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Aug 17, 2022, 6:09:52 PM8/17/22

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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]]

<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]]

Aug 19, 2022, 3:16:20 AM8/19/22

to

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]]

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