Flaws in a Young-earth Cooling Mechanism


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:





            In a space-time with uniform flat expanding space sections (as ours now seems to be) the line element in co-moving coordinates is


                                    ds2 = - dt2 + R2(t)[dx2 + dy2 + dz2]


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


                        d2xi/ds2  + 2{j  i o}(dxj/ds)(dt/ds) = Fi/m 


where Fi 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


                        d2x/dt2  + 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/R2 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 = -mw2x so that the restoring force is proportional to the coordinate x.  Then we have


                                    d2x/dt2  + 2H dx/dt + w2x = 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 = (w2 – H2)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 + H2/W2)1/2 cos(Wt + b + tan-1H/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)max2  ,


which works out to (1/2)mW2(1 + H2/W2)A2 = (1/2)mw2A2  .  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 = -mw2X rather than F = -mw2x.  In this case we would have


                        d2x/dt2  + 2H dx/dt + w2X= 0


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


            dX/dt = R dx/dt   , d2X/dt2 = R d2x/dt2 + (dR/dt)(dx/dt)


to get


            dx/dt = (dX/dt)/R and d2x/dt2 = (d2X/dt2)/R – H(dX/dt)/R .


Substitution then gives


            d2X/dt2 + H dX/dt + w2RX = 0.


We now have damped oscillations (though with half the damping constant of the previous model) and an effective frequency R1/2w 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(R1/2wt + 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, D2exp(-Ht), multiplied by the square of the effective frequency, Rw2.  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 ~ t1/2, 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. 


Baumgardner JR, 2000.  Distribution of radioactive isotopes in the earth.  In: Vardiman L, Snelling AA, Chafin E. editors. Radioisotopes and the Age of the Earth. El Cajon (CA): Institute for Creation Research and Creation Research Society.