Question

Be sure to show all calculations clearly and state your final answers in complete sentences. Schwarzschild Radii. Calculate the Schwarzschild radius (in kilometers) for each of the following. a. ${10}^8{M}_{\text{Sun }}$ black hole in the center of a quasar b. $5M_{\text{sun }}$ black hole that formed in the supernova of a massive star c. $A$ mini-black hole with the mass of the Moon d. A mini-black hole formed when a superadvanced civilization decides to punish you (unfairly) by squeezing you until you become so small that you disappear inside your own event horizon

Short answer

a) 295,200 km, b) 14.76 km, c) 0.0001104 km, d) Negligibly small.

Step-by-step solution

Understanding the Schwarzschild Radius Formula

The Schwarzschild radius is the radius of the event horizon of a black hole and is calculated using the formula $R_s=\frac{2GM}{c^2}$, where $G=6.674\times {10}^{-11}$ m${}^3$ kg${}^{-1}$ s${}^{-2}$ is the gravitational constant, $c=3\times {10}^8$ m/s is the speed of light, and $M$ is the mass of the object in kilograms.

Converting Units for Masses

For accurate calculation, we must convert the given masses into kilograms. The solar mass $M_{\text{Sun}}=1.989\times {10}^{30}$ kg. The mass of the Moon is approximately $7.35\times {10}^{22}$ kg. Convert the mass in part (d) to kg based on personal weight (taking 70 kg as an average human mass for demonstration).

Calculating Schwarzschild Radius for Quasar's Black Hole

The mass $M={10}^8{M}_{\text{Sun}}={10}^8\times 1.989\times {10}^{30}$ kg. Substituting into the formula: $$R_s=\frac{2\times 6.674\times {10}^{-11}\times 1.989\times {10}^{38}}{(3\times {10}^8{)}^2}$$.Solving this gives $R_s\approx 2.952\times {10}^{11}$ meters, or approximately $295,200$ kilometers.

Calculating Schwarzschild Radius for Stellar Black Hole

The mass $M=5M_{\text{Sun}}=5\times 1.989\times {10}^{30}$ kg. Substituting into the formula gives: $$R_s=\frac{2\times 6.674\times {10}^{-11}\times 9.945\times {10}^{30}}{(3\times {10}^8{)}^2}$$. This results in $R_s\approx 14,760$ meters, or approximately $14.76$ kilometers.

Calculating Schwarzschild Radius for Mini-Black Hole with Moon's Mass

Using the Moon's mass $M=7.35\times {10}^{22}$ kg: $$R_s=\frac{2\times 6.674\times {10}^{-11}\times 7.35\times {10}^{22}}{(3\times {10}^8{)}^2}$$.This gives $R_s\approx 0.0001104$ meters, or approximately $0.0001104$ kilometers.

Calculating Schwarzschild Radius for Human-Sized Mini-Black Hole

Assuming the mass of a person $M=70$ kg: $$R_s=\frac{2\times 6.674\times {10}^{-11}\times 70}{(3\times {10}^8{)}^2}$$. Calculation yields $R_s\approx 1.04\times {10}^{-25}$ meters, which is negligibly small in kilometers.

Key concepts

black holes

When we talk about black holes, we're diving into some of the most mysterious and fascinating objects in the cosmos. These are regions in space where the gravitational pull is so intense that nothing, not even light, can escape from them. This is because a black hole compresses a vast amount of mass into a tiny space.

• A black hole forms when a massive star dies and collapses under its own gravity, leading to a point of infinite density called a singularity.
• Surrounding the singularity is the event horizon, marking the boundary where the escape velocity equals the speed of light.
• Because no light can escape, black holes appear black, thus their name.
• Black holes can vary in size from just a few kilometers to several billion kilometers across, depending on their mass.
• Supermassive black holes, like the ones found at the centers of galaxies, including our own Milky Way, often influence their surroundings significantly with their immense gravity.

gravitational constant

The gravitational constant, denoted as $G$, is a fundamental constant in physics that plays a crucial role in the calculations involving gravity. It appears in Newton's law of universal gravitation, which explains the forces that two bodies exert on each other.

• The SI unit for $G$ is meters cubed per kilogram per second squared (m${}^3$ kg${}^{-1}$ s${}^{-2}$).
• In the equation for the Schwarzschild radius $R_s=\frac{2GM}{c^2}$, $G$ helps us determine how size correlates with mass in black holes.
• It's measured to be approximately $6.674\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$.
• The constant shapes our understanding of gravitational interactions not only in our solar system but across the entire universe.
• In essence, $G$ helps describe how massive objects like Earth, the Sun, and black holes warp space around them, influencing nearby objects.

speed of light

The speed of light, symbolized by $c$, is one of the most widely known constants in physics, characterizing how fast light travels in a vacuum. It's an essential component in many formulas, especially in the realm of Einstein's theory of relativity.

• The speed of light is precisely $299,792,458$ meters per second, though it's often rounded to $3\times {10}^8$ m/s for simplicity.
• In the formula for calculating the Schwarzschild radius, the speed of light helps define the critical boundary of the event horizon.
• The concept that nothing can move faster than the speed of light is a fundamental limit in our universe and affects how information and matter travel.
• It's pivotal not only in cosmological calculations but also in everyday technologies, impacting GPS satellite synchronization and transmitting signals.
• In black hole physics, light's speed determines the ability for radiation and signals to escape a gravitational field, defining the boundary beyond which they can't return.

event horizon

In the study of black holes, the event horizon is a crucial concept. It's the boundary surrounding a black hole beyond which nothing can return once it crosses.

• This boundary defines the point of no return, governed by the escape velocity needed to break free from the black hole's gravity.
• Once an object passes the event horizon, it inevitably moves toward the black hole's center, or singularity, with no chance of escape.
• The event horizon's size is determined by the Schwarzschild radius, which increases with the mass of the black hole.
• Importantly, the event horizon marks a fundamental break in the fabric of spacetime as understood under general relativity.
• Light emitted from just outside the event horizon can still escape, which is why we can calculate and observe the effects of black holes indirectly.
• Moreover, it's a unique surface, with no solid form but crucial implications for the universe's structure and information theory as per concepts like Hawking radiation.