Brennan Dwyer
Disclaimer: Please know that the author of this paper has no background in physics, and the content of this paper may be incorrect or seen as unprofessional.
Abstract
Black holes are certainly an interesting topic in physics. A black hole is a region in spacetime that is very strong, so strong to the point that nothing can escape it. Not even the speed of light can escape one. Black holes are from the collapse of massive objects like neutron stars. There are many important reasons for studying black holes, and there are also many unsolved problems regarding them. This paper provides interesting relationships between the entropy, mass, area and ‘time’ of a black hole. This paper also shows interesting relationship between the universe and black holes, including how if the mass of the universe is plugged into the area of a black hole equation, it results in an area that is very close to the actual area of the universe. What this paper also shows is how if our universe is the lower dimensional boundary region of the event horizon of a black hole, then perhaps that could then be used to explain why the universe has the fundamental constants as they are, and perhaps explain the theory of everything.
Introduction
Black holes are certainly an interesting topic in physics. A black hole is a region in spacetime that is incredibly strong. Not even the speed of light can escape one. Usually massive bodies like neutron stars are what end up becoming a black hole. There are many important reasons for studying black holes, and there are also many unsolved problems regarding them. This paper will attempt to further investigate the relationship between the entropy, mass, area, and ‘time’ of a black hole. This paper will also provide some interesting relationships between the universe and black holes.
If any object becomes dense enough, it can become a black hole. As stated from astronomy.swin.edu.au “Any object with a physical radius smaller than its Schwarzschild radius will be a black hole.”[12] The Schwarzschild can also be defined as the radius of a black hole. Black holes have three specific parameters to them. According to astronomy.swin.edu.au, “Black holes are completely characterized by only three parameters: mass, rotation and charge”.[3] According to the paper “The Simplicity of Black Holes” on physics.aps.org “in realistic environments, black holes can be distorted by the mass surrounding them.”[1] This source also stated that “mass distortions can be described by a sum of multipole moments, similar to the ones used in electromagnetism to calculate the electric field outside a region containing charges”.[1]. One final fact stated by “The Simplicity of Black Holes” is that, “no simple relation exists between the horizon moments (the intrinsic black hole’s hair) and the field moments (the hair seen by an external observer)”.[1]
Something important the understand about black holes is that they become less dense as they become more massive. As stated by Living in a Low Density Black Hole, Non-Expanding Universe – Perhaps a Reflecting Universe, “the larger and more massive black holes have significantly smaller average densities” (McBryan, Bernard 2013).[10]
It is interesting to note that “If the average density of the universe matches the critical density of just 5.67 hydrogen atoms per cubic meter, it would form a Schwarzschild low density black hole of approximately 13.8 billion light years, matching the big bang model of the universe” as stated by Living in a Low Density Black Hole, Non-Expanding Universe – Perhaps a Reflecting Universe, (McBryan and Bernard 2013).[10]
Not all black holes may be massive as often imagined. Some sources even brought up the idea of incredibly small black holes. According to astronomy.swin.edu.au, “primordial black holes have masses comparable to or less than that of the Earth. These purely hypothetical objects could have been formed through the gravitational collapse of regions of high density at the time of the Big Bang”.[3] According to a Scientific American article about quantum black holes, “For a particle to be both energetic enough and compact enough to form a black hole, it must have the Planck energy.”[2] The Planck energy is about 1019 Gev.
