[Leejohnston]

## The Spacetime Interval — The Geometry of Time

### 1. The Equation

Spacetime interval:

s² = c²t² − x² − y² − z²

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### 2. What Each Symbol Means

- s = spacetime interval
- t = time difference
- x, y, z = spatial differences
- c = speed of light

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### 3. What the Equation Is Telling Us

Different observers disagree on space and time separately.

But **the spacetime interval is invariant**.

Everyone agrees on it.

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

Space and time are not separate.

They are components of a single geometric structure: spacetime.

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### 5. Why This Matters

This equation:
- replaces absolute time
- unifies motion and causality
- underlies all of relativity

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### 6. Real-World Applications

- Relativity
- Particle physics
- Cosmology

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### 7. Common Misconceptions

- Time is just another dimension → incomplete
- Space and time are independent → false
- Geometry is abstract → false (it governs reality)

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### Try It Yourself

Why can two observers disagree on time but agree on causality?