In this installment, I intend to present more detail about the differential equation that characterizes special relativity. It is an isomorphism between circular rotation and hyperbolic rotation. Not every isomorphism describes a physical situation. But the experimental data makes it clear that in this case, both isomorphisms have physical significance. To recap, if the hyperbolic rotation angle is rapidity, w, then its gudermannian is a circular angle, tilt, which we will label θ. Then the diffeq is dw/dθ = γ, the Lorentz factor. This can be rewritten as dw = γdθ = sec(θ) dθ. This can now be integrated:
w = ln(sec(θ)+tan(θ)) + C
When w=0, θ=0, so the constant of integration must also be 0, since sec(0)=1 and tan(0)=0, and ln(1) = 0.

While rapidity is a legitimate physical property, physics does not seem to care about the gudermannian. They use it to define relative velocity, v = c sin(θ), but generally ignore the fact that it is a function of rapidity. Instead of being a parametric definition of velocity, it is the circular rotation that is isomorphic to the hyperbolic rotation that is the Lorentz Transform. While this may seem unfamiliar, it is actually known to virtually every grade school student. And I am not referring to relativity here, but a much, much older application. In difference form, it is the algorithm used by Mercator in the 1500's to create his world-famous map.

In his application, the angles have different names, but the relationship between them is the same. I can't speak about Mercator's intentions, and I don't know if any historical records exist. So, my comments address his invention, the Mercator Projection. The whole purpose of a map is to represent the curved surface of the Earth by a flat drawing. Many different kinds of map were in use in Mercator's time, but using them was challenging. It took an expert to convert map readings into a course and a compass heading. And each kind of map used a different algorithm. I can only presume that Mercator set out to make a better map. The first obstacle to just unrolling the skin off the globe is that the parallels are all different arclengths, depending on the latitude. The radius of a parallel is R cos(latitude). So, circumference is 2 Pi R cos(latitude). The arclength at the equator is maximum, and latitude is 0, so cos(latitude) = 1. To make the Mercator map, each latitude circle must be scaled by sec(latitude). This gives us a rectangular map, but it distorts local coordinates so that the angle of a straight line to the grid is different at every latitude. To make local coordinates which preserve the angle of the curve to latitude and longitude lines, the vertical axis must also be scaled by the same factor as the horizontal. So, as we change latitude, it just changes the local magnification factor. Although the transform is only intended to keep local geometry orthogonal, since the algorithm applies to every point, a straight line on the map, known as a rhumb line, transforms to a spiral with the same tilt at every point on the globe, known as a loxodrome. A 0 degree tilt refers to a longitude line, or meridian. A loxodrome has the same tilt angle to a meridian at every point on the curve. It is known as a constant-compass course, because the angle of the spiral to the meridians is exactly the same angle as the rhumb line to vertical.
Courtesy of:
https://encyclopediaofmath.org/wiki/Loxodrome

For a sphere with first fundamental form
ds² = R²(du²+cos²(u)dv²)
the equation of a loxodrome is
vcotan(α) = Rln(tan(π/4+u/2R))
The reference is based on coordinates in the surface of a sphere of radius R. If we change variables from arclength to central angle, we can eliminate R from the equation. This derivation is valid for all R, and both are independent, a characteristic of spherical coordinates,(R,λ,φ). Then u = Rλ and v = Rφ. If we make these substitutions, then the equation becomes
Rφ cotan(α) = Rln(tan(π/4+λ/2))
φ = tan(α)ln(tan(π/4+λ/2))
= tan(α)ln((tan(π/4)+tan(λ/2))/(1-tan(π/4)*tan(λ/2)))
= tan(α)ln((1+tan(λ/2))/(1-tan(λ/2)))
= tan(α)ln((cos(λ/2)+sin(λ/2))/(cos(λ/2)-sin(λ/2)))
= tan(α)ln((cos(λ/2)+sin(λ/2))²/(cos²(λ/2)-sin²(λ/2)))
= tan(α)ln((cos²(λ/2)+sin²(λ/2)+2sin(λ/2)cos(λ/2))/cos(λ))
= tan(α)ln((1+sin(λ))/cos(λ))
= tan(α)ln(sec(λ)+tan(λ))
e^φ = e^(tan(α)ln(sec(λ)+tan(λ)))
= (sec(λ)+tan(λ))^tan(α)
When λ = gudermannian(φ), sec(λ) = cosh(φ) and tan(λ) = sinh(φ). Then:
e^φ = (cosh(φ)+sinh(φ))^tan(α)
= (e^φ)^tan(α)
tan(α) = 1
α = π/4
The 45 degree loxodrome has coordinates which are related by the gudermannian function. To say 1 is physical, and the other is not, is contradictory, because they are both properties of the same, single radius vector. Special relativity maps relative velocity to latitude angle. As a gudermannian, it also maps a corresponding Lorentz boost to the longitude angle. Consequently, all Lorentz boosts are associated with a phase rotation. The phase rotation is not a hyperbolic rotation, because that is rapidity. It is not rotation around any of the space axes, either, because we use inertial frames. It's a rotation into complex dimensions. Or higher dimensions. Whatever. There's an isomorphism between hypercomplex numbers and ordered n-tuples of real coordinates, so it's all the same. It's a rotation out of the box that special relativity confines us to. Some people will reflexively put this in the category of pink unicorns. The geometry identifies the source of the excess relativistic momentum of high speed particles to be imaginary velocity components, not relativistic mass. And to the smart-asses out there smirking, that's mathematically imaginary (thanks to Descartes sneering at complex numbers). The i*i = -1 kind of imaginary. The momentum that it carries is physical enough when the particle inelastically collides with a target. And, it isn't wishful thinking or an analogy. It is merely the sine projection of a vector, following the rules of geometry. Special relativity simply discards this dimension, and invents self-contradictory fairy tales to rationalize the cut. More details to follow.

The irony of the Mercator map is that Greenland appears to be larger than Australia because of the same transform that fooled physicists into believing in relativistic mass. It is an illusion of the transform, the same transform, in both cases. Einstein, himself, was an amateur sailor. I wonder how many times he stared at Mercator's map, without realizing that his special relativity was based on it, an artifact from 3 centuries before he was born..

On Tuesday, October 19, 2021 at 7:52:08 PM UTC-7, tgca...@gmail.com wrote:
> Special relativity ... invents self-contradictory fairy tales...

