QCEphysics

Special relativity

describe an example of natural phenomena that cannot be explained by Newtonian physics, such as the presence of muons in the atmosphere

Problems with Newtonian physics

• Light appeared to be a wave, but the medium for its propagation was undetectable
• The equations describing electricity and magnetism were inconsistent with Newton's descriptions of space and time

Muon decay

- A muon is an elementary particle similar to an electron but with greater mass and travels near the speed of light (>0.99c)

• Formed through the decay of pions - a subatomic particle produced in the atmosphere as a result of the collisions of cosmic ray protons with nitrogen and oxygen atoms

• Average life span - 2.2 microseconds and then decay into an electron and two neutrinos

- Physicists David Frisch and James Smith counted the number of muons detected at the top of Mount Washington and the bottom (time to reach bottom - 6.4 microseconds)

○ Top of mountain - 563 muons.

○ Bottom of mountain - 409 muons

○ Newtonian calculations predicted 27 per hour

- This confirmed Einstein's theory of relativity

○ To the observers, distance from mountaintop to sea level - 1910 (scientists' frame of reference)

○ For the muon, the distance was 183m (muon's frame of reference)

▪ Due to length contraction for objects travelling at relativistic speeds

○ Muon's lifetime changed from 2.2 microseconds to 23 microseconds

▪ Due to time dilation for objects travelling at relativistic speeds

define the terms frame of reference and inertial frame of reference

- Frame of reference is an arbitrary set of axes with reference to which the position or motion of something is described, or physical laws are formulated

• Relative motion (va relative to B = va rel stationary observer - vb rel stationary observer)

- Inertial frame of reference is a non-accelerating frame of reference in which Newton's laws of motion hold

• Are stationary or moving at a constant velocity
• e.g. at rest on Earth, in a car at constant velocity
• Although the surface of Earth undergoes centripetal acceleration, this is considered small enough to be disregarded

- Non-inertial frame of reference are accelerating

○ Accelerating upwards in an elevator

○ Going around a corner

recall the two postulates of special relativity

1. The laws of physics are the same in all inertial frames of reference
2. The speed of light has a constant value for all observers, regardless of their motion or the motion of the source

recall that motion can only be measured relative to an observer

- Motion can only be measured relative to an observer

• e.g. vball-Earth = vball-car + vcar-Earth

explain the concept of simultaneity

- Simultaneity is the relation between two events assumed to be happening at the same time in a frame of reference

- Relativity of simultaneity states that events that are simultaneous in one frame of reference are not necessarily simultaneous in another frame of reference, even if both frames are inertial

• e.g. if two school bells ring, and you are positioned directly between them, they will be heard simultaneously. However, if you are positioned closer to one of the bells, you will hear one before the other, meaning the event is not simultaneous

recall the consequences of the constant speed of light in a vacuum, e.g. time dilation and length contraction

- Time dilation and length contraction is as a result of the constant speed of light in a vacuum

• The motion of the object does not increase the speed of light, so with increased distance, the time taken increases, and the length travelled decreases (s= d/t)

The simplest way of resolving this
paradox is to change the frame of
reference. Instead of the boy imagining he
is running and the Earth is stationary, he
should consider himself at rest and that
the Earth is moving. He can then say that
he can see himself in the mirror because it
doesn’t matter how the Earth moves
under his feet.

define the terms time dilation, proper time interval, relativistic time interval length contraction proper length, relativistic length, rest mass and relativistic momentum describe the phenomena of time dilation and length contraction, including examples of experimental evidence of the phenomena

- First diagram has Observer 1 on a train watching a beam of light bounce from the floor to a mirror on the ceiling and back again

- Second diagram has Observer 2 watching the train go past. The path taken by the light is much greater.

- To the first observer, the light travels a distance of 2D in a time t o (up to the mirror and back to the source).

• This time is called the proper time interval - the time between two events measured by an observer at rest to the events (t0)

- To the second observer, the light has travelled a triangular path in time t (Figure C)

• As speed of light is the same, light has travelled a longer path from the viewpoint of the observer, so must have taken a longer time

▪ Called relativistic time interval which is the time between two events measured by an observer moving with respect to the events; also known as dilated time

- Time dilation is the difference in the time interval between two events as measured by observed moving with respect to each other

⭐ Look through mathematical proof for time dilation formula (pg 263)

Remember, relativistic speed is >0.1c

Length contraction

- Scenario

○ For man 2, the train is the same length as the platform

• The speed of light is constant for all observers. He sees the two lights turning on at the same time, as he is standing in the middle - the events are simultaneous

○ For man 1, the front and rear light turns on, however the train moves further forward

