 "inertial index" of the quadratic form (1.1.1) occurs. These features are of fundamental importance from the point of view of the physical interpretation, and will be encountered again in the solutions to be considered later." 1.2.Remarks on Mӧller ubnormal famous paper from 1943 Recall that the classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent bundle with a symmetric bilinear form which is allowed to become degenerate (singular). But if the fundamental tensor is allowed to be degenerate (singular), there are some obstructions in constructing the geometric objects normally associated to the fundamental tensor. Also, local orthonormal frames and co-frames no longer exist, as well as the metric connection and its curvature operator [2]. As an important example of the geometry with the fundamental tensor which is allowed to be degenerate, we consider now Mӧller's uniformly accelerated frame given by Mӧller's line element (1.2.1). Recall that Möller dealing with the following line element [3]: ds2  xdt2  dx2  dy2  dz2, 1. 2. 1 where x  a  gx2. Remark 1.2.1. Of course Mӧller's metric (1.2.1) degenerate at Mӧller horizon xhor  a/g. However in contrast with A.Einstein paper [1],in famous but ubnormal paper [3] Möller mistakenly argue that metric field (1.2.1) is an global vaccuum solution of the A.Einstein field equations (1.1.4), i.e. the g of this field for all values of t, x, y, z satisfy the equations R ik  12 gikR  0. 1. 2. 2 Remark 1.2.2. In physical literature this Möller's ubnormal mistake holds from Möller's time until nowadays. Remark 1.2.3. Note that formally corresponding to the Mӧller's metric (1.2.1) classical Levi-Civit'a connection is 44 1 
" inertial index " of the quadratic form (1.1.1) occurs. These features are of cardinal importance from the item of view of the physical rendition, and will be encountered again in the solution to be considered by and by. " 1.2.Remarks on Mӧller ubnormal celebrated paper from 1943 Recall that the classical Cartan's structural equations show in a compendious way the relation between a connection and its curvature, and reveals their geometric rendition in terms of moving human body. In order to study the numerical properties of singularities, we need to learn the geometry of manifolds endowed on the tangent package with a symmetric bilinear kind which is allowed to become degenerate (singular). But if the fundamental tensor is allowed to be degenerate (singular), there be some obstructions in constructing the geometric objects normally associated to the fundamental tensor. besides, local orthonormal skeleton and co - frames no longer exist, as well as the metric connection and its curvature operator [ 2 ]. As an important example of the geometry with the fundamental tensor which is allow to be degenerate, we consider immediately Mӧller's uniformly accelerated frame of reference given by Mӧller's line element (1.2.1). echo that Möller dealing with the following line element [ 3 ]: ds2  xdt2  dx2  dy2  dz2, 1. 2. 1 where x  a  gx2. Remark 1.2.1. Of naturally Mӧller's metric (1.2.1) degenerate at Mӧller horizon xhor  a / g. However in contrast with A.Einstein paper [ 1],in famous but ubnormal paper [ 3 ] Möller mistakenly argue that metric field (1.2.1) is an global vaccuum solution of the A.Einstein field equations (1.1.4), i.e. the g of this field for all value of t, x, y, omega meet the equation R ik  12 gikR  0. 1. 2. 2 Remark 1.2.2. In physical literature this Möller's ubnormal mistake holds from Möller's meter until nowadays. Remark 1.2.3. Note that formally corresponding to the Mӧller's metric (1.2.1) authoritative Levi - Civit'a connection is 44 1 
 "infrtial index" of the quadvatic form (1.1.1) occuts. These featudes are uf fundamental importance frlm the kjint of view of thd physicap interpeetauion, and will be xhcountevzd agzln in vhe solutions tp be consigered later." 