Principle of Equivalence: Einstein 1907

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Principle of Equivalence Einstein 1907
Box stationary in gravity field
Box accelerates in empty space
Box falling freely
Box moves through space at constant velocity
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Equivalence Principle

• Special relativity all uniformly moving frames
  are equivalent, i.e., no acceleration
• Equivalence principle
• Gravitational field acceleration
• freely falling frames in GR
• uniformly moving frames in SR.

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Tides

• Problem
• Gravity decreases with distance gt stretch

r2
r1
moon
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Tides

• Tides gravity changes from place to place

not freely falling
?
?
?
?
freely falling
not freely falling
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Light rays and Gravity

• Remember gravity bends light

accelerating observer
gravity
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Light Rays and Gravity II

• In SR light rays travel on straight lines
• gt in freely falling fame, light travels on
  straight lines
• BUT to stationary observer light travels on
  curved paths
• gt Maybe gravity has something to do with
• curvature of space ?

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Curved Spacetime

• Remember Gravity warps time

BUT in spacetime, time and space are not
separable
fast
gt Both space and time are curved (warped)
This is a bit hard to vizualize (spacetime
already 4D)
slow
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GR Einstein, 1915

• Einstein mass/energy squeeze/stretch spacetime
  away from being flat
• Moving objects follow curvature (e.g.,
  satellites, photons)
• The equivalence principle guarantees
  spacetime is locally flat
• The more mass/energy there is in a given volume,
  the more spacetime is distorted in and around
  that volume.

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GR Einstein, 1915

• Einsteins field equations correct action at a
  distance problem
• Gravity information propagates at the speed of
  light
• gt gravitational waves

r?
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Curvature in 2D

• Imagine being an ant living in 2D
• You would understand
• left, right, forward, backward,
• but NOT up/down
• How do you know your world is curved?

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Curvature in 2D

• In a curved space, Euclidean geometry does not
  apply
• - circumference ? 2? R
• - triangles ? 180
• - parallel lines dont stay parallel

2?R
R
??180?
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Curvature in 2D
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Curvature in 2D
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Geodesics

• To do geometry, we need a way to measure
  distances
• gt use ant (lets call the ant metric), count
  steps it has to take on its way from P1 to P2
  (in spacetime, the ant-walk is a bit funny
  looking, but never mind that)
• Geodesic shortest line between P1 and P2
• (the fewest possible ant steps)

ant
P1
P2
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Geodesics

• To the ant, the geodesic is a straight line,
• i.e., the ant never has to turn
• In SR and in freely falling frames, objects move
  in straight lines (uniform motion)
• In GR, freely falling objects (freely
  falling under the influence of gravity only, no
  rocket engines and such objects apples,
  photons, etc.)
• move on geodesics in spacetime.

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Experimental Evidence for GR

• If mass is small / at large distances, curvature
  is weak
• gt Newtons laws are good approximation
• But Detailed observations confirm GR
• 1) Orbital deviations for Mercury (perihelion
  precession)

Newton
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Experimental Evidence for GR

• 2) Deflection of light

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Experimental Evidence for GR
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Black Holes

• What happens as the star shrinks / its mass
  increases? How much can spacetime be distorted by
  a very massive object?
• Remember in a Newtonian black hole, the escape
  speed simply exceeds the speed of light
• gt Can gravity warp spacetime to the point where
  even light cannot escape its grip?
• That, then, would be a black hole.

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Black Holes
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Black Holes

• Time flows more slowly near a massive object,
• space is stretched out (circumference lt 2?R)
• Critical the ratio of circumference/mass of the
  object.
• If this ratio is small, GR effects are large
  (i.e., more mass within same region or same mass
  within smaller region)

1) massive
2) small
???
???
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The Schwarzschild Radius

• GR predicts If mass is contained in a
  circumference smaller than a certain size
• space time within and around that mass
  concentration qualitatively changes. A far away
  observer would locate this critical surface at a
  radius
• Gravitational time dilation becomes infinite as
  one approaches the critical surface.

gravitational constant
critical circumference
speed of light
mass
Schwarzschild radius
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Black Holes

• To a stationary oberserver far away, time flow at
  the critical surface (at RS) is slowed down
  infinitely.
• Light emitted close to the critical surface is
  severely red-shifted (the frequency is lower) and
  at the critical surface, the redshift is infinite.

From inside this region no information can escape
red-shifted
red-shifted into oblivion
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Black Holes

• Inside the critical surface, spacetime is so
  warped that objects cannot move outward at all,
  not even light.
• gt Events inside the critical surface can never
  affect the region outside the critical surface,
  since no information about them can escape
  gravity.
• gt We call this surface the event horizon
• because it shields the outside completely from
  any events on the inside.

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Black Holes

• Critical distinction to the Newtonian black hole
• Nothing ever leaves the horizon of a GR black
  hole.
• Lots of questions
• What happens to matter falling in?
• What happens at the center?
• Can we observe black holes anyway?
• And much, much more

Newton
Einstein