Re: Incorrect mathematical integration

Den 21.07.2024 23:06, skrev Richard Hachel:

Le 21/07/2024 à 21:26, "Paul.B.Andersen" a écrit :

Den 20.07.2024 23:55, skrev Richard Hachel:

 

So: in an inertial system K, a clock C is in inertial motion.
>
t is the time in K.

Per DEFINITION of MY scenario.
Don't insist on putting your indexes on MY coordinates!

 

A and B are two stationary, synchronous clocks in K.
>
At t = t₁
  C
  A                     B
--|---------------------|------> x
  0                     L
Clock A is showing t₁, clock C is adjacent to A and is set to zero.
At t = t₂
                         C
   A                     B
--|---------------------|------> x
   0                     L
Clock B is showing t₂, clock C is adjacent to B and is showing τ
Let L = 0.0001 light second = 29979 m
Let (t₂ - t₁) = T = 125 μs
v = L/T = 239833966.4 m/s = 0.8c
w = L/τ
What kind of time is T ?

 T=To (observable time in the laboratory, time observable but not real mesured by two anisochronic clocks A and B).

The clocks are synchronous per definition!
T is obviously the time measured in the in the inertial system.
Of course the time T = 125 μs is a real time.
Look.
All clocks in France show the time GMT+2 hours.
If you travel from Paris when a clock in Paris show 12.00
and you arrive at Nice (distance 937 km) when a clock in
Nice shows 22.00, then the duration of your journey is T = 10 h.
What do you mean with your claim that these 10 hours aren't real?
Does it mean that the 10 hours are imaginary? Are you dreaming them?

 

What kind of speed is v ?

 Observable speed Vo.

Which is not a real speed?
The speed v = 0.8c is the speed measured in the inertial system.
Of course it is a real speed.
Your speed from Paris to Nice, measured to 93.7 km/h
is  a real speed.
You claim that the speed of your car relative to the ground
is not a real speed! Good grief!

What is τ ? (equation, value and type of time)

 Proper time of C (real time).

Quite.
τ = T/γ = 75 μs
It is the time shown on clock C when it is adjacent to clock B.
Of course it is a real time.

What kind of speed is w ?

 Real speed (Vr)

w = L/τ = 399723277 m/s = (4/3)c
This is the "proper velocity".
But is it "a real speed"?

 

What is the physical significance of w?

It is simply a distance in an inertial system divided
with the proper time of a clock to travel the distance.
It is a speed, but not the speed of a physical object relative
to another physical object.
Clock C is moving with the speed v relative to clock A and B,
and clock A and B are moving with the speed v relative to clock C.
The notion "proper" used on a frame dependent entity
is problematic because "proper entities" are usually
invariant (proper time, proper length).
I have seen "proper velocity" defined in only one book:
J.H. Smith "Introduction to Special Relativity" from 1965.
It was the first (non popular) book I read about SR.
But it isn't mentioned in more modern books such as:
Carrol:           Spacetime geometry,
Taylor & Wheeler: Spacetime physics
D'Inverno:        Introduction to Einstein's relativity
Misner, Thorne and Wheeler: Gravitation
BUT this "proper velocity" is the spatial component of
the four-velocity.
(Which is very different from the three velocity.)

   Vr=AB/τ
 It is a physical notion, but not very important in Galilean relativistic physics, on the other hand, which becomes fundamental in the physics of accelerated mobiles and which we can no longer do without as soon as we leave basic relativistic physics.

"Galilean relativistic physics"! Good grief. :-D
The "proper velocity" is never used in SR.

 I recognize that it is quite strange to say that it is in the laboratory that we measure the real distance traveled, and at the level of the particle that we measure the real time to travel it.

It isn't only strange, it is nonsensical.

 

Can it say anything about the position of C at the time τ ?

 x = Vo.To = Vr.Tr = Vapp.Tapp
 x=Vr.τ

x = (L/τ)⋅τ = L
Very useful equation indeed :-D
-----------------
Let's look at the LHC again.
The length of the circuit is L = 27 k, γ = 7460
The real speed of the proton in the lab frame is
v = 0.999999991·c
The real time measured in the lab frame for the proton to go
around the circuit is
  T = L/v ≈ 90 μs
The proper time of a proton to go around the circuit is
  τ = T/γ ≈ 12 ns
"proper speed" = L/τ = 6947c
This "speed" is not the speed of a proton or anything else.
The proton is moving at the speed 0.999999991·c relative to the lab,
and the lab is moving at the speed 0.999999991·c relative to the proton.
And you thought that the real speed speed of the proton in the lab frame
was 6947c and therefore you "tell them [the physicists at CERN]  that
the proton rotates 78 million times per second".
Which is 6933 times the real number,  ≈ 11.25 thousand times per second.
But you are too stupid to understand that your nonsense is just that.
Right? :-D
--
Paul
https://paulba.no/