On Thursday, September 29, 2022 at 7:35:14 PM UTC-7, rotchm wrote:
> dono incited me to ask such a question.
>
> In the recent thread entitled "addition of velocities", the op
> inquired about closing speeds.
>
> An aggressive debate ensued.
>
> I (and others) maintained that closing speed (of two things)
> is simply "v1 ± v2" [depending of the sign conventions].
> Or, the rate of change of the separation (distance) between the two things.
> [We were discussing trajectories along the x_axis only btw].
>
> Dono however, maintains that the above are not the meaning of closing speed as used here (relativity and kinematics in general). And he reasons that:
>
> "v1+v2 is the closing speed of two objects that STARTED SIMULTANEOSLY. "
>
> " If the two objects don't start simultaneously, one cannot define closing speed."
>
> I am wondering as to what is the consensus here concerning the concept of
> "closing speed".

The objects meet or pass.

Seems logic dictates "space contraction".

On Thursday, September 29, 2022 at 7:44:29 PM UTC-7, Dono. wrote:
> On Thursday, September 29, 2022 at 7:35:14 PM UTC-7, rotchm wrote:
> > dono incited me to ask such a question.
> >
> > In the recent thread entitled "addition of velocities", the op
> > inquired about closing speeds.
> >
> > An aggressive debate ensued.
> >
> > I (and others) maintained that closing speed (of two things)
> > is simply "v1 ± v2" [depending of the sign conventions].
> Well, you are an idiot, Stephane.
> Time has made this only worse.
> Closing speed is a convention, used to express the rate at which two objects, starting simultaneously, cover a distance L between them:
>
> v1*t+v2*t=L (see, "t" needs to be the SAME)
> t=L/(v1+v2)
>
> By CONVENTION, UNDER the above conditions, u=v1+v2 is called closing speed
> The concept is very useful when one of the "objects" is a pulse of light, say v1=c. Then the closing speed of covering a distance "L" is c+v. But cranks, of the type of Thomas Heger and Stephane Baune, call this "relative speed of light". This causes infinitely long debates about light speed not being constant. Most of such debates center around Einstein derivation of the Lorentz transforms in "On the Electrodynamics..." but also in the derivation of the phase difference in the Sagnac experiments where cranks like Stephane Baune will claim that light goes around at c+v in one direction and c-v in the opposite direction. Ergo, the Sagnac effect is "proof" that light speed is "variable".

Or, ..., that it produces higher order objects as waves, ..., that reflect in eventual disposition in
real wave collapse, ..., that c+v and c-v are "imaginary v".

It's called resonance these days some guys just got a Nobel prize for it.

On Thursday, October 6, 2022 at 6:57:27 PM UTC-7, Ross A. Finlayson wrote:
> On Thursday, September 29, 2022 at 7:44:29 PM UTC-7, Dono. wrote:
> > On Thursday, September 29, 2022 at 7:35:14 PM UTC-7, rotchm wrote:
> > > dono incited me to ask such a question.
> > >
> > > In the recent thread entitled "addition of velocities", the op
> > > inquired about closing speeds.
> > >
> > > An aggressive debate ensued.
> > >
> > > I (and others) maintained that closing speed (of two things)
> > > is simply "v1 ± v2" [depending of the sign conventions].
> > Well, you are an idiot, Stephane.
> > Time has made this only worse.
> > Closing speed is a convention, used to express the rate at which two objects, starting simultaneously, cover a distance L between them:
> >
> > v1*t+v2*t=L (see, "t" needs to be the SAME)
> > t=L/(v1+v2)
> >
> > By CONVENTION, UNDER the above conditions, u=v1+v2 is called closing speed
> > The concept is very useful when one of the "objects" is a pulse of light, say v1=c. Then the closing speed of covering a distance "L" is c+v. But cranks, of the type of Thomas Heger and Stephane Baune, call this "relative speed of light". This causes infinitely long debates about light speed not being constant. Most of such debates center around Einstein derivation of the Lorentz transforms in "On the Electrodynamics..." but also in the derivation of the phase difference in the Sagnac experiments where cranks like Stephane Baune will claim that light goes around at c+v in one direction and c-v in the opposite direction. Ergo, the Sagnac effect is "proof" that light speed is "variable".
>
>
> Or, ..., that it produces higher order objects as waves, ..., that reflect in eventual disposition in
> real wave collapse, ..., that c+v and c-v are "imaginary v".
>
> It's called resonance these days some guys just got a Nobel prize for it.

Figuring the simplest machine's the lever, there's still the ramp, ....

Which provides free work given potential.

Here for example the classical, "is" potential, inverse-squared one way and inverse-cubed the other.

Just like a perfect spherical geometry into Kruskal, or, Sczekeres, Kerr and Schwarzschild making a Chandrasekhar.

On Thursday, October 6, 2022 at 6:57:27 PM UTC-7, Ross A. Finlayson wrote:
> On Thursday, September 29, 2022 at 7:44:29 PM UTC-7, Dono. wrote:
> > On Thursday, September 29, 2022 at 7:35:14 PM UTC-7, rotchm wrote:
> > > dono incited me to ask such a question.
> > >
> > > In the recent thread entitled "addition of velocities", the op
> > > inquired about closing speeds.
> > >
> > > An aggressive debate ensued.
> > >
> > > I (and others) maintained that closing speed (of two things)
> > > is simply "v1 ± v2" [depending of the sign conventions].
> > Well, you are an idiot, Stephane.
> > Time has made this only worse.
> > Closing speed is a convention, used to express the rate at which two objects, starting simultaneously, cover a distance L between them:
> >
> > v1*t+v2*t=L (see, "t" needs to be the SAME)
> > t=L/(v1+v2)
> >
> > By CONVENTION, UNDER the above conditions, u=v1+v2 is called closing speed
> > The concept is very useful when one of the "objects" is a pulse of light, say v1=c. Then the closing speed of covering a distance "L" is c+v. But cranks, of the type of Thomas Heger and Stephane Baune, call this "relative speed of light". This causes infinitely long debates about light speed not being constant. Most of such debates center around Einstein derivation of the Lorentz transforms in "On the Electrodynamics..." but also in the derivation of the phase difference in the Sagnac experiments where cranks like Stephane Baune will claim that light goes around at c+v in one direction and c-v in the opposite direction. Ergo, the Sagnac effect is "proof" that light speed is "variable".
>
>
> Or, ..., that it produces higher order objects as waves, ..., that reflect in eventual disposition in
> real wave collapse, ..., that c+v and c-v are "imaginary v".
>
> It's called resonance these days some guys just got a Nobel prize for it.

"It's going in a circle", ....

