Relativity: Time, Length, and Momentum

Relativity of Time, Length, and Momentum

• Extends the exploration of Einstein's postulates and their impact on time and distance.
• Challenges common-sense notions of time and space.

Time Dilation

• Addresses the debate on whether time flows uniformly across different locations.
• Establishes that time is relative to the observer.
• Undermines the concept of absolute time.
• Uses a thought experiment with two observers measuring time intervals within a spaceship containing parallel mirrors to discuss Time Dilation.
• An astronaut inside the spaceship measures a time interval of 2∆ts for a light pulse to travel between the mirrors and back using a clock.
• An observer on Earth sees the spaceship moving at speed v, observing a longer time interval Atm for the same pulse to travel a greater distance.
• Light's speed is constant for both observers, according to Einstein's second postulate.
• Mathematical derivation, based on the Pythagorean theorem, is (cAtm)² = (vAtm)² + (cats)2 (see image).
• Leads to the equation Atm = Ats / √(1 - v²/c²), which relates the time intervals measured by the two observers.
• The time interval (2Atm) perceived by the Earth observer is greater than the interval (2∆t) seen by the astronaut, demonstrating time dilation.
• Emphasizes that time intervals are measured between the emission of light from the bottom mirror (tick) and its detection back at the mirror (tock).
• In the spaceship's frame, the tick and tock occur at the same position.
• From Earth's perspective, the events are separated by a distance of 2vAtm.
• The time interval between events occurring at the same position in a reference frame is shorter than in any other frame.
• "One position time" is less than "two position time" by a factor of √(1 - v²/c²).
• Proper time (∆t₀): Duration of a process measured by an observer who sees the beginning and end occurring at the same location.
  • Time runs slower on a clock moving relative to an observer compared to a stationary clock, known as time dilation.
• The expression √ 1/(1 - v²/c²) is given the symbol $Y$, so the equation Atm = Ats / √(1 - v²/c²) becomes Atm = yAts.
• In the equation At, can only be a real number if v < c, establishing c as the "speed limit".
• No material object can reach or exceed the speed of light.
• Atomic clocks flown around the world in 1971 validated time dilation effects by showing time differences compared to a stationary clock.
• Predicted loss for eastward journey: 40 ± 23 ns.
• Predicted loss for westward journey: 275 ± 21 ns.
• Actual loss on eastward journey: 59 ± 10 ns.
• Actual loss on westward journey: 273 ± 7 ns.
• Muons, unstable subnuclear particles, exhibit time dilation when accelerated to 0.9994c, living 30 times longer than expected.
• The mean life of the muons had increased by 30 times relative to Earth; the time elapsed relative to Earth was dilated.
• Passengers in a spaceship moving near light speed would experience a life expectancy of hundreds of years relative to Earth.
• However, their experience of time would remain normal.
• The effect from Earth's perspective is that the spaceship occupants are living life at a slower rate, extending their life cycle enormously.

Sample problem 1

• Pulse frequency of an astronaut remains constant at 72 beats/min during a voyage
• Determine the pulse beat relative to Earth when the ship's speed is at 0.10c and 0.90c
• Use the time dilation formulas to calculate the pulse frequency at each speed

The Twin Paradox

• Thought experiment: one twin travels to a star and back at near-light speed, while the other stays on Earth.
• Addresses whether the traveling twin ages less, and whether each twin perceives the other to be aging slower.
• Resolves the paradox by stating that special relativity applies to Earth
• Travelling twin changes inertial frame, making the situation asymmetrical.
• The twin in the spaceship returns younger.

Length Contraction

• Addresses that length, previously considered absolute, is perceived differently by different observers.
• Considers a spaceship traveling from planet A to B at speed v, with a distance Ls, relative to the planets.
• The spaceship captain measures proper-time duration At, and finds the distance to be Lm, using a single clock
• Observers on the planets measure, using synchronized clocks, a different measure Atm over two locations
• It follows that Lm = vAts and L₁ = vAtm, and subsequent rewriting that leads to Lm = Ls * √(1 - v²/c²)
  • Ls represents the actual distance between A and B.
  • Lm represents the distance seen in an inertial frame where A and B are in motion.
• Length contraction occurs in the direction of motion only.
• If a cylindrical spaceship is moving past Earth very fast, its diameter stays the same.
• The expression for calculating this change in length is known as Lorentz contraction or Lorentz-Einstein contraction.
• Occupants on the spaceship observe the distance between planets as being less than what Earth observers measure.
• Observers on the planets measure this length without moving relative to the planets
• Moving through this space, the spaceship measures the distance as Lm.

Sample problem 2

• A UFO heads towards the center of Earth at 0.500c, and passes an orbiting satellite at 3.28 × 103 km
• Find the altitude of the UFO as determined by its pilot
• The formula accounts for which lengths or durations are proper

Sample problem 3

• A spacecraft travels at 0.87c and observers on Earth measures it to be 48.0 m long
• Calculates proper length
• The proper length of the spaceship is calculated to be 97.4 m

Relativistic Momentum

• Addresses the need to consider relativistic momentum due to subatomic particles traveling at high speeds.
• Presents Einstein's formula p = mv / √(1 - v²/c²) for relativistic momentum.
• Defines m as the rest mass, measured by an observer at rest relative to the mass.
• Uses Newtonian physics to measure rest mass because inertial and gravitational masses are equivalent at low speeds
• Physicists primarily use rest mass because mass cannot be defined uniquely when an object is accelerated
• The nonrelativistic expression p = mv is valid for speeds much smaller than c.

Sample problem 4

• Finds the magnitude of relativistic momentum of a proton accelerated to 0.999 994c
• Compares it to the nonrelativistic momentum

Relativistic momentum

• Anything travelling faster than about 0.1 c can be called relativistic
• According to the equation, as v approaches c, the quantity √(1 - v²/c²) approaches 0
• Relativistic momentum becomes infinite at the speed of light, but photons have no mass and travel at the speed of light