Is Velocity Addition in Special Relativity Contradictory?

[Austin0]

Waht is your definition of a composite object. My intuitive and very very loose idea of such a body would be one whose "component" parts are constrained to remain at rest relative to each other when referred to an inertial frame in which one (any one) of the component parts is at rest. Is there an accepted definition?

Surely an object is moving inertially if the resultant forces acting upon it are zero. In this case the resultant of frictional forces and forward acceleration cancel out to give a constant forward speed for the body of the tanks. As in most cases of course we consider the earth, for our purposes, to be moving inertially

Matheinste

Are you saying that the tank is inertial because the constant acceleration is being evenly countered by friction and inertia to maintain a steady velocity??

Do you think that there is no difference between a tank suspended by Mike 's crane with no resistence on the bottom track and the same tank on the ground where the bottom track is under great force, between the friction and acceleration of the ground and the acceleration of the drive which is suffcient to overcome that and propel the mass forward??

DO you think the actual length contraction could be uneffected by the counter force of the Earth and the stress involved?
I.e. the air tank could have exactoly the same contraction of the bottom track as the ground tank where there is great force in effect, stretching or resisting contraction?

 

[matheinste]

Are you saying that the tank is inertial because the constant acceleration is being evenly countered by friction and inertia to maintain a steady velocity??

Do you think that there is no difference between a tank suspended by Mike 's crane with no resistence on the bottom track and the same tank on the ground where the bottom track is under great force, between the friction and acceleration of the ground and the acceleration of the drive which is suffcient to overcome that and propel the mass forward??

DO you think the actual length contraction could be uneffected by the counter force of the Earth and the stress involved?
I.e. the air tank could have exactoly the same contraction of the bottom track as the ground tank where there is great force in effect, stretching or resisting contraction?

Basic physics says that if the velocity of a body is constant the the resultant sum of the forces applied is to it zero and vice versa. Constant velocity of an object implies that the object is at rest in some inertial frame. The top and bottom tracks are not at rest in the same frame as each other or at rest in the same rest frame as the tank.

As for stressed contactions and stretchings, that's more of an engineering problem and too complicated.

Matheinste

 

[Austin0]

Basic physics says that if the velocity of a body is constant the the resultant sum of the forces applied is to it zero and vice versa. Constant velocity of an object implies that the object is at rest in some inertial frame. The top and bottom tracks are not at rest in the same frame as each other or at rest in the same rest frame as the tank.

As for stressed contactions and stretchings, that's more of an engineering problem and too complicated.

Matheinste

In an elevator rising at constant velocity wrt the Earth do you think you are in an inertial frame??

In the research I have done so far on the Ehrenfest question I found that they definitely took into consideration the centrifugal forces inherent in acceleration up to speed.
That the conclusion was that the force of expansion would override contraction until the cessation of acceleration. In the tank question acceleration never stops but is an ongoing application of force.
SO it would seem to be more than simply an engineering question and while certainly complicated is that reason enough to ignore it and assume that the effects are not relevant?

You did not answer the main point: DO you think the contraction that would take effect could be the same in a tank suspended without stress or resistence or acceleration from the ground effecting the track and the same tank on the ground with the bottom in contact with the ground and the whole track also being subjected to the stress between the acceleration of the drive wheel and the inertia of the tank body?

 

[yuiop]

In an elevator rising at constant velocity wrt the Earth do you think you are in an inertial frame??

In this case the elevator is not an inertial frame, but then again, even when the elevator is stationary with respect to the Earth, it is still not in an inertial frame. This is an effect due to the gravity and in SR problems we normally ignore gravity. In the elevator (moving with constant velocity or stationary with respect to the Earth) a vertical accelerometer would always show a non-zero reading. The tank in the OP is not moving vertically. A vertical accelerometer in the tank would also show a non-zero (but constant) reading, but a horizontal accelerometer in the tank body would always read zero and it is the horizontal motion we are mainly concerned with.

In the tank question acceleration never stops but is an ongoing application of force.
...
You did not answer the main point: DO you think the contraction that would take effect could be the same in a tank suspended without stress or resistence or acceleration from the ground effecting the track and the same tank on the ground with the bottom in contact with the ground and the whole track also being subjected to the stress between the acceleration of the drive wheel and the inertia of the tank body?

