SR Issue

[Aqueous Id]

The Lorentz transformation can be thought of a hyperbolic analogue of rotation (points move on hyperbolas, not circles).
Alternately it can be thought of stretching one diagonal by a factor of $$\sqrt{\frac{c + v}{c - v}}$$ and shrinking the other diagonal by the same factor.

[followed by those nice illustrations]

That's in a nutshell what all the cranks are missing. If they understood what is meant by a projection of an object in rotation they might be in the home stretch. The simplest case is pretty easy to visualize: a right triangle sitting on its base can be thought of as a physical object with girth, which, when a light is shone above it, will project a shadow of the hypotenuse onto the base, and it will be equal in length to the base. Thus, the "transformation" for the 1-D case is merely b = h cos(θ). Even to get chinglu that far would be like pulling teeth, but to get from the 1D case to 3D to 4D, and then to account for the twist on this, that the correct representation is the hyperbolic cosine (cosh) is really just a very small step further, one that chinglu simply can't manage. He just doesn't have the chops.

Mainly I just wanted to say again that this was a nice explanation. It helps the folks who haven't taken trig to begin to glean what you're talking about. Of course, chinglu is essentially in denial of cartography.

 

[rpenner]

Of course, chinglu is essentially in denial of cartography.

Well, there are vaguely interesting posts like [post=3198449]the OP[/post] (which doesn't apparently go off the rails until the "Conclusions") and really terrible broken-record posts without sign of cognition like the [post=3209204]most recent[/post]. I simply don't see how one admits:

$$t'=(t-vx/c^2)\gamma$$

and absolutely meaningful simultaneity without acknowledging what t and t' mean.

But even worse than denial of cartography is his posting style of insisting that people have the responsibility to prove claims that exist only in chinglu's head or have a duty to respond to his empty taunts without corresponding duty to respond to the month-old [post=3198606]Post #2[/post]. That's more like denial of the humanity of the rest of the world, which seems like pathological solipsism.

 

[Aqueous Id]

Well, there are vaguely interesting posts like [post=3198449]the OP[/post] (which doesn't apparently go off the rails until the "Conclusions") and really terrible broken-record posts without sign of cognition like the [post=3209204]most recent[/post]. I simply don't see how one admits:

​

and absolutely meaningful simultaneity without acknowledging what t and t' mean.

But even worse than denial of cartography is his posting style of insisting that people have the responsibility to prove claims that exist only in chinglu's head or have a duty to respond to his empty taunts without corresponding duty to respond to the month-old [post=3198606]Post #2[/post]. That's more like denial of the humanity of the rest of the world, which seems like pathological solipsism.

And of course you generously availed yourself with the impeccable answers you are famous for. From where I sit, that makes his remarks all the more ludicrous. But you point is well taken. If there is a transformation as he admits, then the rest of the claims are rendered not only plain wrong but moot. But I did like this recent answer you gave in particular. It gave me a sense of what's going through your mind as you're pitching pearls to swine. If the cranks could simply understand what a projection is, I think they might even go through some sort of catharsis.

 

[chinglu]

SR says the light flash is located at those two different locations at two different times. It is not a contradiction, because substituting two different times into the light postulate equation x'=ct' mathematically requires that the light flash must be located at two different locations.
All you have to say is, "At two different times," and everything will make sense.

SR says the light flash is located at those two different locations at two different times.

Your statement is false. The OP proves if C' and M are co-located, then the lightning is located at 2 different primed frame positive x-axis corrdinates.

In the primed frame, if the clock at C' reads $$t'=\frac{d'}{c}$$, then the lightning is at primed $$(d',0,0,d'/c)$$.

In the unprimed frame, if the clock at C' reads $$t'=\frac{d'}{c}$$, then the lightning is at primed $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$.

So, if the clock at C' reads $$t'=\frac{d'}{c}$$, then the lightning is at 2 different places on the positive x-axis in the primed frame.

Notice, this is but one time on the C' clock.

 

[paddoboy]

Well, there are vaguely interesting posts like [post=3198449]the OP[/post] (which doesn't apparently go off the rails until the "Conclusions") and really terrible broken-record posts without sign of cognition like the [post=3209204]most recent[/post]. I simply don't see how one admits:

​

and absolutely meaningful simultaneity without acknowledging what t and t' mean.

But even worse than denial of cartography is his posting style of insisting that people have the responsibility to prove claims that exist only in chinglu's head or have a duty to respond to his empty taunts without corresponding duty to respond to the month-old [post=3198606]Post #2[/post]. That's more like denial of the humanity of the rest of the world, which seems like pathological solipsism.

And of course if chinglu is so sure SR is invalidated, despite you falsifying his conclusions, [which is the purpose of this thread] then he would get it peer reviewed [which you have done anyway]through more official channels.
But he doesn't have the intestinal fortitude to do that.

 

[chinglu]

Well, there are vaguely interesting posts like [post=3198449]the OP[/post] (which doesn't apparently go off the rails until the "Conclusions") and really terrible broken-record posts without sign of cognition like the [post=3209204]most recent[/post]. I simply don't see how one admits:

​

and absolutely meaningful simultaneity without acknowledging what t and t' mean.