On Thursday, October 6, 2022 at 7:42:59 PM UTC-7, Ross A. Finlayson wrote:
> On Thursday, October 6, 2022 at 6:57:27 PM UTC-7, Ross A. Finlayson wrote:
> > On Thursday, September 29, 2022 at 7:44:29 PM UTC-7, Dono. wrote:
> > > On Thursday, September 29, 2022 at 7:35:14 PM UTC-7, rotchm wrote:
> > > > dono incited me to ask such a question.
> > > >
> > > > In the recent thread entitled "addition of velocities", the op
> > > > inquired about closing speeds.
> > > >
> > > > An aggressive debate ensued.
> > > >
> > > > I (and others) maintained that closing speed (of two things)
> > > > is simply "v1 ± v2" [depending of the sign conventions].
> > > Well, you are an idiot, Stephane.
> > > Time has made this only worse.
> > > Closing speed is a convention, used to express the rate at which two objects, starting simultaneously, cover a distance L between them:
> > >
> > > v1*t+v2*t=L (see, "t" needs to be the SAME)
> > > t=L/(v1+v2)
> > >
> > > By CONVENTION, UNDER the above conditions, u=v1+v2 is called closing speed
> > > The concept is very useful when one of the "objects" is a pulse of light, say v1=c. Then the closing speed of covering a distance "L" is c+v. But cranks, of the type of Thomas Heger and Stephane Baune, call this "relative speed of light". This causes infinitely long debates about light speed not being constant. Most of such debates center around Einstein derivation of the Lorentz transforms in "On the Electrodynamics..." but also in the derivation of the phase difference in the Sagnac experiments where cranks like Stephane Baune will claim that light goes around at c+v in one direction and c-v in the opposite direction. Ergo, the Sagnac effect is "proof" that light speed is "variable".
> >
> >
> > Or, ..., that it produces higher order objects as waves, ..., that reflect in eventual disposition in
> > real wave collapse, ..., that c+v and c-v are "imaginary v".
> >
> > It's called resonance these days some guys just got a Nobel prize for it.
> "It's going in a circle", ....

"Bianchi and Darboux: very close"

Hmm, the Bianchi identities are:

Del R^phi mu = 1/2 Del mu R^0.

Here I'm most interested in it where it's 1/2 instead of 1, that it is covariant differentiation,
that it is not, "re-co-variant", why it would be 1 instead of 1/2 this tensor convention, or
statement as according to formula that the common gradient is one-half one-way.

Del and Grad, here I always write Del and Grad together.

What is this mu here in Ricci tensor, it's associated to the covariant derivative the tensor
analysis, and I think a tensor is after a matroid, here that it's not satisfied, "what was
the tensor connection", that it's only "non-linear", then calling it one half.

It's a convention there are others, and probably it's pretty particular, but mostly
here about the equivalency function its integral is one half, but defines one.

Then there's the right-associator bit, here the connection's shifting out the kernel.

(Of its connection.)

Tensor analysis, usually vectors, ..., "simple tensors".

Tensors are like, "hey thanks for writing all this linear state in a vector space,
yes it's all covariant and what is this covariant derivative of a tensor what?"

Torsion on the kernel, singularities.

I like to write my road to reality in like these layers that are like "Road to Reality", ....

Fundamentally, ....

Yeah, then it's for the twistor theory, there is Penrose and there is the guy,

Rindler, ..., "Road to Reality, II"

"... In1961 Rindler used the Fitzgerald contraction
as the premise of his article "Length contraction paradox".
The thought experiment is now called the "ladder paradox". ...."

Restitutive, dissipative, ....

"... and derives the Einstein tensor, ", ....

On Thursday, October 6, 2022 at 3:26:16 PM UTC-7, tjrob137 wrote:
> On 10/1/22 12:58 PM, patdolan wrote:
> > On Saturday, October 1, 2022 at 10:02:56 AM UTC-7, tjrob137 wrote:
> >> On 9/30/22 10:17 PM, rotchm wrote:
> >>>>>> Since you talked about closing speeds, can you explain
> >>>>>> what you mean by such an expression?
> >>>> ... in this post he says..."Closing speed is the rate of
> >>>> change of the distance between two objects as they both move.
> >>>
> >>> Yes, that's basically it. The closing speed between two entities
> >>> is the rate of change of the distance (separation) between them,
> >>> as measured by a given frame (system of coordinates).
> >> Yes, but only for inertial coordinates. The closing speed for a
> >> given pair of objects varies, depending on which inertial frame is
> >> used to calculate/measure it.
> >>> Its "v1 ± v2" depending of sign conventions.
> >> Only when the two objects are moving along a single axis in the
> >> inertial frame used. If that isn't so, the calculation is MUCH
> >> more complicated.
> >>
> >> Tom Roberts
> > Now, Tom Roberts, provide this forum with the proper and coordinate
> > closing speeds, as calulated by observers on each closing object.
> Consider inertial frame S with coordinates (x,t) -- ignore y and z as
> they don't appear in this problem. Object A is located at x>0 and is
> moving inertially along the x axis of S with velocity Va, Va<0. Object B
> is located at x<0 and is moving inertially along the x axis of S with
> velocity Vb, Vb>0. These two objects are clearly approaching each other.
>
> In frame S, the closing speed of these two objects is |Va|+|Vb| = Vb-Va..
>
> (Remember my V's are 3-velocities along the x axis, with
> Va<0 and Vb>0.)
>
> In the rest frame of A, object B has velocity Vba such that
> Vb = (Va+Vba)/(1+Va*Vba/c^2)
> so
> Vba = (Vb-Va)/(1-Va*Vb/c^2)
> In the rest frame of A, their closing speed is |Vba| = Vba.
>
> In the rest frame of B, object A has velocity Vab such that
> Vb = (Va+Vab)/(1+Va*Vab/c^2)
> so
> Vab = (Va-Vb)/(1-Va*Vb/c^2)
> In the rest frame of B, their closing speed is |Vab| = -Vab.
>
> While the expressions are different, |Vab| = |Vba|, and the closing
> speeds in the two object's rest frames are the same.
>
> Note that if Va<<c and Vb<<c, the closing speeds in all three frames are
> approximately the same, |Va|+|Vb| = Vb-Va.
> > I will state up front that you won't/can't do it.
> As usual, you are wrong.
>
> Tom Roberts

Thank you Tom Roberts for deigning to respond to my challenge. So far we have been making our way up a 5.4 - 5.6 route. Time to find some 5.11 stuff.. In your pedestrian implementation of the Einstein velocity addition formula, these are the two most interesting lines that you type:

1) "While the expressions are different, |Vab| = |Vba|, and the closing
speeds in the two object's rest frames are the same."

