# Special Relativity SPH4U.

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Special Relativity SPH4U

Review of Scientific “Theories”
Recall discussion from the first day of class A scientific “theory” is a proposed explanation/description for observed facts It is possible for a theory to be a good approximation or have some usefulness even if it is not fully correct One of the best examples is “Newtonian” Physics vs. Relativity & Quantum Mechanics

Newtonian Physics Physics principles as explained by Newton and others
Newton’s 3 Laws and Law of Gravity Maxwell’s Equations of Electromagnetism Equations for motion, momentum, kinetic energy, etc. discussed earlier in this class Underlying foundations of space and time as absolute

Relativity and Quantum Mechanics
New physics as described by Einstein and others, most of the work done in the early 1900s Time dilation, length contraction Uncertainty principle Bohr Theory of the Atom Different fundamental assumptions about the Universe

The Special Theory of Relativity
Aimed to answer some burning questions: Could Maxwell’s equations for electricity and magnetism be reconciled with the laws of mechanics? Where was the aether?

The Conflict Newtonian physics seems to describe the world as we are used to it However, several experiments as well as some hypothetical arguments signaled some problems Relativity and Quantum Mechanics improve upon Newtonian physics

Newtonian Physics Newtonian physics accurately describes the Universe when… Speeds are not too large Gravity is not too strong You are at a macroscopic level, i.e. not dealing with individual molecules/atoms

Newtonian Physics, cont.
Under the conditions of the previous slide, there is no reason to use anything other than Newtonian physics Equations give the same results to high accuracy Example: Trajectories of satellites and space probes use Newtonian physics

Relativity Relativity is a set of physics concepts and laws deduced by primarily by Albert Einstein Special Relativity Published by Einstein in 1905 “Special” case with no forces/acceleration General Relativity Published by Einstein in 1915 Extension of previous theory to include forces

What ISN’T Relativity? Relativity does not simply mean “everything is relative” On the contrary, relativity says certain things are relative, and other things are absolute Relativity also tells us by how much those certain things are relative and in what way

Experimental and Theoretical Need for Relativity
Michelson-Morley Experiment Speed of light is the same regardless of the Earth’s motion through the aether (“absolute space”) Maxwell’s Equations of Electromagnetism Predict very unusual things, like magnetic fields with “loose ends”, when speeds are extremely large

Michelson-Morley Experiment
For a long time, scientists believed in an “aether”—absolute space In the Michelson-Morley experiment, the speed of light was measured “with” and “across” the “flow of the aether” as the Earth moved through it

Michelson-Morley Experiment
Flash Contrary to expectations, however, the speed of light was the same both “with” and “across”!

Theoretical Foundations of Relativity
To explain all of these things, Einstein came up with new laws of physics based on two assumptions The laws of physics are the same in all inertial (non-accelerating) frames The speed of light is the same as measured by all observers in all inertial frames Einstein took these principles “on faith” The principles and their implications have passed subsequent experimental testing

Relatively Speaking What do Einstein’s two assumptions imply?
All motion is relative Relativity of simultaneity Relativistic velocity addition Time dilation Length contraction Relativistic mass increase E = mc2

Who is moving?

All Motion is Relative, cont.
You and your friend Jackie like to travel in bizarre spherical spaceships Who is moving? Who is stationary?

All Motion is Relative, cont.
In spite of our everyday intuition, the only velocities that can be measured are relative velocities Examples: Relative to the surface of the Earth Relative to the Sun Relative to a distant galaxy

Galilean Relativity 1,000,000 ms-1 1,000,000 ms-1
How fast is Spaceship A approaching Spaceship B? Both Spaceships see the other approaching at 2,000,000 ms-1. This is Galilean or Classical Relativity. VT = V1 + V2 Mention the Ether

Einstein’s Special Relativity
0 ms-1 300,000,000 ms-1 Moving with respect to the Ether 1,000,000 ms-1 Both spacemen measure the speed of the approaching ray of light. How fast do they measure the speed of light to be?

Nothing Can Go Faster Than The Speed of Light

In normal circumstances, if you are moving and throw an object, an outside observer will see the object at a different velocity Straight-forward velocity addition But all observers measure the speed of light to be the same

Velocity Additions Do Not Apply to Light
Even if you are moving away from your friend at a very high velocity, you will both see a light beam moving at c. Nice to know formula

A formula for adding velocities exists, but it is not required for the course. The formula works such that you can never get velocities greater than c For small velocities, is approximately the same as just adding the velocities

Relativity of Simultaneity
Two lights an equal distance from M go off A passing train carries M’ M’ sees the light from B first M see the light flashes at the same time M’ is moving in the direction of B This relativity is determined by the speed of light and the relative motion of the objects/observers

Relativity of Simultaneity
Events which are simultaneous in one frame may not be in another! Each observer is correct in their own frame of reference

