Time Dilation & Twin Paradox

Refutation of Special Relativity for Dummies (Time Dilation, Twin Paradox)

By Wolfgang G. Gasser

Original discussion – Comments or criticism can be posted there or sent to: info@pandualism.com

#1 – 2014-12-05

An electron and a positron at rest can annihilate into two photons, each with a frequency corresponding to mass/energy of one electron (or positron).

In order to refute SR, we simply ask what happens in this situation:

Electron and positron annihilate while both moving at (relativistic) speed v.

From energy conservation we conclude:

The sum of the two frequencies [1] of the emerging photon-pair is higher than the sum calculated by applying classical Doppler shift [2] to the annihilation event with rest mass, as kinetic energy of the electron-positron-pair increases the sum of the photon frequencies.

From time dilation (or relativistic Doppler effect) we conclude:

The sum of the two frequencies [1] of the emerging photon-pair is lower than the sum calculated by applying classical Doppler shift [2] to an annihilation event with rest mass, as time dilation reduces photon frequencies.

Using common sense, we can easily recognize that a real experiment can only confirm energy conservation (i.e. higher frequencies) and refute time dilation.
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[1] Two photons with freq₁ and freq₂ have the same energy as a single photon with freq_sum = freq₁ + freq₂, i.e. total frequency is proportional to total energy.
[2] The relativistic Doppler effect is the combination of the classical Doppler effect [3] with time dilation of SR.
[3] with observer (=receiver) at rest relative to the medium, and source moving at -c < v < c

#15 – 2014-12-07

Sorry, the title should rather be: Refutation of Special Relativity from Idiocy

When dealing with mass-energy equivalence and center of mass/energy considerations in another thread, I got the impression that it is possible to derive relativistic time dilation (according to the Lorentz factor). Because of an unfortunate series of events/mistakes, I came to the opposite conclusion, namely that Lorentz-factor time-dilation must be wrong.

Reasoning "mass-energy conservation"

An electron-positron pair moves at v and disintegrates into two photons, propagating along the line of v with c and -c.

The center of mass of the two emerging photons must continue to move at v. This can only happen if the relativistic mass m₁ of the forward moving photon is higher than m₂ of the backward moving photon:

p_tot / m_tot = v
(m₁ c - m₂ c) / (m₁ + m₂) = v
(m₁ c - m₂ c) = (m₁ v + m₂ v)
m₁ (c-v) = m₂ (c+v)
m₁ = m₂ (c+v)/(c-v)

In the case of photons, relativistic mass is proportional to frequency. So we conclude:

f₁ = f₂ (c+v)/(c-v)

The energy of the electron-positron pair was 2 y E, where E = 511 keV is the rest-energy of an electron or proton, and y the Lorentz factor of v. As f = E/h we get:

f₁ + f₂ = 2 y E / h

Reasoning "relativistic Doppler effect"

Now, let us apply the relativistic Doppler effect with observer velocity v to this situation: An electron-positron pair at rest gives rise to two photons moving (along the line of v) in opposite directions with each frequency f. In the rest-frame, we get:

f = E/h = 511 keV / h

For the moving observer, the relativistic Doppler effect leads to two photons of different frequency and energy:

f₁ = f ∙ √[c+v] / √[c-v]
f₂ = f ∙ √[c-v] / √[c+v]
E₁ = h f₁ = h f ∙ √[c+v] / √[c-v]
E₂ = h f₂ = h f ∙ √[c-v] / √[c+v]

We get:

f₁ = f₂ (c+v) / (c-v)
E₁ + E₂ = 2 h f ∙ (√[c+v] / √[c-v] + √[c-v] / √[c+v] ) = 2 h f / √[1-v²/c²] = 2 E y
f₁ + f₂ = 2 y E / h = 2 f / √ [1-v²/c²]

Conlusion

Both reasonings lead to the same frequencies of the emerging photon pair. Therefore it is not possible to refute SR in this way.

The error in post #1

My loose concept "sum of frequencies of the emerging photon pair calculated by applying classical Doppler shift" is the main culprit. If we do the calculation (taking into account "observer (=receiver) at rest relative to the medium, and source moving at -c < v < c"), we get

f₁ = f / (c-v)
f₂ = f / (c+v)
f₁ + f₂ = (c+v)+(c-v) / (c-v)(c+v) = 2 f / (1-v²/c²) = 2 f y²

Concerning "energy-conservation" we conclude:

The sum (f₁+ f₂ = 2 f y) of the two frequencies of the photon-pair (emerging from the moving electron-positron pair) is ~~higher~~ lower (by Lorentz factor y) than the sum (f₁+ f₂ = 2 f y²) calculated by applying classical Doppler shift to the annihilation event at rest (~~because~~ despite kinetic energy of the electron-positron-pair seeming to increase the sum of the photon frequencies).

In any case, this shows that velocity-dependent time dilation can be derived from mass-energy equivalence, conservation of momentum and conservation of mass/energy alone, without the necessity of the Lorentz transformation. So experimental proof for relativistic time dilation can only be considered evidence, but not proof for the Lorentz transformation, as time dilation follows from conservation laws simpler than the Lorentz transformation. By the way, many experimental results rather show what experimentalists think nature does than what nature actually does.

Until now I have left open the question, whether I should believe in Lorentz-factor time-dilation or not. Now I have to start to believe in it. (I'm careful and arrogant enough to only believe in what I can understand myself.)

#26 – 2014-12-16

By wogoga in #15:

For the moving observer, the relativistic Doppler effect leads to two photons of different frequency and energy:

f₁ = f ∙ √[c+v] / √[c-v]
f₂ = f ∙ √[c-v] / √[c+v]
E₁ = h f₁ = h f ∙ √[c+v] / √[c-v]
E₂ = h f₂ = h f ∙ √[c-v] / √[c+v]

By Darwin123 in #16:

There is no u+c or u-c anywhere in the frequencies.

