Special Theory of Relativity. Relativity of Motion “Relativity” refers to the way measurements are made in a given reference frame (RF) compared to another.

Special Theory of Relativity

Relativity of Motion

“Relativity” refers to the way measurements are made in a given reference frame (RF) compared to another reference frame.

We usually take the ground RF to be at REST.

Depending on your RF, 2 people can get different answers for the same motion.

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

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

Speed of LightSpeed of light (c) has a value of 3x108m/s or 186,000miles/sec in a vacuum. ‘c’ is finite.

Relative to the TRAIN, how fast is light beam moving?

Relative to the WOMAN, how fast is light beam moving?

Relative velocity for LIGHT

Train is moving at 40mph in relation to the ground

In the late 1800’s, Michelson and Morley investigated theapplication of relativity to the Speed of Light. Hypothesizedthat Light from the sun should measure to have different relative speeds depending on our orbital path through a proposed aether substance.

c

Light does NOT appear to obey the Laws of Sir Issac!

In summary

Had been thinking about the “light” question since age 16. Proposes his theory of Special Relativity while working as a Swiss Patent Clerk.

Einstein's 1905 still evokes awe. Historians call it the miracle year or annus mirabilis.

MIRACLE YEAR 1905 – ANNUS MIRABILISWhile still a patent clerk working 40hrs/wk, Einstein published 4 scientific papers, ALL Nobel Prize worthy.1) Einstein invented a new method of counting and determining the size of the atoms or molecules in a given space and explained the phenomenon of Brownian motion. The net result was a proof that atoms actually exist - still an issue at that time 2) Einstein created the quantum theory of light, the idea that light exists as tiny packets, or particles, which he called photons. Alongside Max Planck's work on quanta of heat Einstein proposed one of the most shocking idea in twentieth century physics: we live in a quantum universe, one built out of tiny, discrete chunks of energy and matter.

3) Special Relativity – Challenged Newton’s views on space and time. We shall soon see!

4) Special Relativity – Addition to the original paper which derived his most famous result, E = mc2, the beginning of nuclear energy and the atomic age.

If this already wasn’t enough…

Postulates of Special Theory

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….

SIMULTANEITY: Must 2 events that are simultaneous to one observer ALSO be simultaneous to another?

Einstein said that if the speed of light is the SAME for 2 observers in relative motion, then _____________________

Imagine 2 parallel mirrors separated by distance h.

This represents a ‘light’ clock where the time between a ‘ticks’ (one way trip) for the clock is:

Imagine a ‘pulse’ of light that bounces back and forth between the mirrors with no NRG loss.

The time measured by this clock is called the proper time, to. The clock is stationary relative to observer standing next to it. (ticks (events) occur at the same place according a person next to clock.

What happens if the same clock MOVES past YOUwith velocity v?

RF where observer is at rest relative to clock. He measures time equal to to = h / c.

TIME DILATION derivation

Lorentz factor (γ)

If you were moving in a spaceship at a high speed relative to Earth, would you notice a difference in your pulse rate or the pulse rate of people on Earth?

A. Yes, you would notice a difference in both pulse rates.

B. You would notice a difference in your pulse rate, but not the pulse rate of people on Earth.

C. You would notice a difference in the pulse rate of people on Earth, but not in your own pulse rate.

D. You would not notice a difference in either pulse rate.

Time DilationCHECK YOUR NEIGHBOR

Will observers A and B agree on measurements of time if A moves at half the speed of light relative to B?

A. Yes, they would agree completely.

B. No, they would disagree completely.

C. They would agree half of the time and disagree the other half of the time.

D. None of the above.

Time DilationCHECK YOUR NEIGHBOR

Will observers A and B agree on measurements of time if both A and B move together at half the speed of light relative to Earth?

A. Yes, they would agree completely.

B. No, they would disagree completely.

C. They would agree half of the time and disagree the other half of the time.

D. None of the above.

Time DilationCHECK YOUR NEIGHBOR

If v = 0.50c

If v = 0.87c

If v = 0.995c

Let us run some numbers for Lorentz factor, γ

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.

Example - Lifetime of a Muon v = 0.99c lifetime in muon RF, to = 2.2x10-6s

Find lifetime of muon from Earth frame:

Any Clock

GPS can only function properly using time dilation equation

~31 satellites orbit the earth with clocks that are synchronized with earth clocks

Example: 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?

Example: A boy travels to Vega (5th brightest star in our sky) leaving 35yr old twin sister behind. He travels at 0.990c and Vega is 26.4ly from Earth.

a) How long does the trip take according to the Girl?

b) How long did the trip take for Boy according to his clock?

c) How old is Boy and Girl when he reaches Vega?

TIM

ESPEED THROUGH SPACE

Imagine a light clock at rest, where a flash is emitted at A and moves to B. The vertical arrow represents the time it would take flash to move relative to RF outside of clock.

A

B

Vertical arrow indicates an object at REST. It moves through time, but does not move through space. Red arrow indicates moving through space at a slower rate of time!

IF clock starts to move through space, say at ½c, its time will be affected as seen by diagonal arrow. When light flash has covered original distance AB (now AE but same due to circle), the time it took is less than before.

½c c

E

SPACE TIME DIAGRAM

LENGTH is Relative Too!