The Lorentz Factor Calculating length contraction, time dilation, and other quantities requires calculating the Lorentz factor  = v/c If v = 99% of c, then  = 0.99  is always < 1    1

The Lorentz Factor, cont.
Some examples: v = 0.1% of c   = v = 1% of c   = v = 10% of c   = 1.005 v = 50% of c   = 1.155 v = 90% of c   = 2.294 v = 99% of c   = 7.089 v = 99.9% of c   = 22.37

Time Dilation Distance = Velocity x Time A clock using light pulses to keep time. Every time the pulse returns, a unit of time has passed D Note: Consider a clock that uses a light pulse to tick . . . This time is known as Proper Time. Because the clock is rest the frame of the occurring event. The Proper Time interval between two events is always the time interval measured by an observer for whom the two events take place at the same position.

Time Dilation L=vDt ½ cDt D ½ vDt V We are now watching the clock move horizontally with velocity v. We will examine one cycle, more specifically one-half of one cycle. During a cycle of the light photon the clock will have moved horizontally a distance L, and if we calculate the distance travelled by the light in this one cycle (a upside down V), the distance would be c times the time we measured for the cycle, that is ct. So for one-half cycle the distance travelled by the light is ½ ct. Consider a clock that uses a light pulse to tick . . . Now since This says a moving clock run slow. If ts =1 then you watching it move would notice it taking more than 1s (tm >1) on your clock, so ts runs slow.

Time Dilation Example You and a friend are having a eating contest. Your friend is on a train traveling at speed v=0.9 c. By her watch, she finishes her food in 5 seconds. Determine the time you measure, if you are standing still at the train station. Since eating is happening on the train, that is the “proper” time, ts=5.

Time Dilation Example 2 Both people think they won!
Now it is your turn to eat. According to your watch you finish your food in 5 seconds. How long does your friend think it took you to finish the food? Now eating is happening at the station, so that is the “proper” time, again ts=5. Your friend would consider you to be moving. Remember the proper time is where the event and clock are together Both people think they won!

Space Travel Alpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri). How long would people on earth think it takes for a spaceship traveling v=0.95c to reach A.C.? How long do people on the ship think it takes? People on ship have ‘proper’ time since they see earth leave, and Alpha Centauri arrive. Dts Dts = 1.4 years

Space Travel Another approach that solves any special relativity problem by treating space and time as spacetime. The only requirement is that both separated units are recorded in the same units. (i.e.: light seconds, light minutes, light years, …)

Space Travel Alpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri). How long would people on earth think it takes for a spaceship traveling v=0.95c to reach A.C.? How long do people on the ship think it takes? An amazing technique is to place time and space in the same units then use the following relativistic formula:

Time Dilation Review Time flows more slowly in a moving frame as observed by an outside observer But remember motion is relative If you and I are moving past each other I see your clock moving more slowly But you also see mine moving more slowly…!!!

Length Contraction Objects moving relative to an outside observer appear contracted in the direction of their motion as measured by the observer

Length Contraction, cont.
If you and I move past each other in some sweet sports cars I measure your sports car as being shorter You measure my sports car as being shorter Only applies to the direction of motion We see our sports cars as still being the same height

Length Contraction v=0.1 c v=0.8 c v=0.95 c

Length Contraction Example
People on ship and on earth agree on relative velocity v = 0.95 c. But they disagree on the time (4.5 vs 1.4 years). What about the distance between the planets? Earth/Alpha: d0 = v t = .95 (3x108 m/s) (4.5 years) = 4x1016m (4.3 light years) Ship: d = v t = .95 (3x108 m/s) (1.4 years) = 1.25x1016m (1.3 light years) Length in moving frame Length in object’s rest frame

Twin Paradox Traveling twin will be younger!
Twins decide that one will travel to Alpha Centauri and back at 0.95c, while the other stays on earth. Compare their ages when they meet on earth. Earth twin thinks it takes 2 x 4.5 = 9 years Traveling twin thinks it takes 2 x 1.4 = 2.8 years Traveling twin will be younger! Note: Traveling twin is NOT in inertial frame!

Question You’re eating a burger at the interstellar café in outer space - your spaceship is parked outside. A speeder zooms by in an identical ship at half the speed of light. From your perspective, their ship looks: (1) longer than your ship (2) shorter than your ship (3) exactly the same as your ship Ls > LM In the speeder’s reference frame In your reference frame Always <1

Comparison: Time Dilation vs. Length Contraction
Dto = time in same reference frame as event i.e. if event is clock ticking, then Dto is in the reference frame of the clock (even if the clock is in a moving spaceship). Lo = length in same reference frame as object length of the object when you don’t think it’s moving. Time seems longer from “outside” Dt > Dto Length seems shorter from “outside” L0 > L