My above formulas are simply the formulas of the relativistic Doppler effect: Motion along the line of sight.

f[v] = f₀ c / (c+v)

Premise two is time dilation of the moving source (i.e. source time running slower) by the Lorentz-factor y[v] = (1-v²/c²)^-1/2 >= 1. We derive:

f[v] = f₀ ∙ c/(c+v) / y[v]
= f₀ ∙ c/(c+v) ∙ √[1-v²/c²] = f₀ ∙ √[c²-v²] / (c+v)
= f₀ √[(c+v) (c-v)] / √[(c+v)(c+v)] = f₀ √[c-v]/√[c+v]

The reasoning is not only formal, but also very concrete and intuitively accessible, e.g. by counting the wave periods received by an observer at rest.

Assume that a radio source, having a constant frequency f = 1 GHz at rest, moves at v = 0.8 c, passing observer 0 at x₀ = 0 and then observer 1 at x₁ = L = 1 LS (1 Light Second = 300 000 km). The "events" of crossing observer 0 and of crossing observer 1 are clearly defined in all inertial frames. In our rest frame, we also know that it takes for the source t = L/v = 1 LS / 0.8 c = 1.25 seconds to move from x₀ to x₁. (Light needs 1 second for the distance of 1 LS).

Let us now at first assume that frequency f of the source is not affected by the propagation speed v:

During the passage from x₀ to x₁, the source emits n = f t = f L/v = 1.25 ∙10⁹ periods. The wave periods in front of the source are obviously compressed from c/f to (c-v)/f (i.e. by factor (c-v)/c = 0.2). Thus frequency of forward moving waves is increased to f[v] = c/(c-v) f = 5 GHz. To get the frequency of backward propagating radiation we use –v instead of v (as movement of source is opposite to wave propagation): f[-v] = c/(c+v) f = 1/1.8 GHz

From the case where a source is completely converted into radiation (see #15) we conclude that the frequencies f[v] and f[-v] must be reduced by the Lorentz-factor y[v] in order to satisfy energy conservation. This frequency decrease can be explained by time running slower by the same factor.

In order to confirm or refute the Lorentz transformation, we must check the opposite case: moving observer and stationary sources. According to common sense:

For the moving observer, the time of the sources runs faster than proper (own) time.

And according to the Lorentz transformation:

For the moving observer, the time of the sources runs slower than proper time, as time dilation is mutual (leading to the so called twin paradox).

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A refutation actually satisfying the thread title: Light clock refutes constancy of c

#33 – 2014-12-16

By wogoga in #26:

The relativistic Doppler formulas can be derived from the classical Doppler effect under two premises. Premise one: receiver velocity relative to the medium v_receiver = 0. … Premise two is time dilation of the moving source (i.e. source time running slower) by the Lorentz-factor y[v] = √[1-v²/c²)^-1/2 >= 1.

There is a second possibility. The relativistic Doppler formula can be derived from the classical Doppler effect under two premises. Premise one: source velocity relative to the medium v_source = 0. Thus v = v_receiver, and we get:

f[v] = f₀ (c-v) / c

Premise two is time contraction of the source at rest (i.e. source time running faster) by the Lorentz-factor y[v] = (1-v²/c²)^-1/2 >= 1. We derive:

f[v] = f₀ ∙ (c-v)/c ∙ y[v]
= f₀ (c-v)/c (1-v²/c²)^-1/2 = f₀ (c-v) (c²-v²)^-1/2
= f₀ (c-v)^1/2 (c-v)^1/2 (c+v)^-1/2 (c-v)^-1/2 = f₀ ∙ √[c-v]/√[c+v]

So instead of derivation 1

· classical Doppler shift: observer at rest, moving source

· time dilation of moving source (with respect to receiver at rest)

we can also use derivation 2

· classical Doppler shift: source at rest, moving observer

· time contraction (i.e. running faster) of source at rest (with respect to moving observer)

in order to get the relativistic Doppler effect (in longitudinal direction, receiver = observer).

At least in case of time-dilation of objects moving around closed tracks in particle accelerators, derivation 1 cannot be applied to radiation emitted from accelerator-parts (assumed at rest) and received resp. observed by such a high-speed object. The reason is simple:

Insofar as time of the high-speed object runs slower with respect to the particle-accelerator, particle-accelerator time runs faster with respect to the high-speed object.

The application of the classical Doppler shift source at rest, moving observer in derivation 2 then shows that light propagation with respect to the moving high-speed observer cannot be c, but must be c+v or c-v in the longitudinal case.
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Lorentz-Ether-Theory and SR Simultaneity, Contraction & Expansion

#43 – 2015-08-17

By Reality Check in #39:

Do you really think that SR ignores its postulate that the speed of light is c according to all inertial observers?

I only want to show that also from the alternative premise "speed of light is c only for observers in a privileged frame assumed at rest", (non-mutual) time-dilation can be derived. In general I try to understand why Special Relativity is such an efficient and useful theory despite being fundamentally flawed.

Take for instance the twin paradox:

"In 1911, Paul Langevin gave a 'striking example' by describing the story of a traveler making a trip at a Lorentz factor of y = 100 (99.995% the speed of light). The traveler remains in a projectile for one year of his time, and then reverses direction. Upon return, the traveler will find that he has aged two years, while 200 years have passed on Earth. During the trip, both the traveler and Earth keep sending signals to each other at a constant rate, which places Langevin's story among the Doppler shift versions of the twin paradox."

We can understand why such a Doppler-shift resolution actually seems to work. For the twin at rest we get the relativistic shift using these premises:

· signals move at c (relative to himself)

· time of travelling twin runs slower

From this we derive by "normal" logic that the following is valid for the travelling twin:

· signals move at c relative to twin at rest

· time of twin at rest runs faster

The SR derivation of the Doppler shift for the travelling twin however is based on these premises:

· signals move at c (relative to himself)

· time of twin at rest runs slower

Both derivations for the travelling twin lead (by math-magic) to the same Doppler shift. The first derivation (implying a signal-speed from c-v to c+v relative to himself) obviously violates Special Relativity. Yet also the second derivation with "normal" application of SR leads to a serious problem which becomes evident by introducing a light-clock on Earth.

