, regarded from the standpoint of the original theory. Thus, in the framework of the original theory one has the gravitational equations R ik  12 gikR  Tik, 1. 1. 4 where Tik is the tensor of mass or energy density. To interpret (1.1.1) in the framework of this theory we must approximate the line element by a slightly different one which avoids the singularity g  0. Accordingly we introduce a small constant  and let ds2  dx1 2  dx2 2  dx3 2  2x12  dx42 1. 1. 1. a the smaller  ( 0) is chosen, the nearer does this gravitational field come to that of (1.1.1). If one calculates from this the (fictitious) energy tensor Tik one obtains as nonvanishing components T22  T23   2 2x12   2. 1. 1. 4. a We see then that the smaller one takes  the more is the tensor concentrated in the neighborhood of the hypersurface x1  0.From the standpoint of the original theory the solution (1.1.1) contains a singularity which corresponds to an energy or mass concentrated in the surface x1  0; from the standpoint of the modified theory, however, (1.1.1) is a solution of (1.1.3.a), free from singularities, which describes the "field-producing mass, " without requiring for this the introduction of any new field quantities. It is clear that all equations of the absolute differential calculus can be written in a form free from denominators, whereby the tensors are replaced by tensor densities of suitable weight. It is to be noted that in the case of the solution (1.1.1) the whole field consists of two equal halves, separated by the surface of symmetry x1  0, such that for the corresponding points x1, x2, x3, x4 and x1, x2, x3, x4 the gik are equal. As a result we find that, although we are permitting the determinant g to take on the value 0 x1  0, no change of sign of g and in general no change in the
, regarded from the standpoint of the original theory. Thus, in the model of the original hypothesis one own the gravitational equation gas constant ik  12 gikR  Tik, 1. 1. 4 where Tik is the tensor of mass or energy concentration. To interpret (1.1.1) in the framework of this theory we must estimate the credit line element by a slenderly different one which avoids the singularity g  0. Accordingly we precede a small constant  and lease ds2  dx1 2  dx2 2  dx3 2  2x12  dx42 1. 1. 1. a the smaller  ( 0) is chosen, the nearer does this gravitational battlefield come to that of (1.1.1). If one calculates from this the (fictitious) department of energy tensor Tik one obtains as nonvanishing components T22  T23   2 2x12   2. 1. 1. 4. a We interpret then that the smaller one remove  the more is the tensor concentrated in the neighborhood of the hypersurface x1  0.From the standpoint of the original theory the solution (1.1.1) contains a singularity which correspond to an energy or mass concentrated in the surface x1  0; from the standpoint of the modified theory, however, (1.1.1) is a solution of (1.1.3.a), free from singularities, which describes the " field - producing mass, " without requiring for this the introduction of any new field measure. It is clear that all equations of the absolute differential tartar can be publish in a form free from denominators, whereby the tensor are replaced by tensor densities of suitable weight. It is to be note that in the case of the solution (1.1.1) the whole field consists of two equal halves, distinguish by the surface of symmetry x1  0, such that for the corresponding points x1, x2, x3, x4 and x1, x2, x3, x4 the gik are equal. As a result we witness that, although we are allow the determinant g to take on the value 0 x1  0, no variety of sign of g and in general no change in the
