NEW RESEARCH IN RELATIVITYAND IT’S CONSEQUENCES

By

The Mathematical Physicist Dr. Nikias Stavroulakis

Université de Limoges, Faculté des Sciences de Limoges,U. E. R. des Sciences de Limoges, Département de Mathématique,

Limoges, France

AN EXPOSITION COMPOSED

By

Dr. Ioannis, Neoklis Philadelphos, M. Roussos Professor of Mathematics

*A Critique Against

(1) The “Birkhoff Theorem in Relativity” and the indiscriminate use of the spherical coordinates

(2) The “Black Holes” Theory

(3) The “Black Holes” Theory is so closely related to the metaphy-sical “Big Bang” Theory that claims the creation of the Universe out of nothing, so that any damage done to the “Black Holes” Theory inevitably causes an equivalent damage to this kind of “Big Bang” Theory and vice versa. Hence, the Dr. Stavroulakis’ results against the “Black Holes” Theory immediately apply against the so much in mode “Big Bang” Theory too.

All information provided here has been composed byDr. Ioannis, Neoklis Philadelphos, M, Roussos, Professor of Mathematics,

under the supervision and by the kind permission of

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Dr. Nikias Stavroulakis, Professor of Mathematical Physics.

A BIOGRAPHICAL NOTE

NIKIAS STAVROULAKIS was born at the village Thronos Rethymnes of the island of Crete, Greece, the year 1921. He entered the National Technical University (E. M. Polytechnion), Athens, Greece, in 1938, where he studied Civil Engineering.

Although World War II interrupted the smooth course of his studies, destroyed his country, and he escaped execution by the Nazis for just a few days, he continued his studies after the war was over in 1945. He graduated from the National Technical University (E. M. Polytechnion), Athens, Greece, in 1947.

During the years 1949 – 1963 he worked as a civil engineer in Greece. His work was very trying and under bad conditions, great difficulties, political turmoil and pressure.

The year 1963 he went to France to pursue graduate studies in mathematics. He eventually received “Doctorat d’ Etat” from the “Faculté des Sciences” of Paris in 1969. His dissertation was on: “Substructure of Differentiable Manifolds and Riemannian Spaces with Singularities”.

Then, he was immediately hired as a professor of mathematics by the University of Limoges, France, from which he retired the year 1990.

He is the author of numerous papers related to the subjects of: Geometry, algebraic topology, differential geometry, optimization problems, mathematical physics and general relativity.

Although he has retired for several years, he still continues (2009 at the ages of 88) his scientific and mathematical research. His main purpose is to restore the theory of gravitational field by pointing out the misunderstandings and thus rejecting them, and correcting the mathematical errors committed by relativists from the beginnings of general relativity.

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MAIN SYNOPTIC ARTICLE

TITLE: GENERAL RELATIVITY AND BLACK HOLES By

Dr. Nikias Stavroulakis

Solomou Street 35, Chalandri, Athens 15233, Greece

March 2009

ABSTRACT: In this expository paper Dr. Nikias Stavroulakis, based on the results derived by various specialists since the emergence of Relativity Theory and his own works, composes a detailed enough historical retrospection about the static solutions of Einstein’s equations, given by Einstein only in local coordinates, of the gravitational field of a spherically symmetric body. He presents the solutions provided by Bondi, Schwarzschild, Droste, Hilbert, Eddington-Finkelstein, Kruskal and indicates some of the wrong points in all of them along with the arbitrary and unjustified steps (such as: use of spherical coordinates, change of manifold, use of manifolds with boundaries, Hilbert’s transformation, Droste-Hilbert parameter, Birkhoff’s “Theorem”, and more) taken in order to derive them and to eventually come up with the mysterious, exotic and controversial object called black hole together with all of its schizoid properties. At the end Dr. Stavroulakis presents his corrections, suggestions, static solutions and his result that “Einstein’s equations exclude the creation of black holes and matter cannot be shrunk beyond a certain lower bound”.

1. INTRODUCTION

In the article [8] “Mathématiques et Trous Noirs” published in 1986 in the magazine “Gazette Des Mathématiciens” of the French Mathematical Society, various violations of mathematical principles were pointed out that had appeared since the first attempts of solving Einstein’s equations, which later led many theoreticians to attribute natural existence to a hypothetical object that was named black hole. This has to do with an arbitrary interpretation of those equations which is related to erroneous assumptions that concern the mathematical theory of the gravitational field of a spherical body in General Relativity. This phenomenon is due to the fact that during the time that General Relativity was formulated, the mathematical concepts used, and especially that of manifold, had not been adequately elucidated. The mathematical formalism was not used accurately, as this has been explained in the articles [8], [9], [10].

In the present article we come back to the article in the “Gazette Des Mathémati-ciens” for two reasons:

First: To clarify the basic mathematical concepts that enter into the theory of the gravitational field of a spherical body. Second: To expose with greater clarity the unacceptable points of the classical method, which are supported by the static solutions in order to justify a hypothetical dynamical process that leads to the black hole.

We shall see that this process is excluded by the equations of the theory themselves. 3

With this present chance we will restore a historical inaccuracy in the article of the “Gazette Des Mathématiciens”, an inaccuracy that is stereotypically repeated in all the bibliographical references and was also recorded bona fide in the article under discussion. This has to do with the so called Schwarzschild solution, to which all refer when they deal with the gravitational field of a spherical body. In reality the Schwarzschild solution appeared first, but was abandoned very quickly. However, the name of Schwarzschild remained. Thus the solution which today is called the Schwarzschild solution is in reality the Droste – Hilbert solution. There is an unbelievable confusion on this subject which was nurtured by Droste and Hilbert themselves, who communicated that they had found all over again the Schwarzschild solution. We will not expand on the reasons that have caused this confusion but from now on we will keep the correct terminology distinguishing one solution from the other.

2. THE MANIFOLD OF THE PROBLEM AND THE SPHERICAL COORDINATES

The space that enters into the problem is the ordinary vector space , which is also considered as a topological space (topological product of three real straight lines). In relation to time, various assumptions lead us to portray it as a variable that runs over the real straight line .

Consequently the mathematical space that enters into the theory of the gravitational field of a spherical body is the vector space considered also as a topological space (topological product of four real straight lines).

This simple and clear algebraic and topological characterization was already altered from the inception of General Relativity by the systemic usage of spherical coordinates of , which leads to the alteration of the underlying manifold.

In the frame of Euclidian Geometry, when we say that a point is defined by three Cartesian coordinates we refer to an orthogonal system of three axes. But when we say that the point is defined by the spherical coordinates ρ, φ, θ, then what is the system of reference? We refer again to the system of axes with respect to which we define the distance ρ and the angles φ and θ. To say that the point is defined by the spherical coordinates ρ, φ, θ, without referring to the orthogonal system of three axes has no meaning. Therefore and according to their original definition, the spherical coordinates do not have autonomous existence and moreover they cannot be defined on the axis of .

To define an “autonomous manifold in spherical coordinates”, we must bring in the canonical homeomorphism between

and

If we denote the unit sphere with , i.e.,4

then every point is written in a unique way as the product with

and , because from the relation

with

it follows that and .

Therefore the homeomorphism

is defined if we set

The spherical coordinates are defined over the manifold , if we introduce two parameters in order to define the points y of the sphere . Usually we introduce the parameters

and ,

and then

Since this coordinate system does not cover the whole sphere , we must also use a second coordinate system

, ( and ) ,

and to set (for example)

This coordinate system is always omitted, but certainly both systems (2.1) and (2.2), with , are indispensable in defining the homeomorphism F.

