✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 1 — #1
✐
✐
Part I: Real or apparent?
1.-FitzGerald-Lorentz contraction
Introduction
1 The visualization of a physical object is accomplished by means of the rays of light that, reflected
by the object, reach the corresponding optical system. Due to the finite speed of light and to the different distances light must traverse from the differOptical
ent parts of the object, not all reflected rays reach
device
the system at the same instant. In consequence,
1.1: Each reflected ray
FitzGerald-Lorentz contraction cannot be visually Figure
follows a different trajectory toperceived, or photographed, as such a contraction wards the optical device. This
in the direction of the relative motion but as a sort is why for relative velocities approaching the speed of light, Fitzof rotation [9], [11], [8]. Although the appropri- Gerald-Lorentz contraction is not
ate correction for these effects will reveal it to be seen as a simple contraction in the
direction of the relative motion.
present.
2 In this book we have assumed the results of all observations and measurements
performed by any observer in any inertial reference frame are properly corrected
by a computer whose screen displays the corrected images. In consequence, all
objects observed in relative motion will be considered and represented as actually
contracted in the direction of the relative motion, in agreement with FitzGeraldLorentz contraction.
3 In a previous chapter we had the opportunity to examine the role of the
relativistic factor γ = (1 − v 2 /c2 )−1/2 in Lorentz transformation. As we saw
there, γ can also be reinterpreted as a factor for the translation between the analog
and the digital languages. Length contraction, time dilation, phase difference in
synchronization and mass increment with relative motion can, all of them, be
expressed in terms of that relativistic factor.
4 As we will see in the next section, the real or apparent nature of FitzGerald1
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 2 — #2
✐
✐
2 —— FitzGerald-Lorentz contraction
Lorentz contraction remains open to discussion, although the discussions are not
very popular. Most of the introductory and university textbooks on the special
theory of relativity pay little attention, if any, to it. The discussions on the real
or apparent nature of time dilation, phase difference in synchronization and mass
increment with relative motion are practically ignored in the literature on the
special theory of relativity. And this is shocking because be it real or apparent,
the nature of these four consequences of relative motion must be the same, just
because all of them derive from the same Lorentz transformation.
5 The discussion about the real or apparent nature of the FitzGerald-Lorentz
contraction is further complicated by other external discussions that put into
question the very existence of an objective reality beyond human observers.
6 Apart from reviewing some opinions on the real or apparent nature of FitzGerald-Lorentz contraction, this chapter poses some problems derived from that
relativistic contraction, and then from the analog interpretation of Lorentz transformation.
FitzGerald-Lorentz contraction
7 In the year 1889 G. F. FitzGerald [4] and in 1892 H. A. Lorentz [7] proposed
independently a real length contraction of moving objects in the direction of motion through the luminiferous aether in order to explain the negative results of
the Michelson-Morley experiment (see Chapter ?? on Michelson-Morley). According to FitzGerald and Lorentz, the contraction was caused by changes in the
intermolecular forces of moving bodies (where motion has to be understood as
absolute motion). Since there were no reason to such changes, the proposal was
considered as an ad hoc hypothesis.
8 FitzGerald-Lorentz contraction is a consequence of the principles of the special
theory of relativity. In fact, assume in a frame RFo a photon traverses a vertical
distance Lo in a time t0 at its universal speed c. In other inertial frame RFv ,
coincident with RFo at a certain instant and from which RFo moves from left to
right in the direction of the Xo axis with a velocity v, the photon traverses the
hypotenuse of a right angled triangle, one of whose legs is Lo (Figure 1.2). Since
the hypotenuse (in the spacetime continuum) is greater than both catheti and the
speed of light is constant, the photon lasts a time tv greater than to to complete
its trip. So, the hypothenuse of that triangle will be ctv and its legs Lv = Lo and
vtv (see ?? in Chapter ??):
c2 t2v = L2o + v 2 t2v = c2 t2o + v 2 t2v
t2v c2 − v 2 = c2 t2o
(1)
(2)
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 3 — #3
✐
✐
FitzGerald-Lorentz contraction —— 3
t2v =
=
c2 t2o
c2 − v 2
(3)
t2o
1 − v 2 /c2
(4)
= γto > to
RFo
Lo=cto
cto
RFv
photon
Lv =c
tv
photon
Lo=cto
(5)
RFo stick
RFv stick
vtv
Xo
ctv
Xv
Figure 1.2: Time dilation and length contraction as a consequence of the principles of
the special theory of relativity
9 Consequently the interval of time (to ) elapsed in RFo is less than the interval
(tv ) elapsed in RFv . This means that if in the frame RFo a photon traverses the
length Lo of a horizontal metric stick in a time to (Figure 1.2), this time is less
than the corresponding time tv in RFv , which in turn means that from the RFv
perspective the length of the RFo -stick has to be seen contracted by a factor γ −1
to make it possible the photon traverses it in a time to at its universal speed c.