Relativistic Mass Increase
Einstein made two other surprising discoveries… Mass must increase with speed, as viewed by an outside observer Due to conservation of momentum There is “leftover” energy even when the object is at rest Due to conservation of energy E = mc2

Relativistic Mass Rest mass Rest mass Actually written

E = mc2 E = mc2 = m0c2 This E is the total energy of an object
When the object is at rest… v = 0  = 1 E = m0c2 (“rest mass energy”) The reason that energy can be released through fusion/fission

Total Energy Relativistic kinetic energy is the extra energy an object with mass has as a result of its motion: We can solve this for the Kinetic energy of an object:

Relativistic Momentum
Note: for v<<c p=mv Note: for v=c p=infinity Relativistic Energy Note: for v=0 E = mc2 Note: for v<<c E = mc2 + ½ mv2 Note: for v=c E = infinity (if m<> 0) Objects with mass can’t go faster than c!

Question Calculate the rest energy of an electron (m=9.1x10-31 kg) in joules. Calculate the electron’s Kinetic energy if it is moving at 0.98c.

Simultaneous? Simultaneous depends on frame!
A flash of light is emitted from the exact center of a box. Does the light reach all the sides at the same time? At Rest YES Moving NO Simultaneous depends on frame!

Simultaneous? Many times, questions are concerned with the determination of the spatial interval and/or the time interval between two events. In this case a useful technique is to subtract from each other the appropriate Lorentz contraction describing each event.

Three Other Effects 3 strange effects of special relativity
Lorentz Transformations Relativistic Doppler Effect Headlight Effect We will consider three consequences of a constant speed of light

Lorentz Transformations
But what happens if the beam is moving?

Lorentz Transformations
Light from the top of the bar has further to travel. It therefore takes longer to reach the eye. So, the bar appears bent. Weird!

Doppler Effect The pitch of the siren: The same applies to light!
Rises as the ambulance approaches Falls once the ambulance has passed. The same applies to light! Approaching objects appear blue (Blue-shift) Receding objects appear red (Red-shift) When the ambulance is approaching, the sound waves are bunched up. Likewise, once the ambulance has passed, the waves are spread out.

V Counter intuitive result Beam becomes focused. Same amount of light concentrated in a smaller area Torch appears brighter!

Warp Program used to visualise the three effects Notes for the demo.
Demonstrate Doppler shift. Demonstrate Lorentz Transformations – including being able to see the back of objects when they are in front of you! Demonstrate Headlight effect by turning it on/off.

Fun stuff Eiffel Tower Stonehenge

Summary

Understanding An observer has a pendulum that has a period of 3.00 seconds. His friend who happens to own a spaceship (with cool engines), zooms by the stationary pendulum. If the speedometer of the spaceship says 0.95c, what will the friend measure are the period of the pendulum? Since I am with the pendulum, my measured time is the Proper Time. This makes sense because a moving clock would run slower from my perspective. So the pendulum would have a period of 9.6s.

Understanding Vega is 25 light-years away Travel to Vega at 0.999c
The length would appear contracted to you About 1 light-year Make the trip in ~1 light-year (each way) as measured by you Earth would measure 25 years each way You would spend 2 years (your time) travelling and arrive 50 years in the future Earth time.

Understanding Strange but True!
You throw a photon (3x108 m/s). How fast do I think it goes when I am: Standing still Running 1.5x108 m/s towards Running 1.5x108 m/s away 3x108 m/s 3x108 m/s 3x108 m/s Strange but True!

Understanding A 1.0 m long object with a rest mass of 1.0 kg is moving at 0.90c. Find its relative length and mass Use length contraction formula: Mass increase formula:

Understanding For a 1.0 kg mass moving at 0.90c. Find the rest energy and kinetic energy of the object For rest energy, Use energy formula: For Kinetic energy, Use relativistic energy formula: Now: Therefore

Understanding A person’s pulse rate is 65 beats per minute.
If the person is on a spaceship moving at 0.10c, what is the pulse rate as measured by a person on Earth? What would the pulse rate be if the ship were moving at 0.999c? a) Use time dilation: a) Use time dilation: We need time for a heart beat

Understanding A muon at rest has an average lifespan of 2.20 x 10 -6 s
What will an observer on Earth measure as its lifespan if it travels at 0.990c? What distance would we observe it travel before disintegrating? What distance would it travel if relativistic effects were not taken into account? a) Time dilation: b) Distance formula c) Distance formula To Show consistency This is the distance the muon measures it travels before disintegrating

Understanding You measure the length of an object as 100m when it passes you at 0.90c. What is its length when at rest? Use length contraction formula:

Understanding As a rocket ship sweeps past the Earth with speed v, it sends out a light pulse ahead of it. How fast does the light pulse move according to the people sitting on the Earth?

Understanding A train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train? Units: We are given that tb-ta=0 and what we want to determine is tb’-ta’

Understanding A train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train? The negative sign reminds us that even a occurred after event b