Findley 1

MAT 272

Black Hole Topology A Calculus III Version

Christina Findley

Section Number 72375

Professor Kathleen Brewer

December 10 2012

Findley 2

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 = ( ) = 4πG where G is the

gravitational constant (6674x10-11 Nm2Kg2) and is the density of the object the formula for

the force translates to F=GMR2 where M is the mass and R the distance of separation (Waner

2002) After Newton an English astronomer named John Michell utilized Newtonrsquos laws to

hypothesize that ldquoan object massive enough to have an escape velocityrdquo the velocity required for

an object to escape the gravitational force present on the star ldquogreater than the speed of lightrdquo

which is about 299793x108 ms 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 ldquoinvisible starsrdquo (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 (Falcke amp Friedrich

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

z2

c2=x2

a2 +y2

b2

Findley 3

(Falcke amp Friedrich 2003 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 direction2

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 mathematically3 Using

cylindrical coordinates Schwarzschild constructed one of the first models of a black hole as a

hyperboloid of one sheet of the form

1= x2

a2 +y2

b2 minus z2

c2

(Falcke amp 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 direction5

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 escape6 This radius is defined by the following equation where G is the

1 See figure 1 in the Appendix2 See figure 2 in Appendix3 See figure 3 in Appendix4 See figure 4 in Appendix5 See figure 5 in Appendix6 See figure 6 in Appendix

Findley 4

gravitational constant m is the mass c is the speed of light in a vacuum and Rs is the

Schwarzschild radius7

R s=2Gm

c2

This radius holds significance since a sphere of radius r = a changes according to

Schwarzschildrsquos physical metric of a black hole that was derived using spherical coordinates

d s2=(1+ ar )

4

(d r2+r2 dθ2+sin2θd ϑ2 )

(Falcke amp Friedrich 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 (Falcke amp Friedrich 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

7 See figure 7 in Appendix8 See figure 8 in Appendix9 See figure 8 in Appendix

Findley 5

Works Cited

Falcke Heino and 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

NASA Blogs ldquoMonster Black Holesrdquo 2012

httpnasa-spacestationinfoblog spotcom2012_02

_01_archivehtml

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 lthttplibrarythinkquestorg25715discoveryconceivinghtmgt

Powell Richard ldquoInside a Black Holerdquo Web 18 Apr 2000

lthttpnrumianofreefrEstarsint_bhhtmlgt

Steinberg Daniel No Escape The Truth About Black Holes Space Telescope Science

Institute Mar 2005 Web 5 Apr 2012

lthttpamazing-spacestscieduresourcesexplorationsblackholesteacher

sciencebackgroundhtmlgt

Waner Stefan The Einstein Field Equations and Derivation of Newtons Law HOFSTRA

University Jan 2002 Web 5 Apr 2012

lthttppeoplehofstraedustefan_wanerdiff_geomSec14htmlgt

Wolfram Alpha Research Company ldquoWolfram Alphardquo 2012 Web

httpwwwwolframalphacom

Findley 2

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 = ( ) = 4πG where G is the

gravitational constant (6674x10-11 Nm2Kg2) and is the density of the object the formula for

the force translates to F=GMR2 where M is the mass and R the distance of separation (Waner

2002) After Newton an English astronomer named John Michell utilized Newtonrsquos laws to

hypothesize that ldquoan object massive enough to have an escape velocityrdquo the velocity required for

an object to escape the gravitational force present on the star ldquogreater than the speed of lightrdquo

which is about 299793x108 ms 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 ldquoinvisible starsrdquo (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 (Falcke amp Friedrich

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

z2

c2=x2

a2 +y2

b2

Findley 3

(Falcke amp Friedrich 2003 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 direction2

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 mathematically3 Using

cylindrical coordinates Schwarzschild constructed one of the first models of a black hole as a

hyperboloid of one sheet of the form

1= x2

a2 +y2

b2 minus z2

c2

(Falcke amp 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 direction5

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 escape6 This radius is defined by the following equation where G is the