But even worse than denial of cartography is his posting style of insisting that people have the responsibility to prove claims that exist only in chinglu's head or have a duty to respond to his empty taunts without corresponding duty to respond to the month-old [post=3198606]Post #2[/post]. That's more like denial of the humanity of the rest of the world, which seems like pathological solipsism.

I am going to put the problem in terms of propositional logic. That means, if you disagree, you will need to indicate exactly which step is not logical.

Isn't this fair?

Now, we have accepted that event P is the co-location of C' and M. Instead, we will define P(M) to mean C' and M are co-located in the M frame and P(M') means C' and M are co-located in the M' frame.

But, from the OP, if P(M) is true, then the clock at M is $$t=\frac{d'}{c\gamma}$$. Also, if P(M) is true, the clock at C' is $$t'=\frac{d'}{c}$$.

However, from the OP, if the clock at M is $$t=\frac{d'}{c\gamma}$$ and the clock at C' is $$t'=\frac{d'}{c}$$, then P(M') is true.

Hence, we have established the logical implication, $$P(M)\Rightarrow P(M')$$

Then, from the primed frame light postulate, it was proven if P(M') is true then the lightning is located at $$(d',0,0,d'/c)$$. Define $$Loc'(Lightning)$$ to mean the space-time location of the lightning on the positive x-axis in primed frame measurements. So, the logical implication is below.

$$P(M')\Rightarrow Loc'(Lightning)=(d',0,0,d'/c)$$

Finally, by the light postulate in the unprimed frame M and then translation by LT, we have $$P(M)\Rightarrow Loc'(Lightning)=(d'(1-v/c),0,0,d'(1-v/c)/c)$$

Therefore, the three proven logical implications are as follows:
(1) $$P(M)\Rightarrow P(M')$$
(2) $$P(M')\Rightarrow Loc'(Lightning)=(d',0,0,d'/c)$$
(3) $$P(M)\Rightarrow Loc'(Lightning)=(d'(1-v/c),0,0,d'(1-v/c)/c)$$.

Now, to boot SR into a contradiction, we simply assume $$P(M)$$ is true. Therefore, by (1) $$P'(M)$$ is true.

But, if $$P(M)$$ is true then by (3) $$Loc'(Lightning)=(d'(1-v/c),0,0,d'(1-v/c)/c)$$ is true.

Also, since $$P(M')$$ is true then by (2) $$ Loc'(Lightning)=(d',0,0,d'/c)$$ is true.

Therefore, by simple propositional logic, $$P(M) \Rightarrow Loc'(Lightning)=(d',0,0,d'/c) \wedge Loc'(Lightning)=(d'(1-v/c),0,0,d'(1-v/c)/c)$$, which is a logical contradiction.

So, I have put the argument in the language of propositional logic and I hope you can follow it so that you can understand why the argument is logically correct.

 

[paddoboy]

SR says the light flash is located at those two different locations at two different times. It is not a contradiction, because substituting two different times into the light postulate equation x'=ct' mathematically requires that the light flash must be located at two different locations.




All you have to say is, "At two different times," and everything will make sense.

It's only one person that needs convincing.......One person with a hell of an agenda, and that cannot even recognise the observed fact that time is not an absolute thing and can and is dilated.
Sad.

 

[rpenner]

I am going to put the problem in terms of propositional logic. That means, if you disagree, you will need to indicate exactly which step is not logical.

Or point out how you have abused language and notation.

Isn't this fair?

"Fair" would require that you understood the 1905 Einstein paper you cited and did not contradict the assumptions of special relativity in the OP. At best, I have hope that this will bring enlightenment to you.

Now, we have accepted that event P is the co-location of C' and M.

Events can be named, specifically because they have existence independent of the coordinates used to describe them. Events are the constituents of space-time, $$\mathcal{M}$$, so every non-empty subset of space-time contains events. Frames are imaginary, man-made coordinate systems so can only be used to describe events.

Instead, we will define P(M) to mean C' and M are co-located in the M frame and P(M') means C' and M are co-located in the M' frame.

Can you not see that you have severe language and notation difficulties?

• M is defined both as the spatial origin of a frame and the frame itself. It is better to have two different symbols for two different things. That's why I introduced Σ and Σ' as symbols for frames.
• By assumptions made in the OP, C' and M have non-zero relative velocity and have trajectories through space-time that meet at event P. Saying event P exists in space-time is saying there is a portion of space-time common to both trajectories. So P exists in space-time. Thus it is possible (in Special Relativity) to assign it coordinates in any frame, because in special relativity every event in space-time has four coordinates in every frame.
• It's pointless to try and "define" P(Σ) as meaning "C' and M are co-located in the Σ frame" because unlike a picture frame, a coordinate frame doesn't have contents. A coordinate frame, Σ, is a one-to-one linear homeomorphism, $$\phi_{\Sigma}$$ , between space-time (which contains P) and ℝ⁴. Because it is a one-to-one linear homeomorphism, it has an linear inverse and thus $$\phi_{\Sigma'} \, \circ \, \phi_{\Sigma}^{\tiny -1} $$ is a linear transformation of coordinates in ℝ⁴, (x,y,z,t) to different coordinates in ℝ⁴ (x', y', z', t') which happens to correspond to a Poincaré transform (a superset of Lorentz transforms), while $$ \phi_{\Sigma'}^{\tiny -1} \, \circ \, \phi_{\Sigma} $$ is an analogue transform of space-time itself, also called a Poincaré transform. That's why a change of coordinates can be legitimately referred to in physical terms as a "translation", "rotation" or "boost" because the coordinate transformation and the physical change to space-time are two sides of the same coin.