The light-pulse in the clock on Earth is assumed to move orthogonally to the movement of the travelling twin. Without time dilation of this clock by Lorentz-factor 100, with respect to the travelling twin, the speed of this light-pulse would be not c but around 1.4 c (hypotenuse of right triangle with c and 0.99995 c). Thus, with respect to the travelling twin, during 1.00005 year (proper time) before reversing direction, only 0.01 year can pass in the light clock at rest (otherwise the light-pulse would be superluminal). Also during the return trip of 1 year, only clock cycles corresponding to 0.01 year can be accomplished in the light clock at rest.

Thus we have to conclude: During reversal of direction, with respect to the travelling twin, clock-cycles corresponding to 199.98 year must be accomplished in the light-clock. During these 199.99 year (200.01 - 0.02 year) the light-pulse in the clock moves a distance of 199.99 light-year. And a light-pulse moving 199.99 light-year during an arbitrarily short time of direction-reversal quite obviously violates the very principle SR is based on.

By Reality Check in #39:

Do you really think that scientists are so stupid that they do not test SR - including the speed of light from moving sources?
What is the experimental basis of Special Relativity? 3.3 Tests of Light Speed from Moving Sources

Please choose the experiment which according to you most convincingly contradicts what I have written. Then I can demonstrate that the experiment is either irrelevant or simply wrong.
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The assumption that a theory as complex as Special Relativity is either true or false is unduly simplistic!

#50 – 2015-08-19

By wogoga in #43:

Please choose the experiment which according to you most convincingly contradicts what I have written. Then I can demonstrate that the experiment is either irrelevant or simply wrong.

By Reality Check in #44:

Please show how "Operation of FLASH, a free-electron laser" is wrong.

This experiment is irrelevant, as my arguments have nothing to do with any form of ballistic theory of light.

Quote from the summary of the experiment:

"... relativistic electron beam … generates the X-rays … speed of light emitted from the moving electrons is c+kv … observed … upper limit on k of 2.5×10⁻⁷…"

In the same way as Einstein, I assume that k is exactly zero. Nevertheless, I assume that Einstein has been fooled into believing that this is caused by the Lorentz transformation. In reality however, this is caused by ether dragging according to the inverse-square-distance law.

The refutations of this ether-drag hypothesis are as wrong as Einstein's "resolution" of the twin-paradox.

#53 – 2015-08-21

By Kid Eager in #51:

Oh really? So how does your interpretation of aether dragging reconcile the failure to accommodate stellar aberration, whilst relativity does?

All aberration-based arguments against dragged ether start with the premise that light is a continuous phenomenon analogous to the propagation of sound, where propagation direction is orthogonal to the wave-front.

If the photon hypothesis already had been generally accepted, then such a strange hypothesis as "length contraction" probably would not have been taken seriously in the first place. The alternative hypothesis "wave-fronts breaking-apart" is simpler and much less mind-boggling.

A photon emitted by the sun simply follows the direction of inertial movement when it adapts its speed more and more to the average velocity of the particles constituting the mass of the Earth. This complicates a little bit further the already complicated question of coherence-of-light, but it fully explains the fact that sun-light is "aberrated".

Ether dragging according to the inverse-distance-square law means that every particle in the universe drags the ether according to its mass (in the same way as every particle generates gravitational attraction). In order to calculate the movement of the ether at a given point, we simply use weighted averages of these effects (if interested see). Gravitational time dilation due to lost gravitational potential, and light bending due to gravitation show that masses are capable of influencing photon behavior.

#56 – 2015-08-22

By Reality Check in #44:

To derive time dilation you need to use all of SR. That post [#43] was silly - it starts with the relativistic Doppler formula to arrive at relativistic time dilation.

As long as you do not understand the simple reasoning shown in post #33, you cannot judge #43.

And what do you consider wrong with my treatment of Langevin's twin paradox in #43, with Lorentz-factor 100 and one-way travel distance of 100 LY)?

Let us use two light-clocks (with light-pulse moving orthogonally to direction of relative movement) instead of twins: one clock at rest (R-clock) on Earth and the other clock (T-clock) travelling at v = 0.99995c.

1. Distance of 100 light-year is reduced for T-clock to 1 LY during both outward and return journey.

2. Both trips need each 100 LY / 0.99995c = 100.005 year, resp. 1 LY / 0.99995c = 1.00005 year.

3. During 1.00005 year of one way, pulse in T-clock moves 1.00005 LY relative to T-clock.

4. During 1.00005 year of one way, pulse in R-clock moves 1.00005 LY relative to T-clock.

5. This 1.00005 LY path of R-clock pulse relative to T-clock is only a 0.0100005 LY path relative to R-clock.

6. During 1.00005 year of inertial motion, 0.01 year pass in R-clock (corresponding to the 0.01 LY).

7. During two times 1.00005 year of inertial motion, 0.02 year pass in R-clock.

8. When both clocks meet again, 200.01 year (corresponding to light-pulse path of 200.01 LY) have passed in R-clock.

9. During direction-change, pulse in R-clock moves 200.01 LY – 0.02 LY = 199.99 LY.

Which point do consider invalid? Or do you think that the light-clock principle itself is not consistent with SR, or that SR-simultaneity cannot be applied to the inertial motion of the travelling clock?

The contradiction is obvious:

o For the travelling clock, time of the clock at rest is running slower by factor 100 during both inertial movements.

o For the travelling clock, time of the clock at rest runs on average faster by factor 100 over the whole round trip.

o Yet running faster by factor 100 is the opposite of running slower by factor 100!

o The confusion between "slower by 100" and "slower by 0.01" as the opposite of "faster by 100" has helped SR to prosper.

#62 – 2015-08-25

By fuelair in #60:

Show your research and your math or give up. You won't win this because you are wrong.

Insofar as your comment represents modern "science", it shows that "science" essentially is based on faith (in authoritative orthodoxy).

By AdamSK in #59:

You're not paying attention to what each of the two observers will actually see.

