Having read a number of books on cosmology and particle physics, I found my-self raking through 5 or 6 books or looking on the web as I tried to remember some tangible fact that had interested me. In the end, I decided to gather this info and post it under various headings as blogs on MySpace. With the introduction of LaTeX at Physics Forums, I decided to move a couple of them over here. Some are a year old, some are more recent. MySpace blogs
Black Holes - Planck Unit?
The following is an attempt to establish simple parameters for the ring singularity* in a rotating black hole based on Planck density.
*Regardless of how compact matter is, if it has density then technically it's not a singularity but in theory, the energy that makes up both matter and space combines at Planck dimensions, creating a quantum 'foam' supported by either Loop Quantum Gravity or strings so even though it has dimensions, it might still be described as a singularity as it is a combination of both matter and space (and probably time) rather than just collapsed matter in space. For ease of reference, the term singularity is used when referring to the collapsed core of a black hole.
Planck density* = 5.155x10^96 kg/m^3 (or 5.155x10^87 tonnes/cm^3) (Based on Planck mass, 2.176x10^-8 kg and Planck length, 1.616x10^-35 m).
'This is a unit which is very large, about equivalent to 10^23 solar masses (the observable universe) squeezed into the space of a single atomic nucleus. At one unit of Planck time (5.391x10^-44 s) after the Big Bang, the mass density of the universe is thought to have been approximately one unit of Planck density.' http://en.wikipedia.org/wiki/Planck_units
*Planck units are sometimes (humorously) referred to as 'God's Units' as they are based on the properties of free space and not on the properties of any prototype, object or particle (that could be thought of as arbitrarily chosen). They are also referred to as natural units because the origin of their definition comes only from properties of nature and not from any human construct. There are 5 basic Planck units, the above shows Planck length, mass and time; the other two are charge (1.876x10^-18 C) and temperature (1.417x10^32 K). They were established in the early 20th century by Max Planck, who is considered the founder of quantum theory. It's also postulated that Planck matter is the constitute energy of both matter and space itself and that under extreme gravitational collapse, in theory, will be sustained from any further collapse (i.e. into a singularity) due to loop quantum gravity.
The Schwarzschild radius of a 3 sol mass is 8861.099 m. Based on an overall mass of 5.967x10^30 kg and a volume of 2.914x10^12 m^3, the average density at the point a 3 sol mass collapses into a black hole would be 2.048x10^18 kg/cm^3. As the core collapses into something beyond quark matter, pulling the rest of the star beyond the Schwarzschild radius, light would begin to free fall towards the surface of the collapsing mass, unable to escape from the gravity (the escape velocity for a sphere with this mass and radius would exceed 300,000 km/s, the speed of light), hitting the surface of the sphere, compacting the sphere further.
It seems to be an accepted fact that photons are massless but have energy due to their high momentum (E=pc), the light coming into the black hole would be highly blueshifted (x-rays, gamma rays) and the photons energy would be converted to mass as it collided with the collapsing star, contributing further to the black holes mass and compactness (though the mass of the black hole might more or less stay the same as Hawking radiation in contrast slowly evaporates the black hole). So where does the energy for the highly blueshifted light come from? Light trying to escape the black holes gravity well is 'drained' and highly redshifted, it's as if it's put through a press, the energy seemingly extracted and added to light falling into the gravity well which becomes highly blueshifted.
In general relativity, the source of black holes are considered geometric singularities, in quantum mechanics, they are speculated as having Planck density (5.155x10^90 kg/cm^3), the maximum energy density allowing in current physics.
Theoretical fundamental particles such as preons or strings are approx. 10^-33 m in size which are close to the Planck length, the smallest measurement currently used in physics (1.616x10^-35 m). If strings or preons are at the heart of all quarks and leptons, they would normally be at their closest (in ground state) ~10^-15 m (a Fermi), the distance between quarks within a nucleon. Possibly under great pressure the quarks would break down (approx. 10^20 kg/cm^3) and under greater pressure, the preons/strings would break down also and the pure quanta of energy that reside at the very core of fundamental particles might compact to something in the region of Planck density.
If the core of a star about the mass of 3 of our suns collapsed beyond the Schwarzschild radius (8861.099 m) then it's possible it could collapse all the way to Planck density. For a mass the size of the sun (1.989x10^30 kg) this would result in a sphere with a radius of 4.516x10^-23 m or ~45 yoctometre (a yoctometre or ym is 10^-24 m). For the supermassive black hole at the centre of our galaxy which is predicted to have a mass of 3.6 million solar masses (7.161x10^36 kg), the core radius at Planck density would be 6.922x10^-21 m or ~7 zeptometre (a zeptometre or zm is 10^-21 m), the Schwarzschild radius (event horizon) would be 1.063x10^10 m or ~10.6 million km (our Sun has a radius of 0.696 million km).
