1996MNRAS.282..206T
Mon. Not. R. Astron. Soc. 282, 206-210 (1996)
Newtonian cosmology revisited
Frank J. Tipler*
Department of Mathematics and Department of Physics, Tulane University, New Orleans, LA 70118, USA
Accepted 1996 April 16. Received 1996 February 23; in original form 1995 July 14
ABSTRACT
I show that if Newtonian gravity is formulated in geometrical language, then
Newtonian cosmology is as rigorous as relativistic cosmology. In homogeneous and
isotropic universes, the geodesic deviation equation in Newtonian cosmology is
proven to be exactly the same as the geodesic deviation equation in relativistic
Friedmann cosmologies. This equation can be integrated to yield a constraint
equation formally identical to the Friedmann equation. However, Newtonian
cosmology is more general than Friedmann cosmology: by generalizing the flatspace Newtonian gravity force law to Riemannian metrics, I show that everexpanding and recollapsing universes are allowed in any homogeneous and isotropic
spatial geometry.
Key words: gravitation - relativity - cosmology: theory - large-scale structure of
Universe.
1 INTRODUCTION
The fully relativistic Friedmann equation for homogeneous
and isotropic universes is
1
R2
(dR)2
dt
8reG p k
=-3-- R 2'
(1.1)
where p is the density of matter, and k = ± 1 or O.
As first noted by Milne (1934) and McCrea & Milne
(1934), the Newtonian equation governing the evolution of
a particle with mass m and total energy E located a distance
R from the centre of a homogeneous and isotropic sphere of
matter with density p (t) is
1
R2
(dR)2
dt
8reGp (-2Elm)
=-3-R2
.
(1.2)
The similarities are striking. In fact, if we set ( - Elm) = k,
then the equations are identical. Because Newtonian mechanics requires less mathematical knowledge than general
relativity, one usually finds (1.2) 'derived' in elementary
discussions of modern cosmology. However, it is also well
known (e.g. Layzer 1954; McCrea 1955a,b; Bondi 1961;
Harrison 1981) that the derivations are defective. Among
other problems, the 'energy per unit mass' Elm is not
defined in an infinite Newtonian universe with constant
density.
*Bitnet address: [email protected]
I shall show how the difficulties can be overcome in Section 2. The idea is to use the reformulation of Newtonian
gravity theory into geometric language achieved by Cartan
(1923, 1924). In brief, Cartan showed that the orbits of
particles in a Newtonian gravitational field should be
regarded as geodesics of an affine space, and therefore
gravity should be considered as curvature of the affine connection, not as a force. Although Cartan's ideas have been
developed by a number of authors (Trautman 1965, 1966;
Penrose 1968; Misner 1969; Ellis 1971; Stewart 1990), none
have explicitly shown how the Cartan formulation can be
applied to make Newtonian cosmology as rigorous as Einsteinian cosmology, a task I shall accomplish in Section 2. I
shall do so by deriving the equation of geodesic deviation
for the 'fundamental observers' in Newtonian cosmology,
and show that it is exactly the same as the equation of
geodesic deviation for the observers normal to the hypersurfaces of homogeneity and isotropy in the Friedmann
universe. Using the equation of continuity - which is the
same for Newtonian and Einstein cosmology - the equation
of geodesic deviation can be integrated in both cases to yield
the Friedmann equation. However, this equation is more
general in Newtonian cosmology, where any spatial topology is consistent with both ever-expanding and recollapsing
models. I shall demonstrate this by generalizing Newtonian
gravity theory to Riemannian manifolds in Section 3.
I shall discuss in Section 4 the physical significance of
Newtonian cosmology. For example, putting Newtonian
cosmology on a rigorous foundation permits cosmological
perturbation theory to be carried out rigorously in the
©1996 RAS
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.282..206T
Newtonian cosmology revisited
Newtonian framework. Thus any perturbation problem that
does not involve truly relativistic phenomena (e.g., relativistic shocks) can be confidently solved in Newtonian
mechanics even in cosmology. I shall point out also that the
Cartan viewpoint suggests how to impose boundary conditions at infinity in a homogeneous but anisotropic
Newtonian universe.