But you haven't pointed out anything self-contradictory in special relativity, so your assertion is baseless. Indeed, you've admitted that special relativity (i.e., local Lorentz invariance) is perfectly correct.

> You ask "What missing parts?" ... Special relativity's "entire magnitude" is less than
> the same measurement in the rest frame of the object. The "missing part" is the difference.

No, that is a misunderstanding. Every distinct time and place has unique coordinates in terms of one system of inertia-based coordinates, and it has unique coordinates in terms of another such system, and the relationship between them is one-to-one. There are manifestly no "missing parts". Look, if someone says you are 2 meters tall, and someone else says you are 200 centimeters tall, this does not mean that the first person thinks you have 198 "missing parts". They are describing your height in terms of two different coordinate systems. Of course, higher-dimensional coordinates systems are more complicated than one-dimensional coordinate systems, but the concept is the same. There is nothing "missing".

> To claim that things physically contract and deny that they are shorter than they
> should be is plainly a contradiction.

Your statement is garbled and inaccurate. The relationship between inertial coordinate systems (given by the Lorentz transformation) doesn't entail any contradiction. You have to distinguish between active and passive transformations to understand this fully.

> I have read numerous books on the subject including several of Einstein's.. None
> distinguished between so-called "active and passive" transformations.

That is not true, it is discussed in all (decent) books on relativity, although with different terms and names. It just refers to the difference between (1) the descriptions, in terms of one system of coordinates, of an object in two different states of motion (after undergoing acceleration), and (2) the descriptions of an object in a single state of motion in terms of two different coordinate systems.

> Relativity asserts that the effects of high velocity are based on "relative" velocity.

To be accurate, relativity asserts that all the laws of physics take the same homogeneous and isotropic form in terms of any system of inertia-based coordinates. That's the principle of relativity, which special relativity shares with Newtonian physics. They differ in the realization that all forms of energy have inertia (which Newton didn't know), and that's why the inertia-based coordinate systems are related by Lorentz transformations rather than Galilean transformations.

> It is well established that special relativity denies the existence of objective reality.

Again, that's categorically false. What is well known is that special relativity gives a completely objective account of all phenomena. It is not a subjectivist theory. It does not entail "different realities". Of course, we can describe phenomena in terms of different systems of coordinates, like you being 2 meters tall and 200 cm tall, but this does not imply different realities. Your height is what it is. Likewise the fact that a given object has speed 5 mph in terms of one system of coordinates and 200 mph in terms of another does not imply any alternate realities.

> Two observers moving at different relative velocities get different measurements
> of the same object...

Stated accurately, the descriptions of phenomena in terms of different coordinate systems are different (duh), but this doesn't mean the phenomena are different. Just like your height, all the phenomena are what they are, and we can describe them in terms of different systems of coordinates. The important symmetry is that there is a class of coordinate systems -- called the inertia-based or inertial coordinate systems -- in terms of which the equations of physics take the same simple homogeneous and isotropic form. The relationship between these systems is given by Lorentz transformations (consistent with the fact that every quantity E of energy has inertia of E/c^2)..

> Tell me how a stationary object "knows" how much to shrink when it has no
> way to determine relative velocity. Better, explain how it can shrink different
> amounts for different observers at the same time.

Again, you need to distinguish between active and passive transformations, but for a passive transformation, it's like your height being 2 and 200, depending on the system of coordinates. The height h' in cm is related to the height h in meters by h'=100h. Inertial coordinate systems are related by Lorentz transformation, i.e., x'=(x-vt)g, t'=(t-vx)g. There's nothing magical or mysterious about this.

> Based on your last comment, Lorentz's Aether Theory is just another interpretation of
> special relativity.

Right, it's well known that there is a Lorentzian interpretation of special relativity, that is empirically indistinguishable from the more common interpretation. It is actually just semantically different, using different systems of coordinates to describe the same phenomena.

> I am not impressed that physicists choose to corrupt the rules of logic rather than
> recognize a contradiction.

Special relativity doesn't violate any rules of logic. It is perfectly logical and free of any contradictions. You simply have a garbled understanding of it. That's not entirely your fault, because special relativity is often badly taught.

On Wednesday, 20 October 2021 at 06:10:19 UTC+2, Townes Olson wrote:

> Special relativity doesn't violate any rules of logic. It is perfectly logical and free of any contradictions.

An idiot has said!!! Must be true.
In the meantime in the real world, however, GPS clocks keep measuring
t'=t, just like all serious clocks always did.

Op 19-okt.-2021 om 04:52 schreef Tom Capizzi:

> Townes Olson: so many misconceptions. In order:
> You ask "What missing parts?" Are you purposely being
> obtuse? If you never heard of length contraction or
> time dilation, you >don't belong here.

Ha... that's what you mean with missing parts:
I measure an object at rest in my reference frame to have
length L. You are moving w.r.t. me with speed v, and measure
my object to have length L/gamma. You call
L - L/gamma
the "missing part" by length contraction, induced by the
the Lorentz transformation, right?

Two points about that:

1)
I measure a clock at rest in my reference frame to have
time interval T between two ticks. You are moving w.r.t.
me with speed v, and measure my clock ticks to have a
time interval T*gamma between them as measured on your
clock.
In your lingo, you should call
T*gamma - T
an "added part" by time dilation, induced by the
the Lorentz transformation.
There's no reason to call that a "missing part".

2)
You are moving with speed v w.r.t. me. I shoot
a bullet at speed 3v to you after you have passed me.
For me the bullet has speed 3v. For you the bullet
has speed 2v.
Do you call
3v - 2v = v
the "missing part" by speed decrease, induced by
the Galilean transformation?

Dirk Vdm

On Wednesday, 20 October 2021 at 17:52:37 UTC+2, Dirk Van de moortel wrote:
> Op 19-okt.-2021 om 04:52 schreef Tom Capizzi:
>
>
> > Townes Olson: so many misconceptions. In order:
> > You ask "What missing parts?" Are you purposely being
> > obtuse? If you never heard of length contraction or
> > time dilation, you >don't belong here.
>
> Ha... that's what you mean with missing parts:
> I measure an object at rest in my reference frame to have
> length L. You are moving w.r.t. me with speed v, and measure
> my object to have length L/gamma. You call

In the meantime in the real world, however,
GPS clocks keep measuring t'=t, just like all serious
clocks always did.

Dirk Van de moortel:

>> Townes Olson: so many misconceptions. In order:
>> You ask "What missing parts?" Are you purposely being
>> obtuse? If you never heard of length contraction or
>> time dilation, you >don't belong here.

