Naked Science Forum
On the Lighter Side => New Theories => Topic started by: MikeFontenot on 06/03/2023 16:19:13
-
Two perpetually inertial observers (IO1 and IO2), perpetually mutually stationary with one another, are initially co-located with two separated observers (AO1 and AO2), with separation "L". AO1 and AO2 are about to begin a constant (according to them) acceleration "A" (with the separation in the direction of their acceleration). AO1 and AO2 KNOW that their acceleration is "A", because they each are carrying an accelerometer that confirms it. IO1 and IO2 will conclude that AO1 and AO2 maintain the separation "L" during the accelerations. And AO1 and AO2 will agree with that: AO1 and AO2 conclude that their separation remains constant at "L" during the acceleration. But two other inertial observers, IO3 and IO4, who are momentarily co-located with AO1 and AO2 at any time later in the trip, will NOT agree that the separation "L" is constant: they will say that it has increased since the start of the trip.
NOTE: My use of the phrase "Proper Separation" in the title of this submission means that it is the separation of the two people undergoing the acceleration, ACCORDING TO THOSE TWO PEOPLE THEMSELVES. And the REFERENCE FRAME of an accelerating observer, say AO1's reference frame, is constructed in the same manner as an inertial observer constructs his reference frame (as Einstein explained to us). AO1's helpers just lay out yardsticks, end-to-end, in the direction of the acceleration. To keep the yardsticks in place, they each are attached to an accelerometer-controlled rocket, so they each are accelerating at "A" lightyears/year/year. And between every pair of yardsticks, there is a clock. The clocks were initially synchronized before the the constant acceleration started. Once the acceleration starts, the clocks don't remain synchronized ... clocks located farther in the direction of the acceleration tic faster, by an amount originally given incorrectly by Einstein (his exponential equation), but which has now been corrected. (END NOTE)
Each accelerometer directs its attached rocket to accelerate at exactly "A" lightyears per year per year. NEWTONIAN physics would say that the velocity of AO1 and AO2 would increase linearly with time, forever:
v = A * t. (incorrect)
But that means that "v" would go to infinity as "t" goes to infinity, which we know can't be true in special relativity. So the above equation is clearly wrong. Special relativity says the quantity "(A * t)", which it calls the " rapidity" (denoted by the variable "theta"), is related to the velocity "v" by
velocity = v = tanh(rapidity) = tanh(theta) = tanh(A * t). (correct)
That says that with a constant acceleration "A", "v" approaches (but never equals) the speed of light, "c", as "t" goes to infinity.
The distance "D" each rocket moves, according to AO1 and AO2, is
D = integral {from 0 to tau} [ v dt ] .
= integral {from 0 to tau} [ tanh(A*t) dt ] .
The integral of tanh(x) is equal to log[cosh(x)], so
D = log[cosh(tau)] - log[cosh(0)] .
D = log[cosh(tau)] .
So "D" grows forever, but it's RATE of growth decreases as tau increases.
The distance "D" each rocket moves during the acceleration is EXACTLY the same, so the separation "L" between AO1 and AO2, according to THEM, can't change during the acceleration.
The INERTIAL observers (IO1 and IO2) also conclude that the separation "L" between AO1 and AO2 stays constant during the acceleration. But two inertial observers (IO3 and IO4) who are momentarily co-located with AO1 and AO2 at some later instant in the trip will conclude that the separation between AO1 and AO2 is LARGER than it was when the acceleration started. And that larger separation continues to increase as the trip progresses, according to the INERTIAL observers momentarily co-located with AO1 and AO2 later in the trip.
So the accelerating observers (AO1 and AO2) say that their separation is CONSTANT during their trip. The inertial observers (IO3 and IO4) say that the separation of AO1 and AO2 INCREASES during the trip. Those two groups of observers DISAGREE. That's just the way special relativity IS.
But it's normal in special relativity for an accelerating observer to agree with the inertial observer who is momentarily co-located with him at at some instant ... that's what the CMIF simultaneity method IS. The inertial observer IO3 is momentarily co-located with AO1, and IO3 tells AO1 that the separation between AO1 and AO2 is larger than it was when the separation began. Does that contradict my above argument? No, it doesn't, because the scenarios are themselves different: the actual accelerations are slightly different. How do the people producing the scenario with all the inertial observers achieve the acceleration "A"? It's based on the CALCULATIONS by the inertial people: they measure positions of AO1 and AO2 versus the time on their own watches, and COMPUTE the acceleration. It is NOT based on any accelerometer, and it differs from what AO1 and AO2 read on their accelerometers.
