The Peripatetic Albert, Round 3

[OneEye]

...And so, I was driving north from McPherson toward Salina, explaining Special Relativity to my wife and oldest daughter (and however many of the other children were actually listening). As I did so, I used our Suburban as an example, explaining how that, in the absence of any absolute coordinate system, there was no real way to tell whether the Suburban was moving over the road, or the road was moving under the Suburban.

About that time, we turned onto the highway - and I realized that, I had either used the steering wheel to turn the car in relation to the earth, or I had used the steering wheel to turn the Earth in relation to the car - and that, motion being purely relative, either could be as true as the other.

Now, here was a remarkable fact: Assuming the car to be at rest (which Special Relativity requires me to be able to do), it seems that I was actually able to change the orientation of the entire Earth simply by turning the steering wheel!

Now, that's power steering!

How does Special Relativity explain this curiousity?

 

[humanino]

That does not seem to be a paradox to me : when you turn, you are not an inertial observer anymore ! You have been accelerated.

 

[humanino]

Sometimes when I walk, I like to think I am having to Earth roll under my feet :shy:

 

[geometer]

...And so, I was driving north from McPherson toward Salina, explaining Special Relativity to my wife and oldest daughter (and however many of the other children were actually listening). As I did so, I used our Suburban as an example, explaining how that, in the absence of any absolute coordinate system, there was no real way to tell whether the Suburban was moving over the road, or the road was moving under the Suburban.

Actually, you left out a critical detail. If you were undergoing completely unaccelerated motion, there is no experiment that will tell you whether you are moving or the Earth is moving in your example. However, in a car, it's just about impossible to be completely unacclerated; bumps are acclerations, the swaying of the car is an acceleration, obviously, even minute changes in your speed are accelerations.

 

[OneEye]

Okay, then...

How does GR explain this?

(I can't help but think that we've missed the point, though. Essentially, we see what appears to be a blatant abrogation of fundamental laws - the energy required to change the Earth's direction is many orders of magnitude greater than the energy required to change the car's direction - so how can we say that the principle of relativity is upheld in this scenario? I don't think that promoting the problem to a GR question helps this observation. Further, I have heard several board members, much more knowledgeable than me, make SR analyses of non-linear circumstances not unlike this. But if we are going to say that "SR only applies to straight-line motion," then fine. In that case, I welcome a GR explanation.)

 

[humanino]

when you turn, you are not an inertial observer anymore ! You have been accelerated.

What is wrong with this ?
I don't think

bumps are acclerations, the swaying of the car is an acceleration, obviously, even minute changes in your speed are accelerations

has any relevance for the problem. However, the whole thing looks solved to me by the previous one.

 

[humanino]

the energy required to change the Earth's direction

Contrary to you, the Earth has not been accelerated ! Besides, you would have to change the direction of the entire Universe

 

[OneEye]

Contrary to you, the Earth has not been accelerated ! Besides, you would have to change the direction of the entire Universe

But this is exactly my point: What reason have I to believe that it is I, not the earth, which has been accelerated - other than resorting to classical mechanics? Observationally, by the principle of relativity, neither idea should have any advantage over the other. Is it me moving, or the earth, or the entire universe? Why would one observational perspective have any advantage over another?

 

[humanino]

No, you can measure a force. There was no force on the earth.

Well, strictly speaking, there was the exact same and opposite force, and it did move. But its inertia or mass is so huge, that it did not really affect it much.

 

[a1_phy]

well i suggest u perform this thought experiment..keep a glass of water i n ur car and on Earth beside and turn ur car...u will easily see which body is being accelerated by observing the movement in the water in the glass...
so it is u who is accelerating since a device to measure force records one in ur frame(accelerated that is) and not in the frame of reference of the earth

 

[humanino]

Welcome a1_phy ! Very good argument.

 

[gonzo]

One-eye, relativity applies to velocity not acceleration. Acceleration is not relative.

 

[OneEye]

No, you can measure a force. There was no force on the earth.

Well, strictly speaking, there was the exact same and opposite force, and it did move. But its inertia or mass is so huge, that it did not really affect it much.

