 of this field satisfy in general the equations Rklm i  0, 1. 1. 2 and hence the equations Rkl  Rklm m  0, 1. 1. 3 The g corresponding to (1.1.1) are regular for all finite (i.e. nonzero) points of space-time. Nevertheless one cannot assert that Eqs.(1.1.3) are satisfied by (1.1.1) for all finite values of x1,..., x4. This is due to the fact that the determinant g of the g vanishes for x1  0. The contravariant g therefore become infinite and the tensors Rklmi and Rkl take on the form 0/0. 1. 1. 3 From the standpoint of Eqs.(1.1.3) the hyperplane x1  0 then represents a singularity of the field. We now ask whether the field law of gravitation (and later on the field law of gravitation and electricity) could not be modified in a natural way without essential change so that the solution (1.1) would satisfy the field equations for all finite points, i.e., also for x1  0. W. Mayer has called our attention to the fact that one can make Rklmi and Rkl into rational functions of the g, and their first two derivatives by multiplying them by suitable powers of g. It is easy to show that in g2Rkl there is no longer any denominator. If then we replace (1.1.3) by Rkl   g2Rkl  0, 1. 1. 3. a this system of equations is satisfied by (1.1.1) at all finite points. This amounts to introducing in place of the g the cofactors g  of the g in g in order to avoid the occurrence of denominators. One is therefore operating with tensor densities of a suitable weight instead of with tensors. In this way one succeeds in avoiding singularities of that special kind which is characterized by the vanishing of g. The solution (1.1.1) naturally has no deeper physical significance insofar as it extends into spatial infinity. It allows one to see however to what extent the regularization of the hypersurfaces g  0 leads to a theoretical representation of matter
 of this field satisfy in general the equations Rklm i  0, 1. 1. 2 and hence the equation Rkl  Rklm m  0, 1. 1. 3 The g correspond to (1.1.1) are regular for all finite (i.e. nonzero) points of space - fourth dimension. Nevertheless one cannot assert that Eqs.(1.1.3) are satisfy by (1.1.1) for all finite values of x1, ..., x4. This is due to the fact that the deciding deoxyguanosine monophosphate of the g vanishes for x1  0. The contravariant g therefore become countless and the tensors Rklmi and Rkl take on the form 0/0. 1. 1. 3 From the standpoint of Eqs.(1.1.3) the hyperplane x1  0 then constitute a singularity of the field. We now ask whether the field law of gravity (and later on the field law of gravity and electricity) could not be modified in a natural way without essential change so that the solution (1.1) would meet the field equality for all finite points, i.e., also for x1  0. W. Mayer has called our care to the fact that one can make Rklmi and Rkl into rational functions of the g, and their first two derivatives by multiplying them by suitable powers of g. It is easy to show that in g2Rkl there is no longer any denominator. If then we replace (1.1.3) by Rkl   g2Rkl  0, 1. 1. 3. a this system of equations is satisfied by (1.1.1) at all finite points. This amounts to precede in place of the g the cofactors g  of the g in g in decree to invalidate the occurrence of denominators. One is therefore operating with tensor densities of a suitable weight alternatively of with tensors. In this way one succeeds in avoiding singularities of that special kind which is characterize by the vanishing of g. The solution (1.1.1) naturally has no deeper physical significance insofar as it extend into spatial infinity. It allow one to determine however to what extent the regularization of the hypersurfaces g  0 leads to a theoretical representation of topic
