Date post: | 09-May-2015 |
Category: |
Technology |
Upload: | praveens |
View: | 908 times |
Download: | 4 times |
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 11
SPECIAL RELATIVITYPresented by: PRAVEEN MOHAN
TSO (Electronics)
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 22
Relative Motion Always specify the point from which motion is being observed and
measured.
To measure the speed of an object: Choose a frame of reference Frame of reference standing still.
Frames of ReferencePhysical surroundings from which you observe and measure the world around you.
Two Frame of Reference Inertial Non Inertial
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 33
Reference Frames
Inertial Frame of Reference: one in which the Newton's first laws of motion apply.
Example: A train moving at a Constant velocity.
Non- Inertial Frame of Reference: An accelerating or decelerating objects.
Example: Ferris wheel (accelerating inwards)
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 44
Theories before Einstein
Prior to Einstein two great theories of physics
Newtonian Mechanics and Gravity.
Maxwell’s unified theory of electricity and magnetism.
These two theories imply different views of time.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 55
Theory before Einstein……
Newton Theory: Time is an absolute, universal measure, same for all
observers, no matter whether they are at rest or travelling at high speed.
Maxwell’s Theory: people and objects cannot travel at speeds greater than the
speed of light.
Einstein explored the consequences of Maxwell’s theory.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 66
Words of Einstein (Thoughts…)
Can you catch up with light?Can you catch up with light?
What would happen if I rode a light beam?What would happen if I rode a light beam?
If you were travelling at the speed of light and looked If you were travelling at the speed of light and looked in a mirror - would you see your reflection?in a mirror - would you see your reflection?
“Questions keep me awake at night. If you were in your vehicle travelling at the speed of light and you turn your headlights on, what would they do?”
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 77
Thought Experiment
“What would happen if I rode a light beam?”
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 88
Electromagnetic radiation requires changing E and B fields.
Would see static electric and magnetic fields with no understandable source.
(Cont.)
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 99
Can be intuitive...
Moving apart; what is the green car’s velocity relative to yellow car?
v = 70 km h-1 v = 50 km h-1
We know the answer intuitively (120 km h-1)
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1010
Same idea with light From the examples above, we would expect the relative
c = 3 x 108 ms-1 c = 3 x 108 ms-1
velocity to be 2c = 6 x 108 ms-1. This is in fact wrong!
Then what will be the answer??
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1111
Theory of Relativity
Theory provide relation between:
1. Space-Time (4 coordinates-3 spatial coordinates(x,y,z),4th is
time.)
2. Matter-Energy
3. Electricity-Magnetism
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1212
Theory of Relativity
Special relativity(1905)
Deals with consequences of LACK OF UNIVERSAL inertial reference frame.
General relativity(1915)
Describe the relationship between GRAVITY and geometrical structure of SPACE and TIME.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1313
Postulates of Special Relativity
Two postulates:
The Relativity Principle:
The laws of motion are the same in every inertial frame of reference.
Constancy of the Speed of Light:
The speed of light in a vacuum is the same independent of the speed of the source or the observer.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1414
Consequences of the Special Theory Of Relativity
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1515
4-Vector and Lorentz Transformation
Two reference frames K and K’, with K’ moving away from K in the x positive direction at constant velocity v.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1616
(Cont.)
2
1
2
2
2
1where
'
'
'
'
c
v-
zz
yy
vtxx
c
vxtt
z
y
x
ct
c
vc
v
z
y
x
tc
1000
0100
00
00
z
y
x
ct
is the
Position 4-vector
Or ’= where
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1717
Simultaneity
An interesting consequence of Einstein's second postulate occurs with the concept of simultaneity.
Two events are simultaneous if they occur at the same time.
Consider, for example, a light source in the exact center of the compartment of a rocket ship.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1818
From the point of view of the observer who travels with the compartment, light from the source travels equal distances to both ends of the compartment and therefore strikes both ends simultaneously.
(Cont.)
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 1919
The events of light striking the front and back of the compartment are not simultaneous from the point of view of an observer in a different frame of reference.
Light that strikes the back of the compartment doesn’t have as far to go and strikes sooner than light strikes the front of the compartment
(Cont.)
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2020
Light Clock Light pulse bouncing between two mirrors Light pulse bouncing between two mirrors perpendicular to direction of perpendicular to direction of
possible motionpossible motion
A one way trip is one unit of time Dt = d/cA one way trip is one unit of time Dt = d/c
Clearly moving light clock has longer interval between light round tripsClearly moving light clock has longer interval between light round trips
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2121
Handy Light Clock
Consider pulse of light bouncing between two mirrors (retroreflectors)
dd
ttoo = d / c = d / c
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2222
Thought
Experiment
• Consider an inertial frame of Consider an inertial frame of reference.reference.