In my opinion, you try to rescue Special Relativity by confusing world-map with world-picture. Quotes from Essential Relativity, Wolfgang Rindler, 1977, World-picture and world-map:

"In relativity it is especially important to distinguish between the set of events that an observer sees at one instant and the set of events that the observer considers to have occurred at that instant. What an observer actually sees or can photograph at one instant is called a world-picture. It is a composite of events that occurred progressively earlier as they occurred farther away. For our present purposes it is irrelevant."

"The concept that plays a pervasive role in special relativity is that of the world-map. As the name implies, this may be thought of as a (3-dimensional) map of events, namely those constituting an observer’s instantaneous 3-space t = t₀. It could be produced by having auxiliary observers at the coordinate lattice-points all map their immediate neighborhoods at a pre-determined time t = t₀, and then joining all these local maps into a single global map."

"The world-map is generally what matters."

By Reality Check in #58:

There is no contradiction about observers measuring that each other's clocks tick slower. That is what SR states!

For the travelling clock, the clock at rest is running slower during any inertial movements.
This is standard SR.
The clock at rest after the whole round trip reads less than the travelling clock.
This is the twin paradox - the at home twin is younger than the travelling twin!

Your first point is correct:

· For the travelling clock, the clock at rest is running slower during any inertial movements.
This is standard SR.

Your second point however, despite seemingly being a common-sense conclusion from the first point, is wrong:

· The clock at rest after the whole round trip reads less than the travelling clock.
This is the twin paradox - the at home twin is younger than the travelling twin!

Correct within " standard SR " is the exact opposite:

The clock at rest after the whole round trip reads more than the travelling clock.
This is the twin paradox - the at home twin is older than the travelling twin!

Actually your error is a salient incarnation of the Lorentz-factor confusion I mentioned at the end of post #56.

#66 – 2015-08-30

By Reality Check in #52:

Show that Einstein's resolution of the twin paradox is wrong. Start with citing his paper that you should have read.

[The following analysis is wrong. See #109]

The article Dialog about Objections against the Theory of Relativity, 1918 is quite revealing and shows that poor Einstein really had to struggle with the many serious objections against Relativity. The Critic of the dialog at least somehow represents also Einstein's own "reservations about the theory" and insecurity. For instance (first paragraph):

"We have no wish to dwell on whether this neglect [to deal with objections] was due to arrogance, or a sense of weakness, or laziness – maybe it was a particularly effective mixture of these afflictions of the soul".

Interestingly, Einstein also uses his Critic to criticize other critics (second paragraph):

"I am not so full with the status of my guild so as to make me act as a superior being with superhuman insight and certainty (…). On the contrary, I talk as a human being, since I am aware that it is not rare for criticism to originate from lack of own thoughts."

I've compiled a short text with the essence of Einstein's resolution of the twin paradox, Essence of Einstein's Resolution of the Twin Paradox, using only extracts from his original (translated) text.

Here I deal with the last paragraph of this compilation. At first, I quote Einstein, and then, I show the implications for my treatment of Langevin's twin paradox in #43 and #56.

Einstein: "According to both descriptions the clock U2 is running a certain amount behind clock U1 at the end of the observed process."

Implication: When both clocks meet again, travelling T-clock is behind R-clock, because only 2.0001 year have passed in T-clock whereas 200.01 year have passed in R-clock.

Einstein: "When relating to the coordinate system K' the behavior explains itself as follows: During the partial processes 2 and 4 [inertial motion] the clock U1, going at a velocity v, runs indeed at a slower pace than the [in K'] resting clock U2."

Implication: With respect to T-clock, R-clock runs indeed slower according to Lorentz-factor 100 during inertial motion of both forward and return trip of the journey. This results in 0.020001 year in R-clock versus 2.0001 year in T-clock.

Einstein: "However, this is more than compensated by a faster pace of U1 during partial process 3."

Process 3: "A homogenous gravitational field appears, that is directed towards the positive x-axis. Clock U1 is accelerated in the direction of the positive x-axis until it has reached the velocity v, then the gravitational field disappears again."

Implication: During direction reversal, R-clock runs faster by 199.99 year. In this way, the time delay of 2.0001 - 0.020001 = 1.98 year of T-clock with respect to R-clock during inertial motion is transformed for the whole journey into a time delay of 200.01 - 2.0001 = 198.01 year of R-clock with respect to T-clock (implying a two-time change by factor 100).

I'm genuinely flabbergasted by Einstein's continuation:

"According to the general theory of relativity, a clock will go faster the higher [weaker] the gravitational potential of the location where it is located, and during partial process 3 U2 happens to be located at a higher [weaker] gravitational potential than U1."

I had to read this several times on different days before noticing and becoming (almost) certain that already this statement stems from confusion and wishful thinking. Einstein must have confused "higher gravitational potential" with "stronger gravitational potential" or "[in K'] resting clock U2" with "clock at rest U1". If clock U1 is accelerated by a gravitational field, then U1 is located at a lower (stronger) gravitational ~~field~~ potential. And this makes clock U1 running slower, and not faster as it would be necessary to resolve the paradox.

"The calculation shows that this speeding ahead constitutes exactly twice as much as the lagging behind during the partial processes 2 and 4."

Even if instead of gravitational time dilation we had its opposite, gravitational time contraction, this statement still would be untenable, and Einstein should have written:

Under the premise that Relativity Theory is consistent, a calculation must show that this speeding ahead of clock U1 constitutes on a logarithmic scale exactly twice as much as the clock's lagging behind during the partial processes 2 and 4.

I know from personal experience that all too willingly one accepts the result of a superficial calculation or reasoning, if it agrees with one's expectation, for whatever reason.

In any case, even if I do not agree with every detail of On Einstein's resolution of the twin clock paradox, C. S. Unnikrishnan, 2005, I have to subscribe to Unnikrishnan's conclusion:

"It is concluded that Einstein's resolution using gravitational time dilation suffers from logical and physical flaws, and gives incorrect answers in a general setting."

#70 – 2015-09-02

By wogoga in #66:

I'm genuinely flabbergasted by Einstein's continuation:

By Reality Check in #67:

is easily understood.

