Three Experiments Challenging SRT

[Masterov]

rpenner, why $$\Delta t=0$$?

Your math not going helps you to think.
You do used math to complicate and to have perplex.

 

[Masterov]

If I'm right, you would be banned X days, if I am wrong you are banned X days, if the moderator agrees. You get to pick X and the moderator. Is this fair enough?

1. A judge - who? Who will decide who of us two is right?

2. I'm not a gambler. I do not bet.

3. I do not have X days.

 

[Masterov]

MT Velocity transformation

$$x'=(x-vt)(1-v^2/c^2)$$
$$y'=y\sqrt{1-v^2/c^2}$$
$$z'=z\sqrt{1-v^2/c^2}$$
$$t'=t-vx/c^2$$

visual ccordinates have up-commas $$\ '\ $$

$$x$$ - real coordinate
$$x'$$ - visual coordinate

$$dx'=(dx-vdt)(1-v^2/c^2)$$
$$dy'=dy\sqrt{1-v^2/c^2}$$
$$dz'=dz\sqrt{1-v^2/c^2}$$

$$\vec{V}=(\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt})=(V_x,V_y,V_z)$$ - real speed
$$\vec{V'}=(\frac{dx'}{dt},\frac{dy'}{dt},\frac{dz'}{dt})$$ - visual speed

$$\vec{V'}=\left((V_x-v)(1-v^2/c^2),\ V_y\sqrt{1-v^2/c^2},\ V_z\sqrt{1-v^2/c^2}\right)\ \wedge\ V\ll v$$

 

[rpenner]

rpenner, why $$\Delta t=0$$?

Because objects A and B may be in motion, so we need to compare their positions at the same time and that means at the same value of the t-coordinate when we are working with coordinates t, x, y, z and it means at the same value of the t'-coordinate when we are working with coordinates t', x', y' z'. If A and B are in motion and you don't measure at the same times then $$\Delta t \neq 0$$ and $$(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 \neq L_x^2 + L_y^2 + L_z^2$$ as these excerpts illustrate.

Let A & B be objects moving in parallel world lines separated only by a fixed amount in the x coordinate direction.

Then
$$\begin{pmatrix} \rule{0pt}{20ex} 0 & 0 & 0 & 0 \\ \rule{0pt}{20ex} -u_x & 1 & 0 & 0 \\ \rule{0pt}{20ex} -u_y & 0 & 1 & 0 \\ \rule{0pt}{20ex} -u_z & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \rule{0pt}{20ex} t_A \\ \rule{0pt}{20ex} x_A \\ \rule{0pt}{20ex} y_A \\ \rule{0pt}{20ex} z_A \end{pmatrix} = \begin{pmatrix} \rule{0pt}{20ex} 0 \\ \rule{0pt}{20ex} x_0 \\ \rule{0pt}{20ex} y_0 \\ \rule{0pt}{20ex} z_0 \end{pmatrix} $$
and
$$\begin{pmatrix} \rule{0pt}{20ex} 0 & 0 & 0 & 0 \\ \rule{0pt}{20ex} -u_x & 1 & 0 & 0 \\ \rule{0pt}{20ex} -u_y & 0 & 1 & 0 \\ \rule{0pt}{20ex} -u_z & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \rule{0pt}{20ex} t_B \\ \rule{0pt}{20ex} x_B \\ \rule{0pt}{20ex} y_B \\ \rule{0pt}{20ex} z_B \end{pmatrix} = \begin{pmatrix} \rule{0pt}{20ex} 0 \\ \rule{0pt}{20ex} x_0 - L \\ \rule{0pt}{20ex} y_0 \\ \rule{0pt}{20ex} z_0 \end{pmatrix} $$
Subtracting one from the other, we find:
$$\begin{pmatrix} \rule{0pt}{20ex} 0 & 0 & 0 & 0 \\ \rule{0pt}{20ex} -u_x & 1 & 0 & 0 \\ \rule{0pt}{20ex} -u_y & 0 & 1 & 0 \\ \rule{0pt}{20ex} -u_z & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \rule{0pt}{20ex} t_A - t_B \\ \rule{0pt}{20ex} x_A - x_B\\ \rule{0pt}{20ex} y_A - y_B\\ \rule{0pt}{20ex} z_A - z_B \end{pmatrix} = \begin{pmatrix} \rule{0pt}{20ex} 0 & 0 & 0 & 0 \\ \rule{0pt}{20ex} -u_x & 1 & 0 & 0 \\ \rule{0pt}{20ex} -u_y & 0 & 1 & 0 \\ \rule{0pt}{20ex} -u_z & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \rule{0pt}{20ex} \Delta t \\ \rule{0pt}{20ex} \Delta x \\ \rule{0pt}{20ex} \Delta y \\ \rule{0pt}{20ex} \Delta z \end{pmatrix} = \begin{pmatrix} \rule{0pt}{20ex} 0 \\ \rule{0pt}{20ex} L \\ \rule{0pt}{20ex} 0 \\ \rule{0pt}{20ex} 0 \end{pmatrix}$$
so $$\Delta x = L + u_x \Delta t$$ and $$\sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2} = \sqrt{(L + u_x \Delta t)^2 + (u_y^2 + u_z^2 ) (\Delta t)^2 } = \sqrt{L^2+ (u_x^2 + u_y^2 + u_z^2 ) (\Delta t)^2 - 2 L u_x \Delta t}$$.
The point that this emphasizes is that $$\Delta t = 0 \; \Rightarrow \; \Delta x = L \, \wedge \, \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2} = \left| L \right|$$ or for parallel world-lines, what we call their separation in space at the same time is constant.

That's a simple consequence of working in Cartesian coordinates for time and space.

Look at the separation between two events, A and B, on two distinct world lines with the same velocity. The separation at the same time in one coordinate system, $$\Delta t = 0$$, is not the same as the separation in the other coordinate system, $$\Delta t' = 0$$.

