[ (-) 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. Deriving the Captured Velji Equations for Aderived Length, Aderived Time, and Aderived Velocity In the previous sections in Time Dilation and Length Contraction Thought-Experiments, we came up with the equations time dilation and length contraction. > l _{O} = l_{I}/γ_{c}• where γ _{c} is a Captured Lorentz factor for the outsider l _{O},l_{I} are lengths measured by the outsider and insider> t _{O} = γ_{c}*t_{I}_{c} is a Captured Lorentz factor for the outsider t _{O},t_{I} are times measured by the outsider and insiderWe 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.-
__the insider and outsider exist in a peaceful system,__ -
__the insider is a proper observer and the outsider is some given observer,__and -
__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. >(_{d}L_{c})=(Aderived Length Contraction for Captured Lorentz factor)> _{d}l_{O} = / γ_{d}l_{I}_{cO}• where γ _{cO} = 1/(1-v_{cO}²/c²)^½ is the captured Lorentz factor by peaceful outsider v _{cO} is the captured Lorentz velocity by peaceful outsider_{d}l_{I} is an aderived length by in-frame insider_{d}l_{O} is an aderived length by peaceful outsider>(_{d}T_{c})=(Aderived Time Dilation for Captured Lorentz factor)> _{d}t_{O} = γ * _{cO}_{d}t_{I}_{}• where γ _{cO} = 1/(1-v_{cO}²/c²)^½ is the captured Lorentz factor by peaceful outsider v _{cO} is the captured Lorentz velocity by peaceful outsider_{d}t_{I} is an aderived time by in-frame insider_{d}t_{O} is an aderived time by peaceful outsiderWe 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 P. Thus, 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. (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!)_{ d}l_{p}, _{d}t_{p}, _{d}l, _{d}t, 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.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 a A^, 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 ExplainedDeriving 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: >(_{d}X_{c})=(Aderived Distance Augmentation for Captured Lorentz factor)> _{d}x_{O} = γ * _{cO}_{d}x_{I}• where γ _{cO} = 1/(1-v_{cO}²/c²)^½ is the captured Lorentz factor by peaceful outsider v _{cO} is the captured Lorentz velocity by peaceful outsider_{d}x_{I} is an aderived distance by in-frame insider_{d}x_{O} is an aderived distance by peaceful outsiderRatifying 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) _{d}l_{O} = _{d}l_{I }/ γ_{cO}We assume this to ratify the equations, that ameasured lengths augment, we have: >(2) _{m}l_{O} = γ_{cO} * _{m}l_{I} Observe: Starting with outsiders: > _{d}l_{O} * _{m}l_{O} = (_{d}l_{I}/γ_{cO}) * (γ_{cO}*_{m}l_{I}) = _{d}l_{I} * _{m}l_{I}Starting with insiders: > _{d}l_{I} * _{m}l_{I} = (γ_{cO}*_{d}l_{O}) * (_{m}l_{1}/γ_{cO}) = _{d}l_{O} * _{m}l_{O}Ratifying the equations for length one gets: > _{d}l_{O}*_{m}l_{O} = _{d}l_{I}*_{m}l_{I}.ratified". Thus, when we have equations ratified we can sit back and observe the Universe using either the equations for (_{d}l_{I},_{m}l_{I}) or the equations for (_{d}l_{O},_{m}l_{O}). So notice now that we have, alongside the aderived quantities _{d}L, _{d}T, _{d}X, we now include the ameasured counterparts _{m}L, _{m}T, _{m}X, which successfully ratify all captured morph equations.