Findley 1
MAT 272
Black Hole Topology: A Calculus III Version
Christina Findley
Section Number: 72375
Professor Kathleen Brewer
December 10, 2012
Black Hole Topology: A Calculus III Version
Sir Isaac Newton published his laws of universal gravitation in 1687, stating that F=, where F is the gravitational force required to escape gravity and is the gravitational potential (Waner, 2002). Using that Divergence Theorem that 2=.() = 4G , where G is the gravitational constant (6.674x10-11 Nm2/Kg2) and is the density of the object, the formula for the force translates to F=GM/R2, where M is the mass and R the distance of separation (Waner, 2002). After Newton, an English astronomer named John Michell utilized Newton’s laws to hypothesize that “an object massive enough to have an escape velocity,” the velocity required for an object to escape the gravitational force present on the star, “greater than the speed of light,” which is about 2.99793x108 m/s, may exist within space in 1783 (Oracle Think Quest Education Foundation, 1999). This hypothesis led to the discovery of black holes; however, it is sometimes accredited to Pierre LaPlace who concluded the same hypothesis in France a century later, calling them “invisible stars” (Steinberg 2005). Further analysis on black holes from a topological perspective consists of the formation of a black hole due to photon movement, the hyperbolic nature of a black hole,and the spheres within a black hole.
A stellar black hole emerges after the death of a star due to photon movement. This star collapses when its massive center of mass collapses upon itself, resulting in a black hole and an exploding star called a supernova that disperses parts of the star into space (FalckeFriedrich 2003). As the star collapses, incoming and departing photons of the following conical form represent massive point-like particles that cause a black hole to emerge throughout the space time continuum:
(FalckeFriedrich2003; Briggs and Cochran)[1] The size of the photon depends on the values of a, b, and c. As the value of c increases, the cone stretches in the z direction; as the value of a increases, the cone stretches in the x direction; and as the value of b increases, the cone stretches in the y direction.[2]
Further analysis on the movement of the photons demonstrates that a stellar black hole represents a hyperboloid of one sheet. As black holes can only be seen using infrared or ultraviolet waves, the shape of a black hole was difficult to model mathematically.[3]Using cylindrical coordinates, Schwarzschild constructed one of the first models of a black hole as a hyperboloid of one sheet of the form:
(Falcke & Friedrich 2003; Briggs and Cochran)[4] Over time, the model of a black hole was studied further using space time curvature to yield the full representation of a black hole in both the positive and negative portions of the z axis. The size of the black hole depends on the values of a, b, and c. As the value of c increases, the hyperboloid stretches in the z direction; as the value of a increases, the hyperboloid stretches in the x direction; and as the value of b increases, the hyperboloid stretches in the y direction.[5]
Within the black hole, spheres exist. Schwarzschild defined the Schwarzschild radius of a black hole about the event horizon, which is a boundary along space time curvature where no light or radiation can escape.[6]This radius is defined by the following equation, where G is the gravitational constant, m is the mass, c is the speed of light in a vacuum, and Rs is the Schwarzschild radius:[7]
This radius holds significance since a sphere of radius r = a changes according to Schwarzschild’s physical metric of a black hole that was derived using spherical coordinates:
(FalckeFriedrich 2003)[8].In other words, the radius of the sphere tends to approach infinity in the + z axis direction, while the radius of the sphere tends to approach zero in the - z axis direction(FalckeFriedrich 2003).[9]
Studying the formation of a black hole due to photon movement, the hyperbolic nature of a black hole, and the spheres within a black hole from a topological perspective allows one to understand the composition of black holes. In order for society to progress within these fields, it is crucial that people study the phenomenon of black holes in order for a broader understanding of their physical properties and behavior to be understood. As of right now, the knowledge scientists have discovered about black holes is limited. The topological arguments presented represent only the beginning towards a complete understand of black holes.
Works Cited
Falcke, Heinoand Friedrich W. Hehl.The Galactic Black Hole: Lectures on General Relativity and Astrophysics. Institute of Physics Publishing.Bristol and Philadelphia: 2003.
Briggs, William and Lyle Cochran.Calculus Early Transcendentals.Arizona State University. Pearson Education Inc. Boston: 2006.
NASABlogs.“Monster Black Holes.” 2012. spot.com/2012_02
_01_archive.html
Oracle Think Quest Education Foundation. "Event Horizon-Shedding the Light on the Discovery of Black Holes."Thinkquest. Oracle Think Quest Education Foundation, 1999. Web. 5 Apr. 2012. <
Powell, Richard. “Inside a Black Hole.”Web. 18 Apr. 2000.
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Steinberg, Daniel. "No Escape: The Truth About Black Holes." Space Telescope Science Institute, Mar. 2005. Web. 5 Apr. 2012. <
Waner, Stefan."The Einstein Field Equations and Derivation of Newton's Law." HOFSTRA University, Jan. 2002. Web. 5 Apr. 2012. <
Wolfram Alpha Research Company. “Wolfram Alpha.” 2012. Web.
Appendix
Figure 1 is a diagram that allows one to visualize the movement of conical photons towards the point of singularity in order to create a black hole as the star collapses (Powell 2000).
Figure 2 is a graphical representation of the conical nature of photons (Falcke & Friedrich 2003; Wolfram Alpha Research Company 2012).
Figure 3 is a computer enhanced picture taken by NASA of a black hole using an infrared camera on a satellite; notice that it looks similar to a hyperboloid of one sheet (NASA Blogs 2012).
Figure 4 is a graphical representation of the geometry of Schwarzschild spacetime that demonstrates the hyperbolic nature of black holes in regards to the analysis of photons (FalckeFriedrich 2003).
Figure 5 is a graphical representation of the hyperbolic nature of black holes (FalckeFriedrich 2003; Wolfram Alpha Research Company 2012).
Figure 6 is a graphical representation of the Schwarzschild radius of black holes (Powell 2000).
Figure 7 is a graphical representation of the spheres within black holes (Powell 2000).
Figure 8 is a graphical representation the changing radii within a black hole (Falcke & Friedrich 2003).
[1] See figure 1 in the Appendix
[2] See figure 2 in Appendix
[3] See figure 3 in Appendix
[4] See figure 4 in Appendix
[5] See figure 5 in Appendix
[6] See figure 6 in Appendix
[7]See figure 7 in Appendix
[8] See figure 8 in Appendix
[9] See figure 8 in Appendix