ctv
x
x v =ct
o =c v
ot
cto
Xo
x o2
x o1
xv1
Xv
xv2
Figure 1.3: Spacetime diagram of a metric stick whose proper space coordinates are
xo1 and xo2 .
10 On the other hand, FitzGerald-Lorentz contraction can be immediately derived from Lorentz transformation. In effect, if xo1 and xo2 are the space coordi-
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 4 — #4
✐
✐
4 —— FitzGerald-Lorentz contraction
nates of the two endpoints of a metric stick in RFo parallel to Xo , in the frame
RFv , from which RFo moves from left to right in the Xo direction with a constant
velocity v, the corresponding coordinates xv1 , xv2 will be such that:
xo1 = γ(xv1 − vtv1 )
(6)
xo2 = γ(xv2 − vtv2 )
(7)
And being tv1 = tv2 in the measurement performed1 in RFv , we will have:
xo2 − xo1 = γ(xv2 − xv1 )
(8)
xv2 − xv1 = γ −1 (xo2 − xo1 )
(9)
Or what is the same:
Real or apparent?
11 We have just seen that FitzGerald-Lorentz contraction is an inevitable consequence of the principles of relativity. Now then, is that contraction real2 or
apparent? Most of the authors of books on the special theory of relativity avoid
dealing with this ’notorious controversy,’ as Max Born called it [3]. On this controversy Anthony P. French wrote [5, pp. 113-114]:
This discussion should make it clear that the question ”Does the FitzGerald-Lorentz contraction really take place?” has no single, unequivocal answer from a relativistic point of view. The whole emphasis
is on defining what actual observations we must make if we want to
measure the length of some object that may be in motion relative to
us. And the prescription is simply that we measure the positions of its
ends at the same instant as judged by us. What else could we possibly
do? Thus the contraction, when we observe it, is not a property of
matter but something inherent in the measuring process.
Could we say the same of time dilation, phase difference in clock synchronization
and increment of mass with relative motion?
12 In his now classical book on Einstein’s relativity Max Born wrote [3, pp.
254-55]:
If we slice a cucumber, the slices will be larger the more obliquely we
cut them. It is meaningless to call the sizes of the various oblique
slices ’apparent’ and call, say, the smallest which we get by slicing
perpendicularly to the axis, the ’real’ size. In exactly the same way
1 To
measure a moving stick we would have to measure the position of its endpoints at
the same instant, otherwise one side will be displaced with respect to the other and we
would get an erroneous measure of the stick.
2 As some contemporary authors (for instance [1]) claim.
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 5 — #5
✐
✐
Real or apparent? —— 5
Figure 1.4: Born’s cucumber and Bohm shrinking men.
a rod in Einstein’s theory has various lengths according to the point
of view of the observer. One of these lengths, the statical or proper
length, is the greatest but this does not make it more real than the
others.
13 On the same issue, David Bohm wrote [2, Loc. 1253-71 Kindle edition]:
One may perhaps compare this situation to what happens when two
people A and B separate, while still in each other’s line of sight. A
says that B seems to be getting smaller, while B says that A seems
to be getting smaller. Why then does not B say that A seems to be
getting larger? The answer is that each is seeing something different,
i.e. the image of the world on his retina. There is no paradox in the
fact that the image of A on B’s retina gets smaller at the same time
that the image of B on A’s retina gets smaller. Similarly, there is no
paradox in saying that A will ascribe a contraction to B’s ruler, while
B ascribes a contraction to A’s simply because each is referring to
something different when he talks about the length of an object.
14 And finally, in a contemporary university textbook of physics we can read
[6, p. 1032]:
Does a moving object really shrink? Reality is based on observations
and measurements; if the results are always consistent and if no error
can be determined, then what is observed and measured is real. In
that sense, the object really does shrink.