Assume the Lorentz transform applies since we are exploring special relativity. If you don't see a K in an equation, then I have not used any part of special relativity for that result.
$$K=c^{\tiny -2}$$ (eqn 1)
So the coordinate separation between events in one system is:
$${\huge \Delta S} \equiv \begin{pmatrix} \rule{0pt}{20ex} \Delta t \\ \rule{0pt}{20ex} \Delta x \\ \rule{0pt}{20ex} \Delta y \\ \rule{0pt}{20ex} \Delta z \end{pmatrix} \equiv \begin{pmatrix} \rule{0pt}{20ex} t_A - t_B \\ \rule{0pt}{20ex} x_A - x_B\\ \rule{0pt}{20ex} y_A - y_B\\ \rule{0pt}{20ex} z_A - z_B \end{pmatrix} $$ (eqn 2)
The following equation contrains events A and B to be on parallel world lines both moving at velocity $$(u_x, \; u_y, \; u_z)$$ :
$$\begin{pmatrix} \rule{0pt}{20ex} 0 & 0 & 0 & 0 \\ \rule{0pt}{20ex} -u_x & 1 & 0 & 0 \\ \rule{0pt}{20ex} -u_y & 0 & 1 & 0 \\ \rule{0pt}{20ex} -u_z & 0 & 0 & 1 \end{pmatrix} {\huge \Delta S} = \begin{pmatrix} \rule{0pt}{20ex} 0 \\ \rule{0pt}{20ex} L_x \\ \rule{0pt}{20ex} L_y \\ \rule{0pt}{20ex} L_z \end{pmatrix} $$ (eqn 3)

This is just a convenient way to write:
$$x_A - x_B = L_x + u_x ( t_A - t_B ) \\ y_A - y_B = L_y + u_y ( t_A - t_B ) \\ z_A - z_B = L_z + u_z ( t_A - t_B ) $$ which is what you get when you subtract equations describing linear (inertial according to Newton) motion at the same speed from each other.
Example:
$$x_A = (x_0 + L_x) + u_x t_A \quad \wedge \quad x_B = (x_0) + u_x t_B \quad \Rightarrow \quad x_A - x_B = (x_0 + L_x - x_0) + u_x t_A - u_x t_B = L_x + u_x ( t_A - t_B )$$

So the spatial separation between events A and B is ONLY equal to $$(L_x, \; L_y, \; L_z)$$ if $$(u_x = 0, \; u_y = 0, \; u_z = 0)$$ or if $$\Delta t = 0$$:
$$u_x = 0 \, \wedge \, u_y = 0 \, \wedge \, u_z = 0 \; \vee \; \Delta t = 0 \quad \Leftrightarrow \quad \begin{pmatrix} \rule{0pt}{20ex} \Delta x \\ \rule{0pt}{20ex} \Delta y \\ \rule{0pt}{20ex} \Delta z \end{pmatrix} = \begin{pmatrix} \rule{0pt}{20ex} L_x \\ \rule{0pt}{20ex} L_y \\ \rule{0pt}{20ex} L_z \end{pmatrix}$$ (eqn 4)

Here I am stating the the arguments up to this point (remember, there are no K's yet, so I am just talking about linear motion, not any type of Relativity), allow me to claim a prove. I'm saying
IF the worldlines of A and B are motionless THEN $$\Delta x = L_x$$ and $$\Delta y = L_y$$ and $$\Delta x = L_x$$
AND IF the coordinates of A and B have the same t-coordinate THEN $$\Delta x = L_x$$ and $$\Delta y = L_y$$ and $$\Delta x = L_x$$
AND IF $$\Delta x = L_x$$ and $$\Delta y = L_y$$ and $$\Delta x = L_x$$ THEN EITHER the coordinates of A and B have the same t-coordinate OR the worldlines of A and B are motionless OR BOTH.

As you see, this is a simple consequence of studying motion in Cartesian coordinates and has nothing to do with the details of special relativity -- only motion.

I have been using more math than you because it is obvious that you have difficulty with my English and math is a language of logic and you wrote:

I speak by mathematics as well as you

Is this claim true?

-----

Your math not going helps you to think.

Generally, I do my thinking before I start my math. The math is there to show that my thinking is self-consistent, logical and precise -- the qualities of good thinking in physics.

You do used math to complicate and to have perplex.

I am not using mathematics to obfuscate or intimidate. I am using math to elucidate the meaning of length contraction in special relativity, a theory about coordinate transformations with which you are unfamiliar. We have many physicists and mathematicians here, so I am confident that any of them would say I am using exactly the right level of mathematics to calculate general length contraction.

Your repeated defamatory claims have no basis in fact.

-----

Am I just tired or is that kind of like, "Heads I win, tails you lose?"

Corrected proposal for a bet:

Oops. “If I'm right, you would be banned X days, if I am wrong I would be banned X days, if the moderator agrees. You get to pick X and the moderator. Is this fair enough?” was meant.

1. A judge - who? Who will decide who of us two is right?

2. I'm not a gambler. I do not bet.

3. I do not have X days.

1. I was letting you pick who will decide, but it would have to be a moderator since only moderators have the power to ban. 2. OK, so you don't agree to bet. I still stand by my claim that the level of mathematics I have used is wholly appropriate for the discussion of general length contraction in special relativity. 3. I was letting you pick what X is -- you have picked X=0 days. Game over. 4. You did not discuss if my offer was fair or not.