, reharded from the standpoikt of the originco theocy. Thus, in the wramework of the original thxory one yas the gravitational dquations R ik  12 tikR  Tik, 1. 1. 4 whxde Tik lf ths tenvir of mass or gnergy densidy. To interpred (1.1.1) iu the framework of this theory we muft apprpxlmate the line elekqnt gj c slightly different one which zvoids uhe singularity g  0. Accordingly we introduce a slall constant  and let ds2  dx1 2  dv2 2  dx3 2  2x12  db42 1. 1. 1. a the smaller  ( 0) js chosen, the nearer does this eravicational figls fmme to that of (1.1.1). Pf one calculates frok this yhe (fictitious) enxrgy tensor Tik one obtaiis as nonvanishing cjmponents T22  T23   2 2x12   2. 1. 1. 4. a Qe see jhen dhat rhe smzlked one hakxs  the mors is the tebsor concentrated im eye neighborhokd of ehq hypersurface x1  0.From the standpoint ox tge original theory the wolution (1.1.1) contains a dingulariey which corresponds to an energy or mass concentsated kn uhc ryrvace x1  0; from the standpoint of the modified egeprj, however, (1.1.1) is a sjlution of (1.1.3.s), vrrg from singulatities, cgidh describes the "fleld-proqucint mass, " wythoit requiring for this the ibtroduction jd any new field qucntities. It ns clesr thst all equations of the absklute diffegential czuculus can be wrkttvn it a form free from denominwtors, wheceby che tensurs sre re[laced by hensov densities of suitwble ceighd. It is to be noted that in the case of tix solution (1.1.1) tne whmle fielb conslsts of two equwl halves, sepatated by che sufface of sjmmetry x1  0, such thae for the corsgsponding poiits x1, x2, x3, x4 abd x1, x2, x3, x4 gfe gik are equsl. As a rvsblt we fibd that, although wc are lermitting the bttetninant g to tale un ehv velue 0 f1  0, no changa of siem of e and in gekerxl np change in the
, regarded from the standpoint of the Thus, the framework the original theory R  12 gikR Tik, 1. 1. where Tik is the tensor of or energy density. To interpret (1.1.1) in the framework of this theory we approximate the line element by a slightly different one which avoids the singularity  Accordingly introduce small constant  and let ds2  dx1 2  dx2 2  dx3 2   dx42 1. 1. 1. a the smaller ( 0) is chosen, nearer does this gravitational field to of (1.1.1). one from the (fictitious) energy Tik one obtains as nonvanishing components T22  T23   2 2x12   2. 1. 4. a then that smaller takes the more is concentrated in the neighborhood of the 0.From the standpoint of the original theory the (1.1.1) contains singularity which corresponds to an energy mass concentrated in the surface x1  0; the standpoint of the modified theory, however, (1.1.1) is a solution of (1.1.3.a), free from describes the "field-producing mass, without requiring for the of new quantities. It clear that all equations of the absolute differential calculus can be in a form free from denominators, whereby the tensors are tensor of suitable weight. is to be noted in case of the solution whole consists halves, by surface of symmetry x1 0, such that for the points x1, x2, x3, x4 the gik are equal. As a result find that, although we are permitting the g to take on the value 0 x1  0, no change sign of in general no change in the
, regarded from the standpoint Of the origiNal thEorY. ThUs, In thE fraMework of the oriGInal Theory one has the gravitaTionaL eQUatiONs r ik  12 giKR  Tik, 1. 1. 4 whERe tIK is ThE tEnsOr OF mAss or EneRgy densIty. To interPreT (1.1.1) iN the frameworK Of This theory We mUst approximaTe tHe line ElEmeNT by a sLigHtly dIffereNT one whIch avoids ThE SingulARity g  0. AcCORdInglY we introduce a smalL CoNStant  and let ds2  dX1 2  dx2 2  dx3 2  2x12  Dx42 1. 1. 1. A ThE SMalLer  ( 0) Is chosen, thE nEarer DOes this GRaVITAtiONal field come tO that of (1.1.1). If onE CalCulateS fRom THis the (FictiTiOUs) eNergy tensor tik oNe obtains As nonvANishing COmponenTs T22  T23   2 2x12   2. 1. 1. 4. a we sEe tHen tHAt ThE smAlLEr oNE tAkeS  The More is thE tEnSor coNcenTRATEd in The NeigHborhOod of the hyperSurFace X1  0.froM the sTandpOint Of The orIginal TheorY tHe solution (1.1.1) contaIns a SingulariTy wHiCh cOrRespoNDs to an EneRgy Or mass cOncentrATed In THE SuRface x1  0; from the standPoINT oF the modiFied thEOrY, hOWever, (1.1.1) is a SoLutIon oF (1.1.3.A), Free fRom sINgUlaritieS, which DEsCrIbes the "FiEld-proDuCinG maSs, " witHOut rEquiriNg for thiS the iNTroduction of anY New field quantITiES. it IS cleAr tHat all equatIons OF the AbsoLUtE diFFerenTial cAlCUlUS can be written in a forM fRee froM denoMinators, whereBy the tensoRS ARe replacEd by TEnSOr densities of sUitabLe weight. It IS to be notEd thaT in the caSe of the soLUTion (1.1.1) the wHolE fiEld ConSIStS of two equal haLVEs, sePaRated by The Surface Of sYmmEtrY x1  0, sUcH that for tHe corresPoNdInG pOinTs x1, x2, x3, X4 And x1, x2, x3, x4 tHe Gik ArE eqUal. As A Result We finD thaT, aLtHOugH we are pERmITTing ThE dEterMinAnT g to tAke oN The Value 0 x1  0, nO change of SigN Of g aNd In General No change in the
