Transcript Relativity - University of Houston

Relativity: History
• 1879: 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: Einstein 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).
Relativity
Thought Experiment - Gedanken
The Special Theory of Relativity
Einstein asked the question
“What would happed if I rode a light beam?”
• Would see static electric and magnetic fields
with no understandable source.
• Electromagnetic radiation requires changing
E and B fields.
• Einstein concluded that:
• No one could travel at speed of light.
• No one could be in frame where speed
of light was anything other than c.
• No absolute reference frame
Einstein’s Postulates
• All inertial frames of reference are equivalent with respect
to the laws of physics
or
• No experiment one can perform in a uniformly moving system
in order to tell whether one is at rest or in a state of uniform
motion. (No dependence on absolute velocity.)
• The speed of light in a vacuum always has the same value c,
independent of the motion of the source or observer.
or
• Nothing can move faster than the speed of light in a vacuum,
which is the same with respect to all inertial frames
Space-Time Diagram
Requirement for 4 Dimensions
Definitions:
• Event: characterized by location
(e.g., x,y,z) and time (t) at that location
• Space-time diagram:
a coordinate system in which
every point represents an event.
4 dimensions required.
• World line: trajectory of an event in
the space-time diagram
ct
World line
A (ct,x)
O
x
Description of Motion
In a spacetime diagram, the motion of an object traces out a
world line.
For an object that moves at a constant velocity, a simple way
of measuring the velocity is to measure the positions of the
object at two different times. Assume that the object moves
from r1 at t1 to r2 at t2, the velocity of the object is then
 
r2  r1

v 

t2 (r2 )  t1 (r1 )
We need a way of synchronizing the clocks
at different locations!
Synchronization of Clocks
According the Einstein’s second postulate, no information can
be transmitted at a rate greater than the speed of light in vacuum.
Since the speed of light is independent of inertial frames,
it provides a natural (and ideal) way of sychronizing clocks.
The procedure can be described as follows:
• Choose a reference clock and reset it to zero
• Generate a light pulse from the location of the
reference clock
• Set a local clock to the time that it takes for the
light pulse to propagate from the location of the
reference clock to the current location.
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 lightning bolts strike A,B
Right bolt seen first at C’
Two bolts seen simultaneously at C
Left bolt seen second at C’
Relativity of Simultaneity
Two events simultaneous in one inertial frame are not simultaneous
in any other inertial frame moving relative to the first
OR
Clocks synchronized in one inertial frame are not synchronized
in any other inertial frame moving relative to the first
v0
v0
ct
x  xi  vt
ct ct’
x’
x
O
A
B
C
x
O
A
B
C
Light Clock
• Light pulse bouncing between two mirrors
perpendicular to direction of possible motion
• A one way trip is one unit of time Dt = d/c
• Clearly moving light clock has longer interval
between light round trips
Handy Light Clock
Consider pulse of light bouncing between two
mirrors (retroreflectors)
d
to = d / c
Now Observe Same Clock moving
Thought Experiment
Gedanken Experiment
Consider an inertial
frame of reference:
Elevator moving upward
at a constant velocity, v.
Moving Light Clock
Consider path of pulse of light in moving
frame of reference: Light Clock
ct
d
vt
to = d / c
Time Dilation calculated
Use Pythagorean Theorem:
ct
(ct) = d + (vt)
d
d 2 = (ct) 2 - (vt) 2
d 2/ c 2 = t 2 - (v 2/ c 2)t 2
d / c = t [1 - (v 2/ c 2)]1/2
But d = cto ,
So
to = t [1- (v 2/ c 2)] 1/2
2
2
2
The clock in the moving frame runs slower.
vt
Time Dilation Observed!
Does this really work?
to = t [1- (v 2/ c 2)] 1/2
t =gto
1. Mu-Mesons last longer before decaying if
they are moving very fast.
by factor g = 1/ [1- (v 2/ c 2)] 1/2
2. Atomic Clocks run slower when moving.
1 sec/1 000 000 sec at 675 mph.
Time Dilation: Derivation
Analyze laser “beam-bounce” in two reference frames
In S’ frame, light travels up or down a distance D.
In S frame, light travels a longer path along hypotenuse.
• Solve for Dt
• Substitute Dt’ = D/c
(proper time)
D' D
Dt' 
c
c
D
D
1
Dt 