In actuality spherical coordinates are also used for . For this to take place, we must accept a violation of the already accepted mathematical principles, that is, to use an extension of F, which is not a homeomorphism. This extension is defined in a

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unique way: We attach the set to the open manifold , that is, we

replace it with the manifold , and we let as before

Then for every .

But the mapping is not a homeomorphism. The set in its totality is mapped onto the origin of . Hence the phrase that is usually said

“the origin ”

means the “boundary ”, which is introduced in our computations to cover the whole space , but without having any natural meaning.

The spheres with center the origin of are not defined in spherical coordinates.

The distinction between F and is never mentioned, with severe confusion as a result. To see this, let us consider the case of the transformation of the Euclidian metric

via F and via .

The transformation via F, that is, via the transformations (2.1) and (2.2), gives respectively the positive definite forms

and

which together define a positive definite metric on the manifold , that does not represent the totality of the space .

The transformation via again gives the above two positive definite forms, but with , which means the transformed metric on the manifold degenerates

on the boundary since:Firstly: Its signature becomes (1, 0, 0).Secondly: The induced metric on the boundary vanishes.

The injection of spherical coordinates into General Relativity is characterized by the tacit use of the transformation , which causes the alteration of the manifold. Instead of the space the manifold with boundary is used, and instead of the

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space we use the manifold , which is tacitly identified with .

This error has grave consequences. The space-time metrics that are written directly on the manifold come from generally not admissible metrics on the space

. A simple example is the so called Bondi metric

which is on the manifold . To this a metric on the space corresponds, which is obviously written as

and which develops a discontinuity at . So an anomaly is hidden in the Bondi metric deprived of any natural meaning.

Such discontinuities generally appear when the metrics are not defined with the required mathematical rigor on the space .

Another case in which spherical coordinates lead to wrong results is the case of the functions of the norm.

3. THE SMOOTH FUNCTIONS OF THE NORM

A function of the norm

is expressed in spherical coordinates as a function of one variable, , and it is tacitly accepted that it is sufficient to assume that the latter one is differentiable on the half-line in order to deal with various problems. However, the differentiability of does not imply the differentiability of with respect to the variables at the origin of the coordinates. Consequently, if the functions of the metric tensor are functions of the norm , the curvature tensor will expose discontinuities at the origin of the coordinates that do not any natural meaning. To avoid these discontinuities, the functions of the norm that are involved in the problem

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of the gravitational field ought to be smooth functions of the norm, in accordance with the following definition.

DEFINITION. A function of the norm, , will be called a smooth function of the norm, if

a) is with respect to the variables in the space

b) For any set of indices the derivative

At the points tends to a certain limit when .

REMARK. In previous articles, [11], [12], a smooth function of the norm is considered in . But this condition alone does not imply that the derivatives of

exist at the origin. To deal with problems related to the functions of the norm,

we must keep in mind from the beginning that the notation presumes the function of one variable to be defined over the half-line .

THEOREM. Let us assume that the function of one variable is over the half-line , where naturally its derivatives at u = 0 are right-derivatives. In such a case, the function is a smooth function of the norm if and only if the derivatives of the odd order of at u = 0 are vanishing.

Proof: The restriction of to an axis, let us say, is the function and so

for

for

By definition, the derivatives of any order s of these two functions, i.e.,

and

must tend to the same limit when . Then

This relation is an identity when s = even = 2k, but when s = odd = 2k +1, we will have

8

from which

Hence the condition of the theorem is necessary.

We shall show that the condition is also sufficient.The function

( for , and for )

is on the half-lines and . But because

for

for

it follows

and

These relations prove that the function is differentiable also at the origin if

So is an even function on the real line .

The vanishing of the derivative leads to the relation

and since we can differentiate under the integral sign, the function

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is an even function on the real line .

Now, induction shows that we can define a sequence of even functions on the real line . Indeed, if the even function , , has already been defined and we have , then

and the next even function is defined if we let

Returning now back to the functions of the norm , we see that the functions

are with respect to on the half line , where the derivatives of

with respect to the variable , i.e.,

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tend to certain limits when

COROLLARY. The set of smooth functions of the norm is an algebra over the real line .

Proof: Indeed, if the functions and , , have derivatives of odd orders vanishing at , the same is true for the derivatives of odd orders of their product

because in this sum each term contains a derivative of odd order of one of the two functions.

REMARK: In applications the function is not required to be on the half-line . The order of differentiability of depends on the required order of differentiability for the partial derivatives .

4. S Θ (4) – INVARIANT METRICS

A Riemann metric on the space is said to be spherically symmetric if it remains invariant under the action of the group SO(3) in . Intuitively such a metric is written as

where the functions and must naturally be smooth functions of the norm.

It is of course obvious that the metric (4.1) remains invariant not only by the action of the group SO(3), but also by the action of the group Ο(3). The converse proposition, that is, a metric invariant by the action of the group SO(3), (or the action of the group O(3) ), is of the form (4.1), is usually referred to without proof as evident.

H. Weyl [13] simply says that a metric with spherical symmetry must be of the form (4.1) with an appropriate choice of coordinates (bei Benutzung geeigneter Koordinaten), but without any further explanation. Levi – Civita in his monograph

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“Lezioni di Calcolo differetiale assoluto” presents a method due to Palatini, which refers to spherical coordinates. However, this is not a real proof because the action of the group SO(3) is not defined in spherical coordinates.

In reality the proof of the converse proposition is not evident and lies in the general frame of SO(3) – invariant tensor fields on the space . For the case of the space –time metrics there is an additional difficulty, because these are defined on the space

, whereas the groups SO(3) and Ο(3) act on the space . To end up with clear definitions, it is necessary to introduce the group SΘ(4), which consists of the matrices

with , , and SO(3) . It is also necessary to consider the

larger group Θ(4), which consists of the matrices of the same type but with O(3).

It is surely understood that instead of the simple smooth function of the norm, we will now have functions of the form , whose derivatives with respect to the

variables for , tend to certain limits when

. The set of these functions is also an algebra and we denote it with Γ0 .

The proof of the next Theorem is given in the article [11].

THEOREM. Let , be a covariant, SΘ(4) – invariant, symmetric tensor field of degree 2 on the space . Then there are 4 functions

that belong to the algebra Γ0 and such that

with and

Moreover the tensor field is Θ(4) – invariant. □

If the tensor field is a space-time metric, that is it has signature (1, -1, -1, -1), we usually write it as a quadratic form

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, (4.2)

This general form of the space-time metric has never been used in Einstein’s equations for the gravitational field of the spherical body. Only stationary or static metrics, that is metrics that do not depend on time, enter into the classical solutions of Einstein’s equations. During the first years of General Relativity, this restriction seems to be standard due to technical reasons for avoiding the difficulties involved in considering a time-dependent tensor. But later, with the emergence of the so called “Birkhoff Theorem”, it became believable that the exterior field of the spherical body is always static regardless of the transformations of the body as long as it preserves its spherical shape. As it is analyzed in previous articles and especially [10], we have got in hand a pseudo-theorem that has disoriented the evolution of ideas in Relativity. We will not especially deal here with this issue, since our critique is mainly addressed to the classical solutions of Einstein’s equations. Into the latter another “simplifying assumption” has entered: it is assumed that the function q01 is zero. This admission was most probably suggested for technical reasons, but it also seems that there was the erroneous view (based on abusive usage of implicit transformations) that this admission does not influence essentially the solutions that result. In reality, however, the vanishing of q01 has not allowed the study of the time contribution to the space-time metrics.

