The Calculus of Black Holes

1
The Calculus of Black Holes

• James Wang
• Elizabeth Lee
• Alina Leung
• Elizabeth Klinger

2
What is a Black Hole?

• A region from which even light cannot escape
• Thus the black hole itself cannot be seen
• Detected through gravitational distortion of
  nearby planets and stars, and radiation
• Has infinite gravitational pull and density

3
What is the Event Horizon?

• An area around the singularity of the black hole
  where no particle can escape its pull
• No outside influences can affect the particles
  descent towards the black hole

4
What are Stationary Limits?

• Stationary limit area around black hole (outer
  border)
• Particles in area are in constant motion
• Rotating black hole (Kerrs) distortion of
  space
• Doesnt apply to Schwarzchild black hole
  doesnt rotate
• Gravity infinitely intense
• Limit between this and event horizon ergosphere
  
• Limit at which light can escape

5
Diagram of a Black Hole
6
Pictures of Black Holes
7
How are Black Holes Modeled?

• Black holes create indentations in space/time
  continuum
• Curvature is the only logical way to model black
  holes
• Black holes follow the no-hair theorem
• Only three characteristics distinguishing black
  holes from one another are mass, angular
  momentum, and electric charge

8
Black Holes and Einsteins Theory of Relativity
Black Holes and Einsteins General Theory of
Relativity

• Gravity curved space time
• Caused by mass and radius of an object, as well
  as energy
• Strong gravitational field more curvature
• Applies to light light gets curved
• Space affects movement of object
• No material object can move faster than speed of
  light
• Black hole area where space time curved so much
  that objects fall out of the universe
• Escape velocity speed of light

9
Maxwells Equations

• 1st equation

• 2nd equation

• Determines total flow of electric charge out from
  closed surface
• Cover surface with patches of area of dA
  (represented as vectors), use dot product to find
  component of field that points in outward
  direction (only component that matters)

• Net magnetic flux is 0
• Magnetic flux product of magnetic field and
  area it goes through integral of vector quantity
  (magnetic force) over surface

10
Maxwells Equations (contd)

• 3rd equation

• Line integral products of vector functions of
  electric and magnetic field
• Equation says line integral of electric field
  around closed loop is equal to negative rate of
  change of magnetic flux

11
Maxwells Equations (contd)

• 4th equation

• Light was in form of electromagnetic wave

12
How are Maxwells Equations Related to Black
Holes?

• Moving electric field creates magnetic vortex
• Electromagnetic radiation from charged
  particles that move towards black hole
• Light affected by extremely strong gravity
• Black hole is large magnetic field b/c electric
  field created when charge falls into black hole

13
Using Riemannian Manifolds to Describe Curvature

• Manifolds describe complex structures of
  non-Euclidian space within the context of
  Euclidian space using mathematical equations
• Riemannian manifolds are real differentiable
  manifolds that use angles
• Black holes are mapped into more simple
  structures using Riemannian manifolds

14
Equations Modeling Black Hole Curvature
The Schwarzschild Metric Equation
15
Equations Modeling Black Hole Curvature
The Schwarzschild Metric Equation (Continued)
16
Equations for Escape Velocity and Gravitational
Force

• Gravitational Energy would have to equal kinetic
  energy

• Force as mass becomes infinite and radius 0

17
Significance of Change in Radius in Relation to
Curvature

• Curvature is the deviation of an object from
  being flat
• A smaller radius has more curvature and vice
  versa
• Therefore, black holes with smaller radii have
  more curvature

18
Behavior and Emissions of a Black Hole

• Electromagnetic radiation comes from charged
  particles that move towards black hole
• Black hole is large magnetic field b/c electric
  field created when charge falls into black hole

19
Photon and Gamma Particle Radiation from Black
Holes

• Black holes emit thermal radiation at temperature
• reduced Planck constant
• c speed of light
• K Boltzmann constant
• G gravitational constant
• M mass of black hole
• Unlike most objects, the temperature of a black
  hole increases as it radiates away mass

20
Gravitational Force Considerations

• Black holes become impossible to escape as it
  approaches the event horizon as the escape
  velocity required, regardless of mass, equals the
  speed of light
• Relativity, as c is constant, in order for energy
  to increase towards infinite, mass infinite

21
Bibliography

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• Ando, David. "An Introduction to Black Holes." 7
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22
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23
Bibliography continued

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24
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25
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