**Flaws in a Young-earth Cooling
Mechanism**

by

George Murphy and Glenn Morton

The main body of
this paper is in another page.

This page contains a supplementary APPENDIX

(it wasn't in the paper published
by NCSE)

that
is
described
in
the
published paper:

One of us has developed* a
simple classical model for a harmonic oscillator *(like a particle oscillating
in a crystal), and** in this model the particle does not lose energy to the
cosmic
expansion**. While other force terms could be used in the equation of
motion to give different results, the one used here seems to be the simplest
and most
natural generalization of the ordinary linear restoring force. **The fact
that energy is not lost here suggests that Humphreys's qualitative argument
is incorrect**. {emphasis added}

**MODELS OF A CLASSICAL OSCILLATOR
IN AN EXPANDING UNIVERSE**

ds^{2}
= - dt^{2} + R^{2}(t)[dx^{2} + dy^{2} + dz^{2}]

with R the scale factor and c
= 1. The equation of motion for a
particle is

d^{2}x^{i}/ds^{2} + 2{_{j }^{i}_{ o}}(dx^{j}/ds)(dt/ds) =
F^{i}/m

where F^{i} is the
force due to non-gravitational interactions. For sufficiently slow motions and weak fields we can replace
the proper time s by coordinate time.
The equation for motion in the x direction (with F the x-component of
force) is

d^{2}x/dt^{2} + 2H dx/dt = F/m ,

where H = (1/R)(dR/dt) is the
Hubble parameter. For an
inflationary de Sitter state, which is of the most interest in several
situations, R ~ exp(Ht) with H a constant.

For
a free particle (F = 0) we have dv/dt + 2(dR/dt)v/R = 0, with v = dx/dt. This gives v = C/R^{2} with C a
constant. x, however, is only a
coordinate, and actual spatial distances are given by X where dX = Rdx. We can write V = dX/dt = Rv so V = C/R.

For both v and V we see the
"stopping dead" phenomenon noted by Tolman and Schroedinger. Since our coordinates are co-moving,
this means that all free particles are eventually swept along with the cosmic
expansion (as long as R is increasing).

What
about a particle that is not free – i.e., for which F is not zero? How might we represent a harmonic
oscillator, which provides a useful model of many systems? There are a couple of possibilities
which are easy to deal with mathematically, though neither of them is
perfect. But they are, of course,
only mathematical models.

First,
we can write F = -mw^{2}x so that the restoring force is proportional
to the __coordinate__ x. Then
we have

d^{2}x/dt^{2} + 2H dx/dt + w^{2}x = 0

which, when H is constant, is
the standard equation for a damped oscillator. The general solution for the underdamped case (w > H)

x
= A exp(-Ht) cos(Wt + b)

where the damped frequency is
W = (w^{2} – H^{2})^{1/2} and A and b are
constants.

But
again we must remember that displacements are actually given by X, not x.

We have

dX/dt
= exp(Ht) dx/dt = -HA cos(Wt + b) – WA sin(Wt + b)

which can be integrated to
give (after some trigonometric manipulations)

X
= A(1 + H^{2}/W^{2})^{1/2} cos(Wt + b + tan^{-1}H/W) .

Cosmic expansion thus results
in an undamped oscillation with a lower frequency and larger amplitude than
would be the case with no expansion.
The energy is given by

E
= (1/2)m(dX/dt)_{max}^{2
},

which works out to (1/2)mW^{2}(1
+ H^{2}/W^{2})A^{2} = (1/2)mw^{2}A^{2} .
Thus the energy is not only constant but has the same value as in a
static universe.

The
restoring force in this model is proportional to the coordinate x, and one
might argue that it would be better to write it as F = -mw^{2}X rather
than F = -mw^{2}x. In this
case we would have

d^{2}x/dt^{2} + 2H dx/dt + w^{2}X= 0

as the equation of
motion. We can put this entirely
in terms of X by writing

dX/dt
= R dx/dt , d^{2}X/dt^{2}
= R d^{2}x/dt^{2} + (dR/dt)(dx/dt)

to get

dx/dt
= (dX/dt)/R and d^{2}x/dt^{2 }= (d^{2}X/dt^{2})/R
– H(dX/dt)/R .

Substitution then gives

d^{2}X/dt^{2
}+ H dX/dt + w^{2}RX = 0.

We now have damped
oscillations (though with half the damping constant of the previous model) and
an effective frequency R^{1/2}w that increases with time. If this
increase is slow (as it will be if the vibrational period is much greater than
the Hubble time 1/H) we can write
approximately

X
= D exp(-Ht/2) cos(R^{1/2}wt + h)

with D and h constants. This is no longer a simple sinusoidal
oscillation, but to a first approximation the energy will be proportional to
the square of the amplitude, D^{2}exp(-Ht), multiplied by the square of
the effective frequency, Rw^{2}.
Since R ~ exp(Ht) the time dependence cancels out and the energy will be
constant over many oscillation periods.

We
have assumed here that H is a constant, a situation to which our present
universe seems to be approaching and which would hold in any inflationary
phase. In this case both our
models seem to indicate that there is no dissipation of energy of an oscillator
due to the expansion. The
situation will be different if R(t) has a different form. If R ~ t^{1/2}, as for a
radiation filled universe with no cosmological constant, then the oscillator
equation for our second model can be solved in terms of Bessel functions, and
use of the asymptotic form of those functions indicates that there would be a
decrease in energy.

REFERENCE:

Baumgardner JR, 2000. Distribution of radioactive isotopes in the earth. In: Vardiman L, Snelling AA, Chafin E. editors.