I shall follow the notation and postulates of Misner,
Thome & Wheeler (1973, hereafter MTW), Chapter 12, for
the geometric formulation of Newtonian gravity. The MTW
postulates are equivalent to those of Cartan (1923, 1924)
and Trautman (1965, 1966).
equation,
Roo = 41tGp.
(2.6)
With the expression (2.5) for the curvature, the equation
for geodesic deviation
D 2n"
dx P
dA.
dx'
dA.
--+R" -nP-=O
dA.2
pP'
(2.8a)
(2.8b)
Cartan's reformulation of Newtonian gravity theory begins
by recalling that the Newtonian equation of motion for a
particle of arbitrary mass in a gravitational field generated
by potential <I> is
(2.1a)
which can be written
(2.1b)
Affinely propagated Newtonian clocks carried by the test
particles measure some linear mUltiple A. = at + b of
Newtonian time t. This means the equation of motion (2.1)
can be written as two equations:
d 2t
Let me now show what the equation of geodesic deviation
becomes in Newtonian cosmology. Pick the fiducial geodesic to lie at the origin of coordinates in Euclidean space
R 3 , which is covered by a constant, time-independent 'Galilean' coordinate grid with basis vectors 0/&; satisfying
(0/&;)' (o/&j) = (jij' Then, since all physical quantities
depend only on the time and are independent of spatial
position, the position as a function of time of any geodesic in
the congruence with constant coordinate position x = (x, y,
z) == (xl, x 2 , x 3 ) is
n (t) =R (t)x.
(2.9)
In particular, if initially we have n (to) = (n 1, 0, 0), then at
any time t we must have n (t) = [R (t)x, 0, 0]. However, since
the equation of geodesic deviation (2.8b) implies for
n 2 =n 3 =0 at t=to that
(2.2a)
d2
x + 0<1> (dt)2 = O.
i
dA.2
(2.7)
becomes (MTW, chapter 12)
2 A RIGOROUS DERIVATION OF
NEWTONIAN COSMOLOGY
-=0
dA.2 '
207
OX' dA.
for i i= 1, we would contradict n (t) = [R (t)x, 0, 0] unless
for i i= 1 at all times t. Repeating the argument with
n (to) = (0, n 2, 0) implies that n (t) = [0, R (t)y, 0] and
n (to) = (0, 0, n 3) implies that n (t) = [0, 0, R (t)z], and so the
only non-vanishing components of the Riemann tensor are
R~iO (no sum over i). Repeating the argument for an arbitrary initial direction n (t) =R (t)x,
R~lO = 0
(2.2b)
If equations (2.2) are compared with the geodesic equations
(2.3)
then the connection coefficients are seen to be
r;X) = 0<1>/&;,
(2.4)
with all other r p" being zero.
If the connection coefficients (2.4) are inserted into the
standard formula for the Riemann tensor, one obtains
(2.5)
and all other R~p" are zero.
The only non-vanishing component of the Ricci tensor is
RIM) ==R~"o =R:lio = V2 <1>, and so Poisson's equation for the
gravitational potential becomes Cartan's (1923, 1924)
which in tum implies that R~iO = (1/3)Roo. Combining this
with Cartan's equation (2.6) shows that for the deviation
between any two geodesics in the x = constant congruence,
equation (2.8b) can be written as
d 2R
41tG
-=--pR.
dt 2
3
(2.10)
Equation (2.10) can be integrated if we make the
standard assumption about the relation between the density
of matter p (which, by isotropy and homogeneity, can
depend only on the time t) and the scalefactor R (t):
© 1996 RAS, MNRAS 282, 206-210
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
(2.11a)
1996MNRAS.282..206T
208
F. 1. Tipler
which is just the usual equation of continuity. The integral
of (2.lla) is
(2.llb)
Putting (2.11b) into (2.10) and multiplying the result by
dR/dt gives
dR)2
d (d( =
-
equation of geodesic deviation (2.10). This extraneous solution appears because (2.12) was derived by first multiplying
(2.10) by dR/dt. Whatever the value of the constant K, a
Newtonian universe is necessarily an evolving universe.