>(_{m}L_{c})=(Ameasured Length Augmentation for Captured Lorentz factor)> _{m}l_{O} = γ * _{cO}_{m}l_{I}• where γ _{cO} = 1/(1-v_{cO}²/c²)^½ is the captured Lorentz factor by peaceful outsider v _{cO} is the captured Lorentz velocity by peaceful outsider_{m}l_{I} is an ameasured length by in-frame insider_{m}l_{O} is an ameasured length by peaceful outsider>(_{m}T_{c})=(Ameasured Time Constriction for Captured Lorentz factor)> _{m}t_{O} = _{m}t_{I} / γ_{cO}• where γ_{cO} = 1/(1-v_{cO}²/c²)^½ is the captured Lorentz factor by peaceful outsider v _{cO} is the captured Lorentz velocity by peaceful outsider_{m}t_{I} is an ameasured times by in-frame insider_{m}t_{O} is an ameasured time by peaceful outsider>(_{m}X_{c})=(Ameasured Distance Contraction for Captured Lorentz factor)> _{m}x_{O} = _{m}x_{I} / γ_{cO}• where γ _{cO} = 1/(1-v_{cO}²/c²)^½ is the captured Lorentz factor by peaceful outsider v _{cO} is the captured Lorentz velocity by peaceful outsider_{m}x_{I} is an ameasured distance by in-frame insider_{m}x_{O} is an ameasured distance by peaceful outsider- As the outsider's v
_{c}tends to 0,_{m}t_{O}tends to_{m}t_{I}, and_{m}x_{O}tends to_{m}x_{I} - As the outsider's v
_{c}tends to c,_{m}t_{O}tends to zero and_{m}x_{O}tends to zero
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 γ _{1} and γ_{2}. We
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 γ_{1} with respect to their proper observer. The proper observer then passes these times and lengths to dilate and contract at the rate γ_{2} for the second improper observer. So:> _{d}l_{2} = Γ_{c12 d}l_{1}> _{m}l_{2} = Γ_{c21 m}l_{1}_{d}t_{2} = Γ_{c21 d}t_{1}> _{m}t_{2} = Γ_{c12 m}t_{1}_{d}x_{2} = Γ_{c21 d}x_{1}> _{m}x_{2} = Γ_{c12 m}x_{1}• where Γ _{c12} is the captured Essen factor "from 1 to 2" ( Γ_{c12 }= γ_{c1}/γ_{c2 }) Γ _{c21} is the captured Essen factor "from 2 to 1" ( Γ_{c21 }= γ_{c2}/γ_{c1 }) γ _{c2} is the captured Lorentz velocity by peaceful observer 2_{m}l_{1},_{d}l_{1} is an ameasured/aderived improper lengths by peaceful observer 1 _{m}l_{2},_{d}l_{2} is an ameasured/aderived improper lengths by peaceful observer 2 - if Γ
_{AB}< 1, then the Essen factor is said to be "slow" - if Γ
_{AB}= 1, then the Essen factor is said to be "constant" - if 1 < Γ
_{AB}, then the Essen factor is said to be "fast"
Using General Lorentz Factors From the previous section: > _{d}t_{I} = γ_{eI} * _{d}t_{P}> _{m}t_{I} = _{m}t_{P} / γ_{eI}_{ }• where γ _{eI} = 1/(1-g_{w}²/c²)^½ is the escape Lorentz factor for the insider_{d}t_{P} is aderived time for the proper insider_{d}t_{I} is aderived time for the insider_{m}t_{P} is ameasured time for the proper insider_{m}t_{I} is ameasured time for the insider From this section: > _{d}t_{O} = γ_{cO} * _{d}t_{I}_{m}t_{O} = _{m}t_{I} / γ_{cO}• where γ _{cO} is the captured Lorentz factor for a peaceful outsider_{d}t_{I} is an aderived time by an in-frame insider_{d}t_{O} is an aderived time by a peaceful outsider_{m}t_{I} is an ameasured time by an in-frame insider_{m}t_{O} is an ameasured time by a peaceful outsider
> _{d}t_{O} = γ_{cO} * _{d}t_{I}> = γ _{cO} * (γ_{eI} * _{d}t_{P})> = (γ _{cO} * γ_{eI}) * _{d}t_{P}> = γ _{g} * _{d}t_{P}_{m}t_{O} = _{m}t_{I} / γ_{cO}> = ( _{m}t_{P} / γ_{eI}) / γ_{cO}> = _{m}t_{P} / (γ_{cO} * γ_{eI})> = _{m}t_{P} / γ_{g}• where γ _{g} = γ_{eI} * γ_{cO} is the general Lorentz factor by outsiderγ _{eI} is the escape Lorentz factor by insiderγ _{cO} is the escape Lorentz factor by outsider |