----

MT Velocity transformation

$$x'=(x-vt)(1-v^2/c^2)$$
$$y'=y\sqrt{1-v^2/c^2}$$
$$z'=z\sqrt{1-v^2/c^2}$$
$$t'=t-vx/c^2$$

visual ccordinates have up-commas $$\ '\ $$

This isn't what you wrote before in this thread. Your new transform is a distorted Lorentz transformation:

$$\begin{pmatrix}t' \\ x' \\ y' \\ z'\end{pmatrix} = {\huge \textrm{MT2}(-v) } \; \begin{pmatrix}t \\ x \\ y \\ z\end{pmatrix} = \begin{pmatrix} 1 & \frac{-v}{c^2} & 0 & 0 \\ (-v)(1 - \frac{v^2}{c^2}) & 1 - \frac{v^2}{c^2} & 0 & 0 \\ 0 & 0 & \sqrt{1 - \frac{v^2}{c^2}} & 0 \\ 0 & 0 & 0 & \sqrt{1 - \frac{v^2}{c^2}} \end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z\end{pmatrix} = \sqrt{1 - \frac{v^2}{c^2}} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 - \frac{v^2}{c^2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} & \frac{-v}{c^2} \times \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} & 0 & 0 \\ \frac{-v}{\sqrt{1-\frac{v^2}{c^2}}} & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z\end{pmatrix} = \begin{pmatrix} \sqrt{1 - \frac{v^2}{c^2}} & 0 & 0 & 0 \\ 0 & \left( 1 - \frac{v^2}{c^2} \right) ^{\tiny \frac{3}{2}} & 0 & 0 \\ 0 & 0 & \sqrt{1 - \frac{v^2}{c^2}} & 0 \\ 0 & 0 & 0 & \sqrt{1 - \frac{v^2}{c^2}} \end{pmatrix} { \huge \Lambda(-v) } \begin{pmatrix}t \\ x \\ y \\ z\end{pmatrix}$$

$$x$$ - real coordinate
$$x'$$ - visual coordinate

"real coordinate" is a contradiction in terms -- all coordinate systems are man-made inventions to help them order and describe the natural universe. In relativity x and x' are on equal footing -- one way to think about it is x is my coordinate system and x' is your coordinate system, that doesn't make x better than x' -- just different.

When explaining special relativity and the Lorentz transforms, I wrote:

Special relativity is NOT a rescaling of the coordinate axes -- it is much more akin to a rotation of the space and time axis.

That which is rotated can be rotated back to the original state.
That which is Lorentz-transformed can be Lorentz-transformed back to the original state.
That is just one of many ways the the Lorentz transformation is like a rotation -- it is one of the main reasons why neither Masterov's summary of Special Relativity or the MT transform is physically meaningful.

So lets see if MT2 is like a rotation.

So it follows that
$$\left( {\huge \textrm{MT2}(-v) } \right)^{-1} = \left( { \huge \Lambda(-v) } \right)^{-1} \begin{pmatrix} \sqrt{1 - \frac{v^2}{c^2}} & 0 & 0 & 0 \\ 0 & \left( 1 - \frac{v^2}{c^2} \right) ^{\tiny \frac{3}{2}} & 0 & 0 \\ 0 & 0 & \sqrt{1 - \frac{v^2}{c^2}} & 0 \\ 0 & 0 & 0 & \sqrt{1 - \frac{v^2}{c^2}} \end{pmatrix}^{-1} = { \huge \Lambda(v) } \begin{pmatrix} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} & 0 & 0 & 0 \\ 0 & \left( 1 - \frac{v^2}{c^2} \right) ^{\tiny -\frac{3}{2}} & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \end{pmatrix} = \begin{pmatrix} \frac{1}{1 - \frac{v^2}{c^2}} & \frac{v}{c^2 \left( 1 - \frac{v^2}{c^2} \right)^2} & 0 & 0 \\ \frac{v}{1 - \frac{v^2}{c^2}} & \frac{1}{\left( 1 - \frac{v^2}{c^2} \right)^2} & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \end{pmatrix}$$
And since the inverse is not a MT2 transform, the MT2 transform is not a symmetry. What is MT2-transformed cannot be MT2-transformed back.

Nor does it preserve the consistency of light speed in the X direction:
$${\huge \textrm{MT2}(-v) } \begin{pmatrix}t \\ ct \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix}(1 - \frac{v}{c}) t \\ (1 - \frac{v^2}{c^2}) (1 - \frac{v}{c}) ct \\ 0 \\ 0 \end{pmatrix}$$
So it can't explain the null-result of the Michelson-Morley experiment.

It is a worthless definition, not suitable for any purpose in physics.

 

[Masterov]

$$\Delta t \neq 0$$

$$\Delta t$$ - arbitrary value.

$$\Delta x, \Delta y, \Delta z, \Delta t$$ - arbitrary values and have not well-defined.

 

[Masterov]

Relativistic transformations (type LT) are not important in MT, because relativistic effects are visual (type - acoustic).

In MT are valid Galilean transformation.

 

[Masterov]

I speak by mathematics as well as you

Is this claim true?

Yes.
Argumrnt: my monogrphy.

My specialty: nonlinear dynamics (auto-wave, auto-oscillations, attractors ...).
I'm a programming-mathematician (algorithmist) and a physicist.
Wrote his first program in 1979.
I do program for everything: from cell phones to clusters.

Web-master from 1995.

 

[Masterov]

Your math not going helps you to think.

Generally, I do my thinking before I start my math. The math is there to show that my thinking is self-consistent, logical and precise -- the qualities of good thinking in physics.

I see your math.
Your math does not make sense.
Your math is this: the formula for formulas.

Are you trying to scare me by "complicated" mathematical calculations.
This is ridiculous.

I recently thrown a few pounds of paper (where I scribbled my formulas for half a long while) in trash.

I have not write formulas almost ten years, and therefore lost many of the skills, and (nevertheless): your attempts to scare me your primitive mathematics make me smile.

 

[Masterov]

Your math not going helps you to think.

Generally, I do my thinking before I start my math. The math is there to show that my thinking is self-consistent, logical and precise -- the qualities of good thinking in physics.

I see your math.
Your math does not make sense.
Your math is this: the formula for formulas.

Are you trying to scare me by "complicated" mathematical calculations.
This is ridiculous.

I recently thrown a few pounds of paper (where I scribbled my formulas for half a long while) in trash.

I have not write formulas almost ten years, and therefore lost many of the skills, and (nevertheless): your attempts to scare me your primitive mathematics make me smile.
==================================

ß âèæó âàøó ìàòåìàòèêó.
 âàøåé ìàòåìàòèêå íåò íèêàêîãî ñìûñëà.
Âàøà ìàòåìàòèêà, ýòî: ôîðìóëû ðàäè ôîðìóë.

