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However, it is quite remarkable that many of these Spacetime concepts have precise analogues in geometry. Thus, rather than reason with our unrefined physical intuition, it has been quite useful to reason with our better-refined geometrical intuition. Thus, it is often useful to think in geometrical terms when discussing concepts in relativity.
For the sake of the reader, we distinguish "public" universally-agreed-upon notions from "private" observer-dependent notions.
I highly recommend that you refer to this page often.
ENGLISH | SPACETIME CONCEPT | GEOMETRICAL CONCEPT |
---|---|---|
History of the Universe | Spacetime | certain 4-dimensional space (Lorentz manifold) |
Finger-snap | Event | point |
History of an Observer | Worldline | Timelike curve |
History of an Inertial Observer | Freely-falling Worldline | Timelike geodesic curve (a curve that is as straight as it can be) |
History of a Quick flash of light | Light-Ray | Lightlike geodesic curve |
The elapsed-time you measure between two events. | Private-Time elapsed between two events. | Do a radar measurement. One-half the difference of your clock-time. |
The separation-distance you measure between two events. | Private-Distance between two events. | Do a radar measurement. One-half the sum your clock-times. (i.e., the average of your clock-times> |
Time read off a Wristwatch | Proper-Time along Worldline | Spacetime-length along a timelike curve |
Your speed and direction of travel (through spacetime) | Velocity Vector | Unit timelike tangent-vector |
How fast the other guy seems to be moving | Relative-velocity of one observer with repsect to another observer | Spacetime-Angle between two lines. Decompose his velocity-vector into a spatial-part and a temporal-part. Divide the spatial-part by the temporal-part. |
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