Biking at near the speed of light

[Cosmobrain]

Biking at near the speed of light

If a biker is going at 0.99c, an observer standing still would notice his clocking moving slower.Fine. But then how would the wheels of the bicycle match the ground? Would the bike hover over the ground? thanks

 

[Drakkith]

Biking at near the speed of light

If a biker is going at 0.99c, an observer standing still would notice his clocking moving slower.Fine. But then how would the wheels of the bicycle match the ground? Would the bike hover over the ground? thanks

The velocity of the wheels, according to the stationary observer, varies from 0c at the point where they touch the ground, to faster than the bike is moving at the apex of the wheel's arc.

 

[Cosmobrain]

The velocity of the wheels, according to the stationary observer, varies from 0c at the point where they touch the ground, to faster than the bike is moving at the apex of the wheel's arc.

Sorry but that didn't explain much to me. I don't know what you're talking about.

I am going to attempt to answer my own question. Perhaps because of the effects of length contraction, even with a slower spin of the wheel, the tire can match the ground and not hover over. I'm not quite sure

cb

 

[phinds]

Why do you think the tires would hover as opposed to touching the ground? I just can't figure any basis for even HAVING that thought, so please share yours.

 

[Cosmobrain]

Why do you think the tires would hover as opposed to touching the ground? I just can't figure any basis for even HAVING that thought, so please share yours.

sorry, that's not what I meant. What I'm saying is that the bike would seem to be sliding on the ground, because of time dilation making the wheels spin less fast.

cb

 

[phinds]

sorry, that's not what I meant. What I'm saying is that the bike would seem to be sliding on the ground, because of time dilation making the wheels spin less fast.

cb

Ah, I see. You have some misconception that the wheels would exhibit time dilation in a way differently from the bike and rider, which is not the case. The whole thing would actually appear pretty much motionless, depending on how close to c we are talking about. Far from looking like it was sliding over the ground, it would appear to be amazing that the guy could keep the bike upright while going so slow

 

[Drakkith]

Sorry but that didn't explain much to me. I don't know what you're talking about.

I am going to attempt to answer my own question. Perhaps because of the effects of length contraction, even with a slower spin of the wheel, the tire can match the ground and not hover over. I'm not quite sure

cb

You have two frames to consider this from. The ground, and the bike. From the Bike's frame, each point on the tread of the tire is moving at a steady speed but with a constantly changing direction. However, from the ground's frame of reference this is not so.

The part of the wheel that is in contact with the ground is stationary with respect to the ground. As the tire rotates, that particular point increases with velocity until it reaches maximum at the very top of the wheel. This is because the top of the wheel has to move faster than the bike in order to get in front of the bike. Remember that the part of the wheel in contact with the ground is applying a force in order to keep the bike moving against friction, air resistance, etc. It is only when the bike begins to skid or when it is "burning rubber" that the point of contact is no longer stationary with respect to the ground.

 

[Drakkith]

Ah, I see. You have some misconception that the wheels would exhibit time dilation in a way differently from the bike and rider, which is not the case. The whole thing would actually appear pretty much motionless, depending on how close to c we are talking about. Far from looking like it was sliding over the ground, it would appear to be amazing that the guy could keep the bike upright while going so slow

Not true. Different parts of the wheel are moving at different velocities with respect to the ground, and the wheels would have to spin very, very quickly indeed to keep up with how fast the bike is moving.

 

[craigi]

sorry, that's not what I meant. What I'm saying is that the bike would seem to be sliding on the ground, because of time dilation making the wheels spin less fast.

cb

You can't treat a wheel with SR, because every point on it is accelerating. You need GR for that and I doubt that a bicycle is a good introduction to the subject.

 

[Nugatory]

Biking at near the speed of light

If a biker is going at 0.99c, an observer standing still would notice his clocking moving slower.Fine. But then how would the wheels of the bicycle match the ground? Would the bike hover over the ground? thanks

First, think about the easy situation where the speed is nowhere near the speed of light, just somebody pedaling their bicycle down the road at the everyday speed of 20 km/hr. I'm going to describe this situation in detail because it's not clear from your response to Drakkith that you understand this case completely - if you do, and I'm telling you something that you already know, I apologize.

How fast is the bottom of the wheel, the part that's touching the ground, moving relative to the ground? It's not moving relative to the ground at all - if it were moving relative to the ground, the rubber would be rubbing against the ground, causing tire-squeal noises and smoke and leaving a black skidmark on the ground.