So the phrase "C' and M are co-located in the M frame" betrays your lack of understanding about special relativity and likely cripples your attempt to explain your point of view as valid.

But, from the OP, if P(M) is true, then the clock at M is $$t=\frac{d'}{c\gamma}$$. Also, if P(M) is true, the clock at C' is $$t'=\frac{d'}{c}$$.

Here you have $$\phi_{\Sigma}(P) = \left(x_P,\;y_P,\;z_P,\;t_P \right)$$ and $$\phi_{\Sigma'}(P) = \left(x'_P,\;y'_P,\;z'_P,\;t'_P \right)$$. Your algebra and mine agree that $$t_P = \frac{d'}{c\gamma}$$ and $$t'_P=\frac{d'}{c}$$. But you seem to be confusing "if" and "when" again. By assumption the trajectories of C' and M cross at a particular event, event P, so there is no "if" -- the existence of P is true -- it's just the time of the existence of P has to be singled out for the purposes of the OP. And since coordinate-time is a frame-dependent concept, the set of events, E, where $$t_E = t_P$$ is true need not correspond to the set of events where $$t'_E = t'_P$$ is true. Do you follow?

However, from the OP, if the clock at M is $$t=\frac{d'}{c\gamma}$$ and the clock at C' is $$t'=\frac{d'}{c}$$, then P(M') is true.

Why did you use "however?" Why is this an unexpected result? If you think correctly about space-time and coordinate frames, you have simply restated: $$\phi_{\Sigma}(P) = \left(x_P,\;y_P,\;z_P,\;t_P \right)$$ and $$\phi_{\Sigma'}(P) = \left(x'_P,\;y'_P,\;z'_P,\;t'_P \right)$$, so no "however" is necessary. It's not the least bit surprising.

Hence, we have established the logical implication, $$P(M)\Rightarrow P(M')$$

You only think that you have established this. Actually, what you have established that linear homeomorphisms are faithful representations of geometry and two lines intersect in space-time if and only if the images of those lines under $$\phi_{\Sigma}$$ intersect. (This is, of course, why the field of analytic geometry exists and why we bother with coordinates at all.) By the transitive property, it follows two lines intersect in Σ if and only if the images of those lines in Σ', the images of those lines under $$\phi_{\Sigma'} \, \circ \, \phi_{\Sigma}^{\tiny -1}$$, intersect. But since in our case, $$\phi_{\Sigma'} \, \circ \, \phi_{\Sigma}^{\tiny -1}$$ is $$(x', \;y', \;z', \;t') = ((x - v t) \gamma , \;y, \;z, \;(t - v x /c^2) \gamma )$$ this happens to the point made previously in [post=3198606]post #2[/post] and [post=3198777]post #4[/post].

The rest of your mistakes follow from the ones described above. The "when" of event P is a frame-dependent concept, analogous to how the "where" of event P is a frame-dependent concept. You have established that the space-time lines corresponding to $$x=x_P$$ and $$x'=x'_P$$ (when $$y=z=y'=z'=0$$) intersect in space-time event P, but have failed to realize that lines corresponding to $$t=t_P$$ and $$t' = t'_P$$ (again, when $$y=z=y'=z'=0$$) also intersect in space-time event P. This cripples your attempt to interpret coordinate time as absolutely meaningful and thus cripples you attempt to use existence of event on spatial slices of space-time corresponding to a particular time as logical predicates.

Instead of saying "P(M)" you wanted to say $$\left{ E \in \mathcal{M} | t_E = t_P }$$ which was already given to you in [post=3198606]post #2[/post] as line j, thus the proper logic is not simple predicate logic but first-order logic or alternately existential predicate logic.

