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In article <MATT.92Jul22142345@physics.berkeley.edu> matt@physics.berkeley.edu (Matt Austern) writes:

[..]

>That is: I define the inertial mass of a body as the "m" appearing in
>the equation
>               F = m a,                                        (1)
>and I define the gravitational mass of a body as the "m" appearing
>in the equation
>               F = G m M / r^2.                                (2)
>I believe that these definitions are customary; I don't know what
>other meaning could be attached to the phrases "inertial mass" or
>"gravitational mass."

[..]

That are excellent definitions of inertial and gravitational mass. I'd like
to add a comment on *gravit'l* mass: 

In principle, it is not guaranteed a priori that the field generating 
"active" gravit'l mass *M* which occurs in the gravit'l field eq (Poisson eq)
(U:      Newtonian gravit'l potential, 
 \Delta: Laplace op (assume most of the audience speek TeX))

       \Delta U = - 4 pi G M                                    (2a)

is identical to the interaction perceiving "passive" gravit'l mass *m* 
appearing in the force eq/momentum law

       F = m * \nabla U .                                       (2b)

Identity can be simply established for interactions based on a Lagrange
density on a very general level (in Newtonian continuous media phys, SR
and GR).

(in GR the eq's (2a) and (2b) read, respectively:
      G_ = 8 pi T_         (Einstein eq)                        (2a')
      
      T_mu^nu_;nu = 0      (eq of continuity)                   (2b')
)

This feature is identical in theories involving charges 
(e.g. electromagnetism, electro-weak theory, chromo dynamics, or thelike;
just put M -> Q, m -> q for the e.m. case.)
It results from the simple fact that the gravit'l mass is, from the 
theoretical point of view, the charge corresponding to gravit'l interaction,
i.e. gravit'l charge.

[Side remark 1:]
So if one could prove that any interaction had different active and passive 
charges for at least one particle, one had found that nature cannot be 
described by a Lagrangian based theory. All nice features of the Lagrange,
and Hamilton, theory would no more be present (i.e. automatic conservation
laws, conserved currents, etc.).

[Side remark 2:]
For gravity to be interpreted as a manifestation of a dynamic, non-trivial
geometry (e.g. curvature) there's also *need* for the equivalence of
inertial and (passive) gravit'l mass since (2b) is then replaced by the 
covariant eq of continuity of the energy-momentum tensor (Einstein 
equivalence principle). 
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