Inertial mass and gravitational mass

[Luigi Fortunati]

In my animation
<https://www.geogebra.org/m/kmssvz3t>
there is the mass of the ocean stationary in the gravitational field
and, therefore, it is a gravitational mass.

Its surface is certainly a spherical cap.

By clicking on the appropriate button, you switch to inertial mass,
imagining that you can eliminate gravity to replace it with external
forces that accelerate the ocean upwards.

In this case, however, the surface of the ocean would be flat and not
curved.

How does one reconcile this different conformation of the gravitational
mass with respect to the inertial one, if the equivalence principle
states that an observer is unable to distinguish an acceleration due to
an external force from that generated by a gravitational field?

Could it be due to the fact that external forces neither converge nor
diverge, while the forces of the gravitational field all converge
towards the center of gravity?

[[Mod. note -- The resolution requires a correct statement of the
equivalence principle (EP), namely, that a observer making only "local"
measurements (i.e., ones confined to a laboratory, not "looking out the
window" at the outside world) is unable to distinguish between
(a) the entire laboratory being accelerated with some (constant)
acceleration with respect to an inertial reference frame, and
(b) the entire laboratory being in a *uniform* gravitational field "g".
The qualifiers "local" and "uniform" are important here!

As we've discussed before in this newsgroup, real-world gravitational
fields are invariably non-uniform, so we need to introduce a tolerance
for how much non-uniformity we're willing to tolerate, i.e., for how
accurately we want to measure the accelerations and/or the "g". That
tolerance then sets an upper limit on the size of our laboratory, and
on the duration of our measurements, such that within that limit we
can approximate the gravitational field as uniform to within our
measurement tolerance.

In your animation you've chosen a "laboratory" large enough that the
gravitational field is strongly non-uniform, so it's not surprising
that this (non-uniform) gravitational field is readily disginguishable
from any non-gravitational acceleration.
-- jt]]

[Luigi Fortunati]

Post by Luigi Fortunati
In my animation
<https://www.geogebra.org/m/kmssvz3t>
there is the mass of the ocean stationary in the gravitational field
and, therefore, it is a gravitational mass.
Its surface is certainly a spherical cap.
By clicking on the appropriate button, you switch to inertial mass,
imagining that you can eliminate gravity to replace it with external
forces that accelerate the ocean upwards.
In this case, however, the surface of the ocean would be flat and not
curved.
How does one reconcile this different conformation of the gravitational
mass with respect to the inertial one, if the equivalence principle
states that an observer is unable to distinguish an acceleration due to
an external force from that generated by a gravitational field?
Could it be due to the fact that external forces neither converge nor
diverge, while the forces of the gravitational field all converge
towards the center of gravity?
[[Mod. note -- The resolution requires a correct statement of the
equivalence principle (EP), namely, that a observer making only "local"
measurements (i.e., ones confined to a laboratory, not "looking out the
window" at the outside world) is unable to distinguish between
(a) the entire laboratory being accelerated with some (constant)
acceleration with respect to an inertial reference frame, and
(b) the entire laboratory being in a *uniform* gravitational field "g".
The qualifiers "local" and "uniform" are important here!
As we've discussed before in this newsgroup, real-world gravitational
fields are invariably non-uniform, so we need to introduce a tolerance
for how much non-uniformity we're willing to tolerate, i.e., for how
accurately we want to measure the accelerations and/or the "g". That
tolerance then sets an upper limit on the size of our laboratory, and
on the duration of our measurements, such that within that limit we
can approximate the gravitational field as uniform to within our
measurement tolerance.
In your animation you've chosen a "laboratory" large enough that the
gravitational field is strongly non-uniform, so it's not surprising
that this (non-uniform) gravitational field is readily disginguishable
from any non-gravitational acceleration.
-- jt]]

I understand.

The equivalence is true if the masses are small and it is not true if
they are large.


[[Mod. note -- It's not really the size of the masses that determines
whether or not (a) and (b) are distinguishable, rather, it's the choice
of tolerance compared with the non-uniformity of the gravitational field.

The "thing" that the tolerance applies to is measured acceleration
(with respect to a local inertial reference frame).

For a given (fixed) laboratory acceleration with respect to an inertial
reference frame, if we compare different attracting masses which could
produce that same acceleration (which is known as the Newtonian "little g"
if it's due to gravitation), a less-massive attracting mass would have
to be closer than a more-massive attracting mass. This means that the
smaller attracting mass would result in a more *non*-uniform field.
I.e., for the same-sized laboratory, the equivalence might be true (to
within some fixed tolerance) for a large mass far away, but false for
a small mass nearby.
-- jt]]