The Twins Paradox and Flat Earth Paradox

An Effort in Understanding Special Relativity (1)

The Twins Paradox and Flat Earth Paradox

Ziaedin Shafiei 26 September 2015

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Overview of Einstein’s Light Clock --------------------------- 3 The Twin Paradox ------------------------------------------------- 4 There is no Paradox ---------------------------------------------- 5 A Universe with Only Two Observers ------------------------ 6 A Revised Twin Paradox ----------------------------------------- 9 Triplet Paradox ---------------------------------------------------- 13 My Clock or Your Clock? ----------------------------------------- 14 Flat Earth Paradox ------------------------------------------------ 15 Size Shrinking is an Illusion ------------------------------------ 16 There Should be no Pick and Choose Reality -------------- 17 Does Time Dilation Apply to Photon? ----------------------- 19 Reality of Time dilation ----------------------------------------- 20 The Root of the twins Paradox ------------------------------- 21 Michelson-Morley (MM) Setting ----------------------------- 22 Null Result ---------------------------------------------------------- 27 Throwing a Ball to the Wall Inside a Train ----------------- 30 Ball Path Observed from the Railway Platform ---------- 31 Inertial Reference Frame vs MM Setting ------------------ 33 Boat Clock vs Light Clock --------------------------------------- 34 Comments --------------------------------------------------------- 36 Notes ---------------------------------------------------------------- 37

05/12/2015

Inquiry Into the Speed of Light

2

Overview of Einstein’s Light Clock • The light clock is a thought creation • It is simply a pulse of light being bounced between two parallel mirrors 𝑳 meters apart in vacuum

• Unit of time can be defined as the duration of a return light travel 𝟐𝑳 between the two mirrors,𝑇0 = (c is the speed of light in vacuum) 𝒄

• This unit is called proper time in special relativity and is obviously constant in any inertial reference frame (IRF) for the same 𝑳 • Unit of time for a moving clock is assumed different from proper time and 2 varies with its relative speed from its observer 𝑇𝑣 = 𝑇0 / 1 − 𝑣 Τ𝑐2 ½ • In these slides the clock is represented by • For more details about the light clock see “Tuning Einstein's Light Clock” 𝑣

Mirror

𝑐 Tv/2

𝑐 T0/2

𝐿

26-Sept-2015


𝑣Tv/2

3

The Twin Paradox • • • • •

•

•

•

Alex started a space adventure on his 20th birthday His twin brother Bob stayed behind on earth Each carried his own light clock Alex travelled with a constant high speed, 𝒗, during the journey After an arbitrarily long duration of travel, say 15 years, Alex turned the spaceship round towards the Earth to be able to celebrate his 50th birthday with Bob However, based on the theory of special relativity they should face the fact that they will be of different ages when they meet again Bob should accept that Alex would be younger than him because he observed Alex’s light clock slowed down due to the movement of the spaceship at constant speed Thus Bob would be older than Alex or simply Bob > Alex Alex, on the other hand, should admit that his brother would be younger than him as he also observed that Bob’s light clock slowed down as the earth and Bob moved away from him at constant speed Alex > Bob 26-Sept-2015


Alex

𝒗 Bob

4

There is no Paradox • Almost all physicists think that • There is no paradox in the theory of relativity • The contradictions is between the theory and common sense. “That is relativity. Weird? Yes. Contradiction? No.”1 • Generally, it is argued that the space-time paths of the twins are not symmetrical and the twin who went through acceleration and deceleration during taking off, turning around and landing would be the younger twin2 • Accurate atomic clocks were also put on commercial aircrafts to fly towards east and west proving this interpretation and resolving the paradox3 26-Sept-2015


5

A Universe with Only Two Observers (1/3) • The theory of relativity, for simplicity, has initially been thought of in a world with only two observers in two inertial reference frames, (IRF), moving relative to each other with a constant speed, 𝒗 • Each observer thinks he/she is stationary and the other is moving • Each observer is provided with the facility to measure the time in: – his own frame according to his own clock - 𝑻0 is the unit and equal for both observers – the moving frame according to the moving clock - 𝑻𝒗 changes between 𝑻0 and infinity depending on the value of 𝒗 suggesting that time goes slower in the moving frame

• Any paradox in relativity is due to the difference between these times for both observers which only becomes tangible when 𝒗 approaches closer to the speed of light Bob 26-Sept-2015