>Ha... that's what you mean with missing parts:
>I measure an object at rest in my reference frame to have
>length L. You are moving w.r.t. me with speed v, and measure
>my object to have length L/gamma. You call
>L - L/gamma
>the "missing part" by length contraction, induced by the
>the Lorentz transformation, right?
>
>
>Two points about that:
>
>1)
>I measure a clock at rest in my reference frame to have
>time interval T between two ticks. You are moving w.r.t.
>me with speed v, and measure my clock ticks to have a
>time interval T*gamma between them as measured on your
>clock.
>In your lingo, you should call
>T*gamma - T
>an "added part" by time dilation, induced by the
>the Lorentz transformation.
>There's no reason to call that a "missing part".
>
>2)
>You are moving with speed v w.r.t. me. I shoot
>a bullet at speed 3v to you after you have passed me.
>For me the bullet has speed 3v. For you the bullet
>has speed 2v.
>Do you call
>3v - 2v = v
>the "missing part" by speed decrease, induced by
>the Galilean transformation?

Dirk Vdm

0) The “missing part” is a vector, not a scalar. So, while it is a difference, your formula is not correct. The part measured while moving is the cosine projection. The part measured while stationary is the hypotenuse. Their vector difference is the sine projection, which is the cosine projection times the tangent of the projection angle. Alternately, we could just apply the Pythagorean identity and subtract the square of the moving measurement from the square of the hypotenuse. This difference is the square of the missing part. This geometry puts it firmly in an imaginary dimension where it cannot be seen or measured. Second, the Lorentz Transform doesn't induce anything.

1) There is no reason to introduce fake terminology to describe time dilation. Time intervals don't get bigger. And, first, you make the same error with time as with space. The missing part is a vector projection. Second, it is missing, which is why the clock runs slow. Certainly nothing is “added”.

2) This one just makes no sense. At least you got the arithmetic right on this one. Both velocities are parallel, so their combination is by linear addition (no bullet can reach a velocity where relativistic effects are measurable). But nothing is missing. My reference origin is -v from my frame, and the bullet is +2v. But, relative to my origin, the bullet is 3v. Nothing is missing.

On Tuesday, October 19, 2021 at 7:52:08 PM UTC-7, tgca...@gmail.com wrote:
> Special relativity ... invents self-contradictory fairy tales...

But you have not pointed out anything self-contradictory in special relativity, so your assertion is baseless. Indeed, you've admitted that special relativity (i.e., local Lorentz invariance) is perfectly correct.

> You ask "What missing parts?" ... Special relativity's "entire magnitude" is less than
> the same measurement in the rest frame of the object. The "missing part" is the difference.

No, that's a misunderstanding. Every distinct time and place has unique coordinates in terms of one system of inertia-based coordinates, and it has unique coordinates in terms of another such system, and the relationship between them is one-to-one. There are manifestly no "missing parts".

> To claim that things physically contract and deny that they are shorter than they
> should be is plainly a contradiction.

Your statement is garbled and inaccurate. The relationship between inertial coordinate systems (given by the Lorentz transformation) does not entail any contradiction. You're failing to distinguish between active and passive transformations.

> I have read numerous books on the subject including several of Einstein's.. None
> distinguished between so-called "active and passive" transformations.

That's not true, it is discussed in all (decent) books on relativity, although with different terms and names. It just refers to the difference between (1) the descriptions, in terms of one system of coordinates, of an object in two different states of motion (after undergoing acceleration), and (2) the descriptions of an object in a single state of motion in terms of two different coordinate systems.

> Relativity asserts that the effects of high velocity are based on "relative" velocity.

To be accurate, relativity asserts that all the laws of physics take the same homogeneous and isotropic form in terms of any system of inertia-based coordinates. That is the principle of relativity, which special relativity shares with Newtonian physics. They differ in the realization that all forms of energy have inertia (which Newton didn't know), and it follows by elementary reasoning that inertia-based coordinate systems are related by Lorentz transformations rather than Galilean transformations.

> It is well established that special relativity denies the existence of objective reality.

Again, that is categorically false. What is well known is that special relativity gives a completely objective account of all phenomena. It is not a subjectivist theory as you imagine. It does not entail "different realities".. Of course, we can describe phenomena in terms of different systems of coordinates, but this does not imply different realities. For example, a given object may have speed 5 mph in terms of one system of coordinates and 200 mph in terms of another, but that does not imply any alternate realities. Also, if you are first at rest in inertial coordinates in which the object has speed 5 mph, and then you accelerate to rest in a system of coordinates in terms of which the object has speed 200 mph, this does not mean anything has happened to the object. Understand?

> Two observers moving at different relative velocities get different measurements
> of the same object...

Stated accurately, the descriptions of phenomena in terms of different coordinate systems are different (obviously), but this doesn't mean the phenomena are different. All the phenomena are what they are, and we can describe them in terms of different systems of coordinates. The important symmetry is that there is a class of coordinate systems -- called the inertia-based or inertial coordinate systems -- in terms of which the equations of physics take the same simple homogeneous and isotropic form. The relationship between these systems is given by Lorentz transformations (consistent with the fact that every quantity E of energy has inertia of E/c^2).

> Tell me how a stationary object "knows" how much to shrink when it has no
> way to determine relative velocity. Better, explain how it can shrink different
> amounts for different observers at the same time.

Again, you need to distinguish between active and passive transformations. For a passive transformation, it's like your height being 2 meters and 200 centimeters, depending on the system of coordinates. The height h' in cm is related to the height h in meters by h'=100h. Inertial coordinate systems are related by Lorentz transformation, i.e., x'=(x-vt)g, t'=(t-vx)g. There's nothing magical or mysterious about this.

> Based on your last comment, Lorentz's Aether Theory is just another interpretation of
> special relativity.

Right, it is well known that there is a Lorentzian interpretation of special relativity, that is empirically indistinguishable from the more common interpretation. It is actually just semantically different, using different systems of coordinates to describe the same phenomena.

> I am not impressed that physicists choose to corrupt the rules of logic rather than
> recognize a contradiction.

Special relativity doesn't violate any rules of logic... if you think it does, please mention which rules of logic it violates. If you can't, then your allegation is baseless. Right?

On Thursday, 21 October 2021 at 07:10:49 UTC+2, Townes Olson wrote:

> Special relativity doesn't violate any rules of logic... if you think it does, please mention which rules of logic it violates.