I had no intention to dispute the simplicity and correctness of this statement. However, this statement does not support the proposition "U1 running faster" in support of which it has been introduced. "U2 located at a higher [weaker] gravitational potential than U1" means: "U2 running faster". Thus Einstein uses "U2 running faster" in order to support a necessary premise for his resolution of the twin paradox, namely the premise "U1 running faster than U2".

I explain such a catastrophic mistake by confusion based on the belief in the validity of Relativity Theory. Yet in this case, I can even understand those who assume that Einstein used con tricks.

Once again Langevin's twin paradox with Lorentz-factor 100 and travel-distance 2 * 100 LY, with respect to the travelling clock U2:

                Travelling clock U2    Resting U1
  Inertial motion      2.0001 year    0.02 year
  Direction reversal         0 year 199.99 year
  Total time            2.0001 year 200.01 year

In this limit case of negligible acceleration time, direction reversal must add 199.99 year to clock U1 while clock U2 remains unchanged. And Einstein explains this "speeding ahead" of clock U1 by 199.99 year with gravitational time dilation of U1!

#75 – 2015-09-04

By ben m in #74:

You are doing *something* to a time coordinate t, but you are not using Special Relativity to do it. What you are doing instead is nonsense and wrong.

Please state one concrete example of what you consider wrong. Then I will explain why you are wrong in thinking that I am wrong.

With the premises

U1 = resting clock or twin; U2 = travelling clock or twin
travel speed v (with Lorentz factor 100) in direction of x-coordinate
negligibly small acceleration times (or "synchronization of clocks during inertial encounter")
implicit use of year and LY

the Lorentz transformation for the outward trip of Langevin's twin paradox results in:

[1] x' = 100 x – 99.995 t [3] x = 100 x' + 99.995 t'
[2] t' = 100 t – 99.995 x [4] t = 100 t' + 99.995 x'

The only reasonable way to link two inertial frames F and F' is to use the event {x=0, t=0} because it transforms to event {x'=0, t'=0}. The choice of e.g. event {x=0, t = 2015 year} would transform by [1] and [2] to {-201490 LY, 201500 year} of F'. (Therefore I normally prefer to use the Lorentz transformation with coordinate-differences, e.g. for dealing with Bell's spaceship paradox.)

In frame F, the event of U2 reaching the target 100 LY away, is {100 LY, 100.005 year}. This transforms to {0 LY, 1.00005 year} of F'. This result is obvious, as in F' U2 remains at x' = 0, and 1.00005 year proper time are needed to reach the reversal point.

At t' = 1.00005 year, just before changing direction (or synchronization with a clock moving in opposite direction), U1 is 1 LY (=100 LY / 100) away from U2 (in direction of the negative x'-axis). This means, U2 attributes x' = -1 LY to U1. If we transform this F'-event {-1 LY, 1.00005 year}, we get the F-event {0, 0.01 year}.

Thus, also the use of the Lorentz transformation demonstrates that in F only 0.01 year have passed, whereas 1.00005 year have passed in F'. This result agrees with both Einstein and me. So instead of the table of post #70 we get this more detailed table:

                Travelling clock U2    Resting U1
  Inertial Motion 1    1.00005 year    0.01 year
  Direction reversal        0 year 199.99 year
  Inertial Motion 2    1.00005 year    0.01 year
  Total time           2.0001 year 200.01 year

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Also a false premise (Lorentz transformation) can lead to a correct conclusion (time dilation according to Lorentz factor)

#79 – 2015-09-09

By wogoga in #75:

Please state one concrete example of what you consider wrong. Then I will explain why you are wrong in thinking that I am wrong.
With the premises

...
negligibly small acceleration times (or "synchronization of clocks during inertial encounter")

By Darwin123 in #76:

Okay, this is wrong. The acceleration can't be negligible if it reverses the direction of the space ships motion as seen from the earth.
It the acceleration is small, then it has to act over a long time span in the rocket ship frame in order to bring the space ship back home.
If the span of time is short, then the acceleration has to be huge to bring it back home. However, a large acceleration can't be negligible.

I must admit that what you write here seemed at first completely misguided to me, because it is possible to make the acceleration period of direction-reversal arbitrarily short in comparison with the two periods of inertial motion, by increasing the distance from resting U1 to the reversal point of travelling U2. Therefore the effect of time contraction due to acceleration can be made negligibly small compared to the effect of time dilation due to inertial motion.

Yet from Wheeler's formula of your post #78 I had to learn that my above conclusion is based on a premise, namely that time contraction due to acceleration is independent of the location where it takes place. In any case, I'll comment on Wheeler's formula.

We also have the possibility of inertial encounter mentioned by Unnikrishnan. In our case, this means that at reversal-point 100 LY away, clock U2 (v = +0.99995 c) synchronizes with an identical U2 moving in opposite direction (v = -0.99995 c). In this way, acceleration of U2 is fully kept out of the game.

By wogoga in #75:

Also a false premise (Lorentz transformation) can lead to a correct conclusion (time dilation according to Lorentz factor)

By Reality Check in #77:

That the Lorentz transformation is a "false premise" reveals an abysmal ignorance of SR, since the Lorentz transformation is not a premise!
The Lorentz transformation is a derivation of SR from its postulates that is backed up by overwhelming empirical evidence.

The conclusion (resp. child) of one reasoning can be a premise (resp. parent) of another reasoning. Thus, insofar as we derive Lorentz-factor mass-energy-increase from the Lorentz transformation, the transformation is the premise, and mass-energy-increase is the conclusion.

#82 – 2015-09-13

By wogoga in #79:

Yet from Wheeler's formula of your post #78 I had to learn that …

This is Wheeler's time-dilation formula presented by Darwin123 in post #78:

dt' = dt (1-a∙x/c²) / √[1-v²/c²]

where (according to Darwin123)

dt': small interval of time measured by the observer in the rocket.
dt: corresponding interval of time measured by observer on earth.
x: distance of rocket from earth.
a: the dynamic acceleration of the observer.