The above simplifying admissions were considered self-evident by Schwarzschild, who in order to define, as he believed, the gravitational field of a mass point, suggested the first static metric of spherical symmetry

where F, G, H are introduced as functions of the norm without any further explanation. This metric, which is referred to as obvious, could have provided the basis for the correct investigation of the gravitational field of a spherical static body, but it was never used. Schwarzschild himself immediately transforms the metric to spherical coordinates and from then on different metrics are considered over the manifold with boundary .

We will make here a short critical analysis of the first two static solutions of Einstein’s equations on account of the influence that they have had in the evolution of ideas in Relativity.

5. THE SCHWARZSCHILD SOLUTION [7]

The Schwarzschild solution involves a positive constant α (which later was identified

with the value ) and after we set it is written in the classical

expression

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Let us consider the space metric (on the manifold ), which is a part of this space-time metric

or .

This is differentiable on the space . But for R = α (that is, r = 0) it develops discontinuities. Nevertheless the integral

has meanings and so the lengths of the geodesics (φ = constant, θ = constant), or (φ΄ = constant, θ΄ = constant), measured from the boundary are well defined. Additionally, the metric induced on the boundary, that is , defines the topology of a sphere. The observer’s space is then identified topologically with a three dimensional semi-cylinder, and so with a space different than that was the space of the originally formulated the problem.

Schwarzschild has assumed that the point mass was placed at the origin of , which, by his assumption, was the only irregular point of the metric. However now, we do not have just an irregular point, but an infinitude of irregular points that make up the boundary , which has the cardinality of the continuum. The content of the problem is found to be fundamentally modified.

The Schwarzschild solution is incompatible with the topological and metric presuppositions of the problem and for this reason it was rightly abandoned. Except, the reasons of its abandonment were not rightly and clearly formulated. On this Hilbert [2] writes: “In my opinion, the identification of the space r = α with the origin, as Schwarzschild did, is not recommended. On the top of it, the Schwarzschild transformation is not the simplest for this purpose.” Hilbert obviously hints at the transformations r΄ = r + constant, which, as assumed, had as a consequence the change of the origin of coordinates and the transformation of a sphere to a point.

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6. THE DROSTE SOLUTION [1]

Droste thinks that the gravitational field of a “center” is defined by the static metric in spherical coordinates

where ω, u, v are functions of the variable r. Henceforth he says, we have the freedom to pick the variable r in such a way that u = 1, and therefore we must determine the metric

He does not clearly state the conditions of this change and he does not observe that the variable r represents the radial geodesic distance in the space . He finds that the function v = v(r) satisfies the equation

but without observing that the latter can be immediately solved: Multiplying both sides by v΄ we find

or

hence (Α = constant)

and so

The last equation determines the function and since he also finds that the solution of the problem is over.

However, Droste does not follow this conspicuous method, but he abandons the metric (6.1) and introduces a new metric that depends on the variable (instead of r). The transformed metric that he finds is not simple enough and for this reason he makes one more transformation, , which gives

(α = constant > 0)

Finally, Droste additionally uses the transformation , and finds the metric

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in which the new coordinate r appears just by chance. We insist, just by chance, because if Droste had not changed the original formulation of the problem, he would have found a different result, which indeed would be interesting from the realistic point of view. Droste does not try to determine the constant α at all. He simply exempts the values r < α .

7. THE HILBERT TRANSFORMATION

We come back to the space metric (4.1) in order to expose its evident geometrical properties.

The directed semi-lines are geodesics for this metric and also, for every real ρ > 0, the Euclidian sphere

is also a sphere for the metric. Two basic quantities of the metric correspond to these two geometrical properties:

First: The geodesic distance between the origin and another point is

Second: The radius of curvature of the sphere is given by the formula

Certainly, whereas the function is a smooth function of the norm, the same property is not true for the function , but, because of the geometrical properties of , it is useful to introduce this function, without forgetting that the

differentiability conditions will be checked for the function . Now, if we set

,

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the metric is written in a more convenient form, which is immediately connected with the two basic geometric quantities

As long as we stay within the frame of action of the group Ο(3) on the space , and there is certainly no reason to abandon it, we must keep the metric (7.1) as the metric of reference. But this form has been excluded from the theory of General Relativity. As we know, the underlying space was rejected and superseded by the manifold with boundary .

Let be the induced metric on the unit sphere

which is derived from the Euclidian metric of . Then passing to spherical coordinates alters the restriction of the metric (7.1) on the space to a

metric on the space :

In General Relativity, the metric (7.2) is introduced directly on the space and then the value ρ = 0 that defines the boundary of is considered to represent the “center”, that is the origin of . This is the first false step that was followed by a second one much more serious and with irreparable consequences. This is the inadmissible transformation that standardized the replacement of the radial coordinate ρ by the “coordinate r” by means of the equation

This transformation that was first done by Hilbert [2] presupposes that, no matter what the function in the metric (7.2) is, the above equation determines the radial coordinate via the “coordinate r” and that the corresponding inverse function is well defined and . As we shall see, this assertion is not true, but at the present time we will deal with the reasons that, regardless of the computations, do not permit the introduction of the variable r.

Since the function is unknown and in order to find it we must use Einstein’s equations, the relation introduces an implicit function whose determination is impossible. In reality, Hilbert’s transformation consecrates a tactic that allows the usage of implicit functions that derive from equations that contain the unknown functions of the metric tensor. The (hypothetical) derived transformations, even though they do not admit a real definition, are used in order to impoverish the metric as much as possible (that is in order to minimize the number of the unknown functions as much as possible) before expressing and solving the equations of the

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problem under consideration. Under these situations the derived solutions cannot be adjusted to the boundary condition of the problem.

Despite these simple observations, Hilbert’s “coordinate r” rushed into General Relativity during the decades that elapsed after the publications of Einstein’s first articles in the years 1915-1916.

The abusive introduction of the “coordinate r” into (7.2), through the implicit function , leads to the mutilated metric

which was systematically used as the metric of space within the space-time metric. There is no article or book related to the gravitational field of a spherical body that is not founded on the metric (7.3).

In our previous article [8], we had named the “coordinate r” as the Levi – Civita parameter on the basis of a reference by Levi-Civita himself, who in his monograph “Lezioni di Calcolo differentiale assoluto”, writes that he had already introduced the metric (7.3) in an earlier article of his (published in the: Atti della R. Acc. dei Lincei, 2 sem., Vol. 5, pp. 164-171, 1896). This reference is inaccurate and is due to some confusion. The responsibility for the introduction of the “coordinate r” belongs exclusively to Hilbert, but because this “coordinate” had appeared in the Droste solution, by chance as we said before, from now on we will call it as: the Droste – Hilbert parameter.

It is surprising to comprehend the haste with which the function was uprooted from the metric (7.4). If this metric needed modification, even though there was no reason for that to begin with, then the geodesic distance

should have been chosen as new radial coordinate, whose derivative is strictly positive everywhere.

As for the Droste – Hilbert parameter is concerned, even if it is considered as radial coordinate, it has nothing to do with a coordinate. Indeed, when we refer to the metric (7.1), we see that every sphere can be located, i.e., it can determined inside the

space by its radius , which is the radial coordinate with respect to the metric (7.1), and which allows to possibly use the geodesic distance as radial coordinate. But the sphere is a non-Euclidian object and consequently its radius of curvature does not play any role in locating it in space on account of the following reasons:

First: We cannot find the radial coordinate ρ by using a function whose determination we do not know. Indeed the function admits an infinitude

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of determinations and in the case of General Relativity we seek to find a special determination (or special determinations) by solving the system of Einstein’s equations. Second: Every determination by means of Einstein’s equations depends on the initial conditions that are set on the basis of appropriate values of the radial coordinate ρ.Third: The direct measurement of is very difficult if not impossible. But even if we manage to measure the radius of curvature of some sphere , this would not give us any information about its radius ρ [or the corresponding geodesic distance ].