2
2
c 1 v 2 /c 2
c v
1
 Dt'
 gDt'
2
2
1 v /c
Time Dilation/Length Contraction:
Muon Decay
• Why do we observe muons
created in the upper
atmosphere on earth?
Muon’s frame
Length
Contraction
Earth’s frame
Proper lifetime is only  = 2.2 s
 travel only ~650 m at 0.99c
• Need relativity to explain!
Time
Dilation
– Time Dilation: We see muon’s
lifetime as g = 16 s.
– Length Contraction: Muon sees
shorter length (by g = 7.1)
Length Contraction
• Necessary consequence of postulates and for
consistency of effects
• Can also derive in four dim. (ct, x, y, z) as rotation
in a space-time plane preserving 4-D length,
like rotation in a space-space plane preserve length
3-D
4-D
Dl 2  Dx 2  Dy 2  Dz 2
Pythagorean
Theorem
Ds  cDt  Dx  Dy Dz  cDt  Dl
2
2
2
2
2
2
2
Relationship between Inertial Frames
ct’
ct
x’
O
O
x
Light Cone Unchanged
• If the speed of light is identical
for all inertial frame observers,
then the light cone must be
unchanged.
Aberration of Light
• Discovered by Bradley in 1725 after seeing
pennant on sailboat having direction
intermediate to wind and boat motion.
Doppler Effect
Relativity
Warp 0.92 (0.75c)
Relativistic Increase in Mass
• E = gm0c2 = m0c2
E
• m = gm0
m
m0
1  v2 / c2
E = m c2
v
v=c
Energy & Momentum
3-D Case
4-D Momentum
p  mv  gm0 v
p  m0u  (E /c, px, py , pz )
E  mc  gm0c
2
2
2
2
0
2
2 2
0
p m u m c
Energy and Momentum are separate in 3-D and
have separate conservation laws.
In 4-D are part of same vector and rotations preserve length (norm).
dp
F
dt
dE
P
 F v
dt
dp
F
d
Rest Mass
• The rest mass m0 of a particle is an invariant.
It is the length of the 4-D momentum vector.
p  E /c, px , py , pz 
2
2
2
2
x
2
y
2
z
2
2
p  E /c  p  p  p  E /c  p
E  p c  m 0 c
2 2
E  p c  m0c