15 The question: does a moving object really shrink? reveals a certain intellectual dishonesty: motion can only be relative because absolute motion is incompatible with relativity. Thus, all we can say is that an object, when observed
(measured) in relative motion, is observed contracted in the direction of the relative motion. Relative motion makes a moving object appear as contracted in the
direction of the relative motion, and in such a way that if, in those conditions, we
measures its length it will be shorter than if we do it when the object is at rest.
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 6 — #6
✐
✐
6 —— FitzGerald-Lorentz contraction
16 In its proper reference frame the object, in fact, is not contracted. And,
obviously, no object can be really contracted and really non-contracted at the
same time. It can be observed simultaneously as contracted and as non-contracted
by two different observers that observe it in two different ways: in relative motion
in the first case (contracted) and at rest in the second one (non contracted). But
it cannot be really contracted and really non-contracted at the same time. Saying
things differently means to add unnecessary confusion to this issue.
Apparent deformation
(b)
Real deformation
r
e
las
(c)
la
s
er
(a)
air
water
air
water
Figure 1.5: Apparent and real deformation. Note that in (b) the laser beam is parallel
to the rod even if the laser source is submerged.
17 As a counterpoint, consider now a straight rod partially and obliquely submerged in water. Due to the refraction of light, the rod seems to be bent, but
evidently it is not, as can be immediately proved by removing it from the water,
or even without removing it (Figure 1.5 (b)). Consider also other rod identical to
the first one but mechanically and irreversibly deformed by the application of an
appropriate mechanical effort. There is no controversy here: in the first case the
deformation is only apparent; in the second the deformation is real.
18 Something changes in the atomic structure of the actually deformed rod by a
mechanical effort, and that change can be experimentally proved, for instance, by
means of X-ray diffraction. This is not what happens in the apparently deformed
rod submerged in water, as could also be experimentally proved, for instance by
the laser beam method suggested in Figure 1.5. We have then an experimentally
testable asymmetry between an apparent and a real deformation. So it makes
sense to speak of real and apparent deformations.
19 Assume that in a certain number of reference
Xo frames there is an identical rod (FL rod) equipped
with a digital length meter that measures periodFigure 1.6: FL rod.
ically the length of the rod and displays the numerical result in the corresponding display. Assume all FL rods in all reference
frames are placed parallel to their corresponding X axes, and that all of them are
always at the same standard physical conditions. Consequently the display will
always show the same numerical result, for instance 1.000 m. Let RFo , RFu , RFv ,
RFw . . . be reference frames coincident at a certain instant and in relative motion
1.000
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 7 — #7
✐
✐
Real or apparent? —— 7
with respect to RFo with a velocity u, v, w . . . parallel to Xo . All observers in
all reference frames will observe the same measurement in the display of all FL
rods, although those measurements are not in agreement with their own measurements, except in the case of their own FL rod. Assume finally all observers meet
in RFo to discuss their corresponding measurements. Would not it be reasonable
to conclude all measurements performed in relative motion are distorted by relative motion in the same way the shape of the partially submerged rod is distorted
by the refraction of light?
20 Let us now compare FitzGerald-Lorentz contraction with the deformation of
the submerged rod:
⊚ FitzGerald-Lorentz contraction is real in the same sense it is real the bending
of the partially submerged rod: both perceptions are not hallucinations of
the observers. And both are perfectly explainable in physical terms.
⊚ Both deformations are consequences of two particular ways of observing an
object: in relative motion in the first case, and partially submerged in water
in the second one.
⊚ If we observe a partially submerged rod we can easily reconstruct its actual
shape and size by a simple application of Snell law of light refraction. In
the same way, if we observe a FitzGerald-Lorentz contracted object we can
also reconstruct its real (proper) shape and size by means of the relativistic
factor γ.
⊚ Both deformations are reversible in the sense that by removing the rod from
the water and by decreasing the relative velocity to a null value both rods
recover their original (proper) size and shape.
⊚ By changing the inclination of the partially submerged rod, the level of
deformation will also change. Similarly, by changing the relative velocity at
which an object is observed, the degree of its contraction in the direction of
the relative motion will also change.
⊚ Both deformations occur without a mechanical effort acts on the deformed
objects.
