Special Relativity: Note 1

Introduction to Modern Physics & Frames of Reference From Classical to Modern Physics By the end of the 19th century (1800’s), physicists thought they had accomplished almost all understanding of the physical world. • Newton’s laws described motion of objects on Earth and in the Heavens -Everything from the motion to a thrown ball to the orbit of the moon could be explained. -Electromagnetism was well understood. Light was mostly considered to be an electromagnetic wave. -Thermodynamics was fully described by its own laws. • Remaining puzzles: Structure of the atom & the orbit of Mercury. -Many scientists still regarded the atom as a solid “billiard ball” or “raisin bun.” -Mercury’s orbit slowly perturbed every rotation. -At this time, two revolutionary ideas began to surface to solve these problems: 1. Quantum Mechanics. 2. Relativity. Frames of Reference -An arbitrary origin from which to view motion. 1. Inertial Frames The Law of Inertia holds in any inertial frame. That is “if no net force acts on an object at rest, it remains at rest, or if in motion, it continues to move in a straight line at constant speed.” Eg. Constant velocity of a train, car, boat, space ship. A house, cat, etc. at rest. 2. Accelerating Frames of Reference: An accelerating frame of reference is a non-inertial frame. That is, the laws of Newtonian Mechanics DO NOT apply! Eg. Accelerometer. This is like some fuzzy dice hanging on a rear view mirror. Thought Experiment: Say the classroom was a space ship with no windows. How could we tell if it was at rest or moving at constant velocity? Due to different frames of reference, any observer can always assume that they are at rest, while everything else is in motion relative to them. This allows any observer to only be able to absolutely measure two speeds: 1. 2.

Special Relativity: Note 2

Postulates of Relativity The Special Theory of Relativity Consider the speed of light: A ball is thrown forward in a moving train. An expression to describe the velocity of the ball relative to the ground would be: However, this relationship of adding relative speeds does NOT hold with light. Light emitted from a flashlight in a moving train will have the speed of: The speed of light in a vacuum is constant regardless of the ______________________! This leads to profound changes in the way we understand time. Consider the wave properties of light: At the turn of the century light was considered to be an ____________________ wave. It was assumed that light, like all other known waves would require a _____________ to travel through. The proposed medium was called ETHER: i) it was at absolute rest ii) it permeated all space iii) it had no mass, and therefore no density Michelson and Morley proposed an experiment that was to detect the “ether wind” caused by the earth’s rotation through the “either” in space. They built an interferometer which would compare how light was effected on an east-west trajectory with light on a north-south trajectory. The experiment yielded a NULL result! This meant that the “ether wind” was not detectable, and more importantly, the speed of light was constant regardless of its direction of travel through the “ether wind.” Proposing an answer to this big puzzle: In 1905, Albert Einstein (1879-1955) proposed the Special Theory of Relativity. He concluded that the “ether” (and hence the absolute reference frame) did NOT exist. Relativity is based on two postulates: 1. All laws of physics are valid in all inertial frames of reference. 2. The speed of light, denoted as c, is equal to 2.99792458 x 108 m/s (exactly), regardless of the speed of the source or of the observer. *c stands for the Latin word “celeritas” which means “swiftness” In grade 12 physics, we can round c to __________________ m/s These two results lead to strange results involving time, mass, and length.

Special Relativity: Note 2 Relative Velocities With Regard To The Speed of Light Due to the second postulate above (each observer will always measure the speed of light to be 3.0x108 m/s), this makes inference to the concept that no physical object may travel faster than the speed of light. *Note: Not being able to travel faster than the speed of light is not an actual physical law, it is a consequence of the second postulate. But what if two spaceships are travelling toward each other at speeds that approach c? Would they not perceive the other ship to be going faster? No! This is compensated through the effects of special relativity. The following formula will allow us to calculate perceived velocities: v1 + v2 where vp is the perceived velocity between the observers in m/s vp = v1v2 v1 is the velocity of the first object in m/s 1+ 2 c v2 is the velocity of the second object in m/s c is the speed of light in m/s Unit Analysis: Ex. Two spaceships, A and B, are heading in opposite directions. An observer on Earth measures the speed of A to be 0.750c and the speed of B to be 0.850c. Find the velocity of ship B as observed by the crew on ship A. Ex. The speed of a motorcycle relative to a stationary observer is 0.80c. The speed of a ball flying in front of the motorcycle is 0.70c (relative to the motorcycle). Therefore, the speed of the ball relative to a stationary observer is:

Special Relativity: Note 3

Simultaneity & Time Dilation Simultaneity Two events that are simultaneous to one observer are not necessarily simultaneous to a second observer, moving relative to the first. The occurrence of two or more events is a ________ ____________. If an observer sees two events occurring at the same time, it does not necessarily mean that they _____________ ____ _____ __________ __________! Einstein’s Gedunken Experiment: Imagine riding in a train that can travel near the speed of light (vo). While riding, two bolts strike the train at points A and B and leave marks on the boxcar and ground. There are two observers, one on the train (male – M) equidistant from both ends of the car, and one on the ground also at equidistant (female – F). The lightning marks on the boxcar are labeled A’ and B’, while the ones on the ground are A and B. The female has noticed that the lightning has traveled at the same speed over equal distances, so concludes that A and B happened simultaneously. By the time these events have reached the female, the male has moved over some distance to the right, thus the signal from B’ has swept past the male, but the signal from A has not reached him yet. He knows that light must always travel at the same speed so then he concludes that lightning struck B before A.

Special Relativity: Note 3 Time Dilation Consider Bob standing in a box of height H. As he stands in the box he shines a beam of light from the floor, it reflects off the top and comes back down. Now lets say the box begins to move to the right with velocity, v. Alice is standing on the ground (at rest) and watches it go by. She also sees the beam of light reflect off the roof: The beam of light makes the shape of a ________________ according to Alice. If the lengths of the shape are represented by velocities, then we can make an expression to relate the time that Bob observes and the time that Alice observes. If Bob experiences a vertical line (velocity) in Alice’s’ frame of reference, it can be represented as: And putting this all together:

Special Relativity: Note 3 In General: where: t is the dilated time (in s) to t= 2 to is the rest time (in s) 1− v 2 c v is the object’s velocity (in m/s) c is the speed of light (in m/s) Unit Analysis: To a stationary observer, moving clocks appear to run slow. For example, if there was one clock on the spaceship and one clock on the earth, the interval between “ticks” on the clock in the spaceship would occur much more slowly than the interval between “ticks” on the clock on Earth. Physically, an object moving near the speed of light experiences LESS time in order to keep Einstein’s second postulate true. We would see the spaceship go by really quickly, but the motions of everything on the spaceship would look like they were in slow motion. Ex.

The period of a pendulum (as measured by a stationary observer) is 3.0 s. What 8 is the period measured by an observer moving at a speed of 2.8x10 m/s relative to the pendulum?

Explain what this result means:

Special Relativity: Note 3 Experimental Evidence of Special Relativity and Time Dilation



Muons:

Unstable elementary particles that have a charge equal to that of an electron and a mass 207 times larger. They are produced by the collision of cosmic radiation with atoms high in the atmosphere.

Muons have a rest time of 2.200 µs, which is measured in a reference frame that is at rest or moving slowly. After this time, the muon will decay into other subatomic particles. If we take 2.200 µs as the average lifetime of a muon and assume that its speed is close to the speed of light, we find that these particles travel about 600 m before they decay. Hence, they cannot reach the Earth from the atmosphere (about 4-5 km up) where they are produced. However, experiments show that a large number of muons do reach the Earth. How is this possible? The phenomenon of time dilation explains this effect. Relative to an observer on Earth, the muons have a lifetime equal to: to t= where to = 2.200µs v2 1− 2 c For example, if a muon was traveling at 0.99c, we would calculate the dilated time to be 16 µs. This means that the average distance traveled by a muon is 4 800 m, which is more than enough distance to reach the ground. €

Flying Clocks

In 1972, time intervals measured with four cesium atomic clocks in jet flight were compared with time intervals measured by Earth-based reference atomic clocks. The results were in good agreement with the predictions of special relativity and can be explained in terms of the relative motion between the jet and the Earth. It was measured that the clocks gained (273 +/- 7) ns during a westward trip and lost (59 +/- 10) ns during an eastward trip.

