Pondering the conditions of a black hole, considering its Mass/Radius, and what it might look like inside one.
Black Holes - What does it look like inside them?
By S. K. Smith
I will actually have an intelligent answer for this. Really! So keep reading!
As far back at 1798, the French mathematician Pierre-Simon Laplace (1749 - 1827)postulated the existence black holes, using Newton’s theory of gravitation. He calculated that if a star were massive enough and if its radius were small enough, the force of gravity near the star would be so strong that not even light could escape from it.
Fast forward to the 20th century. In 1939, Robert Oppenheimer (1904 – 1967), “the father of the atomic bomb,” and one of his students Hartland Snyder (1913 – 1962) made similar calculations using the more refined formulas of relativity. A star of sufficient mass would undergo a final catostrophic collapse to a tremendously high density and to a size less than or equal to the “Schwarzchild radius.” (That is 2GM/c2, where G is the gravitational constant, M the mass of the star, and c is the speed of light in a vacuum.) The result would be a black hole, which any light emitted from the star would be dragged back by the gravitational field. Hmmmm. Didn’t Laplace come to a similar conclusion about two centuries earlier?
According to Einstein’s theory of special relativity, nothing in the physical universe can travel faster than the speed of light. So if light is not fast enough to escape a black hole, how can anything else escape? It’s been said, there are either holes like this in the universe, or there are holes in the theory of relativity. So the search began in the heavens for black holes, yielding many likely candidates.
In the constellation Cygnus, Cygnus X-1 was discovered in 1964 as one of the strongest X-ray sources seen from Earth. As the invisible member of a binary star, Cygnus X-1 sucked material gravitationally from its visible companion. This material formed a rotating disk, which astronomers detected by its X-rays. In 1990, after losing a bet to physicist Kip Thorne, Professor Stephen Hawking conceded the observational evidence made a strong case for a gravitational singularity at Cygnus X-1, that is, a black hole.
Cygnus X-1 was one of the first evidences to show that black holes were not just creations in mathematics or science fiction, but actual physical entities. Since then, the evidence for black holes has become so overwhelming that astronomers now treat them as normal fixtures in the universe. Millions of ordinary black holes are believed to be peppering each of the estimated 50 billion to 100 billion galaxies in the universe. Giant black holes are believed to punctuate the centers of most galaxies, even our own.
So what does it look like inside a black hole? To answer this, I will focus on one of its classical definitions - its escape velocity is the speed of light or greater.
The total Energy Etotal of a particle of mass m interacting with a planet of mass M is the sum of its Kinetic Energy and its Potential Energy. Conservation of energy dictates that this sum is a constant. We have,
Etotal = Kinetic Energy + Potential Energy = constant
In Newtonian Mechanics, particle m’s Kinetic Energy can be defined as ½mv2, where m is its mass, and v its velocity. The Potential Energy of particle m and planet M is -GMm/R. (G is the gravitational constant, M the mass of the planet, and m the mass of the particle. R is the distance between the center mass of planet M and the center of mass of particle m.) Hence,
Etotal = ½mv2 - GMm/R = constant.
If Etotal < 0, the particle m is bound to the planet M (such as the moon is bound in its orbit about the earth and most of us are stuck on this earth). The particle m escapes planet M’s orbit (such as the Mars Phoenix space probe escaped Earth), if Etotal ≥ 0. The minimum velocity, the escape velocity vescape, for particle m to break from the bonds of planet M is when Etotal = 0.
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Etotal
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Comments
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< 0
(less than zero)
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Particle m is bound
by planet M
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≥ 0
(greater than or equal to zero)
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Particle m is not bound
by planet M
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= 0
(equal to zero)
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Minimum velocity in which particle m escapes planet M
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Hence,
Etotal = 0 = ½mvescape2 - GMm/R.
If we solve for vescape, we get
vescape = √2GM/R.
The escape velocity does not depend at all on the mass of the particle m. If we set the escape velocity for the speed of light c, (which is about 300,000 kilometers per second) then
vescape = c =√2GM/R.
Solving for M/R, we see this ratio is a constant:
M/R = c2/2G
So this classical condition of a black hole is solely dependent on its M/R ratio, the mass of the planet M and the distance R between the centers of mass between particle m and planet M. In MKS units (meter (m) –kilogram (kg) –second (s)), c2/2G = 6.75 x 10 26 kg/m.
If we solve for R, we have the “Schwarzchild radius”, RSchwarzchild, as mentioned earlier in this article:
RSchwarzchild = 2GM/c2
Note: For non-rotating black holes, the Schwarzchild radius defines the area of its event horizon, the boundary in which events inside the black hole cannot affect an observer outside the black hole.
Black holes can be very, very dense. The tidal forces (the difference in gravitational forces across a body) can rip an object apart as it approaches the event horizon. Yet, the M/R ratio shows that black holes need not be extremely dense, only sufficiently massive. The more massive the black hole, the less dense it has to be. Below are some simple calculations demonstrating that.
Let M be the mass of the black hole, R its Schwarzchild radius. Its mean density Mass/Volume is proportional to M/R3. For the sake of clearing up the clutter, let the constant K = c2/2G. Therefore, the M/R ratio = K and the minimum mass required for a black hole of radius R is M = KR.