How fast is the the center of the wheel moving relative to the ground? Well, the center of the wheel is pretty obviously moving at the same speed as the bicycle because they aren't moving relative to one another. So if the bicycle is moving at 20 km/hr, that's the speed of the mounting point for the axle and therefore for the center of the wheel.

There's no contradiction here because the wheel itself is turning; if it's turning clockwise, the bottom of the wheel is moving from right to left and the top of the wheel is moving from left to right relative to the center of the wheel, which is moving at the same speed as the bicycle.

So in the easy non-relativistic case, the bottom of the wheel and the road are moving at 0 km/hr, the center of the wheel and the bicycle are both moving at 20 km/hr, and the top of the wheel is moving at 40 km/hr, all relative to the road. Relative to the bicycle, the bottom of the wheel and the road are moving at -20 km/hr, the center of the wheel and the bicycle are moving at 0 km/hr, and the top of the wheel is moving at 20 km/hr.

STOP HERE AND SATISFY YOURSELF THAT THE ABOVE MAKES SENSE BEFORE PROCEEDING (and as I said above, if this part is obvious and you already understand it, I apologize).

Now, what happens as the speed of the bicycle becomes relativistic, say .99c as you suggest?

As the far as the bicycle rider is concerned, the bottom of the wheel and the ground are both moving at -.99c relative to him; they're moving at the same speed so there's no skidding/smoking/skid marks being laid down on the road. The center of the wheel is at rest relative to him, and the top of the wheel is moving at a speed of .99c relative to him.

Relative to someone standing at the side of the road? The bottom of the wheel is still at rest relative to the road and this observer. The center of the wheel and the bicycle are moving at .99c. And the top of the wheel is moving at...? Well, it's moving at .99c relative to the rider, and the rider is moving at .99c relative to the road, so we have to use the relativistic velocity addition formula (google will find it if you're not already familiar with it) to calculate that the top of the wheel is moving at .99995c relative to the ground.

(It would be a good exercise to calculate the tension in the spokes of the wheel as a result of the centrifugal force at these speeds. Make the reasonable assumptions that the wheel has a mass of 5 kg, a diameter of one meter, and there are 100 wire spokes in the wheel, and you will conclude that this is not a situation in which you should trust your intuition).

 

[Vanadium 50]

Why do people keep saying that? SR handles accelerations just fine. It's gravity that it doesn't handle.

 

[Nugatory]

You can't treat a wheel with SR, because every point on it is accelerating. You need GR for that...

That's not true, although it's one of the more common misconceptions and repeated so often by so many pop-sci treatments that you can be forgiven for believing it up to now.

In fact SR works just fine for accelerations. The limitation of SR is that it only applies to flat spacetimes, which is to say no gravitational fields.

 

[Nugatory]

sorry, that's not what I meant. What I'm saying is that the bike would seem to be sliding on the ground, because of time dilation making the wheels spin less fast.

Different points on the circumference of the wheel are moving at different speeds relative to the ground (I just did a huge long post on this) so there is no one time dilation formula that you can apply to all points on the circumference. In particular, the point that is touching the ground is always at rest relative to the ground.

 

[jartsa]

Biking at near the speed of light

If a biker is going at 0.99c, an observer standing still would notice his clocking moving slower.Fine. But then how would the wheels of the bicycle match the ground? Would the bike hover over the ground? thanks

The maximum velocity of a bike is k * c.

I have always thought k is 0.5.

But if we consider the maximum velocity of a caterpillar, that kind of seems to be 0.5 c. (a piece of caterpillar track moves half of the time, is still half of the time)

So maximum speed of a bike may be less than 0.5 c.EDIT: k is actually 1. It's 1 for bikes and caterpillars, and also for the average velocity of a person that does interval training: runs very fast 1 minute according to his clock, and then rests 1 minute according to his clock.

 

[phinds]

Not true. Different parts of the wheel are moving at different velocities with respect to the ground, and the wheels would have to spin very, very quickly indeed to keep up with how fast the bike is moving.

Yes but my point was that time dilation would make the entire thing appear to be standing still, or very close. That isn't changed by the fact that the top of the wheel is moving faster than the bike.

 

[Nugatory]

The maximum velocity of a bike is k * c.

I have always thought k is 0.5.

But if we consider the maximum velocity of a caterpillar, that kind of seems to be 0.5 c. (a piece of caterpillar track moves half of the time, is still half of the time)

So maximum speed of a bike may be less than 0.5 c.