Âû ïûòàåòåñü íàïóãàòü ìåíÿ ãðîìîçäêîñòüþ ìàòåìàòè÷åñêèõ âûêëàäîê.
Ýòî ñìåøíî.

ß íåäàâíî âûáîðîíèë â ìóñîð íåñêîëüêî êèëîãðàìì (èñïèñàííîé ôîðìóëàìè) áóìàãè.
Òàì âûêëàäêè, íà êîòîðûå ÿ ïîòðàòèë ãîäû çàòâîðíè÷åñòâà.

ß ïî÷òè äåñÿòü ëåò íå ïèñàë ôîðìóë, à ïîòîìó ðàñòåðÿë ìíîãèå íàâûêè, è (òåì íå ìåíåå): âàøè ïîòóãè íàïóãàòü ìåíÿ âàøåé ïðèìèòèâíîé ìàòåìàòèêîé âûçûâàþò ó ìåíÿ óëûáêó.

 

[AlphaNumeric]

Your math does not make sense.
Your math is this: the formula for formulas.

The fact you don't understand them doesn't make them nonsense.

Are you trying to scare me by "complicated" mathematical calculations.

Complicated? The stuff Rpenner has been posting is basic special relativity. Special relativity can be grasped with basic understanding of vectors and matrices, it is the application of simple linear algebra. If you think what he's posting is to be considered 'complicated' you don't know the level of complexity difficult mathematical physics can be on. You might have no grasp of it but a proper discussion of the details of special relativity will involve the use of some mathematics. Denying it because you think it's complicated doesn't make your ignoring of it valid.

I recently thrown a few pounds of paper (where I scribbled my formulas for half a long while) in trash.

Not surprising.

I have not write formulas almost ten years, and therefore lost many of the skills, and (nevertheless): your attempts to scare me your primitive mathematics make me smile.

Hacks might think the use of mathematics is meant to scare but people who can actually do physics know that mathematics is the language which allows for precise things to be modelled and predicted.

 

[Masterov]

Your math does not make sense.
Your math is this: the formula for formulas.

The fact you don't understand them doesn't make them nonsense.

I don't understand?

I understand the written.
And I understand that written nonsense.

you don't know the level of complexity difficult mathematical physics

I know the level of complexity difficult mathematical physics.

Look again at my monograph, to be sure.

There I tried to write a complex by simple language.

rpenner tries to express simple things by cumbersome calculations, and he can not cope with it.
I can clearly see his attempts to confuse my mind.

 

[Masterov]

I recently thrown a few pounds of paper (where I scribbled my formulas for half a long while) in trash.

Not surprising.

Not surprising?
That is not surprising?

My monograph are garbage?
That?

You read it?

 

[Masterov]

I have not write formulas almost ten years, and therefore lost many of the skills, and (nevertheless): your attempts to scare me your primitive mathematics make me smile.

Hacks might think the use of mathematics is meant to scare but people who can actually do physics know that mathematics is the language which allows for precise things to be modelled and predicted.

Is meant?
No.

Can used to scare.

rpenner writes nonsense by cumbersome formulas in the hope that I do not understand them, and - I have to get scared and I have to stop debate.

rpenner wants it.

And rpenner wants that the debate was drowned in the analysis of those stupid things that he wrote.

======================

Предназначена?
Нет.

Математика может быть использована, чтобы напугать оппонента.

rpenner пишет ерунду громоздкихми формулами в надежде, что я не понимаю их, и - я должен испугаться, и я должен остановить дискуссию.

rpenner так хочет.

И rpenner хочет, чтобы дискуссия утонула в анализе глупостей, которые он написал формулами.

 

[Masterov]

Let's go back to the topic.

I continue to look forward a repetitive (Liangzao FAN's) experiments, which will be shown (or denied) that the temperature of the target does not cease to grow in proportion to the potential difference of the accelerating field under the influence of super-relativistic electrons.

And I'm surprised by the fact that so important for science an experiments, which repeatedly conducted in many laboratories around the world, has never been put in print for a century.

For it could be only one explanation: the temperature of the target ceases to grow along with the speed of super-relativistic electrons, that refutes the SRT.

This fact of that the results of these experiments contradict SRT, is the reason for the absence of the results of these experiments in the press.

=================

Я продолжаю с нетерпением ожидать повторного эксперимента, в котором будет показано (ли опровергнуто), что температура мишени не прекращает свой рост пропорционально разности потенциалов ускоряющего поля под воздействием супер релятивистских электронов.

И меня удивляет тот факт, что столь важные для науки эксперименты, многократно проводимые во многих лабораториях мира, ни разу не попали в печать за целое столетие.

Этому может быть лишь одно объяснение: температура мишени прекращает свой рост вместе со скоростью супер релятивистских электронов, что опровергает SRT.

Именно тот факт, что результаты этих экспериментов опровергают SRT, является причиной отсутствия результатов этих экспериментов в печати.

 

[Masterov]

Topic 2:

1. The traveller immigrated to another planet, which runs away from his home planet. Therefore traveler lives in slow time. (So ​​says SRT.)

2. The traveller decided to return to their homeland. His time has accelerated and become normal. Time of the traveller has slowed (yet again) with respect to the second planet.

So there we have a conflict: time of the traveler are accelerated and slowed down at the same time.

How is this possible?

If STR allows such findings, it is evident: STR - lying.

If LT allows such findings, it is evident: LT - lying.

====================

1.Путешественник иммигрировали на другую планету, которая убегает от его родной планеты. Поэтому путешественник живет в замедленном времени. (Так говорит SRT).

2.Путешественник решил вернуться на родину. Его время ускорились и стало нормальным. В то же время путешественник замедлился (еще раз) по отношению ко второй планете.

Так что у нас есть конфликт: время путешественника ускоряются и замедляются одновременно.

Как это возможно?

