### 2 Deriving Velji Morph Equations (regarding Captured Lorentz factor)

[ (-) All the equations in the this section are inward bound. ]

In this section we analyze morph effects imposed on the outsider, who observers an insider.  The outsider is traveling at velocity relative to the insider, who may be accelerating relative to a proper insider.

In the previous sections in Time Dilation and Length Contraction Thought-Experiments, we came up with the equations time dilation and length contraction.

>          lOlI/γc

• where γc is a Captured Lorentz factor for the outsider
lO,lI are lengths measured by the outsider and insider

>          tOγc*tI

• where γc is a Captured Lorentz factor for the outsider
tO,tI are times measured by the outsider and insider

We will now "liberate" (and re-use) Einstein's Time Dilation and Length Contraction Thought-Experiments because they are correct, when we have use the Einstein Assumption:

The Einstein Assumption assumes that given a set of observers, the set is at peace with itself, that is, all observers agree with the center of force points, cfps.  In particular, one observer is a proper observer and the others are improper observers.

Thus we have 3 caveats to get Einstein's Morph Effect Thought-Experiments to work:
1.  the insider and outsider exist in a peaceful system,
2.  the insider is a proper observer and the outsider is some given observer, and
3.  the times and lengths within those thought-experiments were all aderived.
Thus, we have two correct equations, one for aderived time dilation, the other for aderived length contraction.

>(dLc)=(Aderived Length Contraction for Captured Lorentz factor)
>          dlO = dlI / γcO

• where γcO = 1/(1-vcO²/c²)^½ is the captured Lorentz factor by peaceful outsider
vcO is the captured Lorentz velocity by peaceful outsider
dlI is an aderived length by in-frame insider
dlO is an aderived length by peaceful outsider

>(dTc)=(Aderived Time Dilation for Captured Lorentz factor)
>          dtO = γcO * dtI

• where γcO = 1/(1-vcO²/c²)^½ is the captured Lorentz factor by peaceful outsider
vcO is the captured Lorentz velocity by peaceful outsider
dtI is an aderived time by in-frame insider
dtO is an aderived time by peaceful outsider

We now discuss exactly how to "liberate" Einstein's TDLC Thought-Experiments here updating it to the Velji's TDLC Thought-Experiment for Captured Lorentz Factors.

Velji's Morph Thought-Experiment for Captured Lorentz Factors

Let us do this experiment in space instead, this time in cubes, not trains on the Earth as Einstein had.  (Recall a cube is a 1m-by-1m-by-1m empty box.)  Let one cube be inhabited by an insider, while the other cube contain an outsider.  Let the insider have unity Lorentz factors, while the outsider is traveling at velocity v relative to the insider, and has general Lorentz factor γ.  Let us assume that his escape Lorentz factor is 1 so that it does not contribute anything to the following calculations.  Let us do two thought-experiments at once, that is, we have 2 two-way light-clocks, both within the insider's cube.  We will call the light-clock that derives time dilation, the Time Dilation light-clock, while the other will be called the Length Contraction light-clock.  Let the vector X be the vector from the center of the insider's cube to the center of the outsider's cube.

In the standard setup, the Time Dilation light-clock and the Length Contraction light-clock is set up so that they lie in a plane that is orthogonal to X, thus making X a normal vector to the plane.  Let us call this plane PThus, the Time Dilation light-clock is set up perpendicular to the Length Contraction light-clock.  Let us name these directions to have unit vectors A^ and B^ respectively.

Now, we can do both thought-experiments in tandem.  Notice that when both experiments are done in tandem they all agree with their proper lengths, proper times and proper velocities, i.e. dlp, dtp, dl, dt, are the same for all experiments.  Why is this true?  Because they are all at rest with the proper observer, the insider, and we believe in a Trickster System.

(In the end, we will see that the fact that time dilates makes it possible for light to be measured at the speed c within the Time Dilation light-clock!  The fact that length contracts makes it possible, in the end, for light to be measured at the speed of c within the Length Contraction light-clock!)

We've been considering the light-clocks as being at rest while the outsider is in motion.  I'd like to switch the two, but we *cannot* readily do this even though velocities are relative and invariant.  This is because the insider is at proper rest with the light-clock, that is, it has a general Lorentz factor of unity.  If we state that the light-clock is moving then the light-clock will have TDLC effects relative to the insider who is still at rest with the proper frame.

