Chapter 27: Relativity

Special Relativity

Einstein’s theory that relates space and time, consisting of two main assumptions.

Time Dilation

The slowing of clocks that are moving relative to an observer’s frame of reference.

Length Contraction

The decrease in the linear dimension aligned with the direction of motion of an object moving relative to an observer in a different frame of reference.

Lorentz Factor (γ)

A factor that occurs frequently in relativity equations, defined as γ = 1 / √(1 - v²/c²).

Rest-Mass Energy (E₀)

The energy of an object when it is not moving; the energy equivalent of its mass if that mass were entirely converted into energy.

Assumptions of Special Relativity

• All the laws of physics have the same mathematical form in all inertial reference frames.
• The speed of light in a vacuum is the same in all reference frames.
• The first postulate extends Galilean relativity to include thermodynamics, electromagnetism, and optics.
• No physics experiment can distinguish between a reference frame moving at a constant rate and one that is stationary.
• The relative motion of the light source and the observer does not affect the observer’s measurement of light’s speed.

Lorentz Transformations

• Consider two light sources moving relative to each other.
• Source 1: S(x, y, z, t)
• Source 2: S′(x′, y′, z′, t′)
• According to the Galilean transformation: x′ = x + (–v)Δt, y′ = y, z′ = z, t′ = t
• The new (Lorentz) transformation is as follows (v = speed of object measured within its frame of reference):
• x′ = (x - vt) / √(1 - v²/c²)
• y′ = y
• z′ = z
• t′ = (t - vx/c²) / √(1 - v²/c²)

Predictions of Special Relativity

• Stationary and moving are relative to the observer, who always considers his own frame of reference to be the stationary one.
• There are three key predictions: Simultaneity, Time Dilation, and Length Contraction.
• Simultaneity: Events that appear to be simultaneous to one observer will not be simultaneous to the other.
• Time Dilation: A clock running at a relativistic speed with respect to a stationary observer will appear to be running slow.
• Length Contraction: The length of an object measured by an observer moving with the object is its proper length (l₀).
• The length l when measured between frames of reference moving relative to each other is less than l₀.

Example: Determining Elapsed Time

Two astronauts are in separate spacecraft as one conducts an experiment that he says takes 275 s. If the two spacecraft are moving at 0.850c relative to each other, what elapsed time does the other astronaut record?

Δt′ = 275 s

v = 0.850c

Δt = Δt′ / √(1 - v²/c²)

Δt = 275 s / √(1 - (0.850c/c)²)

Δt = 275 s / √(1 - 0.7225)

Δt = 275 s / √(0.2775)

Δt = 275 s / 0.5267

Δt = 522 s

Example: Determining Length

Assume that both spacecraft from the previous example have a proper length of 96.7 m. What is the observed length of one craft according to the other astronaut?

l₀ = 96.7 m

v = 0.850c

l = l₀ √(1 - v²/c²)

l = 96.7 m √(1 - (0.850c/c)²)

l = 96.7 m √(1 - 0.7225)

l = 96.7 m √(0.2775)

l = 96.7 m × 0.5267

l = 50.9 m

Questions for Students

1. Define special relativity and its two main assumptions.
2. What is time dilation and how does it occur?
3. Explain the concept of length contraction and provide an example.
4. What is the Lorentz factor (γ) and how is it used in relativity equations?
5. Describe the relationship between rest-mass energy and mass-energy equivalence.