johanfprins wrote:When Tom posts sloppy mathematics like this:
Sure: The LT says it all. For simplicity set c = 1, so v < 1.
gamma = sqrt[1- (v^2)]^-0.5
x = gamma x' +vt
t = gamma t' + gamma v x'
Then I lose interest. Except for setting a physically real parameter equal to unity so that it drops out of the equations, which is a dangerous thing to do in physics,
Using units appropriate for the calculation is a very usual thing in physics. If you can't abstract away from SI units you are an engineer, not a physicist.
In this case just choose "natural" units of time and space so that c=1, e.g.:
time measured in years
space measured in light-years
etc.
c of course is not a physically real parameter (like fine structure constant) it is a constant dimensioned quantity. Therefore it defines physically real units. SI units don't respect this, but others do.
the first of his last two equations is wrong. I do not have the time to try and figure out the meanderings in Tom's head are, since so far I have found him to be confused.
typos will abound in anything I post here. And possibly errors. But Johan has not yet bothered himself with any substantive math.
In this case gamma x' + vt should be gamma x' + vt'
If you want a math and logic consistency check, the real one is comparing the clock time difference as calculated in P rest frame:
where if the P clock time is Tp, the Q clock time (from time dilation) is T/gamma, so the difference is Tp(1-1/gamma) = Tq(gamma-1)
with that calculated (in a much more complex way) in the Q frames as above. I'll do this myself when I have time since I may well have made a mistake in precisely how I've done the Q frames math.
So let me post a manifesto:
MANIFESTO
1. A stationary clock does not have time-dilation.
Time dilation only exists between two time rates. This manifesto statement is therefore meaningless without qualification! Time-dilation relative to what?
2. All stationary clocks must thus keep time at the exact same rate.
What does this mean? I think Johan means that every clock measured time in its rest frame (proper time). That is the definition of a clock. But why does he keep on viewing this as something that must be repeatedly stated?
3. A moving clock is stationary within the inertial reference frame that moves with it.
We all agree this
4. All moving clocks must thus keep the same time rate within the inertial reference fames that move with them respectively.
Of course
5. A time dilation occurs within a reference frame relative to which a clock moves.
This is loose. I think Johan means that time dilation is the difference in rate between a clock measured in its rest frame, and the clock measured in some other frame.
6. The clock moves with zillions of relative speeds relative to zillions of inertial reference frames, and thus causes zillions of time-dilations.
Or, more properly, every different reference frame, measuring a given clock, will see a possibly different time dilation as determined by the relative speed of the frame and the clock rest frame. But I don't think we have any disagreement here.
7. Zillions of time dilations cannot simultaneously manifest on the same clock within its own inertial reference frame, and do not do so since this clock, like all other clocks, is keeping the exact stationary time rate within its own inertial reference frame like ALL clocks do.
Whoaa! What is "time dilation manifesting on a clock"? It sounds mystical to me, not physics. The physics is simply that clock time, when measured using another clock, depends on the frame within which it is measured.
8. Since the clocks with both twins are stationary relative to the twins respectively, they keep the SAME stationary time rate, and therefore the twins must remain the exact same age.
Whoaa! What does "the twins must remain the exact same age" mean? It has no meaning except when the meet again. But then the statement is only true if proper time path length is the same for two different (non-inertial) paths in spacetime. That is a Johan special assumption. Never proven by him, nor implied by SR postulates. It slips past most readers because we are conditioned to think in Newtonian terms, where it would automatically be true.
subject to a time dilation which is also accompanied by length stretching.
I don't understand what this means. Both time dilation and length stretching happen but are properties of measurement in one frame of a clock or distance in another. They precisely correspond to the LT which transforms length and time coordinates between frames.
I'm just not sure how Johan will apply these things. If he does so consistently he will have no diference from me!
10. The latter perceived time dilation is the same as observed by both twins, and is the same whether the two twins move away from one another or towards one another.
OK. Here he is specifying frames, so the statement is meaningful, and correct. On both outbound and inbound journeys measuring of one twin's clock in the other twin's frame results in a time dilation of gamma.
11. Whether the twins move away or towards one another still does not change the fact that their respective clocks keep the exact time rate within the respective reference frames of the twins.