Some papers have suggested that our universe is actually inside a black hole. Universe In a Black Hole in Einstein-Cartan Gravity, is an example of a paper that theorizes this. For instance, the paper stated: “Gravitational repulsion induced by spin and torsion, which becomes significant at extremely high densities, prevents singularities in black holes and at the big bang. Because of this repulsion and particle production, every black hole may create a new universe on the other side of its event horizon” (N. Popławski 2016).[11] As quoted in Universe In a Black Hole in Einstein-Cartan Gravity, “Accordingly, each spatial point in the interior of a black hole locally evolves toward the singularity as an independent, spatially homogeneous, and isotropic universe” (Landau & Lifshitz 1975; Lord 1976; N. Popławski 2016).[8][9][11] The paper Living in a Low Density Black Hole, Non-Expanding Universe – Perhaps a Reflecting Universe also suggests that the universe may be inside a black hole (McBryan, Bernard 2013).[10]
Entropy, Mass, and Area of Black Holes
A black hole must have a specific amount of entropy based on its mass and area. The reason why a black hole would ‘require’ a specific amount of entropy with a given volume is so that it ‘satisfies’ the laws of thermodynamics. This is considering what is stated by Jacob D. Bekenstein: “The Bekenstein-Hawking entropy or black hole entropy is the amount of entropy that must be assigned to a black hole in order for it to comply with the laws of thermodynamics as they are interpreted by observers external to that black hole.”[6]
The entropy of a black hole is given as follows in Equation 1, which was also taken from Jacob D. Bekenstein’s article from scholarpedia.org.[6] This equation gives the unitless value of the entropy of the black hole. However, we multiplied Boltzmann constant to it.
Equation 1
Where S is the entropy of a black hole, k is the Boltzmann constant, c is the speed of light, A is the area of the black hole, G is the Gravitational constant, and h is the reduced Planck constant.
So, to increase the entropy of a black hole, the area must increase. However, to increase the area of the black hole, there needs to be an increase in its mass. Equation 2 is the area of a black hole, which was also taken from Jacob D. Bekenstein’s article from scholarpedia.org.[6]
Equation 2
Where A is the area of a black hole, G is the Gravitational constant M is the mass of a black hole, and c is the speed of light.
So, an increase in the mass of a black hole from bringing in other masses such as other stars and planets increases the area, which in turn increases the entropy of the black hole.
The radius of a black hole would obviously increase from more mass like the area does. The radius of a black hole can be seen in Equation 3.
Equation 3
Where r is the radius of a black hole, G is the gravitational constant, M is the mass of a black hole, and c is the speed of light.
Something to take note is that the time (age) of our universe can be approximately calculated as the radius of the universe divided by the speed of light. This gives 4.65×1010 years, which is close to the real age of the universe which is 1.38×109 years. The reason for this is the size of the observable universe is based on how far in the universe we can observe, specifically how far light has traveled.
So, what is interesting is that if that same math is applied to a black hole, then you get Equation 4.
Equation 4
Where T is the ‘time’ (age) of a black hole, r is the radius of a black hole, c is the speed of light, G is the gravitational constant, and M is the mass of a black hole.
So perhaps Equation 4 may be a way to calculate the ‘time’ (age) of a black hole. The reason Equation 4 may work for a black hole, is because of the similarities found between the universe and a black hole, which are discussed in the next section.
Similarities between the Universe and Black Holes
There were some interesting similarities between the universe and black holes that were found. Perhaps some of the similarities that will be presented in this section may even give more evidence towards the papers that suggest that the universe is ‘inside a black hole’
One major finding was that if the mass of the universe was plugged into the equation of the area of a black hole (Equation 2 in the previous section), then it would get approximately the actual area of the universe. Though there are different estimates of the total mass of our universe, the estimation of 3×1055 g was taken from curious.astro.cornell.edu.[7] The value from Equation 2 using the mass of the universe got about 2.5×1052 m2. That actual area of the universe (in terms of surface area) is about 2.43×1054 m2.
Considering the area of a black hole equation (Equation 2) seems to work very closely for our universe, that means that the equation for the radius of a black hole (Equation 3 in the previous section) can work very closely as well for our universe. Also, that means that Equation 4 would work out as well. The equation for the radius of a black hole is used to solve for what is called the Schwarzschild radius. The value from Equation 3 using the mass of the universe gets about 4.5×1025 meters. The actual radius of the universe is about 46.5 billion light years, or 4.4×1026 meters (which ‘pops up’ from a quick Google search).
If the equation for the radius of a black hole (Equation 3) is solved for the speed of light c, then that gets Equation 5.
Equation 5
Where c is the speed of light, G is the gravitational constant, M is the mass of a black hole, and r is the radius of a black hole.