21 Unfortunate expressions like ’motion modifies the length of the moving objects’ are more frequent than expected in the literature on the special theory of
relativity. They are indeed unfortunate because ’motion modifies’ makes one to
think in terms of absolute motion, which is obviously inconsistent with the principles of relativity. In addition it is taken for granted that the deformation is real.
Although less spectacular, it would be more accurate and less confusing to say
that, when observed in relative motion, an object is observed contracted in the
direction of the relative motion. And, in consequence, the same should be said
on time dilation, phase difference in synchronization and mass increment with
relative motion.
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 8 — #8
✐
✐
8 —— FitzGerald-Lorentz contraction
22 The idea, on the other hand, that specific realities cannot exist without human observers makes it impossible the existence of human observers, because the
specific history of life from which they have evolved could not have been possible
without human observers. That said, and recalling Ockham’s razor, should not
we use the word ’real’ only in the cases as the rod mechanically deformed and
the word ’apparent’ in the cases as the partially submerged rod? In this sense, is
FitzGerald-Lorentz contraction real as the mechanically deformed rod or apparent as the submerged one? Or perhaps it is real (or apparent) in another sense.
And if this were the case, in which sense?
23 As with length contraction, inertial time dilation, phase difference in synchronization and increment of mass with relative motion are also consequences
of Lorentz analog transformation. Consequently, the status on their apparent or
real nature should be the same as that of FitzGerald-Lorentz contraction.
Mirror
c
Mirror
Photon
Rod
Figure 1.7: The apparent? length of a rod seen in relative motion can be corrected
only if we also correct the corresponding inertial time dilation and phase difference in
synchronization.
24 Imagine a photon that is continually reflected by two vertical mirrors placed
at the ends of a horizontal rod (Figure 1.7). An observer in relative motion with
respect to the rod in a direction parallel to it, will see the rod contracted by a
factor γ −1 . If he corrects FitzGerald-Lorentz contraction he must also correct
time dilation and phase difference in synchronization for a correct calculation of
the photon speed.
25 Let me end this section by paraphrasing the following quote from a university
text ([10, p. 42]):
(Original) I need to warn you about language. I have said that a rod
with length Lo as observed from its own frame has a shorter length
Lv as observed from another frame. Often this result is stated as ’A
rod with length Lo as observed from its own frame, appears to have a
shorter length Lv as observed from another frame.’ This statement is
true: the rod appears to have shorter length Lv because it does have
shorter length Lv . Using the term ’appears’ gives the false impression
that, when the rod is observed from a frame in which it moves, the
rod really is of length Lo and only appears to be of length Lv . No.
As observed from a frame in which it moves, the rod really does have
the shorter length Lv .
(Paraphrase) I need to warn you about language. I have said that
a straight rod is bent when observed partially submerged in water.
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 9 — #9
✐
✐
Changing shapes —— 9
Often this result is stated as ’A straight rod partially submerged in
water appears bent’ This statement is true: the rod appears bent
because it is bent. Using the term ’appears’ gives the false impression
that, when the rod is observed partially submerged, the rod really
is straight and only appears to be bent. No. As observed partially
submerged, the rod is really bent.
Changing shapes
26 Consider now a square plate P made of the strongest steel whose side has a proper length Lo , and assume
it is oriented in such a way that one of its sides is parallel to the Xo axis of its proper reference frame RFo .
As expected, after a clockwise rotation of 45◦ around
its geometrical center (Figure 1.8), P continues to be
a square plate in RFo .
Yo
RFo
Xo
Yv
RFv
v
27 However, in other inertial frame RFv from which
Xv
RFo moves in the Xo direction, before the rotation P
−1
is seen as a rectangle whose sides are Lo and γ Lo . Figure 1.8: In RFv a single
After a rotation of 45◦ the shape of P changes to a rotation suffices to change
the shape of a metal plate.
rhombus whose side is Lo ((1 + γ −2 )/2)1/2 .
28 That a single free rotation suffices to change the shape of a metal plate goes
against several mechanical, physicochemical and crystallographical laws. On the
other hand, and according to the First Principle of the special theory of relativity,
physical laws are the same in all reference frames. This incongruence should be a
sufficient reason for the RFv observers to conclude their observations are distorted
by relative motion in the same way the observation of a partially submerged rod
is distorted by the refraction of light.
Yo
RFo
Yv
RFv
v
Xo
Xv
Figure 1.9: Deformation without mechanical efforts. By rotating the structure, its two
arms contract and expand alternatively without any force acts on them.
