# Special Theory of Relativity Albert Einstein Microworld Marcoworld

- Slides: 76

Special Theory of Relativity Albert Einstein Micro-world Marco-world Lect 11

watching a light flash go by v c The man on earth sees c = (& agrees with Maxwell) 2 k k

watching a light flash go by v c If the man on the rocket sees c-v, he disagrees with Maxwell

watching a light flash go by v If Maxwell’s theory works for this guy, It doesn’t works for this guy, c or vice versa

Do Maxwell’s Eqns only work in one reference frame? If so, this would be the rest frame of the luminiferous Aether.

If so, the speed of light should change throughout the year downstream, light moves faster “Aether wind” upstream, light moves slower

Michelson-Morley No aether wind detected: 1907 Nobel Prize

Einstein’s hypotheses: 1. The laws of nature are equally valid in every inertial reference frame. Including Maxwell’s eqns 2. The speed of light in empty space is same for all inertial observers, regardless of their velocity or the velocity of the source of light.

All observers see light flashes go by them with the same speed No matter how fast the guy on the rocket is moving!! v c Both guys see the light flash travel with velocity = c

Even when the light flash is traveling in an opposite direction v c Both guys see the light flash travel past with velocity = c

Gunfight viewed by observer at rest He sees both shots fired simultaneously Bang !

Viewed by a moving observer

Viewed by a moving observer He sees cowboy shoot 1 st & cowgirl shoot later Bang !

Viewed by an observer in the opposite direction

Viewed by a moving observer He sees cowgirl shoot 1 st & cowboy shoot later Bang !

Time depends of state of motion of the observer!! Events that occur simultaneously according to one observer can occur at different times for other observers

Light clock

Seen from the ground

Events y (x 1, t 1) x t x 1 (x 2, t 2) x x 2 x

y Prior to Einstein, everyone agreed the distance between events depends Same events, different observers upon the observer, but not the time. y’ y’ (x 2, t 2) (x 1, t 1) x x (x 1’, t 1’) t’ t t’ (x 2’, t 2’) x 1’ dist’ x 1 dist x 2’ x’ x’ x 2 x

Time is the 4 th dimension Einstein discovered that there is no “absolute” time, it too depends upon the state of motion of the observer Newton completely different concepts Space & Time Einstein Space-Time 2 different aspects of the same thing

How are the times seen by 2 different observers related? We can figure this out with simple HS-level math ( + a little effort)

Catch ball on a rocket ship Event 2: girl catches the ball v= w=4 m w =4 m/s t t=1 s Event 1: boy throws the ball

Seen from earth V 0=3 m/s Location of the 2 events is different Elapsed time is the same The ball appears to travel faster t=1 s )2 4 m ( + )2 = d m 5 = m 3 ( w=4 m v 0 t=3 m v= d = 5 m/s t

Flash a light on a rocket ship Event 2: light flash reaches the girl c= w wt 0 Event 1: boy flashes the light

Seen from earth V V Speed has to Be the same 2 Dist is longer Time must be longer t=? d= ( w 2+ ) vt w vt c= d 2 2 = (vt) +w t t

How is t related to t 0? t= time on Earth clock (vt)2+w 2 t c = ct = t 0 = time on moving clock c = wt 0 ct 0 = w (vt)2+w 2 (ct)2 = (vt)2+(ct 0)2 t 2 (ct)2 -(vt)2= (ct 0)2 c 2 = 2 t 02 2 c – v t= this is called g 1 t 1 – v 2/c 2 0 t 2 = (c 2 -v 2)t 2= c 2 t 02 1 t 02 2 2 1 – v /c t = g t 0

Properties of g = Suppose v = 0. 01 c (i. e. 1% of c) 1 g = 1 – (0. 01 c)2/c 2 1 g = 1 – (0. 01)2 1 1 – v 2/c 2 = 1 – (0. 01)2 c 2/c 2 1 1 = 1 – 0. 0001 0. 9999 g = 1. 00005

1 1 – v 2/c 2 Properties of g = Suppose v = 0. 1 c (cont’d) (i. e. 10% of c) 1 g = 1 – (0. 1 c)2/c 2 1 = 1 – (0. 1)2 c 2/c 2 1 g = 1 – (0. 1)2 1 1 – 0. 01 = g = 1. 005 = 1 0. 99

