Physics 3 for Electrical EngineeringPhysics 3 for Electrical Engineering

Ben Gurion University of the Negevwww.bgu.ac.il/atomchip, www.bgu.ac.il/nanocenter

Lecturers: Daniel Rohrlich, Ron Folman Teaching Assistants: Daniel Ariad, Barukh Dolgin

Week 1. Special relativity – crisis of the aether • equivalence of inertial reference frames • speed of light as maximum signaling speed • definition of an event • Lorentz transformations • relativity of simultaneity • spacetime 4-vectors • timelike, lightlike and spacelike intervals Sources: Feynman Lectures I, Chap. 15 Sects. 1-7; Tipler and Llewellyn, Chap. 1 Sects. 1-4;

4, פרק 1פרקים בפיסיקה מודרנית, יחידה

The crisis of the aether

Maxwell’s equations (1865) predict that electromagnetic waves (including light waves) travel at speed c in vacuum. The accepted value of c today is 299,792,458 m/s.

EEE

E

BEE

BE

BE

22

2

2002

2

1

)()(

0

0

tctt

t

t

t

Faraday’s law

Vector identity

Gauss’s law in a vacuum: •E = 0

Ampère’s law, as modified by Maxwell

The crisis of the aether

Maxwell’s equations (1865) predict that electromagnetic waves (including light waves) travel at speed c in vacuum. The accepted value of c today is 299,792,458 m/s.

BUT…

this speed is relative to…what?

The nineteenth century answer: c is relative to the “aether”. All wave move in a medium: ripples on water, sounds in the air, etc. The aether is the medium for light.

We can detect water, air, etc. Can we detect the aether?

The experiment of Michelson and Morley (1887)

direction of earth's motion through the “aether”

down2earth

2up / tvcDt

)/( earthright vcDt

)/( earthleft vcDt

detector

Michelson and Morley rotated the experiment by 90º to cancel experimental errors.

D

D

The experiment of Michelson and Morley (1887)

direction of earth's motion through the “aether”

detector

Expected path difference:

2(tright+ tleft – tup – tdown) c ≈ 2(vearth /c)2 D = 1 · 10–8 · D

= 1 · 10–8 · (2 · 107 λ)

= 0.2 λ

D

D

)/( earthright vcDt

)/( earthleft vcDt

down2earth

2up / tvcDt

The experiment of Michelson and Morley (1887)

direction of earth's motion through the “aether”

detector

D

D Expected fringe shift:

0.4 Observed fringe shift: <

0.02 Result for vearth: vearth < 8 km/s instead of 30 km/s

)/( earthright vcDt

)/( earthleft vcDt

Expected fringe shift (after rotation):

2(tright+ tleft – tup – tdown) c ≈ 2(vearth /c)2 D = 2 · 10–8 · D

= 2 · 10–8 · (2 · 107 λ)

= 0.4 λ

down2earth

2up / tvcDt

The experiment of Michelson and Morley (1887)

direction of earth's motion through the “aether”

detector

D

D

If there is no aether, then Maxwell’s electromagnetism and Newton’s

mechanics are incompatible!

If there is no aether, then Maxwell’s electromagnetism and Newton’s

mechanics are incompatible!

Einstein’s bold line of attack was to combine an old principle – relativity, true in Newton’s mechanics – with a new principle: the speed of light c is a universal constant.

Einstein’s first postulate:

The equivalence of inertial reference frames

Definition: an inertial reference frame is a system of coordinates moving with constant velocity (constant speed and constant direction).

Example: the system of coordinates of a passenger in a train that moves at constant velocity is an inertial reference frame.

The principle of relativity:

The laws of physics are the same in all inertial reference frames.

A reference frame made of meter sticks and clocks:

Einstein’s second postulate:

The speed of light is the maximum signaling speed

There is a maximum signaling speed, and it is the speed of light c. Since this is a law of physics, the speed of light c is the same in every inertial reference frame. The speed of light is a universal constant.

Einstein’s second postulate:

The speed of light is the maximum signaling speed

There is a maximum signaling speed, and it is the speed of light c. Since this is a law of physics, the speed of light c is the same in every inertial reference frame. The speed of light is a universal constant.

Einstein’s first and second postulates seem incompatible. He was the first to understand that they are not!

Events

Definition: an event is any physical process that occurs at a definite time and at a definite point in space.

