SR Issue

[chinglu]

After you apply the LT you get a different location for the light, but you also get a different time, (as rpenner has already explained in his response to this post). Using my example where v=0.500c and d'=1.000 the following results:

According to the Σ frame, the co-location of C' and M event has a time coordinate of t=0.866, and because c=1.000 that means the light flash must be located at x=ct=0.866 at that time. The event described by the light flash being located at (x,t)=(0.866,0.866) can be Lorentz transformed to the Σ' frame and found to have a time coordinate of t'=0.500, so that event is not simultaneous with the co-location event which had a time of t'=1.000.

According to the Σ' frame, the co-location of C' and M event has a time coordinate of t'=1.000, and because c=1.000 that means the light flash must be located at x'=ct'=1.000 at that time. The event described by the light flash being located at (x',t')=(1.000,1.000) can be Lorentz transformed to the Σ frame and found to have a time coordinate of t=1.732, so that event is not simultaneous with the co-location event which has a time of t=0.866.

Agreed, you get a different time and position.

Now, in M' frame coordinates what is this time and position of the lightning in M' frame coordinators when M' and C' are co-located?

See, when you leave the fact above off, you simply get a different space-time coordinate for the lightning. But, The fact is, C' and M are co-located. Is that the correct coordinate of the lightning position when C' and M are co-located in M' frame coordinates?

The answer is no. If the answer is yes, explain why.

 

[chinglu]

Aether theory is a topic from 19th century physics, and therefore consists of pre-quantum and pre-relativity (and thus strictly classical) models purporting to describe phenomena for which we have experimental evidence since 1859 don't obey predictions of Newtonian absolute space and absolute time.

Currently, a few child molesters have attempted to re-purpose the term to describe quantum excitations of a relativistic quantum field, but that physics already has a perfectly reasonable name of quantum field theory and doesn't need a old-and-misleading one.
If you link to a specific source, I will tell you if it is child-molesting or physics, and why.

Currently, the simplest physical theories that agree with experiments are Lorentz-covariant and are far simpler than classical models or wildly at odds with 150 years of experimental results.

Can you explain why the earth's rotational sagnac shows up in GPS but the earth's orbital saganc does not? Sagnac is sagnac no?

 

[chinglu]

Aether theory is a topic from 19th century physics, and therefore consists of pre-quantum and pre-relativity (and thus strictly classical) models purporting to describe phenomena for which we have experimental evidence since 1859 don't obey predictions of Newtonian absolute space and absolute time.

Currently, a few child molesters have attempted to re-purpose the term to describe quantum excitations of a relativistic quantum field, but that physics already has a perfectly reasonable name of quantum field theory and doesn't need a old-and-misleading one.
If you link to a specific source, I will tell you if it is child-molesting or physics, and why.

Currently, the simplest physical theories that agree with experiments are Lorentz-covariant and are far simpler than classical models or wildly at odds with 150 years of experimental results.

You did not answer.

What is the distance of the lightning from C' and M in M' frame coordinates when calculated by the M frame.

What is the distance of the lightning from C' and M in M' frame coordinates when calculated by the M' frame.

Why are they different and how can lightning be 2 different distances in M' frame coordinates from C' and M when the are co-located?

 

[Neddy Bate]

Agreed, you get a different time and position.

Correct. And that can be seen at a glance on this Minkowski diagram:

Now, in M' frame coordinates what is this time and position of the lightning in M' frame coordinators when M' and C' are co-located?

The answer to that question is:
x' = 1.000
t' = 1.000

See, when you leave the fact above off, you simply get a different space-time coordinate for the lightning. But, The fact is, C' and M are co-located. Is that the correct coordinate of the lightning position when C' and M are co-located in M' frame coordinates?

The answer is no. If the answer is yes, explain why.

The answer is yes. The M' frame finds the co-location of C' and M to occur at t'=1.000 so naturally the light pulse must be located at x'=ct'=1.000.

Now you might ask what these other M' frame coordinates represent:
x' = 0.500
t' = 0.500
That a different event which represents where the the light pulse was located at an earlier time.

 

[arfa brane]

More stuff about the lightcone and coordinates

In special relativity, light-cone coordinates is a special coordinate system where two of the coordinates, x+ and x− are null coordinates and all the other coordinates are spatial. Call them $$\, x_\perp $$.

Assume we're working with a (d,1) Lorentzian signature.

Instead of the standard coordinate system (using Einstein notation)

$$ ds^2=-dt^2+\delta_{ij}dx^i dx^j$$​

,

with $$\, i,j=1,\dots,d$$ we have

$$ ds^2=-2dx^+dx^- + \delta_{ij}dx^i dx^j$$​

with $$\, i,j=1, \dots,d-1, x^+=\frac{t+x}{\sqrt{2}}$$ and $$x^-=\frac{t-x}{\sqrt{2}}$$.

Both x+ and x- can act as "time" coordinates.

--http://en.wikipedia.org/wiki/Light-cone_coordinates
Ok. So I read something about how there is no Lorentz transform from (x+, x-, x[sup]2[/sup], x[sup]3[/sup]) to (x[sup]0[/sup], x[sup]1[/sup], x[sup]2[/sup], x[sup]3[/sup]). Or, you can't take these 'special' coordinates to a frame of reference.

 

[rpenner]

This is a thought experiment

In the OP, the author agrees to rigorously apply special relativity, which states times and locations of events are not absolute but are frame-dependent and the Lorentz transformation relates inertial coordinates given one standard of rest to inertial coordinates with different standard of rest. For convenience, the OP assumes that the two coordinate systems, Σ and Σ',are set up such that $$x' = \gamma ( x - v t), \; y' = y, \; z' = z, \; t' = \gamma( t - v x /c^2 ) $$. Further, for all events of interest, the OP assumes $$y' = y = z' = z = 0$$, so even though every event has 8 coordinates (x, y, z, t, x', y', z', t'), we only have two degrees of freedom at play. Therefore specifying any of the following pairs of coordinates for an event tells you everything about the event: (x,t), (x,x'), (x, t'), (t,x'), (t, t'), (x', t').

