Tuning Einstein's Light Clock?

An Effort in Understanding Special Relativity (2)

Tuning Einstein's Light Clock?

Ziaedin Shafiei 10 October 2015

Einstein’s Light Clock (1/2) Mirror

𝐿

10/10/2015

• 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 • It is assumed that different observers can easily detect/follow the plain working of the clock. Thus this appliance has been the clock of choice in special relativity (SR) • For accurate, high resolution, time measurement the distance between the two mirrors, 𝑳, should be as short as possible • The question is, can we accurately and consistently measure time with this apparatus? Tuning Einstein's light clock

2

Einstein’s Light Clock (2/2) B

𝐿

A

10/10/2015

• To study this clock let us consider a more elaborated version • Two lasers A and B are fixed in the centre of two parallel circular plates in a cylindrical structure with the height of 𝑳 • Two photocells are also fixed on the insides of the circular plates • We assume that when the upper plate receives a red light pulse from laser A, laser B immediately emits a blue light pulse and vice versa, ad infinitum Tuning Einstein's light clock

3

Basic Time Measurement • Suppose the clock is at rest, 𝒗 = 𝟎 in all three perpendicular Cartesian directions (𝒙, 𝒚 and 𝒛). In other words we assume the clock is put in an inertial reference frame (IRF) and observed from the same frame • The time for the pulse of light to travel between the two plates 𝑳 is (c being the speed of light in vacuum) 𝒄

y B

𝐿

O

x A

z 10/10/2015

Tuning Einstein's light clock

4

The Effect of Observer Position (1/4) Case 1 B

•

• O1 𝐿 •

Suppose the observer is positioned at point O1 inside the clock which is the middle point on the line between A and B 𝐿 Any light pulse from A or B reaches the observer with a time delay of 2𝑐 The duration between any two consecutive light pulses spotted by the observer is

A

𝑇𝐴𝐵

• •

10/10/2015

𝐿 = 𝑇𝐵𝐴 = 𝑐

where 𝑇𝐴𝐵 is the time duration measured between observing a flash from A followed by a flash from B. We can call this a tick 𝑇𝐵𝐴 is the time duration measured between observing a flash from B followed by a flash from A. We can call this a tock


5

The Effect of Observer Position (2/4) Case 2 B 𝑑

𝐿

O2

• Suppose the observer moves to point O2 on the middle of a side line • The observer is now outside of the light beam and practically cannot see anything to measure. But if he could see the flashes, they could be observed with a delay of 𝐿 2𝑐

A

•

(1 + 𝑑2/𝐿2) 1/2

which is the duration for light travel along the length of O2 A or O2 B line However, the duration between the two consecutive light pulses detected by the observer in this new position is 𝑇𝐴𝐵 = 𝑇𝐵𝐴 =

𝐿 𝑐

which is the same as observed in case 1

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6

The Effect of Observer Position (3/4) Case 3 • •

B Tick=Tock Line

𝐿

𝑇𝐵𝐴 = 0

O A • •

•

10/10/2015

Suppose the observer moves to point A The time between two flashes detected by the observer in this new position is 2𝐿 𝑇𝐴𝐵 = 𝑐

In general the time between two flashes seen by the observer in any 2𝐿 point inside or outside the clock varies between 0 and 𝑐 In other words, the duration of a tick is not the same as the duration of a tock in Einstein’s clock unless the observer is positioned on a perpendicular line to the laser beams bisecting it at the middle of AB line. Let us call this line the Tick=Tock Line. The addition of the two measurements, tick+tock, however, is constant from any observing point and equals to 2𝐿 𝑇0 = 𝑇𝐴𝐵 + 𝑇𝐵𝐴 = 𝑐 Tuning Einstein's light clock

7

The Effect of Observer Position (4/4) It can be concluded that • An observer positioned anywhere, within the IRF of the clock, can only accurately measure time by: 1. Observing the duration of two consecutive flashes from the same source (A or B) as a unit of time 2. Staying on the Tick=Tock Line • This option is only theoretically possible • The existence of the Tick=Tock Line does not mean other observing points are not correct, rather highlighting a limitation of the clock

• The measurement comes with a delay between a flash and its observation which depends on the distance between the observer and the source of the flash 10/10/2015


8

The Effect of the Clock Movement (1/3) y

y’ 𝑣

S O

z

S’ x

O’

x’

z’

