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Black Holes: Basic Mathematics

L. David Roper, roperld@vt.edu, roperld.com/personal/RoperLDavid

What is a Black Hole? “A black hole is a region in space where the pulling force of gravity is so strong that light is not able to escape. The

strong gravity occurs because matter has been pressed into a tiny space. This compression can take place at the

end of a star's life.”

Types of Black Holes 1. Primordial Black Holes: They are possibly formed shortly after the Big Bang and are the size of nuclei of atoms.

a. A primordial black hole would “evaporate” by Hawking Radiation by now, 13.82-billion years after the

Big Bang, if its mass is less than 19 1110 2 10 kg ,M where 302 10 kg mass of Sun .M

(Mass of the moon is 7.35 x 1022 kg. Mass of Tesla Model S car is 2 x 103 kg.) So, the minimum

primordial-black-hole mass is very large! No evidence exists for the existence of primordial black holes.

b. Later it will be shown that the radius of a black hole is proportional to the mass. The minimum radius is

19 1910 3 10 kmR , where 3 km radius of a black hole having the Sun's mass .R (Radius

of the proton is 198.41 10 km. ) So, the minimum primordial-black-hole radius is very small!

2. Stellar Black Holes: “A black hole formed by the gravitational collapse of a massive star. They have masses

ranging from about 5 to several tens of solar masses. The process is observed as a hypernova explosion or

as a gamma ray burst.” They are formed by red-supergiant stars with mass greater than 25M exploding. See

below for evidence of the existence of stellar black holes. The smallest-mass stellar black hole known is 3.8M

with a diameter of 24 km.

3. Supermassive Black Holes (SMBH): “The largest type of black hole, on the order of hundreds of thousands to

billions of solar masses ( M ) and is found in the centre of almost all currently known massive galaxies. In

the case of the Milky Way, the SMBH corresponds with the location of Sagittarius A and has a mass of about64.1 10 M and an event horizon of about 612 10 km . The smallest know SMBH is 45 10 M in the

nucleus of the dwarf galaxy RGG 118.

Gravitational waves due to ten pairs (binaries) of stellar black holes colliding have been detected by the Laser

Interferometer Gravitational-Wave Observatory (LIGO); the pairs’ and remnant masses ( M ):, energy radiate ( 2c M )

and spin ( /J cM ) were calculated to be:

Primary Mass 35.6 23.3 13.7 31.0 10.9 50.6 35.2 30.7 35.5 39.6

Secondary Mass 30.6 13.6 7.7 20.1 7.6 34.3 23.8 25.3 26.8 29.4

Remnant Mass 63.1 35.7 20.5 49.1 17.8 80.3 56.4 53.4 59.8 65.6

Energy Radiated 3.1 1.5 1.0 2.2 0.9 4.8 2.7 2.7 2.7 3.3

Spin 0.69 0.67 0.74 0.66 0.69 0.81 0.70 0.72 0.67 0.71

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The mathematics of black holes requires knowledge of the mathematics of Einstein’s theory of gravity, called General

Relativity.

Einstein’s Theory of General Relativity Knowledge of the special relativity theory is necessary to understand General Relativity.

Special Relativity Albert Einstein published his theory of Special Relativity in 1905. Its two postulates are:

Principle of Relativity: Fundamental laws of physics have the same mathematical form in all reference frames

that are moving with constant velocity relative to each other. Such reference frames are called “inertial frames”.

The speed of light, and all electromagnetic radiation, is a constant, 82.99792458 10 meters/second , for all

observers. This causes space and time to be related as shown below.

For two inertial frames moving at speed v relative to each other, the mathematics of the relation between space and

time, now called “spacetime”, is given by:

' ' '

' ' '

' '

' '

t t x t t x

x t x x t x

y y y y

z z z z

where2

1 and .