The "ongoing application of force" is mainly to overcome air resistance. In these idealised SR thought experiments we normally consider everything to be taking place in a vacuum (because the speed of light and the Lorentz transformations assume a vacuum). A car moving at 180 mph would not require any fuel to maintain that speed in a vacuum (if we ignore the rolling/bearing friction). Over 90% of the horsepower and fuel consumption of a supercar cruising at 180 mph is being used to overcome air resistance. However, air or no air, if the tank body has constant velocity relative to the Earth surface then the tank will have zero proper acceleration in the horizontal direction which is the motion we are mainly concerned with. An accelerometer attached to a tank track segment will show acceleration as it travels around the curved sections of the track, but when moving along the top section, or along the bottom section in contact with the ground, the track segment would show zero horizontal acceleration and we can consider the tank body and horizontal sections of the track to have inertial motion for our purposes here. If you are analysing a problem, with a view to elucidating some physical fact it is best to keep things as simple as possible.

If we want to obfuscate everything as much as possible, perhaps to cover a blunder, then we keep adding complications to cover our "tracks" to make the problem intractable so that no one can figure out what is going on. For example, we could say that at 0.45c the tank is exceeding the escape velocity of the Earth and as it travels along the curved surface of the Earth, it will probably take off and go into orbit. You have probably not noticed until now that we are assuming a flat Earth in this problem, which we know is not true. Anyway, if we remove the atmosphere, assume a flat Earth and negligible rolling resistance, then in this idealised thought experiment the answer to "DO you think the contraction that would take effect could be the same in a tank suspended without stress or resistance or acceleration from the ground effecting the track and the same tank on the ground with the bottom in contact with the ground and the whole track also being subjected to the stress between the acceleration of the drive wheel and the inertia of the tank body" is (to a good approximation) yes.

No amount of obfuscating, such as considering air resistance or curvature of the Earth's surface is going to make it a universal truth, that the top track moves at exactly twice the velocity of the tank body in the Earth frame and that is the question posed in the OP, no?

 

[matheinste]

In an elevator rising at constant velocity wrt the Earth do you think you are in an inertial frame??

In the research I have done so far on the Ehrenfest question I found that they definitely took into consideration the centrifugal forces inherent in acceleration up to speed.
That the conclusion was that the force of expansion would override contraction until the cessation of acceleration. In the tank question acceleration never stops but is an ongoing application of force.
SO it would seem to be more than simply an engineering question and while certainly complicated is that reason enough to ignore it and assume that the effects are not relevant?

You did not answer the main point: DO you think the contraction that would take effect could be the same in a tank suspended without stress or resistence or acceleration from the ground effecting the track and the same tank on the ground with the bottom in contact with the ground and the whole track also being subjected to the stress between the acceleration of the drive wheel and the inertia of the tank body?

As regards your first question, although it sounds simple I am not confident that I can give a correct answer as my knowledge as yet is limited to SR but when gravity is involved, as implied by the elevator, we are in the realms of GR.

As regards stresses and strains on the tank and tracks I still feel this is an engineering problem. You must of course be careful to distinguish between SR length contraction, which involves no stress, and mechanical contractions which are stressed but not caused by relativistic effects. I am no expert but I suspect that the relativistic effects involved look after themselves.

Matheinste.

EDIT After reading kev's response, I see that my response is probably redundant.

 

[Austin0]

In this case the elevator is not an inertial frame, but then again, even when the elevator is stationary with respect to the Earth, it is still not in an inertial frame. This is an effect due to the gravity and in SR problems we normally ignore gravity. In the elevator (moving with constant velocity or stationary with respect to the Earth) a vertical accelerometer would always show a non-zero reading. The tank in the OP is not moving vertically. A vertical accelerometer in the tank would also show a non-zero (but constant) reading, but a horizontal accelerometer in the tank body would always read zero and it is the horizontal motion we are mainly concerned with.

So you agree that constant motion in this context is not the only consideration for inertial motion.
I was thinking more of the track itself but then again the normal acceleration on the mass of the tank is a relevant factor.