The formula implies that time dilation depends on distance x between U1 and travelling clock or twin U2. And this implication is impossible for the following reason:

Let us assume some space stations on the way from U1 on Earth to the target of travelling U2. All these space stations are at rest relative to the Earth, and time runs identically (apart from small gravitational effects) with identical simultaneity. Therefore, the trip of U2 from the Earth to the target must take the same time with respect to all of them. This is only possible if the factor of time dilation of U2 is the same. According to the above formula however, this factor depends on distance x from U2, which is different for every space station.

There are further problems. In the case of an acceleration of a = c²/x the formula reduces to dt' = 0 dt. This means: Time of U1 runs infinitely fast with respect to U2. And if acceleration a is bigger than c²/x (or smaller than -c²/x) then time of U1 runs from the future to the past!

By the way, acceleration a = c²/x, necessary for such strange effects, is very small if distance x is big enough. In case of a distance of 100 LY we get:

a = c² / (100 year ∙ c) = c / 100 year = 0.095 m/s²

By Darwin123 in #78:

Note that my formula reduces to the traditional 'Lorentz time dilation formula' when ... a = 0. ...

I first got a simple version of it in the book, 'General Relativity' by Wheeler. However, he used both Greek letters and units where c=1. He also didn't really make the role of forces very clear. ...

I found the formula easy to derive from the Lorentz transform and the definition of force common to both Newtonian physics and special relativity. It just took a little calculus. Perhaps you can show how I am wrong without calculus using your incredible insight.

Actually your formula is very interesting. I've never seen it before and unfortunately I cannot find it elsewhere, so I fully rely on your presentation. The formula could result from an attempt to resolve the twin paradox along the lines of Einstein 1918 with a significant difference:

According to Einstein, acceleration of travelling clock U2 has no effect on time dilation. Therefore he uses a fictitious gravitational field acting on resting clock U1. The inventor of the formula seems to replace Einstein's fictitious field acting on U1 with U2's own acceleration.

During inertial motion acceleration a is zero, and as you have stated, the formula reduces to "normal time dilation":

dt' = dt /√[1-v²/c²] = y dt

where y ("gamma") is the Lorentz factor. In the special case discussed here (see #56), where v = 0.99995 c and y = 100, we get:

dt' = 100 dt

This means that a time interval dt' = 100 sec of the travelling twin corresponds to an interval dt = 1 sec passing on Earth. This is time dilation (slowing) of resting U1 with respect to travelling U2. We conclude that this relates to Einstein's:

"During the partial processes 2 and 4 the clock U1, going at a velocity v, runs indeed at a slower pace than the [in K'] resting clock U2."

Thus, also this formula agrees with Einstein and my analysis: With respect to U2, during inertial motion of totally 2.0001 year, only 0.02 year pass in U1. (Otherwise, if U1 were a light-clock then its inside light-pulse would not move at c relative to U2, see #43).

Now let us deal again with the whole of the formula presented by you:

dt' = dt (1-a∙x/c²) / √[1-v²/c²]

A formal analysis shows that, by replacing a∙x with v², the formula results in

dt' = dt ∙ √[1-v²/c²] = dt / y

which is the exact opposite of "normal time dilation". In this way, the inventor of the formula pays tribute to the fact that insofar as time of A runs slower with respect to B, time of B runs faster with respect to A, and insofar there is time dilation of B with respect to A, there is also time contraction of A with respect to B (see SR Simultaneity, Contraction & Expansion).

In any case, the formula presented by you in no way refutes my analysis of the twin paradox. The formula cannot even be a direct consequence of the Lorentz transformation, as this transformation does not deal with acceleration (maybe apart from proper acceleration). The formula is probably only a failed attempt to resolve the twin paradox or similar SR problems.

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Exaggeratedly worded: Insofar as Einstein is overestimated, he was wrong; insofar Einstein was right, he is underestimated

#89 – 2015-09-16

By wogoga in #82:

According to Einstein, acceleration of travelling clock U2 has no effect on time dilation.

By Darwin123 in #88:

That is absolutely not true. In fact, section 4 of his 1905 article uses a formula for centripetal acceleration. He derives the 'time dilation' of clocks rotating with the earth on the assumption that the clocks on the surface are subject to centripetal acceleration.

For quite some time now, I have been astonished at your ability to twist truth and reality. That other "skeptics" normally won't correct you as long as you fight for orthodoxy and against heresy, only shows that "skeptics" are more interested in fighting heterodoxy than interested in truth.

Einstein in "section 4 of his 1905 article" even tries to avoid acceleration by arriving at a "continuously curved line" via a "polygonal line" to which acceleration cannot be applied.

Einstein in §4. Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks:

"What is the rate of this clock, when viewed from the stationary system?

Between the quantities x, t, and t', which refer to the position of the clock, we have, evidently, x = v t and

t' = (1–v²/c²)^-1/2 (t – v x/c²)

Therefore,

t' = t (1–v²/c²)^1/2 = …

… It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide.

If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: … Thence we conclude that a balance-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions."

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Exaggeratedly worded: A typical "skeptic" is a reincarnated religious zealot confounding science with religion

#94 – 2015-09-19

By wogoga in #89:

Einstein in "section 4 of his 1905 article" even tries to avoid acceleration by arriving at a "continuously curved line" via a "polygonal line" to which acceleration cannot be applied.

Clarification: On a "polygonal line", acceleration is not defined (or infinite) at the edge points, and zero in between.

By wollery in #90:

Actually that's one way that centripetal acceleration is derived, by looking at circular motion as a series of straight line segments and looking at what happens to the velocity vectors, then taking the limit as the line segments decrease in length to form a circle rather than a polygon.

In my opinion, you somehow confuse acceleration with infinitesimal. From Wikipedia on Infinitesimal:

"The 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular calculation of area of a circle by representing the latter as an infinite-sided polygon."

The problem, Cusanus and Kepler had dealt with was to approximate a curved line by a "polygonal line".

In the Wikipedia article on acceleration however, polygon is not even mentioned. Instead, a somehow related concept, namely the parallelogram law is mentioned. This law is used to explain movement on a curved line by continuous acceleration, as done e.g. by Newton. The two basis vectors involved are v∙dt and a/2 ∙ dt².

