Phys 320 - Baski Relativity I Page 1 Topic #9: Special Relativity I • Transformation of Variables between Reference Frames – Non-relativistic Galilean Transformation. – Relativistic Lorentz Transformations. • Time Dilation and Length Contraction – Time interval measured by observer in moving reference frame (w/respect to event) is longer than “proper” time. – Length of object measured in moving reference frame (w/respect to object) is shorter than “proper” length. • Simultaneous Events – Two events are only simultaneous in one reference frame. – All other frames will see events at different times. • Doppler Shift – Red shift for receding object; blue shift for approaching.
Transcript
Phys 320 - Baski Relativity I Page 1
Topic #9: Special Relativity I
• Transformation of Variables between Reference Frames– Non-relativistic Galilean Transformation.
– Relativistic Lorentz Transformations.
• Time Dilation and Length Contraction– Time interval measured by observer in moving reference frame
(w/respect to event) is longer than “proper” time.
– Length of object measured in moving reference frame (w/respect to object) is shorter than “proper” length.
• Simultaneous Events– Two events are only simultaneous in one reference frame.
– All other frames will see events at different times.
• Doppler Shift– Red shift for receding object; blue shift for approaching.
Phys 320 - Baski Relativity I Page 2
Special Relativity: Postulates
• Principle of Relativity
– All the laws of physics are the same in all inertial reference frames, i.e. frames moving at constant velocities w/respect to each other.
• Constant Speed of Light
– The speed of light, c = 3×108 m/s, is equal in all inertial frames, regardless of the velocity of the observer or the light source.
• Consequences of Special relativity
– “Slowing” down of clocks (time dilation) and length contraction in moving reference frames as measured by an observer in another reference frame.
Phys 320 - Baski Relativity I Page 3
Special Relativity: History• 1879: Albert Einstein born in Ulm, Germany.
• 1901: Worked at Swiss patent office.– Unable to obtain an academic position.
• 1905: Published 4 famous papers.– Paper on photoelectric effect (Nobel prize).– Paper on Brownian motion.– 2 papers on Special Relativity. – Only 26 years old at the time!!
• 1915: General Theory of Relativity published.
• 1933: Left Nazi-occupied Germany.– Spent remainder of time at Institute of Advanced
Study in Princeton, NJ.– Attempted to develop unified theory of gravity
and electromagnetism (unsuccessful).
Phys 320 - Baski Relativity I Page 4
Galilean Transformations
• Newton’s laws of mechanics are invariant (i.e., the same) in inertial reference frames connected by a Galilean transformation (v << c).
• Inertial reference frame S’ moves at constant velocity v with respect to stationary frame S.
• Maxwell’s equations are not invariant under Galilean transformations.
• Example:– Infinite line charge and point charge q located above it.
– Observer S sees static charge and observer S’ see moving charge!
• Electric Force: FE = 2kq/y1 for observers in both S and S’.
• Magnetic Force: FB = -oqv2/2y1 only according to observer S’.
– Observer S’ sees a moving line charge and moving charge leads to magnetic fields!
S’S v
Phys 320 - Baski Relativity I Page 7
Galilean Transform: Predicts Preferred Ref. Frame
• According to the Galilean transform,the travel time t1 across & back a river is shorter thanthe travel time t2 up & down a river.
• Light moving in an “ether” is an analogous problem.– Can light have different travel times?
t1
t2
2
22 22
2
2
12 2
12
1L L Lc L v
c v c v cc v c
L vt
c c
2
2
2
21 22
12 2 1
12
2 21
L L v
c cc v
L vt
c c
Phys 320 - Baski Relativity I Page 8
Galilean Transform: NO Preferred Ref. Frame for Light!
• In 1800’s, scientists thought that light propagated through some type of “ether.”
• Michelson-Morley Experiment (1887)
– Test if ether exists and sets “preferred” reference frame.
– Analogous to rowboat in river.
– Measure light speed relative to earth’s motion (// and ) using an interferometer (fringes).
• Result: No detection of “ether”
– No detectable shift in interference fringes occurred, indicating that light speed DID NOT depend on direction.
Phys 320 - Baski Relativity I Page 9
Steep rise for v c
/ 0.85v c
/ 0.95v c
/ 0.99v c
2
2 2
,
2
2
,
,
1
1 /
1 /
1 /
x
x
y z
x
x
y z
vt
vxt t
c
v c
v
v c
x
y z
u
u
x
y z
u
c
u
u
u v
NEW Lorentz Transformations
• RELATIVISTIC transformations.– Reduce to Galilean transforms
for v << c (i.e., = 1).– Consistent with Maxwell’s equations.
• Reference frame S’ moves at velocity v to right w/respect to stationary frame S.
v/c Note: Reverse signs for inverse transform.
Phys 320 - Baski Relativity I Page 10
2 2 2
2 2 2
2 2
2
2
2
2 2
2
2
1
1 1
11
x t
x
xt
x y
x
x t
t
x y z c
v
v
v
v
c
t
z c t
Lorentz Transforms: Short Derivation
• Begin with Galilean transforms having a velocity-dependent
• Write down two “invariant” equations in both frames that describe the arrival of a spherical light wave coming from the origin at t = t’ = 0.
