LECTURE # 2RELATIVITY II

NEWTONIAN RELATIVITY- GALILEAN TRANSFORMATIONS - EINSTEIN

PHYS 420-SPRING 2006

Dennis Papadopoulos

Acknowledge contributions from Chris Reynolds http://www.astro.umd.edu/~chris/Teaching/teaching.html and Nick Strobel's Astronomy Notes

I: SPECIAL RELATIVITY WHY AND WHAT

“ SR AROSE FROM NESSECITY, FROM SERIOUS AND DEEP CONTRADICTIONS IN THE OLD THEORY FROM WHICH THERE SEEMED NO ESCAPE” EINSTEIN 1905

RELATIVITY POSTULATES:

1. THE LAWS OF PHYSICS ARE INVARIANT IN INERTIAL REFERENCE FRAMES.

2. SPEED OF LIGHT IN VACUUM IS CONSTANT INDEPENDENT OF MOTION OF SOURCE AND OBSERVER

FRAME OF REFERENCEWHAT IS A FRAME OF REFERENCE ?

THE PROSPECTIVE FROM WHICH A SYSTEM IS OBSERVED

Car moving to left for one observer and to the right for the other. Both agree that is moving south. To translate their observations we need a transformation. In this case a rotation by 180 degrees

A set of axis relative to which an observer can measure at any time the position, motion and orientation of all points in a system

INERTIAL FRAME OF REFERENCE

A COORDINATE SYSTEM DEFINED BY THE NON-ACCELERATING MOTION OF OBJECTS THAT HAVE A

COMMON DIRECTION AND SPEED

x

y

V=0

Red arrow constant velocity

Green inertial framesRed non-inertial frameIs the Earth an inertial frame ?

THE SPEED OF LIGHT PROBLEM

c

c

TRUTH AND CONSEQUENCES

WHAT IS SPEED ?

WHAT DISTANCE AN OBJECT WILL TRAVEL IN A GIVEN DURATION OF TIME V=DX/DT DISTANCE IS A NOTION ABOUT SPACE – HOW MUCH SPACE IS BETWEEN TWO POINTSDURATION IS A NOTION ABOUT TIME – HOW MUCH TIME ELAPSES BETWEEN EVENTS

SPEED IS A SPACE-TIME NOTION – CONSTANCY OF SPEED OF LIGHT REQUIRES THAT WE MODIFY CONVENTIONAL CONCEPTS OF SPACE AND TIME

NEWTON’S WORRIES ABOUT GRAVITY

• Newton’s had big concerns about his own gravitaional theory!

• Gravitation force somehow (mysteriously) could reach across large distances – it was an “action at a distance”. Newton’s didn’t like that.

• Newton’s static universe– Newton imagined that the Universe was infinite and full of

stationary stars, each exerting a gravitational force on the others.– Turns out that this configuration (and any other that Newton could

think of) is unstable… the smallest disturbance and it will collapse.

II:FRAMES OF REFERENCE

We have already come across idea of frames of reference that move with constant velocity. In such frames, Newton’s law’s (esp. N1) hold. These are called inertial frames of reference.

Suppose you are in an accelerating car looking at a freely moving object (i.e., one with no forces acting on it). You will see its velocity changing because you are accelerating! In accelerating frames of reference, N1 doesn’t hold – this is a non-inertial frame of reference

III:NEWTONIAN PRINCIPLE OF RELATIVITY

• THE LAWS OF MECHANICS ARE INVARIANT IN INERTIAL REFERENCE FRAMES

THE LAWS OF PHYSICS ARE COVARIANT IN INERTIAL REFERENCE FRAMES.

EG: Play ping-pong on a train moving with constant velocity same as playing on the ground.No mechanical experiment can detect motion at constant speed

LAWS THAT EXHIBIT THE SAME MATHEMATICAL FORM FOR ALL OBSERVERS ARE CALLED COVARIANT

Fig. 1-1, p. 4

Experiment at rest Experiment in moving frame

Same result. Ball rises and ends up in the thrower’s hand. Ball in the air the same length of time.Experiment looks different from ground observer (parabolic trajectory, speed as a function of time) and observer on the truck. However, they both agree on the validity of Newton’s laws.

Fig. 1-1a, p. 4

x

y x(t)=0y(t)=vyt-1/2gt2

Fig. 1-1b, p. 4

x’

y’

x’(t)=ut

2' 2/1)( gttvty y

u

Assumed t invariant

Fig. 1-2, p. 4

CLOCKS SYNCHRONIZED

AT t=0 (t=t’=0) AND ORIGINS

COINCIDE

GALILEAN TRANSFORMATIONS

From http://hyperphysics.phy-astr.gsu.edu

Photo by Andrew Davidhazy

V. LIGHTWe all have enough common sense to know that waves need a medium, such as water, in which to propagate.

That medium is a reference frame which may be:At rest with respect to an

observer Moving with respect to an observer

-or-

Assume Galilean velocity transformation holds:

isotropic

If light is wave in what medium propagates ?

From T. Ferris : “Coming of Age in the Milky Way”

ETHER

MAXWELL’S EQUATIONS

Maxwell’s highly successful equations…

Contain a constant velocity!

In the 19th century, these equations were thought to hold only in the luminiferous ether!

Gauss’ law

no magnetic monopoles

Faraday’s law

Ampere’s law

tc

t

O

O

/)/1(

//

0

2 EJBBE

EB

2222 )/1(

tc

from wikipedia

Light must travel through a medium: hypothesize that a “luminiferous ether” exists

Earth is moving with respect to the ether (or the ether is moving with respect to the earth), so there should be some directional/season dependent change in the speed of light as

observed from the reference frame of the earth.

Fig. 1-3, p. 7

Fig. 1-3a, p. 7

Fig. 1-3b, p. 7

Fig. 1-3c, p. 7

Fig. 1-4, p. 8

Wavelength Distance from one crest to the nextFrequency Number of crests per secondPeriod T=1/ v the speed of the wave

Waves interfere, reflect refract and exhbit Doppler Shift

+ =

Destructive interferenceZero amplitude and energy

+ =

Constructive interferenceDouble amplitude and4 times energy

Doppler Effect

Doppler shift z = (rec-em)/em=V/cNegative z blue shift, positive z redshift

Wavelength

Interference in two slit experiment

Fig. 1-6, p. 9

Original Apparatus

Q: Can anyone guess why the invention of lasers made the Michelson-Morley experiment more convincing?

.

Light travels very fast so you are looking at very subtle difference-use several passes to multiply the effect.Need to select light of a particular frequency to detect any shift.

The speed of light in vacuum has the same value, c=300000000 m/s, in all inertial reference frames, regardless of the velocity of the observer or the velocity of the source emitting the light.

All the laws of physics have the same form in all inertial reference frames.

The Solution???

Oh my goodness…

how can that be right???

Alright…we know that Newtonian mechanics worked in all inertial reference frames under Galilean transformations, but does the same hold true for

Maxwell’s equations of electromagnetism?

The radical consequences

Speed =

distance traveledtime elapsed

If the speed of light is a constant…then…length and time must be variables??

speed of light = 670 616 629 miles per hour

These effects are known as length contraction and time dilation. How come you never noticed this before, and how come

most of the time I can get away with Galilean transformations in your calculations?

Most of the time the speed of the object whose motion you are calculating is do slow relative to the speed of light that the

discrepancy due to relativity is negligible. (Most, but not all of the time)