Newtonian relativityMichelson-Morley ExperimentEinstein’s principle of relativitySpecial relativityLorentz transformationRelativistic momentum and Newton's lawsRelativistic energyMass and energyGeneral relativity

Relativity (chapter nine)

• When glass breaks, the cracks move faster than 3,000 miles per hour.

• Most lipstick contains fish scales!

• A 'jiffy' is an actual unit of time for 1/100th of a second!

• Windmills always turn counter-clockwise. Except for the windmills in Ireland!

• A sneeze travels out your mouth at over 100 m.p.h.!

• Slugs have 4 noses!

• Bats always turn left when exiting a cave!

• Most dust particles in your house are made from dead skin!

Principles of Newtonian Relativity

Inertial frame = objects experience no acceleration if no forces act on it

Newtonian relativity: laws of mechanics same in all inertial reference frames

For example: ball thrown straight up in moving truck

Truck frame: motion straight up and down

Earth frame: parabolic trajectory

Galilean transformation of coordinates

Newtonian Relativity

Both described by Newtonian mechanics Connection between the two: Galilean transformation of coordinates

vuu

tt

zz

yy

vtxx

xx

Galilean addition of velocities

Michelson Morley experiment

Simple mechanics obey Newtonian relativity at low speeds, but some laws of E&M do not.

Speed of light, measured by Michelson and Morley showed no difference parallel vs. perpendicular to the direction of the earth’s motion.

M-M interferometer – measures phase shift of beam in one arm with respect to the other – phase shift should change as interferometer is rotated through 90°.

No change was observed.

No ether, no absolute inertial frames.

Einstein's principle of relativity

1. All laws of physics are same in all inertial reference frames

2. The speed of light in vacuum has the same value in all reference frames

These are the basic postulates of special relativity

Consequences of special relativity

The consequences of these two postulates are:1. There are no absolute standards of length or time.2. Simultaneity depends on the reference frame.

Thought experiment:• Two bolts of lightning strike ends of boxcar• Simultaneous in ground frame• Observer on boxcar sees light from front of boxcar first –

concludes that this event occurred first.• Both are correct – the simultaneity of two event depends on

the reference frame!

Consequences of special relativity

How many relativists does it take to change a light bulb? Two. One holds the bulb, while the other rotates the universe.

Second thought experiment: Light emitted vertically in moving train, reflected from mirror.In train reference frame, time interval equals 2d/c (proper time).In ground frame, light appears to travel a greater distance. Since the speed is the same in both frames, the time interval in the ground frame must be longer.

2

2

22

1

1

2

2

cv

tvc

dt

c

dt

p

p

Length contraction

Since time intervals now depends on the reference frame, length will also. The length measured by an observer at rest relative to the object is called the proper length

p

p

LtvL

Third thought experiment: spaceship traveling a distance d at speed v. The observer at rest measures a (proper) length Lp. A person in the spaceship sees the distance pass with speed v in time t=tp, and therefore calculates a length

Length contraction

Example: You are traveling in a spaceship which you measure to be 100 m long. What length does an observer measure for the spaceship if you pass by at 0.9c?

mm

c

vm

LL p

6.439.011000.1

11000.1

22

2

22

Lorentz transformations

The Lorentz transformations relate the relative position in space-time of an event in two reference frames

For an observer in

frame S’ moving at

a speed v along the

x-axis

xc

vtt

zz

yy

vtxx

2'

'

'

'

Lorentz transformations

From the Lorentz transformations for space-time, one can derive the Lorentz velocity transformations

22 1'

''

cvuvu

dxcv

dt

vdtdx

dt

dxu

x

xx

Likewise along the y and z directions

Limiting cases:

1) v<<c

(Lorentz)

2) ux=c

(speed of light same)

vu

cvuvu

u xx

xx

21'

c

vcc

vc

ccvvc

ux

11

'

2

Relativistic momentum

Generalize Newton’s laws (2nd law) –in particular, need conservation of momentum independent of reference frame

Just using the Lorentz transformation to find the velocities does not conserve momentum

New definition of momentum:

um

cu

ump

2

2

1

The relativistic force on a particle is (still):dt

pdF

Relativistic energy

As with momentum, we need to redefine energy. Starting with the work KE theorem:

22

2

2

2

2/3

2

2

2

2

)1(11

)/(

1

mcmc

cu

mcdx

cu

dtdum

dx

cu

mu

dt

ddx

dt

dpFdxW

Matches usual equation (½mv2) for v<<c

Relativistic energy

The second, constant term is called the rest energy:2mcER

The total energy is this rest energy plus the kinetic energy

2

2

2

1cu

mcE

Mass is a manifestation of energy

Relativistic energy

We can rewrite the equation for the total energy in terms of the relativistic momentum

22222

2222222222222

2222222

2

2

22

)(

1

1

mccpE

cucmcmcmcE

cucuc

c

cu

Extremely useful formulation – also applies for massless particles, e.g., photons

hcpcE

General relativity

Inertial and gravitational mass - same effects

Einstein proposed two postulates of general relativity:

1. All the laws of physics have the same form for observers in any frame of reference

2. In the vicinity of any point, a gravitational field is equivalent to an accelerated reference frame (principle of equivalence).

General relativity

Like special relativity, unusual consequences: Passage of time altered by gravity (experimentally

verified - radiation frequency in strong gravitational fields)

Accelerating reference frame "transforms away" gravity (free fall)

Light passing mass - deflected by gravitational field

There was a man who entered a local paper's pun contest. He sent in ten different puns, in the hope that at least one of the puns would win. Unfortunately, no pun in ten did.