Thus in first order logic, where $$\mathcal{M}$$ is the set of all events in space-time,
$$j = \left{ E \in \mathcal{M} \; | \; t_E = t_P \; \wedge \; y_E = y_P \; \wedge \; z_E = z_P \right} \\ k = \left{ E \in \mathcal{M} \; | \; t'_E = t'_P \; \wedge \; y'_E = y'_P \; \wedge \; z'_E = z'_P \right} \\ \ell = \left{ E \in \mathcal{M} \; | \; x_E = c t_E \; \wedge \; y_E = y_P \; \wedge \; z_E = z_P \; \wedge \; t_E \geq t_O \right} \\ \ell_2 = \left{ E \in \mathcal{M} \; | \; x'_E = c t'_E \; \wedge \; y'_E = y'_P \; \wedge \; z'_E = z'_P \; \wedge \; t'_E \geq t'_O \right}$$
and it's easy to prove
$$\ell = \ell_2, \; O \in \ell, \; P \not \in \ell$$.
But what your OP explores is

• does there exist a unique event Q such that Q is in j and Q is in ℓ?
• does there exist a unique event R such that R is in k and R is in ℓ?

$$\vdash \exists ! Q \in \mathcal{M} \; \left( Q \in j \; \wedge \; Q \in \ell \right) \\ \vdash \exists ! R \in \mathcal{M} \; \left( R \in k \; \wedge \; R \in \ell \right) \\ \vdash \exists ! P \in \mathcal{M} \; \left( P \in j \; \wedge \; P \in k \right) \\ \vdash \neg \exists E \in \mathcal{M} \left( E \in j \; \wedge \; E \in k \; \wedge \; E \in \ell \right) $$

or in existential predicate logic:
$$j(E) \leftrightarrow t_E = t_P \; \wedge \; y_E = y_P \; \wedge \; z_E = z_P \\ k(E) \leftrightarrow t'_E = t'_P \; \wedge \; y'_E = y'_P \; \wedge \; z'_E = z'_P \\ \ell (E) \leftrightarrow x_E = c t_E \; \wedge \; y_E = y_P \; \wedge \; z_E = z_P \; \wedge \; t_E \geq t_O \\ \ell_2 (E) \leftrightarrow x'_E = c t'_E \; \wedge \; y'_E = y'_P \; \wedge \; z'_E = z'_P \; \wedge \; t'_E \geq t'_O $$
and it's easy to prove
$$\vdash \forall E \; \left( \ell(E) \leftrightarrow \ell_2(E) \right) \\ \vdash \ell(O) \\ \vdash \neg \ell(P)$$.
So
$$\vdash \exists ! E \; \left( j(E) \; \wedge \; \ell(E) \right) \\ \vdash \exists ! E \; \left( k(E) \; \wedge \; \ell (E) \right) \\ \vdash \exists ! E \; \left( j(E) \; \wedge \; k(E) \right) \\ \vdash \neg \exists E \left( j(E) \; \wedge \; k(E) \wedge \; \ell(E) \right) $$
The first three lines justify our speaking of definite events Q, R, and P, respectively, while the third shows that events Q, R and P must all be distinct. Geometrically, Q, R and P form a space-time triangle with two space-like sides and one light-like side.
Hence $$\vdash \neg \forall E \left( j(E) \leftrightarrow k(E) \right)$$ which is relativity of simultaneity in predicate logic.


http://en.wikipedia.org/wiki/Uniqueness_quantification
http://en.wikipedia.org/wiki/Universal_quantification
http://en.wikipedia.org/wiki/Existential_quantifier

 

[Neddy Bate]

Your statement is false. The OP proves if C' and M are co-located, then the lightning is located at 2 different primed frame positive x-axis corrdinates.

In the primed frame, if the clock at C' reads $$t'=\frac{d'}{c}$$, then the lightning is at primed $$(d',0,0,d'/c)$$.

In the unprimed frame, if the clock at C' reads $$t'=\frac{d'}{c}$$, then the lightning is at primed $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$.

So, if the clock at C' reads $$t'=\frac{d'}{c}$$, then the lightning is at 2 different places on the positive x-axis in the primed frame.

Notice, this is but one time on the C' clock.

Do you understand that when you type "the lightning is at primed $$(d',0,0,d'/c)$$," that SR was kind enough to provide you with a time coordinate on the end there? You are identifying the coordinates of the light as $$(x', y', z', t') = (d', 0, 0, d'/c)$$. So the time coordinate is $$t' = d'/c$$ by your own admission.

Likewise, when you type "the lightning is at primed $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$," again SR was kind enough to provide you with a time coordinate. You are identifying the coordinates of the light as $$(x', y', z', t') = (d'(1-v/c), 0, 0, d'(1-v/c)/c)$$ and the time coordinate is $$t' = d'(1-v/c)/c)$$ by your own admission.

So you cannot claim these two events take place at the same time in the primed frame, unless you want to claim $$d'/c = d'(1-v/c)/c)$$ which is false, (except when $$v=0$$, in which case C' and M never become co-located).

If you locate the light at two different times, naturally you will find it in two different places, because of $$x'=ct'$$. And note that the $$x'$$ coordinate of each of your events is equal to $$ct'$$, demonstrating that the light postulate is upheld by the LT's, which is hardly surprising considering that they were derived for that purpose.

 

[chinglu]

Do you understand that when you type "the lightning is at primed $$(d',0,0,d'/c)$$," that SR was kind enough to provide you with a time coordinate on the end there? You are identifying the coordinates of the light as $$(x', y', z', t') = (d', 0, 0, d'/c)$$. So the time coordinate is $$t' = d'/c$$ by your own admission.