We then see that, the replacement of the function by the Droste – Hilbert parameter, or equivalently the replacement of the full metric (7.1) by the mutilated metric (7.3) has an irreparable consequence: Since the metric (7.3) contains exclusively the Droste – Hilbert parameter, we cannot locate the positions of the spheres ρ = constant > 0 in space.

Besides all of these, the specialists in Relativity believe that the Droste – Hilbert parameter is the radial coordinate and this idea leads to an unacceptable confusion. Along these lines I. Novikov [6] writes:

“In the ordinary geometry there are two ways to determine the radius: First as the distance of the points of the circular circumference from the center, and second as the ratio of the length of the circular circumference divided by 2π. As a result of “space-curving”, these two quantities do not coincide in the Non-Euclidian geometry. The second way is easier…For this reason, we will always refer as radius the length of the circle divide by 2π .”

Here I. Novikov suggests to us to reject all the basic concepts and principles of geometry: To define the radius by dividing the length of the circumference by 2π. But he forgets to say, we must first know that the closed curve, whose length we measure, is the circumference of a circle. But the circle is a geometrical object that in every Riemannian geometry is defined by its center and its radius. Also, in the Non-Euclidian geometry, the length of the circumference of a circle divided by 2π is not equal to the radius. With this opportunity we emphasize that the length of the circumference of a great circle, in the case of spherical symmetry, is and hence it depends on the determination of the radius of curvature by means of Einstein’s equations.

8. THE HILBERT SOLUTION

Hilbert following Schwarzschild starts with the static metric of spherical symmetry in spherical coordinates, which we write here in the more convenient notation

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( with or )

Hilbert considers as arbitrary functions (willkürliche Funktionen), obviously positive, and then immediately and without any explanation introduces the transformation

By saying that r, θ, φ can also be considered as spherical coordinates, hence introducing, as he says, the coordinate r instead of ρ in the (8.1), we obtain the metric

Hilbert omits to clarify the mathematical conditions required to express the variable ρ by r:

First: The function is strictly monotonic (in our case strictly increasing).Second: The derivative does not vanish anywhere.

In short, Hilbert’s transformation presumes that for every , without offering any justification.

This simple observation is important, because as we shall see below, from the exterior solution of Einstein’s equations in relation to (8.1), we get that the derivative vanishes for the values of ρ that determine the limit of validity of the exterior solution. For this limit the solution does not hold and the derivative of the function at the limit points is infinite. This phenomenon has as consequence the fact that the function becomes also infinite and this has led to endless discussions about the so called back holes.

As we saw earlier several geometrical reasons do not permit the usage of Hilbert’s transformation. Now, we also see that the solutions of Einstein’s equations exclude this transformation. Hilbert of course believes that his transformation does not change the solution of the problem and simply treats the solution of field-equations in relation to the mutilated metric (8.2). Naturally, for the formulation of the problem the initial conditions must first be determined, which amounts to locating the spherical source of the field by determining its center and radius. But this is impossible if we use the metric (8.2). This latter refers to the radius of curvature of the sphere that contains the matter, and it does not allow finding its radius in the absence of the radial coordinate. Also, the center cannot be found as long as we insist to treat the problem on the manifold . All and all, the sphere of matter has neither center nor radius with respect to the metric (8.2); i.e., it is inexistent for this metric.

A badly set problem has but a bad solution. Hilbert’s computations give

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(Α = constant)

and finally the solution he finds (which, he very curiously attributes to Schwarzschild) coincides with the Droste solution. As Droste did not, so Hilbert does not spend any time with the determination of the constant Α. We will soon see that the solution of this problem leads to a well documented critique on the solution!

9. CRITIQUE ON THE DROSTE – HILBERT SOLUTION

As it is known the constant Α is determined in such a way that the function

has asymptotical approximation the Newton Potential at the points located at great distances from the center. For the determination of the constant Α we then need the a-priori computation of the distances

.

But, we cannot compute these distances because the function α(r) is still unknown.

So we are at an impasse: In order to determine the metric we must first calculate the distances, and in order to calculate the distances we must first know the metric. In front of this impasse we change position against the Droste – Hilbert parameter. This latter was a radius of curvature at the beginning. Then it was transformed to radial coordinate. From then on it would be distance! With such

a pseudo-syllogism, is not anymore asymptotically equal to the Newton

potential, but exactly equal, i.e.,

,

from which we get

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In this way we reach the so called Droste – Hilbert solution, which is determined by the functions:

Every time we deal with a non-linear system of differential equations, the values of the constant(s) that bring about the anomalies, at the same time they define the connected sets over which the corresponding solutions are valid. In this particular case the evaluation of the constant Α is determined by the large values of r and then the solution obtained is extended to be valid for the small values of r as long as continuity is guaranteed, i.e., as long as . This inequality determines the validity domain of the solution and especially implies that the radius of curvature of the sphere that confides the matter is greater than 2μ.

After so many violations of mathematical principles, lo, at last, a conclusion that can be proven to be correct. This however, was ignored by the Relativity specialists who still persist that the solution is valid for all with two anomalies

r = 0 and r = 2 μ.

The first is considered as a “real anomaly”, whereas the second is attributed to the “pathology of the Droste – Hilbert coordinates”. Naturally, since r is considered to be distance, the value 2μ is not considered to be radius of curvature but as a radius styled “gravitational radius of the mass m”.

Nevertheless the violations of mathematical principles do not end here and the Droste – Hilbert parameter is destined to undergo a third deformation. Indeed, for , the roles of the coordinates t and r are switched, and now the Droste – Hilbert parameter represents time. However, they still attribute to it an additional “radial character” so that the moment r = 0 represents the center!

Needless to add that the value r = 0 does not define a point, but the manifold on which the induced metric cannot be defined. In the beginning the

spherical mass had neither center nor radius and nobody dared to attribute a radius of curvature to the sphere that confided it. As for the manifold with boundary that represents the “space” they in excess identified it with . Now the space is deformed into a three dimensional cylinder and we no longer comprehend the object of our problem.

As for the alleged “pathology of the Droste – Hilbert coordinates”, we have in hand an expression deprived of any mathematical content and the explanations given to it are contradictory. So, in accordance with Misner, Thorne and Wheeler [5]:

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“We cannot be sure, without pertinent studying, if this pathology of the line element is due to the pathology of Droste – Hilbert geometry itself {in [5] they permanently write Schwarzschild geometry; very curiously, the mane Droste does not exist anywhere in that whole voluminous book, of 1279 pages} or simply to a pathology of the coordinate system near r = 2μ. As an example of a pathology coming from the coordinates, let us consider α neighborhood of θ = 0 on any of the invariant spheres x0 = constant, r = constant, of . There, vanishes because the coordinate system behaves badly”.

Naturally, there is nothing pathological in this coordinate system, but we simply deal with comparing two different situations: The value 2μ belongs to the interval of definition of , whereas the value θ = 0 is excluded before anything else,

since it does not belong to the interval domain of θ, namely the .

This bizarre concept of pathological coordinates echoes, without being referred explicitly, even in literature with dominant mathematical character. So, in his book “Théories relativist de la gravitation et de l’ électromagnétisme” (“Relativistic theories of gravity and electromagnetism”), A. Lichnerowicz has established a manifold structure such that the second derivatives of the changes of coordinates are piecewise . We do not understand at all the purpose of such a structure , but the author explains a bit later that in this way it is possible to introduce discontinuities into the second derivatives of the components of the metric tensor:

“We see that in this way we can make discontinuities of the second derivatives (of the metric tensor) emerge, discontinuities that are without any intrinsic or natural meaning … Such circumstances are found in studying the Schwarzschild conditions.”