1

v

c

The velocities of a moving particle are different in the two inertial frames, , , and , ,x y z x y z . If the velocity of the

, ,x y z frame relative to the , ,x y z frame is ˆv vi in the x direction and the velocity of the particle relative to the

, ,x y z frame is ˆˆ ˆx y zu u i u j u k , then the particle velocities in the two inertial frames are related by:

21

xx

x

u vu

vu

c

21

zx

x

u vu

vu

c

2

2

1

1

y

y

x

uu

vu

c

2

2

1

1

y

y

x

uu

vu

c

2

2

1

1

zz

x

uu

vu

c

2

2

1

1

zz

x

uu

vu

c

3

General Relativity Albert Einstein published his theory of General Relativity in 1915. Is should be called Relativistic Gravity. Two of its

postulates are:

The theory has to reduce to special relativity when no gravity is present.

The theory has to reduce to Newtonian gravity when the velocity is small compared to c, the speed of light.

The mathematics of satisfying the first postulate is that there is a spacetime line element such that ds

2 2 2 2 2 2 2 2ds c dt dx dy dz c d , where d is the time interval measured by an observer moving relative to

the coordinate frame , , ,ct x y z with velocity 2 2 2

x y zv v v v and dt is the time interval measured by an observer

at rest.

Then 2 2 2 2 2

2 2 2 2 2 2

11 1 or ,

ds dx dy dz vd dt dt d d dt

c c dt dt dt c

as required in special relativity.

Here d is called the “proper time” and dt is called the “coordinate time”. Usually the speed of light, c, is set to be 1 in

this equation.

Short notation is3 3

2

0 0

ds dx dx dx dx

,

where 0 1 2 3, , , , , , and 1,1,1,1 .dx dx dx dx cdt dx dy dz

The repeated indices and imply a summation over the four indices 0,1,2,3 .

Here is called the metric for flat empty space.

0 1 2 3, , , , , , or , , , etc.x x x x ct x y z ct r , depending on the chosen coordinate system, is called the “four-

spacetime vector”.

Einstein postulated that general relativity must be a tensor equation that represents curved space. That is, the presence

of mass causes space to be curved. So, spacetime of special relativity now becomes spacetime of relativistic gravity.

Einstein’s equation is 4

8 GG g T

c

where

1Einstein tensor, cosmological constant, metric tensor,

2G R Rg g

Newton gravitational constant, stress-energy tensor, spacetime metric,

R = Ricci curvature tensor and Ricci curvature scalar.

G T g

R

These are complicated equations that define the Ricci curvature tensor and scalar in terms of the spacetime metric.

Usually and are set equal to 1 in Einstein's equation.c G The cosmological constant, , is very small and is set to 0

for solar and galactic calculations. Relativistic gravity does not have a force that moves particles; instead it moves

particles by curving space.

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Given a stress-energy tensor one must solve for the spacetime metric, g . There are two forms of the spacetime

metric: covariant and contravariant g g

such that1 for

0 for

g g

.

A useful equation is 1 where , the four-velocity vector.dx

g u u ud

Similarly, vectors can be covariant or contravariant .V V

The spacetime metric can be used to convert between

them: and V g V V g V

Geodesic Equation

The geodesic equation yields particle orbits due to the curvature of spacetime.

Once a spacetime metric is calculated, particle orbits (equations of motion) can be calculated using the geodesic

equation:

10 .

2

d dx dx dxg g

d d d d

There are more complicated versions of the geodesic equation. This one is easy to use because it explicitly depends on

the metric, .g

Schwarzschild Black Hole

The first calculation of g was done by Karl Schwarzschild in 1915 for a spherically-symmetric non-rotating mass in

otherwise empty space. The calculated Schwarzschild metric is

12 2 2 2 2 2 2 2

2

1 / 1 / , , sin in the spherical coordinate system , , , ,

where 2 / and mass of the spherical object.

s s

s

g x x r r c dt r r dr r d r d ct r

r GM c M

The coordinates , , ,ct r are the ones used by a far-away observer.

It turns out that 22 /sr GM c is the radius below which electromagnetic radiation cannot escape if the entire object’s

mass is inside that radius. Such an object is called a Black Hole!

Many detailed calculations have been done for a Schwarzschild black hole.

However, probably all black holes in existence have spin, which has a more complicated spacetime metric. For example,

the spin of the remnant black holes of the six binary-black-hole collisions that have been observed have median spin

parameter of 0.685 0.034 out of a maximum possible 1. So, to be realistic one should consider black holes with spin.