Yes, it does; it violates not(p and not p)

Op 21-okt.-2021 om 06:14 schreef Tom Capizzi:
> Dirk Van de moortel:
>
>>> Townes Olson: so many misconceptions. In order:
>>> You ask "What missing parts?" Are you purposely being
>>> obtuse? If you never heard of length contraction or
>>> time dilation, you >don't belong here.
>
>> Ha... that's what you mean with missing parts:
>> I measure an object at rest in my reference frame to have
>> length L. You are moving w.r.t. me with speed v, and measure
>> my object to have length L/gamma. You call
>> L - L/gamma
>> the "missing part" by length contraction, induced by the
>> the Lorentz transformation, right?
>>
>>
>> Two points about that:
>>
>> 1)
>> I measure a clock at rest in my reference frame to have
>> time interval T between two ticks. You are moving w.r.t.
>> me with speed v, and measure my clock ticks to have a
>> time interval T*gamma between them as measured on your
>> clock.
>> In your lingo, you should call
>> T*gamma - T
>> an "added part" by time dilation, induced by the
>> the Lorentz transformation.
>> There's no reason to call that a "missing part".
>>
>> 2)
>> You are moving with speed v w.r.t. me. I shoot
>> a bullet at speed 3v to you after you have passed me.
>> For me the bullet has speed 3v. For you the bullet
>> has speed 2v.
>> Do you call
>> 3v - 2v = v
>> the "missing part" by speed decrease, induced by
>> the Galilean transformation?
>
> Dirk Vdm
>
> 0) The “missing part” is a vector, not a scalar. So, while it is a
> difference, your formula is not correct. The part measured while
> moving is the cosine projection. The part measured while stationary
> is the hypotenuse. Their vector difference is the sine projection,
> which is the cosine projection times the tangent of the projection
> angle. Alternately, we could just apply the Pythagorean identity and
> subtract the square of the moving measurement from the square of the
> hypotenuse. This difference is the square of the missing part. This
> geometry puts it firmly in an imaginary dimension where it cannot be
> seen or measured. Second, the Lorentz Transform doesn't induce
> anything.
>
> 1) There is no reason to introduce fake terminology to describe time
> dilation. Time intervals don't get bigger. And, first, you make the
> same error with time as with space. The missing part is a vector
> projection. Second, it is missing, which is why the clock runs slow.
> Certainly nothing is “added”.
>
> 2) This one just makes no sense. At least you got the arithmetic
> right on this one. Both velocities are parallel, so their combination
> is by linear addition (no bullet can reach a velocity where
> relativistic effects are measurable). But nothing is missing. My
> reference origin is -v from my frame, and the bullet is +2v. But,
> relative to my origin, the bullet is 3v. Nothing is missing.
>

Um.
You probably don't understand how my message was crafted
to show what kind of fool you are, and how your response
confirms that.
Good grief, what a fucking mess :-|

Dirk Vdm

On Tuesday, October 19, 2021 at 7:52:08 PM UTC-7, tgca...@gmail.com wrote:
> In this installment, I intend to present more detail about the differential equation that characterizes special relativity. It is an isomorphism between circular rotation and hyperbolic rotation. Not every isomorphism describes a physical situation. But the experimental data makes it clear that in this case, both isomorphisms have physical significance. To recap, if the hyperbolic rotation angle is rapidity, w, then its gudermannian is a circular angle, tilt, which we will label θ. Then the diffeq is dw/dθ = γ, the Lorentz factor. This can be rewritten as dw = γdθ = sec(θ) dθ. This can now be integrated:
> w = ln(sec(θ)+tan(θ)) + C
> When w=0, θ=0, so the constant of integration must also be 0, since sec(0)=1 and tan(0)=0, and ln(1) = 0.
>
> While rapidity is a legitimate physical property, physics does not seem to care about the gudermannian. They use it to define relative velocity, v = c sin(θ), but generally ignore the fact that it is a function of rapidity. Instead of being a parametric definition of velocity, it is the circular rotation that is isomorphic to the hyperbolic rotation that is the Lorentz Transform. While this may seem unfamiliar, it is actually known to virtually every grade school student. And I am not referring to relativity here, but a much, much older application. In difference form, it is the algorithm used by Mercator in the 1500's to create his world-famous map.
>
> In his application, the angles have different names, but the relationship between them is the same. I can't speak about Mercator's intentions, and I don't know if any historical records exist. So, my comments address his invention, the Mercator Projection. The whole purpose of a map is to represent the curved surface of the Earth by a flat drawing. Many different kinds of map were in use in Mercator's time, but using them was challenging. It took an expert to convert map readings into a course and a compass heading. And each kind of map used a different algorithm. I can only presume that Mercator set out to make a better map. The first obstacle to just unrolling the skin off the globe is that the parallels are all different arclengths, depending on the latitude. The radius of a parallel is R cos(latitude). So, circumference is 2 Pi R cos(latitude). The arclength at the equator is maximum, and latitude is 0, so cos(latitude) = 1. To make the Mercator map, each latitude circle must be scaled by sec(latitude). This gives us a rectangular map, but it distorts local coordinates so that the angle of a straight line to the grid is different at every latitude. To make local coordinates which preserve the angle of the curve to latitude and longitude lines, the vertical axis must also be scaled by the same factor as the horizontal. So, as we change latitude, it just changes the local magnification factor. Although the transform is only intended to keep local geometry orthogonal, since the algorithm applies to every point, a straight line on the map, known as a rhumb line, transforms to a spiral with the same tilt at every point on the globe, known as a loxodrome. A 0 degree tilt refers to a longitude line, or meridian. A loxodrome has the same tilt angle to a meridian at every point on the curve. It is known as a constant-compass course, because the angle of the spiral to the meridians is exactly the same angle as the rhumb line to vertical.
> Courtesy of:
> https://encyclopediaofmath.org/wiki/Loxodrome
>
> For a sphere with first fundamental form
> ds² = R²(du²+cos²(u)dv²)
> the equation of a loxodrome is
> vcotan(α) = Rln(tan(π/4+u/2R))
> The reference is based on coordinates in the surface of a sphere of radius R. If we change variables from arclength to central angle, we can eliminate R from the equation. This derivation is valid for all R, and both are independent, a characteristic of spherical coordinates,(R,λ,φ). Then u = Rλ and v = Rφ. If we make these substitutions, then the equation becomes
> Rφ cotan(α) = Rln(tan(π/4+λ/2))
> φ = tan(α)ln(tan(π/4+λ/2))
> = tan(α)ln((tan(π/4)+tan(λ/2))/(1-tan(π/4)*tan(λ/2)))
> = tan(α)ln((1+tan(λ/2))/(1-tan(λ/2)))
> = tan(α)ln((cos(λ/2)+sin(λ/2))/(cos(λ/2)-sin(λ/2)))
> = tan(α)ln((cos(λ/2)+sin(λ/2))²/(cos²(λ/2)-sin²(λ/2)))
> = tan(α)ln((cos²(λ/2)+sin²(λ/2)+2sin(λ/2)cos(λ/2))/cos(λ))
> = tan(α)ln((1+sin(λ))/cos(λ))
> = tan(α)ln(sec(λ)+tan(λ))
> e^φ = e^(tan(α)ln(sec(λ)+tan(λ)))
> = (sec(λ)+tan(λ))^tan(α)
> When λ = gudermannian(φ), sec(λ) = cosh(φ) and tan(λ) = sinh(φ). Then:
> e^φ = (cosh(φ)+sinh(φ))^tan(α)
> = (e^φ)^tan(α)
> tan(α) = 1
> α = π/4
> The 45 degree loxodrome has coordinates which are related by the gudermannian function. To say 1 is physical, and the other is not, is contradictory, because they are both properties of the same, single radius vector. Special relativity maps relative velocity to latitude angle. As a gudermannian, it also maps a corresponding Lorentz boost to the longitude angle. Consequently, all Lorentz boosts are associated with a phase rotation. The phase rotation is not a hyperbolic rotation, because that is rapidity. It is not rotation around any of the space axes, either, because we use inertial frames.. It's a rotation into complex dimensions. Or higher dimensions. Whatever. There's an isomorphism between hypercomplex numbers and ordered n-tuples of real coordinates, so it's all the same. It's a rotation out of the box that special relativity confines us to. Some people will reflexively put this in the category of pink unicorns. The geometry identifies the source of the excess relativistic momentum of high speed particles to be imaginary velocity components, not relativistic mass. And to the smart-asses out there smirking, that's mathematically imaginary (thanks to Descartes sneering at complex numbers). The i*i = -1 kind of imaginary. The momentum that it carries is physical enough when the particle inelastically collides with a target. And, it isn't wishful thinking or an analogy. It is merely the sine projection of a vector, following the rules of geometry. Special relativity simply discards this dimension, and invents self-contradictory fairy tales to rationalize the cut. More details to follow.
>
> The irony of the Mercator map is that Greenland appears to be larger than Australia because of the same transform that fooled physicists into believing in relativistic mass. It is an illusion of the transform, the same transform, in both cases. Einstein, himself, was an amateur sailor. I wonder how many times he stared at Mercator's map, without realizing that his special relativity was based on it, an artifact from 3 centuries before he was born.

