[Leejohnston]

## The Lorentz Factor — The Core of Relativity

### 1. The Equation

Lorentz factor:

γ = 1 ÷ √(1 − v² / c²)

---

### 2. What Each Symbol Means

- γ (gamma) = Lorentz factor
- v = relative velocity
- c = speed of light

---

### 3. What the Equation Is Telling Us

The Lorentz factor measures **how extreme relativistic effects are**.

As speed increases:
- time dilation increases
- length contraction increases
- energy increases dramatically

---

### 4. Where It Comes From (Intuition)

Relativity preserves the speed of light.

To do this:
- time slows
- lengths shrink
- mass–energy rises

γ quantifies all of it.

---

### 5. Worked Example

If:
v = 0.8c

γ = 1 ÷ 0.6 ≈ **1.67**

That means:
- time dilation = 1.67×
- energy = 1.67× rest energy

---

### 6. Real-World Applications

- Particle accelerators
- GPS corrections
- High-energy astrophysics

---

### 7. Common Misconceptions

- γ only affects time → false
- γ becomes infinite gradually → false (it diverges at c)
- γ allows faster-than-light travel → false

---

### Try It Yourself

What happens to γ at 0.9c? 0.99c?