2.1 The Apparent Need for Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein’s Postulates 2.4 The Lorentz Transformation 2.5 Time Dilation and Length Contraction 2.6 Addition of Velocities 2.7 Experimental Verification 2.8 Twin Paradox 2.9 Space-time

2.10 Doppler Effect 2.11 Relativistic Momentum 2.12 Relativistic Energy 2.13 Computations in Modern Physics 2.14 Electromagnetism and Relativity

CHAPTER 2Special Theory of Relativity-part 1

It was found that there was no displacement of the interference fringes, so that the result of the experiment was negative and would, therefore, show that there is still a difficulty in the theory itself…

- Albert Michelson, 1907

Newtonian (Classical) Relativity

A reference frame is called an inertial frameif Newton laws are valid in that frame.

Inertial Frames K and K’

The Galilean Transformations

For a point P In system K: P = (x, y, z, t) In system K’: P = (x’, y’, z’, t’)

x

K

P

K’ x’-axis

x-axis

v t

The Galilean Transformations

The Inverse Relations

Step 1. Replace withStep 2. Replace “primed” quantities with “unprimed”

and “unprimed” with “primed”

The Transition to Modern Relativity Although Newton’s laws of motion had the same form under the

Galilean transformation, Maxwell’s equations did not keep the same form.

(ex) the speed of light always

In 1892, Hendrik Lorentz and George FitzGerald proposed a radical idea that that solved the electromagnetic problem: Space was contracted along the direction of motion of the body. A precursor to Einstein’s theory.

In 1905, Albert Einstein proposed a fundamental connection between space and time by introducing two postulates to reconcile the problem and define “Special Relativity”.-> He proposed that space and time are not separated

and that Newton’s laws are only an approximation.

0 01/c

2.1: The Apparent Need for Ether

The wave nature of light suggested that there existed a propagation medium called the luminiferous ether or just ether. Ether had to have such a low density that the planets

could move through it without loss of energy

It also had to have an elasticity to support the high velocity of light waves

An Absolute Reference System

The Ether frame was proposed as an absolute reference system in which the ether was stationary and the speed of light was constant c.

In the other systems, the speed of light would indeed affected by the relative speed v of the reference ether frame, such as c + v.

However, Maxwell’s equations don’t differentiate between these two systems, since the speed of light is given by

The Michelson-Morley experiment was an attempt to show the existence of ether.

0 01/c

2.2: The Michelson-Morley Experiment

Albert Michelson (1852–1931) was the first U.S. citizen to receive the Nobel Prize for Physics (1907).

He built an extremely precise device called an interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions.

Differences in speed of light moving in different directions through the ether would be detected as a phase difference at the detector.

The Michelson Interferometer

Accuracy

the time difference can be shown to be (page 23):

15 18 3exp( ) ~ 10 10 10 radiani t

Michelson’s Conclusion

1838-1923

Michelson-Morley experiment

The Lorentz-FitzGerald Contraction

2.3: Einstein’s Postulates Albert Einstein (1879–1955) was only two years old

when Michelson reported his first null measurement for the existence of the ether.

At the age of 16 (1895) Einstein began thinking about the form of Maxwell’s equations in moving inertial systems.

In 1905, at the age of 26, he published his startling proposal about the principle of relativity, which he believed to be fundamental.

With the belief that Maxwell’s equations must bevalid in all inertial frames, Einstein proposes thefollowing postulates:

1) The principle of relativity: The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists.

2) The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

2.3: Einstein’s Postulates

Revisiting Inertial Frames and the Re-evaluation of Time

In Newtonian physics we previously assumed that t = t’ Thus with “synchronized” clocks, events in K and K’ can be

considered simultaneous

Einstein realized that each system must have its own observers with their own clocks and meter sticks Thus, events considered simultaneous in K may not be in K’.

The Problem of Simultaneity

Mary, moving to the right with speed observes events A and B in different order: Mary “sees” event B, then A.

Frank at rest is equidistant from events A and B: Frank “sees” both flashbulbs go off simultaneously.