Gobbledygook.

--
Jan

Sorry, Jan. Haven't got the hang of this editor yet. This reply is for Townes Olson:

This is the claim:
In 1911 Vladimir Varićak asserted that one sees the length contraction in an objective way, according to Lorentz, while it is "only an apparent, subjective phenomenon, caused by the manner of our clock-regulation and length-measurement", according to Einstein.

Einstein's rebuttal:
The author unjustifiably stated a difference of Lorentz's view and that of mine concerning the physical facts. The question as to whether length contraction really exists or not is misleading. It doesn't "really" exist, in so far as it doesn't exist for a comoving observer; though it "really" exists, i.e. in such a way that it could be demonstrated in principle by physical means by a non-comoving observer.[22]
— Albert Einstein, 1911

In Einstein's rebuttal he cites the comoving frame and a non-comoving frame.. He acknowledges that the perception of length contraction is different in the two frames. But he affirms that it “really” exists in the non-comoving frame. I suppose it did not occur to him that his own principle of relativity would make the length contraction just as real for a second non-comoving observer. Just as real, but not equal in magnitude. This is a contradiction. One that is erased by double-talk about objective and subjective reality. Each observer's observations are objective to the observer, but ONLY to that observer. Each observer has their own subjective reality. Not to the extent that it is in quantum mechanics, where the observer influences the outcome, but enough to isolate each observer from every other observer, so that they all can make different measurements at the same time. While this may satisfy some narrow definition of logic, it isn't reality.. Reality is not a Java program with an unlimited number of instances of a class. While there can be an arbitrary number of observers and an assortment of different measurements, there is only one, common target of observation.

Worse for the cultists, eigenvector decomposition takes any observer's raw measurements, and produces an invariant magnitude in the reference frame out of them. It does so regardless of the relative velocity of the moving frame that the raw data came from. This is a reality that all moving observers agree on. The inescapable conclusion is that Einstein was wrong. Length contraction is not real. It is an illusion. As an illusion, any number of differently moving observers can get different measurements of the same object.. We just have to get used to the fact that illusions can be measured. Quite simply, reality is an illusion, too. The difference between reality and relativity is that the phase angle of relativity is not 0 like it is in reality. At 0 relative velocity, the phase angle is also 0, and the projection cosine is 1. The result, in Newtonian physics, is WYSIWYG. Like Newtonian physics itself, it is not valid outside the realm of v << c. At any relative velocity, the phase angle is not 0, and the projection cosine is less than 1. The result is the twin illusions of relativistic time dilation and length contraction.

I posed the question explicitly to Don Lincoln, of YouTube fame. I asked him point blank what happens to the moving meter stick. He dodged that question, instead answering his own question. He removed the meter stick from the scenario and claimed that the space between where the ends of the meter stick “used to be” was contracted. I wanted to know where he stood on the physical nature of length contraction. Inadvertently, he had given me a definition of a displacement, not a distance, and displacement is the cosine projection of total distance. The reason I asked was the argument I use is different in both cases. If the effect is physical, the principle of relativity results in a contradiction. In that case, relativity is patently wrong. If the effect is an illusion, then I don't have to demonstrate that special relativity is wrong, I merely have to demonstrate that geometric relativity is better. Before I respond further, which side of the argument are you on?

To you and all the other glib one-liners, I refuse to respond to such frivolous comments. Where are all the real physicists? I provide inordinate detail to encourage explicit replies, and all I get are wisecracks. I have more important things to do than indulge schoolyard arguments like "Is not!" "Is too!". Maybe you should all learn some mathematics before commenting on it.