Newtonian and Schwarzschild Event Horizon

Event horizon = radius at which an outgoing particle must have a velocity of the speed of light, c, to escape.

Newtonian: 2

2

1 2,

2

mM GMmc G r

r c the Newtonian event horizon.

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Schwarzschild: For an outgoing particle: 12 2 2 2 20 and 0 1 / 1 / .s sd d c d r r c dt r r dr

12 2

2 21 1 .s sr rdt drc c

r d r d

It can be shown that 21 sr dt

c Er d

particle’s relativistic energy

per unit mass as measured at infinity: At 20 0.s

dtr r E c

d

That is, the energy is 0 at ,sr so the particle is stopped from escaping at 2

2,s

GMr

c the spherical event horizon.

The Newtonian and the Schwarzschild event horizons are the same.

Kerr Black Hole

In 1965 Roy Kerr calculated the Kerr metric, ,g for an axially-symmetric rotating black hole in otherwise empty space.

The calculated metric is:

2 22 2 2 2 2 2 2 2 4 sin

1 sin ,s s sr r r ra r rag x x c dt dr d r a d cdtd

where 2 2 2 2 2

2

2, , cos , and spin .s s

GM Jr a r a r a r r J

c Mc

Note that the metric does not depend on and t except through the differentials and that it has a cross term in

and .t

This metric’s coordinates , , ,t r are the oblate-spheroid coordinates:

2 2 2 2

22 2 2 2 2 2 4 2 2 2

sin cos , sin sin and cos .

22 , arccos and arctan .

2

x r a y r a z r

z yr x y z x y z a a x y z

r x

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Kerr Singularities: There are two singularities in the Kerr metric given above when

2 2 2 2 2cos 0 and 0sr a r a r r .

The first one is an actual singularity and the second one is a coordinate singularity which can be eliminated by choosing a

different coordinate system. However, it does have physical significance: It is the event horizon, a sphere below which

light cannot escape the black hole. The radius of the event horizon is

2 2

20

1 24 ( for + sign) Schwarzschild event horizon.

2s s

a

GMr r r a

c

The significance of the physical singularity is that 2 2 coss s

tt

r r r a r rg

changes sign from positive to

negative when passing through the singularity. The sign change occurs at 2 2 cos sr a r r =0 or

2 21 4 cos

2e s sr r r a . The + sign defines an oblate spheroid outside the event horizon, called the

ergosphere. The event horizon and ergosphere touch at the poles cos 1 and the event horizon is inside the

ergosphere at the equator cos 0 . Inside the ergosphere where 0ttg makes it act like a special metric

component, so particles inside the ergosphere must co-rotate with the black hole; this is called “frame dragging”.

Slow and Fast Kerr Spacetime: As we vary the parameter a , a profound change occurs at the extreme value a2 = GM2.

Following O’Neill (1995):

a2 = 0 gives Schwarzschild spacetime, as expected.

0 < a2 < GM2 gives slowly rotating Kerr spacetime, known in short as slow Kerr spacetime. This is the physical

situation.

a2 = GM2 gives extreme Kerr spacetime.

a2 > GM2 gives rapidly rotating Kerr spacetime, or fast Kerr spacetime. As discussed in D’Inverno (1992), the

spacetime singularity in fast Kerr spacetime is “naked”—that is, it is not hidden from the outside universe by an

event horizon. The consequences of such a scenario are so bizarre that Penrose has proposed a cosmic

censorship hypothesis, which forbids the existence of naked singularities.

For 0a the metric is the Schwarzschild metric.

2 2 2 2 2 2 2 2

0 0

0

For r 0 : sin where .s ds dt dr d d r a

2 2 2 2 2 2 2 2For 0 & r 0 : sin .sa ds dt dr r d r d , the flat empty spacetime in spherical coordinates.

Constants of Motion There are four constants of motion:

1. Rest mass of the particle, which I set to be 1, so that all other constants are per rest mass.

2. z component of angular momentum parallel to the spin axis.

3. Carter Constant: 2

2 2 2 2

2cos 1

sin

zQ C p a e

see below.