Yet in case of time-dilation of a movement at v along a "continuously curved line", one has to show that the time-dilation formula derived for inertial motion at v, is still valid.

By wollery in #90:

Einstein was using centripetal acceleration, he just wasn't saying so explicitly.

And you'll note that he derives a time dilation for the clock at the equator. Which is under centripetal acceleration.

Einstein derives time dilation at the equator only from sidereal rotation speed of around v = 465 m/s. Acceleration of a = v²/r = 5.4 mm/s² is totally irrelevant. If acceleration had a relevant effect on time dilation then this effect should show up in curved particle accelerators, where accelerations become extremely high.

Correlation with acceleration does not imply causation by acceleration.

This whole detour in the discussion started with the easily refutable claim (see #82) that the correct time-dilation formula depends on acceleration. If you think that this claim agrees with either SR or GR, then please explain how centripetal acceleration of 5.4 mm/s² affects time dilation at the equator.

#98 – 2015-09-23

By Reality Check in #85:

According to Einstein and every other person who has learned Special Relativity, acceleration of travelling clock U2 has no effect on SR time dilation because SR is for inertial frames of reference!

Yes, that should be obvious!

By Reality Check in #85:

Therefore he uses an equivalent gravitational field as in General Relativity.

In 1918, Einstein still believed in the validity of Special and General Relativity. Therefore he concluded by logical necessity that time-dilation ala General Relativity must be the cause of the time-dilation asymmetry in the twin paradox.

If my refutation (post #66) of Einstein's twin-paradox resolution was wrong then you would be able to show which point is wrong:

1) According to Einstein, reciprocal time dilation during inertial motion "is more than compensated by a faster pace" of resting twin U1 during direction-reversal of U2.

2) This "faster pace" of U1 is explained by a fictitious "homogenous gravitational field" acting on U1.

3) Gravitational attraction implies (stronger, lower) gravitational potential.

4) Gravitational potential leads to gravitational time dilation.

5) Thus, Einstein explains the "faster pace" with gravitational time-dilation.

6) Yet gravitational time dilation of U1 is the opposite of "faster pace of U1".

#101 – 2015-09-26

Apart from rather pointless nitpicking based on biased interpretation of my simplifying explanations, the brave defenders of Einstein's failed twin-paradox resolution come to contradicting conclusions:

By Reality Check in #99:

Gravitational dilation is clocks ticking faster!
The gravitational time dilation of U1 is the same as the "faster pace of U1".

By Darwin123 in #100:

The gravitational potential is higher at point U2 than at point U1. So the clocks at U2 are faster than the clocks at U1.

Wikipedia on gravitational time dilation:

"The stronger the gravitational potential (the closer the clock is to the source of gravitation), the slower time passes."

Reality Check is right insofar as Einstein's reasoning needs a "faster pace of U1". Darwin123 is right insofar as the (lower, stronger) gravitational potential of U1 leads to a slower pace of U1.

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Symmetric time dilation explained

#106 – 2015-09-29

Instead of dealing with inconsistent objections, let us deal with the return trip of the journey of post #75.

We get the Lorentz transformation for the return trip from [1] to [4] of #75 by replacing v with -v:

[5] x' = 100 x + 99.995 t [7] x = 100 x' - 99.995 t'
[6] t' = 100 t + 99.995 x [8] t = 100 t' - 99.995 x'

The linking event {x=0, t=0} transforming to {x'=0, t'=0} is the only event same location, same time. Because at the end of the journey, U1 and U2 meet again at the same location and the same time, we must rename the direction-reversal event from {0 LY, 1.00005 year} of frame F'(+v) to {0 LY, -1.00005 year} of a new frame F'(-v). Using the return-trip equations [7] and [8], the not-renamed event {t' = 0, x' = 1.00005} would transform to event {x = -100, t = 100.005}, where the x-coordinate -100 LY of twin U2 would have the wrong sign with respect to U1, as U2 remains at the relative position x = 100 LY during direction reversal.

The renamed event {x' = 0, t' = -1.00005 year} of F'(-v) transforms as needed to {100 LY, -100.005 year} of F. This result is reasonable, as with respect to resting twin U1, U2 will move during t = -100.005 year and t = 0 back from the direction-reversal position x = 100 LY to x = 0.

At direction-reversal, the moving twin U2 attributes x' = -1 LY to resting twin U1. The renaming of the direction-reversal time from t' = 1.00005 year of F'(+v) to t' = -1.00005 year of F'(-v) leads to this conclusion:

§ Just before reversal, F'(+v) attributes the event {-1 LY, +1.00005 year} to U1, which transforms to F event {0, 0.01 year}.

§ Just after reversal, F'(-v) attributes the event {-1 LY, -1.00005 year} to U1, which transforms to F event {0, -0.01 year}.

This again (see #75) shows that according to the Lorentz transformation also during the return trip, only 0.01 year of U1 correspond to the 1.00005 year of inertial motion of U2. The 1.00005 year of U2 range from event {0, -1.00005 year} to {0, 0}. The 0.01 year of U1 range from {0, -0.01 year} to {0, 0}.

In this calculation, the "speeding ahead" of 199.9 year of U1 with respect to U2 during direction-reversal does not show up explicitly.

#109 – 2015-09-30

By wogoga in #98:

If my refutation (post #66) of Einstein's twin-paradox resolution was wrong then …

By Darwin123 in #100:

The word, 'homogeneous,' means that it is everywhere the same. The gravitational FIELD is the same for both twins, U1 and U2, as well as all objects in between. The gravitational POTENTIAL which is caused by the FIELD , is different for U1 and U2.
Your error was in not understanding the word 'homogeneous'. The field acts on everything in the universe as observed by U1.

By Darwin123 in #105:

The effective source of gravity is BEHIND the rocket, not in front like you assume.

Ok. You are right. Insofar as Einstein uses a "homogeneous gravitational field", he uses an infinite mass attracting from x = x' = infinite. As the source is closer to direction-changing twin U2 than to U1, potential is stronger (lower) for U2 and time of U1 is less dilated, i.e. faster.

