Relativity

Historical Development 1600s Newton discovered his laws of mechanics Insert joke about apple Late 1800s Maxwell’s equations explained electromagnetism Early 1900s Relativity Quantum Mechanics

Types of Relativity Relativity – moving really fast Special relativity General relativity Observation from different reference frames

Relativity A situation is described from different points of view Describe the motion of the ball Section 27.1

Reference Frames reference frame - a set of coordinate axes Inertial reference frames move with constant velocity principle of Galilean relativity - the laws of motion are the same in all inertial frames Section 27.1

Interpretation by Ted and Alice Both agree the ball’s acceleration is g Both agree the ball’s horizontal acceleration is zero Both agree the only force on the ball is gravity and that Newton’s second law is obeyed Section 27.1

Galilean Relativity and Light speed of light is a constant speed of light is independent of the motion of both the source and the observer Classic vs. Maxwell

Special Relativity Einstein developed special relativity He suspected the speed of light constant Maxwell was correct He then worked out what that implies for all the other laws of physics Explained lots of things Section 27.2

Postulates of Special Relativity Laws of physics are the same in all inertial reference frames The speed of light in a vacuum is a constant, independent of the motion of the light source and all observers (non-Newtonian) Section 27.2

More About Light Light can move through vacuum Unlike any other wave A light wave carries its medium with it In the electric and magnetic fields The lack of a conventional medium was surprising and hard to reconcile with conventional intuition Section 27.2

Inertial Reference Frames Wut is this? inertial reference is one in which Newton’s First Law is true the motion of a particle is zero particle moves with a constant velocity Particle doesn’t accelerate Section 27.2

Earth as a Reference Frame Earth spins - all points on the Earth’s surface have a nonzero acceleration Earthnot in an inertial reference frame acceleration is small enough => ignored Conclusion: We can usually consider the Earth to be an inertial reference frame Section 27.2

Light Clock The two postulates lead to surprising results Imagine this weird clock Time needed for one “tick” is a round trip of light beam: 2ℓ / c Take light clock and put it on the railroad Section 27.3

Light Clock Rail cart moves with constant v From Ted’s reference frame, the light pulse travels up and down between the two mirrors From Alice’s reference frame, follows path of triangle We examine Section 27.3

Light Clock Section 27.3 Ted measures:Alice Measures: Why?

Moving Light Clock, cont. Alice sees the light pulse travel longer distance Postulate says light is same speed in all frames Alice sees the light take longer to travel between the mirrors In other words The moving clock runs slow Section 27.3

Moving Clocks Run Slow Alice’s measures a longer time than Ted Postulate 1 states that all the laws of physics must be the same in all inertial reference frames Therefore the result must hold for any clock Special relativity predicts that moving clocks run slow This effect is called time dilation For human kind speeds, the difference between Δt and Δt o is so small Section 27.3

Time Dilation When the speed is small compared to c, the factor is very close to 1 Section 27.3

Speeds Greater the c Faster than the speed of light – never observed Time dilation experimentally confirmed The result applies to all clocks, even biological ones Section 27.3

Proper Time The time interval Δt o is measured by the observer at rest relative to the clock This quantity is called the proper time Proper time is the shortest time Always Section 27.3

Proper Time Section 27.3 In the situation where we have more clocks the proper time is measured by the clock which is situated in the coordinate system from which we observe.

Proper Time Example 1 What speed must you move to age at a rate half someone who is standing still? Section 27.3

Proper Time Example 2 Astronaut travels into a star that is 5e17m away, at speed 0.95c. She is 25 years old when she leaves. How old will she be when she gets to the star? Section 27.3

Q:Is this for real? Time actually slows down?? Lol wut? Yes its for real. Each GPS satellite contains a very accurate clock The satellite clocks are moving in orbit, so they experience time dilation They run slow by about 7µs per day To accurately determine a position, the effect of time dilation must be accounted for Otherwise your position would be off by hundreds of feet Section 27.3

Concept: Simultaneity simultaneous events occur at the same time Ted is standing the middle of his railroad car He moves at a speed v relative to Alice Two lightning bolts strike the ends of the car and leave burn marks on the ground which indicate the location of the two ends of the car where the bolts strike Section 27.4

