Lecture8 Curved Space

# Lecture8 Curved Space - Curved Space Hawley &amp; Holcomb,...

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3/25/09 Astronomy 309R - Spring 2009 1 Curved Space Hawley & Holcomb, Chapter 8

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3/25/09 Astronomy 309R - Spring 2009 2 Simple Axioms or Postulates Logical Reasoning and Inference Advanced Conclusions, Theorems Intuitive/Cognitive Knowledge
3/25/09 Astronomy 309R - Spring 2009 3 Euclidean Axioms (Postulates) of Geometry Euclid: Greek Mathematician in Alexandria, 325-270 B.C., attempted to derive all of geometry from a simple set of “axioms” 1. A unique straight line can be drawn between any two points 2. A ﬁnite line can be extended inﬁnitely in both directions 3. A circle can be drawn with any center and any radius 4. All right angles are equal to each other 5. Given a line and a point not on the line, only one line can be drawn through the point parallel to the line line parallel line point

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3/25/09 Astronomy 309R - Spring 2009 4 Parallel Postulate • Parallel lines are lines that 1. belong in the same plane, and 2. do not intersect each other • How do we know that two lines that appear to be parallel continue to be parallel when extended to large distances? • Geometry in which Euclid’s parallel postulate holds is called Euclidean Geometry . • There are many other non-Euclidean geometries in which the parallel postulate is invalid! ?
3/25/09 Astronomy 309R - Spring 2009 5 Curved Space • Euclidean geometry is sometimes called “ﬂat geometry.” • The opposite of “ﬂat” is “curved.” Curved geometry is non-Euclidean geometry. • Examples of ﬂat and curved geometries (note, these are examples of two-dimensional geometries, but equivalent, three dimensional geometries are also possible) – The surface of a table is ﬂat – The surface of a sphere is curved • Curved space cannot be described by Euclidean geometry. • In curved space, there is a characteristic length scale: the radius of curvature (think about the radius of the Earth)

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3/25/09 Astronomy 309R - Spring 2009 6 Spherical Geometry All lines intersect: no parallel lines exist! Euclid would have had to extend his “parallel” lines to very large
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## This note was uploaded on 09/14/2009 for the course AST 309 taught by Professor Johnlacy during the Spring '08 term at University of Texas at Austin.

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Lecture8 Curved Space - Curved Space Hawley &amp; Holcomb,...

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