Special Relativity: Note 4

Length Contraction & Mass Increase Length Contraction In a similar derivation manner to time dilation, we can examine the effects of high speeds on length and mass of an object. Length contraction: L is the contracted length in metres v 2 -where: L = L 1− o 2 Lo is the rest length in metres c v is the velocity of the object in m/s c is the speed of light in m/s Unit Analysis: € Eg. An observer on Earth measures the length of a spacecraft travelling at a speed of 0.700c to be 78.0 m long. Determine the proper (rest) length of the spacecraft. What does this mean? Special Note: Length Contraction Length contraction only occurs in the ______________ of _________________. Whatever axis is parallel to the direction that the object is travelling in; it will become contracted. Say, for instance, a spaceship watched Earth pass by its windows at 0.75c. Instead of being a sphere, Earth would look more like an oval with its horizontal axis being contracted while its vertical axis remaining unaffected. Mass Increase Mass increase: m = mo -where: m is the increased mass in kg 2 mo is the rest mass in kg v 1− v is the velocity of the object in m/s c2 c is the speed of light in m/s Unit Analysis: € Eg. At the Large Hadron Collider at CERN in Switzerland, protons (mp = 1.67x10-27 kg) are accelerated up to speeds of 0.995c. What is their apparent mass at this speed?

Special Relativity: Note 4 What does this mean? Special Note: Mass Increase When massive objects approach the speed of light, it is their inertial mass that is increased as they speed up. If we were to weigh these fast objects on a scale, it would not appear that their gravitational mass is any heavier, however, they appear to act as if they are heavier. Consider the Following: The Twins Paradox Say that twins are born on Earth (Alice and Bob). Alice is placed on a spaceship and Bob remains on Earth. Alice blasts off, travels to our nearest star (Alpha Centauri) and comes back. According to Bob, about 20 years have passed before the spaceship has returned, so he is now 20 years old. Alice, however, was travelling near the speed of light, so time would have ticked at a slower pace for her. When she returns, she is only 12 years old. Is this possible? What would Alice observe from her frame of reference? Could she assume that she is 20 years old and that Bob is 12 years old? Who is right? What are the difference(s) (if any) between the two frames of reference? Who is really 20 years old? Who is really 12 years old?

Special Relativity: Note 5

Energy-Mass Interconvertibility Recall the formula for mass increase:

m=

mo v2 1− 2 c



Using the equations of special relativity, Einstein concluded that the total energy for any object with rest mass m moving at speed v is equal to: mc 2 E total = v2 1− 2 c In a special case, where the velocity of the object is 0 m/s, this equation reduces to: € This is Einstein’s famous equation. There are two conclusions from this formula: 1. Rest mass is a form of energy that is associated with all massive objects 2. There might be forces or interactions in nature capable of converting mass into energy and vice versa.



Examples of Energy-Mass Conversion: • E is released in fission of U238 • πo meson decay: All mass disappears and a photon of light is emitted • Fusion (in sun): Huge amount of energy released is accompanied by a corresponding loss of mass, where Δm = E/c2. The Sun loses mass at a rate of 1.98 x 107 kg/s Eg. A nuclear bomb uses 54.7 kg of reactant and it is observed that 54.5 kg of product are collected. How much energy was produced in the nuclear explosion? Eg. If coal is used in a power plant, it contains 3.2x107 J/kg. According to Einstein, if we converted part of coal’s mass to this amount of energy, how much would we need?