, regarded from the standp oint of th e ori gin alth eory . Th us, in the fra m ewor k of the original theo ry on eh as t h egravi tationa l  e q u ati on sR i k  1 2 gik R   Tik,1. 1. 4whe re  Tik is thet en sor of mas s o r energy den sit y. Toin ter p ret ( 1.1 .1) i n thef ramewo rk of thi st heoryw e musta p pr oxim ate the line elem e nt by a slightlydiffer en t  o n e  wh ich  avoids th esingu l arity g 0 . Acc o rdingly we in troduce a s m all  const an t  and le t ds2   dx 1 2  dx2 2   d x3 2   2x12  dx42 1. 1. 1 . a t hesma ller (  0)  i s  ch o se n,t henearer d oe sthisgrav i t a t iona l f ield  come  to that of ( 1.1 .1). Ifone c alcul ates  f rom t his th e (fi ct itious) energytens or Tik on e o bt ain sas no n vanish ing  co mponent s T22  T23    2  2x12   2. 1 .1 .  4 . a Wesee th e nth a t the sm al ler  one t akes th e  m ore is t he ten s or  c oncentr at ed inth e n eig hborh o od o f thehypersur facex 1  0.From the standpoint of th e or i gina l t heory the s olut i on ( 1.1. 1 )con t ainsa sin gu l ar i ty which correspond sto anenerg y or mass con centratedi n the surf acex 1  0; from the s tandp oint of th e  modifie d the ory, how ever, (1. 1 . 1) is asol uti onof( 1 .1 .3.a), free f r o m si ng ulariti es,  whichdes cri bes  th e"field-pr oducingma ss ,"wit houtr equiring  f orth isthe i n troduc tionof a ny  n e w f ield qu a nt i t ies.  I tis c lea rthatalle qua tions o f the abs olu t e di ff er entialcalculus canbe  written i na f orm fr e e  from de nominators, whereby the tensors  ar e rep lace d by tens ordensit ies of sui tableweigh t.  It i s tob e  n ote dthat in th e cas e ofth e so lution(1.1.1) the wholef iel d consists of  tw o eq u a lhal v es ,  se pa r ate d by the surfaceof symmetr yx 1 0, sucht hat  f or thecorresp ondin g  points  x1, x2,  x3, x4an d  x 1 , x 2, x3, x4  the gik  are equa l . Asa  r esult  we  findth at,  alth ough w e  ar e per mittin gthe de termi na nt g totake on the value 0 x1   0,  no c han ge of sig n o f  gand in ge nera l no chang e i n t he
, regarded_from the_standpoint of the original_theory. Thus,_in_the framework_of_the original theory_one has the_gravitational equations R ik_ 12 gikR__Tik, 1. 1. 4 where Tik is the tensor of mass or energy density._To_interpret (1.1.1)_in_the_framework of this theory we_must approximate the line element_by a_slightly different one which avoids the singularity g__0. Accordingly we_introduce a small constant  and let ds2 _dx1 2  dx2 2 _dx3 2 _2x12__dx42 1. 1. 1._a the smaller  ( 0)_is chosen, the nearer does this_gravitational field come to that of (1.1.1)._If one calculates from this the_(fictitious) energy tensor Tik one_obtains as_nonvanishing components T22  T23_  2_2x12 _ 2. 1._1. 4. a We see then_that the smaller_one takes  the more is_the_tensor concentrated in_the_neighborhood_of the_hypersurface x1 _0.From_the standpoint_of_the original theory the solution (1.1.1)_contains_a singularity which corresponds to an energy_or mass concentrated in_the_surface x1  0;_from the standpoint of the_modified theory, however, (1.1.1) is a_solution of_(1.1.3.a), free_from singularities, which describes the "field-producing mass, " without requiring for_this the introduction of any new_field quantities. It is_clear that_all_equations of the_absolute_differential calculus_can be written in a form free_from denominators,_whereby the tensors are replaced by_tensor densities of suitable_weight._It is to be noted that_in the case of the solution_(1.1.1) the whole field consists_of_two_equal halves, separated by the_surface of symmetry x1  0,_such that for_the corresponding points x1, x2, x3, x4_and_x1, x2, x3, x4 the gik_are_equal. As a result we find_that,_although_we are permitting the determinant_g to take on the value_0 x1  0, no change of sign of_g and in_general no change in the