Return to trip to VEGA example:

Boy travels to Vega @ v = 0.99c and Vega is 26.4ly from Earth as measured in Earth RF. Boy measures time to Vega to be 3.77yrs. Determine distance between Earth and Vega in Boy RF.

If moving clock is slower by a factor of 1/ γ, (to = t / γ), Then ________________________________________

Length Contraction – An explanationA stick of length, L, is at rest next to cool stick figure (S.F.).

L

Skateboard man moves past the stick at speed v

In S.F.'s frame, it takes skateborad a time of t = L /v to move length of stick. S.F. is at rest relative to ground.

In Skateboard’s frame, time will advance by a time (to) where to = t / γ . Subbing for t, to = (L /v)(1/ γ ) = L / γv In Skateboard’s frame, the stick moves by at speed v. The time between the two ends passing by is to = L /vγ from above.

To get the length of stick in the SB frame, he multiplies speed x time. He measures the length to be L = vto = v (L / vγ) = L /γ which is the desired contraction.

Length Contraction only along direction of motion

A Meterstick

How fast does a meterstick need to move past you for you to measure it to be 0.50m?

Space travel made possible, 2 viewpoints

Twin Paradox - Who is really younger?

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?

What explanation would observer riding with muon give in order to explain reaching surface before decaying?

SPACE-TIME

Relativistic Velocity AdditionRecall our velocity addition formula from long ago…

It must now be altered for relativistic speeds…

EXAMPLE: A spaceship moves at 0.80c relative to Earth. The spaceship fires a projectile at 0.50c relative to the ship. A) Determine the speed of the projectile relative to the Earth.

B) The spaceship now fires a photon at ‘c’ relative to ship. Determine speed of photon relative to Earth.

Example: Two spaceships leave the earth in opposite directions. The speed of each spaceship is identical and measured to be 0.750c with respect to the earth. Determine the velocity of spaceship 1 relative to spaceship 2 assuming spaceship 1 is moving to the left.

• Accelerators produce radioisotopes for use in medicine/cancer therapy. Some larger hospitals make their own radioisotopes in basement-cyclotrons.

• Particle accelerators like the one at CERN rely on relativistic equations in order to operate.

More applications for Einstein’s relativity equations

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?

What does each RF measure?• Frame of Barn:

• Frame of Runner:

Does the pole ‘fit’ in barn or not?

Scenario: Consider an incompressible metal rod. If we push one end, the entire rod moves. Someone standing at the other end could be signaled this way.

Now imagine the rod is constructed to be 1light-year long. A person on the other end would see that end move a light year away. You have sent an instantaneous signal! It would have taken light at least a year to travel that distance. Have we gotten around the cosmic speed limit? Thoughts?

ENERGY, MASS, & MOMENTUM

Energy and Mass Equivalence• Another 1905 paper was an addendum to the

special theory.

Relativistic Mass

Be careful not to think an object acquires more particles as is its speed increases. It doesn’t. It acquires more energy which is also considered mass.

An ordinary CRT television set is a simple form of particle accelerator

Rest Energy, Eo = moc2

Total Energy, E = mc2

Energy of object at rest relative to you.

Energy of object moving relative to you.

Relativistic Momentum (for particles)

So, how does light move at ‘c’? Doesn’t it have infinite mass?

How does a photon have momentum if it has no rest mass?

There is a particle called a positron which is exactly like an electron except that it has positive charge. 

If a positron and an electron collide at low speed, so there is very little kinetic energy, they both disappear in a flash of electromagnetic radiation (light).  This can be detected and its energy measured.  It turns out to be E=2m0c2 where m0 is the mass of the electron (and the positron).

Thus, “particles” can “vaporize” into pure energy, that is, electromagnetic radiation. They could be thought of as pure energy prior to collision which may make more sense. Total remains unchanged.

2 masses can combine to get pure energy

Example: The Tevatron accelerator at Fermilab in Chicago can accelerate protons to KE =1x1012eV. 1eV (electron*volt) = 1.6x10-19J. How fast is the proton moving if the rest energy of proton is 938MeV?

Example: An unstable particle at rest breaks up into two fragments of unequal mass. The mass of the lighter fragment is 2.50x10-28kg, and that of the heavier fragment is 1.67x10-27 kg. If the lighter fragment has a speed of 0.893c after the breakup, what is the speed of the heavier fragment?

The resulting molecules have very slightly less mass than the separate P and O2 molecules. The whole is slightly less massive than the parts. WHY?

Striking a match causes a chemical reaction... Phosphorus (P) atoms in the match head rearrange and combine with O2 to form new molecules.

E = mc2 not just restricted to reactions…

lightHeThU 42

23490

23892

However, in nuclear reactions, the decrease in mass is considerably MORE noticeable. For example, our SUN bathes us in its loss of mass which is energy. Why is nuclear more evident?

This difference is called binding energy.

He42

Mass per Nucleon Graph

If we take the total mass of class and divide by # of students in class, we arrive at average mass per student.Similarly, if we take the mass of the nucleus of each atom and divide by the number of nucleons in each respective nucleus, it would yield average mass per nucleon.

Fission

When heavy elements (like Uranium) are split into smaller elements and energy is released.

When light nuclei fuse (Hydrogen), making larger elements, energy is also released.

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