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4. Relativistic total energy .te p

Carter Constant

2

2 2 2 2

2cos 1

sin

zC p a e

Carter Constant, which is a function of three conserved quantities

, and .zp e For the Schwarzschild metric 2

20 .

sin

za C p

For equatorial motion:

2/ 2 .zC Some authors use the symbol instead of Q C for the Carter Constant.

Define 2

zL Q total angular momentum for the Schwarzschild metric 0 .a That is, for

2 20 : .x ya Q

Massive Particle Circular Orbits

Use the geodesic equation for the t component, component, component and put their results into the r

component geodesic equation.

Reminder: geodesic equation is1

0 .2

d dx dx dxg g

d d d d

Choose spherical coordinates such that the orbit is on the equator; i.e., 2 or sin 1, 0 and .2

dr

d

That is, consider only orbits around the equator relative to the spin axis.

Then 22

2 2 2 2 2 21 ,s s sr r a r ar

g x x dt dr r a d dtdr r r

where 2 2 and 1.sr a r r c .

When 2 22 2 2 2 2 21 1

0 4 2 2 4 .2 2

s k s s kr a r r r r r a GM GM a GM GM a r

The plus sign is the spherical event horizon; the minus sign is called the spherical Cauchy horizon, about which there are

differing interpretations of its significance.

Note that 0tt kg r at the event horizon and Cauchy horizon.

Kerr black hole has two infinite-redshift surfaces: 2 2 2cos for .

Jr GM GM a a GM

M

At poles: 2 2r GM GM a and at equator: 2 .r GM

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Time Component (= 1, x1 = t)

Note that g is time independent, so 0 constant for fixed .tt t tt t

d dt d dt dg g g g r

d d d d d

1 s stt t

r r adt d dt dg g e r

d d r d r d

1

At , ; = object's 4-velocity at .dt p dt

r e e ud m d

This e is the relativistic energy per unit mass at infinity.

Component (= 4, x4 = )

Note that g is t and independent, so 0 constant for fixed ,t t

d d dt d dtg g g g r

d d d d d

2

2 2 .s st

r a r ad dt d dtg g r r a

d d r d r d

2 2At , .d

r r rd

This z is the angular momentum per unit mass at infinity.

Combined and : t and = ,tt t t

dt d d dte g g g g

d d d d

9

2 2

and .t t tt

t tt t tt

g e g g e gdt d

d dg g g g g g

2 2

22 2 2 21 for / 2.s s s

t tt s

r a r a rg g g r a r a r r

r r r

22 2

2 2 2 2

1

and .

s s s s

s s

r a r a r a rr a e e

r rdt d r r

d r a r r d r a r r

Component (= 3, x3 = )

There is no component for equatorial motion.

r Component (= 2, x2 =r )

All metric components depend on r. Easiest calculation is to use 1 u u g u u

:

2 2 2

1 2tt rr t

dt dr d dt dg g g g

d d d d d

2 2 2222 2 2

1 1s s sr r a r adt r dr d dt dr a

r d d r d r d d

Substitute in the two equations above for and and rearrange :dt d

d d

22 2 22

2

2 3

11 1Or 1 .

2 2 2 2 2

ssa e r aerdr

E e E K Vd r r r

Where

2

2

22 2 2

2

11 effective conserved energy per unit mass, relativistic energy

2

1effective radial kinetic energy per unit mass and

2

11effective potential ener

2

s

s

E e e

drK r

d

a e r aeV r r

r r r

.

gy per unit mass

The Schwarzschild 0a radial equation is:

2 2 22 2

2

2 3 2

1 1 11 ; .

2 2 2 2 2 2

s s ss

r r rdrE e E K V V r r

d r r r r r r

Here are graphs for specific orbit parameters of the Newton, Schwarzschild and Kerr potentials:

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Newton: 21

2sV r r

r r

2

min

22.45 .

s

rr

Schwarzschild (a = 0) & Kerr (for a = 0.3): 22

2

1

2

ss

rV r r

r r r

&

22 2 2

2

11

2

s

s

a e r aeV r r

r r r

11

Schwarzschild (a = 0) & Kerr (for a = 0.3) & Newton:

Schwarzschild: 2 2 2

min

13 1.4s

s

r rr

; 2 2 2

max

13 1.05s

s

r rr

.