• Elevator moving upward at a Elevator moving upward at a constant velocity,constant velocity, v v..
Now Observe Same Clock moving
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2323
Moving Light Clock
Consider path of pulse of light in moving frame of reference: Light Clock
ddctct
vtvt
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2424
Time Dilation calculated
Use Pythagorean Theorem: (ct) 2 = d 2 + (vt) 2
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
ddctct
ddctct
to = t [1- (v 2/ c 2)] 1/2
The clock in the moving frame runs The clock in the moving frame runs slower.slower.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2525
Length of 1 minute w.r.t. Speed of frame
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2626
Muon Experiment Muons are created in upper atmosphere from cosmic ray hits
Typical muon travel speeds are 0.99995c, giving =100
Half-life of muons in their own rest frame (measured in lab) is th= 2 microseconds =0.000002s
Travelling at 0.99995c for th=0.000002s, the muons would go only 600 m
But travelling for th= 0.0002s, the muons can go 60 km
They easily reach the Earth’s surface, and are detected!
Half-life can be measured by comparing muon flux on a mountain and at sea level; result agrees with th
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2727
(Cont.) Muons traveling with a speed of
0.99c travel only about 650 m as measured in the muons’ reference frame, where their lifetime is about 2.2 s.
The muons travel about 4700 m as measured by an observer on Earth. Because of time dilation, the muons’ lifetime is longer as measured by the Earth observer.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2828
Length Contraction
As objects move through space-time, space as well as time changes (as velocity of light is constant in vacuum)
Space is contracted
Objects look shorter when they move at relativistic speeds.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 2929
(Cont.)
v = relative velocity between the observed object and the observer
c = speed of light L = the measured length of the moving object Lo = the measured length of the object at rest.
2
2
1c
vLL o
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3030
(Cont.)
Contraction takes place only in the direction of motion. If an object is moving horizontally, no contraction takes place vertically.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3131
Length of 1 meter w.r.t. Speed of frame
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3232
Fly by a House
Location of objects
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3333
Fly by a House at a speed near the Speed of Light
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3434
Fly sideways by a house at a normal speed
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3535
Fly sideways by a house (at 50% the speed of light)
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3636
Fly sideways by a house at 80% the speed of light
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3737
Fly sideways by a house at 90% the speed of light
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3838
Fly sideways by a house at 99.9% the speed of light
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 3939
Length contraction should be of considerable interest to space voyagers.
The center of our Milky Way galaxy is 25,000 light-years away. The center of our Milky Way galaxy is 25,000 light-years away.
Does this mean that if we traveled in that direction at the speed of light it Does this mean that if we traveled in that direction at the speed of light it would take 25,000 years to get there? would take 25,000 years to get there?
From an Earth frame of reference, yes, but to the space voyagers, From an Earth frame of reference, yes, but to the space voyagers, decidedly not! decidedly not!
At the speed of light, the 25,000-light-year distance would be contracted to At the speed of light, the 25,000-light-year distance would be contracted to no distance at all. no distance at all.
Space voyagers would arrive there instantly! Space voyagers would arrive there instantly!
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4040
Twin Paradox
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4141
Twin Paradox I Suppose two twins, Chat & Abhi, are born on Earth.
Suppose Abhi goes into a spaceship and travels at 0.99c to the nearest star, Alpha Centauri, which is 4.3 light years away.
Abhi stays at Alpha Centauri for 5 years, and then returns home at0.99c.
Chat stays home.
(1 light year (ly ) is the distance light travels in 1 year.) Short hand: 1 ly = 1 [c*yr]
How many years have passed for Chat and Abhi ?
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4242
Twin Paradox II How many years will have passed according to Chat’s clock ?
Chat (on Earth): His clock has been at rest the entire time no time dilation.
Total time for trip = time for Abhi to go to Alpha Cent. + time stayed on Alpha Centauri + time for Abhi to get home Total time for trip = (4.3c*yrs/ 0.99c) + 5 yrs + (4.3c*yrs/0.99c) =4.34 yrs + 5 yrs + 4.34 yrs =13.7 years.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4343
Twin Paradox III
How many years will have passed according to Abhi’s clock?