29 The right angled structure depicted in Figure 1.9 poses a similar problem:
successive free rotations of 90◦ suffices to alternatively contract one of its arms
by a factor γ −1 and, at the same time, expanded the other by a factor γ without
any force act on them.
30 These examples of mechanical deformation without acting forces resemble
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 10 — #10
✐
✐
10 —— FitzGerald-Lorentz contraction
the apparent deformation of a rod when it is partially submerged in water and
when it is removed from the water.
RFv
RFo
31 And finally, let B be a bubble of a certain
v
fluid F2 in hydrostatic equilibrium with other
F
F
1
1
?
fluid F1 . In its proper frame RFo , the bubble
has a spherical shape due to the fact that the
F2
F2
hydrostatic pressure is the same in all direcXv
Xo
tions. In RFv that moves relative to RFo in
the Xo direction, the bubble has an ellipsoidal
shape due to FitzGerald-Lorentz contraction Figure 1.10: A bubble in hydrostatic
equilibrium as seen from its proper
in the direction of the relative motion. Will frame RF (left) and from other refo
the observers in RFv explain the shape of the erence frame RFv that moves relative
bubble in terms of a real or of an apparent de- to RFo with a velocity v in the Xo direction (right).
formation?
The sliding pulley
32 A more intriguing consequence of FitzGerald-Lorentz contraction is the next
case of the sliding pulley, in which several laws of mechanics are (apparently?)
violated.
Yo
Vertical
support
sliding sopport
vertical section
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
elastic cord
aaaaaaaaaaaaaaaaaaaa
fixing screw
horizontal section
aaaaaaaaaaaaaaaaaaaa
single
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
pulley
lo
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
metric scale
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
Tension To lo
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
mo
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa Xo
Figure 1.11: According to the laws of mechanics, at equilibrium all black and white
marks of the elastic cord must of necessity have the same length.
33 RFo is the proper frame of the mechanical device schematically depicted in
Figure 1.11 in which a load of rest mass mo hangs from an elastic cord whose left
end is attached to a fixed vertical support. The cord runs around a pulley which
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 11 — #11
✐
✐
The sliding pulley —— 11
in turn is attached to a horizontal sliding support whose position can be set with
the corresponding fixing screw. A metric scale, that consists of a certain number
of black and white marks of equal length each, is printed on the cord3 (Figure
1.11). At equilibrium, the mechanical tension can only be constant along the
whole cord, in consequence all metric marks must have the same length, whether
they are in the horizontal or in the vertical section of the elastic cord. A conclusion
that, being an immediate consequence of the laws of mechanics, must hold in all
reference frames.
34 Let lo be the proper length of the metric marks, and let us assume the Xo
axis of RFo is parallel to the horizontal section of the elastic cord. By sliding
its horizontal support, the pulley moves in the left-right direction, which has the
effect of changing, in a complementary way, the number of those horizontal and
vertical metric marks (for the sake of simplicity we will assume the pulley always
slides an integer number of marks). As expected, in RFo the length of all metric
marks remains constant and equal to lo , be them in the horizontal or in the
vertical section of the cord.
Yv
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
g -1l o
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
Tvh < Tvv
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
Tension Tvh
aaaaaaaaaaaaaaa
Tension Tvv
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
lv = lo
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
mv
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa Xv
Figure 1.12: In RFv , from which RFo moves from left to right in the Xo direction, the
length of the horizontal metric marks are contracted by a factor γ −1 while the vertical
marks maintain its proper length lo .
35 RFv is an inertial reference frame whose spacetime diagram coincides with
that of RFo at a certain instant, and from which RFo moves from left to right,
in the Xo , direction with a velocity v. In consequence, and according to Lorentz
transformation, in RFv the length of the vertical marks of the metric scale is also
3A
common trick we use in geology in order to illustrate the mechanical deformations of
rocks.
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 12 — #12
✐
✐
12 —— FitzGerald-Lorentz contraction
lo , while the horizontal marks have a contracted length γ −1 lo .