The OP does not give names to the events, but names appeared immediately afterwords in [post=3198606]post #2[/post]: O, P, Q and R. One naively imagines that the OP would have no trouble keeping track of just two degrees of freedom of just four events.

Assume M and M' are the origins of 2 frames

Here the OP introduces line f, defined by $$x = x_O = 0$$, and line g, defined by $$x' = x'_O = 0$$. These are both straight, time-like lines.

and [at rest relative to M'], there is an observer C' located at $$(\frac{-vd'}{c},0,0)$$ with $$d'>0$$.

Here the OP introduces line h, defined by $$x' = x'_P = - v \frac{d'}{c}$$.

When M and M' are co-located, lightning strikes their command location.

Here, for the first time, the OP introduces event O. Three lines meet at event O, lines f and g which have already been described and a new line ℓ, which the OP will define as light-like. Note that the OP realizes to specify both "when" and "location" of the event starting line ℓ.
Event O is known to have coordinates $$(x_O = 0, x'_O = 0)$$, and from $$x' = \gamma ( x - v t)$$ we solve $$t_O = \frac{x_O - x'_O/ \gamma}{v} = 0 $$. Likewise from $$t' = \gamma( t - v x /c^2 )$$ we immediately get $$t'_O = \gamma( t_O - v x_O /c^2 ) = 0$$ so the coordinates are $$(x_O = 0, \; t_O = 0, \; x'_O = 0, \; t'_O = 0)$$.

Likewise, event P is about to be introduced without being named. Event P is defined as the event where both $$x_P = x_O = 0$$ and $$x'_P = - v \frac{d'}{c}$$ are true or equivalently as the event at the crossing of lines f and g. From the above it follows that $$t_P = \frac{x_P - x'_P/ \gamma}{v} = \frac{d'}{\gamma c}$$ and $$t'_P = \gamma( t_P - v x_P /c^2 ) = \frac{d'}{c}$$ so the coordinates are $$(x_P = 0, \; t_P = \frac{d'}{\gamma c}, \; x'_P = - v \frac{d'}{c}, \; t'_P = \frac{d'}{c})$$.

Here is the question, when C' and M are co-located, where is the light[] along the positive x-axis for both frame coordinate systems?

This is an ambiguous question, because "when C' and M are co-located" could mean $$t = t_P$$ or $$t' = t'_P$$ but because there is no absolute time in special relativity, it follows that these two definitions are only both true at event P. That's why [post=3198606]post #2[/post] introduces line j which is defined as $$t = t_P$$, and line k which is defined as $$t' = t'_P$$. Because these are two different space-like lines, it follows that any time-like or light-like line that does not pass through event P must intersect these lines at two different Events. So one ambiguity of this question is which of the two events we are talking about. The easiest way to resolve this ambiguity is to label the lines (or equations) for j and k, as well as the events (or solutions) Q and R.
But there is a second ambiguity, because "where is the light[] along the positive x-axis" is asking for both the x and the x' values. Asking such a confused question suggests not enough time was spent understanding the geometry of special relativity. As we will later see, the OP attempts to answer this question without labeling the results, resulting in further confusion.

We call the event where lines j and ℓ meet event Q. We have $$t_Q=t_P = \frac{d'}{\gamma c}, \; x_Q = x_O + c t_Q - c t_O = \frac{d'}{\gamma}, \; x'_Q = \gamma ( x_Q - v t_Q) = d' \left( 1 - \frac{v}{c} \right), \; t'_Q = \gamma( t_Q - v x_Q /c^2 ) = \frac{d'}{c} \left( 1 - \frac{v}{c} \right)$$
We call the event where lines k and ℓ meet event R. We have $$t'_R = t'_P = \frac{d'}{c}, \; x_R = x_O + c t_R - c t_O = c t_R$$. From $$t'_R = \gamma( t_R - v x_R /c^2 ) = \gamma t_R ( 1- v/c )$$ it follows that $$t_R =\frac{t'_R}{\gamma \left( 1 - \frac{v}{c} \right) } = \gamma \frac{d'}{c} \left( 1 + \frac{v}{c} \right), \; x_R = \gamma d' \left( 1 + \frac{v}{c} \right)$$. Finally $$x'_R = \gamma ( x_R - v t_R) = \gamma^2 d' \left( 1 + \frac{v}{c} \right)\left( 1 - \frac{v}{c} \right) = \gamma^2 d' \left( 1 - \frac{v^2}{c^2} \right) = d'$$.

So the coordinates for Q are $$\left( x_Q = \frac{d'}{\gamma}, \; t_Q = \frac{d'}{\gamma c}, \; x'_Q = d' \left( 1 - \frac{v}{c} \right), \; t'_Q = \frac{d'}{c} \left( 1 - \frac{v}{c} \right) \right)$$. And the coordinates of R are $$\left( x_R = \gamma d' \left( 1 + \frac{v}{c} \right), \; t_R = \gamma \frac{d'}{c} \left( 1 + \frac{v}{c} \right), \; x'_R = d', \; t'_R = \frac{d'}{c} \right)$$

Thus the fully-labelled coordinates for Q and R completely answer the question in an unambiguous manner. The OP does calculate these same quantities, but fails to label them and so the OP becomes confused.

When C' and M are co-located, SR claims the light[] is located at M frame space-time coordinates of $$(d'/\gamma,0,0,\frac{d'}{c\gamma})$$ and $$(d'\gamma(1+v/c),0,0,d'\gamma(1+v/c)/c)$$.

