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Carlos I Calle is a prominent NASA scientist with extensive experience with Einstein's theories. In 2011-12 he wrote a book for the Wiley 'Dummies' edition--Einstein for Dummies.There he says that according to Einstein acceleration and gravity are indistinguishable. For example if you drop a ball on the earth's surface it is not the ball that falls to the Earth's surface but the Earth that accelerates up to meet the ball. But the Earth is configured with the solar system and other stars--how will it accelerate on it's own? He goes on to say that according to Einstein the Earth along with the solar system accelerates to meet the ball. An elegant concept. I would have dismissed it as an oversimplification for the Dummies series but I found the concept repeated in a basic Cambridge A level textbook. (I forget specifically which). Obviously it must be true.

My question is-- during the 2-3 seconds it takes for the ball and earth to meet each other if another ball is dropped on the Earth's surface diametrically opposite to the first point--what happens then? Obviously the Earth cannot at the same instant be accelerating in two diametrically opposite directions. Also objects can be dropped on other planets and stars during that 2-3 seconds. What happens then? I even wrote Mr.Calle an e mail on this issue but it went unanswered.

Can anyone explain this paradox to me?

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In Newtonian mechanics acceleration is an absolute quantity. All observers can calculate the acceleration of some object and all observers will end up with the same result. So we say an object is accelerating this is an unambiguous statement.

In special relativity acceleration is also absolute in the sense that all observers will agree whether an object is accelerating, though different observers will measure different values for the acceleration.

However in GR acceleration is not absolute. By this I mean that I can choose a coordinate frame in which some object is accelerating and I can choose a frame in which the object isn't accelerating and both choices are equally valid.

Take your example of an object falling to Earth, and suppose you are that object. Suppose also that you're falling in a closed spaceship with no windows so you can't see out. Are you accelerating or not? One way to tell if you are accelerating is by whether you feel any force. If you feel no force acting on you then in your coordinate frame you are not accelerating. Technically your proper acceleration is zero. So in your frame the Earth is accelerating towards you.

Now suppose you're floating in a zero G bubble at the centre of the Earth. Once again you feel no force acting on you, so your proper acceleration is zero. In this frame it's the falling object that is accelerating towards the Earth.

The point is that it's incorrect to simply say you are accelerating towards the Earth or the Earth is accelerating towards you. The correct statement would be something like:

  • it's possible to choose a coordinate frame in which you have zero proper acceleration and the Earth is accelerating towards you

or:

  • it's possible to choose a coordinate frame in which the Earth has zero proper acceleration and you are accelerating towards the Earth

So there is no paradox. Which object is accelerating just comes down to a choice of coordinates rather than anything physical.

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The word General in General Relativity refers to the fact that you deal with the note general case of no global inertial frames.

You might be used to Newtonian mechanics or Special Relativity where can can pick an inertial frame and talking about constant velocity in that frame and acceleration and force in that frame and the frame covers everything.

But there is no law of physics that says a single inertial frame has to be able to work for arbitrarily times or for things super far away. And in fact in general relativity it doesn't.

The general idea is that if you had a small low mass,low energy, non spinning particle that wasn't feeling an electromagnetic forces or contact forces and so on. And then you let it move freely, then at each place it was at and each time you could make a really really small inertial frame around it that worked just for a small region and just for a small time interval. And in that frame the particle would be at rest.

It's similar to how you could walk on a great circle around a spherical earth and locally you could find a system where everything looks pretty flat and it looks like you are moving in a straight line in a flat space.

So if you choose a frame near the earth it would see the earth rushing up and accelerating upwards. And same thing if you were in the atmosphere.

In Newtonian mechanics the atmosphere has a pressure gradient, the pressure below a layer of air is a bit bigger than the pressure above that layer and the net difference would accelerate the air upwards except gravity pulls it down and so it doesn't go anywhere.

In general relativity you have a purely local inertial frame and there is still a pressure difference so the air does feel a larger pressure from the air below than from above and so it just accelerates upwards. Nothing mysterious, the air feels an imbalance of forces so accelerates upwards.

But that test particle is, by definition, not feeling any forces so in particular it isn't feeling that pressure. Which means it doesn't accelerate upwards with the air so the to someone that wrongly through the air was at rest they thing the test particle is accelerating downwards.

Now when you try to make a global frame that covers both sides of the earth (inertial frames can't be that large) then you have to use a non inertial frame and so you make up fictitious inertial forces to explain why things don't follow $F=ma$ type equations. Just like when you use a rotating frame.

So the frame with both ends of the earth isn't an inertial frame and so has fictitious inertial forces and says things are at rest when in a true inertial frame they are accelerating. But the true inertial frames are local. If you assumed the inertial frames must be global then you actually wouldn't get gravity.

We see gravity, so the inertial frames must be local and only cover small regions. When your frames are local they are like buying lots of maps, you can work in a map and before you get to the edge of your map you should switch to a different one. And as long as you have a Mao of your starting point and your end point and you can always switch to a new map before you get to the absolute edge of one of your maps then you have enough and it works fine, you don't actually need a global map, its nice but not needed. And for instance with the surface of the earth it is hard to map a global 2d map, even though locally you can.

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To be more precise, Einstein's equivalence principle states that the effect of a gravitational field is locally indistinguishable from an accelerated rest frame. So as you mentioned, when someone drops a ball above the surface of the Earth, there are no local measurements we could make that would be able to distinguish between there being an actual gravitational field induced by the Earth forcing the ball towards the ground, or us simply accelerating our frame of reference towards the ball with no gravitational field. The key point to your example of two balls being dropped on opposite sides is that this principle is a local one, and would not apply to two different balls on opposite sides of the earth, as the measurement of the course of both of their trajectories would not be a local measurement.

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It is true that active gravity in space means all objects theoretically can pull eachother out of their positions. However any object between two objects of equal mass will not move as both forces cancel out. Tying the general principle of acceleration to gravity isn't proper, as there many ways of achieving acceleration, and yes the speed of light can easily be broken with a properly configured Zero Mass Drive, which barring a sub-atomic barrier bow wave can easily achieve almost unlimited acceleration.

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  • $\begingroup$ Not sure if you were serious when writing this, but I like your answer either way. $\endgroup$ – Horus Mar 14 '16 at 9:06

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