Note that the square term is the extra beyond Newton.

Kerr:

22 2 2 4 2 4 2 2 2 2 2 2

min

22 2 2 4 2 4 2 2 2 2 2 2

max

11 1 3 2 1 3 2 2.36

11 1 3 2 1 3 2 0.52

s s

s

s s

s

r a e a e r a e ar e aer

r a e a e r a e ar e aer

.

Note that for 0a Kerr reduces to Schwarzschild.

Kerr Orbits

22 2 2

2

11

2

s

s

a e r aeV r r

r r r

Kerr maxima & minima:

22 2 2 4 2 2 2 2 2 2 2 2

min

22 2 2 4 2 2 2 2 2 2 2 2

max

11 1 3 2 1 3 2

11 1 3 2 1 3 2

s s

s

s s

s

r a e a e r a e ar e aer

r a e a e r a e ar e aer

(This calculation was done using Scientific Workplace.)

For 0a the Kerr result reduces to Schwarzschild.

For the Kerr potential-vs-radius graph above min max2.36 & 0.52r r .

From the graphs it is obvious that minr is a stable orbit and maxr is an unstable orbit.

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0 1 2 3 4 5 6 7 8 9 10

Kerr Potential (L=0.7, e=0.2, rs=0.4)

a = 0 a = 0.3 Newton

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First derivative 0dV

dr locates maxima and minima and second derivative indicates whether concave up

2

20

d V

dr or

concave down2

20.

d V

dr

Newton: 2

min

2.

s

rr

For the Newton potential-vs-radius graph above

22

min

2 0.722.45

0.4s

rr

.

Schwarzschild: 2 2 2 2 2 2

min max

1 13 & 3 .s s

s s

r r r rr r

Note that the square-root term is the extra term beyond Newton.

For the Schwarzschild potential-vs-radius graph above min max1.4 & 1.05r r .

These radii are real only if 3 .sr When 3 sr

2

2

min

33 6 ,

s

s ISCO

s s

rr r GM r

r r

the Schwarzschield innermost stable circular orbit (ISCO).

Innermost Stable Equatorial Circular Orbits (ISCO)

The radius of the Kerr innermost stable orbit is found by solving

2

2

2

11 , 0 and 0. 0

2

eff eff

eff

dV d V dre V r

dr dr d

22 2 2

2 3

1 3.eff

a e GM aeGMV r

r r r

The orbit equation is quartic in r a and quadratic in a r . The physical solution is a root of this equation:

2 26 for 0, the Schwarzschild value

6 3 8 09 for 1, the maximum value

r GM ar GMr a a GMr

r GM a

Let 2 2/ and / : 6 3 8 0 .z r GM a Gm z z z

2 22 3 3

22

3 3 1 1 3 1 1 :

.16

3 3 3 RISCO

w

rz w w

GMw

See Appendix B, p.33 of https://arxiv.org/pdf/0903.3684.pdf and equation 2.26 on p.60 of

https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=1214&context=etd

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The red curve is ISCO for the co-rotating orbit and the blue curve is for the counter-rotating orbit.

The vertical cyan line at 0.685 is the median and average remnant spin of the 6 LIGO black-hole collisions detected.

Marginally-Bound Orbits:

Marginally-bound orbits occur when the potential minimum becomes an inflection point.

Conditions for a particle orbit to be marginal bound are:

1; 0 and 0.dV r

e V rdr

The marginally-bound equatorial radius is given by:

4 3 2 2 5 2 24 4 0 or 4 4 0 2 2 1 .mba a a a a a

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IMCO+ is the counter-orbit; IMCO- is the co-orbit

The vertical cyan line at 0.685 is the median and average remnant spin of the 6 LIGO black-hole collisions detected.

Marginally-Bound Inclined Orbit

Consider orbits inclined at an angle defined by 2cos where .zzi L Q

L

For the Schwarzschild metric L total angular momentum of the orbiting particle.