At any instant of time, the right-hand side of (2.12)
depends only on time, so the left-hand side must also be
only a function of time
3)
(81tG
dR
-3- PoRn R2(t)'
which, if integrated, gives
~(dR)2 _ 81tG
R2 dt
-
3
or, since the proper distance t from the origin of coordinates geodesic to a geodesic at x is R (t)x,
(1
2 [81tGP
dR)2]
t _ Ro -3-- Rd( 0
p()
R2(t)
(2.12)
Equation (2.12) is, of course, Friedmann's equation (1.1).
The constant in the last term of (2.12) is more complicated
than either the k = ± 1 or 0 in the Friedmann equation, or
( - 2E/m) obtained in the incorrect 'Newtonian' equation
(1.2). The physically significant constant is the expression in
parentheses in (2.12), namely
K=r81t~p -(~
:)1.
(2.13)
Both the constant (2.13) and the entire remainder of
equation (2.12) are invariant under the scale change
R (t) ---+aR (t), where a is an arbitrary constant. This invariance is a manifestation of the fact that the change
R (t) ---+aR (t), x---+a -IX does not change proper distance
between points in space, and so is of no physical significance. Exactly the same invariance is present in the
general relativity Friedmann equation for the flat k = 0
case, which is
81tG
-1 (dR)2
=-p(t).
R2 dt
3
(2.14)
The invariance of (2.14) under R(t)---+aR(t) is known
(MTW, p. 722) to be a consequence of the flatness of the
surfaces of homogeneity and isotropy when k = O. The
metric of the spatial sections in Newtonian cosmology is
Euclidean flat space, so it is not surprising that the same
invariance appears in the corresponding first-order equation (2.12).
However, equation (2.12) is more general than (2.14),
and thus allows more qualitatively distinct solutions for
R (t). If p of- 0, then in general relativity dR (t)/dt must never
be zero; if it is expanding at any instant of time, then it must
expand for all future time out of an initial singularity a finite
time in the past. In Newtonian cosmology, if the constant K
in (2.13) is positive, R (t) increases from R (t) = 0 a finite
time in the past to a maximum value, and then recontracts
to R (t) = 0 a finite time in the future; if the constant K = 0,
then R (t) behaves exactly the same as the k = 0 dust-filled
Friedmann universe [R (t) octZ/3]; if the constant K> 0, then
R (t) behaves qualitatively the same as the open Friedmann
universe: R(t)oct as t---+oo.
Equation (2.12) allows the static solution dR/dt=O when
K = 0, but this is an extraneous solution not allowed by the
dt
-=H(t)t(t),
dt
(2.15)
which is Hubble's Law. Hubble's Law in the form (2.15) is,
of course, an exact equation in the relativistic Friedmann
universe also, an immediate consequence of the definition
H(t)=R-IdR/dt of the Hubble parameter in general relativity. The exact validity in both cases is often not appreciated in discussions of Newtonian cosmology. In both cases,
t is the proper distance between two points at one instant of
time. Operationally in Newtonian cosmology, t is the distance one would measure by laying a huge number of metre
sticks end to end at one instant of time, a meaningful procedure in Newtonian mechanics, but of course not in relativity. Connecting (2.15) with the redshift of light requires
additional structure in Newtonian cosmology, since, as
Trautman (1954, 1966) has emphasized, in the Cart an
formalism, there is no theory of light propagation.
3 NEWTONIAN COSMOLOGY ON NONEUCLIDEAN GEOMETRIES
Although the natural geometry for Newtonian cosmology is
flat Euclidean 3-space, other geometries are allowed.
Indeed, since in Newtonian gravity the connection (2.4)
does not arise from a metric, the homogeneity and isotropy
condition (2.9) and hence both the equation of geodesic
deviation (2.10) and its integral (2.14) are also consistent
with either a homogeneous and isotropic 3-sphere, or the
hyperbolic R3 of general relativity. The spatial geometry is
not the determining feature in the dynamics of Newtonian
cosmology as it is in general relativity, a point not clearly
understood in the literature (e.g. Penrose 1968, p. 127
footnote).
