Q:

What exactly is gravitational mass and how many meanings does it have (in general relativity)?

- Adrian Moscani (age 26)

Los Cocos, Cordoba, Argentine

- Adrian Moscani (age 26)

Los Cocos, Cordoba, Argentine

A:

Adrian - Tough question!

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

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)*