𝑻0


𝒗

𝑻0

Alex 6

A Universe with Only Two Observers (2/3) • In general our options are 1. Accept time measurement based on the residence clock with universally equal unit of time, 𝑻0 2. Accept time measurement based on the moving clock, 𝑻𝒗 3. Accept/prefer only one of the observers’ measurements of the residence and moving clocks and reject the same measurements from the second observer. In a world with only two observers this is equivalent to select one of the frames as the absolute frame of reference 4. Accept both observers measurements of both clocks

• Special relativity claims the correctness of the 4th option but has to switch to the 3rd option for refuting many paradoxes option 4 creates Bob 26-Sept-2015

𝑻0


𝒗

𝑻0

Alex 7

A Universe with Only Two Observers (3/3) • We should note that the paradox is the result of thought experiments using thought equipment, i.e. light clock, and then trying to apply it to real scenarios, real experiment • The twins thought experiment in fact tries to highlight the symmetrical effect of slowing down of any moving clock in one IRF observed from a second IRF • The question is: why slowing down of Bob’s clock observed by Alex is not accepted as real and effective as slowing down of Alex’s clock observed by Bob • Trying to deny the paradox by alluding to acceleration, deceleration or any other cause just ignores the main point of the paradox 1 • A few examples will be given which eliminate any difference between the twins’ tracks and show that there indeed exists a paradox Bob 26-Sept-2015

𝑻0


𝒗

𝑻0

Alex 8

A Revised Twin Paradox (1/4) Case 1 • Alice and Kate decided to venture a space travel on their 20th birthdays • They went away in separate spaceships but always alongside each other with the same constant speed 𝒗 • After one year traveling in the same direction they returned towards the earth and celebrated their 22nd birthday on the planet • The earth acts just as a departure and arrival point 26-Sept-2015


Alice

𝒗

Kate 𝒗

9

A Revised Twin Paradox (2/4) Case 2 • Alice and Kate then decided to go for an identical profile space travel but at opposite direction • They went in separate spaceships but alongside each other with the same constant speed 𝒗 • After one year traveling in the same direction they returned towards the earth and they celebrated their 24th birthday on the planet 26-Sept-2015


Alice

Kate

−𝒗

−𝒗

10

A Revised Twin Paradox (3/4) Case 3 • Alice and Kate then decided to go for the identical profile space travels of cases 1 and 2 in separate spaceships but opposite each other with the same constant speed 𝒗 • On their return to earth they did not know what to expect • Based on their own experiment from Cases 1 and 2 they assumed they would be able to celebrate their 26th birthday on the planet • However, based on the theory of special relativity they should find that they would be of different ages • Alice should face the fact that Kate would be younger than her as she observed Kate’s light clock slowed down Alice > Kate • Kate, however, should accept that Alice would be younger than her as she observed Alice’s light clock slowed down Kate > Alice 26-Sept-2015

Kate

−𝒗


Alice

𝒗

11

A Revised Twin Paradox (4/4) Case 4 • One can remove the earth from the picture and again based on the theory of special relativity Alice and Kate should find that they would be of different ages after their identical journey

Kate

−𝒗

Alice

𝒗

– Alice should find Kate younger than her Alice > Kate – Kate, on the other hand, should find Alice younger than her Kate > Alice 26-Sept-2015


12

Triplet Paradox Case 5 •

• • •

Alex and Tom travelled in separate spaceships opposite each other on their 20th birthday. The paths of their travels are identical to Case 3 and they maintained the constant speed 𝒗 during the journey Bob stayed behind on earth When Alex and Tom returned to the planet the triplets could not agree on their ages Based on the theory of special relativity they argued as follows

Alex

Tom

𝒗

−𝒗 Bob

– Alex said that Bob is younger than him but Tom is the youngest as his clock slowed down the most Alex>Bob>Tom – Tom said that Bob is younger than him but Alex is the youngest Tom>Bob>Alex – Bob, on the other hand, maintained that Tom and Alex are of the same younger age as their clocks slowed down equally Bob>Alex = Tom 26-Sept-2015


13

My Clock or Your Clock? Case 6 • Kate and Alice were in deep space on their 20th birthdays when Kate decided to move away and not to observe Alice’s clock • According to Kate’s light clock 2 years has since passed • Alice kept observing Kate’s clock which was moving away with the constant high speed of 𝒗 from Alice