Basics for a 3 sol mass Schwarzschild (static) black hole-
3 solar masses = 3 x 1.989x10^30 kg = 5.967x10^30 kg
Gravitational radius (considered the unit of measurement for black holes)-
Rg = Gm/c^2 = 4430.550
Event horizon (Schwarzschild radius or Rs)-
Rs = 2Gm/c^2 = 8861.099 m
Photon sphere-
Rph = 3Gm/c^2 = 13,291.648 m
Marginally stable orbit (normally the inner edge of the accretion disk)-
Rms = 6Gm/c^2 = 26,583.297 m
where G- gravitational constant, m- mass in kg and c- speed of light
Core radius based on Planck density for a static 3 sol mass black hole-
Planck density = 5.155x10^96 kg/m^3
Sphere volume (volume = mass/density)-
[tex]V_{p}=\frac{m_{bh}}{\rho_{p}}=\frac{5.967\cdot10^{30}}{5.155\cdot10^{96 }}=1.158\cdot10^{-66}\ m^3[/tex]
where VP- volume based on Planck density, mbh- black hole mass in kg, ρP- Planck density
sphere radius-
[tex]V=\frac{4}{3}\pi r^{3}\ \ \ \Rightarrow\ \ \
r=\sqrt[3]{\frac{3V}{4\pi}}[/tex]
incorporating VP-
[tex]r=\sqrt[3]{\frac{3\cdot1.158\cdot10^{-66}}{4\pi}}= 6.514\cdot10^{-23}\,m[/tex]
core radius for a 3 sol mass static black hole = ~65 yoctometre (a yoctometre or ym is 10^-24 m)
Ring singularity
The collapse of a large star with a fast rotation may create a spinning black hole. As the core becomes smaller under pressure, it spins faster. This causes the sphere to flatten. Massive centrifugal forces cancel out the inward force of gravity and create a spinning ring of degenerate matter which eventually collapses into a ring singularity. This is also called a Kerr black hole (or Kerr-Newman black hole if charge is included for). It's believed that most black holes are like this in nature due to the fact that most stars spin.
Based on the above, the Planck matter could take the form of a ring rather than a sphere.
A ring singularity for a rotating (Kerr) black hole could be based on a torus with an absolute minimum cross section of 2.6115x10^-70 m^2 (Planck length^2). Imagine the ring singularity is made up of Planck units, 1.616x10^-35 m in length, each unit weighing Planck mass (2.176x10^-8 kg). Based on a black hole of 3 solar masses, the maximum circumference of the ring would be-
max ring circumference-
[tex]c_{rs}=\frac{m_{bh}\cdot l_{p}}{m_{p}}[/tex]
where mbh- black hole mass, lP- Planck length and mP- Planck mass
[tex]c_{rs}=\frac{5.967\cdot10^{30}\cdot1.616\cdot10^{-35}}{2.176\cdot10^{-8}}=4431.598\ m[/tex]
max ring radius (or reduced circumference)-
[tex]c_{rs}=2\pi r_{rs}\ \ \ \Rightarrow\ \ \ r_{rs}=\frac{c_{rs}}{2\pi}[/tex]
incorporating crs
[tex]r_{rs}=\frac{4431.598}{2\pi}= 705.311\ m[/tex]
the above can be abbreviated to
[tex]r_{rs}=\frac{m_{bh}\cdot l_{p}}{2\pi\cdot m_{p}}[/tex]
The ring singularity would progress from a sphere (as a static black hole) to the maximum ring radius of 705.311 m at a∗=1 and it's progress from sphere to ring should be taken into account. A fair assumption would be that it grows in accordance with the increase in a∗.
ring singularity radius = rrs a∗2
where a∗ is the unitless spin parameter between 0 and 1
therefore for a 3 sol mass black hole with a spin parameter of 0.8, the radius of the ring singularity would be-
ring singularity radius = 705.311 x 0.8^2 = 451.399 m
The spin parameter a∗ is squared to keep the changes in the ring singularity in proportion with the changes of the outer event horizon (R+=(Gm/c^2)(1+(1-a∗^2)^1/2))
The cross section of the torus would also decrease as the spin parameter a∗ increases but is so marginal the information has not been included in the summary.
