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Chapter 18: Problem 57

What prevents light from escaping from a black hole?

Short Answer

Expert verified
The intense gravitational pull at the Schwarzschild radius prevents light from escaping a black hole.

Step by step solution

01

Understand the concept of Escape Velocity

Escape velocity is the minimum speed that an object needs to escape the gravitational pull of a celestial body. This velocity depends on the mass and radius of the object and is given by the formula: \[ v_e = \sqrt{\frac{2GM}{r}} \] where \( v_e \) is the escape velocity, \( G \) is the gravitational constant, \( M \) is the mass of the celestial body, and \( r \) is its radius.
02

Apply Escape Velocity to a Black Hole

For a black hole, the escape velocity at its event horizon is equal to the speed of light. This means that the gravitational pull of a black hole is so strong that even light cannot escape it. The radius at which the escape velocity equals the speed of light is called the Schwarzschild radius.
03

Define the Schwarzschild Radius

The Schwarzschild radius \( r_s \) is given by: \[ r_s = \frac{2GM}{c^2} \]where \( c \) is the speed of light. Within this radius, any object would need to travel faster than light to escape, which is impossible according to the laws of physics.
04

Explain Why Light Cannot Escape

Since the escape velocity at the Schwarzschild radius equals the speed of light and nothing can travel faster than light, even photons (particles of light) cannot escape once they are within this radius. This ultimate gravitational pull prevents light and anything else from escaping the black hole.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Escape Velocity
Escape velocity is the speed an object needs to break free from a celestial body's gravitational pull.
Imagine you want to throw a ball up so high that it never comes back down.
The exact speed to do this depends on the mass of the planet or star and how far you are from its center.
Here is the formula for escape velocity: \( v_e = \sqrt{\frac{2GM}{r}} \).
In this formula:
  • \
Schwarzschild Radius
The Schwarzschild radius is a critical boundary around a black hole.
Any object within this radius cannot escape the black hole's pull.
This distance is defined where the escape velocity equals the speed of light.
To calculate it, use the formula: \( r_s = \frac{2GM}{c^2} \).
In this formula:
  • \
Gravitational Pull
Gravitational pull is the force that attracts objects towards the center of a celestial body.
In a black hole, this force is incredibly strong.
The stronger the gravitational pull, the harder it is to escape.

Here, the gravity increases steeply as you get closer to the black hole.
This means escape velocity also has to be higher.
At the Schwarzschild radius, the required escape velocity matches the speed of light.
Speed of Light
The speed of light is the fastest speed in the universe.
Nothing can move faster than this speed.
Light travels at about 299,792 kilometers per second.

When we calculate escape velocity for black holes, it equals the speed of light at the Schwarzschild radius.
This means that even light can't escape from inside this radius.
That's why black holes appear 'black' - not even light can get out.

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