• Solve for t’ by substituting x’ into x
• Solve for by simultaneously solving equations for x’, t’ and the invariants.
Phys 320 - Baski Relativity I Page 11
Lorentz Transform: Relative Velocities Problem
x
y
S = earth
v = 0.99cS’
ux = – 0.99c
2 2
0.99 1.980.99995
1.98011 1 0.99
0.99
0.99
x
x
u c
u c
v c cc
v c c c
x'u
Two spaceships are approaching each other at the same speeds (0.99c) relative to the Earth. Find the speed of one spaceship relative to the other.
Let frame S be the rest frame of the Earth and frame S’ be the spaceship moving at speed v to the right relative to the Earth. The 2nd spaceship moving to the left is then a “particle” moving at speed u relative to the earth. Now, find the speed u’ of this 2nd spaceship in the S’ reference frame.
Note that the oncoming spaceship approaches at less than the speed of light, as must be true.
Phys 320 - Baski Relativity I Page 12
Time Dilation (or Time Stretching)
• Proper time t’ = is the time interval between two events as measured by an observer who sees the events occur at the same point in space.
• Time Dilation causes time intervals t measured in other reference frames to be longer than the “proper” time interval t’.
• “A moving clock runs slower than a clock at rest.”– All physical processes, including chemical reactions and biological
processes, slow down relative to a stationary clock when they occur in a moving frame.
2 21 /
tt t
v c
Phys 320 - Baski Relativity I Page 13
Time Dilation: Short Derivation
• In S’ frame, light travels up/down.In S frame, light travels a longer path along hypotenuse.
• Solve for t, where t’ = 2D/c (proper time).
• Analyze laser “beam-bounce” in two reference frames.2 2
2
2 2 2 2
2 2
2
2 1
1
2
/
/
2
1
v t
Dt
D
c
Dc t
ct
v
tt
v c
v c
t in S framet’ in S’ frame
Pythagorean Theorem
DD 2
c t
2
v t
Phys 320 - Baski Relativity I Page 14
Time Intervals: Simultaneous Events• Two events simultaneous in one reference frame are not
simultaneous in any other inertial frame moving relative to the first.
Two bolts seen simultaneously at C
Right bolt seen first at C’
Left bolt seen second at C’
Two lightning bolts strike A,B
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Length Contraction
• Length contraction causes length L measured in other reference frames to be shorter than the “proper” length L’.
S Frame
S’ Frame
Length contraction distorts 3D shapes
2
2
11' '
vL L L
c
Phys 320 - Baski Relativity I Page 16
1
2
2 21
11
22
1
1 or ' '
x
x
x x
L L
vt
v
x
x
x x
L L
t
Length Contraction: Short Derivation
• Write Lorentz transformsfor endpoints of objectin “proper” S’ frame.
• Solve for proper length x’of object in S’ frame.(measured at same time, i.e. t1 = t2).
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Time Dilation/Length Contraction: Muon Decay
• Why do we observe muons created in the upper atmosphere on earth? –Given its ~2 s lifetime, it should travel only ~ 600 m at 0.998 c.
• Need relativity to explain!–In muon’s S’ frame, it sees a shorter length. (Length contraction)–In our S frame, we see a longer lifetime of ~ 30 s. (Time dilation)
Proper LifetimeContracted Length
Muon’s S’ frame Earth’s S frame
Longer LifetimeProper Length
~30 s~2 s
Phys 320 - Baski Relativity I Page 18
Time Dilation/Length Contraction: Homework ProblemA spaceship departs from earth (v = 0.995c) for a star which is 100 light-years away. Find how long it takes to arrive there according to someone on earth (t1) and
to someone on the spaceship (t2).
1100 yr
0.995
where x is measured from the earth's ref
e
erence frame
arth 10
(S frame
.5 yr
)
0tcx
v c
For t2, remember that the spaceship sees a “contracted” distance x’.
2 2
2
2
1001 1
10.011' 9.99 yr
ship 10
10.1 0.995
9.99'
0.995.04 yr
01c yrx
v c
c yrx
v c
x c
t
Note that someone on the ship thinks it takes only 10% of the time to reach the star as someone from the earth believes it takes. This is why we say the clock on the ship “runs slow” compared to the clock on the earth.
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Doppler Shift• Doppler shift causes change in measured frequency.
– When a light source moves towards an observer, the light frequency is shifted higher (i.e. blue shift).
– When a light source moves away from an observer, the light frequency is shifted lower (i.e. red shift).
– Only difference with “classical” Doppler shift for sound is the incorporation of time dilation (causes square root factor).
Approaching - blue shift
1 /
1 /obs source
v cf f
v c
Note: For a receding source, switch signs.
Phys 320 - Baski Relativity I Page 20
Doppler Shift: Homework Problem
The light from a nearby star is observed to be shifted toward the red by 5% (f = 0.95 f o). Is the star approaching or receding from the earth? How fast is it
moving?
The star is receding the earth because the frequency is shifted to a lower value.
