*The Myth of "Lorentz Transformation Reducing to Galilean
Transformation"*

*Wolfgang
G. Gasser*

This has been separated from *Refutation
of Special Relativity for Dummies*

*Introduction with core
statement*

*#118
– 2015-10-05*

"For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation in accordance with the correspondence principle." (Wikipedia on Lorentz transformation)

The claim that the Lorentz transformation turns
into the Galilei Transformation if *v << c*, is exposed as a myth by
the time-shift *v∙x/c** ^{2}* resp.

By dropping the Lorentz-factor as a second-order effect, the
Lorentz transformation (here with ** c = 1**)

*x' = **γ** (x - v∙t)**
**t' = **γ** (t - v∙x) *

does not reduce to the Galilean transformation:

*x' = x - v∙t** **t'
= t*

Instead, the Lorentz transformation (again with ** c = 1**)
reduces to:

*x' = x - v∙t** . . . **t' = t - v∙x*

If we drop the first-order effect ** v∙x** of the
time transformation, then we must also drop the first-order effect

*x' = x** . . . **t' = t*

An even more fantastic and *theology-like myth* is the
so-called *Lorentz invariance of Maxwell's theory of Electromagnetism*. (Maxwell's completely inconsistent theory, based on the premise of *naïve
realism* that instantaneous actions-at-a-distance are impossible, is a wild
conglomerate of findings of Maxwell's predecessors, contemporaries and of
himself. Consecration and benediction by and from "Lorentz invariance"
absolved Maxwell's theory from sins and inconsistencies.)

---

*Changing one's world view due to new findings, better
understanding and logical reasoning should not be confused with surrendering
part of one's identity when brutally forced to change religion*

*Invalidation of counter-arguments*

*#122 – 2015-10-07*

By Darwin123 in #119:

The reason
one that the ** v∙x/c^{2}** term disappears is because
for the observer on earth looking at the rocket,

It is difficult to make full sense of your posts #119 and #121,
but it is clear that you try to rescue the derivability of the Galilean
transformation from the Lorentz transformation, by finding a justification for ** x
= 0** in the "reduced" time transformation

In modern physics, the epistemological principle that one single counter-example is enough to refute a theory has been reversed into its opposite: Theories get generally accepted if they are confirmed in very special cases.

By the way: In the future the question whether 'neutrinos' have 'mass' (in the sense of somehow participating in the God-particle constructed by Higgs) will be considered as relevant as past disputes over Holy Trinity (which I do not disdain). Progress is always unfair to the old ways.

By wogoga in #118:

Instead, the Lorentz transformation
(again with ** c = 1**) reduces to:

*x' = x - v
t t' = t - v∙x*

By Perpetual Student in #130:

This is
incorrect. We have: *x' = **γ** (x - vt) and t' = **γ** (t - vx/c ^{2})*

With *c = 1*,
the *v* in *x' = x - vt* is the actual velocity in the *x*
direction. However, because of the *c ^{2}* term in the denominator
with

I agree that "dimensional analysis" is a prerequisite
for every reasonable concept in physics. Yet in our case the situation is so
clear that no danger can arise from the implicit use of a length-unit and a
time-unit connected by ** c = 1** (as applied in #75).

By using ** t' = t - v x** instead of

Some may even argue that ** v/c^{2} x** can be
removed as a term from the first-order limit because the constant

By slyjoe in #127:

Have we seen any calculus? Introduction to limits may have been useful to the OP.

Instead of joining the ranks of clueless pack followers, you could
have used software for symbolic computation and tried yourself to check what
corresponds to the mathematical reality. Here the problem
formulated with *Mathematica*-syntax:

*γ** = 1/Sqrt[1-v^2/c^2]
Series [*

Series [

If *n = 0*, the result is:

*x' = **x ** ** **t' =** **t*

For the first-order series (with *n = 1*) we get:

*x' =** x - t v *

The second-order result:

*x - t v +
x/2/c^{2} v^{2} t -
x/c^{2} v + t/2/c^{2} v^{2}*

The only possibility to get the Galilean transformation is taking the first order Taylor-approximation for the x-coordinate and the zeroth order approximation for the t-coordinate.

Does anybody know who is responsible for spreading the myth that the Lorentz transformation reduces to the Galilean transformation? I assume and hope it was not Einstein.