On Thursday, October 21, 2021 at 7:19:22 AM UTC-7, tgca...@gmail.com wrote:
> I provide inordinate detail to encourage explicit replies, and all I get are wisecracks.

No, what you get is responses exposing you as a crank

On Thursday, 21 October 2021 at 16:24:43 UTC+2, Dono. wrote:
> On Thursday, October 21, 2021 at 7:19:22 AM UTC-7, tgca...@gmail.com wrote:
> > I provide inordinate detail to encourage explicit replies, and all I get are wisecracks.
> No, what you get is responses exposing you as a crank

In the meantime in the real world, however, GPS clcoks keep
measuring t'=t, just like all serious clocks always did.

Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
> To you and all the other glib one-liners, I refuse to respond to such
> frivolous comments. Where are all the real physicists? I provide
> inordinate detail to encourage explicit replies, and all I get are
> wisecracks. I have more important things to do than indulge
> schoolyard arguments like "Is not!" "Is too!". Maybe you should all
> learn some mathematics before commenting on it.
>

Maybe you should realise that physics is not just an
exercise in algebra. If you don't understand the physical
meanings of the variables in the equations, you are a
hopeless victim of your own self-inflicted ignorance.
Thinking that you can either learn something or teach
someone here, is a waste of your own time.

Dirk Vdm

On Thursday, October 21, 2021 at 11:31:01 AM UTC-4, Dirk Van de moortel wrote:
> Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
> > To you and all the other glib one-liners, I refuse to respond to such
> > frivolous comments. Where are all the real physicists? I provide
> > inordinate detail to encourage explicit replies, and all I get are
> > wisecracks. I have more important things to do than indulge
> > schoolyard arguments like "Is not!" "Is too!". Maybe you should all
> > learn some mathematics before commenting on it.
> >
> Maybe you should realise that physics is not just an
> exercise in algebra. If you don't understand the physical
> meanings of the variables in the equations, you are a
> hopeless victim of your own self-inflicted ignorance.
> Thinking that you can either learn something or teach
> someone here, is a waste of your own time.
>
> Dirk Vdm

More worthless opinions. I have found, over the last few years that all my critics fall into basically 3 categories. Pronouncements from authority (mindless regurgitation of dogma), strawman arguments (references to the rules of Minkowski geometry are irrelevant in a discussion of Euclidean geometry), and of course, the ad hominem attacks from those who have no logical comments to make. Apparently, teaching you anything is a waste of time. That hardly means that every other reader is as biased.

On Thursday, October 21, 2021 at 9:28:34 AM UTC-7, tgca...@gmail.com wrote:
> On Thursday, October 21, 2021 at 11:31:01 AM UTC-4, Dirk Van de moortel wrote:
> > Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
> > > To you and all the other glib one-liners, I refuse to respond to such
> > > frivolous comments. Where are all the real physicists? I provide
> > > inordinate detail to encourage explicit replies, and all I get are
> > > wisecracks. I have more important things to do than indulge
> > > schoolyard arguments like "Is not!" "Is too!". Maybe you should all
> > > learn some mathematics before commenting on it.
> > >
> > Maybe you should realise that physics is not just an
> > exercise in algebra. If you don't understand the physical
> > meanings of the variables in the equations, you are a
> > hopeless victim of your own self-inflicted ignorance.
> > Thinking that you can either learn something or teach
> > someone here, is a waste of your own time.
> >
> > Dirk Vdm
> More worthless opinions. I have found, over the last few years that all my critics fall into basically 3 categories. Pronouncements from authority (mindless regurgitation of dogma), strawman arguments (references to the rules of Minkowski geometry are irrelevant in a discussion of Euclidean geometry), and of course, the ad hominem attacks from those who have no logical comments to make. Apparently, teaching you anything is a waste of time. That hardly means that every other reader is as biased.

You forgot the main category: the one that rubs your nose in your shit by exposing you as a crank

On Thursday, October 21, 2021 at 12:30:42 PM UTC-4, Dono. wrote:
> On Thursday, October 21, 2021 at 9:28:34 AM UTC-7, tgca...@gmail.com wrote:
> > On Thursday, October 21, 2021 at 11:31:01 AM UTC-4, Dirk Van de moortel wrote:
> > > Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
> > > > To you and all the other glib one-liners, I refuse to respond to such
> > > > frivolous comments. Where are all the real physicists? I provide
> > > > inordinate detail to encourage explicit replies, and all I get are
> > > > wisecracks. I have more important things to do than indulge
> > > > schoolyard arguments like "Is not!" "Is too!". Maybe you should all
> > > > learn some mathematics before commenting on it.
> > > >
> > > Maybe you should realise that physics is not just an
> > > exercise in algebra. If you don't understand the physical
> > > meanings of the variables in the equations, you are a
> > > hopeless victim of your own self-inflicted ignorance.
> > > Thinking that you can either learn something or teach
> > > someone here, is a waste of your own time.
> > >
> > > Dirk Vdm
> > More worthless opinions. I have found, over the last few years that all my critics fall into basically 3 categories. Pronouncements from authority (mindless regurgitation of dogma), strawman arguments (references to the rules of Minkowski geometry are irrelevant in a discussion of Euclidean geometry), and of course, the ad hominem attacks from those who have no logical comments to make. Apparently, teaching you anything is a waste of time. That hardly means that every other reader is as biased.
> You forgot the main category: the one that rubs your nose in your shit by exposing you as a crank

I see no point in engaging crackpot skeptics.

