Relativity and Black Holes

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Relativity and Black Holes

• Frank Hsia-San Shu
• National Tsing Hua University
• 8 October 2005
• Potential Physics Chemistry Majors

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Outline of Lecture

• The speed of light in a vacuum, c 300,000 km/s,
  is the same no matter how fast an observer moves
  with respect to the source of the light (special
  relativity).
• As a consequence, time appears dilated and
  lengths seem to suffer (Lorentz) contraction for
  moving objects.
• As a further consequence, the
  most famous equation in all of science.
• Locally, it is not possible to distinguish
  between gravitation and acceleration (general
  relativity).
• As a consequence, it is possible for gravity to
  bend light and to warp spacetime. Indeed,
  gravitation is warped spacetime.
• Black holes represent regions where spacetime is
  so warped that fundamental challenges are posed
  to our understanding of physics.

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Einstein at 15 Racing at the Speed of Light
Violates Maxwells Equations
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Theory of Special RelativityTwo Basic Postulates

• Valid laws of physics are the same for all
  inertial observers (people who move at constant
  velocity, maybe zero, relative to the fixed
  stars).
• The speed of light, c 300,000 km/s, is the same
  for all observers, independent of their motion
  relative to the source of light.

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Time Dilation
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Proof

• One tick of light clock according to woman
• One tick of light clock according to man
• Relationship between time intervals
• Solving for , we get
• i.e., time passed for woman is factor
  shorter than time passed for man.

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Lorentz Contraction
Twin paradox Cant woman regard herself to be at
rest? Wont she deduce that her brother ages
less quickly than she does?
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Relativistic Increase in Mass

• M and m have identical mass when they are at
  rest with respect to each other.
• Distance seen by A and A is unaffected by
  motion of A in x-direction at
• velocity chosen to equal , which is why
  
• Time for collision measured by A is
  shorter than measured by A by a factor of
  . This implies
• But by momentum balance and symmetry
• Thus, by the inverse factor
  
• The reason is that as seen by A has
  an extra component of velocity in the
  x-direction compared to seen by the same
  observer A.
• In the limit when and go to zero,
  go to the rest mass , and goes
  to v, the total velocity. Therefore, we conclude
  that the mass of a ball in motion at speed v
  is larger than its rest mass by the factor

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Equivalence of Mass and Energy

• According to Newton, what a moving ball has extra
  compared to a stationary ball of mass is
• According to Einstein, what a moving ball has
  extra compared to a nonmoving ball of mass
  is more mass
• Perhaps energy and mass are the same thing!
• In the above equation, E total energy. For v
  ltlt c,
• where the term is called the rest
  energy and is the kinetic energy as
  defined by Newton. More generally,
• According to Newton, when a ball is placed in a
  gravitational field (or more generally, any
  attractive field of force), what it has is
  (negative) potential energy. For Einstein, what
  it has is less mass. According to Lavoisier when
  hydrogen combines with oxygen to make water, mass
  is conserved but heat is released. No, heat is
  energy, and the water actually contains slightly
  less mass than the original hydrogen and oxygen.

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Possibility of Converting Energy into Mass and
Vice-Versa

• Energy of putty balls before collision
• Energy of putty balls after collision
• Energy is conserved in process
• Rest mass after collision is greater than rest
  mass before collision! There has been conversion
  of energy into mass. (Not more putty molecules,
  but putty weighs slightly more than it did at
  rest.)
• Conversion of mass into energy must also be
  possible.
• Basis of nuclear power. Although less well
  known, also basis of chemical power indeed,
  basis of all power.

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Newtons Greatest Achievement, Einsteins
Happiest Thought

• Nothing can travel faster than the speed of
  light. How then can
• Neither F ma nor F GMm/r2 are
  relativistically correct. What can replace these
  equations?
• Maybe gravitation and acceleration are
  indistinguishable locally.
• Accept Galileos empirical finding, but not
  Newtons theoretical explanation.