Mary is moving (K’)

Frank is at rest (K)

The Lorentz TransformationsThe special set of linear transformations that:

1) preserve the constancy of the speed of light (c) between inertial observers;

2) account for the problem of simultaneity between these observers

Lorentz transformation equations:

MaryFrank

Lorentz Transformation Equations

Relativistic factor

11 when 0v

ct21

Use the fixed system K and the moving system K’ At t = 0 the origins and axes of both systems are coincident with

system K’ moving to the right along the x axis. A flashbulb goes off at the origins when t = 0. According to postulate 2, the speed of light will be c in both

systems and the wavefronts observed in both systems must be spherical.

K K’

2.4 Derivation of the Lorentz Transformations

Derivation (con’t)Spherical wavefronts in K:

Spherical wavefronts in K’:

Note: these are not preserved in the classical transformations with

ct

Let ' ( ) so that ( ' '), where the parameter must be close to 1 for .By Einstein's first postulate that the laws of physics by the same in both systems, '= .By Einstein's second p

x x vt x x vt v c

ostulate that the speed of light is c in both systems, , ' '.x ct x ct

2

2 2

' ' ( ) and ( ' ')' (1 / ) and '(1 / ) ' '(1 / )(1 / )

11 /

ct x ct vt ct x ct vtt t v c t t v c t t v c v c

v c

Finding a Transformation for t’

Recalling x’ = (x – vt) substitute into x = (x’ + vt) and solving for t ’ we obtain:

which may be written in terms of β (= v/c):

Complete Lorentz Transformations Including the inverse (i.e v replaced with –v; and primes interchanged)

1) If v << c, i.e., β ≈ 0 and ≈ 1, Galilean transformation.

2) Space and time are now not separated.

3) For non-imaginary transformations, the frame velocity cannot exceed c.

2

11

vc

2 2 2

2 2

22 2 2

'1 / 1

''

/ /' ( )1 / 1

x vt x vtx x vtv c

y yz z

t vx c t vx c vxt tcv c

2

' '''

'( ' )

x x vty yz z

vxt tc

( is real)

2.5: Time Dilation and Length Contraction

Time Dilation:Clocks (T0) in K’ run slow with respect to stationary clocks (T) in K.

Length Contraction:Lengths (L0) in K’ are contracted with respect to the same lengths (L) stationary in K.

Consequences of the Lorentz Transformation:

Moving clocks appear to run slow

Moving objects appear contracted in the direction of the motion

00 2 21 /

TT Tv c

2 200 1 /LL L v c

Time Dilation

To understand time dilation the idea of proper time must be understood:

The term proper time,T0, is the time difference between two events occurring at the same position in a system as measured by a clock at that position.

Same location (spark “on” then off”)

Not Proper Time

spark “on” then spark “off”

Beginning and ending of the event occur at different positions

Time Dilation

➡ He also measures the round trip time in his frame (K) T0 = t2 – t1 = 2L/c

➡ Note, T’0 = T0

➡ Now, he observes the round trip of the light in moving frame (K’)

➡ Looking at one of the triangles:

002 21 /

TT Tv c

0Note T T moving clocks (K’) run slow

as measured by stationary observers (K).

Length ContractionTo understand length contraction the idea of proper length must be understood: Frank (K) and Merry (K’) have a meter stick at rest in their own system

such that each measures the same length at rest. The length as measured at rest is called the proper length.

Mary

Frank

Frank (K) measures his proper length:

0 r lL x x

Merry (K’) measures her proper length:

0' ' 'r lL x x

What is the moving stick length (L) measured by Frank at rest?

0 0'L L Note that

Length Contraction

Mary

Frank

0'L L

Using the Lorentz transformations of

' 'r l r lx x x x

2 2'

1 /xx xv c

0 0SInce 'L L

2 200 1 /LL L v c

0Therefore L L

Moving objects appear contracted in the direction of the motion to stationary observers.

2

'

' ( )

x x vtvxt tc

34

v = 0.8c

v = 0.9c

v = 0.99c

v = 0.9999c

Moving objects appear contracted

in the direction of the motion to stationary observers.