Likewise, when you type "the lightning is at primed $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$," again SR was kind enough to provide you with a time coordinate. You are identifying the coordinates of the light as $$(x', y', z', t') = (d'(1-v/c), 0, 0, d'(1-v/c)/c)$$ and the time coordinate is $$t' = d'(1-v/c)/c)$$ by your own admission.

So you cannot claim these two events take place at the same time in the primed frame, unless you want to claim $$d'/c = d'(1-v/c)/c)$$ which is false, (except when $$v=0$$, in which case C' and M never become co-located).

If you locate the light at two different times, naturally you will find it in two different places, because of $$x'=ct'$$. And note that the $$x'$$ coordinate of each of your events is equal to $$ct'$$, demonstrating that the light postulate is upheld by the LT's, which is hardly surprising considering that they were derived for that purpose.

OK, you are focused above on $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$ and $$(d',0,0,d'/c)$$. You have correctly noted that these are space-time coordinates and each have a different time.

But, that is not the issue. These 2 space time coordinates were derived from SR using one time only at C' and d'/c.

So, on one hand, if the time at C' is d'/c in M' frame considerations, the lightning is at $$(d',0,0,d'/c)$$.

On the other hand, if the time at C' is d'/c in M frame considerations, the lightning is at $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$ and that is false.

That is the problem. The M frame with LT gets the answer wrong and contradicts the M' frame light postulate.

 

[chinglu]

Or point out how you have abused language and notation.
"Fair" would require that you understood the 1905 Einstein paper you cited and did not contradict the assumptions of special relativity in the OP. At best, I have hope that this will bring enlightenment to you.
Events can be named, specifically because they have existence independent of the coordinates used to describe them. Events are the constituents of space-time, $$\mathcal{M}$$, so every non-empty subset of space-time contains events. Frames are imaginary, man-made coordinate systems so can only be used to describe events.
Can you not see that you have severe language and notation difficulties?

• M is defined both as the spatial origin of a frame and the frame itself. It is better to have two different symbols for two different things. That's why I introduced Σ and Σ' as symbols for frames.
• By assumptions made in the OP, C' and M have non-zero relative velocity and have trajectories through space-time that meet at event P. Saying event P exists in space-time is saying there is a portion of space-time common to both trajectories. So P exists in space-time. Thus it is possible (in Special Relativity) to assign it coordinates in any frame, because in special relativity every event in space-time has four coordinates in every frame.
• It's pointless to try and "define" P(Σ) as meaning "C' and M are co-located in the Σ frame" because unlike a picture frame, a coordinate frame doesn't have contents. A coordinate frame, Σ, is a one-to-one linear homeomorphism, $$\phi_{\Sigma}$$ , between space-time (which contains P) and ℝ⁴. Because it is a one-to-one linear homeomorphism, it has an linear inverse and thus $$\phi_{\Sigma'} \, \circ \, \phi_{\Sigma}^{\tiny -1} $$ is a linear transformation of coordinates in ℝ⁴, (x,y,z,t) to different coordinates in ℝ⁴ (x', y', z', t') which happens to correspond to a Poincaré transform (a superset of Lorentz transforms), while $$ \phi_{\Sigma'}^{\tiny -1} \, \circ \, \phi_{\Sigma} $$ is an analogue transform of space-time itself, also called a Poincaré transform. That's why a change of coordinates can be legitimately referred to in physical terms as a "translation", "rotation" or "boost" because the coordinate transformation and the physical change to space-time are two sides of the same coin.

So the phrase "C' and M are co-located in the M frame" betrays your lack of understanding about special relativity and likely cripples your attempt to explain your point of view as valid.

Here you have $$\phi_{\Sigma}(P) = \left(x_P,\;y_P,\;z_P,\;t_P \right)$$ and $$\phi_{\Sigma'}(P) = \left(x'_P,\;y'_P,\;z'_P,\;t'_P \right)$$. Your algebra and mine agree that $$t_P = \frac{d'}{c\gamma}$$ and $$t'_P=\frac{d'}{c}$$. But you seem to be confusing "if" and "when" again. By assumption the trajectories of C' and M cross at a particular event, event P, so there is no "if" -- the existence of P is true -- it's just the time of the existence of P has to be singled out for the purposes of the OP. And since coordinate-time is a frame-dependent concept, the set of events, E, where $$t_E = t_P$$ is true need not correspond to the set of events where $$t'_E = t'_P$$ is true. Do you follow?

Why did you use "however?" Why is this an unexpected result? If you think correctly about space-time and coordinate frames, you have simply restated: $$\phi_{\Sigma}(P) = \left(x_P,\;y_P,\;z_P,\;t_P \right)$$ and $$\phi_{\Sigma'}(P) = \left(x'_P,\;y'_P,\;z'_P,\;t'_P \right)$$, so no "however" is necessary. It's not the least bit surprising.

