Chapter 37 Special Relativity Relativity of time length

Chapter 37 Special Relativity (Relativity of time, length and Lorentz Transformation)

Before 1905 Albert Einstein was an unknown 25 -year-old clerk in the Swiss patent office. Newton’s laws ruled supreme in the past 200 years. Many people were of the opinions that anything worth discovering in Physics had been discovered.

Annus Mirabilis [The Extraordinary Year] Published several papers: 1. Brownian motion (settled the debate on the existence of atoms) 2. Photoelectric effect (kick-started quantum mechanics, eventually won the Nobel Prize for the work) 3. Special relativity (profound change in the foundation of physics) Ended the year with a paper with the equation E=mc 2, arguably the most famous equation in physics. The only other annus mirabilis in history: 1666 when Newton wrote Principia

The Special Theory of Relativity Special relativity is the study of motion. It led to astonishing discovery about the nature of space and time. It laid the foundation for an even more profound work of Einstein, the General Theory of Relativity. It will take him another 10 years to complete General Relativity. Einstein himself said that compared with General Relativity, Special Relativity was a child’s play.

Some Implications 1. Events that are simultaneous for one observer may not be simultaneous for another. 2. When two observers measure a time interval or a length, they may not get the same results. 3. Newton’s Laws and equations for kinetic energy needs to be revised. 4. The equivalence of energy and mass (E=mc 2) 5. Space and time are no longer considered separate entities, and is combined into four-dimensional spacetime. 6. Expansion or contraction of the universe. 7. The existence of exotic objects like black holes, worm holes.

What is time? You will learn why Einstein is regarded as a great genius. He discovered the most profound things about the universe by asking some of the most deceptively simple questions.

Some questions • What does a beam of light look like if you are traveling at the speed of light? In other words, what does a “stationary” beam of light look like. • If you are holding a mirror as you fly, what will you see in the mirror? Will you see your own face? We will discuss the answer to these questions later.

Some facts about light Light is an electromagnetic wave, described by Maxwell’s equation. According to Maxwell’s equation the speed of light is given by: This has been confirmed by experiment, but some puzzles remains. If light is a wave, what exactly is waving? Water waves travels on water, sound waves travels in the air, what is the medium of light? Ether? ? ? If so, how can we detect ether?

Can light goes faster? We just mentioned c=3× 108 m/s, can we somehow make light go faster?

Can light go faster?

Motion is Relative When you say you are driving at 60 mph, what exactly does that mean? It means your car is moving at 60 mph relative to the surface of the earth. If you are piloting a spaceship, suppose you are moving at 1000 km/s, what exactly does that mean? Relative to earth? What if you are far from earth? What if the earth is destroyed? Perhaps relative to space? ? ? But then what is space?

Motion is Relative Einstein believes there is no absolute space, and hence no absolute motion. It makes no sense to say one is moving at 100 m/s without saying “relative to what”. If that is true, then what does it mean to say the speed of light is c=3× 108 m/s? What is the speed relative to? ? ?

Einstein’s answer What is the speed c=3× 108 m/s relative to? Einstein says the speed is 3× 108 m/s relative to everything! [actually it is relative to any inertia frame, but more about that later] If that doesn’t sound strange to you, suppose someone tells you Einstein is driving a car on the freeway at 50 mph. The odd thing is, whether you are standing on the ground, driving behind him or driving toward him, you will still find his speed to be exactly 50 mph. How is that possible?

The constancy of c Einstein believed Maxwell’s equations dictated the constancy of the speed of light, and he was also influenced by philosophy that there is no absolute space. In fact, around the time Einstein was coming to this conclusion, Albert Michelson performed the famous Michelson-Morley experiment, experimentally proving that c=3× 108 m/s independent of your state of motion.

Inertial frame When you are accelerating in a car (when you are stepping on the gas or making a sharp turn), you will feel a force acting on you. Inertial frames are observers who do not experience such force. Example: An object under free fall in a gravitational field.

Two Postulates 1. The principle of relativity: All the laws of physics are the same in all inertial reference frames. 2. The constancy of the speed of light: The speed of light in vacuum has the same value in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light. From these two simple postulates Einstein constructed special relativity and started a revolution in physics.

So what is wrong here?

The problem with time Two events E 1, E 2: E 1: Sam jumps on the right side of the room E 2: Sally jumps on the left side of the room Somebody says: E 1 and E 2 happened at the same time. What is wrong with this statement?

The Relativity of Simultaneity Simultaneous for Sam but not for Sally. Einstein decided there was something wrong with our understanding of time.

The relativity of simultaneity

The relativity of simultaneity

Einstein’s light clock Since the speed of light is fixed, we can bounce a photon up and down to measure time. Sally put such a light clock in her spaceship. From her point of view:

What happens if the clock is moving? Sam is on the ground watching Sally and her clock flies by.