By wogoga in #135:

Some may even
argue that ** v/c^{2} x** can be removed as a term from the
first-order limit because the constant

An illustrative example of this argument (Perpetual Student in #145):

We have: *x'
= **γ** (x - vt)* and *t' = **γ** (t - vx/c ^{2})*

Using a slow
velocity like *v = 10 m/s* and setting *x = 1 m* for simplicity of
calculation demonstrates the case clearly. I am using only orders of magnitude:

*γ** =
(1 - 10 ^{-15})^{-1/2}*, which we
would (hopefully) agree is infinitesimally close to and indistinguishable from

Now, *x -
vt = x - 10** **m*, which is
obviously the Galilean transformation. End of story.

And *t' = t
- vx/c ^{2} = t - 10^{-16 }s* which is infinitesimally close
to t, to about the same order of magnitude that

No objection can be raised against your choice of ** v = 10 m/s**,
leading to a negligible Lorentz factor, but the choice of

*x' = x - v∙t
= 1 m - 10 m
t' = t - v∙x/c ^{2} = 1 s - 1.11∙10^{-16} s*

*K*eeping *v∙t** =
10 m* and dropping *v∙x/c ^{2} =*

Now let us do the same with the (somehow less arbitrary) units *second*
and *light-second*, i.e. let us use ** x = 1 LS** and

*x' = x - t∙v
= 1 LS - 3.33∙10 ^{-8} LS
t' = t - x∙v/c^{2} = 1 s - 3.33∙10^{-8} s*

Keeping *v∙t =** 3.33∙10 ^{-8} LS*
and dropping

The Taylor expansion of the Lorentz factor ** (1-v^{2}/c^{2})^{-1/2}**
with respect to

*1 v ^{0} +
0 v^{1} + 1/(2c^{2}) v^{2} + 0 v^{3} + 3/(8c^{4})
v^{4} + . . .*

If *meter per second* is used as unit for velocity, and both ** v**
and

*1 + 5.56∙10 ^{-18
}v^{2} + 4.64∙10^{-35 }v^{4} + . . .*

By the way, the zero-order Taylor-expansion term of the Lorentz
factor, *1** *** v^{0} = 1**, becomes after
multiplication with

---

*As long as we remain in the abstract, and do not apply formulas
or propositions to concrete situations, contradictions remain undetected*

By Perpetual Student in #155:

You continue
to confuse yourself by setting *c = 1*.

With your choice of v = *10 m/s* and the units *meter*
and *second*, we get this **time-shift** (resp. deviation from
simultaneity):

*v/c ^{2} =
1.11∙10^{-16} sec per meter *

The analogous **length-shift** (resp. speed) is:

*v = 10 meter per
sec*

You claim that compared to *10 m/s*, the *1.11∙10 ^{-16}
s/m* are negligible and can therefore be discarded when taking a first-order
approximation. Yet the difference of 17 decimal powers is only the result of
the (squared) numerical value of light-speed

The hallmark of a first-order effect is the following:

If we reduce an argument further by a factor, then the effect is also reduced approximately by the same factor. (In the second-order case, the effect would be reduced approximately by the square of this factor.)

In our case:

If we reduce *v*
from *10 m/s* to *1 m/s* then time-shift is reduced only from *1.11∙10 ^{-16}
sec/m* to

When dealing with our galaxy or the whole universe, the units *second*
and *light second*, or *year* and *light-year* are rather better
suited than *second* and *meter*.

In any case, if β*
= v/c << 1*, we get for **length-shift** and **time-shift**:

*v = *β* LS/s = *β* LY/year = (3∙10 ^{8})^{+1}
*β

v/c

Contrary
to time dilation and length contraction, which are higher-order effects of *v*,
relativity of simultaneity is a first-order effect. You should really try to understand what I
wrote in #118:

"The time shift *vx/c ^{2}*
is necessary to explain that e.g. the speed of light from an astronomical
object near the ecliptic does not change from

By Reality Check in #160:

When *v/c
<< 1* the Taylor series for the Lorentz factor can be truncated and we
get Newtonian mechanics.

Do you assume one of the following hypotheses?

· Einstein's simultaneity concept is part of Newtonian mechanics.

· Speed of light is infinite in Newtonian mechanics.

If you do not subscribe to one of these two hypotheses,
then the claim that *Special Relativity* reduces to Newtonian mechanics is
simply wrong.
Ask a mathematician or use a program for symbolic computation as shown in #135.

By wogoga in #161:

You claim that compared to *10 m/s*,
the *1.11∙10 ^{-16} s/m* are negligible and can therefore be
discarded when taking a first-order approximation.

