Stars and the holographic upper bound on gravitational action
Stars and the holographic upper bound on gravitational action
Scott Funkhouser
National Oceanic and Atmospheric Administration, 2234 South Hobson Ave.,
Charleston, SC, 29405-2413
ABSTRACT
The holographic upper bound on entropy is applied to the gravitational action associated with the
non-relativistic contraction of a nebula.  A critical radius is identified, as a function of the initial radius and
mass, for which the number of bits associated with the action would equal the maximum number of bits
allowed to the body.  The gravitational action of a typical star approximately saturates the holographic
bound, perhaps suggesting a physical link between holographic principles and astrophysical processes.
Consider an isolated, cold, spherical nebula of gas whose initial radius is r0 and
whose mass is M.  The nebula may have some non-vanishing, uniform rotational angular
momentum whose magnitude is J.  Let the body be subject only to internal forces, and let
the mutual gravitational attraction among the constituent particles be, initially, the only
significant force.  (At some point the contraction may end when another force or effective
force becomes significant with respect to the gravitational binding force.)  At some initial
time t=0 let the body begin to contract gravitationally, and let the rotational angular
momentum and mass of the body remain constant.  There is associated with the
contraction a characteristic quantity of action.  In this present work the characteristic
action of a contracting nebula is examined in scenarios where relativistic effects are not
important.  (Numerical or geometrical coefficients of order near 100 are also unimportant
here.)
Consider times t in which the characteristic radius r of the nebula is much smaller
than r0.  The magnitude U of the gravitational potential energy of the contracting body
will have changed by an amount
ΔU ~ GM
, (1)
where G is the Newtonian gravitational coupling.  The elapsed time Δt=t between the
beginning of the contraction and any time t when 
r << r0  is approximately
, (2)
where 
g ~ GM /r0
2 is the initial gravitational acceleration.  The action A associated with
the contraction is well represented by ΔUΔt, and it follows from (1) and (2) that A is
given by
GM 3r0
, (3)
r << r0 .
Whereas principles of quantum mechanics establish a minimum possible scale of
action, the thermodynamics of black holes and holographic considerations lead to a
maximum.  The holographic upper bound on the number N(R) of degrees of freedom that
the contents of a sphere of radius R could occupy is given by
N(R) ~
2 , (4)
where lP is the Planck length.  The maximum entropy of the system and the maximum
number of bits of information that could be registered by the system are both proportional
to N(R).  Since the minimum possible quantum of action is the Planck quantum h, there
must be associated with any quantity B of action at least B /h registered bits.  The
maximum action a(R) allowed to the contents of a sphere of radius R is therefore given
a(R) ~ hN(R) ~
R2c 3
. (5)
The holographic limit on action leads to a critical condition for the contracting
nebula.  As the body contracts and r decreases, the action (3) increases while the upper
bound (5) decreases.  The gravitational action of the body would be equal to the
maximum action allowed to the body if the radius r were to become as small as the
critical radius rc that satisfies the condition
GM 3r0
, (6)
and thus
. (7)
The critical action Ac of the body when r=rc is given by
Ac ~ Mr0c . (8)
It is instructive to note that the right side of (8) is the holographic upper bound on the
rotational angular momentum of a mass M  and radius r0 [2].  If rotational angular
momentum is conserved during contraction then the Mr0c represents the maximum
possible rotational angular momentum of the body in its final state as well.
As a nebula of mass M  and initial radius r0 contracts there must be some
mechanism that prevents the body from contracting to radii smaller than r=rc.  Note that
there is no known mechanism by which holographic principles alone may generate some
reaction by which a violation of the holographic bound is prevented. The final,
characteristic radius of a contracting body such as the nebula in the present scenario is
determined, in general, by one of three primary processes:  (a) The body becomes
virialized, whereby further contraction is prevented by centrifugal tendencies.  (b) The
density of the body becomes of the order the characteristic atomic density, whereby
contraction is halted by inter-atomic repulsion.  (c) The thermal radiation pressure within
the body becomes as great as the gravitational pressure.  If violation of the holographic
upper bound is to be prevented by such mechanisms then there must be some implicit
physical connections among astrophysical processes, the physical parameters of
astronomical structures and holographic principles.
Consider a star whose mass is of the order 1030kg that forms from the
gravitational contraction of a nebula whose initial characteristic radius is of the order
1013m, or hundreds of astronomical units.  The critical radius at which the action
associated with the gravitational contraction will saturate the holographic upper bound is
of the order 108m, which is of the order the characteristic radius of a solar-mass star.
Stars may therefore constitute a class of bodies for which the holographic upper bound is
nearly saturated by the gravitational action.  Typical stars are also remarkable since they
represent the unique situation in which all three of the processes (a) – (c), listed above,
are comparably significant in preventing further gravitational contraction.
References:
[1] Lloyd, S., Nature, 406, 1047-1054 (2000)
[2] Hod, S., Phys. Rev. D 61, 024018 (2000)
