[time-nuts] FW: Pendulums & Atomic Clocks & Gravity

James Maynard james.h.maynard at usa.net
Tue May 29 16:26:27 UTC 2007


Didier Juges wrote:
> Bruce,
>
> A lot of the statements that have been made lately on this subject kind of make sense to me in a way taken in isolation, but they do not all agree with each other, and that makes me uncomfortable.
>
> Example:
>
> I do not understand why the frame of reference would matter when you talk about gravity field. There is a gravity field or not, and the frame of reference should not matter. I understand that the frame of reference matters when you talk about displacement, velocity or acceleration. But the magnitude of a field, or a force, does not depend on the observer as it is static, or maybe a better term would be absolute or self-referenced?
>   
The reason that the frame of reference matters is that gravity is 
indistinguishable from acceleration. (This is an assumption that 
Einstein made when deriving his general theory of relativity. It seems 
to work.)

An "inertial" frame of reference is a non-accelerating frame of 
reference. In an inertial frame of reference, Newton's laws of motion 
work -- if you use Newton's gravitational relationship, that the 
gravitational force (weight) that each of two bodies exerts on the other 
is proportional to both their masses, and inversely proportional to the 
square of the distance between them.

In an accelerating frame of reference (either linear acceleration, or 
rotational acceleration, or both) additional forces, technically called 
"fictitious" forces, must be introduced in order to explain the motions 
of bodies with Newtonian mechanics. The "fictitious" forces on a body 
are also proportional to the body's mass. (A body's mass is just a 
measure of its inertia: to accelerate at an acceleration "a", a force 
"F" must be applied, and the mass "m" is just F/a.)

If the frame of reference has linear acceleration (relative to an 
inertial frame of reference), bodies within that frame of reference will 
experience a fictitious force that is proportional to their masses and 
to the acceleration of the frame of reference. Viewed from the frame of 
reference of a car that is accelerating away from a stop light, the 
passengers are pressed back in their seats by a force proportional to 
the acceleration of the car and to their masses. This fictitious force 
disappears when you view the situation from the an inertial frame of 
reference. Viewed from that point of view, the seats are pressing 
forward on the passengers to cause them to accelerate with the car.

Viewed from a rotating frame of reference, we have other fictitious 
forces: centrifugal force and Coriolis force. Both of these are 
proportional to the mass of the body on which they act -- when viewed 
from the rotating frame of reference. Both vanish if you view the 
situation from a non-rotating frame of reference.

Sometimes - usually, even - it's simpler to view the problem from an 
inertial frame of reference. Sometimes, though, it's easier to look at 
the problem in an accelerating frame of reference. If you do that, you 
account for the frame of reference's acceleration by introducing 
fictitious forces.
> Now, it makes sense that an object immersed in gravity fields from several larger objects may not be able to tell the difference between multiple fields, and a unique, "net" field (in the sense of Newton's net force), at least as long as the gradient is small enough that it cannot be observed within the dimensions of the object. So if the "net" field is zero and the gradient small enough to be ignored, the object will behave the same as if there were no field.
>   
When you say "within the dimensions of the object" I assume that you are 
looking at the problem from the frame of reference of the object. That's 
natural if you are, for example, in an orbiting satellite, such as the 
International Space Station. Viewed from an inertial frame of reference, 
the ISS is following an orbit determined by the vector sum of the 
gravitational forces (from earth, moon, sun, etc.) that act upon it. 
Viewed from the frame of reference of the space station, however, these 
forces add to zero.
> However, for an observer on earth, a satellite is in the gravity field of earth (let's assume all other gravity fields from the sun and other planets are negligible), which is not zero at the altitude of the satellite,
Even an observer on earth is on an accelerating frame of reference. (The 
earth rotates on its axis.)
> ... yet for an observer on the satellite, the net field appears to be zero. Where is the counter-field coming from? And why can't we observe it from earth? How can the field be different when observed from different points?
>   
For an observer on the satellite (in the satellite's frame of 
reference), the counter-field is created by the fictitious forces due to 
the satellite's acceleration. For example, "centrifugal" force due to 
the satellite's gravitational acceleration towards the center of mass of 
the earth.
> Could it be that the effect of the gravity field (with is a centripetal force applied to the object in orbit) is compensated by a centrifugal force, (which I was close to admit is not a real force and does not exist) so that the effect of the gravity field, which would be a force of attraction towards the planet, is compensated by another force in the opposite direction so that the net force is zero, as it would be if there were no gravity field? So that the object does not know the difference between two forces that compensate each other and no force at all.
>   
Yes! The "fictitious" forces, however, do exist -- when viewed from the 
frame of reference of the accelerating satellite. I wouldn't say that 
fictitious forces are not real - just that they only exist when viewed 
in an accelerating (non-inertial) frame of reference.





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