You only think that you have established this. Actually, what you have established that linear homeomorphisms are faithful representations of geometry and two lines intersect in space-time if and only if the images of those lines under $$\phi_{\Sigma}$$ intersect. (This is, of course, why the field of analytic geometry exists and why we bother with coordinates at all.) By the transitive property, it follows two lines intersect in Σ if and only if the images of those lines in Σ', the images of those lines under $$\phi_{\Sigma'} \, \circ \, \phi_{\Sigma}^{\tiny -1}$$, intersect. But since in our case, $$\phi_{\Sigma'} \, \circ \, \phi_{\Sigma}^{\tiny -1}$$ is $$(x', \;y', \;z', \;t') = ((x - v t) \gamma , \;y, \;z, \;(t - v x /c^2) \gamma )$$ this happens to the point made previously in [post=3198606]post #2[/post] and [post=3198777]post #4[/post].

The rest of your mistakes follow from the ones described above. The "when" of event P is a frame-dependent concept, analogous to how the "where" of event P is a frame-dependent concept. You have established that the space-time lines corresponding to $$x=x_P$$ and $$x'=x'_P$$ (when $$y=z=y'=z'=0$$) intersect in space-time event P, but have failed to realize that lines corresponding to $$t=t_P$$ and $$t' = t'_P$$ (again, when $$y=z=y'=z'=0$$) also intersect in space-time event P. This cripples your attempt to interpret coordinate time as absolutely meaningful and thus cripples you attempt to use existence of event on spatial slices of space-time corresponding to a particular time as logical predicates.

Instead of saying "P(M)" you wanted to say $$\left{ E \in \mathcal{M} | t_E = t_P }$$ which was already given to you in [post=3198606]post #2[/post] as line j, thus the proper logic is not simple predicate logic but first-order logic or alternately existential predicate logic.

Thus in first order logic, where $$\mathcal{M}$$ is the set of all events in space-time,
$$j = \left{ E \in \mathcal{M} \; | \; t_E = t_P \; \wedge \; y_E = y_P \; \wedge \; z_E = z_P \right} \\ k = \left{ E \in \mathcal{M} \; | \; t'_E = t'_P \; \wedge \; y'_E = y'_P \; \wedge \; z'_E = z'_P \right} \\ \ell = \left{ E \in \mathcal{M} \; | \; x_E = c t_E \; \wedge \; y_E = y_P \; \wedge \; z_E = z_P \; \wedge \; t_E \geq t_O \right} \\ \ell_2 = \left{ E \in \mathcal{M} \; | \; x'_E = c t'_E \; \wedge \; y'_E = y'_P \; \wedge \; z'_E = z'_P \; \wedge \; t'_E \geq t'_O \right}$$
and it's easy to prove
$$\ell = \ell_2, \; O \in \ell, \; P \not \in \ell$$.
But what your OP explores is

• does there exist a unique event Q such that Q is in j and Q is in ℓ?
• does there exist a unique event R such that R is in k and R is in ℓ?

$$\vdash \exists ! Q \in \mathcal{M} \; \left( Q \in j \; \wedge \; Q \in \ell \right) \\ \vdash \exists ! R \in \mathcal{M} \; \left( R \in k \; \wedge \; R \in \ell \right) \\ \vdash \exists ! P \in \mathcal{M} \; \left( P \in j \; \wedge \; P \in k \right) \\ \vdash \neg \exists E \in \mathcal{M} \left( E \in j \; \wedge \; E \in k \; \wedge \; E \in \ell \right) $$

or in existential predicate logic:
$$j(E) \leftrightarrow t_E = t_P \; \wedge \; y_E = y_P \; \wedge \; z_E = z_P \\ k(E) \leftrightarrow t'_E = t'_P \; \wedge \; y'_E = y'_P \; \wedge \; z'_E = z'_P \\ \ell (E) \leftrightarrow x_E = c t_E \; \wedge \; y_E = y_P \; \wedge \; z_E = z_P \; \wedge \; t_E \geq t_O \\ \ell_2 (E) \leftrightarrow x'_E = c t'_E \; \wedge \; y'_E = y'_P \; \wedge \; z'_E = z'_P \; \wedge \; t'_E \geq t'_O $$
and it's easy to prove
$$\vdash \forall E \; \left( \ell(E) \leftrightarrow \ell_2(E) \right) \\ \vdash \ell(O) \\ \vdash \neg \ell(P)$$.
So
$$\vdash \exists ! E \; \left( j(E) \; \wedge \; \ell(E) \right) \\ \vdash \exists ! E \; \left( k(E) \; \wedge \; \ell (E) \right) \\ \vdash \exists ! E \; \left( j(E) \; \wedge \; k(E) \right) \\ \vdash \neg \exists E \left( j(E) \; \wedge \; k(E) \wedge \; \ell(E) \right) $$
The first three lines justify our speaking of definite events Q, R, and P, respectively, while the third shows that events Q, R and P must all be distinct. Geometrically, Q, R and P form a space-time triangle with two space-like sides and one light-like side.
Hence $$\vdash \neg \forall E \left( j(E) \leftrightarrow k(E) \right)$$ which is relativity of simultaneity in predicate logic.


http://en.wikipedia.org/wiki/Uniqueness_quantification
http://en.wikipedia.org/wiki/Universal_quantification
http://en.wikipedia.org/wiki/Existential_quantifier

Your post above did nothing to refute the following logic presented.