• •

Consider two coordinate systems S and S’ S and S’ are inertial reference frames (their movements are non accelerating)

• •

Suppose S is stationary and S’ is moving to the right with constant velocity 𝑣 Suppose that the clock is in S’

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9

The Effect of the Clock Movement (2/3) Case 4 (Standard Position)

𝑣

B

• 𝑐 Tv/2

• 𝐿 •

A

An observer in S sees the light clock moves to the right with constant speed 𝑣 It is suggested that for the observer in S the light has to travel longer for the round trip from A to B and back to A as shown The time for the round trip is1: 2

𝑇𝑣 = 𝑇0 / 1 − 𝑣 Τ𝑐2

𝑣Tv

Lorentz factor 2

𝛾 = 1 / 1 − 𝑣 Τ𝑐2 𝑣=0 ⟹𝛾=1 𝑣=𝑐 ⟹𝛾=∞ 1≤ 𝛾≤∞

10/10/2015

1/2

1/2

= 𝑇0 𝛾 (Eq 1)

which indicates • A moving clock runs slower. This is known as time dilation • As the speed of the clock gets closer to the speed of light, the clock appears to run slower and slower until it freezes at 𝑣 = 𝑐 • Eq 1 is the standard time measurement between two IRFs in SR • Mathematics of SR also relies on 𝛾 for length contraction and relativistic mass variation


10

The Effect of the Clock Movement (3/3) Case 5 (turning Einstein’s light clock)

𝑣 A

1. Suppose the light clock is turned 90 degrees so that the direction of beam from A to B is in line with 𝑣 2. For the observer in S’ the unit of time is still the same, 𝑻𝟎 3. For an observer in S the light has to go longer from A to B and shorter from B to A as shown

B

𝐿 A B A

The time for light to go from A to B, tick, is 𝑐𝑇0 𝑇𝑣1 = tick 2 𝑐−𝑣 and the time for light to go from B to A is

∆𝑥1 ∆𝑥2

𝑇𝑣2 =

𝑐𝑇0 2 𝑐+𝑣

𝑇𝑣 = 𝑇𝑣1 + 𝑇𝑣2 = 𝑇0/(1 − 10/10/2015


tock 𝑣2 ) 𝑐2

= 𝑇0𝛾2 (Eq 2) 11

The Effect of the Clock Orientation Time dilation factor depends on the orientation of the clock • The direction of the clock affects the time measurement from a different inertial reference frame • The difference is between 𝑻𝟎𝛾 and 𝑻𝟎𝛾2 depending on the orientation of the clock

B

𝐿

𝑇𝑣 = 𝑇0𝛾 𝑣

1≤ 𝛾≤∞

A 𝑇0𝛾2

A

B

≥ 𝑇𝑣 ≥ 𝑇0𝛾

2

𝒗/𝒄 0 0.3 0.6 0.9 0.99 0.999 0.9999 0.99999 0.999999

1 1.05 1.25 2.29 7.09 22.37 70.71 223.61 707.11

1 1.10 1.56 5.26 50.25 500.25 5000.25 50000.25 500000.25

𝑇𝑣 = 𝑇0𝛾2

𝐿 10/10/2015


12

Length contraction • George FitzGerald and Hendrik Lorentz wrongly proposed that objects are contracted in the direction of motion relative to aether • Length contraction was put forward to justify the existence of aether despite of Michelson-Morley null result • This idea was accepted into special relativity but between two IRFs • That is a 1m ruler in one IRF must be, not appears to be, 0.5m according to 2nd IRF and 0.5cm, or any other length between 0 to 1m, from a 3rd IRF depending on their relative speeds • It is thus argued that the length of the clock is shrunk by 𝟏/ in the direction of its motion thus the clock timing should be corrected according to its orientation1 • This argument relies on or prefers only one specific orientation of the light clock for time measurement where the durations of ticks and tocks are equal 10/10/2015


13

Widening Gap Between Tick and Tock • Which orientation of the clock is the correct one for measurement of time in S knowing that the speed of S’ can be nonzero in any of the three dimensions of space • Depending on the orientation of the light clock and quantity of 𝑣, the durations of a tick and a tock in S’ can be enormously different and almost be any value if observed from S

y’ 𝑣

S’ O’ y

𝑇0 /2 ≤ 𝑡𝑖𝑐𝑘 ≤ ∞ 𝑇0 /4 ≤ 𝑡𝑜𝑐𝑘 ≤ 𝑇0/2

• Length contraction and the addition of tick and tock practically solves the orientation problem of the clock but it cannot explain the widening difference between a tick and a tock which is nothing to do with length contraction • Here it is shown that there is an obvious reason for these differences

x’

z’