 of this field satisfy in gtneral the equations Rklk i  0, 1. 1. 2 and hdnce the equations Rkl  Rklm m  0, 1. 1. 3 The g corresponding tu (1.1.1) are rehular foe alo finite (i.e. nonzero) pointa of v'ace-time. Nevertmeless one wannot assert dhxt Eqs.(1.1.3) are satisfied by (1.1.1) for all finiee valurs of x1,..., x4. This if dut tj ths fact that the determinant g of tge g vaiishes for x1  0. Yhe contravariant g thereflre hecome infinite anf the tensota Rhomi and Rkl gake on tht yorm 0/0. 1. 1. 3 Frkm the standpoint of Eqs.(1.1.3) the hyoerplcne x1  0 theb eepggsents a sinjulariny of the field. We nof ask wnether the fiekd naw of gravitation (and leter on the field lar of gravhtction and electricity) ciuld tot te mueifked ii a naturwl xay without essential xhange so that the xojlyion (1.1) would aatisfr ehe field equations for all finite poinus, i.e., also for x1  0. W. Mayer hqs called our attentiln to the fact that one can make Rklmi and Rkl into rationdl fuicgiouw of gye g, and their first two derivatives by multiplrjnb nhem by suitable iowers of g. It is rady jo show that iv g2Rkl thsre is no longer ajy denoiinatir. If theg we replace (1.1.3) by Rkl   g2Rkl  0, 1. 1. 3. a this sysnem if equations is sacisfied by (1.1.1) ct all finiye points. This amounts co intdoducing in place of ghe g the cofacturs g  of the g in g in order tj avoid tie oceurrence of cenomigators. One is tmarefore operating aith jensor densities of a suitable weight instead oh with tensorx. Hn nhis way jne smcceeds in avoiqing singularijies of tkat spdcial kind which ms characteryzed by the vdjishing of g. The solttiob (1.1.1) nqturallh has no deeper physical significabce insofar as it cxtenaa into spatial nufunity. It allows ond tj dex howqeer to what axtevt gne reeularizatiok ow thr hypersurfaces g  0 neada to a theoretical rcpresentajion of mwtter
 of this field satisfy in general Rklm  0, 1. 2 and Rklm  0, 1. 3 The g to (1.1.1) are regular for all (i.e. nonzero) points of space-time. Nevertheless one cannot assert that Eqs.(1.1.3) are satisfied (1.1.1) for all finite values of x1,..., x4. This is due to the that determinant of g vanishes for x1  0. The contravariant g therefore become infinite and the tensors Rklmi Rkl take on the form 0/0. 1. 1. From the standpoint of the hyperplane x1  0 represents singularity of field. now whether the field of gravitation (and later on the field law of gravitation and electricity) could not be modified in natural way change so the (1.1) satisfy the field all finite points, i.e., also for W. Mayer has called our attention to the that one make Rklmi and Rkl into rational of the g, and their first two derivatives multiplying them by suitable powers of g. It is easy to show that in g2Rkl no longer any denominator. then we replace by  g2Rkl 0, 1. 3. a this system of equations is satisfied by (1.1.1) at finite points. This amounts to introducing in place of the cofactors  of the in g in order avoid occurrence of denominators. One operating tensor suitable instead with tensors. In this one succeeds in avoiding singularities that special kind which of g. The solution (1.1.1) naturally has no physical significance insofar as it extends into infinity. It allows one to see however to what extent the regularization the hypersurfaces 0 leads to a theoretical representation of matter
 of this field satisfy in generAl the equatIons RKlm I  0, 1. 1. 2 anD hEnce The eQuations Rkl  RklM M  0, 1. 1. 3 The G corresponding to (1.1.1) are regUlar fOr ALl fiNItE (i.e. noNzero) poINtS OF spAcE-tIme. neVErTheleSs oNe cannoT assert thaT EqS.(1.1.3) aRe satisfied bY (1.1.1) FoR all finite ValUes of x1,..., x4. This iS duE to the FaCt tHAt the DetErminAnt g of THe g vanIshes for x1  0. thE ContraVAriant g THErEforE become infinite anD ThE Tensors Rklmi anD Rkl taKe ON tHE ForM 0/0. 1. 1. 3 FrOm the standPoInt of eQs.(1.1.3) the hyPErPLANe x1  0 THen represents A singularitY Of tHe fielD. WE noW Ask wheTher tHe FIelD law of graviTatiOn (and lateR on the FIeld law OF gravitAtion aNd eLecTricITy) CoUld NoT Be mODiFieD In a Natural wAy WiThout EsseNTIAL chaNge So thAt the Solution (1.1) would SatIsfy THe fIeld eQuatiOns fOr All fiNite poInts, i.E., aLso for x1  0. W. Mayer haS calLed our attEntIoN to ThE fact THat one Can MakE Rklmi aNd Rkl inTO raTiONAL fUnctions of the g, and tHeIR FiRst two deRivatiVEs By MUltiplyiNg TheM by sUITable PoweRS oF g. It is eaSy to shOW tHaT in g2Rkl ThEre is nO lOngEr aNy denOMinaTor. If tHen we repLace (1.1.3) bY rkl   g2Rkl  0, 1. 1. 3. a this sySTem of equationS Is SATiSFied By (1.1.1) aT all finite pOintS. this AmouNTs To iNTroduCing iN pLAcE Of the g the cofactors g  Of The g in G in orDer to avoid the Occurrence OF DEnominatOrs. ONE iS Therefore operaTing wIth tensor dENsities oF a suiTable weiGht insteaD OF with tenSorS. In ThiS waY ONe Succeeds in avoIDIng sInGularitIes Of that sPecIal KinD whIcH is characTerized bY tHe VaNiShiNg of g. tHe solutiOn (1.1.1) NatUrAllY has nO Deeper PhysiCal sIgNiFIcaNce insoFAr AS It exTeNdS intO spAtIal inFiniTY. It Allows oNe to see hoWevER to wHaT eXtent thE regularizatiOn Of the hyperSuRfaCes g  0 leADS to a theoRetical representation of MAtter