Note that the above paragraph is relevant to the well-known (and much misunderstood) Bell's Spaceship Paradox:
https://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox
(read the WHOLE thing). Does the string break or not? The answer depends on how the acceleration is measured. If the acceleration is determined by calculations and measurements made by various inertial observers, the string WILL break. But if the acceleration is what accelerometers attached to the spaceships display, the string will NOT break.
-
The Proper Separation of the Clocks Doesn't Change!
By definition, yes. If the separation is changing (such as in the examples in recent posts including this one), then each of the objects is moving relative to the other, and therefore have no proper separation since there's no frame in which both are stationary.
Anyway, thanks for posting in New Theories.
Two perpetually inertial observers (IO1 and IO2), perpetually mutually stationary with one another, are initially co-located with two separated observers (AO1 and AO2), with separation "L". AO1 and AO2 are about to begin a constant [proper] acceleration "A"
...
IO1 and IO2 will conclude that AO1 and AO2 maintain the separation "L" during the accelerations.
This seems to be how you specify an inertial frame reference, in this case frame S (using Einstein's name since you won't name it). Yes, the accelerating objects will maintain constant separation in that frame, but since they're both moving in that frame, it isn't proper separation.
And AO1 and AO2 will agree with that: AO1 and AO2 conclude that their separation remains constant at "L" during the acceleration.
And here you fall apart. They might agree that they're the same separation relative to S, but they didn't say 'relative to S' (or any other frame) so the statement is meaningless at best, and wrong if they're talking about any frame in which one or the other might be momentarily stationary.
If you suggest an accelerating frame, you need to specify which kind since there are several. Rindler coordinates are typical, but those coordinates are not the one Einstein uses.
The distance "D" each rocket moves, according to AO1 and AO2
Distance relative to what? S? Say that then. This cannot be computed without making a frame selection. AO1 has moved nowhere according to AO1 since he started out 'here' and is still 'here' after any amount of time, so in any traditional frame with the observer at the origin, that distance is zero.
The distance "D" each rocket moves during the acceleration is EXACTLY the same
Relative to what? Your confusion mostly stems from refusal to specify frames. The distance each rocket moves relative to the other is certainly not the same.
so the separation "L" between AO1 and AO2, according to THEM, can't change during the acceleration.
Your title say 'proper separation'. You've not computed that at all, and cannot since there is no proper separation between things moving relative to each other.
You seem to be computing separation relative to S, which is trivially constant. But that's not the frame of either accelerating object. Also, don't assume the frame of one object is the same as the other which cannot be the case when the objects are not stationary relative to each other.
Your case this time doesn't suggest that the acceleration ever stops. Just asking to see if the scenario is changed from last time. I also notice you provide no numbers, no coordinates of where everything is in certain frame at certain events. I provided that in my answers, but you never do. Work the numbers and the mistakes will become obvious, and you don't have to keep asking via forum posts if your assertions work. Try working it out with numbers instead of making the assertions in the first place.
But two inertial observers (IO3 and IO4) who are momentarily co-located with AO1 and AO2 at some later instant in the trip will conclude that the separation between AO1 and AO2 is LARGER than it was when the acceleration started. And that larger separation continues to increase as the trip progresses, according to the INERTIAL observers momentarily co-located with AO1 and AO2 later in the trip.
This statement sort of indicates that 'according to AO1' at some event has totally different values than 'according to some comoving inertial observer at the same event'. In that case, I have no idea what coordinate system AO1 might be using. You seem to have no idea either.
But it's normal in special relativity for an accelerating observer to agree with the inertial observer who is momentarily co-located with him at at some instant ... that's what the CMIF simultaneity method IS.
That's not what the rest of the world calls it, but yes. So apparently AO1 is not using what you call CMIF method, but you won't say what he's using instead.
How do the people producing the scenario with all the inertial observers achieve the acceleration "A"?
I don't see how any inertial observer can achieve acceleration at all.
Mike, I'm going to continue to rip on your posts as long as the frame references are absent. You mention 'according to X' sometimes, but then makes statements that are not true in frames where X is stationary, so it gets pretty unclear what 'according to X' means without specification of what coordinate system X is using.
-
Mike, I'm going to continue to rip on your posts as long as the frame references are absent.
The frame I'm defining is the accelerating person's frame. It is analogous to the frame of an inertial person. And, like an inertial reference frame, the accelerating person's frame has a procedure for creating an array of clocks and their attending observers, with fixed separations, that establishes a "NOW-at-a-distance" for the frame . In both cases, that frame guarantees that the observer must consider that "NOW-at-a-distance" to be fully meaningful and real.