Since the equation for the area and radius of a black hole works very closely for the universe, then that means that Equation 5 works very closely for the universe as well, since it is simply taking the radius of a black hole equation and solving for the speed of light.
Equation 5 is the same as the escape velocity equation and shows that the speed of light is approximately the’ escape velocity of the universe’, since the speed of light is very closely given if the mass of the universe is plugged into the equation. With the mass and radius of the universe plugged into the equation, the value is 95,399,256.7 m/s, as opposed to 299,792,458 m/s for the speed of light. It is interesting that the escape velocity of a black hole is the speed of light, which is close to the ‘escape velocity of the universe’.
Since Equation 3 normally represents the radius of a black hole, but also works for the universe, it can be plugged into Newton’s gravitational equation to find the value of the equation with the distance of the radius of the universe plugged into the distance r into the equation. This allows the Gravitational constant to cancel out. This can be seen on Equation 6.
Equation 6
Where G is the gravitational constant, M is the mass of the universe, and c is the speed of light.
It should be noted that the top of Equation 6 has the mass equal to the mass of the universe, so that M can be squared and cancel out. It is interesting that if the distance is equal to the radius of the universe in the gravitation equation, then that is the specific point at which the constant G and the variable M can cancel out. It is interesting that this derivation is equal to speed of light to the fourth power divided by 4 times the gravitational constant. Interestingly, that would be close to the value of the Planck force, which would be c4/G .
Something interesting to take note of that will relate to Equation 6, is that to get a rough approximation of the force of gravity between all objects in the universe, it can likely be found by plugging in the mass of the universe and the radius of the universe into Newton’s gravitational equation (since density is defined as mass divided by volume, and since density gives you a rough estimate of how nearby different objects are from each other on average in the universe, then mass divided by radius would also work). Since the radius of the universe equals approximately the Schwarzschild radius of the universe, this can be plugged into the Newton’s gravitational equation, which would end up getting the same equation and value as Equation 6. If it is always true that the Schwarzschild radius equation works for the universe, and the mass is increasing in the universe if it is a black hole, then the additional mass would cause the radius of the universe to increase, which in turn would still result in the same value for Equation 6. However, if this doesn’t hold true for the universe and only the radius of the universe increases and not proportionally to an increase in mass, then the total energy from the force of gravity between all objects between the universe would decrease with time. Perhaps that would violate the law of conservation of energy, specifically for gravity. (This paragraph may be an oversimplification and may be somewhat inaccurate with just using Newtonian Gravity.)
Again, perhaps the similarities found between the universe and black holes in this section may give more evidence towards the papers that suggest that the universe is inside a black hole.
Interesting Finding with Entropy
Entropy can also be defined as the number of ways a system can be arranged. So perhaps all the ways the mass and energy of the universe can be rearranged can be a measure of the amount of entropy the universe has. So, if all possible mass and energy in the universe can be rearranged in the amount of ‘surface Planck areas’ that can be contained in the universe, then that would be the area of the universe divided by the ‘surface Planck area’. This can be shown in Equation 7.
Equation 7
Where S is the entropy of the universe, A is the surface area of the universe, and l is the Planck length.
Equation 7 equals to about 7.41 × 10122. The entropy of the universe using the entropy of a black hole equation is about 2.33 × 10123 (the surface area of the universe was found by doing 4*pi*r^2). If Equation 1 is multiplied by pi, then it is equal to the entropy of the universe using the entropy of the black hole equation.
Equation 7 equals to the entropy of a black hole equation (with a difference of pi), which can be seen in Equation 8. In Equation 8, the equation for the Planck length is plugged into the variable for Planck length.
Equation 8
If the ratio between the radius and the universe and the Planck length is calculated, there is ratio of about 2.72 × 1061. If the mass of the universe is divided by that value, then it gets about 1.10 × 10-9 kilograms. The Planck mass is about 2.18 × 10−8 kg (which can be found on Wikipedia). Perhaps what that essentially can mean is that if the entire mass of the universe was broken up into ‘Planck masses’ and was spread out into all the amount of ‘Planck lengths’ of the radius of the universe, it would all be evenly filled up with one Planck mass per Planck length. So essentially since the universe has it’s mass and area proportional to each other, perhaps that is what allows the mass to be able to be evenly distributed to just the right amount so that there can be all the right amount of arrangements of that mass so that those arrangements can equal to the entropy of the universe, which would be an entropy proportional to the mass and area of the universe. (The explanation regarding arrangements of the universe may or may not be correct in this paper from a mathematical sense and should be verified.)