Very good. So, then, we have a principle, related to mass and force, which mitigates the principle of relativity: Where force can be calculated, the principle of relativity no longer applies: One can tell with certainty which object is moving when one is able to calculate the force.

Right?

 

[OneEye]

One-eye, relativity applies to velocity not acceleration. Acceleration is not relative.

I don't believe that this is quite right. I believe that the correct statement is, "Special Relativity applies only to velocity; you must use General Relativity to evaluate questions of acceleration."

 

[selfAdjoint]

I don't believe that this is quite right. I believe that the correct statement is, "Special Relativity applies only to velocity; you must use General Relativity to evaluate questions of acceleration."

No, this is an urban legend. You can treat accleration in SR, and people have done it in other threads on this board; look up "relativistic rocket". What is more true is that GR is a complete theory of forces and accelerations whereas SR is a special case. It is correct that accelerations are not Lorentz-invariant or covariant. They follow a different change-of-frame law.

 

[humanino]

Very good. So, then, we have a principle, related to mass and force, which mitigates the principle of relativity: Where force can be calculated, the principle of relativity no longer applies: One can tell with certainty which object is moving when one is able to calculate the force.

Right?

Wrong. Let us sum up what has been collected here : there is an inertial reference frame for the system Car/Earth. In this frame, the acceleration is undergone by the car. The acceleration of the Earth is negligible. If you had a very precise mean of measurement, you could in theory notice the force on Earth, but such an accuracy would probably require to take into account the whole solar system.

You can locally make acceleration disappear by changing the referential. Only locally. The free-fall observer not noticing the Earth gravitational field has a limited spave around him, otherwise he would notice the spherical shape.

You raised the question of energy. Let us assume now that you want to turn very fast. Very very fast. Let us say there is no limit on the energy you can spend for this turn. If you turn too fast, you will really be able to produce a serious force on Earth !

 

[OneEye]

No, this is an urban legend. You can treat accleration in SR, and people have done it in other threads on this board; look up "relativistic rocket". What is more true is that GR is a complete theory of forces and accelerations whereas SR is a special case. It is correct that accelerations are not Lorentz-invariant or covariant. They follow a different change-of-frame law.

selfAdjoint: Thank you for weighing in on this. But let me be sure that I understand what you are saying:

You are saying that SR does apply in the case I originally posited; that the fact that I "accelerated" (not necessarily true, but we will say that it is) or moved in a non-linear fashion does not disallow an analysis of the situation using SR. Is that correct?

I would really like to nail this down, since I have had several questions answered in like manner: "Your scenario does not apply, since you are accelerating, and SR is not useful to consider acceleration." But I have seen other posters who seem to indicate that SR is as useful for non-linear motion (e.g., rotary/orbital motion) as it is for linear motion - and my original case seems to be reducible using the theorem of the addition of velocities from SR. So, I have always doubted the "SR is only for linear velocity" disqualifications which I have heard.

Am I right to doubt?

 

[humanino]

Always right to doubt.

At least I did not tell you SR does apply when acceleration is here. I only wanted to point that there are two inertial frames, before and after the turn, but the interesting moment is in the middle.

 

[geometer]

What is wrong with this ?
I don't think
has any relevance for the problem. However, the whole thing looks solved to me by the previous one.

Nothing wrong with your answer, I was agreeing with you and showing that even if you are moving in a straight line in a car, you are probably not even then an inertial observer.

 

[geometer]

You can locally make acceleration disappear by changing the referential. Only locally. The free-fall observer not noticing the Earth gravitational field has a limited spave around him, otherwise he would notice the spherical shape.

I have to disagree humanino, an observer in an isolated laboratory can detect an acceleration. You can't transform accelerations away.

 

[humanino]

Now I am even more confused. The free falling observer is accelerated, and this is the very important initial idea of Einstein : he would locally not notice this acceleration because the lab falls with him, hence gravity ~ change of ref.

Why might have a misunderstanding here ! If something is accelerated inside the free falling lab, then he would notice it. You cannot make tidal forces disapear, but this is due to different accelerations in different points. This is non-local.

