NEW RESEARCH IN RELATIVITY
AND 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 metaphysical “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 by
Dr. Ioannis, Neoklis Philadelphos, M, Roussos, Professor of Mathematics,
under the supervision and bythe kind permission of
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 NationalTechnicalUniversity (E. M. Polytechnion), Athens,Greece, in 1938, where he studied Civil Engineering.
Although World War II interrupted the smoothcourse of his studies, destroyed his country, and he escaped executionby the Nazis for just a few days, he continued his studies after the war was over in 1945. He graduated from the NationalTechnicalUniversity (E. M. Polytechnion), Athens, Greece, in 1947.
During the years 1949 – 1963 he worked as a civil engineerin 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.
MAIN SYNOPTIC ARTICLE
TITLE: GENERAL RELATIVITY AND BLACK HOLES
By
Dr. Nikias Stavroulakis
Solomou Street 35, Chalandri, Athens15233,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
Inthearticle[8] “Mathématiques et Trous Noirs” publishedin1986 inthemagazine “Gazette Des Mathématiciens” oftheFrench 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 namedblack hole.Thishastodowithanarbitraryinterpretationof 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 phenomenonisduetothefactthatduringthetimethat General Relativity was formulated, the mathematical concepts used, and especially that of manifold, had not been adequately elucidated. Themathematical 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 Mathématiciens” for two reasons:
First: To clarify the basic mathematical concepts that enter into the theory of the gravitational field of a spherical body.
Second: Toexposewithgreaterclaritytheunacceptablepointsoftheclassicalmethod, whicharesupportedbythestatic solutions in order to justify a hypothetical dynamical processthat leads to the black hole.
We shall see that this process is excluded by the equations of the theory themselves.
Withthispresentchancewewillrestoreahistoricalinaccuracyinthearticleof the“Gazette Des Mathématiciens”, an inaccuracy that is stereotypicallyrepeated in all the bibliographical references and was also recorded bona fide in thearticle under discussion.ThishastodowiththesocalledSchwarzschildsolution, to which all refer when they deal with the gravitational field of a spherical body. InrealitytheSchwarzschildsolutionappearedfirst, but was abandoned very quickly.However,thenameofSchwarzschild remained. Thusthesolutionwhichtodayiscalledthe Schwarzschildsolutionis in reality the Droste – Hilbert solution. Thereisanunbelievableconfusiononthissubjectwhich was nurtured by Droste and Hilbertthemselves, who communicated that they had found all over again the Schwarzschildsolution. 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
Thespacethatentersintotheproblemistheordinaryvectorspace, which is also considered as a topological space (topological product of three real straight lines).
Inrelationtotime, variousassumptionsleadus toportrayitasavariablethatrunsovertherealstraightline.
Consequentlythemathematicalspacethatentersintothetheoryofthegravitationalfield of a spherical body is the vector space considered also as a topological space (topological product of four real straight lines).
ThissimpleandclearalgebraicandtopologicalcharacterizationwasalreadyalteredfromtheinceptionofGeneralRelativitybythesystemicusageofsphericalcoordinatesof , which leads to the alteration of the underlying manifold.
Intheframeof EuclidianGeometry, whenwesaythata point is defined by three Cartesian coordinates we refer to an orthogonal system of three axes. Butwhenwesaythatthepointisdefinedbythesphericalcoordinatesρ, φ, θ, then what is the system of reference? Wereferagaintothesystemofaxeswithrespecttowhichwedefinethedistanceρand the anglesφandθ. To say that the point is defined by the spherical coordinates ρ, φ, θ, without referring to the orthogonal system of three axeshas no meaning. Thereforeandaccordingtotheiroriginaldefinition, thesphericalcoordinates do not have autonomous existence and moreover they cannot be defined on the axis of.
Todefinean “autonomousmanifoldinsphericalcoordinates”, we must bring in the canonical homeomorphism between
and
Ifwedenote the unit spherewith, i.e.,
theneverypointiswritteninauniquewayastheproductwith and , because from the relation
with
it follows that and.
Therefore the homeomorphism
is defined if we set
Thesphericalcoordinatesaredefinedoverthemanifold, if we introduce two parameters in order to define the points yof the sphere. Usually we introduce the parameters
and ,
and then
Sincethiscoordinatesystemdoesnotcoverthewholesphere , we must also use a second coordinate system
, ( and ) ,
and to set (for example)
Thiscoordinatesystemisalwaysomitted, butcertainly bothsystems(2.1) and (2.2), with, are indispensable in defining the homeomorphism F.