The situation Sally Sam

Sam’s view

Time dilation

Two points of view Sally Sam

Proper Time “Proper time” is the time interval between two events measured by an observer for whom the events are at the same spatial location. Two events: O: Emission of the photon O’: Returning of the photon To Sally, both events occurred at the same location, so the time interval measured by Sally is the proper time, Δt 0. To Sam, O’ occurred right of O because of the motion of the train, so the time interval he measures Δt is not the proper time. Sally Sam

The γ factor

What if I use a different clock? • The first postulate ensures that there cannot be any difference between a light clock and a “normal” clock, including your biological clock. • If your wrist watch and the light clock do not show the same time, by comparing a light clock with a wrist watch you could tell you are in absolute motion or not (when there really is no such thing). • Conclusion: Time really does slow down! The laws of physics are the same for observers in all inertial reference frames. No one frame is preferred over any other.

Time Dilation Verification – Muon Decays Muon decay in 2. 2μs when it is at rest (relative to the lab). But when it travel fast, it decays much slower (and hence travel much longer than expected). This is time dilation for fast moving muons.

Time Dilation Verification – Muon Decays Muons are unstable particles that have the same charge as an electron, but a mass 207 times more than an electron Muons have a half-life of ∆tp = 2. 2 µs when measured in a reference frame at rest with respect to them (a) Relative to an observer on the Earth, muons should have a lifetime of γ ∆tp (b) A CERN experiment measured lifetimes in agreement with the predictions of relativity

Example Calculate the γ factor for a muon traveling at 0. 9994 c. How much time has passed in the lab for a muon of lifetime 2. 2μs to decay?

Twins Paradox

GPS and Relativity • GPS (Global Positioning System) satellites are moving at 14000 km/h. Each satellite carries atomic clock within 30 ns precision. Fix your position to within 10 m. • Special relativity tells us time on the satellites slows down by 7μs per day. • General relativity (gravitational effect) tells us time on the satellite goes faster by 45μs per day. • Overall clocks on satellites gain 38μs per day, which is 38000 ns. • If not taken into account, GPS fails within minutes, generating an error of 10 km per day.

Calculations with Lightyear In this chapter, you will often encounter the unit of lightyear. A lightyear is defined as the distance light travels in a year. It is a unit of distance, not a unit of time. In a calculation in relativity, however, you will almost never convert lightyear into meters. You are almost always better off writing 1 ly=(1 c)(1 y). For example, how long does it take for a spaceship travelling at 0. 6 c to get to a galaxy 10 ly away (from earth’s view)?

Relativity of length Therefore, the platform is shorter by a factor of γ from the man’s point of view.

Length Contraction The measured distance between two points depends on the frame of reference of the observer The proper length, L 0, of an object is the length of the object measured by someone at rest relative to the object The length of an object measured in a reference frame that is moving with respect to the object is always less than the proper length This effect is known as length contraction

Length Contraction – Equation Length contraction takes place only along the direction of motion

Things shrink as they move

Trip to a Star Planet X is at a distance 10 ly away from Earth (as observed in the Earth’s frame). How long does it take for an astronaut traveling at 0. 95 c (relative to Earth) to get to X?

Trip to a Star Planet X is at a distance 10 ly away from Earth (as observed in the Earth’s frame). How fast does an astronaut need to travel (relative to Earth) to get to X in 1 year (from his view)?

Ladder & Garage Paradox Can the garage trap the ladder?

Ladder’s view

Garage with swing doors

Ladder’s view From the ladder’s point of view, the doors do not close simultaneously. The door on the right closes first.

Will the ladder fall into the hole?

Lorentz Transformation

Time Dilation Rederived (Skip)

Length Contraction Rederived (Skip)

Addition of Velocity (Skip) 3

The way to remember 1 1 2 3

1 The way to remember 2 2 3

Addition of Velocity A fly is moving at velocity vfly/car with respect to a car moving at velocity vcar/you relative to you. What is the velocity of the fly according to you? you car fly

Relative velocity of spacecraft The velocity of A and B are measured to be 0. 75 c and 0. 85 c relative to Earth. Find the velocity of B relative to A.

Relative velocity of spacecraft (Method 2) The velocity of A and B are measured to be 0. 75 c and 0. 85 c relative to Earth. Find the velocity of B relative to A. B E A

Example: Aliens Two civilizations are evolving on opposite sides of a galaxy, whose diameter is L =6 × 104 ly. At time t = 0 in the galaxy frame of reference, civilization A launches its first interstellar spacecraft. Civilization B launches its first spacecraft TB =5 × 104 years later. A being from a more advanced civilization C is traveling through the galaxy at 0. 99 c, on a line from A to B. Which civilization does C judge to have first achieved interstellar travel?

Solution