Do Gravitational Fields Have Mass? Or on the Nature of Dark Matter Ernst Karl Kunst As has been shown before (a brief omment will be given in the text) relativisti mass and relativisti time dilation of moving bodies are equivalent as well as time and mass in the rest frame This implies that the time dilation due to the gravitational field is ombined with inertial and gravitational mass as well and permits the omputation of the gravitational ation of the vauum onstituting the gravitational field in any distane from the soure of the field Theoretial preditions are ompared with experimental results and it is shown that many known astrophysial and gravitational phenomena espeially the so-alled dark or missing matter owe their existene to the gravitational effets of the mass of the field-vauum physis/99007 4 Nov 999 Key words: Equivalene of mass energy and dilated time of moving bodies - mass of the gravitational field Introdution Apparent deviations from the Einstein-Newtonian law of gravitation both on laboratory and astronomial sale have been known long sine Those partly ontroversially disussed gravitational phenomena are: ) Constantly high veloities of individual galaxies within lusters and groups of galaxies departing strongly from the veloities on the strength of the virial law and onstantly high orbital veloities in the viinity of the Milky Way other galaxies and galaxy pairs whih deviate strongly from a Keplerian veloity distribution Both phenomena have led to the urrently aepted onept of non-luminous non-baryoni material in the viinity of large systems on a osmi sale the so-alled "halo of dark unseen matter" [] [2]; 2) An apparent inrease of the universal gravitational onstant G with growing radial distane of test masses measured with the torsion pendulum in the laboratory []; ) A systemati inrease of the gravitional aeleration g as one desends into deep mineshafts or boreholes [4] [5] [6] or derease as one asends towers [7]; 4) A systemati linear deviation of the aeleration of two test masses at the ends of the torsion pendulum in the gravitational field of Earth in proportion to the differene in baryon density (protons plus neutrons per unit mass) whih was found by analytial repliation of the original Eötvös data and led to the suggestion of a omposition-dependent finite range repulsive ( fifth ) fore [8]; 5) A systemati derease of the veloity of spae-probes on their trak
outbound of the solar system as e g Pioneer 0 and [9] 2 In the following we will show that all these experimentally found though - as already stated - partly ontroversially disussed phenomena are due to the gravitational effets of the mass of the gravitational field Connetion between Relativisti Mass and Dilated Time of Moving Bodies Main results of the modified theory of relativisti kinematis among others are inertial motion (veloity) always to be symmetrially omposite and the Lorentz transformation not to predit the Fitzgerald-Lorentz ontration of the dimension (x) parallel to the veloity vetor as invented by Fitzgerald and Lorentz to aount for the null-result of the Mihelson-Morley experiment on moving Earth but rather an expansion of x [0] Aordingly the volume V of an inertially moving body will any observer resting in a frame onsidered at rest seem enhaned by the fator V x y z x 0 y zv 0 () where V means volume and the Lorentz fator based on the omposite veloity v 0 0 Among others it has been demonstrated this expansion of x (or V) to be the ause of the experimentally found inrease of the interation radius respetively ross setion of elementary partiles with rising energy (veloity) as determined in ollider experiments and as is known from studies of osmi radiation From m = V ' = m = 0 V' in onnetion with () follows ' = ' and therewith the fration v / of the 0 0 relativistially dilated time to be the very ause of the relativisti inrease of mass: dt v0 E t m t 2 m t E t dt 2 v0 dx (2) where "'" means density of mass E energy of the produt dt v of a moving t 0 material body and m mass indued by time dilation Furthermore has been t demonstrated mass of the hydrogen (H-) atom and quantum of time 2 / in the rest frame be equivalent and generated by the movement of a fourth spatial dimension of the atom 2 h m where "2 " is the fundamental length in R "h" Plank's onstant and "m" mass of the 4 smallest eltrially neutral and stable piee of matter presumedly of the H-atom [] Analogous to the equivalene of mass and time in the rest frame as well as in the moving one gravitational fields have to be onsidered to be spaes with relativisti mass beause of the time dilation due to the gravitational field Thus we an expet