In reality the introduction of the concept of pathological coordinates is not without ulterior motives, for it is used to bolster the theory of the black holes: The Droste metric is good and the only thing we still need is to change the “system of reference” in order to surpass the anomaly , which is entirely conventional. On the basis of this principle the Relativity specialists search for finding other “systems of non-pathological coordinates”, thing that in reality means other metrics without anomalies and which can be piecewise transformed into the Droste – Hilbert metric. All these metrics are found formalistically starting from the Droste – Hilbert metric and then applying transformations that contain anomalies (in order to eliminate the anomaly r = 2μ) without changing the essence of the problem, which, as before, remains without clear formulation. Thus, the Droste – Hilbert parameter always appears directly or indirectly along with all the hindrances associated with it.

We will get out of our scope, if we refer to the numerous deformations of the Droste – Hilbert solution, but it is pertinent to comment on two: The Eddington – Finkelstein metric and the Kruskal metric.

The Eddington – Finkelstein metric is also defined on the manifold with boundary and its simplest form, in [5], is found by means of the transformation

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,

that gives

.

At a first glance the anomaly r = 2μ has been eliminated, but in reality it remains hidden. First of all the Eddington – Finkelstein metric must be considered with an arbitrary constant Α in the place of 2μ, and the determination of Α requires computation of distances, fact that leads to the same difficulties as the metric (8.2) and the reemerging of the anomaly r = 2μ. In general this anomaly reemerges as soon as we try to execute the tiniest computation (e.g., the determination of the geodesics, etc.). Moreover the value r = 2μ marks a cut in the interpretation of the metric: For

, the coordinate u ceases to have time character and without the possibility to attribute any natural meaning to it. This is indisputably the reason for which Eddington – Finkelstein metric gave up its position to other metrics, no less unclear and gloomy, in the latest edition of the book “Field Theory” by L. Landau and E. Lifchitz. They always consider it as one of the pillars of the back holes theory, but as it seems it does not serve well in this scope. The partisans of the theory [5] use it to describe the motions of particles that traverse the surface r = 2μ with directions towards the so called center r = 0.

Curiously they observe that in the frame of their formalism, Eddington – Finkelstein metric “does not impede” the photons to come out of the black hole contrary to the hypothetical properties of the latter. But in any case, in order to extract the conclusion that “light remains trapped in the black hole”, in the study of the motions of particles with directions toward the outside of the black hole, they use, contrary to elementary principles, e.g. in [5], the different but of the same type metric

.

As for the Kruskal metric, this appears as solution of an implausible problem: “Determine the maximal manifold on which the Droste – Hilbert metric can be extended without anomalies.” In other words, we do not know the manifold entering the problem of the spherically symmetric field and we look for finding it amongst all various possible choices. But every problem in General Relativity is related to a certain manifold and in the examined case this manifold is obviously . For what reason do we have to search a greater one? This question has no meaning evidently.

Kruskal, however, poses this question because he entirely abandons the manifold and thinks that he can arbitrarily transform the half-plane inside

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the manifold with boundary . He considers a partition of this set into 4 subsets which in the sequel he darns back again through some transformations that contain discontinuities, that is, through transformations prohibited by mathematical thinking and consideration. He extends the closed set produced out of these darning processes without justification and by symmetry with respect to the origin of the coordinates u, v. In this way he obtains the “maximal manifold” whose boundary consists of two connected components and defined in the following way:

Let H be the hyperbola with equation v2 – u2 = 1 on the plane of the coordinates u, v and let us denote with U the connected component of that contains the origin. Then , where is the closure of U. Then, the Kruskal manifold is the product and the corresponding metric is written

where the Droste – Hilbert parameter r must be replaced by the function of that is found by solving the equation

.

Contrary to elementary mathematical principles, the “center” is represented by the two branches of the hyperbola Η. As for the “natural space”, it exposes different properties as or . For , it is a three-dimensional cylinder, whereas for it is a non-connected space, union of two three-dimensional semi-cylinders. We do not understand anything anymore and we find ourselves in a complete mythologization.

Despite numerous attempts to improve the Droste – Hilbert solution, its flaws remain conspicuously incurable because they are strongly connected with the Droste – Hilbert parameter, which renders the formulation of the problem impossible. As for the ideas stemming from it, have ended to arriving at a contradiction with the basic principle of General Relativity, according to which, “every deformation of masses has as consequence the transmission of gravitational phenomena from place to place”.

Exactly as in Newton’s classical theory, they corroborate that the field of a spherical mass being been deformed under preservation of its spherical symmetry (contracting, expanding, pulsating, etc.), remains always static according to General Relativity as well. This proposition, that has acquired position of a Theorem styled as Birkhoff Theorem, “is proven” starting from the general stationary metric of spherical symmetry, considered on the manifold with boundary , and in the sequel modifying it (more accurately rendering it poorer) by means of inadmissible transformations that ignore mathematical principles much more than the inadmissible Hilbert’s transformation. But then, as in the case of the static source of the field, the being deformed spherical source is inexistent for the metric. Especially the radius of

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the spherical mass as a function of time is impossible to be defined because there is no radial coordinate. It is also impossible to define the radius of curvature of the sphere that encloses the being deformed mass, as a function of time and with respect to the Droste – Hilbert parameter r, because a function of time represents “a radial motion inside the exterior field” and not a change of the curvature radius itself. As consequence, the formulation of the initial conditions for the exterior non-static field becomes impossible.

Despite all of the above “the proof of Birkhoff theorem” continues and finishes with the integration of Einstein’s equations as they were set forward in terms of this metric, which is deprived of any natural content. The solution found is static, in reality it is the Droste – Hilbert solution, and it could not have been any differently, because the spherical being deformed mass is inexistent ─ as the static spherical mass was also inexistent ─ for the initial conditions of the problem. We now see that the Droste – Hilbert parameter, leads to a vicious circle.

So, the Droste – Hilbert parameter, has led us, via the Droste – Hilbert solution, to the imaginary world of the black holes without contributing anything to our knowledge. But, the thus created ideology remains still deep-rooted. I. Ekeland feels satisfaction about the black holes and the expanded universe that put an end to the “suffocating universe” as Laplace had conceived it. But a universe of fantasies is even more suffocating and the tendency toward mythologization does nothing more than to pull backward our effort in knowing nature. As for Einstein’s equations, the anomalies of their solutions, presented in vague and doubtful forms, were gradually disguised with a mythical face that cause negative reactions even from the part of many specialists in Relativity. As S. Mavridés writes, [4]:

“The cosmic anomaly (Big Bang) is in every part similar to the anomaly found in the interior of the back hole. Such theoretical predictions are repulsive. What makes them even more abhorrent is the fact that by hypothesis they have been realized in a very distant past.”

10. STATIC SOLUTIONS OF EINSTEIN’S EQUATIONS

The situation reveals itself in a new light as soon as we abandon the Droste –Hilbert parameter by thus letting Einstein’s equations to “speak freely”. In order not to extend ourselves to topics beyond the scope of the present article and to also have an immediate comparison with the classical solutions, we will restrict ourselves to static fields, and moreover, by (7.1), the space-time metric on is written without mutilation:

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with .

Supposing moreover, in order to have as simple results as possible, that the matter is not electrically charged, we remind that the integration of Einstein’s equations for the exterior field [12] yields the general static solution by the two following relations

the first of which contains the constant of integration Α.