This result you arrive at is correct of course. But I will demonstrate that it rests on a completely unproved assumption.

2) "Note that if Va<<c and Vb<<c, the closing speeds in all three frames are
approximately the same, |Va|+|Vb| = Vb-Va."

Would you care to speculate on what the ramifications are of closing speeds that are a substantial fraction of c? Say .867c when gamma = 2?

On Friday, October 7, 2022 at 10:38:34 AM UTC-7, patdolan wrote:
> On Thursday, October 6, 2022 at 3:26:16 PM UTC-7, tjrob137 wrote:
> > On 10/1/22 12:58 PM, patdolan wrote:
> > > On Saturday, October 1, 2022 at 10:02:56 AM UTC-7, tjrob137 wrote:
> > >> On 9/30/22 10:17 PM, rotchm wrote:
> > >>>>>> Since you talked about closing speeds, can you explain
> > >>>>>> what you mean by such an expression?
> > >>>> ... in this post he says..."Closing speed is the rate of
> > >>>> change of the distance between two objects as they both move.
> > >>>
> > >>> Yes, that's basically it. The closing speed between two entities
> > >>> is the rate of change of the distance (separation) between them,
> > >>> as measured by a given frame (system of coordinates).
> > >> Yes, but only for inertial coordinates. The closing speed for a
> > >> given pair of objects varies, depending on which inertial frame is
> > >> used to calculate/measure it.
> > >>> Its "v1 ± v2" depending of sign conventions.
> > >> Only when the two objects are moving along a single axis in the
> > >> inertial frame used. If that isn't so, the calculation is MUCH
> > >> more complicated.
> > >>
> > >> Tom Roberts
> > > Now, Tom Roberts, provide this forum with the proper and coordinate
> > > closing speeds, as calulated by observers on each closing object.
> > Consider inertial frame S with coordinates (x,t) -- ignore y and z as
> > they don't appear in this problem. Object A is located at x>0 and is
> > moving inertially along the x axis of S with velocity Va, Va<0. Object B
> > is located at x<0 and is moving inertially along the x axis of S with
> > velocity Vb, Vb>0. These two objects are clearly approaching each other..
> >
> > In frame S, the closing speed of these two objects is |Va|+|Vb| = Vb-Va.
> >
> > (Remember my V's are 3-velocities along the x axis, with
> > Va<0 and Vb>0.)
> >
> > In the rest frame of A, object B has velocity Vba such that
> > Vb = (Va+Vba)/(1+Va*Vba/c^2)
> > so
> > Vba = (Vb-Va)/(1-Va*Vb/c^2)
> > In the rest frame of A, their closing speed is |Vba| = Vba.
> >
> > In the rest frame of B, object A has velocity Vab such that
> > Vb = (Va+Vab)/(1+Va*Vab/c^2)
> > so
> > Vab = (Va-Vb)/(1-Va*Vb/c^2)
> > In the rest frame of B, their closing speed is |Vab| = -Vab.
> >
> > While the expressions are different, |Vab| = |Vba|, and the closing
> > speeds in the two object's rest frames are the same.
> >
> > Note that if Va<<c and Vb<<c, the closing speeds in all three frames are
> > approximately the same, |Va|+|Vb| = Vb-Va.
> > > I will state up front that you won't/can't do it.
> > As usual, you are wrong.
> >
> > Tom Roberts
> Thank you Tom Roberts for deigning to respond to my challenge. So far we have been making our way up a 5.4 - 5.6 route. Time to find some 5.11 stuff.. In your pedestrian implementation of the Einstein velocity addition formula, these are the two most interesting lines that you type:
>
> 1) "While the expressions are different, |Vab| = |Vba|, and the closing
> speeds in the two object's rest frames are the same."
> This result you arrive at is correct of course. But I will demonstrate that it rests on a completely unproved assumption.
>
> 2) "Note that if Va<<c and Vb<<c, the closing speeds in all three frames are
> approximately the same, |Va|+|Vb| = Vb-Va."
> Would you care to speculate on what the ramifications are of closing speeds that are a substantial fraction of c? Say .867c when gamma = 2?

It's they saiy "there's an non-linear impulse", which adds to zero,
then as it resonates, it adds to 1/2.

That's where, "in a circle, the wave resonance observes itself",
"yeah, radial is infinite and non-linear".

So, it adds half up/down what is input.

On Friday, October 7, 2022 at 10:38:34 AM UTC-7, patdolan wrote:
> Would you care to speculate on what the ramifications are of closing speeds
> that are a substantial fraction of c? Say .867c.... ?

There's no need to speculate. If, in terms of inertial coordinates in which you are at rest, one object is approaching you at speed 0.86602c from the left, and another is approaching at that speed from the right, their closing speed in terms of this system is 1.73205c and their relative speed is 0.98974c.

On Friday, October 7, 2022 at 3:53:24 PM UTC-7, Trevor Lange wrote:
> On Friday, October 7, 2022 at 10:38:34 AM UTC-7, patdolan wrote:
> > Would you care to speculate on what the ramifications are of closing speeds
> > that are a substantial fraction of c? Say .867c.... ?
>
> There's no need to speculate. If, in terms of inertial coordinates in which you are at rest, one object is approaching you at speed 0.86602c from the left, and another is approaching at that speed from the right, their closing speed in terms of this system is 1.73205c and their relative speed is 0..98974c.

Trevor, you and Tom Roberts just don't seem to get it. Stare at the LTs for a while and contemplate them as the Dolan does. Apprehend that your proper closing speed v is used both to by you AND the one who is closing on you, to convert between your respective proper and coordinate space intervals and time intervals. How strange. How unsupported by the facts. Why would the one closing on you measure the same proper closing speed that you measure on him??? You measure a different coordinate value of space interval for him. You measure a different coordinate value of time interval for him. Those coordinate values of space and time intervals are the constructors of his coordinate closing speed on you. So why in Dolan's name wouldn't you measure a different coordinate value of closing speed for him?

I'll tell you why: You are sooo steeped in classroom relativity that the foregoing is probably gibberish to you. It is to Jan. It is to Dirk. It is to Paul A#1. Bodkin would have at least tried to unravel its meaning. Tom Roberts has re-read the foregoing several times by now and he still can't make sense of it.

On Saturday, October 8, 2022 at 5:42:18 AM UTC-7, patdolan wrote:
> Your proper closing speed v is used both by you AND the one who is closing
> on you, to convert between your respective proper and coordinate space
> intervals and time intervals.

I think you aren't really meaning to talk about closing speeds, you are talking about what are called relative speeds, meaning the speed of one object in terms of the standard inertial coordinates in which the other object is at rest.