It should be noted that this section of the paper used the values for the mass and radius of the universe that was previously used in the previous section of this paper.
Fundamental Constants and the Theory of Everything
Considering some researchers have theorized that our universe is a black hole, perhaps the universe is more specifically the lower dimensional boundary region of the event horizon of a black hole, then that would involve the holographic principle. Matthew Headrick is a physicist that theorizes that the universe may much like a hologram.[5] As stated in the article “The universe is a hologram and other mind-blowing theories in theoretical physics”, “Headrick works on one of the most cutting-edge theories in theoretical physics—the holographic principle. It holds that the universe is a three-dimensional image projected off a two-dimensional surface, much like a hologram emerges from a sheet of photographic film.”[5] If the holographic principle is true, then the reason why our universe would most likely be the two dimensional surface area of a black hole is because it being that would explain why there are specific fundamental constants, since a black hole would need to have the mass, area, and entropy all proportional to each other. The fundamental constants that would be explained would be the ones in the area and entropy of a black hole equation. Those would specifically be the speed of light, the gravitational constant, the Boltzmann’s constant, and the reduced Planck’s constant. Also, the holographic principle was ‘sprung’ from the theory of black holes.[4] As stated from the summary of the article “Controversial test finds no sign of a holographic universe”, “The holographic principle springs from the theory of black holes, spherical regions where gravity is so intense that not even light can escape.”[4]
One important aspect of this paper to point out is that the hypothesis of this paper involves the concept that the surface area of the cosmic event horizon can be plugged into the entropy of a black hole equation (Equation 1), which other researchers have done to calculate a large part of the entropy of the universe. Charles H. Lineweaver is an example of a researcher that has written papers regarding that idea. This would mean that the area and entropy of the universe are possibly proportional to each other. However, there is additional entropy from things like black holes or dark matter which would add slightly more entropy to the total entropy of the universe then the amount from the cosmic event horizon alone, which would mean there is slightly more entropy than necessary for the mass, area, and entropy to be proportional to each other. However, Perhaps if the universe is the 2D surface area of a black hole and the holographic principle applies to our universe, then perhaps the entropy within the observable universe is also the same information that is encoded on the cosmic event horizon since perhaps what is in the universe is encoded on a 2D surface area with the holographic principle. So perhaps the entropy from things like particles or black holes doesn’t add to the entropy in addition to the entropy of the cosmic event horizon, but rather is part of what is making up the entropy of the cosmic event horizon itself, so the area and entropy of the universe are proportional to each other.
Another important aspect of this paper to add to the previous paragraph, is that it is hypothesized that the ‘force’ behind the mass and area being proportional to each other is dark energy, which causes the expansion of the universe. So, if the universe is a black hole that becomes more massive from absorbing other stars and different forms of matter and energy, then perhaps that ‘enters’ in the form of dark energy so that there is a pressure force that causes the universe to expand to the right (proportional) area based on that additional mass being added. Also, since most of the entropy of the universe aside from the cosmic event horizon is from black holes, perhaps the matter and energy that enters the universe comes also in the form of dark matter so that the black holes in the universe can absorb that dark matter, which in turn causes the black holes in the universe to become more massive and have more entropy, which causes the total entropy of the universe to increase, perhaps to the proportional amount between the area and entropy of the universe. There is obviously more details, mathematics, and evidence that need to be included with this paragraph. Perhaps one piece of evidence that dark energy is increasing over time in the universe, is that it isn’t thinning out, which may mean it is ‘accumulating’ as the universe increases in area.
Regarding the theory of everything, the holographic principle is used to formulate the theory that some researchers have come up with called entropic gravity, which is the theory that gravity is an entropic force. Therefore, entropic gravity is essentially what gravity is if our universe is the lower dimensional boundary region of the event horizon of a black hole. This all fits together in Equation 9.