The "ongoing application of force" is mainly to overcome air resistance. In these idealised SR thought experiments we normally consider everything to be taking place in a vacuum (because the speed of light and the Lorentz transformations assume a vacuum). A car moving at 180 mph would not require any fuel to maintain that speed in a vacuum (if we ignore the rolling/bearing friction). Over 90% of the horsepower and fuel consumption of a supercar cruising at 180 mph is being used to overcome air resistance. However, air or no air, if the tank body has constant velocity relative to the Earth surface then the tank will have zero proper acceleration in the horizontal direction which is the motion we are mainly concerned with. An accelerometer attached to a tank track segment will show acceleration as it travels around the curved sections of the track, but when moving along the top section, or along the bottom section in contact with the ground, the track segment would show zero horizontal acceleration and we can consider the tank body and horizontal sections of the track to have inertial motion for our purposes here. If you are analysing a problem, with a view to elucidating some physical fact it is best to keep things as simple as possible.

Can we not consider a spring a simple accelerometer?
DO you think that springs between the track segments would have equal extension or compression between the top and bottom track?
Do you think that they would have equal extension or compression between the front wheel and the back in the top track??
DO you disagree that there is rapid decceleration of the top segments when they reach the drive wheel and are redirected around to the bottom??
And likewise when moving from the bottom to the top??
DO you think this would continue purely on inertia without a constant input of energy?

If we want to obfuscate everything as much as possible, perhaps to cover a blunder, then we keep adding complications to cover our "tracks" to make the problem intractable so that no one can figure out what is going on. For example, we could say that at 0.45c the tank is exceeding the escape velocity of the Earth and as it travels along the curved surface of the Earth, it will probably take off and go into orbit. You have probably not noticed until now that we are assuming a flat Earth in this problem, which we know is not true. Anyway, if we remove the atmosphere, assume a flat Earth and negligible rolling resistance, then in this idealised thought experiment the answer to "DO you think the contraction that would take effect could be the same in a tank suspended without stress or resistance or acceleration from the ground effecting the track and the same tank on the ground with the bottom in contact with the ground and the whole track also being subjected to the stress between the acceleration of the drive wheel and the inertia of the tank body" is (to a good approximation) yes.

Very subtle kev. Almost creates the impression of a meaningful reductio ad absurdum argument against considering acceleration a valid consideration in this problem.
At the same time implying that I am making similar objections of a whimsical nature such as your suggestions and impugning my motive for offering any suggestions at all.
Good work.

No amount of obfuscating, such as considering air resistance or curvature of the Earth's surface is going to make it a universal truth, that the top track moves at exactly twice the velocity of the tank body in the Earth frame and that is the question posed in the OP, no?

I will say it once again; it was only very late in this thread that I ever suggested that the 0.9c velocity might be correct. After consideration of your contraction workup I put that back on hold and have been considering contraction and it's implications as well as studying the Ehrenfest problem which I was not really familiar with before.
Having gotten a little information on this it is clear that in that problem, which is actually much simpler than the tank question , it took many years of work by many brilliant people to resolve the question and reach concensus. It was not considered a simple question and as far as I know now, was not resolved simply through normal application of contraction but required advanced coordinate systems and perspectives.
And the system in question was completely independant, after spinup there was no acceleration involved. There was no part of the system that was at rest in another frame and there were only two frames involved in the first place.
Here there are a minimum of 4 frames, a complex physical interaction between various combinations and other unique features.
So if you think that taking this complexity into consideration is obfuscation , that questioning a quick simple evaluation derived through absolutely ignoring all these other factors is just contrariness or whatever you think is my motivation I don't know what to tell you.
All I can say is I did not dismiss your work on contraction but have given it a great deal of thought and am still. I really don't care what the actual answer to the question is at this point , in a real sense I am just stuck by my own inability to forget it until SOME really satisfactory , totally compelling resolution is found by somebody.

 

[yuiop]

Can we not consider a spring a simple accelerometer?

Depends on how you use it. If you fix one end to the tank and the other to a test mass, then if the spring bends, the tank is accelerating (which it is not in this thread). If you mean springs connecting track segments, then maybe not. The springs might be stressed due to length contraction at constant speed (as in the Ehrenfest paradox). If there is significant air resistance then the force of the drive required to overcome the air resistance will be measured in the springs, but that measured stress is not a reflection of the acceleration of the tank because it would be there when the tank has constant velocity.

