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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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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)
September 24, 2003NJIT Center for Solar-Terrestrial Research
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!
September 24, 2003NJIT Center for Solar-Terrestrial Research
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.
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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.
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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!
September 24, 2003NJIT Center for Solar-Terrestrial Research
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.
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
Group AssignmentProblem 4.16
Find the value of v/c when a particle’s kinetic energy equals its rest energy.
September 24, 2003NJIT Center for Solar-Terrestrial Research
Class Project
Exhibition
Science
Audience
September 24, 2003NJIT Center for Solar-Terrestrial Research
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!
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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).
September 24, 2003NJIT Center for Solar-Terrestrial Research
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
September 24, 2003NJIT Center for Solar-Terrestrial Research
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!