Lecture 8: Warped Space

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Lecture 8 Warped Space
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Euclidean Geometry
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Euclids Five Axioms

• A line can be drawn from any point to any other
  point
• A finite line segment can be extended to a line
  of any length
• A circle can be drawn with any centre and at any
  distance from that centre
• All right angles are equal to one another
• and lastly ...

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Fifth Axiom The Parallel Postulate

• Given a line, and a point not on this line, there
  is only one line through this point that is
  parallel to the original line
• Many mathematicians tried, unsuccessfully, to
  prove this axiom

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Lobachevskys Non-Euclidean Geometry

• A new geometry built without the parallel
  postulate
• Sum of angles of a triangle is less than 180o
• Related work by Gauss and Bolyai

Nikolai Ivanovich Lobachevski (1792-1856)
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Example of Lobachevskys Geometry
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Actually describes geometry on a surface such as
a saddle
Sum of angles of a triangle is less than 180
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Riemannian Geometry

• Discovered by B. Riemann, who dropped another of
  Euclids axioms (the second one)
• Sum of angles of a triangle is greater than 180o
• Riemann generalised Pythagoras theorem to any
  curved geometry

Bernhard Riemann (1826-1866)
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Example of Riemannian Geometry
Sum of angles of a triangle is more than 180
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The surface of the Earth is curved

• A small region on the surface of the Earth is
  almost flat
• A flat map is adequate to represent this region
• But a flat map cannot accurately represent the
  entire surface of the Earth. Distortions are
  inevitable, since the surface is actually curved.

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Mercator Projection

• There are large distor-tions of area towards
  the polar regions
• The shortest path bet-ween two points is a
  great circle
• Actually appears as a longer curved path in
  this projection

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Geodesics

• A geodesic is the shortest path between two
  given points
• In flat space, geodesics are straight lines
• On the surface of the Earth, geodesics are great
  circles

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Einsteins Theory of General Relativity (GR)

• The effects of gravity can be described in terms
  of curved space-time (Riemannian geometry)
• Free particles follow geodesics in this space-time

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Curvature Mimics a Force(acting between the ants)
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Recall a car moving round a bend undergoes
acceleration
Driver will feel a centrifugalforce pushing him
outwards
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Similarly

• When space-time is curved, particles moving in it
  experience an acceleration
• By the equivalence principle, we interpret these
  particles as being affected by the force of
  gravity

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General Relativity (contd)

• Geometry of space-time is curved by matter in the
  space-time
• On the other hand, the curvature of space-time
  determines how the matter will move

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Einsteins Field Equation
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Imagine Earth as a ball on a rubber sheet
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Experimental Tests of GR

• Bending of light
• Gravitational redshift
• Precession of Mercury
• Gravity waves

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Precession of Mercury

• Mercurys orbit is known to precess by 43
  arcseconds per century
• Newtonian gravity could not explain this, but GR
  is able to!

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Gravity Waves

• Gravity waves are ripples in the space-time
  fabric caused by a violent cosmic event
• Usually caused by a violent cosmic event, such as
  a coalescence of two black holes or neutron stars

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Hulse-Taylor Binary Pulsar

• Two neutron stars orbiting each other
• Slowly spiraling into each other, and at the same
  time emitting gravity waves

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Detection of Gravitational Waves
Laser Interferometer Gravitational-Wave
Observatory (LIGO)
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Detection of Gravity Waves (contd)
In the presence of gravity waves, thedistance
between the mirrors will fluctuate