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From: "Paul B. Andersen" <relativity@paulba.no>
Newsgroups: sci.physics.relativity
Subject: Re: [SR] The traveler of Tau Ceti
Date: Thu, 21 Mar 2024 21:07:10 +0100
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Den 20.03.2024 22:06, skrev Richard Hachel:
> Le 20/03/2024 à 20:18, "Paul B. Andersen" a écrit :
>> Consider an inertial observer in space.
>> She has instruments like clocks and telescopes and computers,
>> so she can measure the speed of a passing rocket relative
>> to herself.
>> Please don't say that this in principle is impossible in the real world.
>>
>> Eleven such observers (O_0 ..O_10) are stationary relative to each
>> other, and are arranged along a straight line with 1 light year
>> between them.
>> A rocket which is accelerating at the constant proper acceleration
>> a = 1 c per year is instantly at rest relative to O_0.
>> The rocket is moving along a line parallel to the line of observers.
>>
>> c = 1 light year per year.
>>
>> Please show what you think the observers O_1 to O_10 would
>> measure the speed of the rocket to be relative to themselves.
>>

This defines Doctor Hachel's  theory:

On 12.03.2024, Richard Hachel wrote:
| In the rocket frame, a is constant. Always.
| The rocket is at rest in its frame of reference,
| and the speed Vr of the surrounding space becomes Vr=a.Tr
| There is no problem, the speed of the rocket, that is to
| say the real speed of movement of the terrestrial frame
| of reference, is indeed Vr.

Below are the inevitable consequences of Doctor Hachel's theory:

The speed of an inertial observer relative to the rocket is
   Vᵣ = a⋅Tᵣ      (1)
where a is the constant proper acceleration of the rocket
and Tᵣ is the proper time of the rocket.

This means that the observer will move a distance x relative to
the rocket:

  x(Tᵣ) = ∫Vᵣ⋅dTᵣ + x(0) = a⋅∫Tᵣ⋅dTᵣ + x(0) = a⋅Tᵣ²/2 + x(0) (2)

Per definition is Vᵣ = 0 when the rocket is passing O₀,
and we define Tᵣ = 0 and x = 0 at the same event.

That means that when an observer has moved the distance x,
the time Tᵣ is:
   Tᵣ = √(2⋅x/a)       (3)

Since a = 1 ly/y² and it is 1 ly between two observers,
observer Oₙ at x = n ly will pass the rocket at the time
   Tᵣ =  √(2⋅n) y.

The speed of observer Oₙ relative to the rocket when she
passes the rockets will be:
  Vᵣ =  √(2⋅n) ly/y

Observer Oₙ will measure the speed of the rocket relative
to herself to be:
  Vᵣ = √(2⋅n) ly/y.

Since all the observers will measure that the speed of the rocket
is faster than the speed of light, we know that Doctor Hachel's theory
is born dead.

> 
> The answers I can give you are very simple as long as you understand 
> correctly what I am saying.

If they are simple, why haven't you tried to calculate
what speeds of the rocket the observers will measure?

> 
> But I repeat again and again, observable speeds are not real speeds. 
> This is very important to understand, because you will realize that 
> things will logically start to go wrong.

The observers measure obviously the real speed of the rocket!
The observers inhabit the real world, not Wonderland.

That they measure an unreal imaginary speed is indeed a weird idea.
Or is "a stupid idea" a more adequate expression?

> 
> If you use real speeds (Vr) you will no longer have any problems, and 
> the equations will remain both simple and true.

"No longer"? I never had any problems.
Measured by observer Oₙ the speed of the rocket is  √(2⋅n) ly/y.
Simple and true.

> 
> If you use traditional observable velocities (v or Vo)
> you will notice that the observable speeds can be different for various 
> observers present in the same frame of reference. Which may seem absurd 
> if we do not understand that, precisely, these speeds are not real but a 
> distortion of what is real.

Of course any measurement of anything will have a limited precision.
But that is not what you are talking about.

> 
> I can easily give you all the equations you need.
> 
> Here you are asking me what is the instantaneous observable velocity for 
> each point placed on the path as the rocket passes in front of it.
> 
> We have :
> Voi/c=[1+c²/2ax]^(-1/2)

So you claim that the rocket passes observer O₅ at the real speed
3.1623 ly/y, but O₅ will measure the speed to be 0.9535 ly/y.

Or generally:
You claim that the speed of an object in an inertial frame
may be several times the speed of light, but will always be
measured to be less than c.

Which is utter nonsense!


Give my regards to Alice.

Case closed.

-- 
Paul

https://paulba.no/