Unit 5: Relativity

Classical Relativity (Frames of Reference, Galilean Invariance)

Frame of Reference

A coordinate system used to measure positions and times of events. As seen in Unit 1, an Inertial Frame is one that is not accelerating (at rest or moving at constant velocity).

Galilean Transformations

This is the "common sense" set of equations for transforming measurements from one inertial frame (S) to another frame (S') that is moving with a constant velocity `v` along the x-axis relative to S.

• x' = x - vt
• y' = y
• z' = z
• t' = t

The last equation, t' = t, is the crucial assumption of classical physics: Time is absolute. Everyone measures the same time interval between two events.

Galilean Invariance

Galilean Invariance (or the Principle of Classical Relativity) states that the laws of mechanics are the same (invariant) in all inertial frames.
If you are in a smooth-moving train (inertial frame), you can play catch, and the ball will behave exactly as it would on the ground. You cannot perform any mechanical experiment *inside* the train to tell if you are moving or at rest.
Problem: Maxwell's equations of electromagnetism (which predict the speed of light `c`) are *not* invariant under Galilean transformations. This implies the laws of E&M are *not* the same in all inertial frames, or that the Galilean transformations are wrong.

Michelson-Morley Experiment (1887)

The "Luminiferous Ether" Hypothesis

In the 19th century, light was known to be a wave. Scientists assumed it *must* travel through a medium, just as sound waves travel through air. They called this invisible, all-pervading medium the "luminiferous ether".

• This ether was assumed to be stationary, forming an "absolute rest frame" in the universe.
• The Earth, in its orbit, must be moving *through* this ether at some speed `v`.
• This should create an "ether wind" (like the wind you feel on a bike).
• The measured speed of light `c` should be `c-v` (upstream) and `c+v` (downstream).

The Experiment

Michelson and Morley used a very sensitive interferometer to detect this tiny difference in the speed of light.

1. A light beam is split in two.
2. Beam 1 travels along an arm (L₁) parallel to the ether wind, and back.
3. Beam 2 travels along an arm (L₂) perpendicular to the ether wind, and back.
4. The two beams are recombined. The difference in their travel times should create a shift in the interference (fringe) pattern.
5. They then rotated the entire apparatus by 90°. The roles of the arms would swap, and a shift in the fringe pattern was expected.

Outcome (The "Null Result")

No fringe shift was ever observed.

The experiment was repeated at different times of day and year. The result was always null (zero).

Conclusion: The speed of light is constant in all directions, regardless of the motion of the observer or the source. The idea of the luminiferous ether was wrong.

Postulates of Special Theory of Relativity (STR)

In 1905, Albert Einstein proposed a new theory based on two simple, radical postulates to explain the Michelson-Morley result and the E&M problem.

Postulate 1: The Principle of Relativity

"The laws of physics are the same (invariant) in all inertial frames of reference."

This extends Galilean relativity to *all* laws of physics, including electromagnetism. There is no "absolute rest frame"; all inertial frames are equivalent.

Postulate 2: The Constancy of the Speed of Light

"The speed of light in a vacuum (c) has the same value for all inertial observers, regardless of the motion of the source or the observer."

This directly accepts the Michelson-Morley result. It is a radical break from "common sense" and Galilean transformations.

Consequence: To make these postulates work together, the classical idea of absolute time (t' = t) must be abandoned. Time and space are now relative and interwoven into "spacetime".

Lorentz Transformations

These are the *new* set of transformation equations that replace the Galilean ones. They are derived directly from Einstein's two postulates. They preserve the laws of E&M and keep `c` constant.

For frame S' moving at velocity `v` along the +x axis relative to frame S:

x' = γ (x - vt)
y' = y
z' = z
t' = γ (t - vx/c²)

Where γ (gamma) is the Lorentz factor:

γ = 1 / √(1 - v²/c²)

Inverse Transformations (from S' to S): Just swap (x, t) with (x', t') and change `v` to `-v`.

x = γ (x' + vt')

t = γ (t' + vx'/c²)

Simultaneity and Order of Events

Relativity of Simultaneity: This is a direct consequence of the `t'` equation: `t' = γ (t - vx/c²)`.