 

[geometer]

Let's assume we are riding on a train and looking out the window at the scenery. Further, let's assume that this train is a perfectly inertial system right now - strictly unacclerated, straight line motion. In this case, there is no way I can tell whether I am moving or the world outside is moving.

But, now, let's assume the train takes a corner. Now I am accelerated, and in fact, I will feel the acceleration and it is possible for me to determine whether I am moving or the world is moving. Since I feel the acceleration, I must be moving. And even I choose myself as the origin of my reference frame, I still feel the acceleration. No matter what coordinate system I choose, I still feel the accleration.

 

[geometer]

Actually, Einstein's great insight was that the person in the freely falling lab couldn't tell whether any acceleration that was imposed on him was due to a gravitational field or some artificial source such as a rocket engine. This argued that gravity wasn't some intrinsic characteristic of a mass itself, but a secondary effect.

 

[pervect]

Now, here was a remarkable fact: Assuming the car to be at rest (which Special Relativity requires me to be able to do), it seems that I was actually able to change the orientation of the entire Earth simply by turning the steering wheel!

Now, that's power steering!

How does Special Relativity explain this curiousity?

Back in the old days, life was simple. You weren't allowed to change the orientation of the Earth by turning your steering wheel, nor were you allowed to change the orientation of the entire universe by turning your head (or swiveling in your swivel chair). You were told, in no uncertain terms, to pick an inertial frame of reference, and stick with it.

Back in the old, simple days, physics was simple, too - bodies at rest remained at rest, bodies in motion remained in motion.

Now, it turns out to be convenient, sometimes, to be able to change the orientation of the Earth, or the universe, with the turn of a head, or the stroke of a pen. You can do physics in such a coordinate system, but don't expect Newton's laws to apply.

If you've internalized the concept of Newton's laws, you might say that changing the orientation of the universe with the stroke of a pen isn't really a "physical" change, it's just a change of coordinates.

If you start focusing on what you calculate, I think a lot of the confusion will disappear. You can calculate what will happen next when you turn the steering wheel - it's just a matter of viewpoint, of how you describe what happens next.

 

[humanino]

But, now, let's assume the train takes a corner.

Non-uniform acceleration ! We are agreeing, I am almost certain of this. The non-uniform acceleration is locally uniform. There is a local change of coordinate making the acceleration vanish at one instant. Even here.

 

[OneEye]

So, then, I am driving down the road... or the road is rolling under me... whichever. Both are equally true, relativistically speaking. I am at liberty to consider the case either way, and neither view has any advantage over the other. In any case, the Earth is turning under me at 100 kph, but I am perfectly still. The sun is shining; the birds are singing; all is right in the world.

And, because I insist on a perfect inertial frame of reference, I do not touch the steering wheel. I lashed it tightly with a bungie cord as soon as the highway got up to 100 kph underneath me.

Unfortunately, geography interferes with my progress: The road curves to the left, and I do not. The road rolls out from under me, and a bridge abutment plows right into the nose of my perfectly stationary vehicle.

How do I calculate the energy of collision?

 

[geometer]

Non-uniform acceleration ! We are agreeing, I am almost certain of this. The non-uniform acceleration is locally uniform. There is a local change of coordinate making the acceleration vanish at one instant. Even here.

I'm afraid not. Inertial coordinate systems are coordinate systems moving at constant velocity with respect to each other. If one system is accelerating, even uniformly, with respect to another system it is no longer an inertial system, and you should be able to detect acclerations and thereby determine who is really moving.

A coordinate system which is in free-fall with respect to the Earth say, is undergoing acceleration and is therefore not an inertial system.

 

[HallsofIvy]

Yes, you can treat acceleration in both SR and GR. The point everyone has been trying to make to OneEye is that acceleration is NOT relative. There is no experiment you make in a completely selfcontained room that will determine uniform velocity. You can determine acceleration. OneEye's original question was about turning which involves acceleration even if done at a constant speed.