I think this is a restatement of 4? Or may be 4 and 10?
12. The clocks can be compared while the twins move relative to one another:
WARNING! Clocks cannot be
uniquely compared in this case. We can compare clocks in either of the two twins' frames (getting different answers) or some completely different frame. If Johan means this, fine. But I think he is implying the existence of some absolute frame within which the comparison must always be done. Apologies if this does him injustice.
It is accepted that they can set both clocks to read zero when their clocks pass one another at the beginning of the journey.
Absolutely.
13. Thus they must be able to compare the clocks when they again pass one another at the end of the journey.
Absolutely
14. Since the clocks kept the exact same time rate within their respective reference frames they must read the same time at the end of the journey when they again pass one another.
No, no, no. That would only be true if the proper time along the two distinct paths the clocks take through spacetime were the same. Now, I have shown precisely using LT that this is NOT the same.
Even if you do not accept this, Johan has nowhere shown that it IS the same. If Johan wishes to propound a version of SR in which space and time are separate, so that by definition all spatial paths between two events have the same proper (clock) time: fair enough. But to pretend that this non-standard idea is a logical consequence of the SR postulates has nowhere been shown. Johan has many times stated this, without proof.
15. Thus the twins will be the same age after the round trip has been completed.
No. See 14.
15. Four dimensional space-time ONLY has physical relevance when stationary clocks at different positions keep different time rates: Only then can one define a Minkowski space-time infinitessimal distance ds that makes physics-sense.
This is just plain wrong. I can't see how Johan derives it. I guess that his confusion over "keeping the same rate" where he thinks that "proper time (ds)" as measured by a clock must be a property of some absolute time rather than simply the (rest frame time) length of the spacetime path through which the clock travels. But that is only a guess.
The LT (which I use) establishes the relationship between times in one frame and times in another. Obviously you can have clocks in the two frames, in which case each clock measures the other as running slower.
Johan is right that the LT is an almost inescapable consequence of the Minkowski space metric. Accept one, you need the other. (I've cited
Lorentzian relativity - which has evolved into something that allows an ether (and hence Newtonian spacetime) but explicitly breaks Einstein's first postulate to do this and remain consistent. It is physically identical to SR for twin's paradox, and can be seen as an alternate interpretation of SR.
Johan, I think, is stuck with a pseudo-Newtonian notion of time where he finds the physical consequences of LT inconsistent. So he tries to construct a physics in which it is not "real".
16. In special relativity ALL stationary clocks keep exactly the same time rate: It is thus stupid to use ds in this case since it has no physics meaning nor relevance.
Again, this extension from local time-keeping (tautologously at same rate) to some sort of global comparison. Johan has not defined how he is comparing the times of clocks in different frames. I have (via LT) and can show all the ins and outs.
You would think that since this concept is so central to his argument Johan would explain how he compares the times of clocks in two different inertial frames to see whether they "keep the same rate". He cannot do this consistently, which is why he does not try. (He did way back propose some way to do this, I corrected it and he has not come back with a correction to my correction).
Note that where clocks can be unambiguously compared we must have one at least changing frames.
17. Only when there is a gravitaional field does clock rates change with position for stationary clocks: Thus ONLY in this case is the use of ds physically relevant, and can the concept of Minkowski space-time be used.
We have no disagreement about GR, and it is not the topic of this debate.
18. Einstein was thus correct to have opposed the use of Minkowski space-time for SR and to only have used it when he modelled gravity.
This is a matter of choice. Minkowski spacetime is consistent with LT. My arguments throughout this thread do not rest on Minkowski spacetime - they rest only on the Lorentz Transformation which is derived from SR postulates. I find it easier to visualise what is happening with Minkowski Spacetime, but it is not compulsory. The point being that MS is consistent with Einstein's understanding of SR.
19. A returning twin can only be younger when he has spent part of his journey near a black hole: But he probably will have died due to the enormous gravity before returning home.
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In summary, Johan's statements here lay out his thinking more completely than before, but it contains gaps and conceptual errors. I've indicated above what I believe these to be. Johan's conclusions are also in violation of many different experiments. If it was just a different interpretation i would not mind, but his reasoning leads to wrong physical predictions.