See how the OP cites unlabeled coordinates: $$\left( x_Q = \frac{d'}{\gamma}, \; t_Q = \frac{d'}{\gamma c} \right)$$ and $$\left( x_R = \gamma d' \left( 1 + \frac{v}{c} \right), \; t_R = \gamma \frac{d'}{c} \left( 1 + \frac{v}{c} \right) \right)$$.

When C' and M are co-located, SR claims the lightning is located at M' frame space-time coordinates of $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$ and $$(d',0,0,d'/c)$$.

See how the OP cites unlabeled coordinates: $$\left( x'_Q = d' \left( 1 - \frac{v}{c} \right), \; t'_Q = \frac{d'}{c} \left( 1 - \frac{v}{c} \right) \right)$$ and $$\left( x'_R = d', \; t'_R = \frac{d'}{c} \right)$$.

when M and C' are co-located, one lightning strike is located at 2 different positions along the positive x-axis in both coordinate systems, which of course is inconsistent with nature.

The OP does not state HOW this is inconsistent with nature. Events Q and R undeniably happen in different positions and times in every frame, but that's true about any two distinct events on a light-like line. That is entirely consistent with the light-postulate which requires the same light at different times to be at different locations.

So if the OP thinks there is a problem with two positions, the real problem must be with existence of two distinct events. But there always were going to be two distinct events Q and R because there were always two different frames describing a frame-dependent concept of "same time as event P". This resulted in lines j and k being distinct space-like lines and as I said above, that leads to two distinct events Q and R.

That the OP now complains that $$x_Q \neq x_R$$ means the OP never understood that $$t = t_P$$ is a different line than $$t' = t'_P$$ and thus never understood Relativity of Simultaneity.

So, where is the error in the calculations?

As I said before, the problem was not in the algebraic calculations, but in asking illegal questions which are predicated on assuming frame-independence to the concept of simultaneity which violates the requirement of a thought experiment to use rigorous mathematical logic. By assuming the opposite of relativity of simultaneity and by using the Lorentz transforms which incorporate relativity of simultaneity, the OP doomed his efforts to do a rigorous thought experiment.

From [POST=3198606]Post #2[/POST] we see that the solution is obvious:

Let f,g,h be time-like inertial world lines. Let j, k be space-like straight lines. Let ℓ be a light-like straight line. Then we have in both coordinate system the following descriptions of these lines:
$$
\begin{array}{c|c|c} \textrm{Line} & \textrm{Frame} \; \Sigma & \textrm{Frame} \; \Sigma' \\ \hline \\ f & x = 0 & x' = -vt' \\ g & x = vt & x' = 0 \\ h & x = vt - \frac{d'}{c} v \sqrt{1 - \frac{v^2}{c^2}} & x' = - \frac{d'}{c} v \\ j & t = \frac{d'}{c} \sqrt{1 - \frac{v^2}{c^2}} & x' = - \frac{c^2}{v} \left( t' - \frac{d'}{c} \left(1 - \frac{v^2}{c^2} \right) \right) \\ k & x = \frac{c^2}{v} \left( t - \frac{d'}{c} \sqrt{1 - \frac{v^2}{c^2} } \right) & t' = \frac{d'}{c} \\ \ell & x = ct & x' = ct' \end{\array} $$ $$
\begin{array}{c|c|c} \textrm{Event} & \textrm{Frame} \; \Sigma & \textrm{Frame} \; \Sigma' \\ \hline \\ O=f \cap g \cap \ell & \left( x=0, \; t=0\right) & \left(x'=0, \; t'=0 \right) \\ P = f \cap h \cap j \cap k & \left( x = 0, \; t = \frac{d'}{c} \sqrt{1 - \frac{v^2}{c^2}} \right) & \left( x' = - \frac{d'}{c} v, \; t' = \frac{d'}{c} \right) \\ Q = j \cap \ell & \left( x = d' \, \sqrt{1 - \frac{v^2}{c^2}}, \; t = \frac{d'}{c} \sqrt{1 - \frac{v^2}{c^2}} \right) & \left( x' = d' \, \left(1 - \frac{v}{c} \right) , \; t' = \frac{d'}{c} \left(1 - \frac{v}{c} \right) \right) \\ R = k \cap \ell & \left( x = \frac{c d' \sqrt{1 - \frac{v^2}{c^2}}}{c - v}, \; t = \frac{d' \sqrt{1 - \frac{v^2}{c^2}}}{c - v} \right) &\left( x' = d' , \; t' = \frac{d'}{c} \right) \end{\array}$$
So by ignoring relativity of simultaneity, you improperly confuse lines j and k and therefore confuse events Q and R.​

A [POST=3204151]later post[/POST] would again emphasize the need to think geometrically about the problem:

Can you show me specifically in the calculations where I asserted absolute time? ... So, please show the specific calculation(s) where I used absolute time.

See above. Absolute time was in the question and the conclusions based on the question.

Assume P is the logical predicate

[POST=3199258]This post[/POST] went nowhere because P is not any type of logical predicate. P is a space-time event which happens to have coordinates in frame Σ and different coordinates in frame Σ'. But $$t = t_P$$ and $$t' = t'_P$$ are only both true at event P.

when P is true, SR claims the lightning is at 2 different places in M frame coordinates.

It's clear that the OP continues to have problems telling the difference between $$t = t_P$$ and $$t' = t'_P$$.

I can put it in your lingo.

When C' and M are co-located, event Q is valid for the frame Σ.

That is not anyone's lingo.

I read through your posts but you did not answer this specific question in the OP. ... So, could you please answer this simple question?

Everyone sees [POST=3198606]Post #2[/POST].

You also need to come to the conclusion that ROS does not apply to one event.

A valid statement of Relativity of Simultaneity is that $$t = t_P$$ is not equivalent to $$t' = t'_P$$ for coordinate time in inertial frames which are in relative motion. Since neither event Q or event R is the same event as event P, it follows that $$t_P = t_Q$$ and $$t'_P = t'_R$$ is perfectly consistent with $$t_Q < t_R$$.