the potential differenes in gravitational fields in dependene on the distane from the soure of the field to be pereptible as physially measurable masses To ompute those masses basially two possibilities exist Global Estimate of the Mass of the Gravitational Field We refer to the lassial definition of mass as the produt of volume and density and then in aord with (2) an write E V m V 2 dt v 0 dydzv v0 wherefrom follows E V m V 2 Vv 0 () if v «and dt/dt where E means energy and m mass of volume of spae 0 V V (vauum) respetively Of ourse () is valid in the ase of moving bodies only with one veloity vetor Suppose we have a spherially symmetri gravitational field in the form of a Shwarzshild-vauole in the Friedmann osmos r 6 M L' 2K N where "r" means the oordinate radius of the vauole and "N" the "radius" "K" means urvature "M" the mass of the entral body under onsideration [2] In this ase the produt of eah of the three dimensions of the vauole and the veloity vetor or (in Newtonian approximation) salar of urvature K r N v 0 2GM 2 R will ontribute to the global energy of the vauole so that aording to () the energy ontent of the spae of the vauole - as seen from "outside" - an be written as M V 2 V N N N x y z K N K N 2GM 2 R (4) whereby "G" is the Newton's gravitational onstant and "R " the radius of the mass of the entral body distributed in the vauole Here we had to onsider that eah of the three geometri dimensions of the gravitational field must be multiplied by the salar of urvature or (in approximation) vetor of veloity aording to the priniple of equivalene On the grounds of the ratio
mass in the osmos mass (of the body) in the vauole ' 0 volume of the osmos volume of the field (vauole) 4 - whereby ' means osmi density of mass - we eventually an approximate the total 0 mass of the gravitational field of the vauole with the expression M V N 2GM 2 R M ' 0 2GM 2 R Aordingly the ratio of the "total mass of the gravitational system" (field + visible mass) to the "visible mass" is M total M MM V M ' 0 2GM 2 R Applying the orresponding values for galaxies and galaxy lusters roughly the right amount of masses results ie greater than the visible masses whih were introdued by astronomers as the so alled "dark or missing matter" to explain the dynamis of large omplexes of gravitational systems The unertainties in the determination of the osmi density ' 0 and espeially of R (boundary between the external and internal Shwarzshild metri) in extended gravitational systems eg galaxies and lusters of galaxies allow but only very global estimates The Mass of the Gravitational Field in Dependene on the Distane from the Soure of Gravitation More exat results an be derived if starting at the boundary between the external and internal Shwarzshild metri of a spherial distribution of mass (soure of the external field or radius of the mass in Eulidian oordinates) the infinitesimal small distanes respetively multiplied by the time dilation at the point of the oordinate in radial diretion are summed Consider the enter of gravity of the field-produing mass to be at rest with T 44 being the essential omponent of the energy momentum tensor T ik so that in a first approximation is valid g 44 2GM 2 GM L 2 8% P T 44 d x x x Furthermore the vauum of the external Shwarzshild metri is onsidered to onsist of thin onentri shells of the thikness dr where R is the distane from the enter of gravity Aording to (4) the mass of eah infinitesimal thin onentri shell measured from the point R radially "within" the field must be proportional to the infinitesimal small distane dr:
dm V R v 0 dr 5 onstant The volume of eah suessive shell inreases as the square of the radius On the other hand the veloity v 0 dereases inversely proportional to the radius ie the ube 2 of the veloity v 0 with inreasing radius as /R As a result the produt of eah suessive shell by the ube of the respetive veloity remains onstant Thus the mass of the vauum of eah suessive shell of the gravitational field remains onstant for all R Beause v 0 v 0 R R onstant for all R - where v means veloity at the point R - and 0R v 0 R 2GM R 2 GM R the proportion of mass of the infinitesimal part dr of the radius R of the gravitational field amounts to dm V R v 0 R R dr GM dr R Integration results in the mass of the field vauum within the radius R M V R P R R GM dr R 2GM ( R R ) (5) measured within the field R means the radius of the internal Shwarzshild metri ("radius" of the mass) R the radial distane from the enter of gravity of the field produing mass measured in Eulidean oordinates The onstant is the mass of the marosopi groundstate of vauo (not disturbed by gravitational fields) whih is null and M the mass of the vauum of the gravitational field in the distane R from VR the enter of gravity whih together with the field-produing mass M or the energy momentum tensor T respetively determines ompletely the behaviour of test bodies i k of the mass m In priniple this result is also valid for the internal Shwarzshild metri beause the spae inside a gravitational body ontributes to the total mass of the body or density of energy T in the distane R from the enter of gravity (in approximation): ik
6 M VR 2GM R Thus if R < R the mass of the gravitational field inside the mass also amounts to the value (5) measured from the point R in negative radial diretion or toward the enter of gravity Comparison of Theoretial Preditions with Experiment ) Aording to (5) the total mass of a gravitational system in the radial distane R from the enter of gravity of the field produing mass M amounts to M total MM V R M 2GM( R R ) (6) or after division by M the quotient is given by M total M MM V R M 2G( R R ) (7) From (6) the orbital Kepler veloity of a body of negligible mass as a funtion of R R and the entral mass M is derived: v 0 R GM R 2G( R R ) (8) Computation results in the flat non-keplerian rotation urves of galaxies and pairs of galaxies established by astronomial observations whereby the morphology of the urve strongly depends on R Calulation of (7) results diretly in the ratio of the total pereptible mass within the distane R of the gravitational field - baryoni plus field - to the amount of the luminous matter whih agrees well with astronomial measurements In the following we ompare theoretially derived values of M /M total aording to (7) with some experimental results for the outer regions of the Milky Way []: M /M (7) Experiment R (kp) R (kp) "m" total 26 0 8 0 arbon monoxide louds 0 90 60 0 louds of Magellan 6 20 75 0 satellite galaxies
7 R = 0 kp is the mean distane of the sun from the galati enter beause astronomial measurements are grounded on the validity of the Newton-Keplerian law within the orbit of the sun (see also 5) below) Equation (7) results also onviningly in the linear inrease of M in vast osmi systems as measured by astronomers [] [2]: M /M (7) Experiment R R "M" total 0 0 00 kp 0 galaxies a f m G 2G( R R ) /M with growing R 25 25 00 kp 0 pairs of galaxies 40 >400 2 mp 0 Coma luster 650 >600 70 mp 0 loal superluster 2) From the preeding is evident that measurements of the R-dependene of the aeleration in loal fields of gravitation must yield apparent disrepanies to Einstein- Newtonian gravity whih usually are interpreted either as a modifiation of the gravitational onstant G or as the effet of an additional (fifth) fore of nature A diret measurement of the gravitational fore f whih a unit of mass M = exerts on a test mass m in the distane R from the enter of gravity results aording to (5) in an additional aeleration: total G R GG G 2( R R ) (9) Evidently G expresses an apparent alteration of G due to the gravitational effet of the R field vauum In 976 Long ompared older measurements at various ranges of R with the results of his own torsion pendulum experiments at R = 45 m and 0 m and found G R G[0002 ln (R)] on laboratory sale [] For an overview we ompare theoretial and experimental results : (9a)