Kate

𝒗 Alice

 How old are Kate and Alice now?  What are the answers if Kate were observing Alice  What are the answers if Alice was not observing Kate as well? 26-Sept-2015


14

Flat Earth Paradox

Muon

Case 7

≈c

• The observation of substantial numbers of Muons on the surface of the earth has been accepted as a definitive prove for the reality of time dilation and length contraction • Muon has the mean lifetime of 2.2 µ𝒔 and moves with the velocity of about 𝟎. 𝟗𝟗𝟗𝟖𝒄

The earth as seen by an observer moving alongside of a Muon

• Suppose a Muon is just created in the upper atmosphere, e.g., 15km from earth, and approaches the earth • The thickness of the earth atmosphere must be reduced to about 𝟔𝟔𝟎𝒎 for the Muon to reach the surface of the earth at high percentage observed. • Based on length contraction the earth also becomes oval with minor radius of about 128 km (about 1/50 of its mean radius) 26-Sept-2015


15

Size Shrinking is an Illusion Case 8 Just to clarify the issue, in case 4 we can also add that • Alice sees Kate and her spaceship getting gradually smaller as Kate moves away from her • Kate likewise sees Alice is getting physically smaller • As this size shrinking is not real but merely an illusion no one has wondered whether Kate is in fact shrinking or Alice • But time dilation and length contraction are being considered as real phenomena with real effect on length, time, mass, existence and even biology1 of entities otherwise no scientist would consider interpreting Muon appearance on the surface of the earth based on mere illusion Alice

𝑿

Kate 26-Sept-2015


16

There Should be no Pick and Choose Reality(1/2) • •

•

•

It should be noted that length contraction was wrongly introduced by FitzGerald and Lorentz as a real physical change of every object in the direction of its movement relative to aether However, length contraction in special relativity is a symmetrical phenomenon meaning length contraction (L = L0 /𝛾) is true for both objects moving relative to each other1,2 If appearance of Muon on the surface of the earth is a proof for reality of length contraction and Muon is easily detectable by any scientists with right equipment then we should also experience other real effects Is the earth and its atmosphere really flattened merely due to its relative movement with a Muon for a few micro seconds and regain its spherical shape again after the Muon decays3 or hits the ground, or this is all an illusion like size shrinking 26-Sept-2015


17

There Should be no Pick and Choose Reality(2/2) •

•

•

•

If length contraction is real then every object, including the whole universe, is constantly subject to considerable contraction and expansion in all directions as the universe are full of particles like Muon busily moving around Muon is deliberately mentioned here to highlight the fact that the size and mass of an observer/mover is not important in special relativity but its speed1 Consequently, one property of even a subatomic particle, speed, is endowed in special relativity with one of the strongest force of the nature, which can contract every celestial body, no matter how huge its size or density The effect should culminate with photon which can almost flatten the whole universe to non-existence 26-Sept-2015


18

Does Time Dilation Apply to Photon? • • •

There are also claims that one can travel to any far away galaxies such as Andromeda within a human lifetime1 This claim and the Muon observation on earth is the cause of another paradox Though everyone agrees that, for example, it takes light about 8 minutes to travel from the Sun to the earth, according to the same reasoning for Muon2 photons should take any galactic journey through vacuum in no time Δt’ = Δt (1 - 1)½ = 0 s *

• Do we see the Sun of 8 minutes ago or 0 minute ago? • Do we see Andromeda galaxy of 3 million years ago or 0s ago? 26-Sept-2015


19

Reality of Time dilation • The point is that if length contraction is an illusion, which our observation of any celestial body confirms, then one can also doubt about time dilation and relativistic momentum ρ = ρ0 𝛾 as real phenomena • In the size shrinking of case 8 we rely on an observer stationary to the object and not the one moving relative to it. What is thus our justification to accept the reality of time dilation and length contraction for a moving object based on observation from outside and ignoring the resident clock in each reference frame which shows the proper time 26-Sept-2015


20

The Root of the twins Paradox • The twins paradox in fact stems from an inherent flaw in special relativity. • To be able to find out the root of the contradiction some introduction is necessary • We start with justification and setting for MichelsonMorley (MM) experiment in 1887 • Michelson and Morley tried to investigate the existence and speed of aether in relation to the earth • Aether was assumed to be the medium through which light wave could propagate and also considered as the absolute frame of reference 26-Sept-2015