(1) $$P(M)\Rightarrow P(M')$$
(2) $$P(M')\Rightarrow Loc'(Lightning)=(d',0,0,d'/c)$$
(3) $$P(M)\Rightarrow Loc'(Lightning)=(d'(1-v/c),0,0,d'(1-v/c)/c)$$.

You said something to the effect that (1) does not hold true.

So, prove it. Prove it is possible that P(M) is true as was defined with P(M') being false. You can't by the way.

We are in logic now, thus, you have to prove your assertions. Therefore, prove any of the above implications are false.

If you cannot, and you can't, then the conclusions are inevitable. We simply assume P(M) is true and SR invokes a logical contradiction.

 

[arfa brane]

These 2 space time coordinates were derived from SR using one time only at C' and d'/c.

That's in spite of the fact that there are two frames, each with a clock.

So, on one hand, if the time at C' is d'/c in M' frame considerations, the lightning is at $$(d',0,0,d'/c)$$.

M' has a clock showing t'.

On the other hand, if the time at C' is d'/c in M frame considerations, the lightning is at $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$ and that is false.

M has a clock showing t. Note how you've been repeating a basic error since post #1: if you describe two separate frames, each has its own clock and the only way to make them show the same time is to synchronise them. That's assuming they are otherwise identical clocks.

Your grasp of the physics involved is infantile.

 

[chinglu]

That's in spite of the fact that there are two frames, each with a clock.

This indicates you do not understand the problem. See, here is the problem. If the clock at C' reads d'/c, where is the lighting in terms of the M' frame on the positive x-axis?

Next, we ask, the M frame, if the clock at C' reads d'/c, where is the lightning on the positive x-axis in the M' frame?

SR gets 2 different answers which is wrong. The lightning is at one place along the positive x-axis if the clock at C' is d'/c.

 

[paddoboy]

SR gets 2 different answers which is wrong. The lightning is at one place along the positive x-axis if the clock at C' is d'/c.

So, you are claiming all those that have ridiculed your claims in this thread, all those that have continued to ridicule and invalidate your claims in past threads, a hundred years of experimental and observational data researched by giants of the present and past, particle accelerators, Muon lifetime observations, GPS systems, etc etc etc, are all wrong and all refuse to see and recognise your revelational genius?

The whole world is wrong except for chinglu?
Did this deity you so blindly hold allegiance to, give you this miraculous insight?
Are you so completely delusional that you think that anyone here really believes your nonsense?

Again, as I have told you in your other old thread where you claim time dilation and length contraction did not happen, if you have any proof of your assertions, then get it peer reviewed.
Because just spreading your nonsense here, is doing nothing at all for this revelational insight you have into the Universe.
You are not changing nor convincing anyone to your weird take on the Universe and nature.

 

[arfa brane]

This indicates you do not understand the problem. See, here is the problem. If the clock at C' reads d'/c, where is the lighting in terms of the M' frame on the positive x-axis?

Next, we ask, the M frame, if the clock at C' reads d'/c, where is the lightning on the positive x-axis in the M' frame?

SR gets 2 different answers which is wrong. The lightning is at one place along the positive x-axis if the clock at C' is d'/c.

This is confusing. You describe what seems to be three different frames: C', M' and M. If there are three frames each has its own clock (or it should if you're discussing special relativity), that's three clocks. The really confusing part is that you also seem to be saying there is only one clock and one x axis.

Like I've said, you can't even describe the theory coherently; you've been attacking your own strawman, and you're still doing it.

 

[arfa brane]

These two diagrams convey this idea of a hyperbolic rotation as being equivalent to 'stretching' spacetime in one diagonal direction, and 'contracting' it transversely. This is something Penrose calls a Poincaré motion, or an inhomogenous Lorentz motion; most other texts substitute transformation for motion.

So, with v = 0.6c as in both diagrams, the 'scale factor' is $$ \sqrt{ \frac {c+0.6c} {c-0.6c} }$$, which is 2. So the (x',ct') = (1,1) intersect at twice the distance from (0,0) as (x,ct) = (1,1). The transverse diagonal in the (x',ct') frame is therefore 1/2 the distance of the (x,ct) frame.

What does that tell us about length contraction?

 

[Neddy Bate]

So, on one hand, if the time at C' is d'/c in M' frame considerations, the lightning is at $$(d',0,0,d'/c)$$.

Yes...

On the other hand, if the time at C' is d'/c in M frame considerations, the lightning is at $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$ and that is false.

But that is not false. That is exactly where the light is located at that time, according to the M frame. Because of ROS, the M frame finds that all of the clocks at rest in the M' frame are out of synch. So even though the M' frame clock at $$x'=0$$ displays the time $$t'=d'/c$$ the M' frame clock which happens to be located at the wavefront of the light at $$x'=d'(1-v/c)$$ displays an earlier time, $$t'=d'(1-v/c)/c$$.