DO you think that springs between the track segments would have equal extension or compression between the top and bottom track?

The extension is equal in the tank frame if we have negligible air and rolling resistance. If you insist of have non negligible resistance, then you would have to figure out how much of the extension is due to resistance and how much is due to length contraction. The question is complicated enough without that. The way to solve things is to make the situation as idealistic and as simple as is reasonable.

DO you disagree that there is rapid deceleration of the top segments when they reach the drive wheel and are redirected around to the bottom?? And likewise when moving from the bottom to the top??

I am not disputing the acceleration. See next comment.

DO you think this would continue purely on inertia without a constant input of energy?

Yes, if the system had perfect bearings. It is the same as a wheel. It will continue to spin forever if it had a perfect bearing despite parts on the rim constantly being accelerated in different directions. In the case of the wheel we can simulate a perfect bearing by spinning the wheel in space without an axle.

Very subtle kev. Almost creates the impression of a meaningful reductio ad absurdum argument against considering acceleration a valid consideration in this problem.
At the same time implying that I am making similar objections of a whimsical nature such as your suggestions and impugning my motive for offering any suggestions at all.

I though my comment was mischievous at the time, but I unreservedly apologise for any impugning implied. I wholeheartedly accept that your motives are good and you have a genuine desire to solve and understand this problem to your own satisfaction. My bad.

I will say it once again; it was only very late in this thread that I ever suggested that the 0.9c velocity might be correct. After consideration of your contraction workup I put that back on hold and have been considering contraction and it's implications as well as studying the Ehrenfest problem which I was not really familiar with before.
Having gotten a little information on this it is clear that in that problem, which is actually much simpler than the tank question , it took many years of work by many brilliant people to resolve the question and reach concensus. It was not considered a simple question and as far as I know now, was not resolved simply through normal application of contraction but required advanced coordinate systems and perspectives.

The Ehrenfest problem is really quite simple if you accept length contraction and the clock postulate. It only took so long to figure out the paradox, because people over complicated it and had ingrained misconceptions that were hard to shake off.

So if you think that taking this complexity into consideration is obfuscation , that questioning a quick simple evaluation derived through absolutely ignoring all these other factors is just contrariness or whatever you think is my motivation I don't know what to tell you.

Well as Einstein said, "physics should be made as simple as possible, but no simpler" or something like that. If the top track is moving at 0.45c in the tank frame then it is moving at (0.45+0.45)/(1+0.45*0.45) in the ground frame and that is the end of the story and what the track does as it rounds the wheel is irrelevant. Trying to analyse what happens at the wheel in the ground frame is horrendously complicated because the wheel is elliptical in the ground frame and translating and as well as rotating at the same time. To get an idea of the complexity, have a look at this post in an old thread https://www.physicsforums.com/showthread.php?t=233399&highlight=rolling+wheel. Why go to all that trouble when it is not even required in this problem? You would have to be some sort of masochist to figure it out and fortunately Dalespam is that sort of masochist and did figure it out for us, but it not helpful here.

All I can say is I did not dismiss your work on contraction but have given it a great deal of thought and am still. I really don't care what the actual answer to the question is at this point , in a real sense I am just stuck by my own inability to forget it until SOME really satisfactory , totally compelling resolution is found by somebody.

I think what is confusing is that the Newtonian analysis is perfectly mathematically correct and consistent on its own terms and so is the relativistic analysis. You can not determine what is a correct theory by mathematics alone. You have to carry out some actual experiments and see if they agree with the theory and the experimental results suggest Newtonian theory is wrong and SR is correct.

The Newtonian solution suggests the top track moves at twice the speed of the bottom track in the ground frame and that is perfectly self consistent, but it means you have to reject:

1) The relativistic velocity addition law.
2) Relativistic length contraction.
3) The relativity of simultaneity.
4) The postulates of relativity.

and everything else about relativity. If you accept any of the above 4 items, then the top track does not move at twice the speed of the bottom track in the ground frame.