21

Michelson-Morley (MM) Setting (1/5) • Consider a river in which water flows at a constant speed of 𝑉 = 3 𝑚/𝑠 to the right • Two toy boats start from point A with the constant speed of 𝑈 = 5𝑚/𝑠 for the following two perfectly straight round trips • The 1st boat aims to reach point B across the river and return • The 2nd boat tries to reach point C in downstream and return

B

V = 3m/s

L=4m 4m A

C

𝐀𝐁 = 𝐀𝐂 = L = 4𝑚

• Both trips are observed from the river bank • This setting was considered by MM to explain the operational principle of the equipment they designed for testing the theory of aether (as a universal fixed reference frame) 26-Sept-2015


22

Michelson-Morley (MM) Setting (2/5) • If the water is still, V = 0 m/s

B

both boats return to point A at the same time. Duration of each trip is

V = 0 m/s

𝑈 = 5m/s

4m

(𝑡0 𝐀𝐁𝐀 = 𝑡0 𝐀𝐂𝐀 = 1.6 𝑠)

• Generally, the duration of any equal distance round trip is the same in any direction when V = 0 and U being constant 26-Sept-2015

𝑈 = 5m/s

C

A


4m

23

Michelson-Morley (MM) Setting (3/5) • With the river current of 3𝑚/𝑠 the first boat must proceed at the direction of line AD (at an angle of 30˚ to the left of vertical line AB) to be able to reach point B in a straight line • The same angle applies to the return trip • Durations of two round trips are

3m V = 3 m/s

D

B U=5 m/s 30˚

vcrossing = 4 m/s

L=4m 30˚

vupstream = 2 m/s

C E

A

vdownstream = 8 m/s 4m

𝑡𝐀𝐁𝐀 = 1 + 1 = 2 𝑠

𝑡𝐀𝐁𝐀 = 𝑡𝐀𝐁 + 𝑡𝐁𝐀 =

2𝐿 / 𝑈

𝑡𝐀𝐂𝐀 = 0.5 + 2 = 2.5 𝑠

𝑡𝐀𝐂𝐀 = 𝑡𝐀𝐂 + 𝑡𝐂𝐀 =

2𝐿 /(1− 𝑈

26-Sept-2015


1− V2/ U2 V2/ U2) 24

Michelson-Morley (MM) Setting (4/5) • In general the formulas for the duration of the two trips are 𝑡𝐀𝐁𝐀 =

2𝐿 / 𝑈

𝑡𝐀𝐂𝐀 =

2𝐿 /(1− 𝑈

1− V2/ U2 V2/ U2)

• The relation between the two durations is 𝑡𝐀𝐂𝐀 /𝑡𝐀𝐁𝐀 = 1/ 1− V2/ U2

• The right hand side of the formula is a familiar expression used in special relativity i.e. time dilation and length contraction, see page 34 26-Sept-2015


25

Michelson-Morley (MM) Setting (5/5) • MM argued that in the presence of streaming aether the speed of light should also be 𝑣𝑊𝑅𝑇 𝐴𝑒𝑡ℎ𝑒𝑟 different in various perpendicular round trip paths, reaching to its maximum difference in Adjustable Mirror certain directions • If the time difference can be measured then light splitter not only the existence of aether is proven but also its direction and relative speed can be identified and calculated1 Fixed mirror • MM experiment was assembled on a rotating stone slab floating on a pool of mercury to be able to detect any speed difference in all horizontal directions as a fringe shift in Interferometer 2 Michelson’s sensitive interferometer • The experiment splits a beam of light into two Michelson Morley simplified setting perpendicular beams which are then reflected by some mirrors and are recombined to form an interference pattern in Michelson Interferometer 26-Sept-2015


26

Null Result (1/3) • MM observed no tangible time difference between any two perpendicular light journeys concluding 𝒗𝑊𝑅𝑇 𝑎𝑒𝑡ℎ𝑒𝑟 ≈ 0 in all directions a)

b)

Adjustable Mirror

Fixed mirror

Detector

No fringe shift observed while the whole equipment was very slowly rotated at different times of the day and different days of the year 26-Sept-2015


27

Null Result (2/3) • Initially, some tried to justify and rationalise the existence of aether despite the null result from MM experiment • Michelson’s explanation was that aether is dragged along as the earth orbits the sun • George FitzGerald and then Hendrik Lorentz proposed length contraction and time dilation in the direction of aether to explain the null result1,2,3

