{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture8 Curved Space

# Lecture8 Curved Space - Curved Space Hawley Holcomb Chapter...

This preview shows pages 1–7. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
3/25/09 Astronomy 309R - Spring 2009 6 Spherical Geometry All lines intersect: no parallel lines exist!
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 20

Lecture8 Curved Space - Curved Space Hawley Holcomb Chapter...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online