The angle i is latitudinal from the equator: Equatorial: cos 1i ; Polar: sin 1.i

Note that for 0 cos 1i i co-rotating (+) and counter-rotating (-) equatorial orbits.

The equation to be solved for the marginally-bound inclined orbit is

4 3 2 2 2 4 2 2 2 24 1 3sin sin 4 cos sin 0r r a r i a i ar i r r a i

For equatorial: 4 3 2 2 2 2 20 4 4 0 4 4 0 2 2 1 .mbi r r a r ar r r r a a r a a

For polar: 4 3 2 2 44 2 02

i r r a r a

No analytical solution.

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This was calculated numerically using the Excel Solver procedure.

The vertical cyan line at 0.685 is the median and average remnant spin of the 6 LIGO black-hole collisions detected.

This was calculated numerically using the Excel Solver procedure.

The vertical cyan line at 0.685 is the median and average remnant spin of the 6 LIGO black-hole collisions detected.

Between the ergosphere equatorial radius and the 45° ergosphere radius the 45° orbit passes through the ergosphere

half of the time. Below the 45° ergosphere radius it is always in the ergosphere.

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Last Stable Equatorial Elliptical Orbit

22 2

0 0

0

2 2 44 2 2

22

0 13

1

perigee of orbit where eccentricity of orbit and

apogee of orbit1

counter-orbit13 16 3 1 3 where

co-orbit

1 3 2 3 1 15 311 3 3

3

lr e

e

el e z z e e e

z

e e e e e ez e e e z

z

3

6 3 32 2 2 4 2 2

1 2

4 3 36 2 3

2

3 1 1 1 3 3 18 459 3 15 3 24 3

1 1 1 3 1 3

z e e e e e e e e e z

z e e e e e

https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=1214&context=etd

counter-orbit apogee; co-orbit apogee

counter-orbit perigee; co-orbit perigee

r r

r r

The vertical cyan line at 0.685 is the median and average remnant spin of the 6 LIGO black-hole collisions detected.

17

Last Stable Inclined Elliptical Orbit An Analytical and Numerical Treatment of Inclined Elliptical Orbits about a Kerr Black Hole (Eqs. 3.62 & 3.B7) derived a

ninth-order radial equation for this system, for which there is no analytical solution.

2

12 13 14

4

15

4 2 2 2 2

12

23 3 2 4

2 2

13

3 25 2

14

1

4

where 1 2 3 2 1 2 3

1 2 3 1 , .

3 1 2 2 ,

1 1 2 2

z z zQ

z

z l e l e e l e e e

l e e e l e

z e e l e l

z l e e l l e

9 8 2 2 2 7

2 2 6

2 2 3 2 2 5

22 2 2 4

24 3 2 3

4 3 36 2 4 6 24 4

4 1 2 7

1 5 3 9 16 8 24

8 1 3 4 2 6

8 1 2 2 1

l e l e e e e l

e Q e l

e e e e Q e e l

Q e e e e l

Q e Q e e l

18

Photon Innermost Orbits

Photon orbits, prograde and retrograde (inclined, equatorial and polar):

5 4 2 3 2 2 2 2 2 4 2 2 2 2 2 2

1

1

Inclined orbit:

3 2 sin 2 2 cos 3 3 sin 0.

2Equatoral orbit : 2 1 cos cos ,

3

2 30 : 2 1 cos cos 0 2 3

3 2

21: 2 1 cos cos

3

p

r r a r a r a i a r ar i r a r a r a i

r a

Ja r GM GM GM r

M

Ja r GM

M

1

1

22 1

3/22

1 2 2 4

2 11: 2 1 cos cos 1 2

3 2

1 1Polar orbit: 1 2 1 / 3 cos cos

3 1 / 3

p

p

polar

GM GM r

Ja r GM GM GM r

M

ar a

a

19

This was calculated numerically using the Excel Solver procedure.

The vertical cyan line at 0.685 is the median and average remnant spin of the 6 LIGO black-hole collisions

detected

20

This was calculated numerically using the Excel Solver procedure.

The orbit equation could not be satisfied when the 45° orbit was to cross the horizon.

The vertical cyan line at 0.685 is the median and average remnant spin of the 6 LIGO black-hole collisions

detected.

