QUESTION: At the very beginning of various books on relativity I have found what looks on its face to be a contradiction, and hope you can clear this up for me. In one such book, for example, the authors make clear that there is no absolute space and that “it only ever makes sense to speak of motion relative to something else.” ( I read this to mean motion relative to a reference frame which is attached to the “something else.”) So far so good.
Then they go on to support the idea of the nonexistence of absolute space by pointing out that no experiment done in a plane, for example, can show whether it is in motion or not, “provided the aircraft is not accelerating.” But to say that a plane is accelerating is to describe its motion, which the first quote indicates is a meaningless exercise unless you describe its motion with respect to a given reference frame.
Any help on this would be greatly appreciated.
ANSWER: Hi David,
You're right. There seems to be a contradiction, and I believe it's the authors' fault. By trying to explain a concept simply, they left out an important qualifier.
In the terms of physics, motion (or "Velocity") must be measured relative to another body in an "Inertial Frame of Reference". Or, Velocity = vector distance traveled (from point X to point Y) in any space (Euclidean or otherwise) divided by the time to travel there. Points X and Y must be defined (those are the "relative" parameters). And we must be in a "Inertial Frame of Reference", or free from any forces. I said "vector", which gives the Velocity its direction, but for completeness, I should say "tensor", which would apply in non-Euclidian spaces. But I don't won't to confuse you - I just want to be entirely accurate.
So what is an "Inertial Frame of Reference"? Think of any physical system without any forces acting on it. Basically, forces cause a change in velocity (termed "acceleration"). Most laws of physics are invariant among systems in a Inertial Frame of Reference. In the plane example, the inside of the plane is in an Inertial Frame of Reference. The outside of the plane, of course, has all sorts of forces acting - there's forces on the wing from the air molecules, providing lift. But in level flight, that balances out gravity trying to bring the plane down. And there's drag on the plane hitting the air. Again, if the plane isn't accelerating or decelerating, that force is balanced by the thrust of the engines. The net force on the plane (the system in question) = 0. Of course, there's still gravity affecting the passengers, but that is ALMOST the same gravity as being on the ground. Not quite, so we could do an experiment which measured our weight on the plane, and deduce that we were high up. But we COULDN'T deduce any motion (relative to the ground or anything else) because we're in an Inertial Frame of Reference. No net forces "on the system" (which is the plane). If we could see the ground, or see stars outside the window, we could, of course, measure the motion (velocity) relative to those.
Now I'll give you an interesting problem. We know that straight line, non-accelerating motion (inertial frame) has no (net) forces acting on the body. And circular motion requires a forces to change the direction (vector velocity) of the motion. Now consider the earth's rotation. How do we know the earth is rotating? It seems obvious - the ancients looked at the stars (and the sun) and saw movement. Some people thought the earth was standing still and everything rotated around it, but pretty soon we realized the earth itself was spinning. The real "proof" came with the Foucault Pendulum - a large suspended pendulum which slowly rotated as the earth turned under it. It actually remains "fixed" in space (relative to the stars), since the only force acting on it is gravity - which points down. It simply can't have sideways motion (causing it to rotate) because there's no sideways forces acting on it.
So that proved the earth itself was rotating. But what if the earth was cloud-covered (like Venus) where we couldn't see the stars or the sun? On a planet like that, would the Foucault Pendulum also work? Sure! Any Venetians (if they exist) can also prove their planet is rotating simply by setting up their own pendulum. So they would know it's rotating - but with respect to what? The usual answer is "with respect to the distant stars" - even if they could never see those stars! There seems to be motion with respect to "the universe" - "Absolute motion" (because we're no longer in an Inertial Frame of Reference)!!
Now, one last thought problem. Suppose the Venetians are looking at their Foucault Pendulum, and we could magically remove the stars from the sky. Venus is the only body in the universe. Would the Foucault Pendulum still work? I would guess it would! We can actually measure accelerating motion (rotation) relative to NOTHING! Non-inertial frames of reference seem to have peculiar properties.
Hope that doesn't confuse the issue too much.
Prof. James Gort
---------- FOLLOW-UP ----------
QUESTION: Thank you! That is fascinating stuff, especially the Foucault pendulum on Venus!! I really appreciate the time you obviously put into your answer. And now I have two follow-up questions:
(1) Would the authors have been correct to say, in the second quote in my letter, “provided the aircraft is not (a) changing speed or direction with respect to any inertial reference frame or (b) rotating?”
(2) Suppose there were a rocket with its engine off, moving in straight line uniform motion with respect to some inertial reference frame (so that the rocket itself would be an inertial reference frame), and all of a sudden all bodies in the universe other than the rocket disappeared. If the engine were then turned on, (a) would it be meaningful to say that it is accelerating and (b) if so, would it then no longer be an inertial reference frame?
I'll first answer your questions concisely:
1) (a) - Yes. (b) - No - because rotation is only one type of acceleration (angular acceleration). In part (a), a change of direction includes both linear and angular acceleration.
2) (a) Yes. (b) Yes.
Part 2(a) might seem strange. After all, After all, velocity cannot be measured, since velocity is the distance between two defined points (A and B) divided by the time it takes to traverse the distance. In our rocket, we'd still have our watches so we can measure time intervals. But we can't measure distance intervals - points A and B don't exist! So we can never measure velocity. But we CAN measure rates of change to velocity!
Think of it this way. Suppose, in our rocket, we had a large fuel tank. Yes, this rocket runs on gas! But our fuel gauge is broken, so we don't know how much fuel is in the tank. But the engines are still working, so we know the tank isn't empty. Instead, we can measure the flow of the fuel into the engines. That's working fine. So we KNOW that our fuel supply is changing (being drained) at the same rate as fuel is flowing into the engines. We don't know the quantity of fuel left, but we know the rate of change of the quantity!
Back to our accelerating rocket. We can quantify the amount of acceleration simply by measuring the weight of the occupants against their seats. We'll call the weight of occupant one "F" (for force), and the mass of occupant one "M" (we measured this on earth, and it doesn't change unless we get relativistic). Then, acceleration = F / M.
If we become relativistic, things become much more complicated. We still can't measure velocity directly, but we know our mass is related to velocity. In that case, we'd have to do an on-board experiment to measure our new masses. Basically, just accelerate them in the rocket, and measure the force needed to accelerate them. Suffice it to say that our acceleration will decrease as we approach the speed of light.
Prof. James Gort