Imagine the bicycle is on a treadmill, so that the bicycle is at rest and the ground under it is moving. This is just looking at the things from an inertial frame in which the bicycle is at rest, instead of one in which the road is at rest.

Why should the speed of the bicycle be constrained? Bicycle at rest, bottom edge of wheel and road can move at any speed less than $c$ in the backwards direction relative to the bicycle, and the top of the wheel moves in the forward direction with the same speed.

 

[jartsa]

Imagine the bicycle is on a treadmill, so that the bicycle is at rest and the ground under it is moving. This is just looking at the things from an inertial frame in which the bicycle is at rest, instead of one in which the road is at rest.

Why should the speed of the bicycle be constrained? Bicycle at rest, bottom edge of wheel and road can move at any speed less than $c$ in the backwards direction relative to the bicycle, and the top of the wheel moves in the forward direction with the same speed.

Ok. Good reasoning.

 

[Drakkith]

Yes but my point was that time dilation would make the entire thing appear to be standing still, or very close. That isn't changed by the fact that the top of the wheel is moving faster than the bike.

But it wouldn't appear to be standing still. If you could watch an object moving at that velocity, the wheels would be turning extremely fast. Are you talking about the rest of the bike and the biker?

 

[bahamagreen]

I think you are all missing the point of the Op's question...

Cosmobrain, are you thinking of the spinning wheel as a clock, and expecting the period of this clock to increase... and so expecting the wheel to slow its rotation, the discrepancy being this slowing of rotation with increasing approach to c?
As in, at the limit, the wheel approaches zero rotation, yet the bike speed approaches c, hence the "hovering" or sliding translation of a non-rotating wheel?

I'm curious to see how this goes.

 

[Drakkith]

I think you are all missing the point of the Op's question...

I'm pretty sure we understand the OP's question. The answer just isn't easy. We aren't dealing with an object moving in a constant linear manner, but a much more complicated scenario involving not only multiple reference frames, but accelerating frames at that. And all of them are attached to each other.

 

[georgir]

Nugatory, I guess a more relevant analysis would not concern itself with speeds of any particular point of the wheels, but rather the wheel circumference length and the number of revolutions per second.

 

[A.T.]

Yes but my point was that time dilation would make the entire thing appear to be standing still, or very close.

No, not the "entire thing". The bottom parts of the wheels are not moving at all, are much slower than c, so they are not subject to significant time dilation.

 

[TumblingDice]

I'm pretty sure we understand the OP's question. The answer just isn't easy. We aren't dealing with an object moving in a constant linear manner, but a much more complicated scenario involving not only multiple reference frames, but accelerating frames at that. And all of them are attached to each other.

I see a part of the problem is in the formation of the thought experiment itself. It includes rotation at relativistic speeds, and this introduces all types of real problems. I think this thread could move in a different direction if the rotational gang, WannabeNewton, PeterDonis, pervect, et al. drop by. WBN would say at best we have to assume the wheels have rotated at that speed forever (i.e., they were never "spun up" to speed) . And we haven't even begun to address stress and strain factors.

The OP asked what wheels would look like to an observer standing still, yet no one has mentioned length contraction. Those wheels aren't going to have anything near to the geometry of a circle to the observer.

I think there are problems with the original premise of the thought experiment that will only drive people crazy trying to explain the whole picture, piece-by-piece.

 

[A.T.]

If a biker is going at 0.99c, an observer standing still would notice his clocking moving slower.Fine. But then how would the wheels of the bicycle match the ground? Would the bike hover over the ground? thanks

No, the speed of the wheel bottom would still be zero, so it would not look like sliding. But it would look very distorted, because different parts of the wheels are time dilated & length contracted by different factors. You also have to differentiate between what happens in your frame (measured), and what you would see with your eyes (with includes signal delay).

Here is what you would measure the rolling wheel to be:
http://www.spacetimetravel.org/tompkins/node7.html#rad1

Here is what you would see:
http://www.spacetimetravel.org/tompkins/node8.html#rad2

 

[phinds]

But it wouldn't appear to be standing still. If you could watch an object moving at that velocity, the wheels would be turning extremely fast. Are you talking about the rest of the bike and the biker?

If you watch something that is traveling near c relative to you, it DOES appear to be standing still. That's what time dilation MEANS.

 

[A.T.]