Let’s make a chart v 0. 01 c 0. 1 c g =1/ (1 -v 2/c 2) 1. 00005 1. 005

Other values of g = Suppose v = 0. 5 c (i. e. 50% of c) 1 g = 1 – (0. 5 c)2/c 2 1 g = 1 – (0. 5)2 1 1 – v 2/c 2 = 1 – (0. 5)2 c 2/c 2 1 1 – (0. 25) g = 1. 15 = 1 0. 75

Enter into chart v 0. 01 c 0. 5 c g =1/ (1 -v 2/c 2) 1. 00005 1. 15

Other values of g = Suppose v = 0. 6 c (i. e. 60% of c) 1 g = 1 – (0. 6 c)2/c 2 1 g = 1 – (0. 6)2 1 1 – v 2/c 2 1 = 1 – (0. 6)2 c 2/c 2 = 1 1 – (0. 36) g = 1. 25 = 1 0. 64

Back to the chart v 0. 01 c 0. 5 c 0. 6 c g =1/ (1 -v 2/c 2) 1. 00005 1. 15 1. 25

Other values of g = Suppose v = 0. 8 c (i. e. 80% of c) 1 g = 1 – (0. 8 c)2/c 2 1 g = 1 – (0. 8)2 1 1 – v 2/c 2 = 1 – (0. 8)2 c 2/c 2 1 1 – (0. 64) g = 1. 67 = 1 0. 36

Enter into the chart v 0. 01 c 0. 5 c 0. 6 c 0. 8 c g =1/ (1 -v 2/c 2) 1. 00005 1. 15 1. 25 1. 67

Other values of g = Suppose v = 0. 9 c 1 1 – v 2/c 2 (i. e. 90% of c) 1 g = 1 – (0. 9 c)2/c 2 1 = 1 – (0. 9)2 c 2/c 2 1 g = 1 – (0. 9)2 1 1 – 0. 81 = g = 2. 29 = 1 0. 19

update chart v 0. 01 c 0. 5 c 0. 6 c 0. 8 c 0. 9 c g =1/ (1 -v 2/c 2) 1. 00005 1. 15 1. 25 1. 67 2. 29

Other values of g = Suppose v = 0. 99 c 1 1 – v 2/c 2 (i. e. 99% of c) 1 g = 1 – (0. 99 c)2/c 2 1 = 1 – (0. 99)2 c 2/c 2 1 g = 1 – (0. 99)2 1 1 – 0. 98 = g = 7. 07 = 1 0. 02

Enter into chart v 0. 01 c 0. 5 c 0. 6 c 0. 8 c 0. 99 c g =1/ (1 -v 2/c 2) 1. 00005 1. 15 1. 25 1. 67 2. 29 7. 07

Other values of g = 1 1 – v 2/c 2 Suppose v = c g = 1 1 – (c)2/c 2 1 1 – 12 g = 1 1 – c 2/c 2 = = 1 0 = Infinity!!! 1 0

update chart v 0. 01 c 0. 5 c 0. 6 c 0. 8 c 0. 99 c 1. 00 c g =1/ (1 -v 2/c 2) 1. 00005 1. 15 1. 25 1. 67 2. 29 7. 07

Other values of g = 1 1 – v 2/c 2 Suppose v = 1. 1 c 1 g = 1 – (1. 1 c)2/c 2 1 = 1 – (1. 1)2 c 2/c 2 1 g = 1 – (1. 1)2 1 1 -1. 21 g = = ? ? ? = 1 -0. 21 Imaginary number!!!

Complete the chart v 0. 01 c 0. 5 c 0. 6 c 0. 8 c 0. 99 c 1. 00 c g =1/ (1 -v 2/c 2) 1. 00005 1. 15 1. 25 1. 67 2. 29 7. 07 Larger than c Imaginary number

Plot results: 1 g= 1 – v 2/c 2 x x x Never-never land v=c

Moving clocks run slower v t 0 t= t 1 t 0 2 2 1 – v /c t = g t 0 g >1 t > t 0

Length contraction v L 0 time=t L 0 = vt man on rocket ! r e t r o Sh Time = t 0 =t/g Length = vt 0 =vt/g =L 0/g

Moving objects appear shorter L = L 0/g V=0. 9999 c V=0. 86 c V=0. 1 c V=0. 99 c Length measured when object is at rest g >1 L < L 0

Length contraction

mass: F=m 0 a = m 0 time t 0 time=t 0 Ft 0 change in v = m 0 Ft 0 m 0 = change in v a m 0 m= Ft change in v = g Ft 0 change in v t=gt 0 = g m 0 mass increases!! m = g m 0 by a factor g

