lördag 24 augusti 2019

Fake Resolution of Twin Paradox

The twin paradox of special relativity SR has traumatised physics ever since it was first formulated 100 years ago, and several attempts to resolve the paradox have been presented over time, none of which has been acclaimed as the correct resolution. Let me here show that a popular variant, which many physicists cling to, is a fake resolution.

We recall that the special theory relativity connects the space-time coordinates $(x,t)$ and $(x^\prime ,t^\prime )$ in two inertial systems moving with constant velocity $v$ with respect to each other by the Lorentz transformation:
  • $x^\prime =\gamma (x - vt)$, $t^\prime =\gamma (t - vx)$,
  • $x =\gamma (x^\prime + vt^\prime )$, $t =\gamma (t^\prime + vx^\prime )$,
where $\gamma = \frac{1}{\sqrt{1-v^2}}$ assuming the speed of light is 1 and $\vert v\vert \lt 1$. We see that the (1d) space coordinate $x$ and time coordinate $t$ appear in symmetric form with an apparent similarity between space and time, which Lorentz viewed to be a formality without physics, but Einstein took as a basis of modern physics with space mixed into time.

Consider now two twins, twin A fixed at the origin $x=0$ of the $(x,t)$-system, and twin B fixed at the origin $x^\prime =0$ of the $(x^\prime ,t^\prime )$-system.  A's clock reads $t$ and B's clock reads $t^\prime$. A will see B follow the trajectory
  • $x=vt$,  
and so at time $t=1$ say, A will see B at $(v,1)$ in the $(x,t)$-system, while the corresponding coordinate in the $(x^\prime ,t^\prime )$-system is $(0,\gamma (1-v^2))=(0,\frac{1}{\gamma})$. A's clock thus reads $t=1$, while B's clock reads $t^\prime =\frac{1}{\gamma}\lt 1$. Twin A (stationary) thus finds B's clock (moving) to run slow compared to A's clock with the factor $\frac{1}{\gamma}$. 

On the other hand, trajectories in the $(x,t)$ system of constant $t^\prime$ take the form
  • $t=vx + constant$   
with the line $t=vx+(1-v^2)$ passing through $(v,1)$. Setting here $x=0$ we find
$t=(1-v^2)=\frac{1}{\gamma^2}$ to be the reading of A's clock on the trajectory of constant $t^\prime$ through the switch point $(v,1)$. B thus views A's clock to read $\frac{1}{\gamma^2}$ when B's clock reads $\frac{1}{\gamma}$, and so B views A's clock to run slow by the factor $\frac{1}{\gamma}$

So far, A considers B's clock to run slow, and B considers A's clock to run slow by the same factor, but since A and B will never meet the contradiction can be viewed to be only apparent and thus not really paradoxical.

To make A and B meet a change of mutual velocity must be made. Assume then that B switches direction at $t=1$, to rejoin A at $(0,2)$ in the (x,t)-system. This means that B changes/jumps to a new inertialsystem $(\bar x,\bar t)$ related to the $(x,t)$-system by the above Lorentz-transformation with $v$ replaced by $-v$. By symmetry, the trajectory of constant $\bar t$ passing through the switch point $(v,1)$ will cross $x=0$ at time $t=2-(1-v^2)=1+v^2$.  

B thus views A's clock to run slow both before and after switch, but can come to a view in agreement with the reading $t=2$ of A's clock at reunion by assuming that A's clock takes a jump of $2v^2$ at the switch.

Note that A and B must agree on the readings of both their clocks at reunion. It is not enough that only A views B to be younger. The only way B can accept this is to assume that A's clock takes a jump at the switch to a new inertial system.
Summary: A considers B's clock to run slow by the factor $\frac{1}{\gamma}$, and vice versa.
Despite the fact that B sees A's clock running slow, B can come to agree with A at reunion that A is older, by assuming that A's clock takes a jump when B switches from one inertial system to another.

Twin paradox: 
  • How it is possible that B's clock at reunion can be seen to run slow compared to A's clock, when B thru the whole round-trip sees A's clock running slow? 
The proposed resolution:
  • This is possible if B assumes A's clock to take a jump forward when switching from one inertial system to another. 
Is this a good resolution within SR? For B to assume that A's clock is reset with a sudden jump forward to give the impression at reunion that A is older? Can you change age by resetting a clock?

Of course not! The resolution is a fake resolution going outside SR by switching inertial systems and thereby resetting A's clock as viewed by B. It is amazing that such an obviously false argument can be put forward by physicists.

What then about going outside SR to general relativity GR, bringing in the idea that at switch B undergoes both retardation and acceleration with unknown effects, such as making A's clock jump forward? This can only make the argument more false and unphysical.

What do you think? Is the proposed resolution true physics or fake physics? Are you a true physicist or a fake physicist?

I have asked a group of physicists to comment the post and its truth value.

Bottom line 1: If you identify your ageing with the reading of your clock, you can by resetting the clock take on any age. If you think this is real, you have a problem.

Bottom line 2 Am I speaking about a petitesse, which physicists can dismiss as crackpot petitesse which can only be met by silence? No, the twin paradox is a real paradox of SR, which if unresolvable will destroy SR as physics. Therefore physicists must come up with a resolution or face the consequence, and that is far-reaching.

Connection to Dingle: Recall that Dingle posed the question to leaders of the physics community how it can be that both twins age more slowly than the other, a tough question which was side-stepped into the above fake physics argument with B switching inertial system and resetting A's clock. Dingle posed his question 50 years after the twin paradox was formulated and now another 50 years has passed with the paradox as glaringly unresolved as ever with silence as the only reaction from leading physicists. No wonder that that modern physics is in a state of deep crisis.

PS The Twins Clock Paradox History and Perspectives by by RL Shuler Jr (2014) identifies more than 200 articles over a period of 108 years seeking to resolve the paradox using at least 10 different approaches, however without ever reaching a conclusive answer:
  • ...there is no doubt at any rate that the twins or clock paradox continues to fascinate and confuse the public and physics students . Much of the literature seeks more effective ways of explaining or visualizing Special Relativity (SR). Though SR does not actually convey longer life experience, the ability to affect clocks, presumably in some explanations remote clocks, seems magical. What causes this effect? Is it the acceleration or the travel at high velocity? Even after 108 years, it can seem to depend on which paper one reads.
This is also my experience from posing the question to physicists and receiving no meaningful answer. Since apparently after 108 years the paradox has not been conclusively resolved, the chance to do so over the next 1000 years appears slim...but of course physicists claim that steady progress towards as resolution is being made, and that therefore the paradox is no real threat to SR, only a nuisance from crackpots seeking to create trouble and mistrust by spreading disinformation.

If there are 10 different suggested resolutions and no agreement, what is the chances that all are correct, or only one? Not big!

Of special concern to Shuler is the difficulty to explain the resolution of the paradox to students witnessed by many physics teachers. It seems that no matter what explanation is tried, students  complain that they cannot follow the argument and say that they understand. Is the reason that students are stupid or that the argument is fake physics? You can test yourself: Is there any of the 10 different resolutions of the paradox, which you can follow and say that you understand to be correct? If you say no, does that mean that you are too stupid to follow even a simple basic argument about physics? 

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