Understanding Einstein's Thought Experiment & Question 10 of the Relativity Quiz

[quantumdude]

The Earth observer says the spaceship is accelerating, and the Earth is an inertial frame, and the spaceship is a non-inertial frame.

The spaceship says the Earth is accelerating, but that doesn't change the fact that the Earth's motion is inertial, while the spaceship's is not.

Initial and "not accelerating" are not necessarily synonymous. Nor are "non-inertial" and "accelerating".

I might be inclined to agree if the "inertial/noninertial" and "accelerating/nonaccelerating" dichotomies were cleanly seperable, but I don't think that they are. The reason for this is that the laws of physics that hold in inertial frames involve acceleration as an integral component.

Everyone here seems to agree that the spaceman can do experiments from a non-inertial frame, and can verify that he is non-inertial by the fact that the laws of mechanics and E+M don't hold for him. But if the spaceman holds that the Earth is inertial, then he holds that the law F=ma holds on Earth. If he holds that the Earth is accelerating then he would find something very wrong if he were to watch a man standing on the Earth, facing the ship, drop a bowling ball, and have it land directly beneath the spot from which it was dropped. If the Earth is inertial, then F=ma holds on Earth. And if the Earth is accelerating with the acceleration of the ship (but in the opposite direction), then the spaceman would expect the ball to land in front of the man.

But that's not what happens.

Here's where it is important to distinguish the physics from the math.

Heh. That's what I was thinking of saying to you. In order for our spaceman to declare himself at rest, he has to rely on a mathematical coordinate system and ignore all the physical telltale signs of acceleration.

 

[learningphysics]

If the Earth is inertial, then F=ma holds on Earth. And if the Earth is accelerating with the acceleration of the ship (but in the opposite direction), then the spaceman would expect the ball to land in front of the man.

The spaceman expects F=ma to hold for observers on the Earth, where acceleration is defined in the Earth coordinate system, not his own.

Acceleration is a kinematic concept... it is defined prior to inertial. It simply refers to d^2s/dt^2 for some object in a coordinate system.

An inertial coordinate system is defined as one where free particles undergo zero acceleration. So acceleration is a concept prior to inertial... in terms of which inertial is defined.

Suppose I'm in some non inertial coordinate system. There's a free particle (no forces acting on it), and I measure d^2x/dt^2 for this particle and I find it to be nonzero. Is it inappropriate for me to say that the particle is "accelerating" in my coordinate system?

 

[quantumdude]

The spaceman expects F=ma to hold for observers on the Earth, where acceleration is defined in the Earth coordinate system, not his own.

OK.

Suppose I'm in some non inertial coordinate system. There's a free particle (no forces acting on it), and I measure d^2x/dt^2 for this particle and I find it to be nonzero. Is it inappropriate for me to say that the particle is "accelerating" in my coordinate system?

No, I don't think it's inappropriate at all. In fact in the past I have argued against this very same point on those very same grounds. When people used to say that acceleration is absolute in SR (that even appears on John Baez' website!), I used to ask them how simply taking the 2nd derivative of a 4-vector (namely, the displacement $x^{\mu}$ could return a Lorentz scalar. Obviously, it can't. But the thing that didn't sit right with me is that any observer can tell whether or not he's inertial.

My mistake here in this thread was equating inertial frame with accelerating frame.

OK, objection withdrawn.

 

[jdavel]

If we cut through all the semantics in this thread, two facts seem clear.

1) the laws of physics aren't the same in stella's and terrence's reference frames

2) the laws of physics aren't constant in stella's reference frame.

So they both understand why stella is younger when she returns to earth.


Edit: I wasn't quite awake when I wrote this, and accidentally said "...stella is older when she returns..."

 

[εllipse]

If we cut through all the semantics in this thread, two facts seem clear.

1) the laws of physics aren't the same in stella's and terrence's reference frames

2) the laws of physics aren't constant in stella's reference frame.

So they both understand why stella is younger when she returns to earth.


Edit: I wasn't quite awake when I wrote this, and accidentally said "...stella is older when she returns..."

That's only true if we assume we're only supposed to take SR into account. But the general principle of relativity states that the laws of physics are the same for all Gaussian coordinate systems, so the only thing that changes is Stella's coordinate system, not the laws of physics that apply to her.

 

[James R]

learningphysics:

I agree with everything you wrote. Thanks for expressing this in a different way.


Tom:

It seems you have come around to the same point of view that learningphysics and I share. Am I right? You say:

My mistake here in this thread was equating inertial frame with accelerating frame.

I think you meant equating "non-inertial frame" with "accelerating frame".

It's not surprising, the concept of "inertial frame" is usually introduced to students as meaning, roughly "accelerating frame". The problem with that definition is that it is only true if you're viewing the "acceleration" from an inertial frame in the first place, so as a definition it is uselessly circular.

It is much better to define an inertial frame as one in which Newton's 1st law holds (i.e. "free" particles do not accelerate.) Under that definition, the frame itself may be accelerating or now accelerating with respect to some other arbitrary frame.

Everyone here seems to agree that the spaceman can do experiments from a non-inertial frame, and can verify that he is non-inertial by the fact that the laws of mechanics and E+M don't hold for him. But if the spaceman holds that the Earth is inertial, then he holds that the law F=ma holds on Earth. If he holds that the Earth is accelerating then he would find something very wrong if he were to watch a man standing on the Earth, facing the ship, drop a bowling ball, and have it land directly beneath the spot from which it was dropped. If the Earth is inertial, then F=ma holds on Earth. And if the Earth is accelerating with the acceleration of the ship (but in the opposite direction), then the spaceman would expect the ball to land in front of the man.

No, because the spaceman would have to introduce additional "inertial" forces. F=ma holds when F and a are measured on Earth, but not when F and a are measured from the non-inertial spaceship. To make the formula work in the spaceship frame, we need to change it to F + Fi = ma, where Fi is an inertial ("imaginary") force.

 

[James R]

Ellipse:

While Stella is accelerating with respect to Earth, the laws of physics DO change forms for her. She experiences "inertial forces", as explained in the post immediately preceding this one.

 

[quantumdude]

Tom:

It seems you have come around to the same point of view that learningphysics and I share. Am I right?

Yes. Acceleration is just the second derivative of displacement, and as such is just how coordinates change with respect to an observer.

I think you meant equating "non-inertial frame" with "accelerating frame".

Yes, it was late when I wrote that.

No, because the spaceman would have to introduce additional "inertial" forces. F=ma holds when F and a are measured on Earth, but not when F and a are measured from the non-inertial spaceship. To make the formula work in the spaceship frame, we need to change it to F + Fi = ma, where Fi is an inertial ("imaginary") force.

Yes I made the mistake of mixing quantities from different frames. I plugged the "a" measured by the spaceman into the "F=ma" of the guy on Earth. I'd never do that with x's and t's, but somehow I managed to confuse myself enough to do with with F's and a's.

Oh well, happens to the best of us I guess.

 

[everneo]

It is possible for an observer in an accelerating frame to clearly distinguish whether it is inertial with uniform gravity or non-inertial. the tidal forces localize the scope of equivalence principle over small distances. whereas in uniformly accelerating frame the there is no tidal force or force gradient. The validity of stella's statement that she is in an 'inertial frame' can still be under scrutiny.