To show explicitly that Newtonian cosmology can be
carried out on any topology that admits a homogeneous and
isotropic metric, and that the time evolution is not
dependent on the spatial topology as it is in general relativity, we must first extend the Cartan theory to an arbitrary
Riemannian space. First we replace the flat space gradient
(2.1a) with the standard expression (Chavel 1984, p. 4) for
the gradient in a Riemannian space:
.. a<l>
V<I>-g IJ
axi a
-
(3.1)
j.
Thus we see that, although the connection does not arise
from a metric, nevertheless in a non-Euclidean background
© 1996 RAS, MNRAS 282, 206-210
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.282..206T
Newtonian cosmology revisited
space the gradient of the potential depends on the spatial
metricg ij . Imposing also the requirement that purely spatial
geodesics must be geodesics of the spatial metric connection
gives us the generalization of the geodesic equations
(2.2a,b):
(3.2a)
(3.2b)
209
which in turn implies that R:,;(}=~R{xl' Combined with (3.5),
equation (3.7) yields (2.10), and from this point on, the
argument for curved homogeneous and isotropic spaces is
identical to the flat -space argument. If the spatial topology
is compact, then the Divergence Theorem (Chavel 1984)
applied to (3.5) implies p == O. The implications of this result
will be published elsewhere.
4 THE PHYSICAL SIGNIFICANCE OF
NEWTONIAN COSMOLOGY
where L(X) = sin X, sinh X, X for k = + 1, -1,0 respectively
for the isotropic and homogeneous spaces. For two geodesics normal to these surfaces of homogeneity and isotropy
separated by a coordinate distance !iX, the separation vector
is
The fiducial geodesic which defines the origin of coordinates is, of course, completely arbitrary; any geodesic at
constant x will do as well. So, from the point of view of any
of these fundamental observers, every other such observer is
seen to be moving away according to Hubble's Law. Since
according to the equation of geodesic deviation, R (t) #- 0, it
is often claimed that ' ... every observer's system is inertial,
although different observers may be accelerated relatively
to each other. This is, of course, not permissible according
to the strictly Newtonian system .. .' (Bondi 1961, p. 78). In
fact, in the Cartan formulation of Newtonian gravity, the
fundamental observers are not accelerated. Just as in
general relativity, a curve is said to be accelerated if and
only if u·u p;. #- O. Since all of the fundamental observers are
geodesics of the Newtonian connection (2.4), they are not
accelerated.
Since the relative 'acceleration' arises from geodesic
deviation, it is really a manifestation of the curvature of
space-time, just as it is in general relativity. Conversely,
both in general relativity and Newtonian mechanics, a gravitational field is present if and only if the Riemann tensor is
non-zero (Penrose 1968; MTW). Since the curvature is a
locally defined mathematical object rather than a global
one, and since it can be determined by the local Cartan
equation and the local symmetry conditions, in the Cartan
formulation gravity is no longer a global 'action at a distance'. As Layzer (1954), McCrea (1955a,b) and Bondi
(1961, p. 79) have pointed out, Newtonian gravity regarded
as a force F is not well defined in an infinite system.
Newtonian gravity regarded as local curvature is well
defined in an infinite system. Newtonian cosmology is just as
rigorous as Einsteinian cosmology.
The underlying physical reason behind the identity
between the relativistic Friedmann equation (1.1) and the
Newtonian evolution equation (2.12) will be made manifest
if we compute the equation of geodesic deviation (2.7) in a
Friedmann universe with metric
n=R(t)!iXex'
ds 2= _dt 2+d0'2,
where ex =R- 1 (t) o/OX is the unit basis vector in the X direction. Thus the equation of geodesic deviation (2.7)
becomes
where d0'2 is given by (3.6). For two geodesics normal to the
surfaces of homogeneity and isotropy separated by a coordinate distance !iX, the separation vector is exactly the same
as in curved Newtonian cosmology:
and hence the connection coefficients are
r~o = gij o<l>/at',
(3.3a)
(3.3b)
with all other connection coefficients equalling zero. The
connection rJk is just the usual metric connection for the
Riemannian metric [fi.
The curvature components corresponding to (2.5) are
now
R~jO = OJ r~o + r~j r~ ,
which yields
Roo =R~iO = Oi(gijOj<l»
+ (g ij Oi<l»(Oj.y1iig),
(3.4)
where the identity r~ = OJ.y1iig has been applied.