To understand the reciprocity, first consider a straight line with two sets of markings, called the x coordinates and the x' coordinates, and suppose they are related by x' = x - 3. This relationship can also be written as x = x' + 3. So the x' coordinates are offset from the x coordinates by 3, and the x coordinates are offset from the x' coordinates by -3.

> How strange. How unsupported by the facts.

It is manifestly neither strange nor unsupported by facts, the reciprocity between the offsets is tautological and self-evident.

> Why would the one closing on you measure the same proper closing speed
> that you measure on him???

Consider two systems of coordinates called the x,t coordinates and the x',t' coordinates, and suppose they are related by x' = 2(x - t*sqrt(3)/2) and t' = 2(t - x*sqrt(3)/2). This relationship is can also be written as x = 2(x' + t'*sqrt(3)/2) and t = 2(t' + x'*sqrt(3)/2).

> How strange. How unsupported by the facts.

It is manifestly neither strange nor unsupported by facts, the reciprocity between these systems of coordinates is tautological and self-evident.

> You measure a different coordinate value of space interval for him. You measure
> a different coordinate value of time interval for him.

The relationship between the space and time coordinates for the two systems is manifestly reciprocal and symmetrical, as shown above.

On Saturday, October 8, 2022 at 5:42:18 AM UTC-7, patdolan wrote:
> Your proper closing speed v is used both by you AND the one who is closing
> on you, to convert between your respective proper and coordinate space
> intervals and time intervals.

I think you aren't really meaning to talk about closing speeds, you are talking about what are called relative speeds, meaning the speed of one object in terms of the standard inertial coordinates in which the other object is at rest.

To understand the reciprocity, first consider a simpler case, e.g., a straight line with two sets of markings, called the x coordinates and the x' coordinates, and suppose they are related by x' = x - 3. This relationship can also be written as x = x' + 3. So the x' coordinates are offset from the x coordinates by 3, and the x coordinates are offset from the x' coordinates by -3. This is neither strange nor unsupported by facts. The reciprocity between the offsets is tautological and self-evident.

> Why would the one closing on you measure the same proper closing speed
> that you measure on him???

Just as in the simple example above, consider two systems of coordinates called the x,t coordinates and the x',t' coordinates, and suppose they are related by x' = 2(x - t*sqrt(3)/2) and t' = 2(t - x*sqrt(3)/2). This relationship is can also be written as x = 2(x' + t'*sqrt(3)/2) and t = 2(t' + x'*sqrt(3)/2).

> How strange. How unsupported by the facts.

It is manifestly neither strange nor unsupported by facts. The reciprocity between these systems of coordinates is tautological and self-evident.

> You measure a different coordinate value of space interval for him. You measure
> a different coordinate value of time interval for him.

The relationship between the space and time coordinates for the two systems is tautologically reciprocal and symmetrical, as shown above.

On Saturday, October 8, 2022 at 9:23:53 AM UTC-7, Trevor Lange wrote:
> On Saturday, October 8, 2022 at 5:42:18 AM UTC-7, patdolan wrote:
> > Your proper closing speed v is used both by you AND the one who is closing
> > on you, to convert between your respective proper and coordinate space
> > intervals and time intervals.
> I think you aren't really meaning to talk about closing speeds, you are talking about what are called relative speeds, meaning the speed of one object in terms of the standard inertial coordinates in which the other object is at rest.
>
> To understand the reciprocity, first consider a simpler case, e.g., a straight line with two sets of markings, called the x coordinates and the x' coordinates, and suppose they are related by x' = x - 3. This relationship can also be written as x = x' + 3. So the x' coordinates are offset from the x coordinates by 3, and the x coordinates are offset from the x' coordinates by -3. This is neither strange nor unsupported by facts. The reciprocity between the offsets is tautological and self-evident.
> > Why would the one closing on you measure the same proper closing speed
> > that you measure on him???
> Just as in the simple example above, consider two systems of coordinates called the x,t coordinates and the x',t' coordinates, and suppose they are related by x' = 2(x - t*sqrt(3)/2) and t' = 2(t - x*sqrt(3)/2). This relationship is can also be written as x = 2(x' + t'*sqrt(3)/2) and t = 2(t' + x'*sqrt(3)/2).
> > How strange. How unsupported by the facts.
> It is manifestly neither strange nor unsupported by facts. The reciprocity between these systems of coordinates is tautological and self-evident.
> > You measure a different coordinate value of space interval for him. You measure
> > a different coordinate value of time interval for him.
> The relationship between the space and time coordinates for the two systems is tautologically reciprocal and symmetrical, as shown above.

You are not to be blamed that you are unable to conceive of two observers disagreeing on their relative velocity (of which closing speed is a subspecies).

Trevor measures Tom Roberts' relative velocity along Trevor's x-axis as 0.8660243c. In Trevor's universe (the only universe(unless SR is a many-worlds theory)) Tom's co-moving x-axis which is parallel to Trevor's x-axis contains two meter marks for every one meter mark of Trevor's. This is not an illusion just because the two axes are moving wrt each other. Tom's meters really are only half as long as Trevor's meters. Therefore Tom must measure in his universe--the only universe--two meters of displacement for every one meter of displacement that Trevor measures.

Furthermore, Trevor observers that Tom's clock is ticking at a rate only half as fast as Trevor's clock. This is not an illusion just because Tom's and Trevor's clocks are moving wrt each other. Tom's seconds really are twice as long as Trevor's seconds. Therefore Tom must measure in his universe--the only universe--one second for every two seconds that Trevor measures.

When Tom divides his displacement by his interval of time in his universe--the only universe--it is not an illusion or mathematically artifact that Tom's brain, operating at half the biological speed of Trevor's brain, concludes that his velocity towards Trevor is 3.4681689c or gama^2 times the proper velocity that Trevor calculates for Tom.

Not as tautological as you though now, is it. The equivalence of coordinate and proper relative velocity is ONLY tautological in the Galilean transforms. It is merely assumed in the LTs. Think about it before you fire off a knee-jerk reply.

