# Q & A: Gravitational Mass

Q:
What exactly is gravitational mass and how many meanings does it have (in general relativity)?
Los Cocos, Cordoba, Argentine
A:

Let me warm up with something I understand better than
General Relativity. In Newtonian gravity, the force is given by

F(gravity) = G*(M1)*(M2)/R^2,

where G is some universal constant, M1 and M2 are the masses of two
objects exerting the force on each other, and R is the distance
between their centers. This force is present between any two objects,
but if the objects are small or a long way apart, the force is very
small.

This law doesn't mean much until you say how the forces
show up as something you can see, which Newton described by:

F=ma.

That says that an object of inertial mass "m" will accelerate at the
rate of "a," when it feels a force "F." The mass, "m" here is called
the "inertial mass."

Newton knew from Galileo's work that any two objects near the same
place and feeling no forces but gravity (from something else, like
the Earth) fall at the same rate- they have the same "a": a_1=a_2
Put that fact together with our two other equations and you find:

m_1/M_1 = m_2/M_2.

That means that gravitational mass and inertial mass are proportional
to each other: if one object has twice the gravitational mass, it
also has twice the inertial mass.

Since nobody had told Newton what value to use for G (it wasn't
printed in some book!) he was free to DEFINE G so that you can use
the same mass units for inertial and gravitational mass. It would be
a pain to say that one inertial gram is 8.73 gravitational grams or
something.

OK, now I'll try to sort of get to your question. Einstein
elevated the equivalence of the gravitational and inertial masses to
a general principle. One result is that you can't tell (from inside)
if you're in a box which isn't accelerating, but are being pulled
down by gravity and held up by some other force, or if there's no
gravity but just some force pushing the box up. When you combine that
with some of the peculiarities that Special Relativity predicts must
be found if you measure things from an accelerating viewpoint, you
find that in the presence of gravity, all sorts of weird stuff must
occur. One example is that identical clocks run at different rates in
the attic and the basement- and this has been confirmed. If there
were some difference between the inertial and gravitational masses,
the whole logical structure of General relativity, with all its
beautiful true predictions, would fall apart.

Mike

(published on 10/22/2007)