Physics, Astronomy & Cosmology / Re: Does time dilation with curved paths vary with the rate of change of separation?« on: 09/04/2020 14:47:16 »
The Archimedes spiral is what you get drawing with a string tied to a pencil and winding it around a central cylinder. The logarithmic spiral has a constant angle at any point with a line drawn from that point to the center. The latter is scale invariant.Quote from: HalcWe're talking about a logarithmic spiral here and not something like an Archimedes spiral.I don't know the difference.
IMaybe I misunderstood the principle. If Mach asserts that the laws of physics are the same in a rotating reference frame, then it is empirically wrong, but I don't see such assertions. This doesn't seem to be a description of our actual universe, but rather a mathematical treatment of the properties of abstract systems. I'm talking about our actual universe in this thread, so all talk about there being only two elements in orbit say is irrelevant. I said add reference stars if you need them to actually consider real physics. The rest of us don't.QuoteIMach0: The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames.Mine do work:
Mach1: Newton's gravitational constant G is a dynamical field.
Mach2: An isolated body in otherwise empty space has no inertia.
Mach3: Local inertial frames are affected by the cosmic motion and distribution of matter.
Mach4: The universe is spatially closed.
Mach5: The total energy, angular and linear momentum of the universe are zero.
Mach6: Inertial mass is affected by the global distribution of matter.
Mach7: If you take away all matter, there is no more space.
Mach8: 'Horseshoe, def above =, 4, pi symbol, funny p, G, T, squared' is a definite number, of order unity, where funny p is the mean density of matter in the universe, and T is the Hubble time.
Mach9: The theory contains no absolute elements.
Mach10: Overall rigid rotations and translations of a system are unobservable.
Acceleration is relative.
Any group of objects treated collectively require a separate point of reference to define any form of motion.
An inertial frame is one without acceleration so you've written into your own definition that the age difference is caused by the fact that one accelerated and one didn't.Two objects cannot separate and rejoin without one accelerating, so that in inherent in the situation at hand, not in my definition.
I'm just saying that all times can be computed purely as a function of the speeds of the participants in one arbitrary but consistent frame. I did not say that the computation must be done that way.
What? You only get that answer in the frame in which Earth is stationary. Both twins will have aged less in any other inertial frame, although the difference on their watches will of course be the same at the end in any frame, not just inertial ones.The difference on their watches is all we care about, so we're good.
You were treating the Earth bound watch as stationary and the other watch as moving.I was, but I don't have to.
I like to use acceleration alone to explain the difference.That's fine. It's your preference. It's actually a reasonable way to explain the difference, whereas my goal was more to compute their ages at the end, not explain the discrepancy.
It gets the correct answers, so it is complete. It doesn't require knowing the one actual correct frame or any absolute speeds of anything, so it doesn't imply a preferred one.Quote from: HalcMy method is simply to pick a frame (any frame) and stick with it.Your method is incomplete and can't explain the difference in age without implying a preferred frame.
Saying 'relative to Bob' identifies the coordinate system being used. In any other coordinate system, it wouldn't be 'relative to Bob', and yes, the difference in their velocities, and their separation, and a bunch of other stuff would be different.Quote from: HalcYes I do, that's my point. Alice's velocity relative to Bob is purely coordinate dependant, because there's no change in separation over time.QuoteBut that would be an absolute velocity relative to Bob.You don't see the contradiction in this statement?
You still seem to be in combat mode.I'm trying to keep the topic on track, and it keeps being diverted to these abstract mathematical universes with different properties than ours.
Your last two posts are very disingenuous, deliberately taking what I say out of context in attempts to obscure the point. This is an example of what you're doing:I said non-inertial observer, bolded above, and I meant non-inertial observer in that statement. You asked if I meant non-stationary, but one can be moving but still inertial, so I didn't mean that. Non-inertial means accelerating.Quote from: HalcBut you said dilation relative to a stationary observer.Quote... no, I meant what I said.Quote from: HalcDilation relative to a stationary observer, and discounting gravity, is entirely a function of speed and not acceleration. The age of the twin in the twin experiment can be expressed as a function of integrating his speed, and acceleration has nothing to do with it. Dilation relative to a non-inertial observer is a function of acceleration, but we're expressing Alice's orbiting clock relative to Bob here.Did you mean dilation relative to a non-stationary observer?
So you meant dilation relative to an observer who is stationary relative to themselves?I didn't follow most of the context around this bit, but I would typically not word things as you ask here, it being redundant. So I'd word it more as "dilation relative to an inertial reference frame". It has nothing to do with observers or actually being observed. Everybody always puts named twins on the ships, but it only seems to serve to give them names, which can be done with a label maker. Put a clock on board and that's enough. It's real easy to get a clock up to .99c so long as you want only a few digits of precision.
Sigh. An accelerating observer takes into account the fact that they are time dilated and length contracted because of their acceleration. They can indirectly measure this by virtue of the fact that they can measure their own acceleration.Not sure what you mean by this. My accelerating ship has a proper length of 10 meters, and not sure what tools I have to show that it measures any other length. I know it is accelerating via various clues like the accelerometer or the fact that water stays in my cup or that time at one end ticks faster than the other end.
OK, you're talking about something other than proper length, but I don't know how an accelerating observer might go about measuring anything except his proper length.
One can compute it: I am a meter tall in a frame where I'm moving fast, but there's no simple way to show that.
They then work out what the speed of light relative to themselves moving away in front of them would be if they were not time dilated and length contracted and they can see that the speed of light relative to themselves moving away in front of them slows as they increase their acceleration, with the reduction in speed of light relative to themselves moving away in front of them lessening with the same increase in acceleration as their overall acceleration increases in such a way that the speed of light relative to themselves moving away in front of them approaches without ever reaching 0.I cannot parse this. Maybe you could illustrate your method with an example. I'll pick one with nice high acceleration.
You, at time 0 shine a pulse of light outward, and immediately go chase it with a ship. You accelerate for 45 days, 7 hours at 20g measured on ship clock. After that much proper time, how far in front of the ship is the pulse of light in the final frame of the ship?
I computed it without any consideration of length contraction. I did the whole thing in the terminal frame of the ship, not in the initial frame. That seemed to simplify the calculations. I got a bit over 2 light years away. Maybe I goofed, but that seems about right. You talk about light slowing in front of the ship, but I computed that it moved 2 light years in an eighth of a year, which is much faster than c.
OK, under SR conditions, there is no horizon in front of an accelerating craft beyond which an outgoing signal will never reach. This is true even of a stationary thing, and yet light seems to move faster than c relative to an accelerating thing.Quote from: HalcAn event horizon is a threshold in space beyond which no event can ever have a causal effect on you. There is no such horizon in front of an accelerating object in the flat spacetime of the special relativity case. Any event that happens in front of you (say a signal sent at you) will get to you, all the faster because you're accelerating towards it instead of just waiting for the light to make the trip.I'm taking about a signal moving away from you.
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