On Thursday, October 21, 2021 at 9:36:32 AM UTC-7, tgca...@gmail.com wrote:
> On Thursday, October 21, 2021 at 12:30:42 PM UTC-4, Dono. wrote:
> > On Thursday, October 21, 2021 at 9:28:34 AM UTC-7, tgca...@gmail.com wrote:
> > > On Thursday, October 21, 2021 at 11:31:01 AM UTC-4, Dirk Van de moortel wrote:
> > > > Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
> > > > > To you and all the other glib one-liners, I refuse to respond to such
> > > > > frivolous comments. Where are all the real physicists? I provide
> > > > > inordinate detail to encourage explicit replies, and all I get are
> > > > > wisecracks. I have more important things to do than indulge
> > > > > schoolyard arguments like "Is not!" "Is too!". Maybe you should all
> > > > > learn some mathematics before commenting on it.
> > > > >
> > > > Maybe you should realise that physics is not just an
> > > > exercise in algebra. If you don't understand the physical
> > > > meanings of the variables in the equations, you are a
> > > > hopeless victim of your own self-inflicted ignorance.
> > > > Thinking that you can either learn something or teach
> > > > someone here, is a waste of your own time.
> > > >
> > > > Dirk Vdm
> > > More worthless opinions. I have found, over the last few years that all my critics fall into basically 3 categories. Pronouncements from authority (mindless regurgitation of dogma), strawman arguments (references to the rules of Minkowski geometry are irrelevant in a discussion of Euclidean geometry), and of course, the ad hominem attacks from those who have no logical comments to make. Apparently, teaching you anything is a waste of time. That hardly means that every other reader is as biased.
> > You forgot the main category: the one that rubs your nose in your shit by exposing you as a crank
> I see no point in engaging crackpot skeptics.
You got one right: we are the skeptics, you are the crackpot

On Thursday, October 21, 2021 at 12:38:47 PM UTC-4, Dono. wrote:
> On Thursday, October 21, 2021 at 9:36:32 AM UTC-7, tgca...@gmail.com wrote:
> > On Thursday, October 21, 2021 at 12:30:42 PM UTC-4, Dono. wrote:
> > > On Thursday, October 21, 2021 at 9:28:34 AM UTC-7, tgca...@gmail.com wrote:
> > > > On Thursday, October 21, 2021 at 11:31:01 AM UTC-4, Dirk Van de moortel wrote:
> > > > > Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
> > > > > > To you and all the other glib one-liners, I refuse to respond to such
> > > > > > frivolous comments. Where are all the real physicists? I provide
> > > > > > inordinate detail to encourage explicit replies, and all I get are
> > > > > > wisecracks. I have more important things to do than indulge
> > > > > > schoolyard arguments like "Is not!" "Is too!". Maybe you should all
> > > > > > learn some mathematics before commenting on it.
> > > > > >
> > > > > Maybe you should realise that physics is not just an
> > > > > exercise in algebra. If you don't understand the physical
> > > > > meanings of the variables in the equations, you are a
> > > > > hopeless victim of your own self-inflicted ignorance.
> > > > > Thinking that you can either learn something or teach
> > > > > someone here, is a waste of your own time.
> > > > >
> > > > > Dirk Vdm
> > > > More worthless opinions. I have found, over the last few years that all my critics fall into basically 3 categories. Pronouncements from authority (mindless regurgitation of dogma), strawman arguments (references to the rules of Minkowski geometry are irrelevant in a discussion of Euclidean geometry), and of course, the ad hominem attacks from those who have no logical comments to make. Apparently, teaching you anything is a waste of time.. That hardly means that every other reader is as biased.
> > > You forgot the main category: the one that rubs your nose in your shit by exposing you as a crank
> > I see no point in engaging crackpot skeptics.
> You got one right: we are the skeptics, you are the crackpot

That's funny. A skeptic who can't present a logical counterargument. Don't like the message, attack the messenger. The rest of this space intentionally left blank.

On Thursday, October 21, 2021 at 9:43:49 AM UTC-7, tgca...@gmail.com wrote:
> On Thursday, October 21, 2021 at 12:38:47 PM UTC-4, Dono. wrote:
> > On Thursday, October 21, 2021 at 9:36:32 AM UTC-7, tgca...@gmail.com wrote:
> > > On Thursday, October 21, 2021 at 12:30:42 PM UTC-4, Dono. wrote:
> > > > On Thursday, October 21, 2021 at 9:28:34 AM UTC-7, tgca...@gmail.com wrote:
> > > > > On Thursday, October 21, 2021 at 11:31:01 AM UTC-4, Dirk Van de moortel wrote:
> > > > > > Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
> > > > > > > To you and all the other glib one-liners, I refuse to respond to such
> > > > > > > frivolous comments. Where are all the real physicists? I provide
> > > > > > > inordinate detail to encourage explicit replies, and all I get are
> > > > > > > wisecracks. I have more important things to do than indulge
> > > > > > > schoolyard arguments like "Is not!" "Is too!". Maybe you should all
> > > > > > > learn some mathematics before commenting on it.
> > > > > > >
> > > > > > Maybe you should realise that physics is not just an
> > > > > > exercise in algebra. If you don't understand the physical
> > > > > > meanings of the variables in the equations, you are a
> > > > > > hopeless victim of your own self-inflicted ignorance.
> > > > > > Thinking that you can either learn something or teach
> > > > > > someone here, is a waste of your own time.
> > > > > >
> > > > > > Dirk Vdm
> > > > > More worthless opinions. I have found, over the last few years that all my critics fall into basically 3 categories. Pronouncements from authority (mindless regurgitation of dogma), strawman arguments (references to the rules of Minkowski geometry are irrelevant in a discussion of Euclidean geometry), and of course, the ad hominem attacks from those who have no logical comments to make. Apparently, teaching you anything is a waste of time. That hardly means that every other reader is as biased.
> > > > You forgot the main category: the one that rubs your nose in your shit by exposing you as a crank
> > > I see no point in engaging crackpot skeptics.
> > You got one right: we are the skeptics, you are the crackpot
> That's funny. A skeptic who can't present a logical counterargument.

I did, I pointed out the imbecilities you spout. Of course, you, in typical crank fashion, deny that the imbecilities were pointed out.

