In this Chapter, we mainly discuss what occurs inside the event horizon of a black hole. Our research has theoretically demonstrated that the maximum velocity that gravity can achieve within the event horizon is four point two four into ten rise to eight. This is much faster than the velocity of light. But the gravitational waves we can detect travel at the speed of light. Let’s explore how this happens in this in this article.
First, let’s examine how gravity reaches its maximum speed inside the event horizon. We know that the escape velocity at the Schwarzschild radius equals the speed of light. All our knowledge so far ends at this point. We don’t know what happens inside the Schwarzschild radius. Let’s explore what happens inside the Schwarzschild radius using mathematics. This will greatly aid in understanding quantum gravity, a field we have been investigating for a long time.
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What we will do here is theoretically examine four familiar objects. Since these four objects represent different masses, the same concept applies to any other mass. Theoretically, we can determine the radius of any object in its smallest possible state. We cannot shrink anything further than that. Now we’re discussing the region containing only matter within the event horizon of a mass that forms a black hole. This is a region where gravity is most turbulent due to the extreme compression of space-time. Since we can determine the radius of matter when it shrinks to its smallest state, we can calculate the escape velocity at its surface. So let’s jump into the mathematical structure.
First, we can use the Planck mass for comparative studies. It is the smallest mass we know of. Here Planck mass is 2.176 \times 10^{- 8} kg. Now we can calculate the Schwarzschild radius of Planck mass. We know that the equation to calculate the Schwarzschild radius is \frac{2GM}{c^2}. When we calculate the Schwarzschild radius of the Planck mass according to this equation, we get the answer as 3.22724978 \times 10^{- 35} meters. We can already understand that the Schwarzschild radius of the Planck mass is approximately twice the Planck length. The radius of an object at its smallest size will be almost half of its Schwarzschild radius. So, if you know the Schwarzschild radius of a black hole, you can easily determine the radius or volume of the mass inside it. We can calculate the Planck length as the radius of the Planck mass. Now, let’s examine the escape velocity at the Planck length. The reason the Planck length is used here is explained on my website or in the books I have written. The equation to calculate escape velocity is \sqrt{\frac{2GM}{r}} . If we replace r with the Planck length in this equation, we get the answer as 4.24 \times 10^8m/ s. The escape velocity at the event horizon will always be the speed of light. The distance from the surface of the Planck mass to the event horizon is approximately equal to the Planck length. When reaching the surface of the Planck mass from the event horizon, the escape velocity increases to f4.24 \times 10^8m/ s from the speed of light. That is, the escape velocity changes by 1.24 \times 10^8m/ s as it traverses the event horizon at a distance equal to the Planck length. Such a huge change in gravity occurs within such a small distance within the event horizon.
Now, let’s consider an electron as the second mass for a comparative study. The mass of an electron is 9.1093837 \times 10^{- 31}kg. The Schwarzschild radius of an electron with this mass is 1.35102282 \times 10^{- 57}m. To proceed further, we need to determine the radius an electron would have if it were to become a black hole. So, we can use the formula \frac{4}{3}\pi r^3 to find the volume of a sphere here. But we know that objects have different masses. Adding each unit of Planck mass will correspondingly add a proportional amount of space-time to the system. To elucidate the nature of larger objects, their mass must be compared to the Planck mass. Therefore, it is necessary to understand the quantity of Planck mass contained within the object. We can quickly comprehend it by using this equation m_{A\rho} = \frac{m}{m_{\rho}}. Here m_{A\rho} represents the ratio of Planck mass within the object, m represents the actual mass of the object, and m_{\rho} represents the Planck mass. m_{A\rho} is variable according to the mass of the object. We can further extend the above mentioned equation to describe objects containing more than one Planck mass and find the volume of the object.
Now the equation to find the volume of a sphere looks like this \frac{4}{3}\pi \left(m_{A\rho}l_{\rho}\right)^3 . Here, l_{\rho} represents the Planck length and m_{A\rho}l_{\rho} is the radius of that object. This is because adding each unit of Planck mass to a system increases its radius by the corresponding Planck length. We already know the Schwarzschild radius two \frac{2GM}{c^2}. This radius m_{A\rho}l_{\rho} will be the actual radius of any mass if it shrinks to its minimum. So, with this equation, we can find the actual radius of any mass. Now we can use the equation m_{A\rho}l_{\rho} to calculate the actual radius of an electron if it were a black hole. We are going to obtain a value similar to 6.672 \times 10^{- 58}m. Surprisingly, we can find that this value is almost half of the Schwarzschild radius of an electron 1.35102282 \times 10^{- 57}m If we calculate the escape velocity at this radius, we will obtain a value equal to the escape velocity of the Planck mass which is equal to f4.24 \times 10^8m/ s. If we look closely, we can see that the gravitational force undergoes large changes within about 6.67902282 \times 10^{- 58}m of the event horizon.
So far we have examined the behavior of two microscopic particles. Now let’s take our own Earth as an example. Mass of Earth is 5.97 \times 10^{24}kg. Ratio between Earth’s mass and Planck mass is 2.74356618 \times 10^{32}. If the Earth were to become a black hole, its Schwarzschild radius would be 8.854 \times 10^{- 3}m. We can now easily calculate that the minimum radius of the Earth is 4.43360294 \times 10^{- 3}m. Here too, we can understand that the minimum radius of the Earth is only about half of the Schwarzschild radius. The escape velocity at the Earth’s minimum radius is equal to the mass of the electron and the Planck mass which is 4.24 \times 10^8m/ s.
Let’s conclude with another comparative study. Here, I am taking our parent star, the Sun, as an example. Mass of the Sun is 1.989 \times 10^{30}kg . Ratio between Sun’s mass and Planck mass is 9.14 \times 10^{37}. If the Sun were to become a black hole, its Schwarzschild radius would be 2.949928 \times 10^3m. Using the equation m_{A\rho}l_{\rho} , we can now calculate the minimum radius of the Sun, which will be approximately 1.477024 \times 10^3m. This will be approximately one and a half kilometers. Now we can understand things quickly. So far, we have understood that the radius of a black hole is its Schwarzschild radius. But, we now know that only about half of the Schwarzschild radius consists of matter, while the other half is a region distorted by the extreme compression of space-time.
Let’s take the Sun as an example. The Schwarzschild radius of the Sun is three kilometers, giving it a total diameter of 6 kilometers. However, Sun’s minimum possible radius, based on our calculations, is 1.5 kilometers, making its corresponding diameter three kilometers. An object that enters the Sun’s event horizon must travel approximately 1.5 kilometers through highly compressed space-time before becoming part of its matter. To be more specific, if an object’s velocity upon entering the event horizon is 3 \times 10^8m/ s , then by the time it becomes part of the matter, its velocity will have increased to 4.24 \times 10^8m/ s. The velocity increases by 1.24 \times 10^8m/ s within one point five kilometers of the event horizon. What is evident here is an unimaginably intense gravitational force. We can never observe this directly because it occurs inside the event horizon, where nothing escapes due to the extremely strong gravity. The velocity of gravitational waves that we can observe will be equal to the speed of light, because the escape velocity at the event horizon is also equal to the speed of light. Even when two black holes merge, we can only observe the events that occur outside the event horizon.
Closing due to time constraints. If you are interested in learning more about this topic, Our book explains this in more detail. This book describes the universe from the initial stage of formation of the Big Bang singularity. Thank you for watching our video.