njit physics 320: astronomy and astrophysics – lecture iv carsten denker physics department center...

NJIT

Physics 320: Astronomy and Astrophysics – Lecture IV

Carsten Denker

Physics DepartmentCenter for Solar–Terrestrial Research

September 24, 2003NJIT Center for Solar-Terrestrial Research

The Theory of Special Relativity

The Failure of the Galilean Transformations

The Lorentz TransformationTime and Space in Special RelativityRelativistic Momentum and Energy


Wave Theory and Ether

Luminiferous Ether transport light waves, no mechanical resistance

Science of early Greek: earth, air, water, and fire heavens composed of fifth element = ether

Maxwell: There can be no doubt that the interplanetary and interstellar spaces are not empty, but are occupied by a material substance or body, which is certainly the largest, and probably the most uniform body of which we have any knowledge.

Measuring absolute velocity? Inertial reference systems (Newton’s 1st law)


Galilean Transformation Equations

Michelson–Morley experiment: c = 3 108 m/s = const. velocity of Earth through ether is zero

Crisis of Newtonian paradigm for v/c << 1

x x ut

y y

z z

t t

and const.x x

y y

z z

v v u

v v v v u u

v v

a a F ma ma Newton’s laws are obeyed

in both inertial reference frames!


The Lorentz Transformations

Einstein 1905 (Special Relativity): On the Electrodynamics of Moving Bodies

Einstein’s postulates: The Principle of Relativity: The laws of physics

are the same in all inertial reference frames The Constancy of the Speed of Light: Light

travels through a vacuum at a constant speed of c that is independent of the motion of the light source.

Linear transformation equations between space and time coordinates (x, y, z, t) and (x, y, z, t ) of an event measured in two inertial reference frames S and S.


Linear Transformation Equations

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

x a x a y a z a t

y a x a y a z a t

z a x a y a z a t

t a x a y a z a t

11 12 13 14

22 33

21 23 24 31 32 34

41 42 43 44

1

0

x a x a y a z a t

y y a a

z z a a a a a a

t a x a y a z a t

ˆ( )u u x i

Principle of Relativity


Linear Transformation Equations (cont.)

11 12 13 14

42 43

41 44

0 ( and )

x a x a y a z a t

y ya a y y z z

z z

t a x a t

Rotational symmetry

11

12 13

11 14

41 44

( )0

0

0

x a x utt t

y y a ax ut

z z a u ax

t a x a t

Boundary conditions at origin

Galilean Transformations

11 44

41

1

0

a a

a


Linear Transformation Equations (cont.)

Spherically symmetric wave front in S and S

2 2 2 2 2 211 44

2 2 2 2 241 11

( ) 1/ 1 /

( ) /

x y z ct a a u c

x y z ct a ua c

2 2

22

2 2

1 /

//

1 /

x utx x ut

u cy y

z z

t ux ct t ux c

u c

Lorentz Transform

2 2

22

2 2

1 /

//

1 /

x utx x ut

u cy y

z z

t ux ct t ux c

u c

Inverse Lorentz Transform

2 2

1

1 /u c


Time and Space in Special Relativity

Intertwining roles of temporal and spatial coordinates in Lorentz transformations

Hermann Minkowski: Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union between the two will preserve an independent reality.

Clocks in relative motion will not stay synchronized

Different observers in relative motion will measure different time intervals between the same two events


Time Dilation

The shortest time interval is measured by a clock at rest relative to the two events. This clock measures the proper time between the two events.

Any other clock moving relative to the two events will measure a longer time interval between them.

22 1

1 2 2 2

( ) /

1 /

x x u ct t

u c

Flashbulbs at x1 and x2 at same time t

rest2 1 moving2 2 2 21 / 1 /

ttt t t t

u c u c

22 1 2 1

2 1 2 2

( ) ( ) /

1 /

t t x x u ct t

u c

Strobe light every t at x1 = x2


Length Contraction

The longest length, called the rod’s proper length, is measured in the rod’s rest frame.

Only lengths or distances parallel to the direction of the relative motion are affected by length contraction.

Distance perpendicular to the direction of the relative motion are unchanged.

2 1 2 1 1 22 2, and

1 /

LL L x x L x x t t

u c

2 2moving rest 1 /L L u c

2 1 2 12 1 2 2

( ) ( )

1 /

x x u t tx x

u c

Rod along x–axis at rest in S


Group Assignment Problem 4.4

A rod moving relative to an observer is measured to have its length Lmoving contracted to one–half of its original length when measured at rest. Find the value of u/c for the rod’s rest frame relative to the observer’s frame of reference.


