Cosmologists have speculated that black holes the size of a proton could have formed during the early days of the Big Bang when the universe began. If we take the diameter of a proton to be 1.0 * 10^-15 m, what would be the mass of a mini black hole?

Verified step by step guidance

1. Identify the Schwarzschild radius formula, which relates the radius of a black hole to its mass: \( r_s = \frac{2GM}{c^2} \), where \( r_s \) is the Schwarzschild radius, \( G \) is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2), \( M \) is the mass of the black hole, and \( c \) is the speed of light in vacuum (approximately 3.00 x 10^8 m/s).
2. Set the Schwarzschild radius equal to the radius of a proton for this problem. Since the diameter of the proton is given as 1.0 x 10^-15 m, the radius will be half of that, which is 0.5 x 10^-15 m.
3. Rearrange the Schwarzschild radius formula to solve for the mass \( M \) of the black hole: \( M = \frac{r_s c^2}{2G} \).
4. Substitute the values for \( r_s \), \( c \), and \( G \) into the rearranged formula to find the mass of the black hole.
5. Calculate the mass using the values substituted into the formula to find the mass of a mini black hole with a radius equivalent to that of a proton.

Key Concepts

Black Hole Mass and Size Relationship

The mass of a black hole is directly related to its size, specifically its Schwarzschild radius, which defines the event horizon. For a non-rotating black hole, the Schwarzschild radius (r_s) is given by the formula r_s = 2GM/c^2, where G is the gravitational constant, M is the mass, and c is the speed of light. This relationship indicates that smaller black holes have significantly lower mass compared to larger ones.

Proton Size and Mass

A proton is a subatomic particle with a diameter of approximately 1.0 x 10^-15 meters. Its mass is about 1.67 x 10^-27 kilograms. Understanding the size and mass of a proton is crucial for calculating the mass of a hypothetical mini black hole, as it provides a reference point for the scale of such objects in the universe.

Big Bang and Primordial Black Holes

The Big Bang theory posits that the universe began from an extremely hot and dense state and has been expanding ever since. During the early moments of the universe, conditions were ripe for the formation of primordial black holes, which could have formed from density fluctuations. These black holes could vary in size, including those as small as a proton, leading to speculation about their properties and implications for cosmology.