View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Cosmological PPN Formalism and Non-Machian Gravitational Theories 1 G. Dautcourt received 1. INTRODUCTION There app ears to b e little agreement what Mach's principle actually is - this was my chief impression from a recent conference on Mach's principle at Tubingen (Germany). Should we relate it to the requirement that the inertial mass of a b o dy is determined by the remaining b o dies in the Universe? Or should we follow one of Barb ours suggestions (Barb our 1995), p erhaps that theories must b e formulated in terms of relative quantities to b ecome Machian, since we observe relative quantities (p ositions, velo cities) only? Mach himself seems to b e of little help in this resp ect since for almost every formulation of a Machian principle one may nd an appropriate citation in his publications. Another item from Mach's General Store (Brill 1995) is the rather vague statement that distant parts of the Universe should have some in uence on the form of lo cal laws of physics. In a non-Machian gravitational theory a lo cal exp eriment should therefore b e independent of the cosmological environment. In particular, no elds of cosmological origin should have an in uence on the lo cal motion of matter, "lo cal" b eing taken to denote a suciently small space region. The purp ose of this note is to show that - within a p ost- Newtonian approximation - metric theories of gravitymay b e divided according to the requirementof time-dependent potentials for a consistent lo cal description of cosmological mo dels. This kind of dep endence on cosmological b oundary conditions is used to de ne Machian and non-Machian theories. To deal with a fairly general class of metric gravitational theories, the Parametrized Post-Newtonian (PPN-) formalism by Kenneth Nordtvedt and Cli ord M. Will will b e used. The notation is taken from Misner et al. (1973), otherwise I follow Will's standard b o ok (Will 1993). The PPN formalism was clearly not designed to deal with cosmology. On the other hand, one has to face the fact that a Newtonian cosmology was develop ed decades ago (see, e.g., Heckmann and Schuecking 1955, 1956, 1959, Trautman 1966), with results very similar to those of General Relativity. The relation of this Newtonian cosmology to the Friedman mo dels remained unclear, however. One exp ects that a Newtonian cosmology should emerge as a rst approximation of the general-relativistic theory. Then one may ask for a p ost-Newtonian cosmology as a second-order approximation. Notice the Newtonian 1 Humb oldt-Universitat zu Berlin, Institut fur Physik, D-10115 Berlin, Germany 1 cosmology as discussed here is based on a Lorentzian manifold, and hence quite di erent from the Newton-Cartan formulation of a transition to Newton's gravity (for the latter see, e.g., Trautman 1956, Ehlers 1981, Lottermoser 1988, Dautcourt 1990, and references therein). Somewhat surprisingly, the PPN formalism as it stands is already able to some extent to deal with cosmological problems, as will b e shown subsequently. The keys are (i) to use di erential relations to avoid the Minkowskian b oundary conditions at spatial in nityin the usual integral formulation of the PPN formalism, and (ii) to take the ratio distance (measured from the center of a selected spatial region) to Hubble distance c=H as the 0 expansion parameter. This leads to a simple lo cal description of cosmological mo dels, which should b e valid for smal l values of the expansion parameter. Section 2 summarizes basic asp ects of Newtonian cosmology. In Section 3 PPN rela- tions as well as the divergence freedom of the matter tensor are used to construct lo cally isotropic cosmological solutions b eyond the Newtonian approximation. Restricting to homogeneous mo dels in Section 4, an inconsistency is found in generic metric theories (with the exception of the non-Machian class) at the p ost-Newtonian level. We suggest that the discrepancy might b e caused by the neglection of time-dep endent comp onents in the Newtonian and p ost-Newtonian p otentials at the origin of the PPN co ordinate system. Section 5 illuminates the problem from a di erent p oint of view: Starting from a co- moving cosmological co ordinate system, in which the cosmological uid with a divergence- free matter tensor is homogeneous and expands isotropically, a transformation into the standard PPN co ordinate system pro duces generically time-dep endent lo cal p otentials. Again the p otentials are absent in a small subset of theories, formed by a non-Machian class of gravitational theories. 