Doppler Shiftobs rest r

rest rest s

v

v

Sound speed vs and radial velocity vr

rest restobs moving light 2 2 2 2

cos1

1 / 1 /

t u tt t t

cu c u c

restobs 2 2

1 ( / ) cos1 /

tt u c

u c

2 2 2 2rest rest

obs

1 / 1 /cos

1 ( / ) cos 1 ( / ) rr

u c u cv u

u c v c

Relativistic Doppler shift

obs rest

0 and 1 /

1 / 180 and rr

r r

v uv c

v c v u


RedshiftSource of light is moving away from the observer:

Source of light is moving toward the observer:

Redshift parameter:

obs rest

rest rest

z

1 / 1 / and 1

1 / 1 /r r

obs restr r

v c v cc z

v c v c

obs

rest

1t

zt

obs rest0rv Redshift

obs rest0rv Blueshift

Radial motion!


Group AssignmentProblem 4.9

Quasar 3C 446 is violently variable. Its luminosity at optical wavelength has been observed to change by a factor of 40 in as little as 10 days. Using the redshift parameter z = 1.404 measured for 3C 446 determine the time for the luminosity variation as measured in the quasar’s rest frame.


Relativistic Velocity Transformations

2

2 2

2

2 2

2

1 /

1 /

1 /

1 /

1 /

xx

x

yy

x

zz

x

v uv

uv c

v u cv

uv c

v u cv

uv c

2

2 2

2

2 2

2

1 /

1 /

1 /

1 /

1 /

xx

x

yy

x

zz

x

v uv

uv c

v u cv

uv c

v u cv

uv c

2

2 2 2 2

( ) (1 / ), , , and

1 / 1 /x x

y z

v u dt uv c dtdx dy v dt dz v dt dt

u c u c

v c v c


Relativistic Momentum and Energy

The mass m of a particle has the same value in all reference frames. It is invariant under a Lorentz tranformation.

f f f f

i i i i

x x p p

x x p p

dp dx dpK Fdx dx dp vdp F

dt dt dt

2 21 /

mvp mv

v c

Relativistic momentum

vector

2

2 2 2 20 0

22 2 2 22

2 2 2 2

1 / 1 /

11 / 1 1

1 / 1( 1)

/

f fv vff f

f

ff

f f

mv mvK p v pdv dv

v c v c

mvmc v c mc

v cmc

v c

Relativistic kinetic energy


Relativistic Energy

2 2 2 2 2E p c m c

22

2 21 /

mcE mc

v c

Total relativistic energy

2restE mc Rest energy

sys1

n

ii

E E

Total energy of a system of n particles

1

n

sys ii

p p

Total momentum of a system of n particles


Group AssignmentProblem 4.16

Find the value of v/c when a particle’s kinetic energy equals its rest energy.


Class Project

Exhibition

Science

Audience


Homework Class Project

Read the Storyline hand–outPrepare a one–page document with

suggestions on how to improve the storyline

Choose one of the five topics that you would like to prepare in more detail during the course of the class

Homework is due Wednesday October 1st, 2003 at the beginning of the lecture!


Homework Solutions Problem 2.3

2

2 2 2

2 2 2

2

2 3/ 2

2 2

(1 ) 2sin and

(1 cos )

2 1 / and 1

2 (1 cos )

(1 )

2 sin 2 (1 cos ) and

1 1

r

r

dr a e d d dA Lv e

dt e dt dt r dt r

L a e P A ab b a e

d e

dt P e

ae d a ev v r

dtP e P e

22 2 2 2

1 2

(1 ) 2 1 and ( )

1 cos r

a er v v v v G m m

e r a


Homework SolutionsProblem 2.9

22 3 6

1 2

4 and 6.99 10 m 96.6min

( )P a a R h P

G m m

73.58 10 m 5.6R R

A geosynchronous satellite must be parked over the equator and orbiting in the direction of Earth’s rotation. This is because the center of the satellite’s orbit is the center of mass of the Earth–satellite system (essentially Earth’s center).


Homework SolutionsProblem 2.11

2 3 17.9 AUP a a

2 330

comet 2

41.98 10 kg

am M M

GP

(1 ) 0.585 AU and (1 ) 35.2 AUp ar a e r a e

0.91 km/s

55 km/s

7.0 km/s

a

p

v

v

GMr a v

a

2

23650p p

a a

K v

K v


Homework

Homework is due Wednesday October 1st, 2003 at the beginning of the lecture!

Homework assignment: Problems 4.5, 4.13, and 4.18

Late homework receives only half the credit!

The homework is group homework!Homework should be handed in as a

text document!

njit physics 320: astronomy and astrophysics – lecture iv carsten denker physics department center...

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