Rpenner, you have many defending you

Are you suing me in Science Court?

all of which are simple minions that cannot follow the math.

The math says minions get paid, so obviously they are not my minions.

Lightning cannot be at 2 different positive x-axis locations at the co-location event of C' and M.

Must you confuse everything? Line ℓ does not pass through event P, the co-location event of C' and M. So the question of what event on line ℓ is simultaneous with event P is a frame-dependent question that has everything to do with man-made coordinate systems and nothing to do with the geometry or physics of space-time.

Can you explain specifically why the co-location event of C' and M are 2 events? Otherwise, agree it is one event.

That would be tantamount to asserting absolute time, which is not consistent with special relativity and therefore disallowed by the rigor required by your OP which purports to do a "Thought experiment."

Can you explain why the earth's rotational sagnac shows up in GPS but the earth's orbital saganc does not? Sagnac is sagnac no?

You don't understand the Sagnac effect, which relates to the area encompassed by the light path and the rotation rate relative to an inertial frame. So you have confused three measures of the Earth's circular motions, the sidereal rotation rate, the mean solar day and the revolution of the planet about the sun.

Why are they different and how can lightning be 2 different distances in M' frame coordinates from C' and M when the are co-located?

It's obvious that [post=3198606]post #2[/post] covers all of this. $$t_P = t_Q$$ and $$t'_P = t'_R$$ and $$t_Q < t_R$$.

 

[paddoboy]

That would be tantamount to asserting absolute time, which is not consistent with special relativity and therefore disallowed by the rigor required by your OP which purports to do a "Thought experiment."

And dear forum, there we have it. chinglu does not accept that space and time are not absolute, despite pages and pages and pages of examples illustrated in other threads he has started on this subject. Threads that this lowly minion also took part in.
As I said from the beginning, it was obvious the mathematical thought experiment/problem, was just a ploy to once again do his best to invalidate something that has already had more then a 100 years of validations.

 

[rpenner]

More stuff about the lightcone and coordinates --http://en.wikipedia.org/wiki/Light-cone_coordinates
Ok. So I read something about how there is no Lorentz transform from (x+, x-, x[sup]2[/sup], x[sup]3[/sup]) to (x[sup]0[/sup], x[sup]1[/sup], x[sup]2[/sup], x[sup]3[/sup]). Or, you can't take these 'special' coordinates to a frame of reference.

Useful things:
$$ \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} ^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} = \begin{pmatrix} \frac{A + B + C + D}{2} & \frac{A - B + C - D}{2} \\ \frac{A + B - C - D}{2} & \frac{A - B - C + D}{2} \end{pmatrix}
\textrm{determinant} \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} = \textrm{determinant} \begin{pmatrix} A & B \\ C & D \end{pmatrix}
\textrm{trace} \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} = \textrm{trace} \begin{pmatrix} A & B \\ C & D \end{pmatrix}
\begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} \begin{pmatrix} \cosh \rho & \sinh \rho \\ \sinh \rho & \cosh \rho \end{pmatrix} \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} = \begin{pmatrix} \cosh \rho \; + \; \sinh \rho & 0 \\ 0 & \cosh \rho \; - \; \sinh \rho\end{pmatrix} = \begin{pmatrix} e^{\rho} & 0 \\ 0 & e^{-\rho} \end{pmatrix}
\begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} \begin{pmatrix} \gamma & \beta \gamma \\ \beta \gamma & \gamma \end{pmatrix} \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} = \begin{pmatrix} \gamma ( 1 + \beta ) & 0 \\ 0 & \gamma ( 1 - \beta ) \end{pmatrix} = \begin{pmatrix} \gamma ( 1 + \beta ) & 0 \\ 0 & \frac{1}{\gamma ( 1 + \beta )} \end{pmatrix}
\begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} \begin{pmatrix} \pm 1 & 0 \\ 0 & \mp 1 \end{pmatrix} \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} = \begin{pmatrix} 0 & \pm 1\\ \pm 1 & 0 \end{pmatrix}
$$

 

[chinglu]

Correct. And that can be seen at a glance on this Minkowski diagram:

The answer to that question is:
x' = 1.000
t' = 1.000




The answer is yes. The M' frame finds the co-location of C' and M to occur at t'=1.000 so naturally the light pulse must be located at x'=ct'=1.000.

Now you might ask what these other M' frame coordinates represent:
x' = 0.500
t' = 0.500
That a different event which represents where the the light pulse was located at an earlier time.

You are showing very well here, when C' and M are co-located, the M frame gets the position wrong for the primed frame.

That is all you are proving. That in fact was the whole point of this thread.

Now, when C' and M are co-located, does the M frame get the correct answer for the primed frame location of the lightning yes or no?

 

[chinglu]

In the OP, the author agrees to rigorously apply special relativity, which states times and locations of events are not absolute but are frame-dependent and the Lorentz transformation relates inertial coordinates given one standard of rest to inertial coordinates with different standard of rest. For convenience, the OP assumes that the two coordinate systems, Σ and Σ',are set up such that $$x' = \gamma ( x - v t), \; y' = y, \; z' = z, \; t' = \gamma( t - v x /c^2 ) $$. Further, for all events of interest, the OP assumes $$y' = y = z' = z = 0$$, so even though every event has 8 coordinates (x, y, z, t, x', y', z', t'), we only have two degrees of freedom at play. Therefore specifying any of the following pairs of coordinates for an event tells you everything about the event: (x,t), (x,x'), (x, t'), (t,x'), (t, t'), (x', t').

The OP does not give names to the events, but names appeared immediately afterwords in [post=3198606]post #2[/post]: O, P, Q and R. One naively imagines that the OP would have no trouble keeping track of just two degrees of freedom of just four events.

