For the time being I will lower myself and answer you since for the first time on this thread you posted something that is not nonsense: Yes I made a typo: It should have been x/=L/. Well done! Although I think that my goldfish could also have picked this up.
Thank you for acknowledging that error with your usual grace.
This is the regular BS that you keep on posting to foul up this thread. I gave you the coordinates, and if you are capable of doing so (which you have demonstrated time and again you are not) you can look up the Lorentz transformation, and if you are capable of doing so (which I sincerely doubt) you can apply the Lorentz transformation to these coordinates: And if you do this ( which I doubt that you are capable of doing) you will find that you, as usual, posted nonsense. Must I send my goldfish to help you?
Although I stated the length contraction correctly in my previous post,
johanfprins wants me to derive that length contraction directly from the Lorentz transformation. Very well.
johanfprins has postulated a rod that is at rest with respect to an inertial frame K´ and lies along the x axis of that frame. At time t´ in that frame, one end of the rod is at x´=0; let's refer to that event as e
1. At the same time t´ (as measured in K´), the other end is at x´=L´; let's refer to that event as e
2.
I now need to use the Lorentz transformation to transform the coordinates of those two events,
including their time coordinates, into the coordinates of a second inertial frame K that is moving along the x axis of K´ with velocity v (as measured in either frame). According to one of my textbooks for freshman physics, the Lorentz transformation is
[latex]
\begin{align*}
x &= \gamma (x^\prime + v t^\prime) \\
t &= \gamma (t^\prime + v x^\prime/c^2)
\end{align*}
[/latex]
where
[latex]
\[ \gamma = 1/\sqrt{1 - v^2/c^2} \]
[/latex]
In inertial frame K, therefore, the coordinates for event e
1 are
[latex]
\begin{align*}
x_1 &= \gamma v t^\prime \\
t_1 &= \gamma t^\prime
\end{align*}
[/latex]
and the coordinates for event e
2 are
[latex]
\begin{align*}
x_2 &= \gamma (L^\prime + v t^\prime) \\
t_2 &= \gamma (t^\prime + v L^\prime/c^2)
\end{align*}
[/latex]
Note that t
1 and t
2 are different. (That difference illustrates the relativity of simultaneity.) If we were to subtract x
1 from x
2, we would get a wrong answer for the length of the rod in inertial frame K. (That wrong answer may be the wrong answer that
Farsight and
johanfprins got, but they haven't shown their work so we can't know for sure where they went wrong.)
To calculate the length of the rod as observed in inertial frame K, we have to calculate the positions of its endpoints at some common time. It's convenient to use t
2 as that common time, because we already know the position of the rod's rightmost end at that time.
[latex]
\[ t_2 - t_1 = \gamma (t^\prime + v L^\prime/c^2) - \gamma t^\prime = \gamma v L^\prime/c^2 \]
[/latex]
so the position of the rod's leftmost end at time t
2 must be
[latex]
\begin{align*}
x_3 = x_1 + v(t_2 - t_1) &= \gamma v t^\prime + v(\gamma v L^\prime/c^2) \\
&= \gamma (v t^\prime + v^2 L^\prime/c^2) \\
&= \gamma (v t^\prime + L^\prime v^2/c^2) \\
\end{align*}
[/latex]
The length of the rod as observed within inertial frame K is therefore
[latex]
\begin{align*}
x_2 - x_3 &= \gamma (L^\prime + v t^\prime) - \gamma (v t^\prime + L^\prime v^2/c^2) \\
&= \gamma (L^\prime + v t^\prime - v t^\prime - L^\prime v^2/c^2) \\
&= \gamma (L^\prime - L^\prime v^2/c^2) \\
&= L^\prime \gamma (1 - v^2/c^2) \\
&= L^\prime \frac{1}{\sqrt{1 - v^2/c^2}} (1 - v^2/c^2) \\
&= L^\prime \sqrt{1 - v^2/c^2} \\
&= L^\prime / \gamma \\
\end{align*}
[/latex]
That's what I said in my previous post. It is not nonsense. It is correct.
sol invictus and most of the other active participants here are very familiar with relativistic length contraction, so that reasoning probably took us less than two seconds.
Do the Lorentz transformation; or get a high school student to do it for you! Please stop being under the impression that you can think and reason. With all due respect, take it from me, you do not have the intellectual capacity to do so.
What are we to make of this?
johanfprins has given us many convincing demonstrations of his incompetence. He's been wrong time after time, but he hopes to distract us from his errors by insulting the intelligence of those who correct him.
He's been doing that for over a year:
I challenge you to an open debate about Maxwell's equations in front of a learned audience; but of course persons like you are always cowards who hide behind anonymity and then throw unfounded insults around. I would not be surprised if my goldfish knows more about Maxwell's equations than you do.
I accepted that challenge, but nothing came of it. His goldfish must be an anonymous coward.
You are the most dishonest person I have ever come along in my life. Not even Joseph Goebbels could have distorted the truth better than you are doing.