On Saturday, October 8, 2022 at 12:35:05 PM UTC-7, patdolan wrote:
> On Saturday, October 8, 2022 at 9:23:53 AM UTC-7, Trevor Lange wrote:
> > On Saturday, October 8, 2022 at 5:42:18 AM UTC-7, patdolan wrote:
> > > Your proper closing speed v is used both by you AND the one who is closing
> > > on you, to convert between your respective proper and coordinate space
> > > intervals and time intervals.
> > I think you aren't really meaning to talk about closing speeds, you are talking about what are called relative speeds, meaning the speed of one object in terms of the standard inertial coordinates in which the other object is at rest.
> >
> > To understand the reciprocity, first consider a simpler case, e.g., a straight line with two sets of markings, called the x coordinates and the x' coordinates, and suppose they are related by x' = x - 3. This relationship can also be written as x = x' + 3. So the x' coordinates are offset from the x coordinates by 3, and the x coordinates are offset from the x' coordinates by -3. This is neither strange nor unsupported by facts. The reciprocity between the offsets is tautological and self-evident.
> > > Why would the one closing on you measure the same proper closing speed
> > > that you measure on him???
> > Just as in the simple example above, consider two systems of coordinates called the x,t coordinates and the x',t' coordinates, and suppose they are related by x' = 2(x - t*sqrt(3)/2) and t' = 2(t - x*sqrt(3)/2). This relationship is can also be written as x = 2(x' + t'*sqrt(3)/2) and t = 2(t' + x'*sqrt(3)/2).
> > > How strange. How unsupported by the facts.
> > It is manifestly neither strange nor unsupported by facts. The reciprocity between these systems of coordinates is tautological and self-evident.
> > > You measure a different coordinate value of space interval for him. You measure
> > > a different coordinate value of time interval for him.
> > The relationship between the space and time coordinates for the two systems is tautologically reciprocal and symmetrical, as shown above.
> You are not to be blamed that you are unable to conceive of two observers disagreeing on their relative velocity (of which closing speed is a subspecies).
>
> Trevor measures Tom Roberts' relative velocity along Trevor's x-axis as 0..8660243c. In Trevor's universe (the only universe(unless SR is a many-worlds theory)) Tom's co-moving x-axis which is parallel to Trevor's x-axis contains two meter marks for every one meter mark of Trevor's. This is not an illusion just because the two axes are moving wrt each other. Tom's meters really are only half as long as Trevor's meters. Therefore Tom must measure in his universe--the only universe--two meters of displacement for every one meter of displacement that Trevor measures.
>
> Furthermore, Trevor observers that Tom's clock is ticking at a rate only half as fast as Trevor's clock. This is not an illusion just because Tom's and Trevor's clocks are moving wrt each other. Tom's seconds really are twice as long as Trevor's seconds. Therefore Tom must measure in his universe--the only universe--one second for every two seconds that Trevor measures.
>
> When Tom divides his displacement by his interval of time in his universe--the only universe--it is not an illusion or mathematically artifact that Tom's brain, operating at half the biological speed of Trevor's brain, concludes that his velocity towards Trevor is 3.4681689c or gama^2 times the proper velocity that Trevor calculates for Tom.
>
> Not as tautological as you though now, is it. The equivalence of coordinate and proper relative velocity is ONLY tautological in the Galilean transforms. It is merely assumed in the LTs. Think about it before you fire off a knee-jerk reply.

Closing speed applies to light converging at light speed... or diverging.
Light obeys the speed limit but moving with respect to itself
it can go to 2C separation or 2C closing additions.

Mitchell Raemsch

On Saturday, October 8, 2022 at 12:35:05 PM UTC-7, patdolan wrote:
> Trevor measures Tom Roberts' relative velocity along Trevor's x-axis as 0..8660243c.
> Tom's co-moving x-axis is parallel to Trevor's x-axis...

The x and x' axes may be spatially aligned, but they are not parallel when both space and time are taken into account, because the x axis is at constant t=0, whereas the x' axis is at constant t'=0, and for standard inertial coordinates these are skewed relative to each other (as required by the inertia of energy), just as are the t and t' axes.

> Tom must measure two meters of displacement for every one meter of
> displacement that Trevor measures.

Not at all. For example, suppose Tom is displaced by 0.866 LY in one year in terms of Trevor's inertial coordinates. But Tom is displaced by 0 LY in terms of his own inertial coordinates. These displacements are not related by a factor of 2. This shows that you need to specify precisely what kind of "displacement" you are referring to, and you can't willy-nilly conflate displacements, distances, and lengths. You must actually understand each of these concepts.

Consider, for example, the markings on Trevor's x axis, and ask yourself how far apart they are in terms of Tom's system. You want to know the spatial distance between the markings *at a single instant* of Tom's system (equal t'), but the systems of coordinates have skewed x and x' axes. In terms of S (Trevors's standard inertial coordinates), the spatial distance between markings "1" and "2" at time t is x2(t) - x1(t), and in S' (Tom's system) the distance between those markings "1" and "2" at time t' is x2'(t') - x1'(t').

The crucial thing you are overlooking is that those are two different pairs of events, so your simplistic reasoning fails. In terms of S', the "L meter" markings of x are a distance sqrt(1-v^2)*L apart, and in terms of S the "L meter" markings of x' are a distance sqrt(1-v^2)*L apart. This is all in accord with the equations I provided in the previous message.

> Furthermore, Trevor observers that Tom's clock is ticking at a rate only half as fast
> as Trevor's clock.

To be precise, each clock is ticking at half speed in terms of the standard inertial coordinates in which the other clock is at rest. This too is exactly in accord with the equations I provided in the previous message.

> Therefore Tom must measure one second for every two seconds that Trevor measures.

That's not quite right. As stated above, each clock runs slow in terms of the inertial coordinates in which the other clock is at rest. The relations between those systems of coordinates (given by the equations in the previous message) are perfectly reciprocal.

> Not as tautological as you though now, is it.

It is all as tautological and self-evident as I said, for the reasons explained above.

> The equivalence of coordinate and proper relative velocity is ONLY tautological
> in the Galilean transforms. It is merely assumed in the LTs.

You are making up terms (like "proper relative velocity") without defining them, so your intended meaning can only be guessed at, but it was fully explained in the previous message that the reciprocity of the stated coordinate transformation is tautological and self-evident.