 

[pervect]

Unfortunately, geography interferes with my progress: The road curves to the left, and I do not. The road rolls out from under me, and a bridge abutment plows right into the nose of my perfectly stationary vehicle.

How do I calculate the energy of collision?

The energy of collision is frame dependent, unfortunately. So what you typically do is go to the center of mass frame between you and the bridge abutment.

You can compute the energy in other frames, but if you want to cmpute how much your care will crumple by the equation

energy = crumple_force * crumple_distance

you want to use the center-of-mass frame energy.

In the case of a car and a bridge abutment, typically this frame is almost exactly the same as the frame of the bridge abutment. And the energy of the bridge abutment in this frame is very close to zero, so the total energy is very close to .5*m_car*v_car^2.

Now, if you were running into a duck, the situation would be different. You would be more massive than the duck, and you'd get a good estimate of the energy of collision by using the car frame as a reference. in this case the energy would be just .5*m_duck*v_duck^2. Here v_duck is measured from the car frame, so it's the total relative velocity of the duck.

Intermediate cases are intermediate. If you were running into a stationary car of the same mass as yourself, you'd wind up with a center of mass frame that was exactly midway between your frame and the stationary car's frame.

This would "waste" about half the energy, which is good when you are colliding cars. Each car has an energy of .5*m*(v/2)^2, i.e. m*v^2/8, so the total energy of both cars in the center-of-mass frame is only m*v^2/4.

But if you were colliding particles, and wanted the energy to be as high as possible, because you spent a lot of money to build your tevatron, this would be bad. Here you'd want to arrange a head-on collision between two particles.

 

[geometer]

Yes, you can treat acceleration in both SR and GR. The point everyone has been trying to make to OneEye is that acceleration is NOT relative. There is no experiment you make in a completely selfcontained room that will determine uniform velocity. You can determine acceleration. OneEye's original question was about turning which involves acceleration even if done at a constant speed.

Exactly! Thanks Halls, obviously I wasn't being too clear!

 

[humanino]

Yes, thank you HallsofIvy. I was probably even less clear that you geometer. I was saying that, in the free fall referential coordinates, there is apparently no acceleration, and in any case, even non-uniform acceleration, the proper referential always make the acceleration disappear. That was really confusing in the context of this discussion, because of course this is not an inertial frame.

In the case of non-uniform acceleration, one can even find an inertial frame in which the acceleration disappear at a single point in spacetime. That was linked to an earlier discussion I had with Pete on GR, curvature and the (speculative but interesting) possibility that curvature does not encompass all gravitational effects, especially when topological defect play in the game. I should have been more careful in keeping my preoccupations out of the current discussion.

Thanks again every one.

 

[ostren]

... If one system is accelerating, even uniformly, with respect to another system it is no longer an inertial system, and you should be able to detect acclerations and thereby determine who is really moving. ...

I think not. Even though a thrusting event produces a sensation in the seat of the pilot's pants, it cannot be differentiated from a gravitational thing and so you are still technically unable to conclude "who is really moving". I find relativity easiest to understand when the observatory is always considered at rest. What power made the Earth and cosmos twist under your car? maybe just some fictitious force, especially considering that GR is full of such forces.

 

[humanino]

Welcome in PF ostren !

It depends ! I agree with you, but we had several misunderstanding during the discussion !
I think geometer was answering the initial question in the context of SR, without gravitational phenomena.

 

[russ_watters]

I think not. Even though a thrusting event produces a sensation in the seat of the pilot's pants, it cannot be differentiated from a gravitational thing and so you are still technically unable to conclude "who is really moving".

Since you can correlate the sensation in your pants (eh...I'll let that one go) with the turning of the steering wheel and the front wheels of the car, you can certainly conclude you are the one accelerating. Same goes for a rocket - you push the "fire" button, hear the engine start, see the flame, and feel the acceleration, you know you are the one accelerating.

 

[ostren]

Well, you might know it's you who is accelerating, or you might equally presume you have encountered a gravitational field; but in either case, you will not be able to make the unequivocal determination that you are "truly in motion". That last was the only conclusion that I was countering.