Here the OP introduces line f, defined by $$x = x_O = 0$$, and line g, defined by $$x' = x'_O = 0$$. These are both straight, time-like lines. Here the OP introduces line h, defined by $$x' = x'_P = - v \frac{d'}{c}$$.
Here, for the first time, the OP introduces event O. Three lines meet at event O, lines f and g which have already been described and a new line ℓ, which the OP will define as light-like. Note that the OP realizes to specify both "when" and "location" of the event starting line ℓ.
Event O is known to have coordinates $$(x_O = 0, x'_O = 0)$$, and from $$x' = \gamma ( x - v t)$$ we solve $$t_O = \frac{x_O - x'_O/ \gamma}{v} = 0 $$. Likewise from $$t' = \gamma( t - v x /c^2 )$$ we immediately get $$t'_O = \gamma( t_O - v x_O /c^2 ) = 0$$ so the coordinates are $$(x_O = 0, \; t_O = 0, \; x'_O = 0, \; t'_O = 0)$$.

Likewise, event P is about to be introduced without being named. Event P is defined as the event where both $$x_P = x_O = 0$$ and $$x'_P = - v \frac{d'}{c}$$ are true or equivalently as the event at the crossing of lines f and g. From the above it follows that $$t_P = \frac{x_P - x'_P/ \gamma}{v} = \frac{d'}{\gamma c}$$ and $$t'_P = \gamma( t_P - v x_P /c^2 ) = \frac{d'}{c}$$ so the coordinates are $$(x_P = 0, \; t_P = \frac{d'}{\gamma c}, \; x'_P = - v \frac{d'}{c}, \; t'_P = \frac{d'}{c})$$.


This is an ambiguous question, because "when C' and M are co-located" could mean $$t = t_P$$ or $$t' = t'_P$$ but because there is no absolute time in special relativity, it follows that these two definitions are only both true at event P. That's why [post=3198606]post #2[/post] introduces line j which is defined as $$t = t_P$$, and line k which is defined as $$t' = t'_P$$. Because these are two different space-like lines, it follows that any time-like or light-like line that does not pass through event P must intersect these lines at two different Events. So one ambiguity of this question is which of the two events we are talking about. The easiest way to resolve this ambiguity is to label the lines (or equations) for j and k, as well as the events (or solutions) Q and R.
But there is a second ambiguity, because "where is the light[] along the positive x-axis" is asking for both the x and the x' values. Asking such a confused question suggests not enough time was spent understanding the geometry of special relativity. As we will later see, the OP attempts to answer this question without labeling the results, resulting in further confusion.

We call the event where lines j and ℓ meet event Q. We have $$t_Q=t_P = \frac{d'}{\gamma c}, \; x_Q = x_O + c t_Q - c t_O = \frac{d'}{\gamma}, \; x'_Q = \gamma ( x_Q - v t_Q) = d' \left( 1 - \frac{v}{c} \right), \; t'_Q = \gamma( t_Q - v x_Q /c^2 ) = \frac{d'}{c} \left( 1 - \frac{v}{c} \right)$$
We call the event where lines k and ℓ meet event R. We have $$t'_R = t'_P = \frac{d'}{c}, \; x_R = x_O + c t_R - c t_O = c t_R$$. From $$t'_R = \gamma( t_R - v x_R /c^2 ) = \gamma t_R ( 1- v/c )$$ it follows that $$t_R =\frac{t'_R}{\gamma \left( 1 - \frac{v}{c} \right) } = \gamma \frac{d'}{c} \left( 1 + \frac{v}{c} \right), \; x_R = \gamma d' \left( 1 + \frac{v}{c} \right)$$. Finally $$x'_R = \gamma ( x_R - v t_R) = \gamma^2 d' \left( 1 + \frac{v}{c} \right)\left( 1 - \frac{v}{c} \right) = \gamma^2 d' \left( 1 - \frac{v^2}{c^2} \right) = d'$$.

So the coordinates for Q are $$\left( x_Q = \frac{d'}{\gamma}, \; t_Q = \frac{d'}{\gamma c}, \; x'_Q = d' \left( 1 - \frac{v}{c} \right), \; t'_Q = \frac{d'}{c} \left( 1 - \frac{v}{c} \right) \right)$$. And the coordinates of R are $$\left( x_R = \gamma d' \left( 1 + \frac{v}{c} \right), \; t_R = \gamma \frac{d'}{c} \left( 1 + \frac{v}{c} \right), \; x'_R = d', \; t'_R = \frac{d'}{c} \right)$$

Thus the fully-labelled coordinates for Q and R completely answer the question in an unambiguous manner. The OP does calculate these same quantities, but fails to label them and so the OP becomes confused.

See how the OP cites unlabeled coordinates: $$\left( x_Q = \frac{d'}{\gamma}, \; t_Q = \frac{d'}{\gamma c} \right)$$ and $$\left( x_R = \gamma d' \left( 1 + \frac{v}{c} \right), \; t_R = \gamma \frac{d'}{c} \left( 1 + \frac{v}{c} \right) \right)$$.
See how the OP cites unlabeled coordinates: $$\left( x'_Q = d' \left( 1 - \frac{v}{c} \right), \; t'_Q = \frac{d'}{c} \left( 1 - \frac{v}{c} \right) \right)$$ and $$\left( x'_R = d', \; t'_R = \frac{d'}{c} \right)$$.

The OP does not state HOW this is inconsistent with nature. Events Q and R undeniably happen in different positions and times in every frame, but that's true about any two distinct events on a light-like line. That is entirely consistent with the light-postulate which requires the same light at different times to be at different locations.

So if the OP thinks there is a problem with two positions, the real problem must be with existence of two distinct events. But there always were going to be two distinct events Q and R because there were always two different frames describing a frame-dependent concept of "same time as event P". This resulted in lines j and k being distinct space-like lines and as I said above, that leads to two distinct events Q and R.