1 See figure 1 in the Appendix2 See figure 2 in Appendix3 See figure 3 in Appendix4 See figure 4 in Appendix5 See figure 5 in Appendix6 See figure 6 in Appendix

Findley 4

gravitational constant m is the mass c is the speed of light in a vacuum and Rs is the

Schwarzschild radius7

R s=2Gm

c2

This radius holds significance since a sphere of radius r = a changes according to

Schwarzschildrsquos physical metric of a black hole that was derived using spherical coordinates

d s2=(1+ ar )

4

(d r2+r2 dθ2+sin2θd ϑ2 )

(Falcke amp Friedrich 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 (Falcke amp Friedrich 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

7 See figure 7 in Appendix8 See figure 8 in Appendix9 See figure 8 in Appendix

Findley 5

Works Cited

Falcke Heino and 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

NASA Blogs ldquoMonster Black Holesrdquo 2012

httpnasa-spacestationinfoblog spotcom2012_02

_01_archivehtml

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 lthttplibrarythinkquestorg25715discoveryconceivinghtmgt

Powell Richard ldquoInside a Black Holerdquo Web 18 Apr 2000

lthttpnrumianofreefrEstarsint_bhhtmlgt

Steinberg Daniel No Escape The Truth About Black Holes Space Telescope Science

Institute Mar 2005 Web 5 Apr 2012

lthttpamazing-spacestscieduresourcesexplorationsblackholesteacher

sciencebackgroundhtmlgt

Waner Stefan The Einstein Field Equations and Derivation of Newtons Law HOFSTRA

University Jan 2002 Web 5 Apr 2012

lthttppeoplehofstraedustefan_wanerdiff_geomSec14htmlgt

Wolfram Alpha Research Company ldquoWolfram Alphardquo 2012 Web

httpwwwwolframalphacom

Findley 3

(Falcke amp Friedrich 2003 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 direction2

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 mathematically3 Using

cylindrical coordinates Schwarzschild constructed one of the first models of a black hole as a

hyperboloid of one sheet of the form

1= x2

a2 +y2

b2 minus z2

c2

(Falcke amp 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 direction5

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 escape6 This radius is defined by the following equation where G is the

1 See figure 1 in the Appendix2 See figure 2 in Appendix3 See figure 3 in Appendix4 See figure 4 in Appendix5 See figure 5 in Appendix6 See figure 6 in Appendix

Findley 4

gravitational constant m is the mass c is the speed of light in a vacuum and Rs is the

Schwarzschild radius7

R s=2Gm

c2

This radius holds significance since a sphere of radius r = a changes according to

Schwarzschildrsquos physical metric of a black hole that was derived using spherical coordinates

d s2=(1+ ar )

4

(d r2+r2 dθ2+sin2θd ϑ2 )

(Falcke amp Friedrich 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 (Falcke amp Friedrich 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

7 See figure 7 in Appendix8 See figure 8 in Appendix9 See figure 8 in Appendix

Findley 5

Works Cited

Falcke Heino and 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

NASA Blogs ldquoMonster Black Holesrdquo 2012

httpnasa-spacestationinfoblog spotcom2012_02

_01_archivehtml

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 lthttplibrarythinkquestorg25715discoveryconceivinghtmgt

Powell Richard ldquoInside a Black Holerdquo Web 18 Apr 2000

lthttpnrumianofreefrEstarsint_bhhtmlgt

Steinberg Daniel No Escape The Truth About Black Holes Space Telescope Science

Institute Mar 2005 Web 5 Apr 2012

lthttpamazing-spacestscieduresourcesexplorationsblackholesteacher

sciencebackgroundhtmlgt

Waner Stefan The Einstein Field Equations and Derivation of Newtons Law HOFSTRA

University Jan 2002 Web 5 Apr 2012

lthttppeoplehofstraedustefan_wanerdiff_geomSec14htmlgt

Wolfram Alpha Research Company ldquoWolfram Alphardquo 2012 Web

httpwwwwolframalphacom

Findley 4

gravitational constant m is the mass c is the speed of light in a vacuum and Rs is the