Если STR позволяет такие выводы, то очевидно: STR - лжет.

Если LT позволяет такие выводы, очевидно: LT – лжет тоже.

 

[OnlyMe]

Topic 2:

1. The traveller immigrated to another planet, which runs away from his home planet. Therefore traveler lives in slow time. (So ​​says SRT.)

2. The traveller decided to return to their homeland. His time has accelerated and become normal. Time of the traveller has slowed (yet again) with respect to the second planet.

So there we have a conflict: time of the traveler are accelerated and slowed down at the same time.

How is this possible?

If STR allows such findings, it is evident: STR - lying.

If LT allows such findings, it is evident: LT - lying.

====================

1.Путешественник иммигрировали на другую планету, которая убегает от его родной планеты. Поэтому путешественник живет в замедленном времени. (Так говорит SRT).

2.Путешественник решил вернуться на родину. Его время ускорились и стало нормальным. В то же время путешественник замедлился (еще раз) по отношению ко второй планете.

Так что у нас есть конфликт: время путешественника ускоряются и замедляются одновременно.

Как это возможно?

Если STR позволяет такие выводы, то очевидно: STR - лжет.

Если LT позволяет такие выводы, очевидно: LT – лжет тоже.

Masterov, it seems to me you are getting stuck in at least part of what is resolved in the Twin Paradox.

I don't think it would add any substance to this thread to get into that, but there are many past examples of discussions that work through the twin paradox, in various levels of complexity.

Try checking the Twin Paradox with a google search, first. You may find an example in Ruusian. There are also many past threads both here and on other sites that go into variations.

You seem to believe that time has some fixed standard in the universe and that SR just ignores that and uses the observer's experience of time (what I think you are calling visual time).

The problem is, we can only define time within the contex of our experience of change and we use clocks to measure that change. This does suggest that our experience of time is subjective, but that does not make it any less real or useful in describing the universe.

SR and Relativity handles this subjective aspect of our experience of time, with Lorentz Transforms and an understanding of the Relativity of Simultaneity. Both of which allow us to accurately project what things look like from frames of reference other than our own.., without assuming that either frame of reference is special, preferred or represents a state of rest relative to the universe.

We have to do that because, in observing the world and the universe around us, we have no way, to determine if there is any frame of reference that can be thought of as at absolute rest, comparred to everything else. Everything appears to be moving....

 

[Masterov]

Masterov, it seems to me you are getting stuck in at least part of what is resolved in the Twin Paradox.

Paradox?

This "Paradox" would be a death sentence for any other theory, but not for SRT.
For some reasons.
You want called these causes?

SRT has such "Paradoxs" no less than a mongrel dog has a fleas.

No need of time dilation.
Enough agree to make all spatial coordinates will be relativistic, and then the need of time dilation will disappear. Twin Paradox and Paradox Ehrenfest longer be relevant after this.

====================================

Этот парадокс был бы смертным приговором для любой другой теории, но не на СТО.
По каким-то причинам.
Вы не хотели бы называть эти причины?

СТО имеет таких парадоксов не меньше, чем беспородные собаки блох.

Нет необходимости в замедлении времени.
Достаточно согласиться, чтобы все пространственные координаты стали релятивистскими, и необходимость замедления времени исчезнет слеолм за эти. Парадокс близнецов и парадокс Эренфеста перестанет быть актуальным после этого.

 

[rpenner]

$$\Delta t \neq 0$$

Not necessarily, but circumstances may cause either $$\Delta t = 0$$ or $$\Delta t \neq 0$$ to be true.

$$\Delta t$$ - arbitrary value.

Here you contradict yourself. If it's arbitrary, then it can be zero, which you deny.

$$\Delta x, \Delta y, \Delta z, \Delta t$$ - arbitrary values and have not well-defined.

Here you are simply wrong. While they are undefined in special relativity, they are defined in any application of special relativity just like I defined them for my exploration of general length contraction.
Here the $$\equiv$$ symbol is used to introduce definitions.

Look at the separation between two events, A and B, on two distinct world lines with the same velocity. ...

So the coordinate separation between events in one system is:
$${\huge \Delta S} \equiv \begin{pmatrix} \rule{0pt}{20ex} \Delta t \\ \rule{0pt}{20ex} \Delta x \\ \rule{0pt}{20ex} \Delta y \\ \rule{0pt}{20ex} \Delta z \end{pmatrix} \equiv \begin{pmatrix} \rule{0pt}{20ex} t_A - t_B \\ \rule{0pt}{20ex} x_A - x_B\\ \rule{0pt}{20ex} y_A - y_B\\ \rule{0pt}{20ex} z_A - z_B \end{pmatrix} $$ (eqn 2)

Moreover, since I defined them, I could use mathematics to describe the constraints on their non-arbirary values.
In high school pre-algebra or algebra, students are taught that the equation of a line is A x + B y + C = 0 or if B isn't zero we can use m = -A/B and b = -C/B and write y = mx + b which is why the equation for linear motion in one spacial dimension is x(t) = u t + x₀.
In three spatial dimensions we write: $$\begin{pmatrix}x \\ y \\ z \end{pmatrix} = t \begin{pmatrix} u_x \\ u_y \\ u_z \end{pmatrix} + \begin{pmatrix}x_0 \\ y_0 \\ z_0 \end{pmatrix}$$ but this form is inconvenient since to talk about coordinate transformations it would be better if all the coordinates were on the same side.
So we write:
$$-t \begin{pmatrix} u_x \\ u_y \\ u_z \end{pmatrix} + \begin{pmatrix}x \\ y \\ z \end{pmatrix} = \begin{pmatrix}x_0 \\ y_0 \\ z_0 \end{pmatrix}$$
$$\begin{pmatrix} -u_x & 0 & 0 & 0\\ -u_y & 0 & 0 & 0\\ -u_z & 0 & 0 & 0\end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z \end{pmatrix} + \begin{pmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z \end{pmatrix} = \begin{pmatrix}x_0 \\ y_0 \\ z_0 \end{pmatrix}$$
$$\begin{pmatrix} -u_x & 1 & 0 & 0\\ -u_y & 0 & 1 & 0\\ -u_z & 0 & 0 & 1\end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z \end{pmatrix} = \begin{pmatrix}x_0 \\ y_0 \\ z_0 \end{pmatrix}$$
Finally, we augment this system of three equations with 0 = 0 so that the matrix on the left will be a 4x4 matrix of rank 3.
$$\begin{pmatrix} 0 & 0 & 0 & 0 \\ -u_x & 1 & 0 & 0\\ -u_y & 0 & 1 & 0\\ -u_z & 0 & 0 & 1\end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z \end{pmatrix} = \begin{pmatrix}0 \\ x_0 \\ y_0 \\ z_0 \end{pmatrix}$$
which is just a convenient way to write:
$$\begin{eqnarray} x & = & u_x t + x_0 \\ y & = & u_y t + y_0 \\ z & = & u_z t + z_0 \end{eqnarray}$$
as a single equation where the coordinates of one event t,x,y,z are treated as a single thing (a vector).