That the OP now complains that $$x_Q \neq x_R$$ means the OP never understood that $$t = t_P$$ is a different line than $$t' = t'_P$$ and thus never understood Relativity of Simultaneity.

As I said before, the problem was not in the algebraic calculations, but in asking illegal questions which are predicated on assuming frame-independence to the concept of simultaneity which violates the requirement of a thought experiment to use rigorous mathematical logic. By assuming the opposite of relativity of simultaneity and by using the Lorentz transforms which incorporate relativity of simultaneity, the OP doomed his efforts to do a rigorous thought experiment.

From [POST=3198606]Post #2[/POST] we see that the solution is obvious:

Let f,g,h be time-like inertial world lines. Let j, k be space-like straight lines. Let ℓ be a light-like straight line. Then we have in both coordinate system the following descriptions of these lines:
$$
\begin{array}{c|c|c} \textrm{Line} & \textrm{Frame} \; \Sigma & \textrm{Frame} \; \Sigma' \\ \hline \\ f & x = 0 & x' = -vt' \\ g & x = vt & x' = 0 \\ h & x = vt - \frac{d'}{c} v \sqrt{1 - \frac{v^2}{c^2}} & x' = - \frac{d'}{c} v \\ j & t = \frac{d'}{c} \sqrt{1 - \frac{v^2}{c^2}} & x' = - \frac{c^2}{v} \left( t' - \frac{d'}{c} \left(1 - \frac{v^2}{c^2} \right) \right) \\ k & x = \frac{c^2}{v} \left( t - \frac{d'}{c} \sqrt{1 - \frac{v^2}{c^2} } \right) & t' = \frac{d'}{c} \\ \ell & x = ct & x' = ct' \end{\array} $$ $$
\begin{array}{c|c|c} \textrm{Event} & \textrm{Frame} \; \Sigma & \textrm{Frame} \; \Sigma' \\ \hline \\ O=f \cap g \cap \ell & \left( x=0, \; t=0\right) & \left(x'=0, \; t'=0 \right) \\ P = f \cap h \cap j \cap k & \left( x = 0, \; t = \frac{d'}{c} \sqrt{1 - \frac{v^2}{c^2}} \right) & \left( x' = - \frac{d'}{c} v, \; t' = \frac{d'}{c} \right) \\ Q = j \cap \ell & \left( x = d' \, \sqrt{1 - \frac{v^2}{c^2}}, \; t = \frac{d'}{c} \sqrt{1 - \frac{v^2}{c^2}} \right) & \left( x' = d' \, \left(1 - \frac{v}{c} \right) , \; t' = \frac{d'}{c} \left(1 - \frac{v}{c} \right) \right) \\ R = k \cap \ell & \left( x = \frac{c d' \sqrt{1 - \frac{v^2}{c^2}}}{c - v}, \; t = \frac{d' \sqrt{1 - \frac{v^2}{c^2}}}{c - v} \right) &\left( x' = d' , \; t' = \frac{d'}{c} \right) \end{\array}$$
So by ignoring relativity of simultaneity, you improperly confuse lines j and k and therefore confuse events Q and R.​

A [POST=3204151]later post[/POST] would again emphasize the need to think geometrically about the problem:

See above. Absolute time was in the question and the conclusions based on the question.

[POST=3199258]This post[/POST] went nowhere because P is not any type of logical predicate. P is a space-time event which happens to have coordinates in frame Σ and different coordinates in frame Σ'. But $$t = t_P$$ and $$t' = t'_P$$ are only both true at event P.
It's clear that the OP continues to have problems telling the difference between $$t = t_P$$ and $$t' = t'_P$$.

That is not anyone's lingo.

Everyone sees [POST=3198606]Post #2[/POST].

A valid statement of Relativity of Simultaneity is that $$t = t_P$$ is not equivalent to $$t' = t'_P$$ for coordinate time in inertial frames which are in relative motion. Since neither event Q or event R is the same event as event P, it follows that $$t_P = t_Q$$ and $$t'_P = t'_R$$ is perfectly consistent with $$t_Q < t_R$$.

Are you suing me in Science Court?
The math says minions get paid, so obviously they are not my minions.

Must you confuse everything? Line ℓ does not pass through event P, the co-location event of C' and M. So the question of what event on line ℓ is simultaneous with event P is a frame-dependent question that has everything to do with man-made coordinate systems and nothing to do with the geometry or physics of space-time.

That would be tantamount to asserting absolute time, which is not consistent with special relativity and therefore disallowed by the rigor required by your OP which purports to do a "Thought experiment."

You don't understand the Sagnac effect, which relates to the area encompassed by the light path and the rotation rate relative to an inertial frame. So you have confused three measures of the Earth's circular motions, the sidereal rotation rate, the mean solar day and the revolution of the planet about the sun.

It's obvious that [post=3198606]post #2[/post] covers all of this. $$t_P = t_Q$$ and $$t'_P = t'_R$$ and $$t_Q < t_R$$.

You have still refused to answer. The OP calculates the position of the lighting when C' and M are co-located in M' frame coordinates using the light postulate in the primed frame.

The OP also calculates the position of the lighting when C' and M are co-located in M' frame coordinates using the light postulate in the unprimed frame.

Now, explain why the unprimed frame gets the answer wrong when C' and M are co-located and that wrong answer is still correct. You continue to refuse to answer this simple question.

 

[chinglu]

And dear forum, there we have it. chinglu does not accept that space and time are not absolute, despite pages and pages and pages of examples illustrated in other threads he has started on this subject. Threads that this lowly minion also took part in.
As I said from the beginning, it was obvious the mathematical thought experiment/problem, was just a ploy to once again do his best to invalidate something that has already had more then a 100 years of validations.

0 to do with absolute. I thought you said you could not follow the math. Is that true?

