A simple interpretation of Mach’s principle implies that an open universe would undergo an accelerating expansion.

Ian C. McKay

University of Glasgow, Macgregor Building, Glasgow G11 6NT, UK.

Evidence emerging from the Supernova Cosmology Project ^{1, 2}
suggests that the Universe is open and that its expansion is accelerating. The
standard explanation depends on some unknown force of repulsion represented by
a positive cosmological constant *L*. Here, it is shown that a simple interpretation of Mach’s principle
would lead us to expect the expansion to accelerate even if the hypothetical
force of repulsion does not exist.

Mach ^{3, 4} believed, partly on
philosophical grounds, that the inertia of a body must depend on the influence
of all the other matter in the Universe. This belief arose because acceleration
of a body is defined relative to other bodies and so resistance to acceleration
can have no meaning or existence in the absence of other matter. Sciama ^{5}
postulated that this influence of other matter obeys an inverse first-power
law: in other words, the inertial mass that a body is endowed with by the
influence of another body is inversely proportional to the distance between the
two bodies. As other authors ^{6, 7} have explained, this appears to be
reasonably compatible with observation, whereas an inverse square law could
easily be refuted by experiment.

These postulates imply that when a homogeneous,
isotropic universe expands (or contracts) uniformly, and every pair of bodies
move apart (or together), the inertial mass of all bodies will change, being
inversely proportional to the cosmic scale factor (loosely, radius) *R* of the universe. Hence the kinetic
energy of expansion or contraction, being proportional to inertial mass
multiplied by the square of velocity, can be written as _{}, where *b* is a
positive constant and *t* is time.

If the potential energy of the universe is measured
on a scale in which zero is arbitrarily chosen to represent the infinitely
expanded state, then on this scale the potential energy will be negative and
inversely proportional to *R*, and may
be written as _{}, where *a* is a
positive constant.

Hence the total mechanical energy *E* of expansion of the universe may be
written as _{}, and will be zero for any universe that has only just enough
energy to continue its expansion indefinitely.

If we solve this differential equation to obtain *R* as a function of *t* we get_{}and so the rate of expansion is _{}and its acceleration is _{}.

In words, this means that if the total energy is positive, i.e. if we have an open universe, then the expansion will accelerate. If the total energy is zero, i.e. we have a flat universe, then the expansion will occur at a constant rate.

This renewed interest in Mach’s principle has obviously been provoked by evidence that the Universe is indeed open and that its expansion has accelerated. Clearly the above equations are only approximations based on instantaneous action at a distance. Attempts to base the equations on the Robertson-Walker metric have so far yielded intractable models. But the formulae as they stand suffice to indicate that if the accelerating expansion is accepted then Mach’s ideas may yet save us from having to reinstate the cosmological constant.

References

1.
Perlmutter, S. *et al.*, 1999. Measurements of Omega and Lambda from 42
High-Redshift Supernovae. Astrophys. J., **517**:
565-586.

2.
Glanz, J. 1998. Cosmic Motion
Revealed. Science **282**: 2156-2157.

3.
Mach, E. (1872). *Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit.
Prag: Calve’sche Buchhandlung*. Translated as “*History and Root of the Principle of the Conservation of Energy. The
Science of Mechanics*”, 6^{th} Edition, 1904.

4.
Mach, E. (1912). *Die Mechanik in ihrer Entwicklung historisch kritisch Dargestellet*.
Leipzig: Broachans.

5. Sciama, D. W. (1953). On the Origin of Inertia. Mon. Not. Roy. Ast. Soc., 113: 34-42.

6.
Berry, M. V. (1989). *Principles of Cosmology and Gravitation*,
pp 37-39. Bristol and Philadelphia: Hilger.

7.
Woodward, J. F., 1990. A new
experimental approach to Mach’s principle and relativistic gravitation.
Foundations of Physics Letters, **3**:
497-505.