Thus, Einstein's resolution actually is (borderline) ingenious: He uses a gravitational field having a stronger effect (time dilation) on U2 in order to accelerate U1. Any concrete source having a stronger time-dilation effect on U2 also leads to a higher acceleration of U2.

In my honest attempt to simplify Einstein's argument and to make it more realistic, I unconsciously replaced a "homogeneous gravitational field" by a field caused by a concrete mass. By placing this concrete mass closer to U1, I could even drop as irrelevant Einstein's:

"An external force, acting upon U2 in the negative direction of the x-axis prevents U2 from being set in motion by the gravitational field."

By Darwin123 in #105:

Thus the earth is effectively above the rocket. The earth is actually falling toward the rocket as though there were a large mass behind the rocket. Anything on the earth has a greater potential energy relative to when it arrives at the rocket.

Because the earth is farther away from the gravitational force than the rocket, the potential at the earth is HIGHER than at the rocket. This is why the rates of clocks on earth are going SLOWER than at the rocket.

This analogy makes the origin of Einstein's twin-paradox resolution, which I still consider untenable, easier to comprehend.

#113 – 2015-10-02

By wogoga in #106:

In this calculation, the "speeding ahead" of 199.9 year of U1 with respect to U2 during direction-reversal does not show up explicitly.

The reason obviously is a consequence of the necessary renaming of the direction-reversal event from the future to the past (see #106). Just before and just after direction-reversal, resting twin U1 is at x' = -1 LY (i.e. at a Lorentz-contracted distance of 1 light-year on the negative x-coordinate-side of travelling twin U2). From equation [4] of #75, valid before direction-reversal, we derive for a given time t' = T':

[4] t = 100 t' + 99.995 x' = 100 T' - 99.995 year

From equation [8] of #106, valid after direction-reversal, we derive for t' = T':

[8] t = 100 t' - 99.995 x' = 100 T' + 99.995 year

This means: Assuming simultaneity of travelling U2, U1 just before direction-reversal is 99.995 years in the past. In the return-trip frame F'(-v) however, an object at rest with the same relative position x' = -1 LY is 99.995 years in the future. Thus, during direction-reversal of U2, the Earth with U1 migrates 199.9 years from the past to the future.

Such changes from the past to the future or the other way round are not more astonishing than huge distance changes inherent in Special Relativity:

Before U2 starts the journey, the target is 100 LY away. After having reached travel speed, the distance to the target is Lorentz-contracted to 1 LY. During the arbitrarily short period of direction-reversal, U1 together with the whole Earth makes a huge journey of 198 LY relative to U2. At first U1 moves away from Lorentz-contracted 1 LY to the normal distance of 100 LY (when U2 is at rest relative to U1) and then comes back to the contracted 1 LY.

By Darwin123 in #108:

Just an infinitesimal amount of time BEFORE the rocket turns the thrust on and off,
t_A' = (t - vx/c²) / √[1-(v/c)²]
…
Just an infinitesimal amount of time AFTER the rocket turns the thrust on and off,
t_B' = (t + vx/c²) / √[1-(v/c)²]

If you replace v with 0.99995 c and c with 1, then your equations transform to [2] of #75 and [6] of #106:

[1] x' = 100 x - 99.995 t . . [3] x = 100 x' + 99.995 t'
[2] t' = 100 t - 99.995 x . . [4] t = 100 t' + 99.995 x'

[5] x' = 100 x + 99.995 t . . [7] x = 100 x' - 99.995 t'
[6] t' = 100 t + 99.995 x . . [8] t = 100 t' - 99.995 x'

By Darwin123 in #108:

The difference in ages is: t_B' - t_A' = -2vx/c²

Thus, the rocket twin at the end of the trip will be younger than the earth twin by 2vx/c².

With v = 0.99995 c and a one-way distance x = 100 LY (see #43) your 2vx/c² results in a "difference in ages" of Dt' = 199.99 year, which comes close to the time difference of standard textbooks: 198.0099 year = 200.01 LY – 2.0001 LY (see table of #75). However, without omission of the Lorentz-factor 1/√[1-(v/c)²] = 100, you get a time difference Dt' = 19999 year when using the event {x=100 LY, t=0}.

The event {100 LY, 0 year} of frame F is not direct part of the solution. Using the forward-trip equations [1] and [2], it transforms this way:

{x = 100 LY, t = 0} --> {x' = 10000 LY, t' = -9999.5 year}

Using the return-trip equations [5] and [6] it transforms this way:

{x = 100 LY, t = 0} --> {x' = 10000 LY, t' = +9999.5 year}

If you are interested in understanding this and why and where Special Relativity works and where not, then see: Reciprocity of SR Length Contraction for Dummies

#118 – 2015-10-05

By Darwin123 in #111:

Thus, the rocket twin at the end of the trip will be younger than the earth twin by 2vx/c².

By wogoga in #113:

However, without omission of the Lorentz-factor 1/√[1-(v/c)²] = 100, you get a time difference Dt' = 19999 year [instead of 199.99] when using the event {x=100 LY, t=0}.

By Darwin123 in #114:

I confess that I am not following this. I can't say you are wrong since I don't understand your words. For example, I don't know what you mean by 'without omission of the Lorentz-factor'.

You had claimed in #111 that the age-difference in the twin-paradox is two times vx/c². This claim is wrong for several reasons, e.g. because the Lorentz-factor of Langevin's twin-paradox is as high as 100 and cannot be ignored. You also somehow confuse time shift vx/c² of the time-transformation with time dilation itself.

If you were right, and twin-paradox time-dilation directly stemmed from the vx/c² time-shift then time-dilation would be a universal effect and not only an effect relevant in case of "relativistic" speeds. If v << c, time-shift vx/c² is a first-order effect whereas time dilation according to the Lorentz factor 1/√[1-v²/c²] is a much smaller second-order effect.

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 in the course of a year. Once in a year, the Earth moves with 30 km/s 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 0.0001∙10⁷ = 1000 year. This 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.

The rest of this post together with the next ones has been transferred to The Myth of "Lorentz Transformation Reducing to Galilean Transformation"