 

[chinglu]

This thread has failed on the following points.

1) When M and C' are co-located, the unprimed frame gets the wrong answer for the location of the lightning along the positive x-axis for the primed frame.

2) I provided calculations for distances from the common location of C' and M to the lightning in primed frame coordinates. SR gives 2 different positions which is impossible. The thread posters have not addressed this either.

 

[arfa brane]

This thread has failed on the following points.

1) When M and C' are co-located, the unprimed frame gets the wrong answer for the location of the lightning along the positive x-axis for the primed frame.

That you think there's a "wrong" answer indicates that you're the one who has failed. Failed to understand, that is.

2) I provided calculations for distances from the common location of C' and M to the lightning in primed frame coordinates. SR gives 2 different positions which is impossible. The thread posters have not addressed this either.

Two different observers is not impossible, it happens every day. Less common is two observers in relative motion where one is moving at 0.5c, but again, not at all impossible.

How likely is it that Einstein got it wrong, everyone has been following a myth for the last century of technological development (a lot of this depends on SR being a "true" theory), then you come along and can show everyone where the mistake got made? You're saying you can overturn decades of science and technology (lasers, semiconductors, particle accelerators) because it was all based on a mistake?

You are out of your mind.

 

[chinglu]

That you think there's a "wrong" answer indicates that you're the one who has failed. Failed to understand, that is.
Two different observers is not impossible, it happens every day. Less common is two observers in relative motion where one is moving at 0.5c, but again, not at all impossible.

How likely is it that Einstein got it wrong, everyone has been following a myth for the last century of technological development (a lot of this depends on SR being a "true" theory), then you come along and can show everyone where the mistake got made? You're saying you can overturn decades of science and technology (lasers, semiconductors, particle accelerators) because it was all based on a mistake?

You are out of your mind.

Your answer is a failure.

Put your answer this way as the OP did or you are talking about different things.

It is OK under SR to conclude the lightning is at 2 different locations in primed frame coordinates along the positive x-axis when C' and M are co-located.

You left off this factual information.

Then explain why you think this is true. This is the issue in the OP, not what you claim above.

 

[paddoboy]

How likely is it that Einstein got it wrong, everyone has been following a myth for the last century of technological development (a lot of this depends on SR being a "true" theory), then you come along and can show everyone where the mistake got made? You're saying you can overturn decades of science and technology (lasers, semiconductors, particle accelerators) because it was all based on a mistake?

You are out of your mind.

Our friend chinglu has overpowering mythical religious concerns he must adhere to.
He is even incapable of accepting the factual reality of time dilation and length contraction...
Yes, a lost cause.

 

[chinglu]

Our friend chinglu has overpowering mythical religious concerns he must adhere to.
He is even incapable of accepting the factual reality of time dilation and length contraction...
Yes, a lost cause.

Can you prove your assertions? Can you also understand any of the math to make such an opinion?

 

[paddoboy]

Can you prove your assertions? Can you also understand any of the math to make such an opinion?

My assertions were proven in the last SR thread you started and you were consequently banned and your crap pseudoscience thread was closed.

 

[paddoboy]

This thread has failed on the following points.

Yes the thread has certainly failed and the object of it as well.

2) I provided calculations for distances from the common location of C' and M to the lightning in primed frame coordinates. SR gives 2 different positions which is impossible. The thread posters have not addressed this either.

Telling lies make little baby Jesus cry.

 

[rpenner]

A conversation is more than two people talking at each other; both parties must take the time to parse the expressions of thought from the other.
I doubt the sincerity and intellectual honesty of someone who perfunctory quotes, en masse, a block of text over 11k long and yet fails to respond to its content. It truly makes it look like the OP isn't even attempting to engage with anyone who points out his misuse of concepts. For example, there has been no response to:

The OP does not state HOW this is inconsistent with nature. Events Q and R undeniably happen in different positions and times in every frame, but that's true about any two distinct events on a light-like line. That is entirely consistent with the light-postulate which requires the same light at different times to be at different locations.

So if the OP thinks there is a problem with two positions, the real problem must be with existence of two distinct events. But there always were going to be two distinct events Q and R because there were always two different frames describing a frame-dependent concept of "same time as event P". This resulted in lines j and k being distinct space-like lines and as I said above, that leads to two distinct events Q and R.

That the OP now complains that $$x_Q \neq x_R$$ means the OP never understood that $$t = t_P$$ is a different line than $$t' = t'_P$$ and thus never understood Relativity of Simultaneity.

The logic of that excerpt is an indictment of a failure of the OP to work within the physical theory being examined in the so-called "thought experiment" and so failure to respond strongly suggests insincerity.

You have still refused to answer.

I don't believe that. I think I have answered the question "when C' and M are co-located, where is the lightning along the positive x-axis for both frame coordinate systems?" precisely

the fully-labelled coordinates for Q and R completely answer the question in an unambiguous manner. The OP does calculate these same quantities, but fails to label them and so the OP becomes confused.

See how the OP cites unlabeled coordinates: $$\left( x_Q = \frac{d'}{\gamma}, \; t_Q = \frac{d'}{\gamma c} \right)$$ and $$\left( x_R = \gamma d' \left( 1 + \frac{v}{c} \right), \; t_R = \gamma \frac{d'}{c} \left( 1 + \frac{v}{c} \right) \right)$$.
See how the OP cites unlabeled coordinates: $$\left( x'_Q = d' \left( 1 - \frac{v}{c} \right), \; t'_Q = \frac{d'}{c} \left( 1 - \frac{v}{c} \right) \right)$$ and $$\left( x'_R = d', \; t'_R = \frac{d'}{c} \right)$$.

and in addition explained why it was a bad question:

This is an ambiguous question, because "when C' and M are co-located" could mean $$t = t_P$$ or $$t' = t'_P$$ but because there is no absolute time in special relativity, it follows that these two definitions are only both true at event P. ... The easiest way to resolve this ambiguity is to label the lines (or equations) for j and k, as well as the events (or solutions) Q and R.
But there is a second ambiguity, because "where is the light[] along the positive x-axis" is asking for both the x and the x' values. Asking such a confused question suggests not enough time was spent understanding the geometry of special relativity.

