Cosmological PPN Formalism and Non-Machian Gravitational Theories

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Cosmological PPN Formalism and

Non-Machian Gravitational Theories

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

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

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)

7 1

g = V W + O ( ); (2)

0i 1 i 2 i

2 2

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:

0 0 0

U (x;t)= dx(x;t)= j x x j; (4)

0 0 0 0

dx(x;t)(x ;t)= j x x j; (5) (x;t)=

0 0 0 0

V (x;t)= dx(x;t)v (x ;t)= j x x j; (6)

i i 2

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)

0 0 0 0 2 0 3

A(x;t)= dx(x;t)[(x x )v (x ;t)] = j x x j ; (8)

where

(x;t)= v + U + p= : (9)

1 2 4 0

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

U = G(t)r ; (10)

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

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 otential (10) gives r 2 G=3 r=L, where L is a typical Hubble distance. The

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

g = 1+ 2U +o( ); (12)

g = o( ); (13)

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)

(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-

stood: One can intro duce comoving cosmological co ordinates ; instead of the lo cal

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.

00 0i

The transformation

2 2 k k

= t r R=(2R)+o(t ); =x=R + o(r) (19)

leads to

00 4

g^ = 1+o( ); (20)

0k 3

g^ = o( ); (21)

ik 2 2

g^ = =R( ) + o( ); (22)

and satis es (i) and (ii) to Newtonian order. It is now easy to see why the Newtonian scale

factor R(t)must b e a solution of the Friedman equation: Consider the Friedman equation

for the correct general-relativistic scale factor S ( ). Writing

2 2

_ _ _

S ( )= S(tr R=(2R)) S (t) Sr R=(2R) (23)

in this equation and expanding, terms (r=L) cancel, and one is left with the Friedman

equation for the Newtonian scale factor R(t) S (t).

Thus it app ears that the applicability of the Newtonian approximations (12) - (18)

with the p otential (10) is con ned to a neighbourhood of the observer, corresponding to

distances smal l compared to the Hubble distance. As a consequence, the matter density

in lo cal co ordinates is a pure time function only on the observer world line x = 0 and

changes with r further out just to ensure homogeneity or a Cop ernican principle. Indeed,

any scalar dep ending only on cosmic time in cosmological co ordinates, dep ends in a lo cal

PPN co ordinate system also on spatial co ordinates (see Eq. (19)).

Manyyears ago, Callan, Dicke and Peebles (1962) have presented similar ideas con-

cerning the relation b etween Newtonian mechanics and cosmological expansion. 4

3. POST-NEWTONIAN EQUATIONS

Beyond the Newtonian approximation we use instead of (5),(6) the corresp onding di er-

ential formulations (G = 1 subsequently,we also write = + , assuming dust):

1 1 2 2

= 4 ; (24)

V = 4v : (25)

k k

Di erential relations also hold for the remaining functions. W may b e written

1 @

0 0 0 0

W = dx (x ;t)(x x )v j

i 0

@x j x x

k k 0 0 0 0 0

= x V + dx (x ;t)x v = j x x j;

or after applying the Laplacian

W = 4v 2V : (26)

i i k;ki

Finally we list some further relationships (see Will 1993, chapter 4.1) already in di erential

form:

V = U ; (27)

k;k ;0

= 2U; (28)

= V W ; (29)

;0k k k

= A : (30)

;00 1

The di erential formulation of the PPN formalism used in this article is based on the

(not all indep endent) relations (24) - (30) and (11). We write down expansions in terms

of p owers of r similar to (10) for the p ost-Newtonian p otentials V ;W ; ;A as well as for

i i

the uid density and velo city, and lo ok for homogeneous and isotropic purely expanding

solutions. The matter density and velo city for isotropically moving dust can b e expanded

= + r ; (31)

0 2

k k 2

v = x (f + f r ); (32)

0 2

where ; ;f ;f are time functions. Using this p ower law expansion, a straightforward

0 2 0 2

calculation yields

1 2

2 4

r r ; (33) U =

0 2

3 5

4 2 2 2

)r ; (34) +2 (3 f =

2 1 0

0 0

2 k

V = f r x ; (35)

k 0 0

W = V ; (36)

k k

4 1

2 4

(3f + )r : (37) A =

0 0

5 3 5

Up to p ost-Newtonian order, the metric tensor can now b e written:

2 4

g = 1+br + cr ; (38)

2 i

g = mr x ; (39)

g = (1 + gr ): (40)

ik ik

The time functions b; c; m; g are determined by the distribution and motion of matter:

; (41) b =

2 4 1

2 2 2

c = + (6 3 10 ) f (4 +3); (42)

2 2 0 1

0 0

5 45 5

m = f (7 ); (43)

0 0 1 2

g = : (44)