The table below shows the relative density based on our black hole’s mass for different radii.
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R
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M = KR
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≈ M/R3 (density)
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1
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K
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K
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2
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2K
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2K/8 = K/4
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3
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3K
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3K/27 = K/9
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4
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4K
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4K/64 = K/16
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5
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5K
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5K/125 = K/25
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…
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…
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…
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n
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nK
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nK/n3 = K/n2
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As we see in the table, as the mass of the black hole increases by a factor of n, its mean density decreases by a factor of 1/n2. Hence, the more massive the black hole, the less its mean density needs to be.
Now, let’s crunch some numbers for familiar objects whose radii have collapsed to the Schwarzchild radius R.
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Object
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Mass
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RSchwarzchild
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M/(4/3πR3)
(density)
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Speck of dust
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10-5 g
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1.5 x 10-33 cm
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7.3 x 1092 g/cm3
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Earth
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6 x 1024 kg
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0.89 cm
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2.0 x 1027 g/cm3
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Sun
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1 M☼
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3 km
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1.8 x 1016 g/cm3
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Star
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2 M☼
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6 km
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4.6 x 1015 g/cm3
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Galaxy
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1011 M☼
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3 x 1011 km
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1.8 x10-6 g/cm3
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100 billion galaxies
(Universe??)
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1022 M☼
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3.1 x 109 light years
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1.8 x10-28 g/cm3
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Note: g is grams; kg kilograms. M☼isa solar mass, the mass of our sun,approximately 2 x 1030 kg. cm is centimeters, km kilometers. A light year is approximately 9.5 x 1012 km. As a reference, water has a density of one g/cm3.
Looking at the table above, if we collapse the matter of a speck of dust to the Schwarzchild radius, its mean density is about 1093 g/cm3. If matter that dense were the size of the nucleus of an atom, it approaches containing all the mass of the known universe. (Now, that is very, very, very dense!) For a mass the size of our galaxy, its Schwarzchild radius is 3 x 1011 km, which about 50 times the radius of Pluto’s orbit about the sun. Its mean density is about 10-6 g/cm3, which is only that of a light gas.
Now, consider 100 billion galaxies. Its Schwarzchild radius is 3.1 x 109 light years, approaching the radius of our known universe, which seems to have the upper limit of 4.6 x 1010 light years. The mean density is about 10-28 g/cm3, which is an exceedingly thin gas, an effective vacuum! If the mass of the known universe were about 10 times more massive as some astronomers have estimated (at 100 billion galaxies), then the condition for M/R ratio would be satisfied for a black hole.
Astronomers have used a formula based the mass-luminosity to estimate the mass of galaxies and clusters. The Virial Theorem (which allows the average total kinetic energy to be calculated for complicated systems)and Kepler’s Laws (which describe the motion of planets in a Solar System) predict missing mass within clusters and galaxies that the mass-luminosity relation does not account for. Theories say dark matter makes up this missing mass, such as non-luminous stars, neutrinos, ultra-dense black holes, and hot, gas like material distributed throughout space. And I’m sure I haven’t even scratched the surface on this topic. That is for another time, perhaps.
What does it look like inside a black hole? Just open your eyes and look around. We may be in one!
There are other factors than the M/R ratio to consider when defining a black hole. In 1979, I had the privilege of talking in person to physicist Kip Thorne. (He autographed his article, “The Search for Black Holes” in my “Readings from Scientific American – New Frontiers in Astronomy.”) We discussed the possibility of the universe being a giant black hole. Dr. Thorne made a case that the universe does not meet the boundary conditions for one. (1) The body must be contracting. Evidence shows the universe is expanding (and at a faster rate than expected.) (2) There must exist a vacuum between the black hole and its event horizon. Does the universe have an event horizon? God only knows – I say this reverently. Also, it does not seem possible to send a satellite in orbit around the universe to determine its total mass – as we would do about a planet or binary star.
Some Conclusions:
A black hole does not need to be super dense and rip us apart, crush us to pieces, and squeeze us out of existence with its tidal forces. A solar system full of water would qualify as a black hole. It is possible to be inside one and everything seems quite normal. As for the universe qualifying as a black, that is a matter of definitions, philosophy, religion, and new science that goes beyond the dark ages when I was in school.
© September 6, 2008, S. K. Smith
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The idea seems to have begun with the Indian astrophysicist, Chandrasekhar, a student of the great Eddington. In 1924 this man studying collapsing stars realized the possibility of the objects we call blackholes today. You may find interesting that when Chandrasekhar decided to present his work at a professional meeting he was followed by his mentor, Arthur Eddington, who proceeded to ridicule the idea. Others believed, so in the thirties men like Oppenheimer continued the work on collapsing stars.
Despite my skepticism I find the mathematics of cosmology fascinating. Recently I found another solution of the Friedmann equations that does seem to say we are inside of a monstrous Black hole. I find that incredible but an interesting thought. If that is true we do know what the inside of a black hole looks like.
Keep up the good work.
Ben Plybon
ETAL: Really I'd Be More Worried About Black Holes In Majority Of Human Brains,i.e.Esp. Greed Mongering POLITCIAN$....
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