By Perpetual Student in #164:

You can stop
right there. Yes, the difference is as negligible as the difference between *1*
and the gamma function. Think! What do you think "reduces to the Galilean
transformation means"? At speeds like *10** **m/s*
and spatial differences like *1** **m*,
time intervals like *10 ^{-16 }s* are irrelevant, and as far as I
know, immeasurable. The rest of your post is nonsense.

We cannot compare apples or *m/sec* to oranges or *sec/m*
in the way you do.

On the one hand we have the "difference between *1* and
the gamma function" (**Lorentz-factor**):

** 1** versus

On the other hand we have the difference in **time-shift**
between Galilean transformation and "first-order reduced" Lorentz
transformation:

** 0
s/m** versus

The change from *1* to *1+10 ^{-16}* is actually
minimal. However, no multiplication-factor is big enough to transform

Whereas a Lorentz-factor of *1+10 ^{-16}* does not
depend on any units, the numerical value of a time-shift

To the mathematical question whether the Lorentz transformation
reduces to the Galilean transformation or not, it is irrelevant that *1 sec
per light-year* may seem negligible to you. On a cosmic scale, the
corresponding length-shift (resp. speed) of *10 m/s ≈ 10 ^{-15}
LY/sec* could be considered negligible too.

By Darwin123 in #162:

I did not
compare *v* and *v/c ^{2}* since they are in different units.
You know that and I know that.

Should
this be an invitation to read between the lines, or do you simply confuse
written-by-*Perpetual-Student* with written-by-*Darwin123*?

By Perpetual Student in #167:

Darwin123, I find it troubling to say that this is a lost cause.

By Darwin123 in #168:

I know that. I have fun with lost causes.

*Arguments from
Baierlein's 'Two myths about special relativity'*

*#184
– 2015-11-01*

A quote from *Two myths about special
relativity* by Ralph Baierlein:

Q. Does the Lorentz
transformation reduce to the Galilean transformation when the ratio *v/c*
is small?

A. No.

One argument of the author:

Consider the usual pair of
inertial reference frames, the primed frame moving with speed *v* along
the *x* axis of the unprimed frame. To avoid any spurious dependence on
the origins of coordinate systems, consider a pair of physical events. The
Lorentz transformation for the time interval between the events takes the form

*Δ**t' = 1/√(1-v ^{2}/c^{2}) (*

Let the ratio *v/c*
be as small as desired (but nonzero). Then it is always possible to find an
event pair for which *Δ**x* is large enough that the term with *Δ**x* dominates over the term with *Δ**t*. This behavior is entirely different
from what the Galilean transformation *Δ**t' = **Δ**t* asserts.

It seems to me that this argument is not complete. Let us create
an **artificial** time-transformation by replacing *v/c ^{2}* of
the Lorentz transformation with

*Δ**t' = 1/√(1-v ^{2}/c^{2}) (*

Any reasonable program for symbolic computation can calculate the first-order
Taylor expansion with respect to *v* of the artificial time
transformation. The result:

*Δ**t' = **Δ**t*

Nevertheless, for every given *Δ**t* it is possible to find a *Δ**x* large enough that

*|v ^{2}/c^{2}
*

Therefore, from Baierlein's argument as quoted above we could
conclude that our artificial transformation does not reduce to *Δ**t' = **Δ**t*. Why is the argument valid for the Lorentz time-transformation,
but not for our artificial transformation?

In the Lorentz case, in order to continue to "dominate"
a given *Δ*** t**, any further reduction from speed

Yet in the analogue situation with the artificial transformation,
length interval *Δ**x* must increase by factor *v ^{2}/w^{2}* in order
to "dominate" a given time interval

In general, when "reducing" formulas, the order-number
of the reduction is essential. E.g. the first-order reduction of kinetic energy
*E _{kin} = m v^{2}/2* is

*E _{kin} =
m c^{2} ( (1-v^{2}/c^{2})^{-0.5} - 1 )*

we assume a second-order Taylor expansion.

A further argument from the same chapter *LOW-SPEED BEHAVIOR OF
THE LORENTZ TRANSFORMATION*
of Ralph Baierlein:

For a sophisticated
justification, note that the composition (the successive use) of two Lorentz
transformations is equivalent to another Lorentz transformation. This
equivalence is the group property of the Lorentz transformation. Moreover, the
Lorentz transformation is differentiable with respect to *v/c*, and the
derivative is nonzero at *v/c=0*. Consequently, any Lorentz transformation
with finite speed can be constructed by iterating a Lorentz transformation with
a small (and ultimately infinitesimal) ratio *v/c*.