Between the ergosphere equatorial radius and the 45° ergosphere radius the 45° orbit passes through the ergosphere

half of the time. Below the 45° ergosphere radius it is always in the ergosphere.

Black Holes A Black Hole is an object that does not have a surface outside of 2GM.

Event Horizon A spherical surface with the Schwarzschild radius, rs=2GM, is also called the Event Horizon. Any object inside cannot

escape because the escape velocity would have to be greater than the speed of light.

Black Hole Density

Inside the Event Horizon the density is

21

33 3 3 2 2

3 3 3 3.

4 / 3 4 32 84 2s s s

mass M M M

volume r r G M GrGM

The larger or more massive the black hole the smaller the density.

Surface Gravity

1.

4 2 s

G

M r

Rotating-Black-Hole Surface Gravity

2

2 2

12 T, where angular velocity of event horizon,

4

where .

MM

a

r a

Black-Hole Thermodynamics

Units

28 30 34 237.426 10 / ; 1.9891 10 ; 1477 ; 1.0546 10 ; 1.3807 10 / . .G m kg M kg GM m J s k J K

Event-horizon temperature: 3

8

cT

kGM

Entropy:

2 2 2

2 3, where 4 16 , Boltzmann's constant & Planck length.

4s P

P

kA GS A r G M k

c

So, 2 33 2 3

2 24; Kerr: .

4

sr kckc GM kcS A S r a

G G

Hawking Radiation

Inside the event horizon a particle with mass can have negative energy/ mass:2

1GM dt

er d

.

22

Vacuum fluctuations create particle-antiparticle pairs. If one occurs near the event horizon, the negative-energy particle

can pass through the event horizon and decrease the black-hole’s energy, the decrease radiating away by means of the

positive-energy particle. Since a photon is its own antiparticle and has zero mass, it is most likely that the particle-

antiparticle pair is two photons. So the black-hole black-body radiation is photons.

Emitted Thermal Black-Body Radiation

2 2

1 1; Kerr: .

8 4 2 2s

rT T

GMk r k r a r

961.7 10, : 60 .

8 8 /

M KT T M M T nK

kGM kGM M M M

23

Black-Hole Lifetime

Stefan-Boltzmann Law for energy radiation from a black-body:

4/ , where Stefan-Boltzmann constant.dE dt A T

So, the mass-loss equation of a black hole is

4 424 2

4 42 2 2

3 4 2

3 4 2 3 4 20

2 3

4 4 0

3

4 4 28 8

4 2 .8 256

256 256So, .

3

256So, he lifetime of a black ho le st i

life

s

lifeM

life

dE dMA T r GM

dt dt kGM kGM

G M MkG k G

k G k GM dM M dt

k

4 3 43 3

4 4

3

67

2048.

3 3

1.095 10

s

life

kGM r

G G

Myr

M

Universe age = 13.82 x 109 years:

9 67 193min

19 30 11

min

13.82 10 / 1.095 10 1.0807 10

1.0807 10 1.98855 10 kg/ 2.149 10 kg

M M M

M M M

Minimum black-hole radius: 19 19

min min3.0 km, 3.0 km 1.0807 10 3.2421 10 km SunBlackHoler r r

24

Gravitational Waves Gravity Wave Events Detected https://en.wikipedia.org/wiki/List_of_gravitational_wave_observat

ions

Ev ent Name

Merger

Distance (Giga light

years)

Energy Radiated

(c2MS)

Primary Mass (c2MS)

Secondary Mass (c2MS)

Remnant Mass (c2MS)

