Re: Poynting-Robertson effect/Perverse Gravity

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

In explaining the Poynting-Robertson effect Leigh hit on a peculiarity of
gravitational interactions.
> ....
> The result is that the pressure of sunlight works to decrease the orbiting
> Earth's energy, and that, perversely, means it speeds up as the orbit
> shrinks!

This is indeed perverse.  It is an example of the incompatibility of the
gravitational interaction with thermodynamics (or at least with the existence
of the thermodynamic limit and with the state of thermodynamic equilibrium).
Normally, systems subject to frictional dissipation cause the relative
motion between the objects frictionally interacting to slow down and both to
approach a common velocity as a stable manifestation of the 2nd law.  In the
case of a gravitationally dominated system the frictional dissipation causes
an unstable increase in the relative motion.  This happens every time a low
orbiting object reenters the earth's atmosphere.  Systems whose main
interaction between internal degrees of freedom is gravitational do not form
usual thermodyamic systems, are unstable, and cannot equilibrate
thermodynamically.  It is the tendency of gravitation to fight off
equilibration that helps make the universe such an exciting place.

An example of this incompatibility (between gravitation and thermal
equilibrium) is in the big bang itself.  Early in the cosmic expansion the
universe was essentially homogeneous and in thermal equilibrium at a very
uniform temperature throughout.  As time went on the unstable nature of
gravitation vis-a-vis thermodynamics caused the system (universe) to
spontaneously fall out of equilbrium as tiny matter inhomogeneities unstably
accumulated.  Normal thermodynamic systems stay in equilibrium once they are
in that state.  [The photon degrees of freedom are still essentially in
equilibrium (as the Cosmic Background Radiation), but the matter is most
decidely not in equilibrium any longer.  (We are a long way off from the
universe's heat death regarding the matter degrees of freedom.)]

Stable thermodynamic systems have an entropy function which is convex wrt
all extensive macroscopic parameters.  This results is such things as the
specific heat that the compressibility of the system being positive.  A
system with a negative specific heat is unstable against thermal fluctuations
and a system with a negative compressibility is unstable against mechanical
flutuations.  A system dominated by gravity doesn't have such a convex
entropy.  For example consider a black hole interacting with its environment
via Hawking radiation.  Such a system is characterized by a negative
specific heat.  That is why black holes must either evaporate (ending in an
explosion) or must gain mass from the environment's radiation flux.  A black
hole is in an unstable equilibrium when it Hawking temperature matches the 
temperature of the environment.  Normally such a temperature match would
result in a stable equilibrium.  The black hole has a negative specific heat
because if it absorbs energy (i.e. heat) from its environment its mass
increases and this increases its proper horizon which decreases the invariant
surface gravity at the horizon which results in a *decrease* in its Hawking
temperature.  This causes the hole to cut back on its outbound radiant flux
which causes an increase in the net absorption rate from the environment.
Similarly, if the hole emits a net thermal energy flux to the environment then
its temperature increases causing it to emit still more energy.

One of the main reasons that gravity is incompatible with thermodyanmic
equilibrium is that gravity is incompatible with the concept of the so-called
thermodynamic limit.  Normally a thermodynamic system has a thermodynamic
limit which says that as the system is scaled up in size in such a way that
the mass increases proportional to the volume so that the density remains
constant, that the relative fluctuations in the thermodynamic quantities
become insignificant and vanish in the limit of an infinite system with an
infinite number of degrees of freedom.  The reason that normal thermodyanmic
objects do not display large statistical flutuations in their properties is
because they are so macroscopic (i.e. close to the thermodynamic limit) that
the central limit theorem insures "good statistics" and we can treat
statistically emergent properties such as temperature, pressure, chemical
potential, etc. as just parameters with thermodynamic state functions
connecting them and not worry about the underlying statistics of the atomic
fluctuations.  In order for the thermodynamic limit to exist all bulk
conserved quantities must be extensive (i.e. they must scale proportional to
the "size" of the system).  Normal (nongravitational) interactions between
degrees of freedom satisfy this constraint (because normal interactions
confine their sources on a large length scale).  Thus the energy of
interaction is effectively finite-ranged and scales proportional to the
number of particles (i.e. mass) present.  With gravity this is not the case.
Nothing screens gravity and there is no negative mass to cancel out long-
ranged gravitational interactions and this makes the gravitational energy of
interaction scale proportional to the *square of* the mass present in the
system.  On large enough length scales the gravitational attractions dominate
over everything and keep growing too fast for the thermodynamic limit to
exist.  Without the thermodynamic limit the statistical mechanics does not
boil down to stable thermodyanmics and thermal equilibrium does not obtain.

David Bowman
dbowman@gtc.georgetown.ky.us