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/c2 resp. v∙x being a first-order effect.

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)     γ = 1 / sqrt[1 - v2]

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 v∙t of the x-transformation, and as result we get the "zero-order" transformation:

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.)

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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/c2 term disappears is because for the observer on earth looking at the rocket, x=v∙t.

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 t' = t - v/c2 x. The center x' = 0 of the moving reference-frame F' obviously remains at the center of itself (despite moving according to x = v t in rest-frame F). So you get t' = t from t' = t - v∙x/c2 by replacing x with x' = 0. This resolves the problem at most in a very superficial way, and in a very special case.

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.

#135 – 2015-10-12

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/c2)

With c = 1, the v in x' = x - vt is the actual velocity in the x direction. However, because of the c2 term in the denominator with vx, where t' = vx/c2, v is the ratio of the actual velocity to c. If you do a dimensional analysis of the two equations, you will see that MUST be the case. To be clear, vx alone is NOT compatible with t, whereas vt is compatible with x.

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 t' = t - v/c2 x I wanted to stress the symmetry between time and length transformation. Exactly this symmetry explains the constancy of c in the Lorentz transformation (see Simple Derivation of the Lorentz Transformation).

Some may even argue that v/c2 x can be removed as a term from the first-order limit because the constant c is as high as 3∙108 m/s. Yet it is obvious that the numerical value of c only depends on the units meter and second. If we use light-year and second then the value of c becomes as low as 3.17∙10-8 light-year per sec.

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 [
γ (x - v t), {v, 0, n}]
Series [
γ (t - v/c^2 x), {v, 0, n}]

If n = 0, the result is:

x' = x       t' = t

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

x' = x - t v       t' = t - x/c2 v

The second-order result:

x - t v + x/2/c2 v2       t - x/c2 v + t/2/c2 v2

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.

#152 – 2015-10-18

By wogoga in #135:

Some may even argue that v/c2 x can be removed as a term from the first-order limit because the constant c is as high as 3∙108 m/s. Yet it is obvious that the numerical value of c only depends on the units meter and second. If we use light-year and second then the value of c becomes as low as 3.1710-8 light-year/sec.

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

We have: x' = γ (x - vt) and t' = γ (t - vx/c2)

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 1.

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

And t' = t - vx/c2 = t - 10-16 s which is infinitesimally close to t, to about the same order of magnitude that γ = 1. So t' = t, also as Galileo would have had it.

No objection can be raised against your choice of v = 10 m/s, leading to a negligible Lorentz factor, but the choice of x = 1 m and t = 1 sec is in principle completely arbitrary, and depends on the units meter and second. With your choices we get:

x' = x - v∙t = 1 m - 10 m
t' = t - v∙x/c2 = 1 s - 1.11∙10-16 s

Keeping v∙t = 10 m and dropping v∙x/c2 = 1.11∙10-16 sec seems justified.

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 t = 1 sec. The Lorentz factor of v = 10 m/s = 3.33∙10-8 LS/s obviously does not change and remains negligible. This time, we get:

x' = x - t∙v = 1 LS - 3.33∙10-8 LS
t' = t - x∙v/c2 = 1 s - 3.33∙10-8 s

Keeping v∙t = 3.33∙10-8 LS and dropping v∙x/c2 = 3.33∙10-8 s seems unjustified.

The Taylor expansion of the Lorentz factor (1-v2/c2)-1/2 with respect to v yields:

1 v0 + 0 v1 + 1/(2c2) v2 + 0 v3 + 3/(8c4) v4 + . . .

If meter per second is used as unit for velocity, and both v and c = 2.998∙108 express unit-less numeric values, then also higher-order terms of the Lorentz-factor expansion turn analogously into seemingly negligible quantities:

1 + 5.56∙10-18 v2 + 4.64∙10-35 v4 + . . .

By the way, the zero-order Taylor-expansion term of the Lorentz factor, 1 v0 = 1, becomes after multiplication with -x∙v/c2 the first-order term of the expansion of the t'-transformation. The term t - x∙v/c2 itself is its own full Taylor expansion t v0 - x/c2 v1.

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

#161 – 2015-10-23

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/c2 = 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 c expressed in meter per second.

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 1.11∙10-17 sec/m.

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∙108)+1 β m/s
v/c2 =
β s/LS = β year/LY = (3∙108)-1 β s/m

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/c2 is necessary to explain that e.g. the speed of light from an astronomical object near the ecliptic does not change from 0.9999 c to 1.0001 c in the course of a year. Once in a year, the Earth moves with 30 km/s [i.e. 0.0001 c] in direction to the object, and half a year later, with 30 km/s away from the object. In case of a galaxy at a distance of 107 light-years, the galaxy makes according to SR every year a time-shift cycle with amplitude of 0.0001107 = 1000 year. Such a galaxy-migration from 1000 years in the past to 1000 year in the future and back during one Earth year is a substantial, first-order effect. Yet length contraction is only a second order effect. A speed of 30 km/s (with Lorentz-factor 1+5∙10-9) reduces a distance of 107 LY only by 0.05 LY."

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.

