Show that the Schwarzschild radius can also be found by taking the escape velocity from an object of mass \(M\) and radius \(R,\) and setting it equal to \(c\)

Escape velocity is the speed needed for an object to break free from the gravitational pull of another object without any additional force.
Imagine throwing a ball up into the air. The harder you throw, the higher it goes before gravity pulls it back down.
Escape velocity is the speed at which the ball goes so high that it never comes back down.
This speed depends on two things:

• The mass of the object creating the gravity (e.g., a planet or star)
• The distance from the center of that object

The formula for escape velocity is given by: \[ v_e = \sqrt{\frac{2GM}{R}} \].
Here, \( v_e \) is escape velocity, \( G \) is the gravitational constant, \( M \) is the mass of the object, and \( R \) is the radius from the center of that object.
The formula shows how a larger mass or a smaller radius results in a higher escape velocity.

The gravitational constant, often denoted as \( G \), is a fundamental constant that appears in the law of universal gravitation.
Isaac Newton first introduced this law, and it describes how two objects in the universe attract each other.
The gravitational constant is a measure of how much force gravity applies to objects.
Its value is approximately \( 6.67430 \times 10^{-11} \text{m}^3\text{kg}^{-1}\text{s}^{-2} \).
In the escape velocity formula \[ v_e = \sqrt{\frac{2GM}{R}} \], \( G \) helps determine how strong the gravitational pull is for a given mass \( M \) and radius \( R \).
Without \( G \), we wouldn't be able to calculate how fast an object needs to travel to escape another object's gravitational pull.
It's crucial for understanding many phenomena in physics and plays a key role in our Schwarzschild radius calculations.

The speed of light, denoted by \( c \), is a fundamental constant in physics.
It represents the maximum speed at which all energy, matter, and information in the universe can travel.
Its value is approximately \( 299,792,458 \text{ meters per second} \).
In our calculation of the Schwarzschild radius, we set the escape velocity equal to the speed of light: \( v_e = c \).
This means an object must travel at the speed of light to escape the gravitational pull of a black hole.
By squaring both sides of the equation \( \sqrt{\frac{2GM}{R}} = c \), we derive the formula for the Schwarzschild radius: \[ R_s = \frac{2GM}{c^2} \].
This shows that the Schwarzschild radius is the radius at which the escape velocity equals the speed of light, creating an event horizon where not even light can escape the gravity of a black hole.