Special Theory of RelativitySpecial Theory of Relativity
Einstein pondered the question, “If I could ride a beam of light, what would I see?”
“If I ran at the speed of light with a mirror held out in front, would I be able to see my reflection?”
Speed of LightSpeed of light (c) has a value of 3x108m/s or 186,000miles/sec in a vacuum.
In order to communicate with astronauts that landed on the moon, it takes at least 2.6 seconds for the radiowave (light) to travel from Earth to Moon and back.
To put that into perspective…
Relativity of Motion as we know it
“Relativity” refers to the way measurements are made in a given reference frame (RF) compared to another frame.
We take the ground to be at REST.Depending on your RF, 2 people can get different answers
Relative to the TRAIN, what is velocity of the man?
Relative to the MAN, what is velocity of the woman?
Relative to the WOMAN, what is velocity of the man?
Relative velocities using Newtonian physics
If the MAN starts walking at 4mph to right, how fast is he moving relative to woman? Relative to train?
Train is moving at 40mph in relation to the ground
Relative to the TRAIN, how fast is light beam moving?
Relative to the WOMAN, how fast is light beam moving?
Relative velocity of LIGHT using Newtonian physics
Train is moving at 40mph in relation to the ground
• In the late 1800’s, scientists tried to apply Newtonian physics principles to the speed of light, where Newton’s Laws were king for 250 years. Tried to measure the aether wind that light travels through, similar to sound moving through air.
c
Houston, we have a PROBLEM!
Newton’s Laws solved every problem for 250 years, but NOT this one.
In summary• No matter how they measured the speed of
light coming from a source…moving away, moving towards, not moving at all,…the speed of light ALWAYS came out to be (c), 186,000 miles per second or 669,600,000mph or 3x108m/s.
In steps EINSTEIN in 1905 at the age of 26.
Proposes his theory of Special Relativity while working as a Swiss Patent Clerk.Rejects 250 yrs of Newtonian relativity as it applies to light
Einstein's 1905 still evokes awe. Historians call it the miracle year or annus mirabilis.
Postulates of Special TheorySee what thinking can do? It ages you. Ignorance really is bliss.
Relative to the TRAIN, how fast is light beam moving?
Relative to the WOMAN, how fast is light beam moving?
Revisit relative velocity of LIGHT problem
(previous answer = c)
(previous answer = v + c)
According to EINSTEIN:
NEW ANSWER = ?
NEW ANSWER = ?
Relative velocity problemHow fast does the beam of light move relative to the person on ground?
How fast does the beam of light move relative to the rocket ship that is moving at 0.9c relative to the ground?How fast would an observer
moving towards light at 0.9c measure light beam to move?
The speed of light is also the speed of information. Suppose the speed of light was relative AND not constant for all observers….
Einstein figured that if the speed of light is the SAME for 2 observers in different RF’s, then something else must be different for each of them.
SIMULTANEITY:
http://www.youtube.com/watch?v=wteiuxyqtoM
Must 2 events that are simultaneous to one observer ALSO be simultaneous to another?
Observer O will see both lightning strikes (events 1 & 2) at the same time.
Imagine 2 parallel mirrors separated by distance h.
This represents our ‘light’ clock where the time between a tick and tock for the clock is:
What happens if the same clock MOVES past YOU with velocity v.
Imagine a ‘pulse’ of light that bounces back and forth between the mirrors
This clock is stationary where the ticks and tocks (events) occur at the same places according to person standing next to clock (proper time, to)
RF where observer is at rest relative to clock.
Earth RF where observer is in motion relative to clock.
Math for time dilation…just a simple right triangle
Lorentz factor (γ)
Einstein’s equations (relativistic mechanics) describes the motion of objects at ANY speed whereas Newton’s Laws (classical mechanics) is only good for slower, everyday speeds.
If v = 0.5c
If v = 0.995c
Means that if you move at 50% speed of light,
Means that if you move at 87% speed of light,
Means that if you move at 99.5% speed of light,
If v = 0.87c
What we know so far…
Muons & Time Dilation
Muons at rest have lifespan of 2.2 millionths of a second (2.2x10-6s) before decaying.