That is the problem. The M frame with LT gets the answer wrong and contradicts the M' frame light postulate.

There is no problem. The M frame gets it right. And the light being located at $$x'=d'(1-v/c)$$ at time $$t'=d'(1-v/c)/c$$ certainly does not contradict the light postulate, because it satisfies $$x'=ct'$$.

Here, look at this diagram:

The time in the M frame $$t$$ progresses up the vertical axis. The purple line shows where the light is located at all times. The light is never located in two places at the same time. The example shown is for $$v=0.500c$$ and $$d'=1.000$$.

The event with coordinates $$(x',y',z',t')=(d'(1-v/c),0,0,d'(1-v/c)/c)$$ is simply earlier than the event with coordinates $$(x',y',z',t')=(d',0,0,d'/c)$$.

 

[chinglu]

This is confusing. You describe what seems to be three different frames: C', M' and M. If there are three frames each has its own clock (or it should if you're discussing special relativity), that's three clocks. The really confusing part is that you also seem to be saying there is only one clock and one x axis.

Like I've said, you can't even describe the theory coherently; you've been attacking your own strawman, and you're still doing it.

M' is the primed frame origin.
C' is at $$(\frac{-vd'}{c},0,0)$$ in primed frame coordinates.
M is the origin of the unprimed frame.

So, there are only 2 frames.

And, 3 clocks are OK as long as they are in 2 frames.

And, there is a shared common x-axis. I already explained this to you with a quote from Einstein.

 

[chinglu]

Yes...




But that is not false. That is exactly where the light is located at that time, according to the M frame. Because of ROS, the M frame finds that all of the clocks at rest in the M' frame are out of synch. So even though the M' frame clock at $$x'=0$$ displays the time $$t'=d'/c$$ the M' frame clock which happens to be located at the wavefront of the light at $$x'=d'(1-v/c)$$ displays an earlier time, $$t'=d'(1-v/c)/c$$.




There is no problem. The M frame gets it right. And the light being located at $$x'=d'(1-v/c)$$ at time $$t'=d'(1-v/c)/c$$ certainly does not contradict the light postulate, because it satisfies $$x'=ct'$$.

Here, look at this diagram:

The time in the M frame $$t$$ progresses up the vertical axis. The purple line shows where the light is located at all times. The light is never located in two places at the same time. The example shown is for $$v=0.500c$$ and $$d'=1.000$$.

The event with coordinates $$(x',y',z',t')=(d'(1-v/c),0,0,d'(1-v/c)/c)$$ is simply earlier than the event with coordinates $$(x',y',z',t')=(d',0,0,d'/c)$$.

Let's focus on this.

But that is not false. That is exactly where the light is located at that time, according to the M frame. Because of ROS, the M frame finds that all of the clocks at rest in the M' frame are out of synch. So even though the M' frame clock at $$x'=0$$ displays the time $$t'=d'/c$$ the M' frame clock which happens to be located at the wavefront of the light at $$x'=d'(1-v/c)$$ displays an earlier time, $$t'=d'(1-v/c)/c$$.

The issue is if C' and M are co-located, where is the lightning in M' frame coordinates?

You have already agreed C' says it is at $$(d',0,0,d'/c)$$

So, if C' and M are co-located, C' will tell M the lightning is at $$(d',0,0,d'/c)$$ in primed frame coordinates.

Now, you say M tells the C' observer that the lightning is actually at $$(d'(1-v/c),0,0,d'(1-v/c)/c)$$ in primed frame coordinates. Then you argue the light beam measures c so all is fine.

But, that is not true. While C' and M are at the same place, the lightning is at $$(d',0,0,d'/c)$$ in primed frame coordinates period. It is not at multiple places in the primed frame on the positive x-axis.

So, while C' and M are together, M give the wrong answer for the location of the lightning in primed frame coordinates. Again, there is only one correct answer for one lightning flash.

You are claiming one lightning flash is at two places in primed frame coordinates while C' and M are co-located and that is a contradiction.

 

[chinglu]

These two diagrams convey this idea of a hyperbolic rotation as being equivalent to 'stretching' spacetime in one diagonal direction, and 'contracting' it transversely. This is something Penrose calls a Poincaré motion, or an inhomogenous Lorentz motion; most other texts substitute transformation for motion.

So, with v = 0.6c as in both diagrams, the 'scale factor' is $$ \sqrt{ \frac {c+0.6c} {c-0.6c} }$$, which is 2. So the (x',ct') = (1,1) intersect at twice the distance from (0,0) as (x,ct) = (1,1). The transverse diagonal in the (x',ct') frame is therefore 1/2 the distance of the (x,ct) frame.

What does that tell us about length contraction?

You need to take into account the simultaneity shift also. If you are looking at moving rods only, SR claims they are length contracted. This works.

If you are looking at light flashes, you have length contraction plus the simultaneity shift caused by the diversion of the origins (the light emission points in the frames).

You have not discovered anything.