If the Lorentz
transformation for infinitesimal *v/c* were to reduce to the Galilean
transformation, then the iterative process could never generate a finite
Lorentz transformation that is radically different from the Galilean transformation.
But the finite transformations are indeed radically different, and so—however
subtly—the infinitesimal Lorentz transformation must differ significantly from
the Galilean transformation

Very interesting! A first-order effect with respect to *v* of
a formula *f[v]* can only disappear, if the first derivative *df[v]/dv*
is *zero* at *v = 0*. However, if we actually calculate the first
derivative of the Lorentz time-transformation *t'* at *v = 0*, we
get:

*dt'/dv = -x/c ^{2}*

This is a clear indication that the Lorentz time-transformation
reduces to *t' = t* only at point *x = 0*.

---

*The Lorentz-transformation is dead. Long live the
Lorentz-factor!!!*

*Pseudo-solution adopted
by Wikipedia*

*#190 – 2015-11-08*

By jmckaskle in #148:

The Lorentz
transformations reduce to the Galilean transformations by selecting an infinite
value for *c* because under Galilean relativity, light travel is
instantaneous:

x' = (x-vt) /
sqrt[1-v^{2}/c^{2}]

t' = (t-vx/c^{2}) / sqrt[1-v^{2}/c^{2}]

Taking the
limit as *c* goes to infinity:

x' = (x-vt) /
sqrt[1-0]

t' = (t-0) / sqrt[1-v^{2}/c^{2}]

simplifying,

x' = x-vt

t' = t

This is the only way to save the claim that the
Lorentz transformation reduces to the Galilean with *v << c*. The
problem that *vx/c ^{2}* does not disappear in a first-order
expansion with respect to

At least in case of a normal reduction resp. series-expansion with
respect to v, this trick only works if we replace *c* by *k*, where *k
→ ∞* if *v → 0*, and *k → c* if *v →
c*. A concrete example is replacing *c* with *k = c∙(c/v) *.
This modified "Lorentz" transformation actually reduces to the
Galilean transformation, yet the "Lorentz" factor becomes *(1 - v ^{4}/c^{4})^{-1/2}*
instead of

The solution of the dilemma by *c → ∞*
has also been adopted by Wikipedia:

"Another important
property is for relative speeds much less than the speed of light, the Lorentz
transformations reduce to the Galilean transformation in accordance with the **correspondence
principle**."

"Mathematically, as v → 0, c → ∞. In words, as relative velocity approaches 0, the speed of light (seems to) approach infinity."

The
chapter *Relativistic
kinetic energy* of *Correspondence principle* presents
the example of relativistic kinetic energy reducing to classical kinetic
energy. In this case *v << c* means:

*E _{kin} = ½ mv^{2}*
is a good approximation for the relativistic formula if e.g.

Increasing a fundamental constant such as light-speed *c* is
quite different from reducing velocity *v*. If *v → 0* and *c
→ ∞* were interchangeable in this respect then it should also be
possible to derive classical kinetic energy from the relativistic *E _{kin}
= mc^{2} ((1-v^{2}/c^{2})^{-0.5}-1)* in such
a way. Yet

The
effect of *c → ∞* on the Lorentz transformation is simple: *xv/c ^{2}*
becomes

"Hence, it is sometimes said that nonrelativistic physics is a physics of 'instantaneous action at a distance'.[10]"

"Nonrelativistic" physics? At least with respect to
"classical" physics, the following is valid: Gravitational, electric
and magnetic interactions were considered instantaneous actions at a distance,
where total energy and momentum of the interacting parts are conserved. Yet
light was considered as something different: Waves (or particles) leave a
source and propagate at a finite speed. Already before Newton laid with his
Principia the "official" foundation of classical mechanics in 1687, the
finiteness of *c*
had been established. Therefore, one cannot
invoke "instantaneous action at a distance" of classical physics in
order to justify *c → ∞*.

Reference
[10] of the Wikipedia quote leads to Einstein's (1916) *Relativity: The
Special and General Theory*. On page 57
we read:

"Let me add a
final remark of a fundamental nature. The success of the Faraday-Maxwell interpretation
of electromagnetic action at a distance resulted in physicists becoming
convinced that there are no such things as instantaneous actions at a distance
(not involving an intermediary medium) of the type of Newton's law of
gravitation. **According to the theory of relativity, action at a distance
with the velocity of light always takes the place of instantaneous action at a
distance** or of action at a distance with an **infinite velocity** of
transmission."

