mu \nu }\ne 0$.

13. Outlook

From [10], the relation between the intrinsic invariants and the extrinsic invariants are studied for a Riemannian submanifold to admit a Ricci soliton.

It is recalled that Ricci solitons are solitons for which

$$R_{\mu \nu }+{\left[\mathcal{H}\left(f\right)\right]}_{\mu \nu }=\rho g_{\mu \nu }$$

(31)

with $H_{\mu \nu }$ being the Hessian matrix operator.

The implication from [6] can now be newly analyzed.

Theorem 19.

Schwarzschild solitons are asymptotic solitons.

The concurrent vector fields of immersed manifolds are discussed in [21]; ibidem, the concurrent vector field of immersed manifolds and with constant length are investigated. The case of Riemannian immersions is specified in [22] for the present purposes.

The presented findings are apt to be inscribed within the research lines outlined in [23]. The phenomenological evidences are searched for in [24] from [25].

Applications of the present findings are proposed in [26]. Ibidem, the presence of spacetime singularities is studied from ibidem; this presence is ascribed to the incompleteness of timelike geodesics or to that of null geodesics. The existence of the (global) Cauchy hypersurface is not requested for these discussions. The modern Astrophysical implementation is proposed in the work of Israel [27].

The possible role of $J^{-}\left({\mathcal{J}}^{+}\right)$ is discussed in the work of Tipler [28].

In the work of Senovilla et al. [29], the perspective studies about particular non-compact Cauchy hypersurfaces are envisaged.

The possible definitions of blackhole are compared in the work of Curiel [27].

The modern perspectives are discussed in [30].

The no-hair theorem can start being discussed in [13]; in the case presented here, the Penrose Theorem is applied to the framework developed here.

Theorem 20.

The inconsistency is removed from [2]. Indeed, the trapped surface does not exist in $(\mathcal{M},g)$ as $\mathcal{T}$ (or the union of all the trapped hypersurfaces) is now newly outlined to be one with null support.

14. Concluding Remarks and Perspective Studies

It is now possible to comment on the results about the cylindricity properties of Ricci solitons discussed in [31]. It is straightforward to conclude that the Definition from ibidem does not apply to Schwarzschild solitons; for this reason, Theorem 1.2 from ibidem does not hold for Schwarzschild solitons, and therefore, the coincidence of Bryant Schwarszchild solitons with Brendle Schwarzschild solitons is not only an improvement in the understanding of the rotational symmetries of the Schwarzschild spacetime, but it also opens the investigation of the properties of the tangent cones from [32].

In contrast, the spacetime structure of the Generalized-Schwarzschild solitons allow for the consideration of implementing curvature estimates from the Definition from ibidem in order to develop the needed new concepts for a comparison with Theorem 1.2 from ibidem.

The properties of gravitational solitons arising from the topological aspects of the spacetime were introduced to be investigated in [33]; ibidem, the results are also suggested to be extended the case of asymptotically non-flat spacetimes: the instance can be therefore taken for Schwarzschild spacetimes, which are asymptotically flat according to the Birkhoff Theorem, and to Generalized-Schwarszchild solitons, which are asymptotically non-flat according to [19].

From [33], the no-hair theorem is recalled to let the found solitons be defined after the three quantities M, Q, and J; the Bekenstein–Hawking [34,35] formula is invoked.

We recall here that the Bekenstein–Hawking entropy formula holds for Schwarzschild solitons, as with the study of the scaling functions from [19]; diversely, the entropy formula will be studied for the Generalized-Schwarzschild cases in future work according to the area formula in General Relativity, with respect to the rotational invariance of the Generalized-Schwarzschild spacetimes.