APPENDIX:
Sign of Roche Acceleration Doesn’t Depend on Particle Size or Mass,
and the Equivalence Principle is Falsified

Figure 1: Particles of radius Δr, density ρ, and a distance R from a mass M

a gravitational field is provided by a mass M, a distance R to the left (not visible)

2 particles of density ρ, and (small) mass m = 4πρ(Δr)³/3

accelerations due to mass M (signed: - = left, + = right):

       g_lM = g_mM - Δr∂g/∂r                g_mM             g_rM = g_mM + Δr∂g/∂r

  ∂g/∂r > 0: g is negative and approaches 0 with increasing r

Roche acceleration apart due to mass M:

       g_rM - g_lM = 2Δr∂g/∂r

gravitational accelerations due to mass of other particle:

       g_lm = 4Gπρ(Δr)³/(3(2Δr)²)= GπρΔr/3

       g_rm = -4Gπρ(Δr)³/(3(2Δr)²)= -GπρΔr/3

acceleration together due to mass of other particle:

        g_lm - g_rm = 2GπρΔr/3

total Roche acceleration apart (if positive):

        2Δr∂g/∂r - 2GπρΔr/3 = 2(∂g/∂r - Gπρ/3)Δr

NOTE: the Δr appearing only as a factor means that the sign of the Roche acceleration apart does not depend on the size of the particles, or on their mass as volume times density; i.e. the particles cannot be made “small enough” to make the Roche acceleration apart “go away”. The Roche acceleration apart may become gedanken “infinitesimal”, but it will always be gedanken greater than zero if the density ρ is close enough to zero, and less than zero (i.e. the particles will accelerate together) if ρ is large enough.

ALSO NOTE: for any value of ∂g/∂r (i.e. > 0), there exist values of ρ that will make the particles accelerate together or apart: if ρ < (3∂g/∂r)/Gπ then the particles will accelerate apart; if ρ > (3∂g/∂r)/Gπ then the particles will stick together here since already in contact. It is obvious that if they are further apart, the g values where the particles are may differ even more, and their mutual gravitational attraction will be even less, increasing the acceleration apart, or allowing a larger density for the same acceleration.

 Falsification of the equivalence principle:

By varying the gedanken density of the 2 particles we can get them to gedanken accelerate either apart or together (even if only gedanken “infinitesimally”) IF our gedanken elevator is in a real world-type gravitational field (as opposed to an unrealistically uniform “uniform gravitational field” — purely a Gedanken Convenience Entity — and even if it is accelerating). We cannot get any gedanken-noticeable change by varying their gedanken density IF our gedanken elevator (accelerating or not) is outside of a gravitational field.

To say that we have gedanken made the particle size so small that the acceleration is “infinitesimal” (and this can be done), and that therefore we have an acceleration that is comparable to the absolute zero acceleration that gedanken pertains in an elevator in a gravity free region is not a competent argument. We gedanken that there exists this difference, that 1) in the case of the elevator in a gravity free region the particles’ acceleration apart-together will always be absolute zero for any 2 elevator-relative-motionless particles in any positions in the elevator, and will remain so indefinitely, and that 2) in the case of an elevator in a gravitational field, we have to make the elevator “infinitesimal” in both space and time, and to make the particles “infinitesimal” and to keep them no more than “infinitesimally” distant from each other, as well. This is already a serious difference between them. And consider:

• For every (“infinitesimal”) ε > 0, we can find an x such that
  (for some constant c):
  f(x) = c + e^x < c + ε
  (and for free we even get all the derivatives
  f'(x) = f''(x) = f'''(x) = ... = e^x < ε),
  but this does not mean that f(x) is not exponentially increasing,
  that it does not increase arbitrarily and arbitrarily quickly as x becomes arbitrarily greater than ln(ε).

So, in a separate gedanken experiment, in a finite size (non-“infinitesimal”) gedanken elevator and in a finite time period, we can gedanken larger particles with appropriate densities that will accelerate apart “indefinitely” (vaguely like our f(x) = e^x) or stick or accelerate together till they make contact — depending only on particle density, on ∂g/∂r, and on Newton’s universal gravitational constant, G — in a way that they would never do in our gedanken elevator in a gedanken gravity free region.

This effectively falsifies the equivalence principle, at least in offering what is obviously a better gedanken approximation to reality.

Einstein should never have missed this “relatively” great... oversight.