S O

x

z 10/10/2015


14

Boat clock (1/4)

B

• Consider a canal in which speed of water can be controlled • Initially water is standing still, V = 0 • Two toy boats repeatedly travel from point A with the constant speed of 𝑈 for the following two straight round trips  The 1st boat aims to reach point B across the river and return  The 2nd boat aims to reach point C in downstream and return • Duration of each trip is 2𝐿 𝑡0 = 𝑡0 𝐀𝐁𝐀 = 𝑡0 𝐀𝐂𝐀 = 𝑈

V= 0 𝐿 𝐿 A

C

B V=0

𝑈

𝐿

𝑈

C

A 𝐿

• Both boats return to point A at the same time • Generally, the duration of any equal distance round trip is the same in any direction when 𝑉 = 0 and 𝑈 being constant • This duration can be assumed as the unit of time 26-Sept-2015

The Twins Paradox and Flat Earth Paradox

15

Boat Clock (2/4)

V t/2

Unit of time in ABA route • 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 unit of time is1 2𝐿 𝑡 = 𝑡𝐀𝐁 = 𝑡𝐁𝐀 = / 1− V2/ U2 𝑈 And its relation to the proper time, 𝑡0, is

V

B

D

𝐿

𝑈t/2

A D

Vt

B

U

ϕ ϕ

L

𝑡 = 𝑡0/ 1− V2/ U2 The above formula is a familiar expressions in special relativity i.e. Lorentz transform, used in time dilation and length contraction

V

vcrossing

A

E

𝑣 𝑐 T/2

𝐿

𝑇 = 𝑇0 / 1− v2/ c2 𝑣T/2 26-Sept-2015


16

Boat clock (3/4)

V

Unit of time in ACA route 𝐿 • The duration of A to C trip is 𝑡𝐀𝐂 = 𝑈+𝑉 • And the duration of C to A trip is • Duration of a return trip

is1

𝑡𝐀𝐂𝐀 = 𝑡𝐀𝐂 + 𝑡𝐂𝐀 =

𝑈+𝑉 𝐿

A

𝑡𝐂𝐀 = 𝑈−𝑉

2𝐿 /(1− 𝑈

𝐿

𝑈−𝑉

V2/

U2)

Which is similar to the formula for time dilation due to the light clock moving with speed v along the direction and then opposite of light 𝑣2 𝑇𝑣 = 𝑇0/(1 −

26-Sept-2015

V

V2/ U2)

• And its relation to the proper time,𝑡0, is 𝑡 = 𝑡0/(1−

C

𝑐2

A

C

𝐿 𝑣 A

B 𝐿

)


17

Boat clock (4/4)

V t/2

Relation between the two units of time

• The relation between the two time durations is

D

𝐿 𝑈t/2

A

𝑡𝐀𝐂𝐀 / 𝑡𝐀𝐁𝐀 = 1/ 1− V2/ U2

V

Which is similar to the relation between units of time of light clock in the two perpendicular directions 𝑇𝟏 / 𝑇𝟐 = 1/ 1− v2/ c2

𝑣 B

V

B

𝑈+𝑉 A 𝐿

C

𝑣 𝐿

2

A

B 𝐿

A 26-Sept-2015

1 The Twins Paradox and Flat Earth Paradox

18

Summary of Boat Clock • The working principle of the light clock used in relativity is the same as the boat clock where a toy boat goes back and forth with the constant speed 𝑈 from bank to bank or between two points along the river (a downstream-upstream route) • Similar to the light clock one can, at least 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) • With water flowing with a constant speed then the duration of the trips changes and this fact is nothing to do with length contraction or time dilation

26-Sept-2015


B

V=0

𝑡0 = 2𝐿/U

L A

Mirror

𝐿

𝑇0 = 2𝐿/c

19

Returning Frame Case 6 • •

𝑣 A

B

For the observer in S’ the unit of time is still the same, 𝑻𝟎 For an observer in S the light has to go shorter from A to B and longer from B to A as shown

