I have been running this through my head for a few days and I understand I am probably missing something very simple here. If a wheel is spinning the outer edge will travel at a speed that is greater than the rotational speed of the center of the wheel as there is more distance to cover to keep up with the center. Should this same concept not apply to all speeds, including the speed of light. If the center is clocked at a rotation that equals the speed of light would the edge be clocked at a speed that is faster? I know this cannot be tested in a practical sense but is there a mathematical answer to this or have I, as I said earlier, missed something fundamental here?
Asked
Active
Viewed 4,150 times
3 Answers
2
The short answer: Not in a physical system . In physical systems nothing can go faster than the velocity of light . The constraints of Lorenz invariance have been validated experimentally innumerable times. The limit of the velocity of light to any motion comes out mathematically from Lorenz invariance.
A physical disk , if enough energy is supplied, [since Lorenz invariance transforms energy to an inertial mass the higher the speed the more energy needed] can approach the speed of light only at its periphery: approach but never reach.
anna v
- 236,935
1
To paraphrase the accepted answer here, the disk becomes subject to non-Euclidean geometry and therefore the ratio of the radius of a point on the disk and the circumference travelled by that point is less than 2$\pi$.
That means in the limit case the entire disk, or rather each chosen point on the disk, is rotating at the same speed of light, even though the various points have different radii, because the normal way (using Euclidean geometry) of calculating the path length traveled by a given point on the disk in a given amount of time doesn't apply.
Put more simply, at relativistic speeds, the difference in speed between the edge of a disk and a point near its center is lower than classical mechanics would predict. As the innermost moving point on the disk (note the very center does not move) approaches the speed of light, the difference in speed between the outermost and innermost points approaches zero.
In reality, such a disk in such motion is not possible.
Todd Wilcox
- 222
1
This type of question is complicated because you are talking about dynamics of a continually accelerating body in a relativistic manner, but it's easy to show that the speed of light can not be surpassed without needing to get into the nasty details.
Let's simplify the problem and look at just a single particle in a pseudorelativistic way. A particle in the disc is moving at a constant speed which we can adjust, and just look at the tangential velocity for a second. We will find that it would still require infinite energy to rotate that single particle s.t. it's speed approached the speed of light. The rotational kinetic energy is going to be.
$T_{rotational}={1\over2} I\omega^2 = {1 \over 2} mv^2 $
$\omega={v\over r}$
$I=mr^2$
The energy of a single particle rotating in the disc is going to be proportional to the. mass m, defined as $m = \sqrt{{m_0}\over{1-{v^2\over{}c^2}}}$
Which will tend to infinity as speed increases, ergo the energy scales to infinity at high c. Now scaling this up its easy to see that each particle has at least this much energy. This idea gets much trickier once you go relativistic in angular momentum, with the introduction of the stress–energy tensor and a much more rigorous treatment is required.
Skyler
- 1,989