Relativistic mass increase m 0 = mass of an object when it is at rest “rest mass” mass of a moving object increases m = g m 0 by the g factor g as v c, m as an object moves faster, it gets harder & harder to accelerate v=c

summary • Moving clocks run slow r o f o g • Moving objects appear shorter t • Moving f c a a object’s y B mass increases

Plot results: 1 g= 1 – v 2/c 2 x x x Never-never land v=c

a-centauri Twin paradox Twin brother & sister 3. 4 t h ig s r ea y l She will travel to a-centauri He will stay home & study Phys 100 (a nearby star on a special rocket ship v = 0. 9 c

Light year distance light travels in 1 year dist = v x time = c yr 1 cyr = 3 x 108 m/s x 3. 2 x 107 s = 9. 6 x 1015 m We will just use cyr units & not worry about meters

Time on the boy’s clock c 9 0. v= . 4 = d 0 r y c 3 c 9 0. v= According to the boy & his clock on Earth: d 0 tout = v = 4. 3 cyr = 4. 8 yrs 0. 9 c d tback = 0 = 4. 3 cyr = 4. 8 yrs 0. 9 c v ttotal = tout+tback = 9. 6 yrs

What does the boy see on her clock? c 9 0. v= 4. = d r y c 3 c 9 0. v= According to the boy her clock runs slower tout 4. 8 yrs = 2. 1 yrs = g = 2. 3 tback = g = 4. 8 yr = 2. 1 yrs 2. 3 tout ttotal = tout+tback = 4. 2 yrs

So, according to the boy: c 9 0. v= 4. = d r y c 3 c 9 0. v= his clock out: 4. 8 yrs back: 4. 8 yrs her clock 2. 1 yrs total: 9. 6 yrs 4. 2 yrs s e g a e h S ç less

But, according to the girl, the boy’s clock is moving &, so, it must be running slower c 9 0. v= According to her, the boy’s clock on Earth says: v= c 9. 0 tout 2. 1 yrs = 0. 9 yrs tout = g = 2. 3 tback 2. 1 yrs tback = = 2. 3 = 0. 9 yrs g ttotal = tout+tback = 1. 8 yrs

Her clock advances 4. 2 yrs & she sees his clock advance only 1. 8 yrs, ? ? n o i t c i d a r t n o A c She should think he has aged less than her!!

Events in the boy’s life: She leaves She arrives & starts turn Finishes turn & heads home She returns As seen by him As seen by her 4. 8 yrs 0. 9 yrs short time ? ? 4. 8 yrs 0. 9 yrs 9. 6+ yrs 1. 8 + ? ? ? yrs

turning around as seen by her According to her, these 2 events occur very, very far apart from each other He sees her start to turn He sees her finish turning Time interval between 2 events depends on the state of motion of the observer

Gunfight viewed by observer at rest He sees both shots fired simultaneously Bang !

Viewed by a moving observer

Viewed by a moving observer He sees cowboy shoot 1 st & cowgirl shoot later Bang !

In fact, ? ? = 7. 8+ years as seen by him She leaves She arrives & starts turn Finishes turn & heads home She returns 4. 8 yrs short time 4. 8 yrs 9. 6+ yrs as seen by her 0. 9 yrs 7. 8+ ? ? ? yrs 0. 9 yrs 1. 89. 6+ + ? ? ? yrs

No paradox: both twins agree The twin that “turned around” is younger

Ladder & Barn Door paradox Stan & Ollie puzzle over how to get a 2 m long ladder thru a 1 m wide barn door ? ? ? 1 m 2 m ladder

Ollie remembers Phys 100 & theory of relativity Stan, pick up the ladder & run very fast 1 m tree 2 m ladder

View from Ollie’s ref. frame Push, Stan! Ollie 1 m 2 m/g Stan V=0. 9 c (g=2. 3)

View from Stan’s ref. frame 1 m/g V=0. 9 c (g=2. 3) Ollie 2 m Stan But it doesn’t fit, Ollie!!

If Stan pushes both ends of the ladder simultaneously, Ollie sees the two ends move at different times: Too late Too soon Stan! Ollie 1 m k n clu Stan clan k V=0. 9 c (g=2. 3)

Fermilab proton accelerator V=0. 9999995 c g=1000 2 km

Stanford electron accelerator v=0. 999995 c m 3 k g=100, 000

status Einstein’s theory of “special relativity” has been carefully tested in many very precise experiments and found to be valid. Time is truly the 4 th dimension of space & time.

test g=29. 3