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# THE LIGHT CLOCK WAS AN ESCHER PICTURE

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## Abstract

The moving light clock is the most prominent example in relativity to introduce the Lorentz factor. According to the constant speed of light the diagonal paths of light are expected to exceed the vertical paths in terms of distance and in terms of time. But the Michelson experiments have been interpreted this kind: Motion against the light path does not change the arrival time of light. The Michelson-observations were used to motivate the introduction of the Lorentz factor. The light clock is applied two distinct reference sizes for velocities. Both denote the motion along the same direction but use different reference magnitudes. The formulas commonly used omit the vector notation and introduced a fatal error in relativity.
Gregor L. Grabenbauer 1/10
THE
LIGHT CLOCK
WAS AN
ESCHER PICTURE
Gregor L. Grabenbauer
gg@grabenbauer,de
September 2016
ABSTRA C T
The moving light clock is the most prominent example in relativity to introduce the
Lorentz factor. According to the constant speed of light the diagonal paths of light are
assumed to exceed the vertical paths in terms of distance and in terms of time as well,
but they do not, the derivation of the Lorentz factor suggests. The light clock is applied
two s for velocities. Both denote the motion along the same direction but
reference size
use different reference magnitudes. The formulas commonly used omit the vector
notation and introduced a fatal error in relativity.
Gregor L. Grabenbauer 2/10
THE LIG H T C L O C K
The light clock
1
installs a virtual object bouncing vertically between two mirrors at the
speed of light. An observer moving with velocity  records the distance
traversed by the light. His readings exceed the strict vertical distance between the
mirrors. Following the principle of invariance of light speed the travel times are
expected to increase, but they don’t have to, as the derivation of the Lorentz factor seems
to show. The observer of the light clock records the following shape. The variables and
denote distinct references of time.



The term  implies a shape having a right angle between lines indicated by
and . For the light clock to operate the velocity has to change reference from to . To
switch reference, e.g. to use inches instead of meters, implies to preserve the meaning of

such that the shapes referring to it do not change.
To keep the magnitude of 
with respect to the direction indicated it is necessary to
convert to such that 

But we have to consider that
the direction of  is not perpendicular to the -axis and
the magnitude of is exceeding the magnitude of .
From this we must conclude that we don’t have some invariant to give a
plausible conversion rule for into .
Gregor
L.
The ratio of
The
transfer of
during mot
By applying the invariant
we
deduc
and
exchange
L.
Grabe
nbauer
The ratio of
and
transfer of
from
during mot
ion
, otherwise both models wo
By applying the invariant
deduc
e the conversion rule
exchange

by
nbauer
and
is given by the shape as:
from
to
, otherwise both models wo
By applying the invariant
e the conversion rule
by

for the primary term,
is given by the shape as:
must ke
ep the
, otherwise both models wo
e the conversion rule
for the primary term,



is given by the shape as:


ep the
whole
, otherwise both models wo
uld
not


The area
given by

If
is set to
for
m.
The value of
As

the
vector
stand alone.
When switching

may not be derived from
If
the area
is kept invariant the shape




for the primary term,
hence



!

whole
area invariant
not
be equivalent:

The area

given by
"
#
"
are equal
is set to
, there
m.
The value of
may be

and 
were given as interde
vector

m
stand alone.
When switching
may not be derived from
the area
is kept invariant the shape

hence
we get:

!


area invariant
that
is
be equivalent:

is equal to the area
 as t
he doubled areas
are equal
.
, there
is
some simplification
may be
provided
were given as interde
m
ay not
be
When switching
from
may not be derived from

is kept invariant the shape
is
affected by
is equal to the area
he doubled areas
some simplification
provided
.
were given as interde
pendent
be
converted
to
, the vector
may not be derived from

only.

is kept invariant the shape
will
not change.
3/10
affected by
light
is equal to the area
he doubled areas
some simplification
pendent
as
, the vector
only.
not change.
Gregor L. Grabenbauer 4/10





 !





Note that
is indicating the speed of the light clock with respect to. As is unchanged
the factors and \$ did not change. The shape did not change and the formula to
describe the geometry of the light clock is still the same.
CONCLUS I O N
If we preserve the area %we preserve the geometrical shape of the light clock.
If we preserve the shape there is preserved the tangens-ratio of to &