Consider two events (A and B) that happen at the same time `t` in frame S, but at different locations `x_A` and `x_B`.

In frame S', the times of the events are:

t'_A = γ (t - vx_A/c²)

t'_B = γ (t - vx_B/c²)

The time difference in S' is: Δt' = t'_B - t'_A = -γv(x_B - x_A)/c²

Since `x_A ≠ x_B`, the time difference Δt' is not zero.

Events that are simultaneous in one frame (S) are NOT simultaneous in another frame (S') that is moving relative to it.

Order of Events:

• If two events are "timelike" separated (one can cause the other), their time order is preserved in all frames (causality is maintained).
• If two events are "spacelike" separated (one cannot cause the other, as a light signal can't connect them), their time order can be *reversed* in different frames.

Lorentz Contraction (Length Contraction)

A moving object appears shorter in the direction of its motion.

Consider a rod at rest in frame S'. Its "proper length" (length in its rest frame) is L₀ = x'₂ - x'₁.

An observer in frame S wants to measure its length. They must measure the positions of its ends (x₁ and x₂) *at the same time* in their frame (t₁ = t₂ = t).

Using the Lorentz transformation `x' = γ (x - vt)`:

x'₂ = γ (x₂ - vt)

x'₁ = γ (x₁ - vt)

Subtracting the two equations: (x'₂ - x'₁) = γ (x₂ - x₁)

L₀ = γ * L (where L is the length measured in S)

L = L₀ / γ = L₀ √(1 - v²/c²)

Since γ ≥ 1, the measured length L is always less than L₀.

• This contraction only happens parallel to the direction of motion.
• Perpendicular lengths (y and z) are unaffected.

Time Dilation

A moving clock runs slow, as measured by a stationary observer.

Consider a clock at rest in frame S'. It ticks at one location (x'₁ = x'₂ = x'). The time interval between two ticks *in its rest frame* is the "proper time", Δt₀ = t'₂ - t'₁.

An observer in S measures this time interval using their own clock. They use the inverse transformation `t = γ (t' + vx'/c²)`:

t₂ = γ (t'₂ + vx'/c²)

t₁ = γ (t'₁ + vx'/c²)

Subtracting the equations: (t₂ - t₁) = γ (t'₂ - t'₁)

Let Δt = t₂ - t₁ (the time interval measured by S, the "moving" observer).

Δt = γ Δt₀ = Δt₀ / √(1 - v²/c²)

Since γ ≥ 1, the measured time interval Δt is always greater than Δt₀.

In words: The observer in S sees the S' clock (which ticks once every Δt₀ seconds) taking a longer time Δt to complete one tick. The moving clock appears to run slow.

Experimental Verification

1. Hafele-Keating Experiment (1971): Atomic clocks were flown on commercial jets around the world. When compared to a reference clock on the ground, the moving clocks ran slightly slower, exactly as predicted by relativity (after accounting for General Relativity effects as well).
2. Muon Decay: Muons are unstable particles created in the upper atmosphere. They have a very short proper lifetime (Δt₀ ≈ 2.2 μs). They travel at ~0.99c.
   • Classically: They should only travel `d = v*Δt₀ ≈ (3e8)(2.2e-6) ≈ 660 m` before decaying. They should not reach the Earth's surface.
   • Relativistically (from Earth's frame): We see the muon's clock running slow. Its lifetime in *our* frame is `Δt = γΔt₀`. For v=0.99c, γ ≈ 7.1. So, `Δt ≈ 7.1 * 2.2 μs ≈ 15.6 μs`.
     In this time, it *can* travel `d = v*Δt ≈ (0.99c)(15.6e-6) ≈ 4600 m`.
   • We detect muons on the ground, which is direct proof of time dilation.

Twin Paradox

This is a thought experiment that highlights the non-intuitive nature of time dilation.

