Free Fall and Inertial Frames?

[kq6up]

Let me set the stage for a thought experiment.

1. A physicist in an elevator at some arbitrary height above Earth holding an accelerometer oriented in a direction pointing toward the center of the Earth.

2. No atmospheric drag.

Drop the elevator car and make a measurement. The accelerometer instantly reads zero, and stays there until the physicist is converted to biological goo at AGL=0.

I have had a bit of trouble explaining to my high school students why this is so. This is very counter intuitive for them (and I admit a tad for me as well).

1. Is not the object in freefall approximately an inertial frame?

2. From an Earth bound observer POV, we see the gravitation force as a real force, and the acceleration a real acceleration. However, the OoE's (Observer on Earth) accelerometer reads 9.8m/s towards the center of the Earth, so the OoE is not in an inertial frame. So am I to conclude that there is no real force acting on the physicist in free fall?

3. It seems to me the easiest way for me to understand this is the equivalence of gravitation and acceleration under GR. However, is there a classical way to understand this? I do understand that objects in orbits are equivalent to objects in free fall. They just have a horizontal component of velocity that allows the object in orbit to miss the Earth before it collides, and it is always in a state of falling and missing.

4. Do you guys/gals have some tips for making this intuitive for students in a non handwaving and factual way?

Thanks,
Chris

 

[UltrafastPED]

You are describing the Einstein elevator experiment. The Einstein story goes: he heard of a painter who had fallen from the gable of a barn, and landed unhurt on a pile of hay. Einstein wondered what the painter felt while he was falling - and when he spoke with the painter he found that the painter was weightless while falling.

The elevator is an idealization of the painter's experience. You can partially replicate it by weighing things on a good, fast scale while going up and down on a multi-story elevator. You can then calculate the elevators acceleration from the changes in the weight readings. We did this for a Science Olympiad in 2005, the Einstein year.

The elevator is falling at the speed consistent with gravitational acceleration ... and since you and the accelerometer are falling at that same rate, there is no force between you and the elevator floor. The accelerometer will likewise feel no force.

The freely falling reference frame is local; it only includes the volume which is all falling at the same rate. It will pass all of the tests required for a Newtonian inertial reference frame.

 

[D H]

1. Is not the object in freefall approximately an inertial frame?

Not at all from the perspective of Newtonian mechanics. You are perhaps thinking of the general relativistic concept of an inertial frame. Inertial frames are local in general relativity, but are global in From Newtonian mechanics.

2. From an Earth bound observer POV, we see the gravitation force as a real force, and the acceleration a real acceleration. However, the OoE's (Observer on Earth) accelerometer reads 9.8m/s towards the center of the Earth, so the OoE is not in an inertial frame. So am I to conclude that there is no real force acting on the physicist in free fall?

Accelerometers cannot sense gravitation. The GR explanation of why this is so is perhaps easier: From a GR perspective, an ideal accelerometer measures acceleration due to the net real forces acting on an the accelerometer. That "net real forces" becomes "net real, non-gravitational forces" when viewed from a Newtonian perspective.

So how does Newtonian mechanics explain why can't accelerometers sense gravitational acceleration? The answer lies in how accelerometers work. They measure the real force needed to keep a test mass stationary with respect to rest of the accelerometer. An accelerometer cannot sense gravitational acceleration because the small extent of the accelerometer means that the test mass and the housing that surrounds the test mass are subject to almost the same gravitational acceleration.

 

[D H]

The freely falling reference frame is local; it only includes the volume which is all falling at the same rate. It will pass all of the tests required for a Newtonian inertial reference frame.

No, it won't, because an inertial frame in Newtonian mechanics is not local. A free-falling observer can see the ground accelerating toward him. That act of looking at the gravitating object that is pulling the observer in is a non-local experiment.

 

[kq6up]

So how does Newtonian mechanics explain why can't accelerometers sense gravitational acceleration? The answer lies in how accelerometers work. They measure the real force needed to keep a test mass stationary with respect to rest of the accelerometer. An accelerometer cannot sense gravitational acceleration because the small extent of the accelerometer means that the test mass and the housing that surrounds the test mass are subject to almost the same gravitational acceleration.

Ok, I need to clarify what I mean by an accelerometer, and if my version of an accelerometer has an issue, we can start there.

1. My accelerometer consist of a mass suspended in a frame with two equal springs on either side. A measurement is made from neutral displacement as compared to (approximately) the displacement as witnessed in intergalactic space -- or even better -- intercluster space.

2. At the surface of the Earth I see my one kg mass has compressed one spring and stretched the other. If these springs are ideal and identical, I believe they are carrying the force of 1/2 Newton each. I conclude there is a 1 Newton force pulling on my test mass.

I do see what you mean by a Newtonian definition of an inertial frame, but first let us hash out the accelerometer issue.

Thanks,
Chris

 

[olivermsun]

Ok, I need to clarify what I mean by an accelerometer, and if my version of an accelerometer has an issue, we can start there.

1. My accelerometer consist of a mass suspended in a frame with two equal springs on either side. A measurement is made from neutral displacement as compared to (approximately) the displacement as witnessed in intergalactic space -- or even better -- intercluster space.

This seems consistent with D H's description. The springs in your accelerometer measure the force needed to keep the mass from moving (further) within the frame.

 

[kq6up]

This seems consistent with D H's description. The springs in your accelerometer measure the force needed to keep the mass from moving (further) within the frame.

I think so too, so I am not certain how such an instrument would differentiate between a gravitation field and an accelerating body.

Thanks,
Chris

 

[A.T.]

However, is there a classical way to understand this?

A force that is proportional to mass (like Newtonian gravity) accelerates all masses within a small area equally, so it can't be picked up by an accelerometer, which is based on differential acceleration of frame and test mass.