I think a rotating rod is similar enough that it fits in this thread. What would happen if we were dealing with a really long rod, say a light year in length, and set it to rotate rapidly around the central point? Would the ends of the rod, where it rotates the fastest, gain mass? Would there be some meaningful difference between the two ends of the rod?
IANOP – so someone please correct any misconceptions here – but I would infer from this and some other comments that uniform circular motion may represent either an inertial or a non-inertial reference frame depending on what is causing the circular motion. An object in orbit around the earth would be an inertial reference frame because there is no net force acting on it and hence it experiences no acceleration. My understanding is that GPS adjustments for time dilation involve a straightforward application of special relativity to account for time dilation due to speed, and general relativity to account for the opposite effect due to diminished gravity compared to clocks on the surface of the earth.
However, putting aside GR effects of gravity for the moment and considering only special relativity time dilation, I would think that the relativistic equivalence of all inertial frames means that just as we see GPS clocks running more slowly due to their speed, the GPS satellites would see clocks on earth also running more slowly by the same amount (again, ignoring gravity for the moment).
That being the case, it would seem that the old meme about astronauts returning from orbit being a few microseconds younger than they otherwise would have been due to the time they spent orbiting at high speed is false, because though we would see them aging more slowly they would also see us aging more slowly. In fact, taking GR into account, they should actually be a few microseconds or so older, because their biological clocks would run faster in lower gravity.
I would think that the person magically able to run in circles at near light speed would be in a non-inertial frame due to centripetal force and due to this asymmetry would therefore age less than the rest of the world and emerge younger than everyone else, as in the twin paradox.
Nah, SR can handle acceleration just fine. Where you need GR is to account for the fact that I and a person in Australia can both be accelerating, in nearly opposite directions, while still staying at rest relative to each other.
Check me on this model. Imagine that the person circling is on the edge of a merry-go-round of radius r, with velocity v. Acceleration is therefore v^2/r. In a non-inertial corotating reference frame there’s no velocity, just an acceleration pervading space. Therefore, relative to the center of the merry-go-round, the person on the edge has a gravitational time dilation that’s the integral of the local acceleration (which decreases closer to the center) times dx (distance to the center). Since the center of the merry-go-round is not accelerating or moving, the center of the merry-go-round is in the same reference frame as someone standing next to the merry-go-round, and so will have the same time dilation relative to the circler. Make sense?
Never mind human-sized. A baseball suffices.
Y’all are doing calculus and shit while I, the OP am over here struggling to remember if it’s PEDMAS or PEMDAS.
Sure. We can spin things on earth fast enough to break.
The reason being the difference between the force at the outside and the center.
No problem doing that with your light year long rod (assuming we can spin it fast enough…probably would not have to go very fast at all but I will leave the math to others).
There is no material that won’t break at some point and it will happen way, way, way before you get anywhere near lightspeed.
Yes uniform circular motion is non-inertial in special relativity. If we just want the time dilation of a clock travelling in uniform circular motion, as observed by an inertial observer it is just dependent on the speed of the clock.
One interesting effect that time dilation would have on a rotating disk is that it makes each second shorter, so the centrifugal acceleration would appear greater for someone in that rotating frame of reference. I think this means that (even if you had a disk made from impossibly strong material) it would fly apart at a slower speed than it would if time dilation did not exist.
Unfortunately I think the flaw in this model is it starts from Newtonian assumptions and not relativistic assumptions. Constructing a rotating frame of reference is a bit more complicated, see Born ccoordinates for the frame of a rigid rotating disk:
You’re both right and wrong: the acceleration of someone in the rotating frame of reference is greater than their cooridnate acceleration in an inertial frame, but the effect is actually to make the disk need to be stronger. The force which accelerates an observer on the disk is the inernal forces of the disk, so these are greater.
You also can’t get out of this by saying “a disc would break before it reached relativistic speed”. All speeds are relativistic, with sensitive enough equipment. In fact, with the right setup, you can measure relativistic effects from something moving at a few centimeters per second, or slower, using only supplies you’d find in an underfunded middle school science lab.
I’m not casting doubt here, but I would love a cite for this as I would be interested in trying out such an experiment/demonstration.
Thanks.
You’ve probably already done it. If you hook a foot of copper wire up to a 1.5 volt battery, the electrons will be moving through the wire at just about exactly 1 cm/s. Wind that copper wire around a nail before you do it, and you’ve made an electromagnet, that you can detect with a compass. And the magnetic field is, entirely and completely, a relativistic consequence of what happens to an electric field, when the charges are moving.
To illustrate further, consider the simplest case for magnetism: Two parallel current-carrying wires (let’s say the current is in the same direction, but it’d work just as well the other way). When no current is flowing, the average spacing between electrons in each wire is the same as the average spacing between protons, and so both wires are neutral, and so there’s no net force between them.
Now turn on the current. From point of view of a proton in wire A, the protons in wire B aren’t moving, so their spacing hasn’t changed. But from that proton’s point of view, the electrons are moving. And hence, according to relativity, the gap between electrons has shrunk, and so the electrons are now slightly denser than the protons. In other words, from the point of view of a proton in wire A, wire B is slightly negatively charged, so the protons in A are attracted to B.
Or do the same thing from the point of view of an electron: The currents are in the same direction, so from the point of view of an electron in wire A, the electrons in B are at rest. But now, from that electron’s point of view, the protons are moving. And so an electron will see the protons as slightly denser than the electrons, and so the electron in A sees wire B as slightly positive. And so the electrons are also attracted to the other wire.
Actually go through and do all of the calculations, and you’ll find that the net force between the wires is exactly the magnetic force.
Thank you.
Isn’t that a pretty long-winded way of saying “witchcraft”?
The physics term for witchcraft is covariant tensor