So, the improper outsider is in motion.  Let's consider the velocity V of the improper outsider to have two (unit) component vectors A^ and
B^.  Defined as above, A^ (parallel to v) and B^ (perpendicular to v) define the plane P orthogonal to X (because A^ and B^ are perpendicular in the plane P).  Thus, these two vectors span our two-dimensional plane P.  Thus, we can write V := aA^ + bB^
, where a and b are some velocities in their respective direction.

So, the two LC and TD light-clocks appear to be moving at velocities aA^, bB^ relative to the improper outsider, respectively.  One can imagine that the component of V in the A^ direction (namely a) is sent to the Length Contraction Thought-Experiment, i.e. let v := a in that thought-experiment.  Meanwhile, the component of V in the B^ direction (namely b) is sent to the Time Dilation Thought-Experiment, i.e. let v := b in that thought-experiment.

Now, of course when the position of the outsider changes, the angles of the three light-clocks potentially change.  Thus, one is required to rotate the light-clocks appropriately just so that the standard setup (defined above) can be maintained.

Note that the fact that the Length Contraction light-clock is in the direction A^ means that only that particular direction of lengths are contracted.  Also, the fact that time is needed to dilate satisfies the Time Dilation light-clock.  The fact that it dilates causes no qualms with the other light-clock.  Thus, with these TDLC effects we now have that the outside improper observer still manages to witness the speed of light to be the constant c no matter that the outsider is traveling at a velocity V relative to the in-frame insider.

Thus, using Einstein's two TDLC Thought-Experiments and with everything above, we have derived the captured Lorentz factor in the Velji equations for TDLC.  Those two thought-experiments somewhat corroborate Einstein's misguided Speed of Light Postulate.   It is wrong, yes, but, as we will discuss below, can one literally *prove* that it is wrong with actual measurements?  We discuss that in a following section, The Apparent Constancy of the Speed of Light Explained

Deriving the Captured Velji Equation for Aderived Distance

Now, when an outsider sees aderived length contract, that means he will see aderived distances augment!  For, consider an outsider who holds a ruler in front of himself and looks at his Universe.  The outsiders length has contracted and thus, the ruler will be contracted.  Thus, when he tries to use his ruler to measure the insider he will see the distances to have augmented because his own rulers have contracted.

So, because at high Lorentz velocities lengths contract, we have that at high Lorentz velocities distance augments (for free!).

Thus, we have a correct equation for distance augmentation:

>(dXc)=(Aderived Distance Augmentation for Captured Lorentz factor)
>          dxO =
γcO * dxI

• where γcO = 1/(1-vcO²/c²)^½ is the captured Lorentz factor by peaceful outsider
vcO is the captured Lorentz velocity by peaceful outsider
dxI is an aderived distance by in-frame insider
dxO is an aderived distance by peaceful outsider

Ratifying the Equations

If we state that outsider's aderived length contracts then, when we allow outsider's ameasured length to augment then, the two equations for lengths have been (what we will now call) "ratified".  With ratified equations the equations can be satisfied by both parties, in this case, the outsider and insider.

From above, aderived lengths contract, we have:

>(1)       dlO = dlI / γcO

We assume this to ratify the equations, that ameasured lengths augment, we have:

>(2)
mlO = γcO * mlI

Observe:

Starting with outsiders:

>          dlO * mlO = (dlI/γcO) * cO*mlI) = dlI * mlI

Starting with insiders:

>
dlI * mlI = cO*dlO) * (ml1/γcO) = dlO * mlO

Ratifying the equations for length one gets:

>          dlO*mlO = dlI*mlI.

When we have this set of equations satisfied in this manner, we say that the equations insider and outsider have been    "ratified".  Thus, when we have equations ratified we can sit back and observe the Universe using either the equations for (dlI,mlI) or the equations for (dlO,mlO).

So notice now that we have, alongside the aderived quantities dL, dT, dX, we now include the ameasured counterparts mLmTmX, which successfully ratify all captured morph equations.