Equation 9
Where {C} is all the physical constants such that M mass is proportional to A area and is proportional to S entropy for the entire universe, s.t. (so that) the next equation is correct where S is the entropy of the universe, k is Boltzmann’s constant, G is the gravitational constant, M is the mass of the universe, c is the speed of light, and is the reduced Planck’s constant. Then, (therefor) QM quatum mechanics is an element (underlying force) of entropy, and G gravitation is a result of entropy (an emergent force of entropy and not an underlying force of it.)
It should be noted that the middle set of Equation 9 is the area of a black hole (which works for the universe) plugged into for A for the equation of the entropy of a black hole (which can also be used for the entropy of universe).
What Equation 9 essentially shows is that if the universe is a black hole and works as one, then there has to be all the physical constants fine-tuned to allow the mass to be proportional to the area and proportional to the entropy for the entire universe. Those constants would have been set up that way so that the entropy of a black hole equation works for the universe. Therefore, if the universe is a black hole and if the holographic principle applies to a black hole, and if the holographic principle is what would allow for there to be entropic gravity, then the universe would have to have entropic gravity which would mean that gravitation would be just a result of entropy, and not an emergent force of it unlike quantum mechanics. In a way if the holographic principle is indeed correct, then the left set of Equation 9 attempts to explain why there are the specific fundamental constants in our universe, the middle term would explain where our universe came from since it would most likely be a black whole if the holographic principle is correct, and the right side would attempt to explain the theory of everything.
Conclusion
This paper attempted to further investigate the properties of black holes, as well as their similarities to the universe. One interesting finding was how if the mass of the universe is plugged into the equation for the area of a black hole, it gets close to the same area so the universe. There were other interesting findings as well. Perhaps the additional finding of there being similarities between the universe and black holes may even points towards more evidence that our universe is part of a black hole. If the universe is a black hole, then that may explain many unsolved problems in physics as explained in this paper. There certainly is more research that needs to be done on black holes, whether it be on the information paradox, and why our universe has similarities to a black hole as found in this paper. There are some questions left unanswered in this paper.
The first question left unanswered is if the entropy that is in the universe is actually what makes up the entropy of the cosmic event horizon if the holographic principle holds true, then how can the entropy of what is within the universe be mathematically related to the cosmic event horizon entropy, since they are different values? How does the entropy in the universe that isn’t from black holes relate to all of this?
The second question is how can it be proven that the rate of dark energy coming into the universe is causing the right amount of expansion for the mass and area of the universe to be proportional to each other?
Finally, the third question is how can it be proven that the rate of dark matter coming into the universe is causing the right amount of mass to be added to the black holes in the universe, which in turn causes the entropy of the universe to be proportional to its area?
Works Cited
[1] A. Ashtekar, Physics (2015).
[2] B.J. Carr, Scientific American (2007).
[3] “Black Hole | COSMOS.” Astronomy.swin, astronomy.swin.edu.au/cosmos/B/Black Hole.
[4] Cho, A. (2015, December 11). Controversial test finds no sign of a holographic universe. Science. https://science.sciencemag.org/content/350/6266/1303.summary.
[5] Goodman, L. (2018, March 6). The universe is a hologram and other mind-blowing theories in theoretical physics. https://phys.org/news/2018-03-universe-hologram-mind-blowing-theories-theoretical.html.
[6] J. Bekenstein, Scholarpedia 3, 7375 (2008).
[7] J.D. Pandian, Home – Curious About Astronomy? Ask an Astronomer.
[8] Landau L. D. and Lifshitz E. M. 1975 The Classical Theory of Fields (Oxford: Pergamon)
[9] Lord E. A. 1976 Tensors, Relativity and Cosmology (New York: McGraw-Hill)
[10] McBryan, Bernard, ArXiv.org (2013).
[11] Nikodem Popławski 2016 ApJ 832 96
[12] “Schwarzschild Radius | COSMOS – Centre for Astrophysics …” Astronomy.swin, astronomy.swin.edu.au/cosmos/S/Schwarzschild Radius.
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