The Calculus of Black Holes
James Wang
Elizabeth Lee
Alina Leung
Elizabeth Klinger
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
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 particle’s
descent towards the
black hole
What are Stationary Limits?
• Stationary limit – area around
black hole (outer border)
– Particles in area are in constant
motion
– Rotating black hole (Kerr’s) –
distortion of space
• Doesn’t apply to Schwarzchild black
hole – doesn’t rotate
• Gravity infinitely intense
• Limit between this and event
horizon – ergosphere
• Limit at which light can escape
Diagram of a Black Hole
Pictures of Black Holes
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
Black Holes and Einstein’s
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
Maxwell’s Equations
• 1st equation:
•2nd equation:
q
 E  dA  0
 B  dA  0
0  electric permittivity
B  magnetic field
q  ch arg e
•
•
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
Maxwell’s Equations (cont’d)
• 3rd equation:
d B
 E  dS   dt
E  line int egral around closed loop
•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
Maxwell’s Equations (cont’d)
• 4th equation:
 B  dl   J  dA   I
  line int egral around contour
c
s
0 enc
c
dl  differential of curve
J  current density through surface enclosed by curve
dA  differential element of surface area
 0  4  10 7
I enc  current enclosed by curve
•Light was in form of electromagnetic wave
How are Maxwell’s 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
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
Equations Modeling Black Hole
Curvature
The Schwarzschild Metric Equation
Equations Modeling Black Hole
Curvature
The Schwarzschild Metric Equation (Continued)
Equations for Escape Velocity
and Gravitational Force
• Gravitational Energy
would have to equal
kinetic energy
U
GmM 1
 mv 2
r
2
mv 2 
2GmM
r
2GM
v 
r
GmM
F
r2
v  lim
M 
2GM

r
v
F  lim
GmM

2
r
F  lim
GmM

r2
M 
2
2GM
r
• Force as mass becomes
infinite and radius 0
v  lim
r 0
2GM

r
r 0
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
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
Photon and Gamma Particle
Radiation from Black Holes
• Black holes emit thermal radiation at temperature
–
c 3
T
8GMk
•
•
•
•
•
 = 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
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
1
E  mv 2
2
v
2GM
c
r
1
E  lim mv 2
vc 2
E  mc 2
F  ma
F  lim ma
m 
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•
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