Theory Experiment R R G = G R - G (9) G G R - G (9a) m 8 005 00077 0 0 0025 0022 0 0 000 0029 50 0 0058 006 70 0-8 - -2 where G = 6656 0 g s aording to Long ) The influene of the mass of vauo onstituting the gravitational field of Earth on the gravitational aeleration g results aording to (8) in: g 2G 2 M E ( R E R ) RE G (ME MV ) G R 2 E GM E 2G 2 M E ( R E R E R ) RE G R E 2R E ( R E R E R ) ME (0) or as an apparent alteration of the gravitational onstant of the amount G (ME + MV) = G + G where M E means mass and R E radius of Earth respetively - R is here the negative radial diretion toward the enter of Earth measured from the point R E Consistently higher values of G from measurements of g in boreholes and mines for some time have been known to point to a deviation from the /R-law of the gravitational potential Therefore a diret omparison of this theory with experimental results is possible In the following we ompare some results of Staey [4] Holding [5] and Hsui [6] from measurements in boreholes and mines with alulations aording to (0): Theory Experiment R -8 - -2 G (ME + MV) (0) 0 m g s m 6674 6724 ± 004 2 0 4
6722 674 ± 0002 0 5 6727 6700 ± 0065 2 0 5 6772 680 ± 0070 4 0 5 ' R ' ' R R ' 0 9-8 - -2 27 8 where G = 6672 0 m g s M = 597 0 g R E = 64 0 m 4 Ekhardt measured in a tower experiment at R 6 0 m above the ground a deviation of g of 500 ± 5 µgal [7] whereas our formula delivers 400 µgal where g = -2 98 m s /2 Of ourse in the ase of the tower experiment ( R E - R ) in (0) must be replaed by /2 ( R E + R ) The oinidene of theory and experiment does not look very impressive - whih easily is explained by the tremendous unertainties on the experimental side - but nevertheless a systemati trend learly shows up 4) A look at (0) shows that the apparent alteration of G due to the mass of the gravitational field should not only be dependent on the distane R but also on the omposition of the material of the attrated mass The reason is that if R = onstant then the fration R /R and thereby the differene (R - R ) differs with the density ' of the material The density ' is the determining parameter and it is lear that ' is appproximately related to the differene in baryoni density In other words we assume the mass of the field vauum also to play a passive role as attrated mass For onveniene we hoose onstituting the density of H2O as the referene value Beause stok density varies inversely to the volume per unit mass relative to H2O the radius R of the unit mass of all materials other than H2O varies as / ' so that (5) attains the form: M V(R 2GM R) 6 ' As ompared with the mass of the gravitational field of an unit mass H2O whih we arbitrarily set zero the mass of the field of the unit mass of a material other than H 2 O 6 varies as ( - / ') Correspondingly the gravitational fore ating upon a test mass in a loally (almost) homogeneous field partiularly that of Earth must vary proportional to the density of the test body as: 2G f g( MM V(R ) M(gg) gm R) 6 ' () whih means: Bodies of equal mass but different density (baryoni density) experiene an apparent omposition - dependent relative gravitational aelerationwhih is due to small differenes of the integrated masses of the respetive gravitational fields
0 Fishbah's analysis of the old Eötvös data inludes among others three pairs of sample material: H2O-Cu asbestos-cu and Pt-Cu [8] The experimental results for these pairs are very onvenient for a omparison with theory Computation of () and 9 omparison with the results of Fishbah-Eötvös in g(g) 0 results in (experiment in brakets): H2O - asbestos = 667 (7±2) H2O - Pt = 69 (4±2) whereby ' asbestos = 28 and ' Pt = 245; ' means density 5) The mass of the field vauum surrounding the sun amounts in any distane R (from the sun) aording to (5) to M V R 2GM ( R R ) Thus the gravitational pull of the mass M in the distane R from the sun must be V GM V R 2G 2 M ( R R ) R 2 R 2 (2) Pioneer 