If you watch something that is traveling near c relative to you, it DOES appear to be standing still.

Traveling near c and standing still? This is not a good way to put it.

That's what time dilation MEANS.

Time dilation means that clocks on the bike run slower in the ground frame, than clocks on the ground. But the clocks on the wheel bottom still run at the same rate as the clocks on the ground. And that is the whole point of the OP:

How can the RPM at the wheel hub be reduced due to time dilation, but the tangential velocity at the wheel bottom still cancel the linear velocity of the bike, so the wheel bottom is stationary relative to the ground?

And the answer to this, is the distortion along the circumference shown in the links a posted above.

 

[craigi]

Ok, so I completely concede the point that SR does indeed handle accelerations. I suspect that it's either a misnomer that I generated myself rather than having anyone else to blame, since I learned SR (or at least some of it!) from a teaching text rather than a pop-science book.

I'm not going to try to claim partial vindication, but the problem of a rotating disk was an important step in creating the theory of GR.

This helps explain the situation:
http://en.wikipedia.org/wiki/Ehrenfest_paradox

 

[Cosmobrain]

Well, I guess it is time for me to say something.

So far I believe that Nugatory, in his long post, has explained it better. Indeed, the wheel is spinning counter clockwise (or clockwise, if the observer is on the other side of the bicycle) and the bottom of the wheel is canceling the forward motion of the vehicle. However, I'm still a little confused. The top of the wheel should be twice as fast. However, it can't be 1.98 c. That'd be an absurd. It would go, more specifically (as Nug said), at 0.99995c or something. Now... wouldn't this make the wheel look distorted? The top of the wheel isn't twice the velocity of the center of it. Again, what would it look like? (let's consider the wheel is resistant enough)

cb

 

[Drakkith]

Did you look at the links in post #24?

 

[Cosmobrain]

Whoops. Totally missed that. Thanks, guys.

cb

 

[phinds]

Traveling near c and standing still? This is not a good way to put it.

I didn't say it WAS standing still, I said it will APPEAR to be standing still.

Time dilation means that clocks on the bike run slower in the ground frame, than clocks on the ground. But the clocks on the wheel bottom still run at the same rate as the clocks on the ground. And that is the whole point of the OP:

How can the RPM at the wheel hub be reduced due to time dilation, but the tangential velocity at the wheel bottom still cancel the linear velocity of the bike, so the wheel bottom is stationary relative to the ground?

And the answer to this, is the distortion along the circumference shown in the links a posted above.

OK, I did get it wrong about the bottom of the wheel being stationary in the ground frame. My bad.

The point I was getting at (and missing the bottom of the wheel thing) is that things moving at high speed relative to you look slow from your perspective but I agree that's not what the OP was getting at.

 

[A.T.]

things moving at high speed relative to you look slow from your perspective

Things moving fast are fast and look fast. What is slow is the rate of clocks attached to these fast things.

 

[phinds]

Things moving fast are fast and look fast. What is slow is the rate of clocks attached to these fast things.

So you are saying that to a remote observer an object falling into a black hole does NOT appear to slow down as it approaches the event horizon? It's the same thing.

EDIT: Or, I suppose, perhaps you believe that objects falling into a black hole are not moving fast?

 

[phinds]

But it wouldn't appear to be standing still. If you could watch an object moving at that velocity, the wheels would be turning extremely fast. Are you talking about the rest of the bike and the biker?

Actually, I'm talking about the whole thing. Although I realize that the top of the wheel is moving even closer to light speed, the whole thing still appears to a remote observer to be moving slowly, just like an object under gravitational time dilation that is falling into a black hole.

 

[phinds]

Things moving fast are fast and look fast. What is slow is the rate of clocks attached to these fast things.

When you say the clocks are slow, what are you comparing them to? Certainly they are NOT slow in the frame of reference of the object they are with, so you must mean that they appear slow to a remote observer. I agree. So does the object they are with.

 

[A.T.]

So you are saying that to a remote observer an object falling into a black hole does NOT appear to slow down as it approaches the event horizon? It's the same thing.

No, it's not. We are talking about relative movement in flat space time here. There is no gravitational time dilation involved here.

 

[A.T.]

When you say the clocks are slow, what are you comparing them to?

Moving clocks run slow compared to resting clocks. That has nothing to do with "remoteness". The clocks can pass very close to each other.