Schwarzschild radius7

R s=2Gm

c2

This radius holds significance since a sphere of radius r = a changes according to

Schwarzschildrsquos physical metric of a black hole that was derived using spherical coordinates

d s2=(1+ ar )

4

(d r2+r2 dθ2+sin2θd ϑ2 )

(Falcke amp Friedrich 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 (Falcke amp Friedrich 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

7 See figure 7 in Appendix8 See figure 8 in Appendix9 See figure 8 in Appendix

Findley 5

Works Cited

Falcke Heino and 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

NASA Blogs ldquoMonster Black Holesrdquo 2012

httpnasa-spacestationinfoblog spotcom2012_02

_01_archivehtml

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 lthttplibrarythinkquestorg25715discoveryconceivinghtmgt

Powell Richard ldquoInside a Black Holerdquo Web 18 Apr 2000

lthttpnrumianofreefrEstarsint_bhhtmlgt

Steinberg Daniel No Escape The Truth About Black Holes Space Telescope Science

Institute Mar 2005 Web 5 Apr 2012

lthttpamazing-spacestscieduresourcesexplorationsblackholesteacher

sciencebackgroundhtmlgt

Waner Stefan The Einstein Field Equations and Derivation of Newtons Law HOFSTRA

University Jan 2002 Web 5 Apr 2012

lthttppeoplehofstraedustefan_wanerdiff_geomSec14htmlgt

Wolfram Alpha Research Company ldquoWolfram Alphardquo 2012 Web

httpwwwwolframalphacom

Findley 5

Works Cited

Falcke Heino and 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

NASA Blogs ldquoMonster Black Holesrdquo 2012

httpnasa-spacestationinfoblog spotcom2012_02

_01_archivehtml

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 lthttplibrarythinkquestorg25715discoveryconceivinghtmgt

Powell Richard ldquoInside a Black Holerdquo Web 18 Apr 2000

lthttpnrumianofreefrEstarsint_bhhtmlgt

Steinberg Daniel No Escape The Truth About Black Holes Space Telescope Science

Institute Mar 2005 Web 5 Apr 2012

lthttpamazing-spacestscieduresourcesexplorationsblackholesteacher

sciencebackgroundhtmlgt

Waner Stefan The Einstein Field Equations and Derivation of Newtons Law HOFSTRA

University Jan 2002 Web 5 Apr 2012

lthttppeoplehofstraedustefan_wanerdiff_geomSec14htmlgt

Wolfram Alpha Research Company ldquoWolfram Alphardquo 2012 Web

httpwwwwolframalphacom

Findley 6

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 amp Friedrich 2003

Wolfram Alpha Research Company 2012)

Findley 7

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 space time that

demonstrates the hyperbolic nature of black holes in regards to the analysis of photons

(Falcke amp Friedrich 2003)

Findley 8

Figure 5 is a graphical representation of the hyperbolic nature of black holes (Falcke amp Friedrich

2003 Wolfram Alpha Research Company 2012)

Figure 6 is a graphical representation of the Schwarzschild radius of black holes (Powell 2000)

Findley 9

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 amp Friedrich

2003)

Findley 7

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 space time that

demonstrates the hyperbolic nature of black holes in regards to the analysis of photons

(Falcke amp Friedrich 2003)

Findley 8

Figure 5 is a graphical representation of the hyperbolic nature of black holes (Falcke amp Friedrich

2003 Wolfram Alpha Research Company 2012)

Figure 6 is a graphical representation of the Schwarzschild radius of black holes (Powell 2000)

Findley 9

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 amp Friedrich

2003)

Findley 8

Figure 5 is a graphical representation of the hyperbolic nature of black holes (Falcke amp Friedrich

2003 Wolfram Alpha Research Company 2012)

Figure 6 is a graphical representation of the Schwarzschild radius of black holes (Powell 2000)

Findley 9

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 amp Friedrich

2003)

Findley 9

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 amp Friedrich

2003)