However, equation (3.4) is just the standard expression
for the Laplacian of a scalar field V2 <1> in a Riemannian
space (Chavel 1984, p. 5). Thus Poisson's equation
V2<1>=4nGp on a Riemannian space is just
Roo=4nGp,
(3.5)
which is, of course, equation (2.6).
Since there are more non-zero components of curvature
in a general Riemannian geometry than there are in (2.5)
(for example, RJkO = - OrJk ot #- 0 in general), the equation of
geodesic deviation will be more complicated than (2.8).
However, the standard arguments still give the 3-metrics
d 20'=R2(t)[dX 2+ I? (x)(d0 2+ sin2 0 dq/)],
(3.6)
(3.7)
One obtains the same equation for the unit vectors eo and
e1>' Homogeneity and isotropy once again imply that
R~iO=Rgoo=Rt1>o,
n=R(t)!iXei'
Thus the equation of geodesic deviation (2.7) becomes
d2R
df + R itxi R (t) = 0,
© 1996 RAS, MNRAS 282, 206-210
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
(4.1)
1996MNRAS.282..206T
210 R J. Tipler
independent of k. (Since Rxw= -R/R, (4.1) is really a
trivial identity.)
If the Friedmann universe is dust-dominated, Gxx=O
implies that Gii =6Rxixi' and so the Einstein equation
Gii =8rcGp gives
4rcG
R XiXi =-3- p.
(4.2)
Combining (4.1) and (4.2) gives
d2R
4rcG
dt2 = --3- pR
(4.3)
for the fully relativistic equation of geodesic deviation.
Equation (4.3) is identical to the Newtonian equation of
geodesic deviation (2.10). Not only are they formally identical, the symbols in the two equations refer to exactly the
same quantities. At the level of the equation of geodesic
deviation, the spatial curvature is irrelevant. Only when
(4.3) is integrated to obtain the Friedmann equation do we
notice the constraint of the spatial curvature on the integration constant (2.13). Even here the conservation equation
(2.11) is exactly the same in the Newtonian and relativistic
cases.
Since from the Cartan viewpoint the effect of gravity is
computed locally, it is possible to do perturbation theory in
Newtonian cosmology in a manner closely analogous to the
way it is done in general relativity. For instance, one could
represent a perturbation as an overdense finite spherically
symmetric region in the overall homogeneous and isotropic
Newtonian universe. In general relativity, one can accomplish this with junction conditions, requiring the spatial
metric and its first derivative to be continuous across the
boundary. In Newtonian cosmology on a Euclidean background, the metric would automatically be continuous, so
we require continuity of the potential cI> and its gradient
gij (ocI>/&j). Requiring continuity of the connection - the
gradient of the potential- (but not its derivative, the curvature) across the spatial boundary would be the analogue of
continuity of first metric derivative. Such junction conditions give, at most, step discontinuities in the curvature,
which are the standard in general relativity.
For Newtonian cosmology in a Riemannian spatial background, the junction conditions would be continuity of the
spatial Riemannian metric and its first spatial derivative the standard relativity junction condition - together with
continuity of the potential and its gradient.
Alternatively, one can simply do perturbation theory
around a Newtonian isotropic and homogeneous background, as is done, for example, in section 15.9 of Weinberg
(1972). Only now, one can be confident that the results will
correctly describe the system intended, small fluctuations of
non-relativistic fluids in an isotropic and homogeneous
cosmology.
One could also generalize Newtonian cosmology to
obtain Newtonian analogues of homogeneous but anisotropic universes, as done, for example, by Heckmann &
Schiicking (1955, 1956). There has been some discussion in
the literature as to how to impose boundary conditions at
infinity in such situations, since the Newtonian potential
does not exist at infinity if the universe is homogeneous. The
connection also diverges at infinity in the Euclidean frame,
but this should not be regarded with alarm, since the affine
connection is not a tensor, and its behaviour in a given
frame is not physically significant. What is significant is the
curvature, and so the boundary conditions at infinity are
appropriately imposed on the curvature.
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© 1996 RAS, MNRAS 282, 206-210
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System