# Spacetime Geometry

# Spacetime Geometry

This chapter shows that the counterintuitive aspects of special relativity are due to the *geometry* of spacetime. We begin by showing, in the familiar context of plane geometry, how a metric equation separates frame‐dependent quantities from invariant ones. The components of a displacement vector depend on the coordinate system you choose, but its magnitude (the distance between two points, which is more physically meaningful) is invariant. Similarly, space and time components of a spacetime displacement are frame‐dependent, but the magnitude (proper time) is invariant and more physically meaningful. In plane geometry displacements in both *x* and *y* contribute positively to the distance, but in spacetime geometry the spatial displacement contributes *negatively* to the proper time. This is the source of counterintuitive aspects of special relativity. We develop spacetime intuition by practicing with a graphic stretching‐triangle representation of spacetime displacement vectors.

*Keywords:*
spacetime geometry, spacetime vector, 4‐vector, proper time, invariant, framedependent, metric, Minkowski metric, spacetime metric

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