Simultaneity, cont. Did the two lightning bolts strike simultaneously? According to Alice She is midway between the burn marks The light pulses reach her at the same time She sees the bolts as simultaneous According to Ted The light pulse from at B struck before the bolt at A Since he is moving toward B The two bolts are not simultaneous in Ted’s view Section 27.4

Simultaneity, final Simultaneity is relative!!! different reference frames see things happening at different times!!! Allow 10 seconds for minds to blow… Different from Newton’s theory: time is absolute It is objective quantity It is the same for all observers All observers agree on the order of the events Section 27.4

Length Contraction Alice marks two points on the ground and measures length L o between them Ted travels in the x-direction at constant velocity v Reads his clock as he passes point A and point B He reads the proper time (shortest time) Section 27.5

Length Contraction, cont. Distance measured by Alice = L o = v Δt Distance measured by Ted = L = v Δt o Since Δt ≠ Δt o, L ≠ L o The difference is due to time dilation and The length measured by Ted is smaller than Alice’s length

Proper Length The proper length, L o, is the length measured by the observer at rest relative to the thing being measured Ted is at rest Alice moves with speed v Ted measures shorter length than Alice Moving things are shortened

Length Contraction Equation Length contraction is described by Small speeds – no effect observed Proper length is longest length Look at this graph Section 27.5

Proper Length and Time, Review Proper time is measured by an observer who is at rest relative to the clock used for the measurement Proper length is measured by an observer who is at rest relative to the object whose length is being measured Extremely experimentally confirmed Section 27.5

Length Contraction Example1 Section 27.5 Rhaego, a very fast horse, just moved past you so fast, but you were ready for him and you measured his length (from head to tail) to be 3m. Drogo, who is riding the horse, measures him and he concludes Rhaegos length is 3.7 meters. How fast is Drogo riding? Who is measuring proper length?

Length Contraction Example2 Section 27.5 You are sitting on the sun, and stare at the earth. Calculate the contraction of the earths diameter. Earth distance from the sun = 93 million miles and earth radius = 6,371 km.

Addition of Velocities Ted on a moving railroad car at constant speed v TA He throws an object with a speed relative to himself of v OT What is the velocity v OA of the ball relative to Alice? Alice is at rest on the ground Section 27.6

Newton’s Addition of Velocities Newton would say: v OA = v OT + v TA The velocity of the object relative to Alice = the velocity of the object relative to Ted + the velocity of Ted relative to Alice Inconsistent with postulates of special relativity For example, if the object’s speed relative to Ted is 0.9 c and the railroad car is moving at 0.9 c, then the object would be traveling at 1.8 c relative to Alice Newton’s theory predicts FTL which is wrong Section 27.6

Relativistic Addition of Velocities The result of special relativity for the addition of velocities is The velocities are: v OT – the velocity of an object relative to an observer v TA – the velocity of one observer relative to a second observer v OA – the velocity of the object relative to the second observer Section 27.6

Relativistic Addition of Velocities, cont. If v OT and v TA are very small compared to c ~ Newtonian addition This works for 10% c and below If the object’s speed relative to Ted is 0.9 c and the railroad car is moving at 0.9 c, then: Newtonian: 1.8 c relative to Alice Relativistic: c relative to Alice Experimentally confirmed Section 27.6

Relativistic Velocities and the Speed of Light In general: if object has a speed less than c for one observer -> its speed is less than c for all other observers If an object moves at the speed of light for one observer -> it moves at the speed of light for all observers Speed of light is “universal speed limit” Section 27.6

Quiz iClicker You’re on a spaceship going 0.8c. You throw a rock at an alien in front of you at speed 0.7c. What velocity does the rock hit the alien with? A) 0.87c B) 0.96c C) 0.79c D) 0.99c E) 1.03c

Relativistic Momentum We saw Δx and Δt can be different for observers in different inertial reference frames Should proper time or proper length be used? Einstein showed that you should use the proper time to calculate momentum Uses a clock traveling along with the particle The result from special relativity is Section 27.7

Relativistic Momentum, cont. Principle of conservation of momentum is obeyed if you use relativistic momentum Applies even for a particle moving at any speed When a particle’s speed is << c, the relativistic momentum becomes p = m o v which is Newton’s momentum Section 27.7 Einstein Newton