The solution will be valid in the half-line where ρ1 denotes the radius of the static spherical mass. Since we have two equations in three unknown functions, the latter seem to be over-determined. Nevertheless we ought to avoid imposing restrictions in the determination of the functions hastily, because there is danger to alter the problem. In this present case we must first solve the problem of determining the constant Α. This problem plays a fundamental role and therefore we will treat it with required precision.

The determination of the constant Α is based on the asymptotic approximation of the metric stemming from Newton’s potential. The latter is defined by the distance from the center, which, in our case, is given by the integral

The function β(ρ) is strictly increasing and tends to when . Therefore the inverse function ρ = γ(δ) is also strictly increasing and tends to when . The function can be considered as functions of the distance δ as follows:

From the identity

it follows

and so

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and the equations (10.1) are written as

The function does not appear anymore because it is replaced via the relation

. Indeed, the introduction of the radial geodesic distance δ presumes a transformation of the initial space-time metric by means of the equations

which, as explained in the article [12], lead to the metric

with

We will not expand into these details; we only note that this transformation is admissible because it uses smooth functions of the norm. The problem is finally reduced to the determination of the function by means of the equation

It is obvious that we must first determine the converse function by integrating the equation

which gives

where Β is the constant of integration. As for the constant ξ, this is not an integration constant, but an arbitrary length that is used in order to have the dimensionless

expressions , and consequently to be able to introduce the logarithm

present. (We can, e.g., take ξ = 1 the unit length.)

From the previous equation it follows that

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tends to 1 when (and so when ).

Therefore

and

where as

As for the function we have

,

hence by putting we find the asymptotic approximation of (10.2)

which, according to a well known syllogism, leads to the identification

and so ,

( k is the constant of the Newtonian gravitational field).

Since from the equation

it follows

we now have

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and so the function of the exterior solution has lower bound the positive number 2μ.

Equation (10.3) is now written as

and since the lower bound 2μ is found when , or

,

by introducing the constant

we get the equation that determines the function in the form

or

which does not contain the ancillary ξ.

Finally the function is defined as the inverse function of a known strictly increasing function. We emphasize that, besides the condition

the function is such that the difference

tends to when .

As for the function , we have

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,

since as we saw above the function of the exterior solution satisfies .

But the equality is impossible. Indeed, if for some value of δ1

of the geodesic distance, we will have

and

which implies the degeneration of the metric, thing naturally impossible. In consequence

in the interval of definition of G.

Finally, the value 2μ does not represent any radius, but a lower bound, impossible to be materialized, of the radius of curvature of the sphere that encloses the matter. But, since the radius of curvature vanishes only at the center, it follows that the contraction of the spherical mass cannot be limitless and hence the problem of the formation of a black hole cannot even be posed. We thus discover that the solution of Einstein’s equations in incompatible with the concept of the mass-point, a concept that was established in the Classical Physics and the Quantum Mechanics. So, General Relativity excludes this concept, which is also found to be in basic antithesis with our intuition. The exterior solution is also characterized by the presence of the new constant δ0, which does not necessarily have just positive values. Thus we distinguish three cases:

α) δ0 < 0. The values of for have no natural meaning, because is defined on the half-line . But even the value δ = 0 is excluded, because from the relation

we get , contrary to the geometrical condition . Consequently, there is a constant δ1 > 0 (the radius of the spherical mass) such that the restriction of to the half-line is considered only.

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β) δ0 = 0. Then

and so contrary to the geometrical condition . Consequently, the solution is valid, as before, in a half-line with δ1 > 0.

γ) δ0 > 0. Then , , and so the metric degenerates, and therefore it has no natural meaning for δ = δ0. Consequently there is a constant δ1 > δ0 such that the solution is only valid on the half-line .

From mathematical point of view, the constant δ0 can be determined via an initial condition, that is, by the radius of curvature of the sphere that encloses the matter. Indeed, if δ1 is the radius of this sphere, and if the value of is known, then

Nevertheless, it is very hard, if not impossible, to find the value with direct measurements.

REFERENCES

[1] J. DROSTE, The field of a single centre in Einstein’s theory of gravitation and the motion of a particle in that field (Communicated in a meeting of May 27, 1916) Proc. Acad. Sci. Amst., 19(i), 197 – 215.

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[2] D. HILBERT, Göttinger Nachr., Zweite Mitteilung (Vorgelect am 23 Dez. 1916).

[3] M. D. KRUSKAL, Maximal Extension of Schwarzschild Metric, Phys. Review, vol. 119, September 1, 1743 – 1745, 1960.

[4] S. MAVRIDÉS, Les concepts de la physique dans la cosmologie contemporaine, dans “La pensée physique contemporaine”, édité par S. Dinner, D. Fargue, G. Lochak, Fondationn Louis de Broglie, éditions Augustin Fresnel, 59 – 79, 1982.

[5] C. W. MISNER, K. S. THORNE, J. A. WHEELER, Gravitation, W. H. Freeman and Company, San Francisco, 1973.

[6] I. NOVIKOV, Chernye dyry i Vselennaia (The Black Holes and the Universe).

[7] K. SCHWARZSCHILD, Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsber. Preuss. Akad. Wissen,. Phys. Math. Kl, 1916, 189-196

[8] N. STAVROULAKIS, Mathématiques et trous noirs. “Gazette des mathemati-ciens”, no 31 – Juillet 1986, 119 – 132.

[9] N. STAVROULAKIS, On the principles of general relativity and the SΘ(4) – invariant metrics. Proceedings of the 3rd Pan-Hellenic Congress of Geometry, University of Athens, Greece 1997, 169 – 182.

[10] N. STAVROULAKIS, Vérité scientifique et trous noirs (premième partie). Les abus du formalisme. Annales de la Fondation Louis de Broglie, Volume 24, no 1, 1999, 67 – 109.

[11] N. STAVROULAKIS Vérité scientifique et trous noirs (deuxmième partie). Symétries relatives au groupe des rotations. Annales de la Fondation Louis de Broglie, Volume 25, no 2, 2000, 223 – 266.

[12] N. STAVROULAKIS Non – Euclidean Geometry and Gravitation. Progress in Physics, Volume 2, April 2006, 68 – 75.

[13] H. WEYL, Zur Gravitationstheorie, Annalen der Physik, 54. 1917, 117-145.

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LIST OF SCIENTIFIC PUBLICATIONSon mathematical physics and general relativity

By

The Mathematical Physicist Dr. Nikias Stavroulakis

Note: The list of these articles is composed in reversed chronological order.

1) On the Stationary Charged Spherical Source. Progress in Physics. Volume 2, April 2009, pp. 66 – 71.

2) Gravitation and Electricity. Progress in Physics, Volume 2, April 2008, pp. 91 – 96.

3) On the Gravitational Field of a Pulsating Source. Progress in Physics, Volume 4, October 2007, pp. 3 – 8.

4) On the Propagation of Gravitation from a Pulsating Source. Progress in Physics, Volume 2, April 2007, pp. 75 – 82.

5) Non – Euclidean Geometry and Gravitation. Progress in Physics, Volume 2, April 2006, pp. 68 – 75.

6) On a paper by J. Smoller and B. Temple. Annales de la Fondation Louis de Broglie, Volume 27 no 3, 2002, pp. 511 – 521.

7) Matière cachée et relatitité générale. Annales de la Fondation Louis de Broglie, Volume 26, no spécial, 2001, pp. 411 – 427.

8) Vérité scientifique et trous noirs (Quatrième partie). Determination de métriques Θ(4) – invariantes. Annales de la Fondation Louis de Broglie, Volume 26, no 4, 2001, pp. 743 – 764.