On Saturday, October 8, 2022 at 1:39:18 PM UTC-7, Trevor Lange wrote:
> On Saturday, October 8, 2022 at 12:35:05 PM UTC-7, patdolan wrote:
> > Trevor measures Tom Roberts' relative velocity along Trevor's x-axis as 0.8660243c.
> > Tom's co-moving x-axis is parallel to Trevor's x-axis...
>
> The x and x' axes may be spatially aligned, but they are not parallel when both space and time are taken into account, because the x axis is at constant t=0, whereas the x' axis is at constant t'=0, and for standard inertial coordinates these are skewed relative to each other (as required by the inertia of energy), just as are the t and t' axes.
>
> > Tom must measure two meters of displacement for every one meter of
> > displacement that Trevor measures.
>
> Not at all. For example, suppose Tom is displaced by 0.866 LY in one year in terms of Trevor's inertial coordinates. But Tom is displaced by 0 LY in terms of his own inertial coordinates. These displacements are not related by a factor of 2. This shows that you need to specify precisely what kind of "displacement" you are referring to, and you can't willy-nilly conflate displacements, distances, and lengths. You must actually understand each of these concepts.
>
> Consider, for example, the markings on Trevor's x axis, and ask yourself how far apart they are in terms of Tom's system. You want to know the spatial distance between the markings *at a single instant* of Tom's system (equal t'), but the systems of coordinates have skewed x and x' axes. In terms of S (Trevors's standard inertial coordinates), the spatial distance between markings "1" and "2" at time t is x2(t) - x1(t), and in S' (Tom's system) the distance between those markings "1" and "2" at time t' is x2'(t') - x1'(t').
>
> The crucial thing you are overlooking is that those are two different pairs of events, so your simplistic reasoning fails. In terms of S', the "L meter" markings of x are a distance sqrt(1-v^2)*L apart, and in terms of S the "L meter" markings of x' are a distance sqrt(1-v^2)*L apart. This is all in accord with the equations I provided in the previous message.
> > Furthermore, Trevor observers that Tom's clock is ticking at a rate only half as fast
> > as Trevor's clock.
> To be precise, each clock is ticking at half speed in terms of the standard inertial coordinates in which the other clock is at rest. This too is exactly in accord with the equations I provided in the previous message.
>
> > Therefore Tom must measure one second for every two seconds that Trevor measures.
>
> That's not quite right. As stated above, each clock runs slow in terms of the inertial coordinates in which the other clock is at rest. The relations between those systems of coordinates (given by the equations in the previous message) are perfectly reciprocal.
> > Not as tautological as you though now, is it.
> It is all as tautological and self-evident as I said, for the reasons explained above.
> > The equivalence of coordinate and proper relative velocity is ONLY tautological
> > in the Galilean transforms. It is merely assumed in the LTs.
> You are making up terms (like "proper relative velocity") without defining them, so your intended meaning can only be guessed at, but it was fully explained in the previous message that the reciprocity of the stated coordinate transformation is tautological and self-evident.

I have extended properness and coordinancy in a direction that should have been accomplished in 1907. This is how science grows and checks itself.

"You are making up terms (like "proper relative velocity") without defining them, so your intended meaning can only be guessed at, but it was fully explained in the previous message that the reciprocity of the stated coordinate transformation is tautological and self-evident."

The above is your most telling statement regarding your prejudicial mindset.. Explain to this forum exactly why there are both proper and coordinate durations in time, proper and coordinate extension in space, but no proper nor coordinate relative velocities.

You, Trevor, are tenaciously holding on to the well trodden paths of an old theory; fearing to explore and expand that theory's own principles, or apply those principles reflexively. Instead you continue to regurgitate and rehearse examples that no longer satisfy. C. S. Pierce categorized four methods of belief. Yours at the moment is the method of tenacity. My advice to you, Trevor, is the same advice that Cool Hand Luke's prison guard gave to him on a similar occasion: "You need to get your mind right."

On Saturday, October 8, 2022 at 2:12:24 PM UTC-7, patdolan wrote:
> On Saturday, October 8, 2022 at 1:39:18 PM UTC-7, Trevor Lange wrote:
> > On Saturday, October 8, 2022 at 12:35:05 PM UTC-7, patdolan wrote:
> > > Trevor measures Tom Roberts' relative velocity along Trevor's x-axis as 0.8660243c.
> > > Tom's co-moving x-axis is parallel to Trevor's x-axis...
> >
> > The x and x' axes may be spatially aligned, but they are not parallel when both space and time are taken into account, because the x axis is at constant t=0, whereas the x' axis is at constant t'=0, and for standard inertial coordinates these are skewed relative to each other (as required by the inertia of energy), just as are the t and t' axes.
> >
> > > Tom must measure two meters of displacement for every one meter of
> > > displacement that Trevor measures.
> >
> > Not at all. For example, suppose Tom is displaced by 0.866 LY in one year in terms of Trevor's inertial coordinates. But Tom is displaced by 0 LY in terms of his own inertial coordinates. These displacements are not related by a factor of 2. This shows that you need to specify precisely what kind of "displacement" you are referring to, and you can't willy-nilly conflate displacements, distances, and lengths. You must actually understand each of these concepts.
> >
> > Consider, for example, the markings on Trevor's x axis, and ask yourself how far apart they are in terms of Tom's system. You want to know the spatial distance between the markings *at a single instant* of Tom's system (equal t'), but the systems of coordinates have skewed x and x' axes. In terms of S (Trevors's standard inertial coordinates), the spatial distance between markings "1" and "2" at time t is x2(t) - x1(t), and in S' (Tom's system) the distance between those markings "1" and "2" at time t' is x2'(t') - x1'(t').
> >
> > The crucial thing you are overlooking is that those are two different pairs of events, so your simplistic reasoning fails. In terms of S', the "L meter" markings of x are a distance sqrt(1-v^2)*L apart, and in terms of S the "L meter" markings of x' are a distance sqrt(1-v^2)*L apart. This is all in accord with the equations I provided in the previous message.
> > > Furthermore, Trevor observers that Tom's clock is ticking at a rate only half as fast
> > > as Trevor's clock.
> > To be precise, each clock is ticking at half speed in terms of the standard inertial coordinates in which the other clock is at rest. This too is exactly in accord with the equations I provided in the previous message.
> >
> > > Therefore Tom must measure one second for every two seconds that Trevor measures.
> >
> > That's not quite right. As stated above, each clock runs slow in terms of the inertial coordinates in which the other clock is at rest. The relations between those systems of coordinates (given by the equations in the previous message) are perfectly reciprocal.
> > > Not as tautological as you though now, is it.
> > It is all as tautological and self-evident as I said, for the reasons explained above.
> > > The equivalence of coordinate and proper relative velocity is ONLY tautological
> > > in the Galilean transforms. It is merely assumed in the LTs.
> > You are making up terms (like "proper relative velocity") without defining them, so your intended meaning can only be guessed at, but it was fully explained in the previous message that the reciprocity of the stated coordinate transformation is tautological and self-evident.
> I have extended properness and coordinancy in a direction that should have been accomplished in 1907. This is how science grows and checks itself.
> "You are making up terms (like "proper relative velocity") without defining them, so your intended meaning can only be guessed at, but it was fully explained in the previous message that the reciprocity of the stated coordinate transformation is tautological and self-evident."
> The above is your most telling statement regarding your prejudicial mindset. Explain to this forum exactly why there are both proper and coordinate durations in time, proper and coordinate extension in space, but no proper nor coordinate relative velocities.
>
> You, Trevor, are tenaciously holding on to the well trodden paths of an old theory; fearing to explore and expand that theory's own principles, or apply those principles reflexively. Instead you continue to regurgitate and rehearse examples that no longer satisfy. C. S. Pierce categorized four methods of belief. Yours at the moment is the method of tenacity. My advice to you, Trevor, is the same advice that Cool Hand Luke's prison guard gave to him on a similar occasion: "You need to get your mind right."

