Transcript Special Relativity - Polson 7-8

Special and
General Relativity
Special Relativity
and
Einstein’s
Physics
General Relativity
Objectives
• Be familiar with the Michelson-Morley
experiment.
• Understand what the results of the
experiment mean in terms of the “ether” and
the speed of light.
Michelson-Morley Experiment
• James Clerk Maxwell (1860):
light is e/m waves traveling at c.
• Waves require a medium, so
light must travel through an
“ether.”
• Michelson and Morley (1880s):
looked for the ether using an
interferometer.
Concept of the
Interferometer
• Two boats will travel 24 m
forward and back at 4 m/s.
The river current is 2 m/s
eastward.
• North-South blue route:
(24 m / 4 m/s) x 2 = 12 s.
• East-West red route: (24 m
/ 6 m/s) + (24 m / 2 m/s) =
16 s.
• Blue boat wins!
• But, if the river flows
northward, the red boat
would win.
Michelson-Morley Experiment
• As the earth moves
through the ether, the
“wind” will act like the
river current, affecting
the motion of the light
waves.
• Rotating the experiment
will cause interference
fringes to change,
proving the existence of
the ether.
Michelson-Morley Experiment
• When they conducted their
experiment, no fringes were
observed to change.
• No ether exists!
• A secondary outcome of the
experiment was that c is
always 3.00 x 108 m/s.
• Lorenz proposed that the
ether wind affected the
distance between the
2
2
1

(
v
/
c
)
mirrors by a factor of
Einstein’s Question
• Light propagates through
space by changing electric
and magnetic fields.
• As a student, Albert
Einstein wondered what
would happen if you could
travel along with a light
wave? Would the changing
fields occur? Would the
light propagate?
• Einstein devoted his life to
understanding light.
Hmm...
Objectives
• Know the two postulates of Einstein’s theory
of relativity.
• Understand how the constancy of the speed
of light affects our concept of time.
• Understand and apply the concept of spacetime.
Einstein’s Postulates of Relativity
1. All the laws of nature are the same in all uniformly
moving frames of reference. You cannot detect
absolute uniform motion (no ether for reference).
2. The speed of light equals c and is independent of
the speed of the source or the observer. C is
absolute.
The evidence for #2:
g
pion
g
detector
measures
energy
g
pion
moving
at 0.99c
g
detector
measures
SAME energy
Simultaneity
• Einstein imagined lightning hitting two poles.
• A stationary observer midway between the poles sees
the light hit the two poles simultaneously.
• A moving observer midway between the poles sees the
light hit the pole that he is moving toward first, and the
other pole afterwards.
• The two observers cannot agree on the order of events:
• Time is relative! Only the speed of light is absolute!
Space-Time
• speed = distance / time.
• Applied to light, c = d / t.
If c is absolute, and time
is relative, then distance
(space) must be relative
too.
• Einstein reasoned that
the concepts of space
and time are woven
together into what he
Think about it: any event
called space-time.
takes place at a specific
time and a specific place
(in 4 dimensions)
Traveling in Space-Time
time
A fast-moving spacecraft
travels through more space
and thus through less time.
We travel mostly through
time, but not through much
space.
As an object approaches c, it
travels mostly through space,
but through little time.
space (distance)
slope = t/d, and 1/v = t/d. As velocity goes up, slope goes down
Objectives
•
•
•
•
Understand the concept of time dilation.
Be able to calculate time dilation.
Be familiar with evidence for time dilation.
Understand the implications of time dilation.
Time Dilation
Imagine two scientists measuring a light-pulse inside a
moving spaceship. One is inside the spaceship, the other is
outside the spaceship…
to = proper time Time and distance
measured by observer
inside the spaceship.
Time and distance
measured by observer
outside the spaceship.
t = dilated time (or td)
Time Dilation
(ct) 2  (cto ) 2  (vt) 2
(ct) 2  (vt) 2  (cto ) 2
c 2t 2  v 2t 2  c 2t o
2
t 2 (c 2  v 2 )  c 2 t o
c·t
c · to
v·t
t is dilated time, clock in motion with
respect to events to is “proper time”,
clock at rest with respect to events
t 2 (c 2  v 2 )
2
 to
2
c
v2
2
2
t (1  2 )  to
c
2
to
2
t 
v2
(1  2 )
c
to
t
v2
1 2
c
2
Calculating Time Dilation
Proxima Centauri is the closest
star to our solar system. If a
spacecraft were sent to Proxima
Centauri traveling at 75% of the
speed of light (0.75 c), the trip
would take 3.72 years according
to the clocks onboard the ship.