Muons are cousins of electrons. They are fast-moving (0.99c), unstable particles created in upper atmosphere & move quickly towards to ground.
Using the muon speed and lifetime, muons should disintegrate at the top of the mountain. HOW DO WE EXPLAIN
MUONS covering this extra distance if at 2.2us they expire at top of mountain?
However, scientists detected many muons reaching surface of earth.
The Lifetime of a Muon v = 0.99c lifetime in muon RF, to = 2.2x10-6s
Find lifetime of muon from Earth frame:
Any Clock•Any device that measures time gives the same effect of time dilation with movement. Heartbeat, digital/analog clocks, a pendulum, etc. •If it happens for a ‘light’ clock, it must happen to ALL clocks.
GPS uses time dilation equation ~31 satellites orbit the earth with clocks
that tick at a different rate then those on Earth because of their fast speeds.
The satellite clocks need to be in perfect synch with those on Earth to allow it to nail down position to a precise degree. Using light waves to communciate with ground GPS unit.
The high speeds of the satellites cause the atomic clocks to tick more slowly than stationary ground clocks by about 7.2x10-6 sec/day. Even though it’s a tiny time difference, it translates into a huge distance error due to high value of speed of light. This translates to about a 7mile/day error.
As a spacecraft moving at 0.92c travels past an observer on Earth, the Earthbound observer and the occupants of the craft each start identical alarm clocks that are set to ring after 6.0 h have passed.
According to the Earthling, what does the Earth clock read when the spacecraft clock rings?
Boy travels to Vega (5th brightest star in our sky) leaving 35yr old twin sister behind. Boy travels at 0.990c and Vega is 26.4ly from Earth.
a) How long does the trip take according to Girl?
Since the 2 events (leaving earth & arriving at Vega) are clearly in different locations for Girl, she does not experience the proper time, to. Boy experiences them both at the spaceship door.
b) How long did the trip take for Boy according to his clock?
c) How old is Boy and Girl when he reaches Vega?
Twin Paradox - Who is really younger?
Consider a rocket ship at rest relative to Earth. If it sends light flashes every 6 minutes, then Earth receives flashes every 6minutes.
TWIN PARADOX, which clock really runs slow?
What would happen to frequency of flashes received by Earth if rocket ship moves towards Earth at high rate? Away from Earth?
Let’s say rocket ship moves away from Earth at a speed that causes flashes to be received on Earth every 12 min.Rocket ship is still sending 1 flash/6min.
Assume rocket ship sends a total of 10 flashes before making sudden turnaround back to earth (ignore acceleration).
Calculate time elapsed for:
a)Rocket ship?
b)Earth?Assume clock time started 6min before 1st flash
Rocket ship now heads towards earth, still sending flashes at 6-minute intervals (trip for rocket would take same time out and back).But now Earth receives MORE frequently, every 3min.
Rocket ship travels for 1 hour, therefore sends 10 flashes.Calculate time elapsed for:a)Rocket ship?b)Earth?
Same would be true if earth twin sent flashes instead of space twin.
Earth twin sends flashes at 6min intervalsSpace twin receives them at 12min intervals moving away.
Space twin now moves towards Earth and receives flashes every 3min
TIM
ESPEED THROUGH SPACE
Imagine a light clock at rest, where a flash is emitted at A and moves to B. The purple arrow represents the time it would take flash to move relative to RF outside of clock.
A
B
Purple arrow indicates an object at REST.
IF clock starts to move through space, say at ½c, its time will be affected as seen by red arrow.
½c c
What does blue arrow represent?
SPACE TIME DIAGRAM
LENGTH & SPACE When you are at rest with respect to an object, you measure
its REST length, Lo .
If you are at rest with respect to a 2 points in space, you measure its rest length.
If you move with respect to an object, the object will be measured to be shorter. The same goes for the space you travel through.
Rest length is ALWAYS the longest length.
