The lifetime of a black hole varies in direct proportion to the cube of the black hole's mass. How much longer does it take a supermassive black hole of \(10^{30} M_{\odot}\) to decay compared to a stellar black hole of \(3 M_{\odot}\) ?

When discussing black holes, mass is a central concept. Mass is the amount of matter in an object. In the case of black holes, this mass is incredibly large. Black holes can vary greatly in mass, ranging from small stellar black holes to supermassive black holes found at the centers of galaxies.
Stellar black holes typically form from the remnants of massive stars that have undergone supernova explosions. They usually have masses up to around 20 solar masses, denoted as \(M_\odot\), where \(M_\odot\) is the mass of our Sun. However, the exercise gives a stellar black hole mass of \(3M_\odot\).
Supermassive black holes are much more massive and can be billions of times the mass of the Sun (\(10^{30} M_\odot\)). These black holes are typically located at the center of galaxies and have a significant impact on their surroundings due to their immense gravitational pull. Understanding the mass of a black hole is crucial when calculating its lifetime and other properties.

Proportional relationships are essential in understanding how different quantities affect each other. In the given exercise, the relationship between the lifetime of a black hole and its mass is a key point.
Specifically, the lifetime \(T\) of a black hole is directly proportional to the cube of its mass \(M\). This relationship is described mathematically as:
\(T \propto M^3\)
This means that if the mass of the black hole increases, its lifetime increases proportionally to the cube of that mass. For example, if the mass of a black hole is doubled, its lifetime would increase by a factor of \(2^3 = 8\).
To express this proportion as an equation, we use:
\(T = k \cdot M^3\)
where \(k\) is a constant of proportionality. Understanding this concept helps to calculate how long it takes for black holes of different masses to decay.

Physics equations provide the mathematical framework to describe the laws of nature. In the context of black holes, specific equations help us understand their behavior and properties. The exercise uses several physics equations to solve the problem.

First, the proportional relationship between the lifetime and mass of a black hole is given by:
\(T \propto M^3\)
This implies that lifetime \(T\) is a function of mass \(M\) cubed, which we write as:
\(T = k \cdot M^3\)
where \(k\) is the constant of proportionality.
Next, these principles are applied to calculate the lifetimes of black holes with different masses. For a supermassive black hole of mass \(10^{30}M_\odot\):
\(T_{supermassive} = k \cdot (10^{30} M_{\odot})^3 = k \cdot 10^{90} M_{\odot}^3\)
For a stellar black hole of mass \(3M_{\odot}\):
\(T_{stellar} = k \cdot (3 M_{\odot})^3 = k \cdot 27 M_{\odot}^3\)
To find the ratio of the lifetimes, we divide these expressions:
\(\frac{T_{supermassive}}{T_{stellar}} = \frac{k \cdot 10^{90} M_{\odot}^3}{k \cdot 27 M_{\odot}^3} = \frac{10^{90}}{27}\)
This cancellation of common factors simplifies our calculations by focusing on the core proportional relationship.