Black Holes and the Einstein-Rosen Bridge: Traversable Wormholes? Chad A. Middleton Mesa State College September 4, 2008

Outline…  Distance in 3D Space & 4D Spacetime  Light Cones  Einstein’s General Relativity  The Schwarzschild Solution & Black Holes  The Maximally Extended Schwarzschild Solution: White Holes & Spacelike Wormholes

3D Euclidean Space Line element in Euclidean space is the line element measuring distance is invariant under rotations

In 1905, Einstein submitted his Theory of Special Relativity  Lorentz Transformations  Length depends on reference frame

4D Spacetime Interval Line element in Minkowski (Flat) spacetime is the line element measuring ‘length’ is invariant under ‘rotations’

Line Elements are Coordinate Independent… The flat line element in Cartesian coordinates perform a coordinate transformation to spherical coordinates Same length, nothing’s changed!  GR is coordinate independent

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 ct y x Particle’s worldline Future Light Cone Past Light Cone

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 ct r Particle’s worldline

In 1915, Einstein gives the world his General Theory of Relativity describes the curvature of space describes the matter & energy

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

Line element in 4D curved space  is the metric tensor  defines the geometry of spacetime  Know, Know Geometry

The empty space, spherically-symmetric, time- independent solution (i.e. spherical star) is Let: Notice: : Coordinate Singularity : Spacetime Singularity

Black Holes  For fixed radius R …  For M star ~3-4 M sun, star collapses to a Black Hole Notice: Black Holes don’t suck! (External geometry of a Black Hole is the same as that of a star or planet)

Consider radial null curves (  &  = const, ds 2 = 0) this yields the slopes of the light cones Light cones “close up” at r = 2GM! t r 2GM

Radial plunge of an experimental physicist.. Notice: It takes a finite amount proper time  to reach r = 2GM and an infinite amount of coordinate time t - Proper time - Coordinate time

In Eddington-Finkelstein Coordinates… The slope of the light cone structure of spacetime is… v rr=2GM r=0 v=const Notice: For r<2GM, light cone tips over!

In Eddington-Finkelstein Coordinates… v Notice:  r = 2GM is the Event Horizon  One-Way Surface  For r < 2GM, all future-directed paths are in direction of decreasing r ! r r=2GM r=0

In Eddington-Finkelstein Coordinates… v r r=2GM r=0  For r < 2GM, r = const., ds 2 > 0 - spacelike interval The r = 0 singularity is NOT a place in space, but rather a moment in time!

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 < 2GM, all worldlines lead to singularity at r = 0!

The Maximally Extended Schwarzschild Solution Consider entire spacetime…. Two r = 0 spacelike singularities (V=  U 2 +1) Upper singularity is a Black Hole Lower singularity is a White Hole Universe appears to emerge from the past singularity!

The Maximally Extended Schwarzschild Solution Two asymptotically flat regions (U   ) Consider V = 0… move from U =  to U = -   Spacelike Wormhole!!

Conclusions General Relativity predicts that…  the ultimate collapse of sufficiently massive stars to Black Holes  an r = 0 ‘moment in time’ singularity  White and Black Holes  Spacelike Wormholes that are NOT traversable.

Coordinate Singularity i.e. 2D Line element in polar coordinates.. Perform a coordinate transformation… 2D Line element becomes.. Notice: y x ds dr rd 

In Kruskal Coordinates… Where Note: For r = const., U 2 - V 2 = const.  Hyperbolas For r = 2GM, U =  V  straight lines w/ m =  1 For r = 0, V=  U 2 +1  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 > 2GM U = 0 for r < 2GM