Expansion in a Newtonian World 19The composition of neutron stars is not known. The density of their cores is a few
times that of matter in terrestrial nuclei, but they contain far more neutrons than
protons, and they are strongly degenerate, thus we have no similar baryonic matter to
study in the laboratories. They could be dominated byquark matteror by excited forms
of baryons such as hyperons which are unstable particles in terrestrial conditions.
Dark components. Nonbaryonic forms of matter or energy which are invisible in
the electromagnetic spectrum are neutrinos, black holes, dark matter and dark energy.
These components will be dedicated considerable space in later Chapters.
1.7 Expansion in a Newtonian World
In this Section we shall use Newtonian mechanics to derive a cosmology without
recourse to Einstein’s theory. Inversely, this formulation can also be derived from Ein-
stein’s theory in the limit of weak gravitational fields.
A system of massive bodies in an attractive Newtonian potential contracts rather
than expands. The Solar System has contracted to a stable, gravitationally bound con-
figuration from some form of hot gaseous cloud, and the same mechanism is likely
to be true for larger systems such as the Milky Way, and perhaps also for clusters
of galaxies. On yet larger scales the Universe expands, but this does not contradict
Newton’s law of gravitation.
The key question in cosmology is whether the Universe as a whole is a gravitation-
ally bound system in which the expansion will be halted one day. We shall next derive
a condition for this from Newtonian mechanics.
Newtonian Mechanics. Consider a galaxy ofgravitating mass푚G located at a
radius푟from the center of a sphere of mean density휌and mass푀= 4 휋푟^3 휌∕3. The
gravitational potential of the galaxy is
푈=−GM푚G∕푟=−^4
3휋퐺푚G휌푟^2 , (1.26)
where퐺is theNewtonian constantexpressing the strength of the gravitational inter-
action. Thus the galaxy falls towards the center of gravitation, acquiring a radial accel-
eration
푟̈=−GM∕푟^2 =−4
3
휋퐺휌푟. (1.27)
This isNewton’s law of gravitation, usually written in the form
퐹=−GM푚G
푟^2
, (1.28)
where퐹(in old-fashioned parlance) is the force exerted by the mass푀on the mass
푚G. The negative signs in Equations (1.28)–(1.30) express the attractive nature of grav-
itation: bodies are forced to move in the direction of decreasing푟.