If *c* is an upper speed limit, then what was considered
infinite velocity in classical physics can only be *c*, and Einstein's
claim that propagation at *c* takes the place of instantaneous actions at
a distance is correct. Yet Einstein's statement does not justify *c → ∞*,
because it does not imply that according to classical physics, instantaneous
action at a distance takes the place of light propagating at *c*.

*Fizeau's experiment as
King's evidence*

*#194 – 2015-11-12*

By Reality Check in #192:

The Lorentz
transformation reduces to the Galilean with *v << c*... In those
limits the actual Lorentz factor (not your imaginary one) **reduces to 1**
and the Lorentz transformation reduces to the Galilean transformation.

Fizeau's experiment concerning relative speed of light in a moving medium is considered a consequence of relativistic velocity addition. This means:

The Lorentz transformation with a **small speed v**
of a transparent medium relative to a laboratory leads to a

In case of water, refraction of *n ≈ 1.33* leads to a
drag of *f ≈ 0.44*. If water in a tube starts moving at *1 m/s*
in direction of light propagation, the speed of this light increases relative
to the laboratory from ** w = c/n ≈ 2.25∙10^{8} m/s**
by

If the Lorentz transformation actually reduced to the Galilean in
case of a velocity as low as *v = 1 m/s* then the application of the
Lorentz transformation with *v = 1 m/s* would increase speed *w = c/n*
by *1 m/s* and not by *0.44 m/s*. Or does anybody deny that the
Galilean transformation with *v = 1 m/s* changes any speed *w* by *1
m/s*?

Light-speed *w' = c/n* in moving water means that in a moving
frame *F'* where the water is at rest, light travels during time-interval *Δ**t'* distance *Δ**x' = w'∙**Δ**t'*. In order to get light-speed *w* with respect to the laboratory frame
*F*, we apply the Lorentz transformation
and calculate *w = **Δ**x /**Δ**t*:

*Δ**x** = **γ** (**Δ**x' + v **Δ**t') = **γ**
(w' **Δ**t' + v **Δ**t') = **γ** **Δ**t' (w' + v)
*

The Lorentz factor ** γ** is irrelevant as it cancels out, and we get:

*w** = **Δ**x/**Δ**t** = (v + w') /(1 + v w'/c ^{2}) = (v + c/n)/(1 + v/(c∙n))*

Series expansion of *w* with respect to *v* leads to:

*w** = c/n + (1 - 1/n ^{2}) v + *

Thus, relativity of simultaneity (or Lorentz's local time of 1892) is assumed to explain Fizeau's experiment, and all the insolvable paradoxes arising from relativity of simultaneity can be adapted to partial dragging of light by a medium.

---

*Human history is full of empiric and experimental confirmations
of wrong beliefs*

Let us assume a closed water flow through four interconnected
straight tubes forming a square. Each edge has a mirror so that light can
change direction from one tube to the next, with a light path of each *3 m*
between two neighboring mirrors.

With water **at rest** in the tubes, speed of light is *c/n ≈
2.25∙10 ^{8} m/s* relative to both the water and the laboratory
(assumed at rest). After the water starts

With respect to the water
flowing on the closed light-path, time at each mirror in front must be *Δ**t'** = - 3m ∙ 1m**/**s / c ^{2} = 3.34∙10^{-17}
sec* in the past with respect to the respective back mirror. Yet
because the light comes back to the same mirror after having changed direction
by means of the three other mirrors, each mirror should be four times

Every atom of both the moving water and the laboratory can be
considered a clock. As the distances between all these clocks remain constant
(apart from the *12 sec* cycle of flowing water), we must apply absolute
simultaneity. (Only if the distance between two clocks continuously decreases/
increases, relative simultaneity does not lead to contradictions, see also *Simple Refutation of Special Relativity by
Light Clock*.)

**Summary:**

o
The low-speed approximation of the Lorentz time-transformation is *t'
= t - vx/c ^{2}* (and not the Galilean

o
This *t' = t - vx/c ^{2}* explains the only partial
drag of light by water in Fizeau's experiment (via relativistic velocity
addition).

o
In a laboratory with a closed, cyclic water flow, we have absolute
simultaneity *t' = t*.

o Thus, the explanation of Fizeau's experiment by the Lorentz transformation is untenable.

---

*Human faculty of reasoning is in principle much better adapted
to the world we experience than modern science makes us believe. Dilemmas and
paradoxes only point to an inadequacy of premises of our beliefs and theories,
and not to an inadequacy of human reason itself*