Curve Space

1
Curve Space

• Perceiving Curvature
• Brian Lynch

2
Thought Experiment

• Elevator experiment takes place in high
  gravitational field.
• Expected Result
• Actual Result
• Why?

3
Two Dimensional Curve

• 2-d Plane filled with 2-d creatures.
• Circles, squares, all 2-d figures.
• Consistency of geometry
• Simple Euclidean test.

4
Triangles

• Two Properties to consider
• Closed
• Sum of the three angles is 180
• 2-d creatures begin their test.

5
On the sphere

• 2-d creatures transplanted to surface of sphere.
• First attempt Results 185
• Eventual Results are a range from 180 to 270.
• If transplanted to saddle, what then?

6
Important Aspects of 2-d Curvature

• Relation between effects of curvature and
  location
• Relation between effects of curvature and size of
  area considered.

7
Curvature of 3-d Universe

• Difficulty of measuring curvature in our system
• Curvature of any space is the effect of the
  masses contained within that space.
• Curvature determines the trajectories and
  interactions of objects.
• Similar to triangle tests for 2-d, tests for
  3-d.

8
Bending Light

• Deflection of light in strong gravitational
  fields.
• Viewed in an early 20th century solar eclipse
• Eddington and Crommlin observed shift in stars.

9
Mercury

• Higher gravitational field because of closeness
  to Sun.
• Supposing the only relevant bodies are Mercury
  and the Sun yields large discrepancy.
• Factoring in other celestial bodies reduces this
  error, but not entirely.

10
Accounting for the Error

• The remaining error after Newtonian mechanics has
  nothing more to account for is 43 arc seconds
  per century.
• Relativity suggests the curvature of space around
  the Sun.
• Curved model eliminates the error almost exactly.
• One of the confirmations of relativity

11
Back to the elevator.

• The curving of the laser light relates to the
  curving of space at that particular location.
• Cutting the cables and allowing it to fall freely
  will correct the problem and allow the light to
  escape from the second hole.