2. NEWTONIAN COSMOLOGY The PPN formalism assumes the existence of a co ordinate system where the metric tensor can b e written as 2 6 g = 1+ 2U 2 U +4 A+O( ); (1) 00 7 1 5 g = V W + O ( ); (2) 0i 1 i 2 i 2 2 4 g = (1 + 2 U )+O( ): (3) ik ik The space-time functions U; ; A;V ;W are de ned in terms of the matter density, i i the pressure p and velo city comp onents v (x;t) of the uid: i Z 0 0 0 U (x;t)= dx(x;t)= j x x j; (4) Z 0 0 0 0 dx(x;t)(x ;t)= j x x j; (5) (x;t)= Z 0 0 0 0 V (x;t)= dx(x;t)v (x ;t)= j x x j; (6) i i 2 Z 0 0 0 0 i 0i 0 3 W (x;t)= dx(x;t)[(x x )v (x ;t)](x x )= j x x j ; (7) i Z 0 0 0 0 2 0 3 A(x;t)= dx(x;t)[(x x )v (x ;t)] = j x x j ; (8) where 3 2 (x;t)= v + U + p= : (9) 1 2 4 0 2 As it stands, the PPN formalism app ears not to b e applicable to cosmology: If a homogeneous matter density (t)isintro duced, the Newtonian p otential U as calculated from (4) diverges - this is the well-known Seeliger-Neumann paradoxon, seen to b e present also in a higher-order correction to the Newtonian p otential. A simple waytoovercome this dicultywas prop osed by Heckmann and Schucking (1959), see also Trautman (1966): In agreement with the equivalence principle, the Newtonian p otential and its rst derivatives are considered as quantities sub ject to changes induced by co ordinate transformations beyond the Galileo group. The second derivatives only, which are determined by the generating mass density, should havea physical meaning. Thus the function 2 2 U = G(t)r ; (10) 3 was taken as the p otential in Newtonian cosmology. It is evidently a solution of the Poisson equation for U , U = 4; (11) generated by homogeneous matter density. The divergence of U for large r can b e consid- ered as co ordinate singularity,now caused by the use of a lo cal (non-cosmological) co or- dinate system. The singularityvanishes, if a transformation to cosmological (comoving) co ordinates is p erformed, see b elow Eq.(19). The p ost-Newtonian approximation as applied to ob jects in the Solar system uses p two expansion parameters = j U j and v=c, where U is the maximal value of the m m Newtonian p otential and v is the velo city of b o dies moving under the in uence of gravity. For the b o okkeeping of the di erent approximations b oth parameters are usually considered as of equal order. This is justi ed also in the case considered here: The cosmological p p otential (10) gives r 2 G=3 r=L, where L is a typical Hubble distance. The 0 radial velo city of galaxies is of equal order as is evident from Hubbles redshift-distance relation. As dimensionless expansion parameter one may therefore take the ratio distance to Hubble distance. To Newtonian approximation the metric tensor is given by 4 g = 1+ 2U +o( ); (12) 00 3 g = o( ); (13) 0i 2 g = + o( ); (14) ik ik and the dynamical equations reduce in the case of dust matter, which will b e assumed for simplicity throughout this note, to i i @ =@ t + @ (v )=@ x =0; (15) (_v + v v )=U ; (16) i i;k k ;i 3 where the p otential U is given by the solution (10) of the Poisson equation (11). For an isotropic expansion with a scale factor R(t) i i i _ v = x f (t)=x R=R; (17) 0 (15) and (16) together with the p otential (10) lead to matter conservation and to the Friedman equation: 3 2 _ R = const; R =2GM=R + const: (18) This similaritybetween Newtonian and general-relativistic cosmology has sometimes b een taken as evidence that Newtonian gravity mighthave a wide range of applicability in cos- mology - in the hop e to discard the mathematical apparatus of Einstein's theory in favour of the much simpler apparatus of Newton's theory. This is to some degree misleading, and the validity of the Friedman equation within Newtonian cosmology is easily under- k stood: One can intro duce comoving cosmological co ordinates ; instead of the lo cal k PPN-co ordinates x ;t used so far by requiring (i) that the space comp onents of the matter 4-velo cityvanish and that (ii)g ^ = 1; g^ = 0 in cosmological co ordinates.