• The speed of light is constant for all observers. As the man is now closer to the front light, he will see the front light turning on first, and then the rear light
• For man 1, he believes that the front of the train passed the platform edge before the rear of the train passed the other edge

- Man 1 believes the train is longer than the platform, while Man 2 believes they are the same length

○ Length has been contracted for an observer watching it

• Proper length (L0) > Relativistic length (L)
• Length contraction is the shorter measurement made by an observer moving relative to the object in the direction of the length being measured

○ Man 1 sees proper length - length as measured by an observer at rest with respect to the object (L0)

○ Man 2 sees relativistic length - length as measured by an observer moving with respect to the object in the direction of motion (L)

- At the Stanford Linear Accelerator (SLAC), particles are accelerated to over 0.99999C over a distance of 3.2km, and for the particles the tube appears to be only 1 metre long

Momentum

- Rest mass is the mass of an object when measured in the same reference frame as the observer

• Mass does not change with speed

- Relativistic momentum is the momentum of an object as measured by an observer moving relative to the object

solve problems involving time dilations, length contraction and relativistic momentum

Time dilation

- How to determine which clock is t and t0?

• Ensure the two clocks must be synchronised

▪ Simplest way to have synchronised clocks at a distance is to synchronise them when they are together and then move them apart

• Proper time (t0) is measured by observer stationary event (requires only 1 clock)
• Relativistic time (t) is the time measured by observers watching the event moving clocks)
• Proper time > Relativistic time

- Rest lifetime - proper time

⭐ Length contraction only occurs in the direction of travel, not in any perpendicular direction.

i.e. for a person travelling at a relativistic speed, their length would contract, but their height and width would stay the same

recall the mass-energy equivalence relationship

- Mass-energy equivalence relationship relates change in mass to change in energy, given by ΔE = Δmc²

○ Mass and energy are two different expressions of the same thing

▪ As the value of c is very high, a very small mass = large amount of energy

▪ A large amount of mass can appear as a very small change in mass

○ Mass defect is the change in mass (in kg)

explain why no object can travel at the speed of light in a vacuum

- Objects with a mass cannot travel at the speed of light in a vacuum

• As velocity approaches the speed of light, the momentum approaches infinity
• The amount of effort required for each increment of velocity gets larger and larger, until the added effort produces no gain in speed

▪ Implication of a cosmic speed limit

explain paradoxical scenarios such as the twins' paradox, flashlights on a train and the ladder in the barn paradox.

Twin's Paradox

- There are two twins, A and B. B goes on a spaceship (at a relativistic speed) to the moon, and returns years later

• In the frame of reference of twin A, twin B has been moving very quickly, so will experience time more slowly (will measure less time). Therefore, twin B will be younger than twin A
• In the frame of reference of twin B, twin A has been moving very quickly, so will experience time more slowly. Therefore twin A will be younger than twin B

- Who is correct?

• Twin B is accelerating when he turns around to come back to Earth (therefore not an inertial frame of reference, and special relativity only applies for inertial frames of reference). Therefore Twin A's statement is correct, and Twin B will be younger than Twin A

Ladder Paradox

- Assume a ladder moves quickly towards a barn, with open doors at the front and back of the barn (at a relativistic speed). The ladder is longer than the length of the barn when both are at rest

• From the barn's perspective, the ladder experiences length contraction and can fit in the barn
• From the ladder's perspective, the barn experiences length contraction and the ladder can't fit

- What happens?

• The relativity of simultaneity needs to be considered. The two events are seeing the front of the ladder in the barn, and seeing the back of the ladder in the barn. These events are simultaneous for the barn, but not simultaneous for the ladder

• Therefore, the ladder fits in the barn

Flashlights on a train paradox

- Man A is on a train travelling at a relativistic speed and Man B is standing on a platform. In the middle of the train is a lightbulb, and on the ends are light-activated doors

• In the frame of reference of Man A, when the bulb switches on, the light travels at constant speed and takes the same time to travel to each door. Therefore, the doors will open simultaneously
• In the frame of reference of Man B, when the bulb switches on, the left door is closer to the light than the right door as the train continues to move forward. The left door opens before the right door.

- Which frame of reference is more useful?

• Both Man A and Man B are correct, it just depends on the frame of reference

Students should be able to define momentum and impulse, solve problems on momentum and impulse, recall Newton's laws of motion, and solve problems using Newton's laws of motion (Unit 2 Topic 1: Linear motion and force)

- Momentum is a vector quantity, being the product of an object's mass and velocity

• p = mv

- Impulse  is a vector quantity defined as the change in momentum of an object

• Δp = Ft

- The law of conservation of momentum states that the momentum before a collision is equal to the momentum after a collision