Curve Space
Perceiving Curvature
Brian Lynch
Thought Experiment
• Elevator experiment
takes place in high
gravitational field.
• Expected Result
• Actual Result
• Why?
Two Dimensional Curve
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•
•
•
2-d Plane filled with 2-d creatures.
Circles, squares, all 2-d figures.
Consistency of geometry
Simple Euclidean test.
Triangles
• Two Properties to
consider:
• Closed
• Sum of the three
angles is 180
• 2-d creatures begin
their test.
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?
Important Aspects of 2-d
Curvature
• Relation between effects of curvature
and location
• Relation between effects of curvature
and size of area considered.
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.
“Bending” Light
• Deflection of light in strong gravitational
fields.
• Viewed in an early 20th century solar
eclipse
• Eddington and Crommlin observed shift in
stars.
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.
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
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.