>(mLc)=(Ameasured Length Augmentation for Captured Lorentz factor)
>          mlO = γcO * mlI

• where γcO = 1/(1-vcO²/c²)^½ is the captured Lorentz factor by peaceful outsider
vcO is the captured Lorentz velocity by peaceful outsider
mlI is an ameasured length by in-frame insider
mlO is an ameasured length by peaceful outsider

>(mTc)=(Ameasured Time Constriction for Captured Lorentz factor)
>          mtO = mtI / γcO

• where γcO = 1/(1-vcO²/c²)^½ is the captured Lorentz factor by peaceful outsider
vcO is the captured Lorentz velocity by peaceful outsider
mtI is an ameasured times by in-frame insider
mtO is an ameasured time by peaceful outsider

>(mXc)=(Ameasured Distance Contraction for Captured Lorentz factor)
>          mxO = mxI / γcO

• where γcO = 1/(1-vcO²/c²)^½ is the captured Lorentz factor by peaceful outsider
vcO is the captured Lorentz velocity by peaceful outsider
mxI is an ameasured distance by in-frame insider
mxO is an ameasured distance by peaceful outsider

• As the outsider's vc tends to 0, mtO tends to mtI, and mxO tends to mxI
• As the outsider's vc tends to c, mtO tends to zero and mxO tends to zero
Because ameasured time constricts and ameasured distance contracts means, as Relativist's say, "moving clocks run slow" and "moving rods contract".

Make note that for Velji's Morph Thought-Experiment for captured Lorentz factors, the outsider experiences morph effects, both aderived and ameasured, whereas the insider is "immune" and so does not require an inclusion of captured Lorentz factors.

#### Using Captured Essen Factors

Let's consider two improper observers with general Lorentz factors γ1 and γ2We use Essen factors when comparing two arbitrary frames of reference that are peaceful non-utopias.  In a sense, the first improper observer has to measure her times and lengths to un-dilate and un-contract by the factor γ1 with respect to their proper observer.  The proper observer then passes these times and lengths to dilate and contract at the rate γ2 for the second improper observer.  So:

>          dl2 = Γc12 dl1
>          ml2 = Γc21 ml1

>          dt2 = Γc21 dt1
>          mt2 = Γc12 mt1

>          dx2 = Γc21 dx1
>          mx2 = Γc12 mx1

• where Γc12 is the captured Essen factor "from 1 to 2" ( Γc12 γc1c2 )
Γc21 is the captured Essen factor "from 2 to 1" ( Γc21 γc2c1 )

γc1 = 1/(1-vc1²/c²)^½ is a captured Lorentz factor by peaceful observer 1
γc2 = 1/(1-vc2²/c²)^½ is a captured Lorentz factor by peaceful observer 2
vc1 is the captured Lorentz velocity by peaceful observer 1

vc2 is the captured Lorentz velocity by peaceful observer 2
ml
1,dl1 is an ameasured/aderived improper lengths by
peaceful observer 1

mt1,dt1 is an ameasured/aderived improper times by peaceful observer 1

mx1,dx1 is an ameasured/aderived improper distances by peaceful observer 1

ml2,dl2 is an ameasured/aderived improper lengths by peaceful observer 2

mt2,dt2 is an ameasured/aderived improper times by peaceful observer 2

mx2,dx2 is an ameasured/aderived improper distances by peaceful observer 2

• if ΓAB < 1, then the Essen factor is said to be "slow"
• if ΓAB = 1, then the Essen factor is said to be "constant"
• if 1 < ΓAB, then the Essen factor is said to be "fast"

Using General Lorentz Factors

From the previous section:

>          dtI = γeI * dtP
>          mtI = mtP / γeI

• where γeI = 1/(1-gw²/c²) is the escape Lorentz factor for the insider
dtP is aderived time for the proper insider

dtI is aderived time for the insider
mtP is ameasured time for the proper insider

mtI is ameasured time for the insider

From this section:

>          dtO = γcO * dtI
>
mtO = mtI /
γcO

• where γcO is the captured Lorentz factor for a peaceful outsider
dtI is an aderived time by an in-frame insider
dtO is an aderived time by a peaceful outsider
mtI is an ameasured time by an in-frame insider
mtO is an ameasured time by a peaceful outsider

Thus, when we consider time, we can use general Lorentz factors γg for the outsider.  Observe:

>          dtO = γcO * dtI
>             = γcO * (γeI * dtP)
>             = cO * γeI) * dtP
>             = γg * dtP

>
mtO = mtI /
γcO
>             = (mtP / γeI) / γcO
>             = mtP / cO * γeI)
>             = mtP / γg

• where γg = γeI * γcO is the general Lorentz factor by outsider
γeI is the escape Lorentz factor by insider
γcO is the escape Lorentz factor by outsider