Gravitational constant and Planck units
The gravitational constant (6.6742x10^-11 N m^2 kg^-2) which is described as a measure of the natural strength of gravity is equal to Plank length^3 / Plank mass x Plank time^2 (which is also reflected in an alternative set of SI units used for G, m^3 kg^-1 sec^-2). By some strange quirk, regardless of the size of the black hole, it appears that the maximum circumference of the ring singularity equals the gravitational radius, Rg (a∗ at 1)-
gravitational radius = maximum ring singularity circumference
[tex]\frac{Gm}{c^{2}}=\frac{m\cdot l_{p}}{m_{p}}[/tex]
or even
[tex]\frac{G}{c^{2}} = \frac{l_{p}}{m_{p}}[/tex]
*Regardless of how compact matter is, if it has density then technically it's not a singularity but in theory, the energy that makes up both matter and space combines at Planck dimensions, creating a quantum 'foam' supported by either Loop Quantum Gravity or strings so even though it has dimensions, it might still be described as a singularity as it is a combination of both matter and space (and probably time) rather than just collapsed matter in space. For ease of reference, the term singularity is used when referring to the collapsed core of a black hole.
Planck density* = 5.155x10^96 kg/m^3 (or 5.155x10^87 tonnes/cm^3) (Based on Planck mass, 2.176x10^-8 kg and Planck length, 1.616x10^-35 m).
'This is a unit which is very large, about equivalent to 10^23 solar masses (the observable universe) squeezed into the space of a single atomic nucleus. At one unit of Planck time (5.391x10^-44 s) after the Big Bang, the mass density of the universe is thought to have been approximately one unit of Planck density.' http://en.wikipedia.org/wiki/Planck_units
*Planck units are sometimes (humorously) referred to as 'God's Units' as they are based on the properties of free space and not on the properties of any prototype, object or particle (that could be thought of as arbitrarily chosen). They are also referred to as natural units because the origin of their definition comes only from properties of nature and not from any human construct. There are 5 basic Planck units, the above shows Planck length, mass and time; the other two are charge (1.876x10^-18 C) and temperature (1.417x10^32 K). They were established in the early 20th century by Max Planck, who is considered the founder of quantum theory. It's also postulated that Planck matter is the constitute energy of both matter and space itself and that under extreme gravitational collapse, in theory, will be sustained from any further collapse (i.e. into a singularity) due to loop quantum gravity.
The Schwarzschild radius of a 3 sol mass is 8861.099 m. Based on an overall mass of 5.967x10^30 kg and a volume of 2.914x10^12 m^3, the average density at the point a 3 sol mass collapses into a black hole would be 2.048x10^18 kg/cm^3. As the core collapses into something beyond quark matter, pulling the rest of the star beyond the Schwarzschild radius, light would begin to free fall towards the surface of the collapsing mass, unable to escape from the gravity (the escape velocity for a sphere with this mass and radius would exceed 300,000 km/s, the speed of light), hitting the surface of the sphere, compacting the sphere further.
It seems to be an accepted fact that photons are massless but have energy due to their high momentum (E=pc), the light coming into the black hole would be highly blueshifted (x-rays, gamma rays) and the photons energy would be converted to mass as it collided with the collapsing star, contributing further to the black holes mass and compactness (though the mass of the black hole might more or less stay the same as Hawking radiation in contrast slowly evaporates the black hole). So where does the energy for the highly blueshifted light come from? Light trying to escape the black holes gravity well is 'drained' and highly redshifted, it's as if it's put through a press, the energy seemingly extracted and added to light falling into the gravity well which becomes highly blueshifted.
In general relativity, the source of black holes are considered geometric singularities, in quantum mechanics, they are speculated as having Planck density (5.155x10^90 kg/cm^3), the maximum energy density allowing in current physics.
Theoretical fundamental particles such as preons or strings are approx. 10^-33 m in size which are close to the Planck length, the smallest measurement currently used in physics (1.616x10^-35 m). If strings or preons are at the heart of all quarks and leptons, they would normally be at their closest (in ground state) ~10^-15 m (a Fermi), the distance between quarks within a nucleon. Possibly under great pressure the quarks would break down (approx. 10^20 kg/cm^3) and under greater pressure, the preons/strings would break down also and the pure quanta of energy that reside at the very core of fundamental particles might compact to something in the region of Planck density.
If the core of a star about the mass of 3 of our suns collapsed beyond the Schwarzschild radius (8861.099 m) then it's possible it could collapse all the way to Planck density. For a mass the size of the sun (1.989x10^30 kg) this would result in a sphere with a radius of 4.516x10^-23 m or ~45 yoctometre (a yoctometre or ym is 10^-24 m). For the supermassive black hole at the centre of our galaxy which is predicted to have a mass of 3.6 million solar masses (7.161x10^36 kg), the core radius at Planck density would be 6.922x10^-21 m or ~7 zeptometre (a zeptometre or zm is 10^-21 m), the Schwarzschild radius (event horizon) would be 1.063x10^10 m or ~10.6 million km (our Sun has a radius of 0.696 million km).
