As the mass of a black hole increases, its Schwarzschild radius a. increases as the square of the mass. b. increases proportionally. c. stays the same. d. decreases proportionally. e. decreases as the square of the mass.

The mass of a black hole is one of its most defining characteristics. Black holes are formed when massive stars collapse under their own gravity.
As the star collapses, its mass gets concentrated into a very small, dense region, creating a black hole.
The mass of a black hole can vary significantly, from a few times the mass of our Sun to billions of times that mass in the case of supermassive black holes located at the centers of galaxies.
The mass has a direct effect on the properties of the black hole, including its Schwarzschild radius — the radius of the event horizon beyond which nothing can escape the gravitational pull.
In our exercise, we found that increasing the mass of a black hole leads to a proportional increase in its Schwarzschild radius, highlighting that more massive black holes exert stronger gravitational forces.

General relativity, proposed by Albert Einstein, is a fundamental theory of physics that describes gravity.
Unlike Newtonian physics, which views gravity as a force between masses, general relativity conceptualizes gravity as a curvature of spacetime caused by mass and energy.
In this framework, massive objects like stars and black holes warp the spacetime around them. This warping is why light bends when it passes near a massive object, a phenomenon known as gravitational lensing.
General relativity predicts the Schwarzschild radius for black holes, which is the critical radius where the escape velocity equals the speed of light.
This means that within this radius, not even light can escape, making the black hole 'black' to an outside observer.
The exercise directly applied the principles of general relativity to understand how the Schwarzschild radius changes with the mass of the black hole.

The gravitational constant (G) is a key figure in Newton's law of gravitation and also plays a crucial role in Einstein's general theory of relativity.
Its value is approximately \(6.67430 \times 10^{−11} \text{m}^3\text{kg}^{−1}\text{s}^{−2}\).
This constant helps quantify the strength of the gravitational force between two masses.
In the context of our exercise, the gravitational constant is part of the formula for the Schwarzschild radius:
\[R_s = \frac{2GM}{c^2}\]
This formula demonstrates that the Schwarzschild radius depends not just on the mass of the black hole but also on fundamental constants like the gravitational constant and the speed of light (c).
It tells us that as the black hole's mass increases, so does the Schwarzschild radius, affirming that the relationship is proportional.
Understanding the gravitational constant helps in grasping why objects with greater mass have stronger gravitational fields and influence spacetime more dramatically.