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The Elements of Relativity$
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David M. Wittman

Print publication date: 2018

Print ISBN-13: 9780199658633

Published to Oxford Scholarship Online: August 2018

DOI: 10.1093/oso/9780199658633.001.0001

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Spacetime Geometry

Spacetime Geometry

Chapter:
(p.132) 11 Spacetime Geometry
Source:
The Elements of Relativity
Author(s):

David M. Wittman

Publisher:
Oxford University Press
DOI:10.1093/oso/9780199658633.003.0011

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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