Inactualitysphericalcoordinatesarealsousedfor. For this to take place, we must accept a violation of the already accepted mathematical principles, that is, to use an extension ofF,which is not ahomeomorphism. Thisextensionisdefinedinauniqueway: Weattachtheset to the open manifold , that is, we replace it with the manifold , and we let as before
Thenforevery.
Butthemappingis not a homeomorphism. The setin its totality is mapped onto the origin of . Hence the phrase that is usually said
“theorigin ”
meansthe“boundary”, whichisintroducedinourcomputationsto cover thewholespace, but without having any natural meaning.
Thesphereswithcentertheoriginofare not defined in spherical coordinates.
ThedistinctionbetweenFandisnever mentioned, with severe confusion as a result. To see this, let us consider the case of the transformation of the Euclidian metric
viaF andvia.
ThetransformationviaF, thatis,viathe transformations (2.1) and (2.2), gives respectively the positive definite forms
and
whichtogetherdefineapositivedefinitemetriconthemanifold, that does not represent the totality of the space .
Thetransformationviaagaingivestheabovetwopositivedefiniteforms, but with , which means the transformed metric on the manifolddegenerates 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. Insteadofthespacethemanifoldwithboundaryisused, and instead of the spacewe use the manifold, which is tacitlyidentified 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
whichison the manifold. Tothisametriconthespacecorresponds, which is obviously written as
and which develops a discontinuity at. Soan anomaly is hidden intheBondimetric deprived of any natural meaning.
Such discontinuities generally appear when the metricsare 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
isexpressedinsphericalcoordinatesasafunctionofonevariable,, 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, thedifferentiabilityofdoes not imply the differentiability ofwith respect to the variablesat 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 of the gravitational field ought to be smooth functions of the norm, in accordance with the following definition.
DEFINITION.Afunctionofthenorm, , will be called a smooth function of the norm, if
a) iswith respect to the variablesin the space
b) For any set of indicesthe derivative
Atthepointstends to a certain limit when.
REMARK. Inpreviousarticles, [11], [12], a smooth function of the norm is consideredin. Butthiscondition alonedoesnotimplythatthederivativesof exist at the origin. To deal with problems related to the functions of the norm, we must keep in mindfrom the beginning that the notation presumes the function of one variable to be defined over the half-line.
THEOREM. Letusassumethatthefunctionofonevariableis over the half-line , where naturally its derivativesatu = 0 are right-derivatives.Insuchacase, thefunctionis a smooth function of the norm if and only if the derivatives of the odd order of atu = 0 are vanishing.
Proof: Therestrictionoftoanaxis, let us say, is the functionand so
for
for
Bydefinition, thederivativesofanyorders of these two functions, i.e.,
and
musttendtothesamelimit when. Then
Thisrelationisanidentitywhens =even= 2k, but whens =odd= 2k +1, we will have
from which
Hence the condition of the theorem is necessary.
We shall show that the condition is also sufficient.
The function
( for , and for )
ison the half-linesand . But because
for
for
it follows
and
Theserelationsprovethatthefunctionis differentiable also at the origin if
So isanevenfunctionon the real line .
Thevanishingofthederivativeleads to the relation
andsincewecandifferentiateundertheintegralsign, thefunction
isanevenfunctionon the real line .
Now, inductionshowsthatwecandefineasequenceofevenfunctions on the real line . Indeed, iftheevenfunction, , has already been defined and we have, then
andthenextevenfunction is defined if we let
Returningnowbacktothefunctionsofthenorm, we see that the functions
arewithrespecttoonthehalfline, where the derivativesof with respect to the variable , i.e.,
tendto certain limits when
COROLLARY. Thesetofsmoothfunctionsofthenormis an algebra over the real line.
Proof: Indeed, ifthefunctionsand, , have derivatives of odd orders vanishing at, the same is true forthe derivatives of odd orders of their product
becauseinthissumeachtermcontainsaderivativeofoddorderofoneofthetwofunctions.
REMARK: Inapplicationsthefunctionis not required to be on the half-line. Theorderofdifferentiabilityofdepends on the required order of differentiability for the partial derivatives.
4. SΘ(4) – INVARIANT METRICS
ARiemann metriconthespaceis said to be spherically symmetric if it remains invariant under the action of the groupSO(3) in . Intuitively such a metric is written as
wherethefunctionsandmustnaturallybesmooth functions of the norm.
Itisofcourseobviousthatthemetric (4.1) remains invariant not only by the action of the group SO(3), but also by the action of the groupΟ(3). Theconverseproposition, thatis, ametricinvariantbytheactionofthegroupSO(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] simplysaysthata 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 – Civitain his monograph“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 ofSO(3) – invariant tensor fields on the space. Forthecaseofthespace –timemetricsthereisanadditionaldifficulty, becausethese are defined on the space , whereas the groups SO(3) andΟ(3) act on the space. Toendupwithcleardefinitions, itisnecessary to introduce the groupSΘ(4), which consists of the matrices
with, , andSO(3) . ItisalsonecessarytoconsiderthelargergroupΘ(4), which consists of the matrices of the same type but withO(3).