From the conservation law of the matter tensor, T = 0, one obtains in Newtonian

;

approximation, in agreement with (18),

f = f ; (45)

0 0

_ = 3f : (46)

0 0 0

The p ost-Newtonian terms lead to equations for the next co ecients f and :

2 2

2 4 4

2 2 2

f = 4f f + k f + k ; (47)

2 0 2 1 0 2 2

0 0

15 45 5

f (2 3 ); (48) _ = 5f 5f +

0 2 0 2 2 0

with

k = 12 9 + 42 6 20 +5;

1 1 1 2

k = 20 +12 6 + 21 3 20 :

2 2 1 2

The system (45) - (48) can b e integrated fairly easily: From (45), (46) one obtains a

second-order linear di erential equation for f alone, is determined algebraically up on

0 0

solution of the f equation. Similarly, if a solution of (45),(46) is known, (47),(48) can

b e transformed into a linear second-order di erential equation for f , leaving an algebraic

equation for .

4. HOMOGENEITY REQUIREMENTS

The mo dels obtained in this way represent - in a p ost-Newtonian approximation -

a large class of lo cally isotropic, but in general inhomogeneous expanding matter dis-

tributions in the neighb ourho o d of the observer. Restrictions for homogeneitymaybe 6

intro duced by several metho ds. Usually one relies on groups of motions. A more intu-

itiveway is to imp ose homogeneity with a Cop ernican principle. The origin of the PPN

co ordinate system may b e asso ciated with a freely falling observer P . The worldline of a

second observer P at rest with regard to the cosmological substratum in his neighb ourho o d

i i

is given by x (t), with the three functions x (t) as solutions of the di erential equations

0 0

i i 2

x_ = x (f + f r ): The co ordinate transformation leading from the P -system to the P -

0 2

0 0 0

system is a generalization of the Chandrasekhar-Contop oulos (1967) transformation, since

the second observer moves with acceleration. A Cop ernican principle should state the

equivalence of all these systems. More sp eci c, in the lo cal PPN system of any comoving

observer the expansion co ecients of matter density and cosmic velo city (Eqns. (31), (32))

should have the same dep endence on the corresp onding co ordinate time. Wemay set up

a Cop ernican principle for the matter density and velo cityaswell as for the metric or

derived quantities such as the expansion rate of the matter congruence. It is not obvious

whether all these principles lead to the same conclusions.

For purp ose of this note we can avoid the construction of a generalized Chandrasekhar-

Contop oulos transformation by refering to Eq.(19): Homogeneity for scalars can b e

de ned by a dep endence on the cosmological time only, hence ( )=(tr f =2)

(t) r f (t)=2, and the lo cal expansion co ecients and are related by =

0 0 2 2

f =2. Applying this to the matter density gives with (46)

0 0

= f : (49)

2 0

It is easy to check that (49) is compatible with the di erential equations (45)-(48) if

and only if the conditions

6 12 +20 + 3( 7 )+26 16 = 0; (50)

1 2 1

9 +12 + 6( 7 )+20 +4 = 0: (51)

1 2 1

for the PPN co ecients are satis ed. Moreover, f is given by

f = f (8 3 ): (52)

2 0 0

(49) - (52) also ensure the homogeneity of the uid's expansion rate up to p ost-Newtonian

order via a relation similar to (49).

The problem is that (50) and (51) hold only for a fairly small class of gravitational

theories. As one easily checks, General Relativity( = 0, all other co ecients = 1) b elongs

to this class. Needless to say that reasonable cosmological solutions are known for many

metric theories, e.g. for the often discussed Brans-Dicke theory with =(1+!)=(2 +

! ); =(3+2!)=(4 + 2! ); =(1+2!)=(4 + 2! ); = (10 + 7! )=(14 + 7! ) (other

1 2 1

parameters as in GR), which do not satisfy (50) and (51).

Obviously, the framework used so far is not adequate. To deal with a p ossible cosmo-

logical time evolution in the Newtonian dynamics of compact b o dies, K. Nordtvedt (1993)

has intro duced time-depending PPN parameters (as well as time-dep endent coupling con-

stants and masses). It is p erhaps not necessary to go so far. Evidently the cosmological 7

solution (19) of the Poisson equation is incomplete, one could have added an arbitrary

time function U (t) to the p otential. This also applies to the p ost-Newtonian p otentials,

giving additional time functions a(t);l(t);d(t) in the metric tensor:

2 4

g = 1+a+br + cr ; (53)

i 2

g = x (l + mr ); (54)

g = (1 + d + gr ): (55)

ik ik

If the added time function vary on scales not smaller than the Hubble time they have

small or negligible in uence on the motion of lo cal b o dies. The equations (53) (55) should

allowtowork out a consistent p ost-Newtonian approximation scheme, as will b e discussed

elsewhere.