Newton’s vs. Relativistic Momentum As v approaches c, relativistic result is very different than Newton’s There is no limit to how large the momentum can be! Even is momentum is very large, the particle’s speed never reaches the speed of light Section 27.7

Rest Mass Rest mass is denoted by m o This is the mass measured by an observer who is standing still relative to the particle The best way to describe the mass of a particle is through its rest mass Section 27.8

Mass and Energy Relativistic effects need to be taken into account when dealing with energy at high speeds From special relativity and work-energy, For v << c, this gives KE ≈ ½ m v 2 which is the expression for kinetic energy from Newton’s results Section 27.9

Kinetic Energy and Speed For small velocities, KE is given by Newton’s results As v approaches c, the relativistic result has a different behavior than does Newton KE can be made very large, the particle’s speed never reaches the speed of light Section 27.9

Rest and Total Energies The initial energy, m o c 2, is a constant called the rest energy A particle will have this much energy even when it is at rest! Einstein’s most famous equation The total energy of the particle is the sum of the kinetic energies and the rest energy Section 27.9

Mass as Energy mass is a form of energy It is possible to convert an amount of energy (m o c 2 ) into a particle of mass m o It is possible to convert a particle of mass m o into an amount of energy (m o c 2 ) The principle of conservation of energy must include rest energy Section 27.9

Rest Energy Example Section 27.9 Biggest nuclear weapon detonated by mankind was 50 megatons of TNT ~ 210 PJ (Petajoules). Peta = 10^15. Imagine there was a way to convert all mass into pure energy. How much energy does your smart phone have? Assume it weighs ~ 0.1 kg an is at rest.

Conservation Principles Conservation of mass Newton’s mechanics - Mass is conserved The total mass of a closed system cannot change Special relativity - Mass is not conserved Conservation of energy must be extended to include mass Momentum is conserved in collisions Use relativistic expression for momentum Electric charge is conserved It is possible to create or annihilate charges as long as the total charge does not change Section 27.9

General Relativity A noninertial reference frame - nonzero acceleration Physics in noninertial frames - general relativity Equivalence principle The equivalence principle states the effects of a uniform gravitational field are identical to motion with constant acceleration Section 27.10

Equivalence Example Ted stands in an elevator at rest (A) He feels the normal force exerted by the floor on his feet He concludes that he is in a gravitational field The elevator is not in a gravitational field and has an acceleration of g (B) Since there is an acceleration, Ted feels the same force on his feet

Equivalence Principle, cont. No way for Ted to tell the difference between the gravitational field and accelerated motion The equivalence principle has the following consequences Inertial mass and gravitational mass are equivalent Light can be deflected by gravity Experiments in 1919 verified light passing near the Sun during an eclipse was deflected by the predicted amount Section 27.10

Black Holes Black holes contain so much mass that light is not able to escape from their gravitational attraction A black hole can be “seen” by its effect on the motion of nearby objects Stars near a black hole move in curved trajectories and so the mass and location of the black hole can be determined Section 27.10

Gravitational Lensing If the black hole is between the star and the Earth, light from the star can pass by either side of the black hole and still be bent by gravity and reach the Earth The black hole acts as a gravitational lens Light from a single star can produce multiple images Analysis of the images can be used to deduce the mass of the black hole Section 27.10

Relativity and Electromagnetism Alice is at rest with the charged line and the point charge Ted sees the line of charge and the point charge in motion The moving charged line acts as a current Section 27.11

Relativity and EM, cont. Ted sees an electric force and a magnetic force on the particle Alice says there is only an electric force Both are correct They will agree on the total force acting on the particle Maxwell’s equations were already consistent with special relativity Section 27.11

Importance of Relativity Mass is not conserved Instead we can only think of energy conservation The three conservation principles in physics are now Conservation of energy Conservation of momentum Conservation of charge All the laws of physics must obey these three Section 27.12

Importance of Relativity, cont. Relativity changes our notion of space and time Time and position are two primary quantities in physics It is not possible to give precise definitions of these quantities Our everyday intuition fails Must remember this when dealing with very fast moving physics Section 27.12

Importance of Relativity, final Relativity plays key role in understanding formation of universe Black holes can’t be understood without relativity Newton’s theory was just approximation Only works very well in some cases (slow) Must understand its limits to know when it cannot be used Section 27.12