Trevor Lange I have come up with a better and time honored way to resolve our differences concerning the uniqueness of proper relative velocity v. Prove to this forum that proper velocity v is always equal to coordinate velocity v' in terms of the LTs. Will you accept this challenge?

On Saturday, October 8, 2022 at 2:32:48 PM UTC-7, patdolan wrote:
> Prove that proper velocity v is always equal to coordinate velocity v' in terms of the LTs.

As I said, you are making up terms (e.g., proper velocity) and nomenclature (e.g., v') without defining them, so your strings of characters have no definite meaning. If you are asking me to explain to you the reciprocity between relatively moving systems of inertial coortdinates, we covered that before.

As a reminder, first consider a simpler case, e.g., a straight line with two sets of markings, called the x coordinates and the x' coordinates, and suppose they are related by x' = x - 3. This relationship can also be written as x = x' + 3. So the x' coordinates are offset from the x coordinates by 3, and the x coordinates are offset from the x' coordinates by -3.

Likewise, consider two systems of coordinates called the x,t coordinates and the x',t' coordinates, and suppose they are related by x' = 2(x - t*sqrt(3)/2) and t' = 2(t - x*sqrt(3)/2). This relationship is can also be written as x = 2(x' + t'*sqrt(3)/2) and t = 2(t' + x'*sqrt(3)/2). Thus the reciprocity between these systems of coordinates is tautological and self-evident.

On Saturday, October 8, 2022 at 3:11:31 PM UTC-7, Trevor Lange wrote:
> On Saturday, October 8, 2022 at 2:32:48 PM UTC-7, patdolan wrote:
> > Prove that proper velocity v is always equal to coordinate velocity v' in terms of the LTs.
>
> As I said, you are making up terms (e.g., proper velocity) and nomenclature (e.g., v') without defining them, so your strings of characters have no definite meaning. If you are asking me to explain to you the reciprocity between relatively moving systems of inertial coortdinates, we covered that before.
>
> As a reminder, first consider a simpler case, e.g., a straight line with two sets of markings, called the x coordinates and the x' coordinates, and suppose they are related by x' = x - 3. This relationship can also be written as x = x' + 3. So the x' coordinates are offset from the x coordinates by 3, and the x coordinates are offset from the x' coordinates by -3.
>
> Likewise, consider two systems of coordinates called the x,t coordinates and the x',t' coordinates, and suppose they are related by x' = 2(x - t*sqrt(3)/2) and t' = 2(t - x*sqrt(3)/2). This relationship is can also be written as x = 2(x' + t'*sqrt(3)/2) and t = 2(t' + x'*sqrt(3)/2). Thus the reciprocity between these systems of coordinates is tautological and self-evident.

definitions:
c = ∆x_0/∆t_0 = ∆x_0'/∆t_0' second postulate
v = ∆x/∆t
v' = ∆x'/∆t'

prove v === v'

On Saturday, October 8, 2022 at 3:18:02 PM UTC-7, patdolan wrote:
> > > Prove that proper velocity v is always equal to coordinate velocity v' in terms of the LTs.
> >
> > As I said, you are making up terms (e.g., proper velocity) and nomenclature (e.g., v') without defining them, so your strings of characters have no definite meaning. If you are asking me to explain to you the reciprocity between relatively moving systems of inertial coortdinates, we covered that before.
> >
> > As a reminder, first consider a simpler case, e.g., a straight line with two sets of markings, called the x coordinates and the x' coordinates, and suppose they are related by x' = x - 3. This relationship can also be written as x = x' + 3. So the x' coordinates are offset from the x coordinates by 3, and the x coordinates are offset from the x' coordinates by -3.
> >
> > Likewise, consider two systems of coordinates called the x,t coordinates and the x',t' coordinates, and suppose they are related by x' = 2(x - t*sqrt(3)/2) and t' = 2(t - x*sqrt(3)/2). This relationship is can also be written as x = 2(x' + t'*sqrt(3)/2) and t = 2(t' + x'*sqrt(3)/2). Thus the reciprocity between these systems of coordinates is tautological and self-evident.

Ignoring (again) everything that was just explained to him, Pat Dolan wrote:
> v = ∆x/∆t ..... v' = ∆x'/∆t' ...... prove v === v'

That's can't be (validly) proven, because it is self-evidently false. We covered this before: If, in terms of S, Tom moves ∆x = 0.866 LY in the time ∆t = 1 year then in terms of S' he moves ∆x' = 0 in any non-zero ∆t', so you have v=0.866 and v'=0.

This is all contained in the equations given to you above.

On Saturday, October 8, 2022 at 3:53:04 PM UTC-7, Trevor Lange wrote:
> On Saturday, October 8, 2022 at 3:18:02 PM UTC-7, patdolan wrote:
> > > > Prove that proper velocity v is always equal to coordinate velocity v' in terms of the LTs.
> > >
> > > As I said, you are making up terms (e.g., proper velocity) and nomenclature (e.g., v') without defining them, so your strings of characters have no definite meaning. If you are asking me to explain to you the reciprocity between relatively moving systems of inertial coortdinates, we covered that before.
> > >
> > > As a reminder, first consider a simpler case, e.g., a straight line with two sets of markings, called the x coordinates and the x' coordinates, and suppose they are related by x' = x - 3. This relationship can also be written as x = x' + 3. So the x' coordinates are offset from the x coordinates by 3, and the x coordinates are offset from the x' coordinates by -3..
> > >
> > > Likewise, consider two systems of coordinates called the x,t coordinates and the x',t' coordinates, and suppose they are related by x' = 2(x - t*sqrt(3)/2) and t' = 2(t - x*sqrt(3)/2). This relationship is can also be written as x = 2(x' + t'*sqrt(3)/2) and t = 2(t' + x'*sqrt(3)/2). Thus the reciprocity between these systems of coordinates is tautological and self-evident.
> Ignoring (again) everything that was just explained to him, Pat Dolan wrote:
> > v = ∆x/∆t ..... v' = ∆x'/∆t' ..... prove v === v'
>
> That's can't be (validly) proven, because it is self-evidently false. We covered this before: If, in terms of S, Tom moves ∆x = 0.866 LY in the time ∆t = 1 year then in terms of S' he moves ∆x' = 0 in any non-zero ∆t', so you have v=0.866 and v'=0.
>
> This is all contained in the equations given to you above.