9) Vérité scientifique et trous noirs (Troisième partie). Équations de gravitation relatives à une métrique Θ(4) – invariante. Annales de la Fondation Louis de Broglie, Volume 26, no 4, 2001, pp. 605 – 631.

10) Vérité scientifique et trous noirs (deuxmième partie). Symétries relatives au groupe des rotations. Annales de la Fondation Louis de Broglie, Volume 25, no 2, 2000, pp. 223 – 266.

11) Vérité scientifique et trous noirs (premième partie). Les abus du formalisme. Annales de la Fondation Louis de Broglie, Volume 24, no 1, 1999, pp. 67 – 109.

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12) On the principles of general relativity and the SΘ(4) – invariant metrics. Proceedings of the 3rd Panhellenic Congress of Geometry, Athens Greece 1997, pp. 169 – 182.

13) Sur la function de propagation des ébranlements gravitationnels. Annales de la Fondation Louis de Broglie, Volume 20, no 1, 1995, pp. 1 - 31.

14) Particules et particules test en relativité générale. Annales de la Fondation Louis de Broglie, Volume 16, no 2, 1991, pp. 129 – 175.

15) Sur quelques points de la theorie gravitationnelle d’ Einstein. Tiré à part de Sinsularité, Lyon, France, Vol. 2, no 7, Aout – Septembre 1991 / 2461 pp. 4 - 20.

16) Solitons et propagation d’ actions suivat la relativité générale. (Deuxième partie). Annales de la Fondation Louis de Broglie, Volume 13, no 1, 1988, pp. 7 – 42.

17) Solitons et propagation d’ actions suivat la relativité générale. (Premième partie). Annales de la Fondation Louis de Broglie, Volume 12, no 4, 1987, pp. 443 – 473.

18) Mathématiques et trous noirs. “Gazette des mathematiciens”, no 31 – Juillet 1986, pp. 119 – 132.

19) Paramètres cachés dans les potentials des champs statiques. Annales de la Fondation Louis de Broglie, Volume 6, no 4, 1981, pp. 287 – 327.

20) A Statical smooth extension of Schwarzschild’s metric. Lettere al Nuono Cimento, Vol 11, no 8, 26 October 1974, pp. 427 -430.

21) More articles by professor emeritus Dr. Nikias Stavroulakis are submitted for publication and or in the making, 2009.

Some Extra Papers on these Subjects

1) More published scientific research on these subjects is found in the references of these papers.

2) Les trous noirs dans l’ obscurité, “Le Monde”, 12-8-1987. (A small article in the French newspaper Le Monde.)

3) La cosmologie: mythe ou science?. La Recherche, Volume 7, No 69, Juillet -Aout 1976, pp. 610 – 616. An article, by the Swedish Physicist and Nobel Prize Laureate: Hannes Alfvén.

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THE CIRCULATING ERRORS AND

SUGGESTIONS, CORRECTIONS, CONCLUSIONS

COMPOSED

By

Dr. Ioannis, Neoklis Philadelphos, M. Roussos Professor of Mathematics

Some of the errors committed by mathematical physicists and relativists in regard with the Birkhoff Theorem in Relativity, the Black Holes Theory and consequently the Big Bang Theory. These and a few more have been pointed out explicitly and in great detail with examples and counterexamples by Professor Dr. Nikias Stavroulakis. They can be studied and checked by anyone throughout his papers above. The referees of all these journals admitted that they have not found even a single minor or major mathematical mistake. However, the main stream mathematical physicists refuse to pay any attention whatsoever, check them out and conform accordingly! Not only have they despised Dr. Stavroulakis, but they have also shown bad and unprofessional behavior toward him and those who have agreed with him, beyond any fit and professional deontology. Another surprising trait of Dr. Stavroulakis’ works and solutions is that his computations are explicit, simple, clear and within the mathematical principles and concepts so that his mathematics and results be accessible even to non-specialists.

1. They have created the following contradiction: One the one hand: they claim that every change in the distribution of matter generates a gravitational effect which is propagated in space according to the law of the null geodesics. On the other hand: they claim that if a distribution of matter is spherically symmetrical, curiously no gravitational waves are produced by its radial pulsations.

2. They have abused manifolds with boundaries instead of using the Euclidian Spaces they had beginned with.

3. They have introduced, abused and wrongly used spherical coordinates with origin singular points or points not necessarily on the manifold, thus giving rise to singularities, frequently incompatible with the data of the problem.

4. They have introduced and abused unknown, implicit and non-admissible transformations which they may not even exist to begin with or in turn they are the cause of the singularities

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5. They have extended solutions of differential equations beyond points of singularities, thus creating poor solutions with singularities.

6. They have used points that they do not know a-priory they belong to the geometrical manifold under consideration.

7. They have used the Bianchi identities as constraints with Einstein’s equations in the vacuum in order to find four degrees of freedom in the solutions, whereas they are universal identities valid for any metric tensor regardless..

8. The conditions defining the so-called harmonic coordinates have lead to inconsistencies.

9. SO(3) acts on R3 in a natural way, thing not valid for R4.

10. They have arbitrarily and without reason removed components of the metric tensor.

11. They conclude things about the solution or impose conditions and discontinuities (singularities) on it, before the solution is available.

12. The Droste – Hilbert parameter r is neither a radial coordinate nor a true distance. It has nothing to do with the coordinates, destroys the boundary conditions and leads to inconsistencies.

13. They think that necessarily the gravitation equations give solutions to static metrics only.

14. They have considered metrics to be equivalent whereas they are not.

15. They have been trapped in hidden vicious circles.

16. Regardless of the state of the field the vacuum solutions related to a spherical mass are inconsistent with the notion of the point source.

17. When mathematics does not work out, then, as they say, they use “intuition” (that is wishful thinking).

18. They have wrongly and falsely called the Droste or Droste – Hilbert solution as Schwarschild solution! (Small issue.)

Dr. Nikias Stavroulakis’ main suggestions, corrections and conclusions

(1) There is no need to switch from the natural Riemannian coordinates to spherical coordinates, or to introduce arbitrary implicit transformations and thus create singularities that in reality do not exist. The use of the Riemannian coordinates works throughout without any problem and does not yield the same results as the use of the spherical coordinates.

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(2) The Birkhoff’s Theorem in relativity is a false theorem which must be rejected together with the Droste and Schwarzschild solutions, in order to reexamine from the outset the problem regarding the SΘ(4) – invariant gravitational fields.

(3) Einstein’s equations have non-stationary (dynamical) solutions without singularities describing the gravitational field outside a spherical mass effecting radial motion. These solutions depend on two fundamental notions, namely the gravitational disturbance and the propagation function.

(4) To determine the non-stationary solutions, we must clarify first the propagation process of the gravitational disturbance and so we must introduce a convenient propagation function.

(5) The value of the symbol 2μ, so-called gravitational radius, does not define any radius and or any horizon, but it is the unreachable lower bound of the values of the curvature radius of the sphere that encloses the considered mass.

(6) The source of gravity is necessarily an extended body of mass.

(7) Shrinking a quantity of a mass ad infinitum is impossible but it always has a lower bound that depends of the quantity of the considered mass. (This fact agrees with Einstein’s intuitive assertion on this issue.)

(8) General Relativity is the first theory in Physics that delivers us from the contradictory or non-common sense concept of the mass-point.

(9) General Relativity not only does not predict the existence of black holes, but it explicitly excludes their formation.

(10) Degenerate space – time metric has no physical meaning.

(11) The astrophysicists invented the black hole as a hypothetical object that traps the light. In Reality the black holes is a myth that keeps trapping, for many years now, the human thinking.