𝐿 A

B

A

B

A

The time for light to go from A to B, tick, is 𝑐𝑇0 tick 𝑇𝑣1 = 2 𝑐+𝑣 and the time for light to go from B to A, tock, is 𝑐𝑇0 tock 𝑇𝑣2 = 2 𝑐−𝑣 Total time is thus 𝑇𝑣 = 𝑇𝑣1 + 𝑇𝑣2 = 𝑇0/(1 −

∆𝑥2 ∆𝑥1

𝑣2 ) 𝑐2

= 𝑇0𝛾2

Which is the same as the clock moving away

Now what is the effect of the direction of the clock movement on varying time delay between any pulse and observer? 10/10/2015


20

Time Delay Between Two Pulses When the Clock is Moving Away • For the observer in S the clock is moving to the right with an speed of 𝑣 for 𝑻𝒗 • Suppose the first pulse is flashed with zero distance from the observer • The second pulse is ∆𝑥 = ∆𝑥1 + ∆𝑥2 away from the observer 𝑣

OA

∆𝑥 = 𝑇𝑣 𝑣

B A

∆𝑥1 ∆𝑥2

and the time for the second pulse to go from A to the observer 𝑣 𝑇𝐷𝑒𝑙𝑎𝑦 = 𝑇𝑣 𝑐

The total duration measured by the observer in S is thus 𝑣

𝑇𝑇𝑜𝑡𝑎𝑙𝐴𝑤𝑎𝑦 = 𝑇𝑣 + 𝑇𝐷𝑒𝑙𝑎𝑦 = 𝑇0 𝛾2(1+ 𝑐 ) 10/10/2015


21

Time Advance Between Two Pulses When the Clock is Returning Back Similarly the time advance for the second pulse is

𝑣 A

B

𝑇𝐴𝑑𝑣𝑎𝑛𝑐𝑒 = −𝑇𝑣

𝑣 𝑐

A O

A

B

And the total time measured by the observer in S when the clock is approaching the observer is

∆𝑥2 ∆𝑥1

𝑣

𝑇𝑇𝑜𝑡𝑎𝑙𝑅𝑒𝑡𝑢𝑟𝑛 = 𝑇𝑣 + 𝑇𝐴𝑑𝑣𝑎𝑛𝑐𝑒 = 𝑇0 𝛾2(1- 𝑐 )

10/10/2015


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Time Delay/Advance in Standard Position • 𝑣 •

𝑐 Tv/2

•

𝐿 A

O

•

In this position no difference has been recognized between the clock getting away or returning The main reason is that in both cases 𝑇𝑣 is also the same. However, no time delay or advance is recognized between the two readings from S In fact if the observer is at point O the actual time measured between two pulses from laser A when the clock moves away is

𝑣Tv 𝑣

𝑇𝑇𝑜𝑡𝑎𝑙𝐴𝑤𝑎𝑦 = 𝑇𝑣 + 𝑇𝐷𝑒𝑙𝑎𝑦 = 𝑇0 𝛾(1+ 𝑐 ) Also the total time measured between the two pulses from laser A by the observer in S when the clock is approaching point O is 𝑣

𝑇𝑇𝑜𝑡𝑎𝑙𝑅𝑒𝑡𝑢𝑟𝑛 = 𝑇𝑣 + 𝑇𝐴𝑑𝑣𝑎𝑛𝑐𝑒 = 𝑇0 𝛾(1- 𝑐 ) 10/10/2015


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Summary

• Time dilation factor depends on 1. Orientation of the clock 2. Direction of the movement of the clock

• Both gravely affect unit of time

B

𝑣

𝑇𝐴𝑤𝑎𝑦 = 𝑇0 𝛾(1+ 𝑐 )

𝐿

𝑣 𝑐

𝑇𝑅𝑒𝑡𝑢𝑟𝑛 = 𝑇0 𝛾(1- )

A

𝑣 𝑣 𝑇0 ) ≥ 𝑇𝐴𝑤𝑎𝑦 ≥ 𝑇0 𝛾(1+ ) 𝑐 𝑐 𝑣 𝑣 2 𝑇0 𝛾 (1− ) ≥ 𝑇𝑅𝑒𝑡𝑢𝑟𝑛 ≥ 𝑇0 𝛾(1− ) 𝑐 𝑐

𝛾2(1+

𝑣

𝑇𝐴𝑤𝑎𝑦 = 𝑇0 𝛾2(1+ 𝑐 ) A

B 𝐿 10/10/2015

𝑣

𝑇𝑅𝑒𝑡𝑢𝑟𝑛 = 𝑇0 𝛾2(1- 𝑐 ) Tuning Einstein's light clock

24

Einstein’s Light Clock with a Counter (1/3) B

• We assume again that when the upper 𝐿 plate receives a red light pulse from laser A, A laser B immediately emits a blue light pulse 00002376948 and vice versa, ad infinitum • For the benefit of an observer, a counter is added to the bottom plate which increments when a light pulse is received from laser B • For an observer in S’, the frame in which the clock is positioned, the unit of time is 𝑻𝟎 10/10/2015