36 Let n be the number of metric marks, and assume nh of them are in the
horizontal section in a certain position of the sliding pulley. Obviously the number
of vertical marks in that position will be n − nh . In RFo the length Lo of the
cord at equilibrium with the pending mass is always the same, independent of the
pulley position:
Lo = nh lo + (n − nh )lo
= lo (nh + n − nh ) = nlo
(10)
(11)
However, in RFv the length Lv of the cord is variable, depending on the pulley
position, i.e. on the number nh of horizontal marks:
Lv = nh γ −1 lo + (n − nh )lo
1
2
g -1l o
3
4
5
1
6
2
mv
3
lo
4
5
6
mv
Figure 1.13: The horizontal mechanical tension of any part of
the cord changes when it changes its position from the horizontal to the vertical section and vice
versa.
(12)
= lo (nh γ −1 + n − nh )
(13)
= lo (n − nh (1 − γ −1 ))
(14)
37 So, as the number nh of horizontal marks increases the cord’s length Lv decreases. Or in other
words, as the pulley moves to the right the length
of the cord decreases; and as it moves to the left its
length increases. Obviously these changes of length
disagree with what is expected from the laws of
mechanics: the same hanging mass cannot originate different mechanical tensions, and then different stretchings, in the same elastic cord.
38 By sliding the pulley towards the left or towards the right we can make any part of the rope
to move from the horizontal to the vertical position
and vice versa. When changing, its mechanical tension also changes, being greater when its position
is vertical (Figure 1.13). These changes of tension
are also incompatible with the laws of mechanics
39 As we have just seen, the mechanical situation
observed from RFv is incompatible with the known physical laws. On the one
hand the mechanical tension of the cord has to be the same along its entire length
and, therefore, the length of all its metric marks has also to be the same, be them
in the horizontal or in the vertical section. On the other hand, the length of the
cord at equilibrium with the pending mass must be always the same, it cannot
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 13 — #13
✐
✐
The sliding pulley —— 13
change as the pulley slides.
40 The observers in RFv would have to conclude they observe a FitzGerald-Lorentz contraction that can only be apparent, as is apparent the bending of the
rod partially submerged in water. Otherwise, they would have to explain the
strange mechanical behaviour of the elastic cord: they would have to explain how
it is possible for an elastic cord not to have the same mechanical tension along its
entire length, and how it can be stretched in different ways by exactly the same
hanging mass.
41 By symmetry, if the contraction of lengths by relative motion were only apparent, so would be for time dilation, phase difference in synchronization and
mass increment with relative motion (as we will see in Chapter ?? on Hooke’s law
there is other non-relativistic source of mass increment derived from the equivalence between mass and energy).
42 In short, all observers in relative motion with respect to RFo should consider
the possibility their observations and measurements are distorted by relative motion in such a way they cannot get conclusions physically acceptable on what
happens in RFo . In this sense, only the observers in RFo may conclude their
observations and measurements agree with what is expected from physical laws.
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 14 — #14
✐
✐
14 —— FitzGerald-Lorentz contraction
✐
✐
✐
✐
✐
✐
“DigitalRelativity” — 2015/5/22 — 7:02 — page 15 — #15
✐
✐
Bibliography
[1] Carlos Barcel´
o and Gil Jannes, A real lorentz-fitzgerald contraction, Foundations
of Physics 38 (2008), 1199–199.
[2] David Bohm, The special theory of relativity, A. Benjamin Inc., London and New
York, 1965.
[3] Max Born, Einstein’s theory of relativity, Dover Publications Inc., New York, 1965.
[4] George Francis FitzGerald, The ether and the earth’s atmosphere, Science 13
(1889), 390.
[5] A. P. French, Special relativity, W. W. Norton and Company Inc., New York, 1968.
[6] David Halliday, Robert Resnick, and Jearl Walker, Fundamentals of physics, John
Wiley and Sons, 2008.
[7] Hendrik Antoon Lorentz, De relatieve beweging van de aarde en den aether,
Zittingsverlag Akad. V. Wet. (1892), 74–79.
[8] N.C. Mcgill, Apparent shape of rapidly moving object in special relativity,
Contemporary Physics Vol. 9. no 1 (1968), 33.
[9] R. Penrose, The apparent shape of a relativistic moving sphere, Proceedings of the
Cambridge Philosophical Society 55 (1959), 137–139.
[10] Daniel F. Styer, Relativity for the questioning mind,
http://www.oberlin.edu/physics/dstyer, 2009.
[11] James Terrell, Invisibility of the lorentz contraction, Physical Review 116 (1959),
1041–1045.
15
✐
✐
✐
✐