Leigh's latest physical argument for calculating the negative
gravitational energy of a finite distribution of spherically symmetric
mass shells is a big improvement over his previous attempt. Although it
is less general than the argument I had in mind, its main virtues are:
1. The finite localized distribution of sources allows one to
legitimately ignore any surface-at-infinity contribution which vanishes
for any localized finite distribution of sources. This is because the
integrand of the surface integral involves the product of the potential
and the field strength. This product has the asymptotic monopole
behavior of ~(1/r)*(1/r^2) = 1/r^3 for a finite localized source
distribution. The area of a closed surface which is receding outward to
infinity in the limit grows as ~r^2. Thus the integral vanishes as
~(r^2)*(1/r^3) = 1/r --> 0 as r -->[infinity].
2. The continuous distribution of the mass sources smears the
singularites which would be present for a system of point masses which
are of necessity always located at points of infinite field strength and
(negative) infinite potential. The point masses have the infinite self-
energy problems that need dealing with. A continuous source distribution
allows the potential and field strength to remain bounded everywhere and
the self-energy contributions automatically vanish.
My main objection with Leigh's argument is that it assumes, without
justification, that *all* of the negative gravitational potential energy
is associated with the pure-field energy density, and none of it is
associated with any interaction energy between the mass sources and the
potential. What I had in mind was a potential energy function of the
form:
where u_g = + (|g|^2)/(8*[pi]*G) is the pure field energy density
(with g = - grad([phi]) being the field strength). Here [rho] is
the mass density and [phi] is the gravitational potential function both
of which are evaluated at the location r corresponding to the integration
measure. The first term in the integrand above represents the energy of
interaction between the sources and the potential field. The second term
represents the energy of the pure field by itself. The negative of the
above integral is added to the kinetic energy of the matter sources to
get the whole Lagrangian for the {matter + field} system.
According to Hamilton's principle the action is extremized to get the
equations that determine the system's behavior. Making the action
stationary w.r.t. the matter degrees of freedom yields the usual
Euler-Lagrange/Newton's equations of motion for the matter, i.e. for
each (pointlike) i-th particle we have m_i*a_i = F_i where m_i is the
mass of the i-th particle, a_i is that particle's acceleration, and the
force F_i on that particle is given by F_i = m_i*g_i where
g_i = -grad([phi](r_i)) is the field strength at the location of that
particle. If such a point particle mass distribution is desired, it can
be modeled with Dirac delta functions as:
[rho](r) = sum{i; (m_i)*delta(r - r_i)}. Otherwise a continuous [rho]
function can be asssumed.
Now, again, making the action stationary, this time w.r.t. the potential
function [phi] yields the equation describing the behavior of the
gravitational potential field itself. Since the Lagrangian involves no
time derivative of [phi] (reflecting the nonrelativistic assumed
instantaneous action-at-a-distance approximation) there is no canonical
momentum conjugate to the [phi] field, and extremizing the action w.r.t.
[phi] results in the instantaneous-in-time Euler-Lagrange equation which
determines [phi] from the source configuration:
0 = - [rho] + (div(grad(phi)))/(4*[pi]*G) .
Rearranging this gives the usual Poisson equation:
div(grad(phi)) = 4*[pi]*G*[rho]
whose solution (for a localized mass distribution) is:
This solution has the potential field respond globally instantly to
changes in the configuration of sources. *If* (as in relativistically
correct treatments) the Lagrangian had included some time deriviative
terms (besides gradient terms) of the fields then the Euler-Lagrange
equation for the field would be a 2nd order (in time) wave equation
describing a mediating role for the field as a time-delayed wave-
propagating medium for which influences do not travel faster than c.
We can use a vector identity and the E-L equation to simplify our
expression for U_tot above. Note:
In the middle line we used the identity (for arbitrary scalar field a and
vector field b):
grad(a)(dot)b == div(a*b) - a*div(b) .
And in the last line for u_g we used g = - grad([phi]) and the Euler-
Lagrange/Poisson equation that [phi] obeys. Using this last expression
for u_g in the integrand for U_tot gives:
Here the -[phi]*[rho])/2 term in u_g cancels against 1/2 of the
interaction energy term. The last divergence term above can be converted
to a surface integral by the divergence theorem. If the mass
distribution is finite and localized then this surface integral vanishes
if it is evaluated at infinity. Thus, the entire total potential energy
of the system is 1/2 of the (negative) interaction energy and is the
negative of the (positive) pure-field energy.
Now let's use this in an example similar to the one Leigh suggested
except I'll choose just one thin spherical shell of matter that initially
has an infinite radius and which is shrunk down to a radius a. Initially
the mass infinitely dispersed at infinity is chosen to have zero
potential energy. Once the shell is shrunk down to radius a the potential
outside is [phi](r) = - G*M/r when r>= a, and - G*M/a when r < a.
Here the pure field energy density is u_g = + G*(M^2)/(8*[pi]*r^4) for
r >=a, and is zero for r < a. If we integrate this energy density over
all space we get: U_field = + G*(M^2)/(2*a). If we integrate the
interaction energy we get:
U_interact = M*[phi](a) = M*(-G*M/a) = - G*(M^2)/a. Adding these two
This result is just the usual potential energy for the shell of
matter evaluated by assembling it piece by piece from infinity against
the attractive Newtonian force law acting between the pieces.
Since we have restricted ourselves to a finite localized and continuous
mass distribution we did not have to deal with either the surface
integral at infinity nor any self-energy terms.
The upshot here is that the pure-field energy is positive and the
interaction energy (between the sources and the potential field) is
negative and the field energy always cancels against half of the
interaction energy. What Leigh has done has attribute all of the
(negative total) gravitational potential energy to the pure field and
ignore any role for the interaction energy.
In electrostatics the same phenomenon occurs, except the sign of the
potential relative to that of the sources is opposite from that of the
gravitational case, and this makes the interaction energy positive for
a collection of like-signed charges, and the contribution from the
pure-field terms negative (after the total divergence surface term is
removed). The reason for the sign change between the gravitational
and E&M cases is that in the electostatic case the pure [phi] field
term in the energy enters the Lagrangian (but not the Hamiltonian) with
the opposite sign as in the gravitational case. This is because in the
E&M case the [phi] field in not a scalar nor a time-time component of a
2-tensor, but the time-component of a *vector* field, and as such, the
|grad([phi])|^2 term in the energy is not really part of a potential
energy-like term that switches signs between the Hamiltonian and the
Lagrangian, but is part of a kinetic energy-like term that has the
*same* sign for the Lagrangian and the Hamiltonian. If the
quadratic-in-field pure-field term enters the Lagrangian with the
opposite sign as before, but the linear-in-field interaction term enters
the Lagrangian with the same sign as before, then the E-L equation that
results from Hamilton's principle causes a sign change between the
potential field and its sources relative to the other case.
In general, if an interaction between sources is mediated by an
even-integer-spin field (e.g. scalar or 2-tensor potential field--like
gravity which has the |grad([phi])|^2 term keep the same sign between
the Hamiltonian and the Lagrangian) then it causes like-signed sources
to *attract* each other, and if it is an odd-integer-spin field (e.g.
vector potential field like E&M which has the |grad([phi])|^2 term
switch signs between the Lagrangian and the Hamiltonian) then it causes
like-signed sources to *repel* each other.