Spin (cJ/GM2) Energy

Emitted

GW170608 BH+BH->BH 1.109 0.85 12 7 18 0.69 GW

GW170608 BH+BH->BH 1.044 0.9 10.9 7.6 17.8 0.69 GW

GW151226 BH+BH->BH 1.435 1.0 13.7 7.7 20.5 0.74 GW

GW170104 BH+BH->BH 3.132 2.2 31.0 20.1 49.1 0.66 GW

GW170814 BH+BH->BH 1.892 2.7 30.7 25.3 53.4 0.72 GW

GW170809 BH+BH->BH 3.229 2.7 35.2 23.8 56.4 0.70 GW

GW170818 BH+BH->BH 3.327 2.7 35.5 26.8 59.8 9.67 GW

GW150914 BH+BH->BH 1.403 3.1 35.6 30.6 63.1 0.69 GW

GW170823 BH+BH->BH 6.034 3.3 39.6 29.4 65.6 0.71 GW

GW170729 BH+BH->BH 8.971 4.8 50.6 34.3 80.3 0.81 GW

GW170817 NS+NS->NS 0.130 0.04 1.46 1.27 2.8 0.89 GW+Kilonova

Future: BH+NS->BH GW+Kilonova

BH = Black Hole; NS = Neutron Star

1 light year = 9.46 x 1015 meters; Age of universe = 13.8 gigayears.

In 5 gigayears the Sun has radiated 0.0003 (c2MS) energy.

Binary and Remnant Black-Hole Masses:

25

26

Solar mass M☉ 1.989 × 1033 g

Earth mass M⊕ 5.974 × 1027 g

electron mass me 9.1094 × 10-28 g 0.511 MeV/c2

proton mass mp 1.6726× 10-24 g 938.272 MeV/c2

neutron mass mn 1.6749× 10-24 g 939.563 MeV/c2

up

u

2.3±0.7 ± 0.5 MeV/c2 = 20.127 × 10-28 g

down d

4.8±0.5 ± 0.3 MeV/c2= 22.627 × 10-28 g

17.827 × 10-28 g/Mev; proton = 2u +1d quarks; neutron = 1u + 2d quarks

The nucleus is unstable if the neutron-proton ratio is less than 1:1 or greater than 1.5.

Average N/P = 1.34 (67/50 Sn)

Average nucleon mass in stable atoms = (50mP + 67mN)/(50 + 67) = 1.6739 × 10-24 g.

Nucleons in the Sun: 1.989 × 1033 g/(1.6739 × 10-24 g = 1.188 x 1057

Quarks in the Sun: 3 x 1.188 x 1057 = 3.6 x 1057

Event horizon for a BH with solar mass = 3 km

Volume of BH with solar mass = 4 x 27 km3/3 = 113.10 km3

Mass density of BH with solar mass = 1.989 × 1033 g/113.10 km3= 1.759 × 1031 g/ km3

Quark density filling a BH with solar mass = 3.6 x 1057/113.10 km3 = 3.183 x 1055 quarks/km3

Average mass of u & d quarks in stable atoms =

(50*(2*20.127 + 22.627)+67*(20.127 + 2*22.627))/3/(50+67) × 10-28 g = 21.438 × 10-28 g

Quark mass density in a BH with solar mass =

27

References The Kerr Spacetime: A Brief Introduction

Black Holes (Schwarzschild and Kerr)

Inside the Schwarzschild Event Horizon

Rotating Black Holes (Bardeen, Press & Teukolsky)

Spherical Null Geodesics of Rotating Kerr Black Holes

Marginally Bound (Critical) Geodesics of Rapidly Rotating Black Holes

Innermost Stable Circular Orbits of Spinning Test Particles in Schwarzschild and Kerr Space-times

Conservative Corrections to the Innermost Stable Circular Orbit of a Kerr Black Hole

Kerr Black Holes: II. Precession, Circular Orbits and Stability

A Study of Elliptical Last Stable Orbits about a Massive Kerr Black Hole

An Analytical and Numerical Treatment of Inclined Elliptical Orbits about a Black Hole

Polar Orbits in the Kerr Space-time

Solutions of Einstein’s Equations and Black Holes

Kerr Black Holes: Precession, Circular Orbits and Stability

Black Holes: A Physical Route to the Kerr Metric

Nearly Horizon Skimming Orbits of Kerr Black Holes

Kerr Metric: Rotating Black Holes

What are the Inner and Outer Horizons of a Black Hole?

White Holes and Blue Sheets; What is the Ring Singularity

Binary Black Hole

https://physics.stackexchange.com/questions/103232/how-can-a-singularity-in-a-black-hole-rotate-if-its-just-a-

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just-a-point