• Apple (and everything else) falls to Earth at g
  9.8 m/s2 (Galileo).
• Moon falls about Earth with centripetal
  acceleration (Huygens) v2/r 0.0029 m/s2
  g/3600.
• Moon is 60 times farther away from center of
  Earth than apple (Erastothenes). 602 3600.
• Maybe gravitation
• Then 2nd and 3rd laws imply
• independent of any property of m.

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Two Views of Gravitation

• Newton
• Gravity is a force which pulls on all things with
  mass.
• Mass acts as the source that generates the force
  of gravitation.
• Einstein
• There is no such thing as the force of gravity.
  Gravitation arises when spacetime has curvature
  indeed gravitation is spacetime curvature.
• Mass-energy and stress (e.g., pressure) act as
  the sources that generate spacetime curvature.

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Theory of General Relativity Two Basic
Postulates

• The concepts of special relativity apply to local
  phenomena.
• Locally, there is no way to distinguish
  physically between gravitation and acceleration.

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Bending of Light
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Gravitational Bending of Light
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Ant Analogy for Bending of Light
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Lensing of Background Galaxies by Galaxy Cluster
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Event Horizon of Black Hole

• Laplace
• Schwarzschild

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Flight Circles about a Black Hole

• 9 km for a 3 solar-mass BH.
• Start with flight circle of circumference 2p90
  km.
• You deduce youre 90 km radially from BH. Dont
  jump to conclusions.
• Lower yourself inward radially by 32 km.
• Fly around measure circumference 2p60 km. (?)
• Lower yourself inward radially by another 33.75
  km.
• Fly around measure circumference 2p30 km.
  (??)
• Lower yourself inward radially by another 19.8
  km.
• You compute 3233.7519.8 85.55. Subtracted
  from 90, wont this bring you inside 9
  km? (!)
• Dont worry lower yourself by 19.8 km as we
  requested.
• Fly around measure circumference 2p15 km.
  Whew!
• All flight circles are in a single plane.
  Clearly, presence of a 3 solar-mass point-mass at
  center has warped our usual (Euclidean) sense of
  geometry.

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Black Holes Are Punctures in Fabric of Spacetime
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Behavior Near Event Horizon
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Reversal of Space and Time Across Event Horizon
of a Black Hole

• Outside event horizon, by exerting enough force
  on the rope, I can hold your position stationary
  with respect to center of BH. But there is
  nothing I can do to stop the forward progression
  of time for you (or, for that matter, for
  myself).
• As I lower you toward event horizon, your
  perception of stars begin to change and blur.
  Are you getting a sinking feeling?
• When you get close enough to the event horizon,
  no rope no matter how strong can stand the
  strain. It will snap and break, and you will
  begin an inexorable fall toward the black hole.
• For you, it takes only a few milliseconds for you
  to reach and cross the event horizon of the BH.
  But for me, it seems that you formally take an
  eternity to reach the event horizon.
• In other words, as you draw near to the event
  horizon, there is nothing I can do to stop your
  forward progression through space. But for me,
  time seems to have stopped moving for you! In
  some sense, for me on the outside, time and space
  seem to reverse roles as you approach the event
  horizon. When you cross it, you will reach a
  different space and time than the one that we on
  the outside occupy. Some people speculate that
  BHs may be portals to other spacetimes and other
  universes!

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Optical Jet Emanating from Nucleus of M87, an
Elliptical Galaxy
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Formation of Magnetized Black Hole in
Self-similar Gravitational Collapse
Cai Shu (2005)
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Speculation 1-- Wormholes Shortcuts through
Space?

• y
• x

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Speculation 2 -- Wormholes Machines through Time?

• x
• t

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Speculation 3 Evaporation of BHs?

• Currently popular theoretical view Proton is a
  long-lived, but ultimately unstable particle,
  which will decay into positron plus other
  particles in some years or so.
• Even BHs may ultimately evaporate completely
  away. According to Hawking, (nonrotating) BHs
  of a mass M have a nonzero surface temperature T
  given by the formula
• This formula has been recently been recovered by
  the methods of superstring theory, which is the
  attempt to reconcile general relativity with
  quantum mechanics.

• Heisenberg uncertainty principle

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Physics as a Community

• An unbroken thread from the beginning of
  modern science to the present age.

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Thank you everyone!