5. COSMOLOGICAL COORDINATES

To obtain di erent lo ok at the problem, wemay search for solutions in cosmological

co ordinates, using symmetries and matter conservation but no other eld equations. In a

second step we transform into a lo cal PPN co ordinate system and compare the resulting

metric with the lo cal one following from (1)-(3) and (33)-(37).

Purely expanding (or contracting) dust matter satis es

T = u u ; (56)

u =(g + u u ); (57)

;

as well as the divergence relation

T =0: (58)

;

One easily veri es the existence of a comoving (u = ) co ordinate system (; ), in

which the line element can b e written

2 2 i 2 k i k

ds = d + S (; ) ( )d d : (59)

(58) reduces to

S = const: (60)

No eld equation has b een used so far. To restrict to homogeneous mo dels, it is assumed

additionally to (56)-(58) that the scale factor S dep ends on the cosmological time only

and that

2 2 2 k k

= =(1 k =4) ; = (61)

ik ik

is the metric of a 3-space of constant curvature. To relate the cosmological co ordinate sys-

tem to the standard PPN gauge up to p ost-Newtonian order, we extend the transformation

(19) by writing

2 4

= u(t)+v(t)r +w(t)r ; (62)

i i 2

= x (1=F (t)+p(t)r ): (63) 8

The function u(t) accounts for the p ossibility that the lo cal PPN time di ers from the

cosmological time even at the origin of the PPN system. Also, the transformation function

F could di er from the scale factor R(t)=S(u[t]). A PPN co ordinate system is de ned

by the condition, that the quantities entering the metric (1) (3) must b e solutions of the

PPN di erential relations. One has furthermore to ensure that up to and including the

order the spatial comp onents g attain no non-diagonal term. This latter condition

restricts the function p(t) in (63) to

2 2

p(t)=Fv =R : (64)

Comparing now the lo cal and transformed metric on obtains to zero order in r from

the g and g comp onents

00 ik

2 2

4 + A ; (65) u_ =12U +2 U

0 0 0 0

F = R= 1+2 U : (66)

These relations cast some light on the meaning of lo cal p otentials. The relation

between comoving and lo cal co ordinates is determined by the lo cal scale factor F (t), which

di ers from the cosmological scale factor R(t), if a lo cal p otential U (t) is present. A time

dep endent p otential in g comes not only from U , but also from the A and terms, as

00 0

seen in (65). If the combination of lo cal p otentials on the rhs of (65) di ers from zero,

the PPN co ordinate time is not equal to the prop er time measured by an observer on the

center world line x = 0 of the PPN system (this world line is still a geo desic, however).

In order to check the conditions for the PPN parameters found ab ovewe exclude all

lo cal comp onents in g .Furthermore, the relation (60) as well as u = can b e trans-

formed into the lo cal co ordinate system up to p ost-Newtonian order. We then compare

the transformed cosmological solution with the lo cal metric sp eci ed by (1)-(3) and (10),

(33)-(37). Additionally to the determination of the transformation functions, all these

relations give the following restrictions for the PPN parameters:

=1; (67)

27(7 ) 48 +12 20 106=0: (68)

1 2 1 2

The domains in the PPN parameter space covered by (50), (51 on the one side and by

(67), (68) on the other do not coincide. (Note however, if we assume = 1 in (50),(51), Eq.

(68) follows from (50),(51)). But coincidence cannot b e exp ected. In deriving (50), (51)

wehave assumed a restricted formulation for the Cop ernican principle (homogeneity for

the matter density). Other formulations for this principle may lead to di erent conditions

for the PPN parameters. On the other side, the discussion in this section is complete with

regard to symmetries but lacks full dynamics.

It is op en if a full consideration of b oth symmetry conditions and dynamics would

con ne the set of theories which are free from lo cal p otentials to General Relativity alone.

6. CONCLUSION

Wehave discussed the question, how cosmological b oundary conditions may in uence

the motion of lo cal matter. There is no in uence for General Relativity (and a small 9

numb er of additional theories of gravity) up to p ost-Newtonian order. For the ma jority

of other gravitational theories there exists an in uence at the p ost-Newtonian level of

approximation. Wehave called this di erent b ehaviour of gravity theories non-Machian

and Machian, resp ectively. In a p ost-p ost-Newtonian approximation the lo cal motion of

matter may dep end on cosmological b oundary conditions also in General Relativity,so

that every metric theory of gravitywould b ecome Machian according to the de nition

given here. - Only cosmological background elds have b een used in this article. A study

of p erturbations up to p ost-Newtonian order may giveachance to determine some PPN

parameters also in a cosmological setting.

7. ACKNOWLEDGEMENTS

I am indepted to C. M. Will and K. Nordtvedt for a useful discussion and to the referee

for valuable suggestions concerning the presentation of the material.

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