• Setup: Twin A stays on Earth (inertial frame). Twin B takes a rocket on a high-speed round trip to a distant star (v ≈ c).
• The "Paradox":
  • Twin A (Earth): Sees Twin B's clock (in the rocket) running slow (Δt = γΔt₀). So, when B returns, A will be much older than B.
  • Twin B (Rocket): From B's perspective, *Earth* is the one moving away. So B should see A's clock running slow, and B should be older.
  This is a contradiction. Who is *really* younger?
• Resolution: The "paradox" is resolved by noting that the situation is not symmetric.
  Twin A (Earth) stays in a single inertial frame the whole time.
  Twin B (Rocket) must accelerate, turn around, and decelerate. B is in a non-inertial frame during these periods.
• The Special Theory of Relativity only applies to inertial frames. The full analysis (using General Relativity or careful application of STR) shows that the non-inertial observer (Twin B) is the one who experiences less time.
• Conclusion: Twin B, the space-traveler, returns to Earth and is genuinely younger than Twin A.

Relativistic Addition of Velocities

This formula replaces the simple Galilean `v_total = u + v`.

Let frame S see an object moving with velocity `u_x`.
Let frame S' move with velocity `v` relative to S (along x-axis).
What velocity `u_x'` does S' measure for the object?

Derive from Lorentz transformations: `x' = γ(x-vt)` and `t' = γ(t-vx/c²)`.

u_x' = dx'/dt' = [γ(dx - v dt)] / [γ(dt - v dx/c²)] = (dx/dt - v) / (1 - (v/c²)(dx/dt))

u_x' = (u_x - v) / (1 - u_x v / c²)

The inverse (what S sees) is more common:

u_x = (u_x' + v) / (1 + u_x' v / c²)

Variation of Mass with Velocity (Relativistic Mass)

(Note: The modern concept is that mass `m` is invariant, and relativistic momentum is `p = γmv`. The "relativistic mass" `m_rel = γm` is an older, but still common, way to teach this.)

To conserve momentum in relativity, the definition of momentum must be changed. This is equivalent to saying that mass increases with velocity.

Let `m₀` be the rest mass (mass of the object in its rest frame).

The mass `m(v)` when it is moving at speed `v` is:

m(v) = γ m₀ = m₀ / √(1 - v²/c²)

• As `v → c`, `γ → ∞`, and the object's mass `m(v) → ∞`.
• This means an infinite force (and infinite energy) would be required to accelerate a massive object to the speed of light.
• This is why no object with mass can ever reach the speed of light.

Massless Particles

What about particles that *do* travel at `v = c`, like photons (particles of light)?

If `v = c`, the mass equation is `m(c) = m₀ / √(1 - c²/c²) = m₀ / 0`.

For this to be non-infinite, the *only* possibility is that the rest mass m₀ must be zero.

• A particle with m₀ = 0 (a massless particle) *must* travel at `v = c` in all frames.
• A particle with m₀ > 0 (a massive particle) *must* travel at `v < c` in all frames.

Mass-Energy Equivalence (E=mc²)

This is the most famous equation in physics. It is a direct consequence of STR.

Relativistic Kinetic Energy (K):

K = (Work done) = ∫ F dx = ∫ (dp/dt) dx = ... (a complex derivation)

The result is: K = (γ - 1) m₀c²

Total Relativistic Energy (E):

E = K + (Rest Energy)

E = (γ - 1)m₀c² + m₀c²

E = γ m₀c² = m(v) c²

This is the full equation. It combines the kinetic energy and the rest energy.

Rest Energy (E₀):

If the particle is at rest (v=0, γ=1), its total energy is:

E₀ = m₀c²

This is the profound conclusion: Mass is a form of energy. A particle at rest has an "in-built" energy content equal to its rest mass times c².

This explains the energy released in:

• Nuclear Fission: A heavy nucleus (Uranium) splits into lighter nuclei. The *total rest mass* of the products is *less* than the original mass. This "lost mass" (Δm) is converted into a huge amount of kinetic energy: E = (Δm)c².
• Nuclear Fusion: Light nuclei (Hydrogen) fuse to form a heavier nucleus (Helium). Again, the final rest mass is less than the sum of the initial masses, releasing energy (this powers the Sun).