A “Gedanken Experiment” to Clarify Length Contraction

At t = t1, the light reaches mirror after travelling the distance of: At t = 0, the flashbulb goes off.

1 1ct L vt At t = t2, the light returns back to the left end of the stick

after travelling the distance of: 2 1 2 1( ) ( )c t t L v t t

2 1( )v t t

L

1 2 1 After rearranging, , L Lt t tc v c v

Total time measured by Frank for light to round trip the stick:

21 2 1

2 T= ( ) L L Lt t tc v c v c

From time dilation of 0T T , and 0 02 /T L c

00 LL L

Length Contraction

Result Moving objects appear shorter to stationary observers.

Equivalently Distances appear shorter to moving objects.

Length Contraction

Result Moving objects appear shorter to stationary observers.

Equivalently Distances appear shorter to moving objects.

2.6: Addition of Velocities

Note: although relative motion of the frames is along x, the y and z components of velocity are also effected.

The Lorentz Velocity TransformationsIn addition to the previous relations, the Lorentz velocity transformationsfor u’x, u’y , and u’z can be obtained by switching primed and unprimed and changing v to –v:

speed does not exceed c. Apparent from Lorentz transforms that c is speed limit.

2.8: Twin Paradox

The Situation

i. One of twins decides to travel out to a distant planet at high speed and return to Earth.

ii. The second remains on the Earth.iii. Which twin is younger when they meet up?

The Paradox

i. When traveling twin returns, stationary twin reasons that the travelers clock has run slow and as such must return younger.

ii. However, traveling twin claims that it is the earthbound twin that is in motion and consequently his clock must run slow i.e. earthbound younger.

iii. Who’s observations are correct?

(P. 48)

2.9: Spacetime Spacetime diagrams were first used by H. Minkowski in 1908. Often called Minkowski diagrams Represent events in terms of the usual spatial coordinates x, y, and z

and a fourth coordinate ict.

To represent events in spacetime use Minkowski diagrams.

Paths in Minkowski spacetime are called worldlines.

1i

Particular Worldlines

(Light line)

(Light line)

Worldlines and Time

World line of Mary (K’)in Frank frame (K)

➡ events which happen simultaneously for one observer (Mary), happen at different times for the other (Frank).

x’

Spacetime diagramsBasics

( )y ct x

' 1 when

cx Vt y ct xV

y V c

45o

V c

V c

' 1

cx Vt y ct xV

cyV

When ;V c

When ;V c

Spacetime diagrams

v c

In Galilean

ct’

A

x’'x x vt

't tt’t

tan vc

B

x

The two events A and B occur at the same time in both frames.

Frank

Mary

In special relativity

ct’

x’

ct

x

In special relativity 00 LL L

0 0T T T

Length contraction:

Time dilation:

Spacetime diagram for Twin Paradox

Distance to star system: 8 light year (ly)Speed of spaceship: 0.8 c

Flank thinks the round trip time = 2 x (8 ly/0.8 c) = 20 years

But, Frank measures Mary’s clock ticking more slowlyHe think Mary’s travel time is only

22 10 1 0.8 12 years

Invariant Quantities: Spacetime Interval

➡ Or, since all observers “see” the same speed of light, then all observers, regardless of their velocities, must see spherical wave fronts.

s2 = x2 – c2t2 = (x’)2 – c2 (t’)2 = (s’)2

Frank

Mary

s

Spacetime Invariants If we consider two events, we can determine the quantity ∆s2 between the two events, and we find that it is invariant in any inertial frame.

The quantity ∆s is known as the spacetime interval between two events.

2s

Spacetime InvariantsThere are three possibilities for the invariant quantity ∆s2:

1) ∆s2 = 0: ∆x2 = c2 ∆t2, and the two events can be connected only by a light signal. The events are said to have a lightlike separation.

2) ∆s2 > 0: ∆x2 > c2 ∆t2, and no signal can travel fast enough to connect the two events. The events are not causally connected and are said to have a spacelike separation.

3) ∆s2 < 0: ∆x2 < c2 ∆t2, and the two events can be causally connected. The interval is said to be timelike.