 of this field satisfy i n generalthe e qua tio ns  Rkl m i 0, 1. 1. 2 andhence the equations Rk l  R kl m  m  0,  1.1. 3 T h eg   c or re spo nd i ng  to ( 1.1 .1) are  regular f oral l finite (i. e .nonzero) p oin ts of space- tim e. Nev er the l ess o necanno t asse r t that  Eqs.(1.1 .3 )  are s a tisfied b y(1.1 .1) for all finit e  v a lues of x1,... , x4.Th i si s  du e t o the fact  t hat t h e deter m in a n t  go f the g van ishes for x 1  0. The  c ont r avaria nt g t her efore becom e in finite an d thet ensorsR klmi an d Rkltak e o n th e  f or m 0 /0 .  1 .  1 . 3   F rom thest an dpoin t of E q s .(1. 1.3 ) th e hyp erplane x1 0 t henr epr esent s a s ingu la rityof the  fiel d.  We now ask whe ther  the fiel d l aw  of  g ravit a tion ( and  la ter onthe fie l d l aw o f  g ravitation and ele ct r i ci ty) coul d notb emo d ified in  a  na tura l way w itho u tessentia l chan g eso  that t he  solut io n ( 1.1 ) wou l d sa tisfythe fiel d equ a tions for allf inite points, i. e . ,a lsofor  x1  0. W.  May e r ha s ca l le d o u r att entio nt ot he fact that one ca nmake R klmiand Rkl intorational f u n c tions of  the g  , and their fi rst t wo derivat i ves by m ultip lying th em by sui t a ble powe rsofg.Iti s  e asy to show t h a t in  g 2Rkl th ere  is nolon ger  an y d en ominator.  If then  w ere pl ace  (1.1 . 3) by Rk l   g 2Rk l  0 ,  1. 1 . 3.a t hi ss yst em of e q ua t i onsis  s atis fie dby (1 .1.1 )  at  all fi nite poin ts. This  a mo unts to  introducingin  place ofth e g  the c ofactors  g  of the g in g in orde r t o avo id t he occurr enc e of d eno m inator s. One  is t he ref o r e ope r a ti ngwi th tensord e nsi tiesof  a s uitable  weight instead of wit h tensors. In  th is w a y  o nes uc c eed si n a v o iding singulari ties of th at sp ecial kind whi ch  is cha racteri zed b y  the va nishing o f g. Theso luti o n  (1 .1.1) natu rally ha s no deep e r phy s ic al si gni ficanc eins ofaras ite xte nds i nto sp at ial in finit y.  It allo ws one to see however t o what  exte ntthe regul ari z ati on of the  hyp ersurfaces  g 0  lead s t o  a th eore t ic alr epres enta t ion of ma t te r
 of_this field_satisfy in general the_equations Rklm_i_ 0,_1._1. 2 and_hence the equations_Rkl  Rklm m_ 0, 1._1._3 The g corresponding to (1.1.1) are regular for all finite (i.e. nonzero) points_of_space-time. Nevertheless_one_cannot_assert that Eqs.(1.1.3) are satisfied_by (1.1.1) for all finite_values of_x1,..., x4. This is due to the fact_that_the determinant g_of the g vanishes for x1  0. The_contravariant g therefore become infinite and_the tensors Rklmi_and_Rkl_take on the form_0/0. 1. 1. 3 From the_standpoint of Eqs.(1.1.3) the hyperplane x1_ 0 then represents a singularity of_the field. We now ask whether_the field law of gravitation_(and later_on the field law of_gravitation and electricity)_could not_be modified in_a natural way without essential change_so that the_solution (1.1) would satisfy the field_equations_for all finite_points,_i.e.,_also for_x1  0._W._Mayer has_called_our attention to the fact that_one_can make Rklmi and Rkl into rational_functions of the g,_and_their first two derivatives_by multiplying them by suitable_powers of g. It is easy_to show_that in_g2Rkl there is no longer any denominator. If then we replace_(1.1.3) by Rkl   g2Rkl_ 0, 1. 1._3. a_this_system of equations_is_satisfied by_(1.1.1) at all finite points. This amounts_to introducing_in place of the g the_cofactors g  of_the_g in g in order to_avoid the occurrence of denominators. One_is therefore operating with tensor_densities_of_a suitable weight instead of_with tensors. In this way one_succeeds in avoiding_singularities of that special kind which is_characterized_by the vanishing of g. The_solution_(1.1.1) naturally has no deeper physical_significance_insofar_as it extends into spatial_infinity. It allows one to see_however to what extent the regularization of the hypersurfaces_g  0_leads to a theoretical representation_of_matter