25

Einstein’s Light Clock with a Counter (2/3) • From an observer in S, the frame from which the clock relatively moves with constant speed, the duration for the counter increment is the same as in S’ • The orientation of the clock has also no effect for any observer at any IRF • The only difference is either a delay or advance, due to the direction of the relative movement of the clock B 𝑣

𝑇𝐴𝑤𝑎𝑦 = 𝑇0 (1+ 𝑐 ) 𝐿

𝑣

𝑇𝑅𝑒𝑡𝑢𝑟𝑛 = 𝑇0 (1- 𝑐 ) A 00002376948 10/10/2015


26

Einstein’s Light Clock with a Counter (3/3) This is acceptable and can be explained by the following examples 1. Suppose a clock moves away from an observer with the speed of light, reaches a distance equal to the distance from the sun to the earth and then stops • It takes about 8 minutes for the clock to reach its destination • When the clock shows 8 minutes the similar clock in the observer frame shows 16 minutes, 𝑇𝐴𝑤𝑎𝑦 = 2 𝑇0 • In its destination the clock is always observed with 8 minutes delay which is also true for the sun as we always see the sun 8 minutes in the past

2. If at some time the clock returns to the observer with the speed of light, the observer only notices this movement when the clock appears in front of his eyes, 𝑇𝑅𝑒𝑡𝑢𝑟𝑛 = 0 10/10/2015


27

Conclusions Einstein’s light clock is not a reliable time measuring apparatus from outside of its inertial reference frame due to the following reasons: 1. The duration of a tick and a tock depends on the observer position 2. The moving clock shows different time unit depending on the orientation of the clock 3. The duration of time unit is also affected by the direction of the movement of the clock Adding a counter eliminates 𝛾 and corrects for orientation 10/10/2015


28

Notes The Effect of the Clock Movement (2/3)

1. (𝑐Tv/2)2 = (𝑣Tv /2 )2 + 𝐿2 ⟹ (𝑐Tv/2)2 = (𝑣Tv /2 )2 + (𝑐𝑇0/2)2 ⟹ 𝑐 2Tv2 = 𝑣2Tv 2 + 𝑐2𝑇02 ⟹ 𝑐 2Tv2 − 𝑣2Tv 2 = 𝑐2𝑇02 ⟹ Tv2 (1 − 𝑣2/𝑐 2) = 𝑇02 ⟹ Tv = 𝑇0/(1 − 𝑣2/𝑐 2) 1/2 Length Contraction

1 - Note that derivation of length contraction is based on reading the time from a light clock at an specific direction (see Mathematics of Special relativity) 𝑥’1 = 𝛾 𝑥1 − 𝑣𝑡1 𝑥’2 = 𝛾 𝑥2 − 𝑣𝑡2 𝐿 = 𝑥2 - 𝑥1 𝐿′ = 𝑥′2 - 𝑥′1 } ⟹ 𝐿′ = 𝐿/𝛾 𝑡1 = 𝑡2 Boat clock (2/4) 2𝐿 1. 𝑈2t2 − 𝑉2t2= 4𝐿2⟹ t2(𝑈2 − 𝑉2) ⟹ 𝑡 = 𝑈 / 1−𝑉2/𝑈2 Boat clock (3/4) 𝐿 𝐿 𝐿𝑈−𝐿𝑉+𝐿𝑈+𝐿𝑉 2𝐿𝑈 2𝐿𝑈 2𝐿 1. 𝑡𝐀𝐂 + 𝑡𝐂𝐀 = 𝑈+𝑉 +𝑈−𝑉 = 𝑈+𝑉 𝑈−𝑉 = 𝑈+𝑉 𝑈−𝑉 = 𝑈 2−𝑉 2 = 𝑈 /(1− V2/ U2) 10/10/2015


29