But the OP did not respond to that. In fact, I'm still waiting for the OP to respond to the obvious correction of "flash" or "light" (which propagates at the speed of light) for the original "lightning".

The OP calculates the position of the lighting when C' and M are co-located in M' frame coordinates using the light postulate in the primed frame.
The OP also calculates the position of the lighting when C' and M are co-located in M' frame coordinates using the light postulate in the unprimed frame.

Both "position" and "when" are frame-dependent concepts. "When" can mean $$t=t_P$$ or it can mean $$t'=t'_P$$ and choosing between them results in a choice of which intersection with the light-like line ℓ we are talking about: Q or R. So we have four possible answers for "location": $$x = x_Q, \; x' = x'_Q, \; x = x_R, \; x' = x'_R$$. If we work strictly with choices that are natural to Σ' coordinates, then the choice of $$t'=t'_P$$ (line k), event R, and answer of $$x' = x'_R$$ have been given many times. If we work with choices that are neither all natural to Σ coordinates or Σ' coordinates, then the choice of $$t'=t'_P$$ (line k), event R, and answer of $$x = x_R$$ have been given many times.
The Lorentz transform has no problem converting the coordinates of event Q in one system to another, because $$\left( x_R = \gamma d' \left( 1 + \frac{v}{c} \right), \; t_R = \gamma \frac{d'}{c} \left( 1 + \frac{v}{c} \right) \right)$$ and $$\left( x'_R = d', \; t'_R = \frac{d'}{c} \right)$$ describe the same event in space-time.

Now, explain why the unprimed frame gets the answer wrong when C' and M are co-located

It's not "wrong" -- it's different. It's different because $$t=t_P$$ is different from $$t'=t'_P$$ everywhere except event P. Thus event Q is a different event in space-time than event R.
What is wrong is to expect $$t=t_P$$ to mean the same thing as $$t'=t'_P$$.

and that wrong answer is still correct. You continue to refuse to answer this simple question.

The OP has the unrealistic expectation that the Lorentz transform is supposed to relate $$\left( x_Q = \frac{d'}{\gamma}, \; t_Q = \frac{d'}{\gamma c} \right)$$ to $$\left( x'_R = d', \; t'_R = \frac{d'}{c} \right)$$ because the OP never understood $$t_Q = t_P$$ and $$t'_R = t'_P$$ doesn't mean $$t_Q = t_R$$. What a waste of human potential and years of labor to wage a campaign against concepts that the OP has never understood.

Telling lies make little baby Jesus cry.

Well said. Likewise refusal to engage is likely to make moderators cry.

 

[brucep]

A conversation is more than two people talking at each other; both parties must take the time to parse the expressions of thought from the other.
I doubt the sincerity and intellectual honesty of someone who perfunctory quotes, en masse, a block of text over 11k long and yet fails to respond to its content. It truly makes it look like the OP isn't even attempting to engage with anyone who points out his misuse of concepts. For example, there has been no response to:

​

The logic of that excerpt is an indictment of a failure of the OP to work within the physical theory being examined in the so-called "thought experiment" and so failure to respond strongly suggests insincerity.

I don't believe that. I think I have answered the question "when C' and M are co-located, where is the lightning along the positive x-axis for both frame coordinate systems?" precisely

​

and in addition explained why it was a bad question:

​

But the OP did not respond to that. In fact, I'm still waiting for the OP to respond to the obvious correction of "flash" or "light" (which propagates at the speed of light) for the original "lightning".
Both "position" and "when" are frame-dependent concepts. "When" can mean $$t=t_P$$ or it can mean $$t'=t'_P$$ and choosing between them results in a choice of which intersection with the light-like line ℓ we are talking about: Q or R. So we have four possible answers for "location": $$x = x_Q, \; x' = x'_Q, \; x = x_R, \; x' = x'_R$$. If we work strictly with choices that are natural to Σ' coordinates, then the choice of $$t'=t'_P$$ (line k), event R, and answer of $$x' = x'_R$$ have been given many times. If we work with choices that are neither all natural to Σ coordinates or Σ' coordinates, then the choice of $$t'=t'_P$$ (line k), event R, and answer of $$x = x_R$$ have been given many times.
The Lorentz transform has no problem converting the coordinates of event Q in one system to another, because $$\left( x_R = \gamma d' \left( 1 + \frac{v}{c} \right), \; t_R = \gamma \frac{d'}{c} \left( 1 + \frac{v}{c} \right) \right)$$ and $$\left( x'_R = d', \; t'_R = \frac{d'}{c} \right)$$ describe the same event in space-time.
It's not "wrong" -- it's different. It's different because $$t=t_P$$ is different from $$t'=t'_P$$ everywhere except event P. Thus event Q is a different event in space-time than event R.
What is wrong is to expect $$t=t_P$$ to mean the same thing as $$t'=t'_P$$.
The OP has the unrealistic expectation that the Lorentz transform is supposed to relate $$\left( x_Q = \frac{d'}{\gamma}, \; t_Q = \frac{d'}{\gamma c} \right)$$ to $$\left( x'_R = d', \; t'_R = \frac{d'}{c} \right)$$ because the OP never understood $$t_Q = t_P$$ and $$t'_R = t'_P$$ doesn't mean $$t_Q = t_R$$. What a waste of human potential and years of labor to wage a campaign against concepts that the OP has never understood.

Well said. Likewise refusal to engage is likely to make moderators cry.

Doesn't seem like moderation is interested in the content of a response beyond it's possible anti social implications. Very informative posts rpenner.