How long would the trip take
according to people on Earth?
Time Dilation: The Evidence
• In 1971, two atomic clocks were
placed on commercial jets and
two “reference” atomic clocks
were placed in a building. The
clocks were synchronized.
• The jets traveled around the world
twice (once east, once west)
• The clocks that traveled through
more space (in jets) recorded less
time than the stationary clocks, as
predicted by Einstein.
The Twin “Paradox”
• One twin travels at relativistic
speeds away from the earth,
turns around, and returns at
relativistic speeds.
• She will be younger than her
twin brother!
• The twin brother experiences
the dilated time.
Twin Paradox: The Evidence
• 1976 at CERN
• Muons normally decay in 2.2 ms
(to) A muon should only be able
to make 15 revolutions around
the accelerator in this time.
• When traveling at 0.9994 c, a
muon will make 432 revolutions
and decay in 63.5 ms (td),
outlasting a twin stationary
muon by a factor of 29.
Length Contraction
• length contraction:
moving objects appear
to contract along the
direction of motion.
• Looking at a clock and
meter-stick inside the
spaceship, you would
see less time pass for a
beam of light to travel
one meter; since c = d/t,
distance must be less.
Lo = proper length
LC = contracted length
Lc  Lo
v2
1 2
c
Length Contraction Calculation
All distances are contracted when
you travel at relativistic speeds.
Thus, Pluto, which is 39 AU away,
would be “closer” if you traveled
at 0.95 c. What is the contracted
distance?
Relativistic Momentum
Newton
p = mv
true only at nonrelativistic speeds
Einstein
p = gmv
particle accelerator
data supports Einstein
What is the momentum of a proton (1.67 x 10-27 kg)
traveling at 0.999c (2.997x108 m/s) according to
Newton? What about to Einstein?
measured value =
Einstein’s value
• Why can’t v > c?
• As v → c, Dp → ∞
•
•
•
•
Momentum (p)
Relativistic Dynamics
Einstein
p = gmv
Impulse-momentum theorem Speed (v)
SF·Dt = m·Dv = Dp
If Dp → ∞, either SF → ∞ or Dt → ∞
It either takes an infinite force or a
finite force applied for an infinite
period of time to reach the speed of
light!
Newton
p = mv
c
The answer to
Einstein’s
question: it is
not possible
to ride a light
beam, so there
is no paradox.
Eo = mc2
• rest energy: the energy an object
possesses due to its mass
• mass ≈ “frozen energy”
• objects gain/lose mass when they
absorb/emit energy
• The sun converts 4 billion kg/s into
energy through the process of
nuclear fusion (4 H → He + energy)
• E = mc2 = (4 x 109 kg)(3 x 108 m/s)2
= 3.6 x 1026 J each second!
= 360 heptillion W light bulb
General Relativity
Equivalence Principle
• Einstein’s “happiest
thought” was that you
don’t feel the force of
gravity when you fall.
• But artificial gravity
exists in an accelerating
spacecraft.
• Gravity and acceleration
are “equivalent.”
• An experiment done on
earth or done when
accelerating at g in a
spacecraft will yield the
same results! (general
relativity).
Light and the Equivalence Principle
• A scientist in an accelerating
spacecraft observes a horizontal
beam of light to curve downward.
• According to the equivalence
principle, gravity should curve
light in a similar manner.
Sun
gravity
acceleration
Astronomical observations after WWI
showed that the sun did indeed bend
starlight, supporting Einstein.
Curved Space
If mass bends light, and
light moves in a straight
line, then mass must warp
or curve space.
Newton’s laws could not
fully explain the orbital motion
of Mercury; however, Einstein
used his general theory to
properly calculate the orbit.
Warped Space and Orbital Motion
Newton (Law #1) said that an object will move in
straight line unless acted on by unbalanced force.
Einstein suggested that the object moves in a
“straight line” through curved space!