Length contraction formula Lo is proper length
where length is measured when at rest with object. L is contracted length
It makes no sense to say what the length of an object really is.
It makes sense only to say what the length is in a given frame.
The situation doesn‘t really look like one thing in particular.
The look depends on the frame in which the looking is being done.
What does object really look like? Do objects really shrink?
Length Contraction only along direction of motion
• Moving observers see that objects contract along the direction of motion.
• Note that there is no contraction of lengths that are perpendicular to the direction of motion
Meterstick
• How fast does a meterstick need to move past you for you to measure it to be 0.50m?
Boy & Girl againBoy travels to Vega. Recall v = 0.990c and Vega is 26.4ly from Earth.
How far does Boy measure the trip to be?
According to Boy, the distance from Earth to Vega is…
Recall from previous problem that…
As Einstein once said, “Common sense is the layer of prejudices put down before the age of eighteen.”
All our intuition about space, time and motion is based on childhood observation of a world in which no objects move at speeds comparable to that of light. Perhaps if we had been raised in a civilization zipping around the universe in spaceships moving at relativistic speeds, Einstein’s assertions about space and time would just seem to be common sense.
NONE of this seems logical. In fact, is seems impossible!
Space travel made possible, 2 viewpoints
Revisit Muon experiment
Recall that muons should have decayed prior to reaching the Earth’s surface, but they didn’t. What was the reason based on the Earth observer?
SPACE-TIME• Special relativity demonstrated that there is a
relationship between space and time where we can no longer reference where without some reference to when. We live in the fabric of space-time where both are woven together.
Greater meaning of ‘c’:
‘c’ is telling us more than just how fast light travels, more importantly,
• Accelerators produce radioisotopes for use in medicine/cancer therapy. Some larger hospitals make their own radioisotopes in basement-cyclotrons.
• Particle accelerators need Einstein’s equations for the correct operation of the machines
More applications for Einstein’s relativity equations
There is simply nothing in our experience that can fit such facts as these into a comfortable mental image. It does not make these phenomena impossible;
It means that for our intuition to be able to accept them we must become sufficiently accustomed to the conditions under which they occur, so that there will be mental images into which these new events can fit.
Why do these concepts seem so weird and fictional?
Barn & the Pole Paradox
• Consider a pole 20m in length and a barn 10m in length at rest. The barn doors can shut & open simultaneously via a switch. You are sitting inside the barn at rest.
• A runner moves with the pole at 0.90c towards the barn.
According to each frame, can the pole fit inside the barn where the doors would be shut for a brief moment?
Does the pole ‘fit’ in barn or not?
ENERGY, MASS, & MOMENTUM
E = mc2
Relativistic Mass
• Technically speaking,
• A ball is
• A flashlight
ΔE = Δmc2
Heating water in a pot on the stove:
An ordinary CRT television set is a simple form of particle accelerator
• Mass Increase with Speed • Electrons in a color TV tube
(moving at about 1/3 c) are about half a percent heavier than electrons at rest. This must be accounted for in determining the strength of the magnetic fields used to guide them to the screen.
• Light or photons (particles of light)
• Although light
So, how does light move at ‘c’?
Relativistic Momentum (for particles)
Rest Energy,
Total Energy,
Lowest energy is rest energy
How does a photon have momentum?
During the burning process, water and other fluids are driven out by converting them to vapor where gases like CO2 are formed when oxygen in the air combines with carbon in the log).
Burning a log and the “mass defect”:
Burning a log is a chemical reaction involving rearrangement of atoms (same elements are present before and after).
However, a nuclear reaction involves a subatomic particle colliding with a nucleus which yields entirely new elements.
Fission
When heavy elements like Uranium are split into smaller elements, energy is released. Why?
When light nuclei fuse, the product nucleus is less massive than the sum of its parts.
Fusion
‘Little Boy’ dropped over Hiroshima (9,000lbs), August 6th, uranium bomb killed 140,000 people
‘Fat Man’ dropped over Nagasaki, August 9th, plutonium bomb killed 70,000