Some Physical and Philosophical Consequencesby

Dr. Ioannis, Neoklis Philadelphos, M. Roussos Professor of Mathematics

Since any quantity of matter can not be shrunk arbitrarily (to point mass or below a certain lower bound) then space and time cannot be annulled and or recreated. Therefore space and time are eternal, without beginning and end, and infinite in size. Within space the total amount of energy and mass is fixed at any

38

instant of the past, present and future. Energy and mass may transform in space and time, i.e., change form, etc., but within certain limits. E.g., the situation of a mass point is impossible just as the contraction of matter beyond a certain lower limit, etc.

Einstein did not believe in the inexistence black holes and mass-points but he has never proved this mathematically. He essentially believed in the same results that Dr. Stavroulakis has proven in his works. That is, matter cannot be shrunk beyond a certain lower limit and that General Relativity not only does not predict the black holes but it also excludes their formation. Einstein has formulated the equations of the gravitational field, but only in local coordinates. He has never announced the whole manifold on which they are valid. Ever since, various bizarre and even schizophrenic “claimed results” have appeared as a result of this vacuum. H. Weyl has tried to give an answer to this question but without any success. Hence the matter remains open, like the “chicken - egg” question. The metric comes before the manifold or the manifold before the metric?!

The space as a whole is an infinite three-dimensional Euclidian space. The known finite universe lies, of course, in this universal infinite space. The laws of Physics, as we discover them, and the disturbances created by local and relatively small clusters of matter are valid within this universe, outside of which we cannot claim or speculate but very few things only. The time is a universal parameter and its laws and disturbances, as we know them, are valid within the known finite universe.

The four dimensional structures, such as the Minkowski space, etc., used in Relativity simply are mathematical abstractions and models, which possibly are locally isomorphic to reality within the known universe but they are not tantamount to reality. [E. g., in mathematics we may face a three (3) dimensional problem with two (2) extra independent parameters, let us say, but for various reasons we may choose to look at it as a five (5 = 3 + 2) dimensional problem. In other situations we may deal with two isomorphic spaces but they consist of completely different elements, etc.] Therefore it is necessary to clear out whether we deal with mathematical or physical dimensions. These two concepts are not always identical.

Thus, on the basis of all the results we have here presented, either we necessarily make a-priori unconfirmed hypotheses about “beginning” and “initial conditions”, most likely biased and of our taste and therefore unscientific, or we admit ignorance and inability at least for now, or we must visit and consider the philosophical concept of world-view (Weltanschauung). The mortals and their mathematics have not so far been able to a-priori determine the whole manifold of cosmos, and probably they will never be able to do so.

In the Ancient Greek world-view, the precept «εκ τού μηδενός, μηδέν», i.e., “from nothing, nothing”, or “ex nihilo, nihil”, in the Latin, finds a physical and mathematical proof of its validity on the basis of the above results! This idea was firstly suggested by the Eleates philosophers and secondly stated by the philosophers Democritus and Diogenes Apoloniates {Testimonia 1319.001, fragment 1, line 8} in the principle: «Ουδέν εκ τού μή όντος γίγνεσθαι ουδέ εις τό μή όν φθείρεσθαι» “neither anything from nothing comes into being nor anything existing into nothing perishes”. This same idea and principle was embraced and sustained by, Anaxagoras, and all the Pre-Socratic philosophers who have also phrased it in various forms but always with the same meaning. This same principle finds scientific and mathematical merit in Einstein himself and Dr. Stavroulakis and in their works.

The father of the Bing Bang Theory, the Roman Catholic priest and cosmologist G. H. J. É. Lemaitre (Belgian, 1894-1966) may have imposed, in his cosmological model in 1927, the initial conditions that in the beginning (whatever that

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is, or how did he know there was a beginning and what that was?) the space, mass, energy and time were zero, that is, in the beginning nothing existed. [Similar ideas we find in the works of A. A. Friedman (Russian, 1888-1925) in 1922, W. de Sitter (Dutch, 1872-1934) in 1932, etc.] But for Lemaitre, God existed before the beginning, as always, and thus he “right in the beginning” committed a logical and philosophical contradiction with the concepts “nothing” and “beginning”. (Rumors have it that the Vatican had exerted certain influence on Lemaitre in order to make a theory similar to the beginning of Genesis. The Russian-Jew Friedman seems to have had such kind of predilections too.) Here, we must not forget that even though Einstein was a German-Jew and thought about the universe with religiosity he had completely abandoned the religion of Abraham notwithstanding!

This same logical and philosophical mistake is made by the whole Biblical-Judeo-Christian world-view, according to which a lonesome God from without the world (whatever that is) created the universe out of nothing in one of his instants and ever since he has not been bored and alone anymore. But according to them God had already, as always, existed and therefore the phrases “nothing” and “in the beginning” are false. This God though existed even before the beginning! It would have been more logical to claim that this God, who had always existed, created the universe out of a part of himself or his energy and or power, etc., in some instant of his!

All these results come to agreement not only with the Ancient Greco-Roman philosophy and worldview, but also with the new discoveries on the “vacuum field” and the energy that contains. These new discoveries agree with the Ancient Greek worldview about the infinite, eternal and without beginning space, filled with a pulsating substance, part of which has continuous form and produces energy and part has dividable form and produces matter. This substance produces everything and this is exactly the vacuum energy. Experiments (in laboratories) have lately proven the existence of this vacuum energy repeatedly, via the measurement of the Casimir forces (Casimir Effect after the Dutch theoretical physicist Hendrick Brugt Gerhard Casimir, 1909-2000), and that the field of the vacuum contains infinite enormously condensed energy. E. g., the fission of 27 mm3 (a cube of side = 3 mm) of the vacuum field can produce all the energy and matter that is found in the known universe.

Moreover it is not just gravity the only cosmological and creating force. There are besides, the incomparably stronger electromagnetic forces. E. g., light is an electromagnetic radiation, etc. Every effort to unite gravity with the other cosmological forces, that is to put all the cosmological forces under the same origin and principle, has failed up to now. Gravity alone cannot explain all creation. The probable discovery of a Riemannian metric that would include not just space and time but also electromagnetism and quanta may be able to perform such unification. Dr. Stavroulakis, in his last papers, has already given hints in this direction but we are still far away from a desired outcome.

This fact along with various, still unexplained, physical paradoxes, e. g., EPR paradox or Einstein – Podolsky – Rosen paradox, the fact that as yet we cannot have any experience of the 96% of the matter and energy of the visible universe ─ we know about this only by computation ─, etc., show that something significant still escapes from the nowadays science. It seems that their solutions and explanations lie in the Ancient Greek worldview about the pulsating continuous substance which is diffused in the infinite universe and the vacuum field that we have quickly referred to above. Besides the experiments of Α. Α. Mickelson (1852-1931) and Ε. W. Morley (1838-1923), is this Ancient Greek continuous substance or the vacuum energy

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possibly tantamount to the “ether” of the famous Scottish physicist James Clerk Max-well (1831-1879), which he never disavowed and was the last word of his life?

Basic Bibliography on the Vacuum Field and the Casimir Forces

1. Peter Miloni: "The Quantum Vacuum" (Acad. Press, 1993). 2. H. B. G. Casimir, and D. Polder, "The Influence of Retardation on the London -

van der Waals Forces", Phys. Rev. 73, 360-372 (1948). 3. H. B. G. Casimir, "On the attraction between two perfectly conducting plates",

Proc. Kon. Nederland. Akad. Wetensch. B51, 793 (1948).4. http://physicsworld.com/cws/article/print/9747

Dr. I. N. Ph. M. Roussos.

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