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The Calculus of Black Holes James Wang Elizabeth Lee Alina Leung Elizabeth Klinger

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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

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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

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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

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Diagram of a Black Hole

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Pictures of Black Holes

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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

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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 Black Holes and Einstein’s Theory of Relativity

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Maxwell’s Equations 1st equation: 2 nd 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

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Maxwell’s Equations (cont’d) 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

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Maxwell’s Equations (cont’d) 4 th equation: Light was in form of electromagnetic wave

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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

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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

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Equations Modeling Black Hole Curvature The Schwarzschild Metric Equation

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Equations Modeling Black Hole Curvature The Schwarzschild Metric Equation (Continued)

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Equations for Escape Velocity and Gravitational Force Gravitational Energy would have to equal kinetic energy Force as mass becomes infinite and radius 0

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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

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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

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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

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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

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Bibliography Andersen, Michael C. "Einstein's Field Equations and Black Holes." Niels Bohr Institute. 14 June Niels Bohr Institute. 14 May Ando, David. "An Introduction to Black Holes." 7 May Cumberlidge, Anne-Marie. "Everything You Need to Know About Black Holes." The Hitch- Hiker's Guide to Black Holes Keele University. 5 May "Curvature." Wikipedia. 7 May 2006.

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Bibliography continued Darling, David. "Euclidean Space." The Encyclopedia of Astrobiology Astronomy and Spaceflight. 12 May Darling, David. "Non-Euclidean Geometry." The Encyclopedia of Astrobiology Astronomy and Spaceflight. 12 May "Escape Velocity." Wikipedia. 7 May "Euclidean Geometry." Wikipedia. 11 May Fowler, Michael. "Maxwell's Equations and Electromagnetic Waves." Physics Dept., University of Virginia. 10 May 2006.

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Bibliography continued "Isomorphism." Wikipedia. 15 May Johnston, Robert. "Black Holes." Relativistic Physics. 14 Oct May Kobes, Randy, and Gabot Kunstatter. "Gravity as Curved Space: Einstein's Theory of General Relativity." Physics Modern Technology. 29 Sept University of Winnipeg. 9 May "Manifold." Wikipedia. 15 May "Maxwell's Equations in Curved Spacetime." Wikipedia. Apr May Nave, C.r. "Maxwell's Equations." HyperPhysics Georgia State University. 9 May 2006.

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Bibliography continued "Q & a: Black Holes." Chandra X-Ray Observatory. 2 Aug Harvard University. 6 May "Radius of Curvature." Wikipedia. 17 May "Riemannian Manifold." Wikipedia. 12 May Salgado, Rob. "The Light Cone: the Schwarzchild Black Hole." Syracuse University. 7 May Seeds, Michael A. Horizons: Exploring the Universe. 7th ed. Pacific Grove, CA: Brooks/Cole, 2002.

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Bibliography continued Smith, Gene E. "General Relativity & Black Holes." Gene Smith's Astronomy Tutorial. 9 Mar University of California San Diego. 17 May "The Mystery of Black Holes." Knowing the Universe and Its Secrets. 7 Feb May Weisstein, Eric W. "Neighborhood." MathWorld. 12 May Wyatt, Ryan, and Rachel Connolly. "From Einstein to Hawking Course Outline." 28 Apr American Museum of Natural History. 10 May 2006.

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