• Eventually the existence of aether was rejected by Einstein who also declared that • All motions are relative, no absolute motion can be verified • Laws of physics are the same in all frames moving with constant speed (inertial reference frame) • Moreover, based on the null result and Maxwell equations, the speed of light is the same in all reference frames • Time dilation and length contraction is valid in absence of aether but are experienced only between two inertial reference frames moving with constant speed relative to each other2 26-Sept-2015


28

Null Result (3/3)

• In the absence of aether: • MM experiment is actually implemented in just one inertial reference frame • Hence, no conclusion can be made for observation from any other reference frame based on this experiment1

• Now consider the following experiment which shows that a round flight measurement can be sometimes misleading 26-Sept-2015


29

Throwing a Ball to the Wall Inside a Train • A train is moving with a constant speed, VTrain • A ball is thrown and then bounces back with the same speed (VBall = 4 𝑚/𝑠) in two following directions: 1. Along the direction of the train movement (A to C and C to A) 2. Vertical to the direction of the train movement (A to B and B to A) B

𝑡𝐀𝐁 = 𝑡𝐁𝐀 = 1𝑠 𝑡𝐀𝐂 = 𝑡𝐂𝐀 = 1𝑠

4 m/s

4m

The duration of the round flight of the ball is the same (2𝑠) irrespective of the speed of the train if observed within the train.

4 m/s

C

A 4m

Ball movement as observed inside the train 26-Sept-2015


30

Ball Path Observed from the Railway Platform (1/2) • • • •

Ball trajectories if observed from a platform are depicted below B1 and B2 are the positions of point B after 1 and 2s, respectively The same applies to A and C For an observer on the railway platform 𝑡𝐀𝐁𝟏 = 𝑡𝐁𝟏𝐀𝟐 = 1𝑠 𝑡𝐀𝐂𝟏 = 𝑡𝐂𝟏𝐀𝟐 = 1𝑠 3m

B1

B

Ball movements as observed from a platform outside the train

VTrain = 3 m/s

B2

5 m/s

7 m/s A

6m

4m

A2

C1

C2

1 m/s

7m 26-Sept-2015


31

Ball Path Observed from the Railway Platform (2/2) 3m B1

B • Again the duration of round trip flights of the balls is the same (2𝑠) irrespective of the speed of the train • The balls arrive in point A2 at the same time if thrown from point A at the same time. • From MM experiment we learned that split rays of light arrive at the same time if originated simultaneously from one source

26-Sept-2015

VTrain = 3 m/s

B2

5 m/s

7 m/s A

6m

4m

A2

C1

C2

1 m/s

7m Ball movement as observed from a platform outside the train


32

Inertial Reference Frame vs MM Setting • It is wrong if one concludes that the balls travel with the same speed if observed from any IRF because they arrive from the round flights at the same time • It is obvious that the speed of the ball are different at different directions varying from minimum (𝑽𝑩𝒂𝒍𝒍 – 𝑽𝑻𝒓𝒂𝒊𝒏) to maximum (𝑽𝑩𝒂𝒍𝒍 + 𝑽𝑻𝒓𝒂𝒊𝒏) • Thus, based on MM type experiment, one even cannot conclude that the speed of light in vacuum is constant in all directions when observed from any IRF • The constancy of the speed of light thus could only be realized from Maxwell equations at the time and now • The round trip of light was essential for MM because they needed a compact equipment for comparing the extremely high speed of light in different directions for a specific reason Moreover, the source and observation of the light is in the same inertial frame even if the equipment was very slowly rotated. After existence of aether was rejected, MM setting is simply reduces to one inertial frame and thus one cannot extend the outcome of the experiment to two inertial reference frames that are in relative uniform motion with respect to each other required in special relativity

26-Sept-2015


33

Boat Clock vs Light Clock (1/2) • It was mentioned earlier that the twins and flat earth paradoxes in fact stem from an inherent flaw in special relativity • It is proposed here that the root of the contradiction is the choice of the clock • One should note that the working principle of the light clock used in relativity is the same as the example used by MM where a toy boat goes back and forth with the constant speed U from bank to bank • Similar to the light clock one can in theory use the repeating travel of the toy boat as a clock • When water in the river is not flowing (V = 0) then one can use the duration of each journey as the unit of time, 𝑡0 (proper time)

