I've been thinking about Einstein's equivalence principle. It applies to inertial mass and gravitational mass. Does anyone know if it applies to angular mass as well? Has anyone ever tried putting an atomic clock in a centrifuge to see if the high centrifigul forces (fictious force) would result in time dilation? I couldn't find it in ARXIV. There are not too many centrifuges that can spin it's contents up to relativistic speeds. At 3600 rpms, and 0.1 meter from the center, the speed would only get up to (60 rps)(0.1m)(6.28) = 37 m/s (which is not very relativistic at all). However, the acceration on the "atomic clock" would be \(a = \frac{v^2}{r} = 13,690 m/s^2 = 1400g's. \) Compared to the centrifuge center which should have a lot lower g force. Would there be any time dilation between the 0.1m arm of the centrifuge and the center of the centrifuge? comments welcome.
I've never heard of "angular mass". But yes, rotation in a centrifuge results in time dilation between the centre and the outside of the centrifuge. There's relative motion between the two, and that's all that is required to get time dilation. Alternatively, you can look at it in a general relativistic context to reach a similar conclusion.
You're right, the term angular mass isn't used. Yeah, there would be relative motion between the centrifuge and the spinning arm with the atomic clock in it, but 37m/s versus 3x10^8 is not going to produce the kind of time dilation I had in mind. I was hoping for a time dilation more like, \(T_D = e^{\frac{gh}{c^2}}\) At, http://en.wikipedia.org/wiki/Gravitational_time_dilation it says that "Any kind of g-load contributes to gravitational time dilation." Oh, wait! The next one is for a rotating disk. \(T_D = \sqrt{1-r^2 \omega^2/c^2}\)
\(r \omega\) is just the tangential velocity of a point on the disc, so that formula just gives the ordinary Lorentz factor for time dilation.
What you are up against is the clock postulate, which says that acceleration does not cause any additional time dilation. This has been tested to as high as 10^18g. Usually it is done with radioisotopes of known half-lives on high speed centrifuges. As James R has noted , the time dilation always comes out to be equal to that caused by the tangential velocity. This is easily verified by varying the length of the centrifuge radius and its speed. This way you can get a number of different combinations of tangential velocity and g-force. So how does this tie in with the equivalence principle? The thing to remember is that gravitational time dilation is due to a difference in gravitational potential and not gravitational force. Gravitational potential is related to the amount of energy it takes to lift a object a distance against gravity. For instance, if I had a clock sitting on a small dense world with a surface gravity of 1g, and one sitting on a more massive but less dense world, also with a surface gravity of 1g, it would take more energy to lift the clock from the second world to space, than it would for the first. This means that the clock on the second world is in a deeper gravity well and has a lower gravitational potential. This causes this clock to run slower than the other clock, even though they are at the same g, So even with gravitational time dilation it is possible to have clocks running at different speeds even when they are both experiencing the same g-force. With the centrifuge, the equivalent potential is the energy needed to move the clock from the end of the centrifuge to the center. This is always going to be related to the tangential velocity. In other words, if you are watching the centrifuge spin, you will measure a time dilation equal to that caused by the velocity. If you were riding the centrifuge and in that rotating frame, you could consider the time dilation as due to the potential difference between the end of the arm and the center, but you would get the same answer for the dilation.
