The classical equations of relative motion are translated by the theory of Special Relativity into relativistic equations. The origin of the Lorentz factor recapitulated.

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FROM GALILEAN TO EINSTEINIAN
Cosmic Adventure 5.1

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what is wrong
with them?
They are not valid
when the speed of
light is considered
You know what is
wrong with your
equations?

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System x:
𝑥′ = 𝑥 − 𝑣𝑡
𝑦′ = 𝑦
𝑧′ = 𝑧
𝑡′ = 𝑡
System x’:
𝑥 = 𝑥′
+ 𝑣𝑡
𝑦 = 𝑦′
𝑧 = 𝑧′
𝑡 = 𝑡′
Because in between
frames, you should have
light speeding along
No Light Involved

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To take the speed of light
into consideration, you need
to have a Lorentz
transformation with the
Lorentz factor! 𝛾 =
1
1 −
𝑣2
𝑐2

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𝑥′′ =
𝑥′ − 𝑣𝑡
1 −
𝑣2
𝑐2
𝑡′′ =
𝑡′
− 𝑣𝑥′/𝑐2
1 −
𝑣2
𝑐2
𝑥′ =
𝑥′′ + 𝑣𝑡
1 −
𝑣2
𝑐2
𝑡′ =
𝑡′′ + 𝑣𝑥′′/𝑐2
1 −
𝑣2
𝑐2
Relativistic Equations

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𝛾 =
1
1 −
𝑣2
𝑐2
Where did you this funky
Lorentz factor come from?

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Normally I would not tell
where I get these equations.
But for this competition, I
have to do so because
Angela already knew it.

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BeamB
Beam A
Viewer
Light Path Geometry [at
Rest] on plan
These equations came from
Michelson’s concept of the
experiment itself.
In the absence of aether wind,
the two lights will recombine
at the viewer in sync. No fringe
will be observed.

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BeamB
Beam A
Viewer
Light Path Geometry
According to Michelson
According to Michelson, the
two lights will pursue
different paths due to the
influence of the aether wind.

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Michelson’s Working Equation
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝐵𝑒𝑎𝑚 𝐴
=
2𝑙 𝑜
𝑐 1 − 𝑣2/𝑐2
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝐵𝑒𝑎𝑚 𝐵
=
2𝑙 𝑜
𝑐 1 − 𝑣2/𝑐2
In the experiment
there are two
durations of time
for the two beams.

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The ratio between the
Times of the Beams
𝑇𝑖𝑚𝑒 𝑓𝑜𝑟 𝐵𝑒𝑎𝑚 𝐴
𝑇𝑖𝑚𝑒 𝑓𝑜𝑟 𝐵𝑒𝑎𝑚 𝐵
=
2𝑙 𝑜
𝑐 1 − 𝑣2/𝑐2
÷
2𝑙 𝑜
𝑐 1 − 𝑣2/𝑐2
=
2𝑙 𝑜
𝑐 1 − 𝑣2/𝑐2
×
𝑐 1 − 𝑣2/𝑐2
2𝑙 𝑜
=
1
1 − 𝑣2/𝑐2
1
1 − 𝑣2/𝑐2

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This factor was later brought
up by Albert Lorentz in his
postulate of time dilation to
explain the experiment. I got
inspired and made use of it
in my theory.
In its turn, the factor
accounts for failure of the
Michelson-Morley
experiment to produce the
desired results.
1
1 − 𝑣2/𝑐2

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Looking at your equations, it is
obvious the major variable is
the relative velocity 𝑣. What if
both systems are at rest?
𝑥′′
=
𝑥′ − 𝑣𝑡
1 −
𝑣2
𝑐2
𝑡′′ =
𝑡′
− 𝑣𝑥′/𝑐2
1 −
𝑣2
𝑐2
Relativistic Equations

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Then 𝑣 will become zero and the
equations will revert to the starting point.
𝑥′′
=
𝑥′
− 𝑣𝑡
1 −
𝑣2
𝑐2
→ 𝑥′
𝑡′′ =
𝑡′ − 𝑣𝑥′/𝑐2
1 −
𝑣2
𝑐2
= 𝑡′

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What about when the observer
and the object are at rest and
are separated by a distance s?
They are not moving, so 𝑣 is
again 0. 0’
𝑥′
P
𝑥′′
0’’ P
𝑠

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VISONIC TRANSFORM WITHOUT MOTION
To be continued on Cosmic Adventure 5.2