26-Sept-2015


B

V=0

𝑡0 = 2𝐿/U

L A

Mirror

𝐿

𝑇0 = 2𝐿/c

34

Boat Clock vs Light Clock (2/2) • With water flowing at constant speed, the unit of time increases similar to the increase in the unit of time in a light clock moving away with a constant speed 𝑉 • The increased time unit is

V t/2

B 𝑈t/2

V

𝐿

A

𝑡 = 𝑡𝐀𝐁 + 𝑡𝐁𝐀 =

2𝐿 / 𝑈

1− V2/ U2 𝑉

And its relation to the proper time, 𝑡0, is

𝑡 = 𝑡0/ 1− V2/ U2

𝑐 T/2 𝐿

Which is similar to the formula for time dilation due to the light clock moving with speed V 𝑉 T/2

𝑇 = 𝑇0 / 1− V2/ c2 See Tuning Einstein’s light clock? for more detail 26-Sept-2015


35

Comments • Based on this observation, no scientist can accept the duration for the repeated back-and-forth movement of the boat from one bank to the opposite side or a length of a river as a unit of time due to its high sensitivity to water flow • It thus can be argued that the light clock is similarly sensitive to movement and hence the cause of the grave contradictions leading to issues like the twins and flat earth paradoxes • One major question that is still outstanding in physics is the meaning/definition of time • It does not help that in the mathematics of relativity time is represented as 𝒄𝒕 which in fact is just length in disguise • Is time really just another space dimension? • For some more thoughts related to light clock see Tuning Einstein's Light Clock? 26-Sept-2015


36

Notes (1/5) 3.

4.

There is no Paradox ---------------------------------------------- 5 1 - Ronald C. Lasky, The Twin Paradox, Dartmouth College's Thayer School of Engineering 2 -“The paradox centres around the contention that, in relativity, either twin could regard the other as the traveller, in which case each should find the other younger - a logical contradiction. This contention assumes that the twins' situations are symmetrical and interchangeable, an assumption that is not correct.“ Robert Resnick, Introduction to Special Relativity, John Wiley, 1965. 3 - In 1972 J.C. Hafele and R.E. Keating put two accurate cesium atomic clocks on two commercial flights and used two clocks in US Naval Observatory in Washington DC as reference. Round the world flights for eastward and westward trips were 41.2 and 48.6 hours, respectively. J.C. Hafele and R.E. Keating, Science 177, 166 – 170. A Universe with Only Two Observers (3/3)------------------------ 8 1 - “This time dilation, like length contraction, is no accident of convention but a real effect. Moving clocks really do go slow. If a standard clock is taken at uniform speed 𝑣 through an inertial frame S along a straight line from point A to point B and back again, the elapsed time 𝑇 indicated on the moving clock will be related to the elapsed time 𝑇0 on the clock fixed at A by 𝑇 = 𝛾𝑇0, except for possible errors introduced when the clock is accelerated to initiate, reverse, and terminate its journey. But whatever these errors are, their contribution can be dwarfed by simply extending the periods of uniform motion. So, at least in theory, the effect is tangible. But it has by now also been amply observed in the real world, as we shall presently recount.” Wolfgang Rindler, Relativity: Special, General and Cosmological, 2nd edition, Oxford University Press, 2006.

05/12/2015


37

Notes (2/5) 5.

8.

A Revised Twin Paradox (4/4) --------------------------------12 “Two twins A and B meet at an event P1, move away from each other, and then meet again at a later event P2. The twin A considers himself as at rest and predicts that B is younger than himself at P2 due to the relativistic time dilation. But according to the principle of relativity B can consider himself as at rest and A as travelling, and he then predicts that A is younger when they meet at P2. “Øyvind Grøn, The twin paradox and the principle of relativity, Oslo University College, Faculty of Engineering, Oslo, Norway. Flat Earth Paradox ------------------------------------------------ 15 • “… we conclude that the phenomena of length contraction and time dilation are real and must be taken into consideration in our (obligatory!) Newtonian measurements in the laboratory and/or in space. The dependence of the spatial distance and the time difference of two events from the velocity of the particles must not worry us in our everyday activities. Indeed, as we have already seen, the effects of relativistic kinematics are appreciable and show up at high (relative) speeds, far beyond the speeds of our sensory capabilities. Problems which refer to cars entering a garage while traveling with speeds, e.g., 0.9c concern “realities” limited at the level of student exercises and not further than that.” Michael Tsamparlis, Special Relativity, 1st Edition, Springer-Verlag, 2010. • “They also underline the fact that the effect does indeed belong to the realm of special relativity, as in this experiment there are no accelerations and yet the lifetime differs in the different frames.