Special Relativity: Note 6

Additional Gedunken Experiments (General Relativity) Ant on a Record Player Imagine an ant on a record player, as shown in the following diagram: B B B A A A The ant is at point A and wants to travel to B. The record player begins to spin, eventually speeding up to 0.99c (length dilation begins to occur). On this turntable, the outer edge is spinning at a faster rate than the inner part, thus more dilation occurs at the outer edge. The ant considers two paths between A and B (draw the paths). Knowing that dilation is occurring, the ant uses metre sticks to measure the distances of the two paths: Path 1 Distance: Path 2 Distance: Notice which path is longer. Here, Einstein is beginning to predict the shape of spacetime, where the shortest distance between two points is NOT NECESSARILY a straight line! Wernher von Braun’s Space Station During the space race in the 1950’s, von Braun created a design for a space station that appeared as a giant wheel. As the wheel spins, the centripetal force (at the outer edge) would act like gravity, so humans could walk around. Consider this space station as the spinning speed continues to increase: As the speed approaches 0.99c, what happens with respect to time and force? Time: Force:

Special Relativity: Note 6 There is some type of connection between the increase of (gravitational) force and the slowing of time. Could this mean that as gravitational force increases, time slows down? YES IT DOES! If we stand on the surface of Jupiter, time moves more slowly than it does on Earth (because it has a stronger gravitational field). Einstein also showed this in his curved space-time theory. This is the essence of relativity: Special Relativity: General Relativity: Space-Time (General Relativity) Space-time is the current definition of gravity. Gravity is not some magical force that every mass has (and is able to attract other masses with). Any massive object will create warps and dips in the space-time fabric. Things with more mass have greater influence compared with smaller objects, as seen below: As these objects warp the fabric of space-time, it influences how other objects act as they travel near these large bodies. It even influences the path that light travels! Gravity is best described as an INTERACTION between object, as opposed to an actual force pulling two things together.

Special Relativity: Note 7

Gravity Equivalency: What Keeps Us Stuck To the Earth? Watch the following: https://www.youtube.com/watch?v=lXG-yoUsVS8 We are presented with two scenarios: Bob Alice Who is right? As it turns out, they are both equivalent! But the Earth is not expanding! Imagine Alice in a car at rest. When she steps on the accelerator, it moves forward and she feels a force into the seat. Now, lets say she is lying on the ground. She feels a force where she is sucked into the ground. Is this not the same? Is the ground accelerating up like the car is accelerating forward? Picture Alice in the middle of deep space with a jetpack and holding a long, flexible rod. At Rest: Jetpacks On: Notice what happens to the rod when the jetpacks turn on. What does the rod look like when you are standing on the ground? Does it do the same thing? Lets look at the following situations, and describe what is happening according to Bob (Newton’s Ideas) and Alice (Einstein’s Ideas): 1.Dropping an Apple 2. Holding an Apple 3. Pogo Stick Graphical Evidence Lets create distance-time of Alice falling off a ladder while Bob is holding the latter from the bottom. We will create two graphs, one for Alice falling down (Newton), and a second for Alice staying still and the ground moving up (Einstein). Newton:

Special Relativity: Note 7 Einstein wants to re-create the same scenario, but with Alice remaining at rest and Bob accelerating up. What can be done to the graph to show this? Einstein: This graph does the trick! Alice’s line is straight (not moving) while Bob’s line is curved (accelerating). The added feature here is that Bob has no motion along the y-axis. He remains at a constant point in space. This means he is accelerating without moving! This may seem weird but it is similar to centripetal motion. In centripetal motion, we are always accelerating, but not moving closer to the centre of the circle. Again, these situations are equivalent! The only difference is something that Einstein’s graph predicts, while Newton’s doesn’t. Einstein’s shows that the time that elapses at the top of the latter is different compared to the bottom of the ladder. Newton’s does not predict this at all. So which one is right? How can we test this? GPS The GPS satellite system depends on time in order to allow it to accurately determine someone’s location on Earth. These satellites are very high above the Earth’s surface (similar to the top of the latter). If we use Newton’s reasoning for the physics in the satellites, they begin to fail within two minutes of being activated! BUT, if the account for some time dilation occurring (due to being far from Earth’s gravitational field) they work perfectly! Einstein was correct! This means that Newton’s theory has failed, and Einstein’s is actually the correct theory. In reality, the ground does accelerate, and objects DO NOT fall down! The Earth’s surface is accelerating up without moving!