 

[DrGreg]

Although I realize that the top of the wheel is moving even closer to light speed, the whole thing still appears to a remote observer to be moving slowly

The speed of the bicycle relative to the ground is already specified to be 0.99c, so you can't say it is going slower than that. That is also the speed of the ground relative to the bicycle and therefore the speed of the wheel rim relative to the wheel centre.

[STRIKE]You are failing to take account of length contraction. According to the observer riding the bicycle, the circumference of the wheel is length contracted and a lot less than 2\pi r, and therefore the angular velocity of the wheel (revs per second) required the make the rim move at 0.99c must be a lot faster than you would expect from Newtonian physics. This counteracts the dilation you refer to above.[/STRIKE]

EDIT: I got that the wrong way round. In the frame of the biker the circumference still has to be 2\pi r, but that value is length-contracted from the "rest length of rubber in the tyre", which is much more than that. The rubber that touches the ground is at rest relative to the ground and so the effective length of rubber that touches the ground over one revolution of the wheel is more than 2\pi r. So to summarise:

In the biker frame the wheel's angular velocity is what Newton would expect.

In the ground frame the wheel's angular velocity is slower than Newton would expect, but this is explained by the rest-length of rubber in the tyre being longer than expected.

 

[jartsa]

The speed of the bicycle relative to the ground is already specified to be 0.99c, so you can't say it is going slower than that. That is also the speed of the ground relative to the bicycle and therefore the speed of the wheel rim relative to the wheel centre.

You are failing to take account of length contraction. According to the observer riding the bicycle, the circumference of the wheel is length contracted and a lot less than 2\pi r, and therefore the angular velocity of the wheel (revs per second) required the make the rim move at 0.99c must be a lot faster than you would expect from Newtonian physics. This counteracts the dilation you refer to above.

The height of the wheel is the same in the bike frame and in the road frame. In the bike frame the width of the wheel is the same as the height of the wheel. But in the road frame that width is contracted.

(I think we have been considering a wheel that does not contract in the bike frame: spokes sturdier than rim)

 

[liometopum]

The center of the wheel and the bicycle are moving at .99c. And the top of the wheel is moving at...? Well, it's moving at .99c relative to the rider, and the rider is moving at .99c relative to the road, so we have to use the relativistic velocity addition formula (google will find it .

Great post! I question, though, the last part:

The top of the tire may have a velocity of .99c relative to the rider, but it is not going anywhere; it is not moving translationally relative to the rider. There is no change in the distance between the two. The top of the tire does not move.

Rotational velocity and translational velocity are different. The use of the relativistic velocity addition equation to add rotational velocity to translational velocity is thus questionable.

 

[Nugatory]

Great post! I question, though, the last part:

The top of the tire may have a velocity of .99c relative to the rider, but it is not going anywhere; it is not moving translationally relative to the rider. There is no change in the distance between the two. The top of the tire does not move.

Rotational velocity and translational velocity are different. The use of the relativistic velocity addition equation to add rotational velocity to translational velocity is thus questionable.

Perhaps I should have said "the piece of rubber that is at the top of the tire right now". That piece of rubber is, right now, moving at .99c relative to the rider, and this is a straightforward movement in the same direction as the rider. Yes, if we wait a moment that piece of rubber will be moving in a different direction, but that just means that we have a different velocity to plug into the velocity addition formula a moment later.

 

[liometopum]

I understand what you are saying (and you say and know it well!). But the velocities are different, like apples and oranges to use the old cliche. I don't think you can use the relativistic velocity addition equation here. The velocities are not of the same type.

 

[Nugatory]

The velocities are not of the same type.

Why not? Velocity is defined to be $\frac{d\vec{r}}{dt}$, where $\vec{r}$ is the position vector. That definition works just fine for things moving in circles like the wheel, in straight lines like bicyclist and the road, or in arbitrarily complicated trajectories.

 

[A.T.]

it is not moving translationally relative to the rider. There is no change in the distance between the two.

Transnational velocity doesn't require change in distance.

Rotational velocity and translational velocity are different.

Yes, and the top of the wheel has both in the frame of the bike:

radius * angular_velocity = linear_velocity

 

[liometopum]

radius * angular_velocity = linear_velocity

That equation gives tangential velocity, not linear velocity. More specifically, it is the tangential velocity of a spinning object at rest (not moving translationally relative to the observer) in your reference frame.

Which leads to maybe the most critical part of this idea. Translational velocity changes the apparent tangential velocity. That is, a spinning object moving relative to an observer appears to have a lower rim speed.