The equation for linear motion in 3 spatial coordinates is written like this for events on world line A:
$$\begin{pmatrix} \rule{0pt}{20ex} 0 & 0 & 0 & 0 \\ \rule{0pt}{20ex} -u_x & 1 & 0 & 0 \\ \rule{0pt}{20ex} -u_y & 0 & 1 & 0 \\ \rule{0pt}{20ex} -u_z & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \rule{0pt}{20ex} t_A \\ \rule{0pt}{20ex} x_A \\ \rule{0pt}{20ex} y_A \\ \rule{0pt}{20ex} z_A \end{pmatrix} = \begin{pmatrix} \rule{0pt}{20ex} 0 \\ \rule{0pt}{20ex} x_0 + L_x\\ \rule{0pt}{20ex} y_0 + L_y\\ \rule{0pt}{20ex} z_0 + L_z\end{pmatrix} $$
And for events on world line B:
$$\begin{pmatrix} \rule{0pt}{20ex} 0 & 0 & 0 & 0 \\ \rule{0pt}{20ex} -u_x & 1 & 0 & 0 \\ \rule{0pt}{20ex} -u_y & 0 & 1 & 0 \\ \rule{0pt}{20ex} -u_z & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \rule{0pt}{20ex} t_B \\ \rule{0pt}{20ex} x_B \\ \rule{0pt}{20ex} y_B \\ \rule{0pt}{20ex} z_B \end{pmatrix} = \begin{pmatrix} \rule{0pt}{20ex} 0 \\ \rule{0pt}{20ex} x_0\\ \rule{0pt}{20ex} y_0\\ \rule{0pt}{20ex} z_0\end{pmatrix} $$
So it follows that we don't need $$x_0, y_0, z_0$$ if we work with coordinate differences.

The following equation contrains events A and B to be on parallel world lines both moving at velocity $$(u_x, \; u_y, \; u_z)$$ :
$$\begin{pmatrix} \rule{0pt}{20ex} 0 & 0 & 0 & 0 \\ \rule{0pt}{20ex} -u_x & 1 & 0 & 0 \\ \rule{0pt}{20ex} -u_y & 0 & 1 & 0 \\ \rule{0pt}{20ex} -u_z & 0 & 0 & 1 \end{pmatrix} {\huge \Delta S} = \begin{pmatrix} \rule{0pt}{20ex} 0 \\ \rule{0pt}{20ex} L_x \\ \rule{0pt}{20ex} L_y \\ \rule{0pt}{20ex} L_z \end{pmatrix} $$ (eqn 3)

Relativistic transformations (type LT) are not important in MT, because relativistic effects are visual (type - acoustic).

This is your misunderstanding of relativity, not found in any college level textbook. Relativity is the mental exercise of putting yourself in the shoes of another and working out how the same universe is described from a different place and different standard of rest. For special relativity, that mental exercise requires use of the Lorentz transform.

---

In MT are valid Galilean transformation.

That is not what you wrote earlier this month:

MT Velocity transformation

$$x'=(x-vt)(1-v^2/c^2)$$
$$y'=y\sqrt{1-v^2/c^2}$$
$$z'=z\sqrt{1-v^2/c^2}$$
$$t'=t-vx/c^2$$

In the language of linear algebra, this is exactly the same as:
$$\begin{pmatrix}t' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix}1 & \quad & \frac{-v}{c^2} & 0 & 0 \\ (-v)(1 - \frac{v^2}{c^2}) & & 1 - \frac{v^2}{c^2} & 0 & 0 \\ 0 & & 0 & \sqrt{1-\frac{v^2}{c^2}} & 0 \\ 0 & & 0 & 0 & \sqrt{1-\frac{v^2}{c^2}} \end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \sqrt{1 - \frac{v^2}{c^2}} & 0 & 0 & 0 \\ 0 & \left(1 - \frac{v^2}{c^2} \right)^{\tiny \frac{3}{2}} & 0 & 0 \\ 0 & 0 & \sqrt{1-\frac{v^2}{c^2}} & 0 \\ 0 & 0 & 0 & \sqrt{1-\frac{v^2}{c^2}} \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} & \frac{-v}{c^2 \sqrt{1-\frac{v^2}{c^2}} } & 0 & 0 \\ \frac{v}{\sqrt{1-\frac{v^2}{c^2}}} & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} & 0 & 0 \\ 0 & 0 & \sqrt{1-\frac{v^2}{c^2}} & 0 \\ 0 & 0 & 0 & \sqrt{1-\frac{v^2}{c^2}} \end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \sqrt{1 - \frac{v^2}{c^2}} & 0 & 0 & 0 \\ 0 & \left(1 - \frac{v^2}{c^2} \right)^{\tiny \frac{3}{2}} & 0 & 0 \\ 0 & 0 & \sqrt{1-\frac{v^2}{c^2}} & 0 \\ 0 & 0 & 0 & \sqrt{1-\frac{v^2}{c^2}} \end{pmatrix} { \huge \Lambda (-v) } \begin{pmatrix}t \\ x \\ y \\ z \end{pmatrix}$$
So since $$\Lambda(-v)$$ is a Lorentz transform and not a Galilean transform, you have made a mistake somewhere.