0 is urrently 7 times as far from the sun as Earth is Aording to (2) the 5 gravitational pull of the field vauum of the sun in this distane R 0626 0 m 2-8 -2 onto the spaeraft must be GM V /R = 05 0 m s - where M = 2 0 g and 4 R = 428 0 m the mean distane of Saturn from the sun - whereas Anderson -8-2 reported an experimentally found aeleration of 85 0 m s toward the sun [9] To hoose the proper value of R in (2) it had to be onsidered that analogous to the ase of the Milky Way astronomial measurements are grounded on the validity of the Newton-Keplerian law within the orbit of Saturn or with other words: in all omputations on the grounds of Einstein-Newtonian gravitation the mass of the field vauum of the sun (and of the planets) is at least till the orbit of Saturn inluded in the mass of the 4-8 sun If R = 2872 0 m - the mean distane of Uranus - (2) yields 797 0 m - 2 s Besides we have to expet that the straightforward appliation of (2) to the gravitational field of the sun is restrited for the following reasons: If the mean distanes between the planets are listed in A U aording to the Titius- Bode law (whih with the exeption of Pluto orrespond roughly to the observed distanes) the following ratios result: Merury - Venus : Venus - Earth = 0 : 0 = : Merury - Earth : Earth - Mars = 06 : 06 = : Merury - Mars : Mars - Ast = 2 : 2 = : Merury - Ast : Ast - Jupiter = 24 : 24 = : Merury - Jupiter : Jupiter - Saturn = 48 : 48 = : Merury - Saturn : Saturn - Uranus = 96 : 96 = : Merury - Uranus : Uranus - Neptune = 92 : 92 = : Merury - Neptune : Neptune - Pluto = 84 : 84 = :
Beause - as shown before - the mass of eah suessive shell (being proportional to the distanes between the planets or the "thikness" of the shells) of the gravitational field of the sun remains onstant the above ratios seem to indiate that in the protoplanetary disk and later the planets positioned more or less exatly between field shells of equal mass This an be desribed as r n 2 n 2 ( r )r () where n = 2 n and r = Inserting r = 04 in () delivers again the Titius-Bode n - 2 law 2 0 + 04 It s lear that all r > r depend on the value of r whih again annot be derived from () Obviously is the simple rule of balane of field mass shells developed above not straightforwardly appliable to the three innermost solar planets But if our hypothesis is orret must their distanes from the sun also depend on the balane of the field masses Hene they should tend to take positions between three shells of equal mass at (in arbitrary units) r = 0 r = 066 and r = On the 2 other hand there must exist a tendeny to form three shells at r = 05 r = 075 and 2 again r = to reah a balane 05 : 05 As a onsequene the planets tend to take position between r = 0 and 05 whih results in r = 04 and r = 066 respetively 2 075 whih results in r = 07 2 Thus if this hypothesis is orret it must be valid for any system where at least three objets (with a similiar genesis as the planets of the sun) orbit a entral mass If always r = (in the ase of Saturn the mean distane of Tethys to Calypso) we find for the two innermost objets e g in the system of the sun r = 09 and r = 072 Jupiter r 2 = 09 and r = 06 Saturn r = 049 (mean distane of Atlas to Epimetheus) and r = 2 2 072 (mean distane of Mimas and Eneladus) Uranus r = 048 and r = 072 the 2 pulsar PSR 257 + 2 r = 04 and r = 077 and the pulsar PSR 828 - r = 044 2 and r = 06 respetively 2
Referenes [] Rubin Vera C Sientifi Amerian 248 96-97 (98) [2] Thuan T X & Montmer T La Reherhe 448 (982) [] Long DR Nature 260 47 (976) [4] Staey FD & Tuk GJ Nature 292 20-22 (98) [5] Holding SC & Tuk GJ Nature 07 74-76 (984) [6] Hsui AT Siene 27 88-882 (987) [7] Ekhardt DH et al Phys Rev Lett 60 2567-2570 (988) [8] Fishbah E et al Phys Rev Lett 56 (986) [9] Anderson J D et al: Indiation from Pioneer 0/ Galileo and Ulysses Data of an Apparent Anomalous Weak Long-Range Aeleration gr-q/980808 [0] Kunst E K: Is the Kinematis of Speial Relativity inomplete? physis/9909059 [] Kunst E K: On the Origin of Time physis/990024 [2] Stephani H Allgemeine Relativitätstheorie 2nd ed Deutsher Verlag der Wissenshaften Berlin 980 p 25-256