On Thursday, October 21, 2021 at 12:48:38 PM UTC-4, Dono. wrote:
> On Thursday, October 21, 2021 at 9:43:49 AM UTC-7, tgca...@gmail.com wrote:
> > On Thursday, October 21, 2021 at 12:38:47 PM UTC-4, Dono. wrote:
> > > On Thursday, October 21, 2021 at 9:36:32 AM UTC-7, tgca...@gmail.com wrote:
> > > > On Thursday, October 21, 2021 at 12:30:42 PM UTC-4, Dono. wrote:
> > > > > On Thursday, October 21, 2021 at 9:28:34 AM UTC-7, tgca...@gmail.com wrote:
> > > > > > On Thursday, October 21, 2021 at 11:31:01 AM UTC-4, Dirk Van de moortel wrote:
> > > > > > > Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
> > > > > > > > To you and all the other glib one-liners, I refuse to respond to such
> > > > > > > > frivolous comments. Where are all the real physicists? I provide
> > > > > > > > inordinate detail to encourage explicit replies, and all I get are
> > > > > > > > wisecracks. I have more important things to do than indulge
> > > > > > > > schoolyard arguments like "Is not!" "Is too!". Maybe you should all
> > > > > > > > learn some mathematics before commenting on it.
> > > > > > > >
> > > > > > > Maybe you should realise that physics is not just an
> > > > > > > exercise in algebra. If you don't understand the physical
> > > > > > > meanings of the variables in the equations, you are a
> > > > > > > hopeless victim of your own self-inflicted ignorance.
> > > > > > > Thinking that you can either learn something or teach
> > > > > > > someone here, is a waste of your own time.
> > > > > > >
> > > > > > > Dirk Vdm
> > > > > > More worthless opinions. I have found, over the last few years that all my critics fall into basically 3 categories. Pronouncements from authority (mindless regurgitation of dogma), strawman arguments (references to the rules of Minkowski geometry are irrelevant in a discussion of Euclidean geometry), and of course, the ad hominem attacks from those who have no logical comments to make. Apparently, teaching you anything is a waste of time. That hardly means that every other reader is as biased.
> > > > > You forgot the main category: the one that rubs your nose in your shit by exposing you as a crank
> > > > I see no point in engaging crackpot skeptics.
> > > You got one right: we are the skeptics, you are the crackpot
> > That's funny. A skeptic who can't present a logical counterargument.
> I did, I pointed out the imbecilities you spout. Of course, you, in typical crank fashion, deny that the imbecilities were pointed out.

Claiming that I am a crank and proving it are not the same thing. Only a crackpot skeptic is so self-important as to think his pronouncements have any logical impact. If my argument is not logical, then it will lead to a contradiction. Instead of insults, present a logical counter-argument. Oooh! That's right. You can't. At least, you won't. You're dismissed.

On Thursday, October 21, 2021 at 9:58:26 AM UTC-7, tgca...@gmail.com wrote:
> On Thursday, October 21, 2021 at 12:48:38 PM UTC-4, Dono. wrote:
> > On Thursday, October 21, 2021 at 9:43:49 AM UTC-7, tgca...@gmail.com wrote:
> > > On Thursday, October 21, 2021 at 12:38:47 PM UTC-4, Dono. wrote:
> > > > On Thursday, October 21, 2021 at 9:36:32 AM UTC-7, tgca...@gmail.com wrote:
> > > > > On Thursday, October 21, 2021 at 12:30:42 PM UTC-4, Dono. wrote:
> > > > > > On Thursday, October 21, 2021 at 9:28:34 AM UTC-7, tgca...@gmail.com wrote:
> > > > > > > On Thursday, October 21, 2021 at 11:31:01 AM UTC-4, Dirk Van de moortel wrote:
> > > > > > > > Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
> > > > > > > > > To you and all the other glib one-liners, I refuse to respond to such
> > > > > > > > > frivolous comments. Where are all the real physicists? I provide
> > > > > > > > > inordinate detail to encourage explicit replies, and all I get are
> > > > > > > > > wisecracks. I have more important things to do than indulge
> > > > > > > > > schoolyard arguments like "Is not!" "Is too!". Maybe you should all
> > > > > > > > > learn some mathematics before commenting on it.
> > > > > > > > >
> > > > > > > > Maybe you should realise that physics is not just an
> > > > > > > > exercise in algebra. If you don't understand the physical
> > > > > > > > meanings of the variables in the equations, you are a
> > > > > > > > hopeless victim of your own self-inflicted ignorance.
> > > > > > > > Thinking that you can either learn something or teach
> > > > > > > > someone here, is a waste of your own time.
> > > > > > > >
> > > > > > > > Dirk Vdm
> > > > > > > More worthless opinions. I have found, over the last few years that all my critics fall into basically 3 categories. Pronouncements from authority (mindless regurgitation of dogma), strawman arguments (references to the rules of Minkowski geometry are irrelevant in a discussion of Euclidean geometry), and of course, the ad hominem attacks from those who have no logical comments to make. Apparently, teaching you anything is a waste of time. That hardly means that every other reader is as biased.
> > > > > > You forgot the main category: the one that rubs your nose in your shit by exposing you as a crank
> > > > > I see no point in engaging crackpot skeptics.
> > > > You got one right: we are the skeptics, you are the crackpot
> > > That's funny. A skeptic who can't present a logical counterargument.
> > I did, I pointed out the imbecilities you spout. Of course, you, in typical crank fashion, deny that the imbecilities were pointed out.
> Claiming that I am a crank and proving it are not the same thing.

Experience says that one can never prove to a crank (i.e. you) that he is a crank. So keep basking in your imbecility.

Dirk Van de moortel wrote:

> Maybe you should realise that physics is not just an exercise in
> algebra. If you don't understand the physical meanings of the variables
> in the equations, you are a hopeless victim of your own self-inflicted
> ignorance. Thinking that you can either learn something or teach someone
> here, is a waste of your own time.

your government is illegitimate. It's capitalist. Your money are in their
bank accounts, they use to kill you.

Op 21-okt.-2021 om 18:28 schreef Tom Capizzi:
> On Thursday, October 21, 2021 at 11:31:01 AM UTC-4, Dirk Van de moortel wrote:
>> Op 21-okt.-2021 om 16:19 schreef Tom Capizzi:
>>> To you and all the other glib one-liners, I refuse to respond to such
>>> frivolous comments. Where are all the real physicists? I provide
>>> inordinate detail to encourage explicit replies, and all I get are
>>> wisecracks. I have more important things to do than indulge
>>> schoolyard arguments like "Is not!" "Is too!". Maybe you should all
>>> learn some mathematics before commenting on it.
>>>
>> Maybe you should realise that physics is not just an
>> exercise in algebra. If you don't understand the physical
>> meanings of the variables in the equations, you are a
>> hopeless victim of your own self-inflicted ignorance.
>> Thinking that you can either learn something or teach
>> someone here, is a waste of your own time.
>>
>> Dirk Vdm
>
> More worthless opinions. I have found, over the last few years that
> all my critics fall into basically 3 categories. Pronouncements from
> authority (mindless regurgitation of dogma), strawman arguments
> (references to the rules of Minkowski geometry are irrelevant in a
> discussion of Euclidean geometry), and of course, the ad hominem
> attacks from those who have no logical comments to make. Apparently,
> teaching you anything is a waste of time. That hardly means that
> every other reader is as biased.
>

You had one shot at properly replying to my first comment and
the painful spectacle of your blowing it does not deserve another
shot. You are effectively unarmed.

Dirk Vdm