1.2.Rekafkd on Mӧller ubnormal famous paper frjm 1943 Recslp that the clafsicsj Cadnak's structural equations show in z compabt way the relatipn between a connection anf itd curvature, and regeals their geoiwtric interpfetation in terms of mkving frames. In order to study ghe mcthematical peopftties of sinjularinies, we need bp studf the grometry of manlfoldv ebdowed on the tangent bundle with a symietric biniuear form which is alooqed tm bewome eegdnedave (aingulwr). Uut if the rundamental tensor is allowed uo fv degenerate (aingulwr), there are some obstructions in construbtinf the geometric objects normally associated jo the funqamental tensor. Also, local orthonormal frames and co-fremds ui lonewr exist, as well as the metric connection and iea vugvature operator [2]. As an impottwny example of tfe geometdy with the fundamfntal tgnsor qhich is wlloeed to be degenerate, we conwider now Mӧljwr's uniformly acceperated fraoe goven ny Mӧller's line element (1.2.1). Reczll that Mölper dealihe with the follodinn lhne elemtvt [3]: ds2  xdt2  dx2  qy2  dz2, 1. 2. 1 whexe x  a  gx2. Temark 1.2.1. Of course Mӧllev'v metric (1.2.1) degenerahe at Mӧnler horizln xhor  a/g. However in contrast with A.Einsteim [apvr [1],in famjus bmt ubnormal papqr [3] Möller mistckenly axgue tfat metric field (1.2.1) is an globwl vaccuum sonotion of the E.Einstein fieod ewuationr (1.1.4), i.e. the g of yhis fielb for all values of t, x, y, e aatisfy the equcuiobs R ik  12 gikR  0. 1. 2. 2 Rqmwrl 1.2.2. Hn physical nitefatjte thks Möller's unnofmal mistake holds from Kölled's time until nowacans. Remark 1.2.3. Note trat formally vorresponding to tje Mӧlner'r mettic (1.2.1) classical Levi-Civit'a connectjon is 44 1 
"inertial index" of the quadratic form (1.1.1) features of fundamental from the point interpretation, will be encountered in the solutions be considered later." 1.2.Remarks on Mӧller famous paper from 1943 Recall that the classical Cartan's structural equations show in compact way the relation between a connection and its curvature, and reveals their interpretation terms moving In order to study the mathematical properties of singularities, we need to study the geometry of endowed on the tangent bundle with a symmetric form which is allowed become degenerate (singular). But if fundamental is allowed be (singular), are some obstructions constructing the geometric objects normally associated to the fundamental tensor. Also, local orthonormal frames and co-frames no exist, as the metric and curvature [2]. As an of the geometry with the fundamental allowed to be degenerate, we consider now Mӧller's accelerated frame by Mӧller's line element (1.2.1). Recall Möller dealing with the following line element [3]:  xdt2  dx2  dy2  dz2, 1. 2. 1 where x  a Remark 1.2.1. Of course metric (1.2.1) degenerate Mӧller xhor a/g. in contrast A.Einstein paper [1],in famous but ubnormal paper [3] Möller mistakenly argue metric field (1.2.1) is an global vaccuum solution of the equations i.e. the g this field for all of x, y, z satisfy R  0. 2. Remark 1.2.2. In physical this Möller's ubnormal mistake holds Möller's time until nowadays. corresponding to the Mӧller's metric (1.2.1) classical Levi-Civit'a is 44 1 
 "inertial index" of the quadratIc form (1.1.1) occuRs. TheSe fEatUrEs arE of fUndamental impoRTancE from the point of view of tHe phySiCAl inTErPretaTion, and WIlL BE enCoUnTerEd AGaIn in tHe sOlutionS to be consiDerEd Later." 1.2.Remarks ON MӦller ubnorMal Famous paper fRom 1943 recall ThAt tHE clasSicAl CarTan's stRUcturaL equationS sHOw in a cOMpact waY THe RelaTion between a conneCTiON and its curvatuRe, and rEvEAlS THeiR geOmetric intErPretaTIon in teRMs OF MOviNG frames. In ordeR to study the MAthEmaticAl ProPErties Of sinGuLAriTies, we need tO stuDy the geomEtry of MAnifoldS Endowed On the tAngEnt BundLE wItH a sYmMEtrIC bIliNEar Form whicH iS aLloweD to bECOME degEneRate (SinguLar). But if the fuNdaMentAL teNsor iS alloWed tO bE degeNerate (SinguLaR), there are some obStruCtions in cOnsTrUctInG the gEOmetriC obJecTs normaLly assoCIatEd TO THe Fundamental tensor. ALsO, LOcAl orthonOrmal fRAmEs ANd co-framEs No lOngeR EXist, aS welL As The metriC conneCTiOn And its cUrVature OpEraTor [2]. as an iMPortAnt exaMple of thE geomETry with the fundAMental tensor wHIcH IS aLLoweD to Be degeneratE, we cONsidEr noW mӧLleR'S unifOrmly AcCElERated frame given by MӧLlEr's linE elemEnt (1.2.1). Recall that möller dealING With the fOlloWInG Line element [3]: ds2  xDt2  dx2  dY2  dz2, 1. 2. 1 where x  a  GX2. Remark 1.2.1. OF courSe Mӧller'S metric (1.2.1) deGENerate at mӧlLer HorIzoN XHoR  a/g. However in cONTrasT wIth A.EinSteIn paper [1],In fAmoUs bUt uBnOrmal papeR [3] Möller mIsTaKeNlY arGue thAT metric fIeLd (1.2.1) iS aN glObal vACcuum sOlutiOn of ThE A.eInsTein fieLD eQUAtioNs (1.1.4), I.e. The g Of tHiS fielD for ALl vAlues of T, x, y, z satisFy tHE equAtIoNs R ik  12 giKR  0. 1. 2. 2 Remark 1.2.2. In phySiCal literatUrE thIs MöllER'S ubnormaL mistake holds from Möller'S Time untIl nOwadaYs. ReMark 1.2.3. Note tHat FormalLy cORrespoNding tO the MӦlLer'S MEtric (1.2.1) CLAsSicAl levi-Civit'a CONneCtion Is 44 1 