05/12/2015


38

Notes (3/5) This can be done because in fact time and distance measurements are made in the lab, which are compared to a time measurement in the frame of the decaying particle (for which the particle itself acts as clock). So time dilation effects as a consequence of special relativity, and of the relativity of simultaneity in particular, are just as real as the law of nature that states that a particle will accelerate when a force is exerted on it.” Sander Bais, Very Special Relativity, An illustrated guide, Harvard University Press, 2007. 9.

Size Shrinking is an Illusion ------------------------------------ 16 1. "If we placed a living organism in a box . . . one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light." Einstein, 1911 – Robert Resnick, Introduction to Special Relativity, John Wiley, 1965. 10. There should be no pick and choose reality -------------- 17 1. “This shows, quite generally, that the length of a body in the direction of its motion with uniform velocity 𝑣 is reduced by a factor (1 - 𝑣 2 /c 2 )1/2,” … “There is one important moral to this story: whatever result we get by correct reasoning in anyone frame, must be true; in particular, it must be true when viewed from any other frame, ” Wolfgang Rindler, Introduction to special relativity, Clarendon Press, 1982, pp 28 &29 2. Ladder Paradox - https://en.wikipedia.org/wiki/Ladder_paradox 3. It decays into an electron (+ or -), a neutrino (v), and an antineutrino. Special Relativity, A. P. French. 05/12/2015


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Notes (4/5) 11. Does time dilation apply to Photon? ----------------------- 19 1. “Einstein’s theory allows us to contemplate the possibility of traveling to distant places in the universe. Speeding within a whisker of the speed of light, we imagined journeying off to the Andromeda galaxy within a human lifetime regardless of the fact that it takes light nearly 3 million years to make the journey.” Why Does E=mc2, Brian Cox and Jeff Forshaw, 2009, P96. 2. “The muon’s decay rate is such that one should expect only about 25 of the original 560 muons to remain after a 6.7 µs interval, so that’s approximately the number we might expect to detect each hour at sea level. However, that 6.7 µs interval is measured in Earth’s reference frame—not the muons’. In the muons’ frame, time dilation should reduce that interval to Δt’ = 6.7 * (1- 0.9942)½= 0.73 µs The muons’ decay is determined by their measure of time, and their decay rate is such that we should expect 414 muons to survive for 0.73 µs. So what happens? Observers count just over 400 muons per hour at sea level. This is no subtle effect. The difference between 25 and 414 is dramatic. At 0.994c, the nonrelativistic description is hopelessly inadequate, and time dilation is obvious.” Essential university physics, Richard Wolfson, 2nd edition, page 592.

13. The Root of the twins Paradox ------------------------------- 21 A.A. Michelson and E.W. Morley, American Journal of Science, 134 – 333, 1887 14. Michelson-Morley (MM) Setting (5/5) --------------------- 26 1 - Exact direction and speed of earth motion in relation to aether was unknown as the scientist of the time could not agree on a fixed coordinate for aether. 2 - Interferometer is an optical instrument invented by Michelson. 15. Null Result (2/3)--------------------------------------------------- 28 1. “Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the aether produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the difference in time mentioned above. “ A. Einstein 1916

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Notes (5/5) 2. T = T0 𝛾, where T0 is the period of the clock at rest in the aether and T is the period of the corresponding clock as it moves through the aether. David Bohm, The Special Theory of Relativity, Routledge, 1996, page 21. 3. Also “a measuring rod of length l0 at rest will, when moving with a velocity v along the direction of its length, have the dimension l0 = l 𝛾” David Bohm, The Special Theory of Relativity, Routledge, 1996, Page 18. 15. Null Result (3/3)--------------------------------------------------- 29

1. Two examples of this type of conclusion are “… the speed of light should be the same for all observers. This conclusion was supported by the experimental work of Michelson and Morley, and taken at face value by Einstein” Brian Cox and Jeff Forshaw, Why Does E=mc2, Da Capo Press, 2009, P91. “This Lorentz invariance is in accordance with the Michelson-Morley null experiment which showed that on the moving Earth light spreads with the same speed in all directions. ” Moshe Carmeli, Cosmological special relativity: The Large-Scale Structure of Space, Time and Velocity, 2nd Edition, World Scientific Publishing Co. Pte. Ltd., 2002, page 43.

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