Relativity of time for two observers, Alice and Bob:First event: flashlight emits flashSecond event: flash returns to flashlightThese two events define one clock cycle.

vt

Lct/2ct/2

Bob, in his frame, measures time t between the two events. The time t must obey (ct/2)2 = (vt/2)2 + L2 so

►(t)2= 4L2/(c2–v2) = (4L2/c2 ) 2 ,

where = (1–v2/c2 )–1/2. On the other hand in Alice’s frame, where the time between the events is t′, we find: (t′)2 = 4L2/c2 (time according to Alice).

Thus ► t = t′.

Observer A (Alice), on a train moving with speed v relative to Bob, shines a flashlight at a mirror on the ceiling of the train.

Observer B (Bob) is outside on the ground.

Note t′ < t since 1 < < ∞. So Bob thinks that for Alice, the “light clock” ticks slower. In fact, Bob will conclude that he ages faster while Alice lives longer!

☺

Relativity of length

train speed = v

L′ – length of ruler in Alice’s “train frame”. In Alice’s frame, the ruler is at rest.

What is L (the length of the ruler in Bob’s frame)?

“Light measurement” of length in Alice’s frame: L′ = ct1′ and L′ = c(t2′ – t1′ ) where t1′ is when the photon hits the right end of the ruler (starting at time 0 for both Alice and Bob) and t2 ′ is when it hits the end it started from. So 2L′ = ct2′.

Now, Bob sees Alice and the ruler moving at speed v, so L + vt1 = ct1 and

L – v(t2 – t1) = c(t2 – t1). Eliminating t1 we get 2L = (c2 – v2) t2/c . Combining with 2L′ = ct2′, we get L′/L = (t2′ /t2) c2/(c2 – v2) = 2(t2′ /t2). But t2 = t2′ so ► L = L′ / (again a consequence of the fact that c is the same in all frames).

L′☺ ruler

Let’s find the “dictionary” (transformation of space-time) for going from one “language” (Alice’s inertial frame) to another (Bob’s inertial frame) while preserving the meaning of the “sentence” (the form of the equations of Nature, e.g. Maxwell’s equations).

Criterion: For any light ray,

x)2 + (y)2 + (z)2 – c2 (t)2 = 0 = (x′)2 + (y′)2 + (z′)2 – c2 (t′)2

If Alice (primed system) moves along the x-axis with velocity v relative to Bob (unprimed system), then

x′ = (x – vt) y′ = y z′ = z t′ = (t – vx/c2)

Lorentz transformations

x = (x′ + vt′) y = y′ z = z′ t = (t′ + vx′ /c2)

Lorentz transformations

Inverse

And by changing v to –v we obtain the

4-vectors

We can write spacetime coordinates as “4-vectors” i.e.

r = (ct, x, y, z)

where the invariant length of any 4-vector v = (v0, v1 , v2, v3

) is

20

23

22

21

2 )()()()( vvvv v

Lorentz transformation of 4-vector:

3

2

1

0

3

2

1

0

1000

0100

00

00

'

'

'

'

v

v

v

v

v

v

v

v

4-vectors

We can write spacetime coordinates as “4-vectors” i.e.

r = (ct, x, y, z)

where the invariant length of any 4-vector v = (v0, v1 , v2, v3

) is

20

23

22

21

2 )()()()( vvvv v

Lorentz transformation of 4-vector:

3

2

1

0

3

2

1

0

1000

00

0010

00

'

'

'

'

v

v

v

v

v

v

v

v

4-vectors

We can write spacetime coordinates as “4-vectors” i.e.

r = (ct, x, y, z)

where the invariant length of any 4-vector v = (v0, v1 , v2, v3

) is

20

23

22

21

2 )()()()( vvvv v

Lorentz transformation of 4-vector:

3

2

1

0

3

2

1

0

00

0100

0010

00

'

'

'

'

v

v

v

v

v

v

v

v

Timelike, spacelike and light-like 4-vectors

Light-like vector: u2 = 0 Timelike vector: u2 < 0 (between causally connected events) Spacelike vector: u2 > 0 (between causally unconnected events)

ct

x

event

future light cone

past light cone

Additional reference:

Experimental tests of time dilation: CERN (1966):

muons at speed v = 0.9997cFactor 12 increase in lifetime (2% accuracy in the prediction)

Daily life: GPS

If we want 30 m accuracy in distance we need 0.1 s accuracy in time. But a satellite in geosynchronous orbit, moving at a speed of 11,000 km/h ≈ 3000 m/s ≈ 10–5 c, loses 4.3 s per day because of time dilation!