OK, that's enough playing with Humber for me for today.
John, I started composing this reply this afternoon. I'm trying to find ways to make things clearer, for myself as well! I have some sort of mad delusion that if I
really understand the concepts of general relativity, I will one day be able to explain them using toy animals on a tabletop. I didn't think my little question about accelerometer readings would get us into such murky waters!
It began because Michael C made a very clear and "obvious" statement that of course an accelerometer in a falling lift would read zero, because it is not accelerating.
Well, that's not exactly what I said. Maybe I wasn't clear enough: what I actually said was "in free fall with no air drag you will experience no acceleration forces". The problem is here: in the Newtonian world if you are in free fall you will indeed be accelerating, but
you won't feel any force. None of your internal organs will be telling you "you're accelerating". You'll only feel the force of gravity when you are being held up by the ground, or the floor, or whatever. I agree that all this can get very confusing, and we can get muddled up in the terminology. I prefer, as you do, to think of the immediate world around me in a "Newtonian" manner, but it's worth remembering from time to time that we see and describe things in a certain way only because we live on the surface of an almost spherical planet with an almost constant gravitational field. We'd describe things differently if we lived inside a hollow planet (no gravity), or in some place where the gravitational field varied greatly over short distances.
However, I feel that from the classical perspective, as wikipedia agrees, there is no acceleration for a body stationary on the earth, but when it falls due to gravity it accelerates with g, which is even called "the acceleration due to gravity". From a classical perspective and a chosen (Earth) frame, the elevator can be seen and measured moving faster over time by that amount.
Agreed. I, as an Earth surface dweller, consider that an object accelerates towards the ground when I drop it. However, the changing acceleration of the elevator dropping through the tunnel in the earth can only be seen and measured from outside it, from the perspective of somebody on Earth. The accelerometer inside cannot measure the changing acceleration.
Yet the link to instructions on using one discusses adding -9.8 m/s/s to get the "true acceleration".
That's a confusing term. In fact they are talking about the acceleration
with respect to the frame of reference of the Earth. If we were using the accelerometer on the Moon, we'd need to add -1.6 m/ s
2 to get the "true" acceleration at the moons' surface. If we were using it on Jupiter, we'd need to add -25.9 m/s
2. We need to make these different adjustments because the frame of reference of the surface of a planet is not an inertial frame.
But wait a minute: didn't we say that a reasonably small section of the surface of the earth could be considered to be an inertial frame? Well yes, we did: it was in fact a cheat, a crafty manoeuvre to serve our purposes (I wonder what Humber will say to that...). We can only consider the place where we're standing on Earth to define an inertial frame if we include gravity as a "
fictitious force". Oh dear! I'm not going to go into a big explanation of fictitious forces here: click on the link if you wish. In fact the surface of the earth qualifies as a constantly accelerating reference frame, whereas an elevator in free fall, whether it is near the surface of the earth, in orbit around the earth, in a tunnel through the earth or somewhere in deep space, defines an inertial frame of reference: throw a ball in it and it will continue in the same direction at the same speed until it hits something (Newton's 1st law). If the elevators are identical and the ball is thrown in the same way in each elevator, its trajectory will be identical in each elevator. Throw a ball in an elevator parked at the surface of some planet and it will describe a particular parabola. Depending on the gravitational field of the planet in question, an identical ball thrown with the same direction and force will not describe the same parabola: you have a way of distinguishing between different non-inertial frames.
There is no way to distinguish between one inertial frame and another (principle of Relativity).
So the straight dope is here: when we're talking about the "frame of the ground", the "frame of the treadmill" or the "frame of the wind" we are not talking about real inertial frames in the strict sense. We can say that they are inertial frames that all have the same fictitious force of 1 g, or we can say that they are non-inertial frames that are all accelerating upwards at the same rate. For the Newtonian discussion, it's best to stick with the idea that they are inertial frames with an identical fictitious force in each frame. Since the force is the same in each frame, we can consider all these frames to be equivalent just as we can consider all real inertial frames to be equivalent.