Whether he's Townes Olsen, Stan Fultoni or Trevor Lange, Legion always balks at proving that the v in the LTs is the same when determining t, t', x or x'

On Saturday, October 8, 2022 at 3:18:02 PM UTC-7, patdolan wrote:
> On Saturday, October 8, 2022 at 3:11:31 PM UTC-7, Trevor Lange wrote:
> > On Saturday, October 8, 2022 at 2:32:48 PM UTC-7, patdolan wrote:
> > > Prove that proper velocity v is always equal to coordinate velocity v' in terms of the LTs.
> >
> > As I said, you are making up terms (e.g., proper velocity) and nomenclature (e.g., v') without defining them, so your strings of characters have no definite meaning. If you are asking me to explain to you the reciprocity between relatively moving systems of inertial coortdinates, we covered that before.
> >
> > As a reminder, first consider a simpler case, e.g., a straight line with two sets of markings, called the x coordinates and the x' coordinates, and suppose they are related by x' = x - 3. This relationship can also be written as x = x' + 3. So the x' coordinates are offset from the x coordinates by 3, and the x coordinates are offset from the x' coordinates by -3.
> >
> > Likewise, consider two systems of coordinates called the x,t coordinates and the x',t' coordinates, and suppose they are related by x' = 2(x - t*sqrt(3)/2) and t' = 2(t - x*sqrt(3)/2). This relationship is can also be written as x = 2(x' + t'*sqrt(3)/2) and t = 2(t' + x'*sqrt(3)/2). Thus the reciprocity between these systems of coordinates is tautological and self-evident.
> definitions:
> c = ∆x_0/∆t_0 = ∆x_0'/∆t_0' second postulate
> v = ∆x/∆t
> v' = ∆x'/∆t'
>
> prove v === v'

Look at a motion black hole...
An atoms movement can leave behind light's constant speed.
That light absolute has to catch up to the fast atom.
Light's absolute competing with the fast atom demonstrates
the atomic motion compares to an absolute.
Does that not show atom has absolute speed compared
to light's faster?

Mitchell Raemsch

On Saturday, October 8, 2022 at 4:55:26 PM UTC-7, patdolan wrote:
> > Ignoring (again) everything that was just explained to him, Pat Dolan wrote:
> > > v = ∆x/∆t ..... v' = ∆x'/∆t' ...... prove v === v'
> >
> > That's can't be (validly) proven, because it is self-evidently false. We covered this before: If, in terms of S, Tom moves ∆x = 0.866 LY in the time ∆t = 1 year then in terms of S' he moves ∆x' = 0 in any non-zero ∆t', so you have v=0.866 and v'=0.
> >
> Trevor balks at proving that the v in the LTs is the same when determining t, t', x or x'

You've asked for proofs of two different propositions, the first being the proposition that relatively moving systems of inertial coordinates are reciprocally related, which is proven by simple grade school algebra, i.e., the relationship x'=(x-vt)g, t'=(t-vx)g where g=1/sqrt(1-v^2) for any constant v has the unique algebraic inverse x=(x'+vt')g, t=(t'+vx')g. Proof complete. The second proposition you asked to be proven was the absurd assertion that velocities are generally invariant, which isn't even true in Galilean relativity, let alone special relativity. In fact, it is insane.. This was patiently explained to you (twice), with an explicit example. You're welcome.

On Saturday, October 8, 2022 at 5:17:11 PM UTC-7, Trevor Lange wrote:
> On Saturday, October 8, 2022 at 4:55:26 PM UTC-7, patdolan wrote:
> > > Ignoring (again) everything that was just explained to him, Pat Dolan wrote:
> > > > v = ∆x/∆t ..... v' = ∆x'/∆t' ...... prove v === v'
> > >
> > > That's can't be (validly) proven, because it is self-evidently false. We covered this before: If, in terms of S, Tom moves ∆x = 0.866 LY in the time ∆t = 1 year then in terms of S' he moves ∆x' = 0 in any non-zero ∆t', so you have v=0.866 and v'=0.
> > >
> > Trevor balks at proving that the v in the LTs is the same when determining t, t', x or x'
>
> You've asked for proofs of two different propositions, the first being the proposition that relatively moving systems of inertial coordinates are reciprocally related, which is proven by simple grade school algebra, i.e., the relationship x'=(x-vt)g, t'=(t-vx)g where g=1/sqrt(1-v^2) for any constant v has the unique algebraic inverse x=(x'+vt')g, t=(t'+vx')g. Proof complete. The second proposition you asked to be proven was the absurd assertion that velocities are generally invariant, which isn't even true in Galilean relativity, let alone special relativity. In fact, it is insane. This was patiently explained to you (twice), with an explicit example. You're welcome.

Legion, stop the constant rephrasing my questions and challenges.

On Saturday, October 8, 2022 at 12:35:05 PM UTC-7, patdolan wrote:

> You are not to be blamed that you are unable to conceive of two observers disagreeing on their relative velocity (of which closing speed is a subspecies).

See? You are wrong right out of the box. It is NEVER the case that 2 observers observing their relative speeds can EVER determine their closing speed. The very definition of closing speed precludes such a situation. It just cannot be done. Get a clue, for once, you are only making a bigger fool of yourself than you usually do.

Read a dang textbook and do try harder.

Den 08.10.2022 21:35, skrev patdolan:
>
> Trevor measures Tom Roberts' relative velocity along Trevor's x-axis as 0.8660243c. In Trevor's universe (the only universe(unless SR is a many-worlds theory)) Tom's co-moving x-axis which is parallel to Trevor's x-axis contains two meter marks for every one meter mark of Trevor's. This is not an illusion just because the two axes are moving wrt each other. Tom's meters really are only half as long as Trevor's meters. Therefore Tom must measure in his universe--the only universe--two meters of displacement for every one meter of displacement that Trevor measures.
>
> Furthermore, Trevor observers that Tom's clock is ticking at a rate only half as fast as Trevor's clock. This is not an illusion just because Tom's and Trevor's clocks are moving wrt each other. Tom's seconds really are twice as long as Trevor's seconds. Therefore Tom must measure in his universe--the only universe--one second for every two seconds that Trevor measures.
>
> When Tom divides his displacement by his interval of time in his universe--the only universe--it is not an illusion or mathematically artifact that Tom's brain, operating at half the biological speed of Trevor's brain, concludes that his velocity towards Trevor is 3.4681689c or gama^2 times the proper velocity that Trevor calculates for Tom.
>

https://paulba.no/pdf/Mutual_time_dilation.pdf

--
Paul

https://paulba.no/