---

Yes[, I speak mathematics as well as you].
Argumrnt: my monogrphy.

There was not much math in those 70 pages, and the very few parts that looked like math were rendered illegible when I opened the document on my computer. It looks like you typed up some common lecture notes that came to the conclusion that analysis of differential equations is hard and not totally solvable at this point in time (1996). I didn't see any original content that made this publishable back in 1996. Possibly you are the A.V. Masterov who co-authored an talk reported in conference proceeding called "Nelineĭnye volny / Nonlinear Waves: Dynamics and Evolution" (1989).

My specialty: nonlinear dynamics (auto-wave, auto-oscillations, attractors ...).

My specialty: Physics and I worked for a professor who specialized in nonlinear systems.

I'm a programming-mathematician (algorithmist) and a physicist.

Neither one should be flustered by linear algebra.

Wrote his first program in 1979.

Nearly the same for me.

I do program for everything: from cell phones to clusters.

Kind of unusual for a practicing physicist, wouldn't you agree? If you know Java, programming for cell phones is almost the same thing a programming for clusters. I created a cluster of computers in the circa 1990 when one machine simply wasn't fast enough.

Web-master from 1995.

Very unimpressive. Web-master is the 1996 career advice I gave my college roommate who had not previously been gainfully employed.

---

None of alleged expertise in math and physics explains why my use of math to convey physics arguments is viewed by you as some sort of scare tactic.

I see your math.
Your math does not make sense.

Math never makes sense to the untutored. As Euclid(Евклид) said to Ptolemy(Птолемей I Сотер): "There is no royal road to geometry."

Your math is this: the formula for formulas.

An equation exists because the left side equals the right side. Algebra is the manipulation of equations in a way that preserves the equality of left and right. Therefore if you are using an argument about precise quantities, algebra is a friend of truth.

Are you trying to scare me by "complicated" mathematical calculations.

I am not trying to scare anyone. I am making a logical argument in detail. I disagree that anything I have written here is complicated. In fact, since I have performed all the calculations that only leaves you with three choices.
1) Trust that my math and logic is sound and accept my conclusions. The truth wins because I am a bearer of the truth.
2) Not trust my math and logic, and perform the calculations yourself to see if you get a different answer. The truth wins because you tested what I was saying.
3) Lie and say my math and logic is wrong even though you don't understand it and thus have no basis to say it is wrong. The truth loses because you refused to hear the truth.

This is ridiculous.
I recently thrown a few pounds of paper (where I scribbled my formulas for half a long while) in trash.
I have not write formulas almost ten years, and therefore lost many of the skills, and (nevertheless):

None of this suggests that you have any math-based or physics-based reason to doubt my posts.

your attempts to scare me your primitive mathematics make me smile.

You contradict yourself. Are the maths "simple" as AlphaNumeric and I say, or "complicated" like you say? If the are simple and "primitive" that doesn't mean that they are wrong. It argues that they are as correct as saying 1+1=2. But this is not the first time you have accused me of trying to win the argument by scaring you.

Are you, rpenner, trying to intimidate me by mathematics, rather than to respond.

You do used math to complicate and to have perplex.

I am not using mathematics to obfuscate or intimidate. I am using math to elucidate the meaning of length contraction in special relativity, a theory about coordinate transformations with which you are unfamiliar. We have many physicists and mathematicians here, so I am confident that any of them would say I am using exactly the right level of mathematics to calculate general length contraction.

Your repeated defamatory claims have no basis in fact.

---

If you wish to "challenge" Special Relativity, it would be nice for you to understand the thing that you are challenging.

Special relativity is NOT a rescaling of the coordinate axes -- it is much more akin to a rotation of the space and time axis.

Russian Wikipedia on hyperbolic rotation

 

[rpenner]

rpenner writes nonsense by cumbersome formulas in the hope that I do not understand them, and - I have to get scared and I have to stop debate.

My hope is that you would understand them and so learn what Special Relativity says.

And rpenner wants that the debate was drowned in the analysis of those stupid things that he wrote.

You claim that they are stupid. In no place have you demonstrated that they are wrong. So we are not having a debate. I am stating correct definitions found in every applicable textbook and wikipedia page, and carefully doing each step of the calculations that such definitions lead to. You are stating incorrect definitions, contradicting yourself, and using rhetorical fallacies like some evil demagogue. You have avoided identifying any point of my math as confusing or wrong, because you know I can refute any specific claim. So you try and claim that because I use more math that you that I am trying to scare you when in fact I am trying to teach you.

---

Let's go back to the topic.

The topic title "Three Experiments Challenging SRT" is untrue because Fan didn't gather the data necessary to demonstrate his point, and because the data Fan did gather favors Special Relativity over Newton and Galileo and Fan, and because the person who started the topic didn't understand what Special Relativity says, what Galilean Relativity says or what Fan's pet theory says. See post 390.

I continue to look forward a repetitive (Liangzao FAN's) experiments, which will be shown (or denied) that the temperature of the target does not cease to grow in proportion to the potential difference of the accelerating field under the influence of super-relativistic electrons.

This would not be a repeat of Fan's experiment because 1) Fan didn't measure how many electrons hit the target and so could not show the point you talk about, and 2) Fan didn't use super-relativistic electrons since v < 0.5 c. Bertozzi, prior to Fan, did both measure how many electrons hit the target and used electrons with speeds of 0.866 c < v < c, so it is more correct that you are asking for a repeat of Bertozzi's experiment.

And I'm surprised by the fact that so important for science an experiments, which repeatedly conducted in many laboratories around the world, has never been put in print for a century.