Moving at enormous velocity relative to Earth, his clock runs slower compared to the Earth clock (time is “stretched out").
time dilation
For every year that passed on Earth, Abhi’s clock (and hence Abhi) only aged by (1/7.1) = 0.14 years!
You only apply the time dilation to the time when he was in moving at 0.99c.
Abhi will only have aged by 4.34/7.1 + 5 + 4.34/7.1 = 6.2 years
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4444
So where’s the paradox?
We assumed it was Abhi who was moving at fast speeds. BUT, Abhi is equally allowed to say:
“I saw you & the Earth zipping away from me at0.99c” ,and so he sees the whole age deal reversed?
As long as they are indifferent inertial reference frames, they are allowed to have “their own time ”, and there’s no conflict…
BUT, after he returns, they’re both in the same reference frame, (on the earth) so one of them must be right. Who is it?
A) Chat B) Abhi C) Neither.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4545
The Solution
Special Relativity only applies to non-accelerating observers.
Abhi cannot apply Special Relativity to his observations.
Chat's reference frame was not accelerating, his observations do apply.
Which means he’s aged by 13.7 years whereas Abhi has aged only 6.2 years.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4646
Momentum and Mass
Analysis of collisions shows that if we want to conserve momentum in relativity theory,
we must redefine it
p m0v
1v2
c2
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4747
(Cont.)
we need to redefine mass as
m0 is the rest mass of the object
This suggests that an object cannot achieve the speed of light since the mass would become infinite!!!
This would require infinite energy to increase the speed to c
m0 / 1 v2 /c2
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4848
Mass and Energy Consider applying a force to an object
As the object has work done on it (force x distance) it increases its kinetic energy
But, its mass also increases.
Not all the work done on the object increases velocity since there is a limit on velocity.
Some of the work is increasing mass
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 4949
(Cont.)
Einstein showed that the correct expression is that
KE = mc2 - m0c2
Einstein called the m0c2 term the rest energy of the object
This means that mc2 is the total energy of the object
This is the origin of the famous
E = mc2
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 5050
Mass and Energy (evidence)
An elementary particle called a pion (composed of a quark/anti-quark pair) is observed to decay into EM waves whose energy is exactly mpionc2
Nuclear energy. (mass defect)
Mass loss in 1kg of dynamite is the order of 10Mass loss in 1kg of dynamite is the order of 10-6-6..
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 5151
Correspondence Principle
Einstein’s Special Relativity reduces to our old Newtonian mechanics for v << c
Theories overlap and Special Theory is the more general case (closer to reality) and that Newtonian Mechanics is a special case for low speeds
The two theories correspond
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 5252
General Theory of RelativityGeneral Theory of Relativity
Einstein's General Theory of Relativity paper. Einstein's General Theory of Relativity paper.
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 5353
Motivation for General Relativity: Einstein’s tower
So far, we have ignored the effects of gravity.
another thought experiment, to test
whether light can be unaffected by gravity.
Consider a tower on Earth Shine a light ray from bottom to
top When light gets to top, turn its
energy into mass. Then drop mass to bottom of
tower, in Earth’s gravity field Then turn it back into energy
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 5454
(Cont.) If we could do this, then we could get energy from nothing!
Original energy in light beam = Estart
mass created at top is m=E/c2
Then drop mass… at bottom of tower it has picked up speed (and energy) due to the effects of gravitational field.
When we turn it back into energy, we have Eend=Estart+Egrav
But, we started off with only Estart – we have made energy! We’re rich!
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 5555
(Cont.) Clearly, our assumption must be wrong…
light must be affected by gravity.
But gravity does not appear in Maxwell’s equations, which govern light
Thus, Maxwell’s equations are not exactly valid in the reference frame of Earth’s surface, where there is gravity.
The Earth’s surface must not be an inertial frame of reference!
Consequences of General Theory of Relativity
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 5757
Bending of Light by the Sun
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 5858
Gravitational Lensing
Galaxies between the earth and a quasar can produce multiple images. From bending one can estimate the mass of galaxy
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 5959
Cont…Cont…
Einstein said….Einstein said….
““When you sit withWhen you sit with
a nice girl for two hours,a nice girl for two hours,
it seems like two minutes.it seems like two minutes.
When you sit on a hot When you sit on a hot stove for two minutes,stove for two minutes,
it seems like two hours.it seems like two hours.
That’s relativity.”That’s relativity.”
31/01/200831/01/2008 NHTS-HyderabadNHTS-Hyderabad 6060
Thank You
Presented By:PRAVEEN MOHANTrainee Scientific Officer (Electronics)