#175 – 2015-10-28

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 1+10-16

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

0 s/m versus 10-16 s/m

The change from 1 to 1+10-16 is actually minimal. However, no multiplication-factor is big enough to transform 0 s/m to 10-16 s/m. The difference between simultaneity and time-shift (linearly depending on distance) is even qualitative.

Whereas a Lorentz-factor of 1+10-16 does not depend on any units, the numerical value of a time-shift 10-16 s/m can be increased by changing units, e.g. to around 1 sec/LY.

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/c2 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

Δt' = 1/√(1-v2/c2) (Δt - v/c2 Δx)

Let the ratio v/c be as small as desired (but nonzero). Then it is always possible to find an event pair for which Δx is large enough that the term with Δx dominates over the term with Δt. This behavior is entirely different from what the Galilean transformation Δt' = Δt asserts.

It seems to me that this argument is not complete. Let us create an artificial time-transformation by replacing v/c2 of the Lorentz transformation with v2/c2:

Δt' = 1/√(1-v2/c2) (Δt - v2/c2 Δx)

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

Δt' = Δt

Nevertheless, for every given Δt it is possible to find a Δx large enough that

|v2/c2 Δx| > |Δt|

Therefore, from Baierlein's argument as quoted above we could conclude that our artificial transformation does not reduce to Δt' = Δt. 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 Δt, any further reduction from speed v << c to w < v must be compensated by increasing Δx with factor v/w. Thus, v Δx → const, if v → 0.

Yet in the analogue situation with the artificial transformation, length interval Δx must increase by factor v2/w2 in order to "dominate" a given time interval Δt. This necessity to square v/w, leading to v Δx → infinity if v → 0, is a clear hint that there is no first-order effect.

In general, when "reducing" formulas, the order-number of the reduction is essential. E.g. the first-order reduction of kinetic energy Ekin = m v2/2 is zero. Thus, when we claim that the classical kinetic energy can be derived as a reduction from the energy-equivalent of the relativistic mass-increase

Ekin = m c2 ( (1-v2/c2)-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/c2

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

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

#190 – 2015-11-08

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-v2/c2]
t' = (t-vx/c2) / sqrt[1-v2/c2]

Taking the limit as c goes to infinity:

x' = (x-vt) / sqrt[1-0]
t' = (t-0) / sqrt[1-v2/c2]

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/c2 does not disappear in a first-order expansion with respect to v is obviously solved by c → ∞.

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 - v4/c4)-1/2 instead of (1 - v2/c2)-1/2.

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:

Ekin = ½ mv2 is a good approximation for the relativistic formula if e.g. v = 0.01 c, a bad approximation if v = 0.5 c, and fully wrong if v= 0.99 c.

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 Ekin = mc2 ((1-v2/c2)-0.5-1) in such a way. Yet c → ∞ instead of v → 0 only leads to Ekin = mc2 (1-1) = 0.

The effect of c → ∞ on the Lorentz transformation is simple: xv/c2 becomes zero, and the Lorentz factor becomes one. By increasing c to infinity we do not get a series expansion but only a limit value.

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

"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  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 relativistic effect, capable of explaining partial drag of light according to the Fresnel drag coefficient f = 1 - 1/n2. This obviously could not be possible if the Lorentz transformation turned for v << c into the Galilean, as generally claimed.

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∙108 m/s by 0.44 m/s (instead of 1 m/s). This only partial drag of light by the movement of the medium is explained by the Lorentz transformation with v = 1 m/s.

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 Δt' distance Δx' = w'∙Δt'. In order to get light-speed w with respect to the laboratory frame F, we apply the Lorentz transformation and calculate w = Δx /Δt:

Δx = γ (Δx' + v Δt') = γ (w' Δt' + v Δt') = γ Δt' (w' + v)
Δt = γ (Δt' + v/c2 Δx') = γ (Δt' + v/c2 w' Δt') = γ Δt' (1 + v w'/c2)

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

w = Δx/Δt = (v + w') /(1 + v w'/c2) = (v + c/n)/(1 + v/(c∙n))

Series expansion of w with respect to v leads to:

w = c/n + (1 - 1/n2) v + (1 - n2)/(c∙n3) v2 +

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.

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

#204 – 2015-11-22

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∙108 m/s relative to both the water and the laboratory (assumed at rest). After the water starts circulating at v = 1 m/s in direction of light propagation, the speed of the light relative to the laboratory will increase to only w = c/n + 0.44 m/s (instead of w = c/n + 1 m/s). This is an empirical fact confirmed by Fizeau's experiment. As the water moves at 1 m/s in the same direction as the light, from absolute simultaneity we conclude that relative to the water, light moves no longer with c/n but with w' = c/n + 0.44 m/s - 1 m/s = c/n - 0.56 m/s. Thus, we need relativity of simultaneity in order get again the original light speed w' = c/n relative to the water (instead of w' = c/n - 0.56 m/s):

With respect to the water flowing on the closed light-path, time at each mirror in front must be Δt' = - 3m ∙ 1m/s / c2 = 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 3.34∙10-17 sec in the past with respect to itself, which obviously is impossible.

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/c2 (and not the Galilean t' = t).

o    This t' = t - vx/c2 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.

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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

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