It is surely understood that instead of the simple smooth function of the norm, we will now havefunctions of the form , whose derivatives with respect tothe variables for, tend to certain limits when . Theset 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, beacovariant,SΘ(4) – invariant,symmetrictensor field of degree 2 on the space. Then there are 4 functions
that belong to the algebraΓ0and such that
with and
Moreoverthetensorfield is Θ(4) – invariant. □
Ifthetensorfieldisaspace-timemetric, that is it has signature (1, -1, -1, -1), we usually write it as a quadratic form
, (4.2)
Thisgeneralformofthespace-timemetrichasneverbeenusedinEinstein’s equations for the gravitational field of the spherical body. Onlystationaryorstaticmetrics, that is metrics that do not depend on time, enter into the classical solutions of Einstein’s equations. DuringthefirstyearsofGeneralRelativity, thisrestrictionseemstobestandarddue to technical reasons for avoiding the difficulties involved in considering a time-dependent tensor. Butlater, withtheemergenceofthesocalled “BirkhoffTheorem”, itbecame 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.Thisadmissionwasmostprobablysuggestedfortechnicalreasons, butitalsoseemsthattherewastheerroneousview (based on abusive usage of implicit transformations)thatthis admission does not influence essentially the solutions that result. In reality, however, the vanishing of q01has not allowed the study of the time contribution to the space-time metrics.
Theabovesimplifyingadmissionswereconsideredself-evidentbySchwarzschild, who in order to define, as he believed, the gravitational field of a masspoint, suggested the first static metric of spherical symmetry
whereF, G, H are introduced as functions of the norm without any further explanation. Thismetric, which is referred to as obvious, couldhaveprovidedthebasisforthe correct investigation of the gravitational field of a spherical static body, but it was never used. Schwarzschildhimselfimmediatelytransformsthemetrictosphericalcoordinatesand from then on different metrics are considered over the manifold with boundary .
Wewillmakehereashortcriticalanalysisofthefirsttwostaticsolutionsof Einstein’s equations on account of the influence that they have had in the evolution of ideas in Relativity.
5. THESCHWARZSCHILDSOLUTION [7]
TheSchwarzschild solution involvesapositiveconstantα (whichlaterwasidentifiedwiththevalue ) and after we set it is written in the classical expression
Letusconsiderthespace metric(onthemanifold ), which is a part of this space-time metric
or .
This is differentiable on the space . ButforR = α (that is, r = 0) itdevelops discontinuities. Nevertheless the integral
hasmeaningsandsothelengthsofthegeodesics(φ = constant, θ = constant), or (φ΄ = constant, θ΄ = constant), measured from the boundary are well defined. Additionally,themetricinducedontheboundary, that is, defines the topology of a sphere. Theobserver’sspaceisthenidentifiedtopologicallywithathree dimensional semi-cylinder, and so with a space different thanthat was the space of the originallyformulated the problem.
Schwarzschildhasassumedthat 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, butan infinitude of irregular points that make up the boundary , which has the cardinality of the continuum. Thecontentoftheproblemisfoundto be fundamentallymodified.
TheSchwarzschildsolutionisincompatible with the topological and metric presuppositions of the problem and for this reason it was rightly abandoned. Except, thereasonsofitsabandonmentwerenotrightly and clearly formulated. OnthisHilbert [2]writes: “Inmyopinion, the identification of the spacer = αwith the origin, as Schwarzschild did,isnotrecommended.On the top of it, the Schwarzschildtransformation is not the simplest for this purpose.”Hilbertobviouslyhintsatthetransformationsr΄ = 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.
6. THEDROSTESOLUTION[1]
Drostethinks that the gravitational field of a “center” is defined by the static metric in spherical coordinates
whereω, u, varefunctionsofthevariabler.Henceforthhesays,wehavethefreedomtopickthevariabler in such a way that u = 1, and therefore we must determine the metric
Hedoesnotclearlystatetheconditionsofthischangeandhedoesnotobservethatthe variabler represents the radial geodesic distance in the space. Hefindsthatthefunctionv = v(r) satisfies the equation
butwithoutobservingthatthelattercan beimmediatelysolved: Multiplying both sides byv΄we find
or
hence (Α = constant)
and so
Thelastequationdeterminesthefunctionand since he also finds thatthe 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)