When was the last time you saw a write up of Galileo's experiment with rolling wooden balls and ramps or Galileo's experiment at the Tower of Pisa? People don't just publish repeats of experiments -- they do publish improvements on repeats of experiments. For example, OPERA measured man-made neutrinos moving with kinetic energies averaging at 17 GeV over a distance of 600 km with a speed indistinguishable from that of light and found no statistically significant relation between speed and kinetic energy which is the prediction of special relativity for a particle that masses under 2 eV. http://arxiv.org/abs/1109.4897v4

For it could be only one explanation: the temperature of the target ceases to grow along with the speed of super-relativistic electrons, that refutes the SRT.

This is not what the evidence shows.

This fact of that the results of these experiments contradict SRT, is the reason for the absence of the results of these experiments in the press.

It is more likely that they are simply bad experiments (poor calibration, poor experimental design, inferior description of equipment, incomplete analysis) not published in any journal of good reputation. Also you say "these experiments" but only focus on the temperature experiment which did not measure how many electrons hit the target and so cannot be used to explain all the results as a per-electron velocity-energy relationship.

---

Topic 2:

1. The traveller immigrated to another planet, which runs away from his home planet. Therefore traveler lives in slow time. (So ​​says SRT.)

That is not what Special Relativity says. Special Relativity says the direction of the time axis of a coordinate system in relative motion to another coordinate system does not point in the same direction.

This is what Special Relativity says:
$${\huge \Delta S} \equiv \begin{pmatrix} \Delta t \\ \Delta x \\ \Delta y \\ \Delta z \end{pmatrix} \\ {\huge \Delta S'} \equiv \begin{pmatrix} \Delta t' \\ \Delta x' \\ \Delta y' \\ \Delta z' \end{pmatrix} \\ {\huge \Lambda (u) } \equiv \begin{pmatrix} \frac{1}{\sqrt{1-\frac{u^2}{c^2}}} & \frac{u}{c^2 \sqrt{1-\frac{u^2}{c^2}}} & 0 & 0 \\ \frac{u}{\sqrt{1-\frac{u^2}{c^2}}} & \frac{1}{\sqrt{1-\frac{u^2}{c^2}}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \\ {\huge \Delta S'} = {\huge \Lambda (v) \; \Delta S} \\ {\huge \Delta S} = {\huge \Lambda (-v) \; \Delta S'} $$

So if on the run-away planet the traveller stands still looks at his watch tick one second:
$${\huge \Delta S'} = \begin{pmatrix} 1 \; \textrm{second} \\ 0 \\ 0 \\ 0 \end{pmatrix}$$
then on the home planet, they see:
$$ \begin{pmatrix} \Delta t \\ \Delta x \\ \Delta y \\ \Delta z \end{pmatrix} = {\huge \Delta S} = {\huge \Lambda (-v) \; \Delta S'} = \begin{pmatrix} \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \; \textrm{second} \\ \frac{-v}{\sqrt{1-\frac{v^2}{c^2}}} \; \times 1 \; {second} \\ 0 \\ 0 \end{pmatrix}$$
So $$\Delta t > \Delta t'$$ but that isn't the whole story since $$(c \Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2 = (c \Delta t')^2 - (\Delta x')^2 - (\Delta y')^2 - (\Delta z')^2$$
In the same way, if we start with one second on the home planet we wind up with $$\Delta t' > \Delta t$$ which is not a contradiction because we aren't talking about rescaling time, we are talking about coordinate intervals on two different axes. Comparing just $$\Delta t$$ and $$\Delta t'$$ is comparing apples and oranges. What special relativity requires you to do is compare $$\Delta S$$ and $$\Delta S'$$.

Minkowski in 1908 explained that Einstein's 1905 theory of special relativity was a marriage of time and the geometry of space into a single geometry of space-time.

2. The traveller decided to return to their homeland. His time has accelerated and become normal. Time of the traveller has slowed (yet again) with respect to the second planet.

No conflict, since
$${\huge \Lambda (-v) \; \Lambda (v)} = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = {\huge \mathbf{1}}$$

So there we have a conflict: time of the traveler are accelerated and slowed down at the same time.

How is this possible?

Perhaps you made a mistake in your math or understanding of special relativity.

If STR allows such findings,

It doesn't.

it is evident: STR - lying.

Which is not the case.

If LT allows such findings,

It doesn't.

it is evident: LT - lying.

which is not the case.

Paradox?

This "Paradox" would be a death sentence for any other theory, but not for SRT.

Just like the word "paranoid" may mean "any mental disease", "paranoid schizophrenia" or "a fear that one has enemies", the word "paradox" means different things. In logic, a paradox is a self-contradiction. In physics, a paradox is a contradiction when two different physical theories are used to analyze the same situation.
In the twin paradox, the two different physical theories are the Newtonian concept that clocks that are started at the same time will read the same when brought together and the Special Relativistic concept that motion is a type of (hyperbolic) rotation of the space and time axes.
But if you use just one physical theory, there is no logical self-contradiction so we let reality cast the deciding vote between the two worldviews.

See Wikipedia: Hafele–Keating experiment
See Russian Wikipedia: Hafele–Keating experiment

No need of time dilation.

Contradicted directly by many experiments.

Enough agree to make all spatial coordinates will be relativistic, and then the need of time dilation will disappear.

Untrue.

 

[Masterov]

Not necessarily, but circumstances may cause either $$\Delta t = 0$$ or $$\Delta t \neq 0$$ to be true.
Here you contradict yourself. If it's arbitrary, then it can be zero, which you deny.
Here you are simply wrong. While they are undefined in special relativity, they are defined in any application of special relativity just like I defined them for my exploration of general length contraction.
Here the $$\equiv$$ symbol is used to introduce definitions.

$$\Delta x, \Delta y, \Delta z, \Delta t$$ - arbitrary values and have not well-defined.
For consideration what or a particular values of $$\Delta x, \Delta y, \Delta z, \Delta t$$ for the variables does not make any sense in this case.
The case of $$\Delta t=0$$ too.
All your math does not make sense.
You have written an empty set of symbols and mathematical operations.
The fact that you wrote makes no sense.