Black Holes and the EinsteinRosen Bridge: Traversable Wormholes

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Black Holes and the Einstein-Rosen Bridge
Traversable Wormholes?

• Chad A. Middleton
• Mesa State College
• September 4, 2008

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Outline

• Distance in 3D Space 4D Spacetime
• Light Cones
• Einsteins General Relativity
• The Schwarzschild Solution Black Holes
• The Maximally Extended Schwarzschild Solution
  White Holes Spacelike Wormholes

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3D Euclidean Space

• Line element in Euclidean space
• is the line element measuring distance
• is invariant under rotations

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In 1905, Einstein submitted his Theory of
Special Relativity

• Lorentz Transformations

• Length depends on reference frame

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4D Spacetime Interval

• Line element in Minkowski (Flat) spacetime
• is the line element measuring length
• is invariant under rotations

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Line Elements are Coordinate Independent

• The flat line element in Cartesian coordinates
• perform a coordinate transformation to
  spherical coordinates
• Same length, nothings changed!
• ? GR is coordinate independent

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Spacetime Diagram Light Cones

• Flat line element in spherical coordinates
• Notice
• For constant time slice, spherical wave front
• Light cone is a one-way surface

Future Light Cone
Past Light Cone
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• Consider radial null curves (? ? const, ds 2
  0)
• this yields the slopes of the light
  cones
• For two events..
• Notice
• r 0 is a timelike worldline

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In 1915, Einstein gives the world his General
Theory of Relativity

• describes the curvature of space
• describes the matter energy

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Space is not an empty void but rather a dynamical
structure whose shape is determined by the
presence of matter and energy.

• Matter tells space how to curve
• Space tells matter how to move

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Line element in 4D curved space

• is the metric tensor

• defines the geometry of spacetime

• Know , Know Geometry

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The empty space, spherically-symmetric,
time-independent solution (i.e. spherical star) is
Let
Notice
Coordinate Singularity
Spacetime Singularity
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• Black Holes
• For fixed radius R
• For Mstar3-4 Msun, star collapses to a Black
  Hole
• Notice
• Black Holes dont suck!
• (External geometry of a Black Hole
• is the same as that of a star or planet)

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Consider radial null curves (? ? const, ds 2
0) this yields the slopes of the light
cones Light cones close up at r 2GM!
t
2GM
r
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Radial plunge of an experimental physicist..
- Proper time
- Coordinate time
Notice
It takes a finite amount proper time ? to reach r
2GM and an infinite amount of coordinate time
t
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In Eddington-Finkelstein Coordinates

• The slope of the light cone structure of
  spacetime is

For rlt2GM, light cone tips
over!
v
vconst
Notice
r0
r
r2GM
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In Eddington-Finkelstein Coordinates
Notice
v

• r 2GM is the Event Horizon
• ? One-Way Surface
• For r lt 2GM,
• all future-directed paths are in direction of
  decreasing r !

r
r0
r2GM
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In Eddington-Finkelstein Coordinates
v

• For r lt 2GM, r const.,
• ds2 gt 0 - spacelike interval
• The r 0 singularity is NOT a place in space,
• but rather a moment in time!

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In Kruskal Coordinates

• Notice
• Kruskal coordinates cover entire spacetime
• r 0 singularity is a spacelike surface
• The slopes of the light cones are
• For r lt 2GM, all worldlines lead to
  singularity at r 0!

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The Maximally Extended Schwarzschild Solution

• Consider entire spacetime.
• Two r 0 spacelike singularities (V?? U21)
• Upper singularity is a Black Hole
• Lower singularity is a White Hole
• Universe appears to emerge from the past
  singularity!

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The Maximally Extended Schwarzschild Solution

• Two asymptotically flat regions (U ? ??)
• Consider V 0
• move from U ? to U -?
• ? Spacelike Wormhole!!

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Coordinate Singularity i.e.

• 2D Line element in polar coordinates..
• Perform a coordinate transformation
• 2D Line element becomes..
• Notice

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In Kruskal Coordinates

• Where
• Note
• For r const., U2- V2 const. ? Hyperbolas
• For r 2GM, U ?V ? straight lines w/ m ?1
• For r 0, V?? U21 ? Hyperbolas
• For t const., U/V const. ? straight lines
  thru origin
• For t ? ??, U ?V ? straight lines w/ m ?1
• For t 0, V 0 for r gt 2GM
• U 0 for r lt 2GM