Basics for a 3 sol mass Schwarzschild (static) black hole-
3 solar masses = 3 x 1.989x10^30 kg = 5.967x10^30 kg
Gravitational radius (considered the unit of measurement for black holes)-
Rg = Gm/c^2 = 4430.550
Event horizon (Schwarzschild radius or Rs)-
Rs = 2Gm/c^2 = 8861.099 m
Photon sphere-
Rph = 3Gm/c^2 = 13,291.648 m
Marginally stable orbit (normally the inner edge of the accretion disk)-
Rms = 6Gm/c^2 = 26,583.297 m
where G- gravitational constant, m- mass in kg and c- speed of light
Core radius based on Planck density for a static 3 sol mass black hole-
Planck density = 5.155x10^96 kg/m^3
Sphere volume (volume = mass/density)-
[tex]V_{p}=\frac{m_{bh}}{\rho_{p}}=\frac{5.967\cdot10^{30}}{5.155\cdot10^{96 }}=1.158\cdot10^{-66}\ m^3[/tex]
where VP- volume based on Planck density, mbh- black hole mass in kg, ρP- Planck density
sphere radius-
[tex]V=\frac{4}{3}\pi r^{3}\ \ \ \Rightarrow\ \ \
r=\sqrt[3]{\frac{3V}{4\pi}}[/tex]
incorporating VP-
[tex]r=\sqrt[3]{\frac{3\cdot1.158\cdot10^{-66}}{4\pi}}= 6.514\cdot10^{-23}\,m[/tex]
core radius for a 3 sol mass static black hole = ~65 yoctometre (a yoctometre or ym is 10^-24 m)
Ring singularity
The collapse of a large star with a fast rotation may create a spinning black hole. As the core becomes smaller under pressure, it spins faster. This causes the sphere to flatten. Massive centrifugal forces cancel out the inward force of gravity and create a spinning ring of degenerate matter which eventually collapses into a ring singularity. This is also called a Kerr black hole (or Kerr-Newman black hole if charge is included for). It's believed that most black holes are like this in nature due to the fact that most stars spin.
Based on the above, the Planck matter could take the form of a ring rather than a sphere.
A ring singularity for a rotating (Kerr) black hole could be based on a torus with an absolute minimum cross section of 2.6115x10^-70 m^2 (Planck length^2). Imagine the ring singularity is made up of Planck units, 1.616x10^-35 m in length, each unit weighing Planck mass (2.176x10^-8 kg). Based on a black hole of 3 solar masses, the maximum circumference of the ring would be-
max ring circumference-
[tex]c_{rs}=\frac{m_{bh}\cdot l_{p}}{m_{p}}[/tex]
where mbh- black hole mass, lP- Planck length and mP- Planck mass
[tex]c_{rs}=\frac{5.967\cdot10^{30}\cdot1.616\cdot10^{-35}}{2.176\cdot10^{-8}}=4431.598\ m[/tex]
max ring radius (or reduced circumference)-
[tex]c_{rs}=2\pi r_{rs}\ \ \ \Rightarrow\ \ \ r_{rs}=\frac{c_{rs}}{2\pi}[/tex]
incorporating crs
[tex]r_{rs}=\frac{4431.598}{2\pi}= 705.311\ m[/tex]
the above can be abbreviated to
[tex]r_{rs}=\frac{m_{bh}\cdot l_{p}}{2\pi\cdot m_{p}}[/tex]
The ring singularity would progress from a sphere (as a static black hole) to the maximum ring radius of 705.311 m at a∗=1 and it's progress from sphere to ring should be taken into account. A fair assumption would be that it grows in accordance with the increase in a∗.
ring singularity radius = rrs a∗2
where a∗ is the unitless spin parameter between 0 and 1
therefore for a 3 sol mass black hole with a spin parameter of 0.8, the radius of the ring singularity would be-
ring singularity radius = 705.311 x 0.8^2 = 451.399 m
The spin parameter a∗ is squared to keep the changes in the ring singularity in proportion with the changes of the outer event horizon (R+=(Gm/c^2)(1+(1-a∗^2)^1/2))
The cross section of the torus would also decrease as the spin parameter a∗ increases but is so marginal the information has not been included in the summary.
Gravitational constant and Planck units
The gravitational constant (6.6742x10^-11 N m^2 kg^-2) which is described as a measure of the natural strength of gravity is equal to Plank length^3 / Plank mass x Plank time^2 (which is also reflected in an alternative set of SI units used for G, m^3 kg^-1 sec^-2). By some strange quirk, regardless of the size of the black hole, it appears that the maximum circumference of the ring singularity equals the gravitational radius, Rg (a∗ at 1)-
gravitational radius = maximum ring singularity circumference
[tex]\frac{Gm}{c^{2}}=\frac{m\cdot l_{p}}{m_{p}}[/tex]
or even
[tex]\frac{G}{c^{2}} = \frac{l_{p}}{m_{p}}[/tex]
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