&
indicating '( which does not imply constraints to velocities  along
the -axis. There is no chance to produce some Lorentz-factor by light clocks.
Gregor L. Grabenbauer 5/10
THE LIGHT CL O C K A S C O M M O N L Y T A U G H T I N RE L A T I V I T Y
The moving light clock is the most prominent example in relativity to introduce the
Lorentz factor. It is used in textbooks, relativity courses and online courses
2
as well.
Examples
3
:
“All moving clocks are slowed by motion.”
“The light blips in both travel at the same
speed relative to us.”
“This means that time vary for frames at
different velocities with respect to each
other.”
4
The light clock implies the unique ratio
%

for towards the -direction.
If ratio  refers to  it gives indication that
 and
 gives the enveloping area of light during motion, by ).
If ratio  refers to % it gives indication that
 and
% gives the enveloping area, by ).
In order to get clarity about the vectors and the relationships to their driving variables
we use in conjunction with and with analogously.
Gregor L. Grabenbauer 6/10
The 1:1 conversion of the -factor, following 

, does not preserve the shape.
The switch to the reference magnitude of  increases the effective speed of *
by the factor .
The area affected by motion is downsized by , the same factor the corresponding
value of  was reduced.
If we compare the area enclosed by "+,+ with that of ", the ratio "
-
-
-
-
"+
-
-
-
-
-
to
acknowledge is enough to give the ratio. The segment ",
-
-
-
-
",+
-
-
-
-
-
, which represents the
1:1 conversion, is common to both.
Despite this obvious discrepancy between both areas, the invariant, given
as
%\$
is fulfilled.
By geometrical construction the distance %+ may be proven to be equal to  easily.
Hence the equation above is fulfilled, if . This identity is satisfied by geometrical
construction and the 1:1 conversion as well.
Gregor
L.
How to resolve this discrepancy?
The invariant
refers
to
gives
the
),
if we apply the
because
Any
impl
To
illustrate th
lectures
L.
Grabe
nbauer
How to resolve this discrepancy?
The invariant
to
the
product
the
'-fold
sum of
if we apply the
because
is perpendicular to
Any
product
s
impl
y
the corresponding
illustrate th
is
ser
lectures
the following examples
nbauer
How to resolve this discrepancy?
product
of two numbers
sum of
..
If we apply the
if we apply the
-
fold to
is perpendicular to
s
of two numbers
the corresponding
ser
i
ous mistake
the following examples
How to resolve this discrepancy?
of two numbers
If we apply the
fold to
%+we
would go astray
is perpendicular to

and not perpendicula
of two numbers
that indicate the magnitude of
the corresponding
vectors
to
ous mistake
built in the light clock
the following examples
may be
%
and %+.
Th
If we apply the
-
fold to
would go astray
and not perpendicula
that indicate the magnitude of
to
be
perpendicular
built in the light clock
may be
useful.
/01
will equal the area
rotate /
to
01.
When
rotat
perpendicular to
012 will
match
\$
Th
e
product
fold to
 we
will
would go astray
.
The
and not perpendicula
r to
%
that indicate the magnitude of
perpendicular
built in the light clock
as t
will equal the area
to
become
perpendicular to
rotat
ing /
to b
perpendicular to
01
match
34.
product
of two numbers
will
have the
equivalent
r
esult gives the
%
+.
that indicate the magnitude of
vectors
perpendicular
to each
other
as t
aught
in com
will equal the area
*34
if
we
perpendicular to
ecome
we
will see that
of two numbers
.
and
equivalent
esult gives the
-
fold of
vectors
other
.
in com
m
on physics
7/10
we
perpendicular to
will see that
and
'
equivalent
area
fold of

on physics
Gregor L. Grabenbauer 8/10
If we change a parallelogram shape such that the width remains constant, we might
suppose to use a simple product of two numbers to get the area. We may do so by short-
cut only, because the transition from rectangles to parallelograms does not change the
height and leaves the width untouched, which is to keep the product of length and height
invariant.
CORRECT I O N
The light clock as common understood implies a speed-up of
The light clock delivers free energy
as speed
gets a free boost of .
,
This energy is free
we some
if first assume to have speed related to
to
and then switch
the same
magnitude
to the newly created reference size 5
5678
98
.
Gregor L. Grabenbauer 9/10
The light clock, when used to derive the Lorentz factor, needs a patch. The indicated
speed may refer to  or to . The angle ( enclosed between the -axis
and  is :. The angle (
5
between the -axis and % is ;<=(
5
>?
@?
. Therefore, we
have to correct
5
the magnitude of
in order to reflect the change of orientation of
with respect to %:
A
B
&:
CDE;A
B
F
BB
G&HICDE;A
BB
DE;;<=
65

%
CDE;A
BB
C
Gregor L. Grabenbauer 10/10
1
https://en.wikisource.org/wiki/The_Principle_of_Relativity,_and_Non-Newtonian_Mechanics
G. N. Lewis and R. C. Tolman: "The Principle of Relativity, and Non-Newtonian Mechanics", in:
“Proceedings of the American Academy of Arts and Sciences”, 1909, 44: 709–726;
The light clock originally was introduced as instructive device for deriving the time dilation formula.
2
See for example: http://galileo.phys.virginia.edu/classes/252/srelwhat.html
3