Special Relativity The Realm of the Big and Fast OR Where Things Do Not Behave As They Should Where Everything You Know Is Not Quite Right

By the end of this unit, you will be able to: Explain the two postulate of Special Relativity Explain and calculate time dilation Explain and calculate length contraction Explain how simultaneous events are viewed in different reference frames Add velocities in relativistic scenarios Calculate relativistic momentum Explain and calculate rest energy of objects Calculate mass defects and binding energies Calculate total energy under relativistic conditions Calculate kinetic energy under relativistic conditions Explain why objects with mass cannot travel faster than the speed of light

Special Relativity Albert Einstein Slow Fast Big Classical Physics Special Relativity Small Quantum Mechanics Quantum Electrodynamics (QED) Albert Einstein In 1905, he published three papers that rocked the scientific world One was on the photoelectric effect Why electrons are shed from metals when exposed to sunlight One was on Brownian motion Confirming the existence of atoms The last was on Special Relativity Anyone know what branch of physics covers the slow and small? (Nick may know this. It hasn’t been discussed before.) Anyone know what branch of physics covers the fast and small? (I doubt anyone knows this. It has never been discussed before)

Inertial Reference Frame System of objects that are not moving with respect to each other Used to record a physical event Consists of a coordinate system (x, y, z, t) Inertial Reference Frame Reference frame where Newton’s First Law is valid If the net force on an object is zero, then the object will maintain its state of motion In an inertial reference frame, the acceleration of the frame is zero What is a reference frame? (Entire class, although most probably won’t remember what it is from September) If I wanted to record an event, say me waking up in the morning, what information do I need? (Everyone) What pieces of information do I need to record the where? Hint: Think math. (Middle and up) What is Newton’s First Law? (Everyone) Can you give me an example of a situation where Newton’s First Law isn’t valid? (Middle and up)

Two Assumptions of Special Relativity The Relativity Postulate A reference frame moving at a constant velocity with respect to an inertial reference frame is an inertial reference frame The laws of physics hold in their simplest form in all inertial reference frames The Speed of Light Postulate Electromagnetic waves travel at c = 3 x 108 m/s in all directions in a vacuum in all inertial reference frames What is the speed of light in a vacuum? (Everyone)

The Relativity Postulate The physical laws are the same in all inertial frames Therefore, it is impossible to design an experiment to distinguish between an inertial frame at rest and an inertial frame that is moving If you are in a car, it is perfectly valid to say that you are at rest in the car and the earth is moving and vice versa Because there is no experiment that will determine one from the other Why is it impossible to design an experiment that can distinguish between the two? (Middle and up)

What is going on here? (Everyone) What is another way to describe what is going on here? (Everyone) What is a third way to describe what is going on here? (Everyone)

The Speed of Light Postulate Imagine you fire a bullet over the cab of a truck that is moving at 20 m/s You, the observer in the truck, will measure the speed of the bullet to be 100 m/s Observer B, on the ground, will measure the speed of the bullet to be 120 m/s The speed of the bullet you measure plus the speed of the truck with respect to the ground Now picture you firing a photon off the back of the truck You measure the speed of light to be c The observer on the ground measures the speed of the photon to be c as well, not as c + the speed of the truck

The Idea of Space-Time An event occurs at a particular place at a particular time in an inertial frame (x, y, z, t) Thought Experiment You are standing in a moving train car. You send a light beam from the floor of the car to a mirror on the ceiling a certain height h above the floor. The light takes Δto to make the trip. If the light travels up and back, the total distance covered is 2h How far does the light travel when between when it is released and when it is detected? (Everyone) What is the definition of velocity? (Everyone) What would the definition of velocity look like if I applied my given information to it? (Middle and up) What do I get when I solve for time? (Everyone) h

What does an outside observer see? The observer outside the car sees the light take a time Δt to return to its starting point s h s L L Train car is moving to the right at a velocity v

From the Pythagorean Theorem If the time to complete the trip is Δt, then the time to complete half of the trip is Δt/2

where Δto = proper time interval, measured by an observer who is at rest with the event Δt = time interval measured by an observer who is in motion with respect to the events and sees beginning and ending of the event occurring in different places v = relative speed between the two observers c = speed of light in a vacuum

Time Dilation (Time Lengthening) Proper Time Interval Proper time interval between events is the time that is measured by the observer who is at rest with the event The other observer will observe a different time interval for the same events The difference between the two measured time intervals is based on the difference in velocities

Example 1 A spacecraft is moving past Earth at a constant speed v that is 92% speed of light. So, v = .92c. The astronaut on the spacecraft measures the time interval between “ticks” on a clock to be 1s. What does the earth bound observer measure the time interval between “ticks” to be? V = .92c Δt0 = 1s

Example 2 Alpha Centauri is 4.3 light years away. If a rocket leaves for Alpha Centauri at a speed of v = .95c relative to the earth, how long will the trip take according to the astronauts clocks? V = .95c d = 4.3 light years

Example 3 Time is measured differently between two different reference frames if they are moving with respect to each other. If you are in a car going 45 m/s and you measure ticks in 1 s intervals, what is the time that an outside observer would measure? What is the difference between Δto and Δt? V = 45 m/s Δt0 = 1 s