2
2
2 2
2 2

2 2
2
Einstein’s General Theory of
Relativity predicts black holes
• Mass warps space resulting in light
traveling in curved paths
Principle of Equivalence
A homogeneous gravitational field is completely
equivalent to a uniformly accelerated reference frame.
mi  mg
It is impossible for us to speak of the absolute acceleration of
the system of reference, just as the theory of special relativity
forbids us to talk of the absolute velocity of a system.
Equivalence Principle
Consider an observer in an elevator, in two situations:
mi = mg
1) Elevator is in free-fall. Although
the Earth is exerting gravitational pull,
the elevator is accelerating so that the
internal system appears inertial!
2) Elevator is accelerating upward.
The observer cannot tell the difference
between gravity and a mechanical
acceleration in deep space!
Uniformly Accelerating Frame
Light in Accelerating Frame of Reference
Gravity?
acceleration
Time Dilation in Gravitational Field
• Clock lower down runs slower
 gh 
 D  
Dt B  1 2 DtA  1 2 DtA
 c 
 c 
2
2
e  B / c DtB  e A / c DtA
Dt B  e
D / c 2
DtA
Is it General Relativity right?
• The orbit of Mercury is explained by
Relativity better than Kepler’s laws
• Light is measurably deflected by the Sun’s
gravitational curving of spacetime.
• Extremely accurate clocks run more slowly
when being flown in aircraft & GPS satellites
• Some stars have spectra that have been
gravitationally redshifted.
If we apply General Relativity to a collapsing
stellar core, we find that it can be sufficiently
dense to trap light in its gravity.
Several binary star systems contain black
holes as evidenced by X-rays emitted
Cygnus
X-1
must have a
mass of
about 7
times that of
the Sun
Other black hole candidates include:
• LMC X-3 in the Large Magallenic Cloud orbits its
companion every 1.7 days and might be about 6 solar
masses
• Monoceros A0620-00 orbits an X-ray source every 7
hours and 45 minutes and might be more than 9 solar
masses.
• V404 Cygnus has an orbital period of 6.47 days which
causes Doppler shifts to vary more than 400 km/s. It is
at least 6 solar masses.
Supermassive
black holes exist
at the centers of
most galaxies
Supermassive
black holes exist
at the centers of
most galaxies
Primordial black holes may have
formed in the early universe
• The Big Bang from which the universe emerged might
have been chaotic and powerful enough to have
compressed tiny knots of matter into primordial black
holes
• Their masses could range from a few grams to more
massive than planet Earth
• These have never been observed
• Mathematical models suggest that these might
evaporate over time.
How big is a black hole?
Matter in a black hole becomes much
simpler than elsewhere in the universe
• No electrons, protons, or neutrons
• Event horizon
– the shell from within light cannot escape
• Schwarzschild radius (RSch)
– the distance from the center to the event horizon
• gravitational waves
– ripples in spacetime which carry energy away from the black
hole
• The only three properties of a black hole
– mass, angular momentum, and electrical charge
Structure of Schwarzschild Black Hole
Structure of Kerr
(Rotating Black hole
Structure of a
Kerr (Rotating)
Black Hole
In the Erogoregion,
nothing can remain at
rest as spacetime here is
being pulled around the
black hole
Falling into a black hole is an
infinite voyage as gravitational tidal
forces pull spacetime in such a way
that time becomes infinitely long
Black Hole Evaporation:
Caused by virtual particles
Black holes evaporate
Virtual particles that appear in pairs near a event
horizon may not be able to mutually annihilate
each other if only one manages to survive a trip
along the event horizon.
Summary
• Special Relativity yields:
–
–
–
–
–
Lost of universal simultaneity
Time dilation of moving systems
Length Contraction of moving objects
Equivalence of Mass and Energy
Integrated 4-Dimensional space-time
• General Relativity / Equivalence Principle
–
–
–
–
–
Curved Space-Time
Time Dilation in gravitational potential (curved time)
Bending of light and all inertial paths (no gravity)
Black Holes
Matter/Energy tells spacetime how to curve,
spacetime tells matter/energy how to move
Appendix: 4-D Vectors
• Summary discussion of four dimensional (4-D) vectors
• Have vector algebra just like 3-D vectors but have 4 components
instead of 3:
3 D : x  (x,y,z)
 D : xarethe
(ct,
x, y,z)
Transformations that leave length4
unchanged
familiar:
1) Translations - displacements in space or time
2) Rotations - angular rotations
3) Velocity boosts (Lorentz transformation) which are equivalent to
rotations in a space-time plane
Lorentz Transformations
Leave 4-D vectors length Invariant
 v 
ct' g ct  x
 c 
 v 
ct  g ct' x'
 c 
 v 
x' g x  ct
 c 
 v 
x  g x' ct'
 c 
y' y
z' z
y  y'
z  z'
Examples of 4-D Vectors
Easiest way to see that 4-D vector transforms like the prototype
under Lorentz transformations is to construct them that way!
x  (ct,x,y,z)
d
dt
u
x
(c,v x ,v y ,vz )  g (c,v x,v y ,v z )
d
d
p  m0 u  m0g (c,v x ,v y ,vz )  (E /c, px , py , pz )
d
d2 x
a
u 2
d
d
Velocity Composition
3-D Velocity
ux  v
ux '
2
1 uxv /c
uy
uy '
2
g (1 uxv /c )
uz
uz '
2
g (1 ux v /c )
4-D Velocity
Lorentz Transformation
2
u c
2
Change of velocity
is simply a rotation
through and angle