 "inertial index" of the q uadratic f orm ( 1.1 .1)  o ccur s. T hese featuresa re o f fundamental importan ce fr om thep oi nt of  view o f  t h e  ph ys ic alin t er preta tio n, andwill be en cou nt ered again i n  t he solutio nsto be consid ere d late r. " 1 . 2.Rem ark s onMӧller ubnorm al famous  p a per fr o m 1943R e ca ll t hat the classical Ca r tan's structur al equ at i on s sho w i n a compac tway t h e relat i on b e twe e n a connectio n and its c u rva ture,an d r e vealstheir  g e ome tric interp reta tion in t erms o f  moving frames.  In or der  to  stu d yth e m at h ema t ic alp rop erties o fsi ngula riti e s , we n eed  tostudy  the geometry  of  man i fol ds en dowed  onth e tan gent b undle  w ith a symmetric  bil inear for m w hi chis  allo w ed tobec ome  degene rate (s i ngu la r ) .  B ut if the fundamen ta l te nsor isallowe d  t ob e degene ra te(sin g u lar),  the r eare some  obstr u ct io ns in c on struct in g t hegeome t ricobject s normal ly as s ociated to the fundamental t e ns o r .A lso,  lo cal orthono rmal fram es a n dco- f rames  no l on g er exist, as well as t he  metri c con nection and i ts curvatu r e operator  [2] .  A s  an importantexamp le of theg eometrywiththe fund amental t e n sor whic h i s a llo wed t obe degenerate , we c on sider n owMӧller' s u nif orm lyac celerated  frame g iv en  b yMӧl ler's line ele me nt(1 .2. 1). R e call t hat M ölle rde a lin g witht he f ollo wi ng  lin e e le ment[3]: ds2   x dt2  dx 2  dy2dz 2, 1.2. 1 where  x   a  g x 2.Remark 1 .2.1. Of  course Mӧller's metric (1.2.1)  de gener ateat Mӧller  ho rizonxho r   a/ g. How everin  co n t rastw i th  A. Ei nstein pap e r  [1 ],infa mous  but ub normal paper [3] M ö lle r mistakenlyarg ue t h a tmet r ic fie ld (1. 2 . 1) is an global  vaccuum s ol u ti on of theA .Ei ns tein fi eld equ ation s  (1.1.4 ), i.e. t he g of  t hisf i eld  for all v alues of  t, x, y, z sat i sf y the  eq uation sR i k  1 2 gikR  0 . 1.  2. 2  R emark1.2.2 .In physi cal literature this Möl ler'subnor mal  mistakehol d s f rom Mölle r'stime until  no wad ays.Rem a rk 1. 2.3. No tet hat f orma l ly corres p on din g to  the Mӧller ' s met ric ( 1.2 . 1) cla ssic al Levi-Civit'a c o nnection is 4 4 1 
 "inertial_index" of_the quadratic form (1.1.1)_occurs. These_features_are of_fundamental_importance from the_point of view_of the physical interpretation,_and will be_encountered_again in the solutions to be considered later." 1.2.Remarks on Mӧller ubnormal famous paper_from_1943 Recall_that_the_classical Cartan's structural equations show_in a compact way the_relation between_a connection and its curvature, and reveals their_geometric_interpretation in terms_of moving frames. In order to study the mathematical_properties of singularities, we need to_study the geometry_of_manifolds_endowed on the tangent_bundle with a symmetric bilinear form_which is allowed to become degenerate_(singular). But if the fundamental tensor is_allowed to be degenerate (singular), there_are some obstructions in constructing_the geometric_objects normally associated to the_fundamental tensor. Also,_local orthonormal_frames and co-frames_no longer exist, as well as_the metric connection_and its curvature operator [2]. As_an_important example of_the_geometry_with the_fundamental tensor which_is_allowed to_be_degenerate, we consider now Mӧller's uniformly_accelerated_frame given by Mӧller's line element (1.2.1)._Recall that Möller dealing_with_the following line element_[3]: ds2  xdt2 _dx2  dy2  dz2, 1._2. 1_where x_ a  gx2. Remark 1.2.1. Of course Mӧller's metric (1.2.1)_degenerate at Mӧller horizon xhor _a/g. However in contrast_with A.Einstein_paper_[1],in famous but_ubnormal_paper [3]_Möller mistakenly argue that metric field (1.2.1)_is an_global vaccuum solution of the A.Einstein_field equations (1.1.4), i.e._the_g of this field for all_values of t, x, y, z_satisfy the equations R ik__12_gikR  0. 1. 2._2 Remark 1.2.2. In physical literature_this Möller's ubnormal_mistake holds from Möller's time until nowadays._Remark_1.2.3. Note that formally corresponding to_the_Mӧller's metric (1.2.1) classical Levi-Civit'a connection_is_44_1 