Length Contraction Going back Alpha Centauri, we know that, from Earth, we would observe the time it takes to get there to be 4.5 years moving at a speed of .95c. The astronauts on the ship observe that only 1.4 years have past The spaceship is moving at the same speed in all reference frames, therefore the distance must be different Using the below relationship and time dilation, you can derive the following

where Lo = proper length, measured by an observer who is at rest with respect to the two points L = length measured by an observer who is in motion with respect to the two points v = relative speed between the two observers c = speed of light in a vacuum

Note: Length contraction only occurs along the dimension of motion The dimensions that are perpendicular to the direction of motion are not shortened v = 0 m/s v = .866c v = .9999c

Example 1 Back to Alpha Centauri. We would measure the proper length of the distance between earth and Alpha Centauri to be 4.3 light-years. What is the distance that the astronauts measure during their trip? Remember the astronauts are traveling at a speed of .95c.

Example 2 A relativistic snake is moving at a speed of .82c. If you, the outside observer, measure the length of the snake to be 1.2 m long, how long does the snake measure himself to be?

Consequence of Time Dilation and Length Contraction Simultaneous events in one reference frame are not simultaneous in all others Picture a train car To the observer inside the train car, the car is struck simultaneously at both ends with lightning bolts To the observer outside the car, the back lightning bolt strikes just before the front one

Reference frame with respect to the observer inside the train car Reference frame with respect to the outside observer

Simultaneous events in one reference frame are not simultaneous in all others Picture a snake To the outside observer, a knife drops in front of and behind him at the same time From the perspective of the snake, what happens?

Reference frame with respect to the observer watching the motion Reference frame of the snake, who is at rest with himself

Relativistic Addition of Velocities Suppose you are in a truck moving at 15 m/s and you shoot a bullet off of the front of the truck with a speed of 100 m/s Normally, you would say that the bullet would be traveling at a speed of 115 m/s with respect to the ground Relativity says that this isn’t quite right When two objects are moving in the same direction, their velocities are related by the following equation

u = the speed of the object with respect to the reference frame that the event does not take place in u' = the speed of the object in the reference frame where the event takes place v = relative speed between reference frames c = speed of light in a vacuum

Example 1 Imagine a hypothetical situation in which a truck is approaching an observer on earth at a relative velocity of 80% the speed of light. A person riding in the back of the truck throws a baseball at 50% the speed of light relative to the truck towards the observer. At what velocity does the observer on earth see the ball approaching?

Example 2 During an intergalactic space battle, our heroes fire a laser beam at an enemy ship. The heroes ship is traveling at .7c when they fire the beam. How fast does the enemy ship see the impending death approach them?

Example 3 A tank that is being developed on earth can travel 10,000 m/s and fire a projectile at 20,000 m/s with respect to the tank. How fast would someone on the ground observe the projectile to be?

Relativistic Momentum Momentum, from earlier, is p = mv At speeds approaching the speed of light, momentum defined as such, was not conserved There must be a correction to classical momentum to account for the high speeds So, according to relativity, the momentum of an object is

Relativistic Energy Rest Energy When doing kinetic energy calculations with special relativity, Einstein realized that the total energy did not go to zero when v = 0 m/s He called this left-over energy the rest energy The rest energy of an object is given by his now famous equation

Implications of the rest energy Mass is a form of energy Mass can be created or destroyed during a reaction but the total energy remains the same Example A .046 kg golf ball is lying on the green. Find the rest energy of the golf ball. If 100% of the rest energy could be used to operate a 75 W bulb, how many years could the bulb stay on?

Example A .046 kg golf ball is lying on the green. Find the rest energy of the golf ball. If 100% of the rest energy could be used to operate a 75 W bulb, how many years could the bulb stay on?

Binding Energy The binding energy is the amount of energy needed to pull apart the nucleus of an atom Because energy and mass are equivalent, then if energy is added to the system the mass of the system must increase As a result, the mass of parts of the nucleus when separated is greater than the nucleus itself This change in mass is known as the mass defect This energy is the source of the energy that is generated through nuclear fusion and fission

Example 1 42He, the most abundant isotope of helium, has a mass of 6.6447 x 10-27 kg. If the mass of a proton is 1.6726 x 10-27 kg and the mass of a neutron is 1.6749 x 10-27 kg, find the mass defect and the binding energy.

Total Relativistic Energy If I wanted to find the total energy of an object, I would take its rest energy and add it to the object’s kinetic energy To find the kinetic energy, once again we need to know that the total energy is equal to the rest energy plus the kinetic Solving for K reveals the kinetic energy of an object

Consequences of Special Relativity Objects with mass cannot reach the speed of light As v approaches c, γ approaches infinity According to the work-energy theorem, an infinite kinetic energy can be supplied by an infinite amount of work

Particles that travel at the speed of light must be massless Photons move at the speed of light, therefore they must have no mass Anything that travels faster than light must have a mass given by the form This particle is called a tachyon (theoretical) Has an imaginary mass Travel backwards in time Breaks cause and effect Travel faster as it loses energy