Calculate the Schwarzschild radius of a \(10^{9}\)-solar-mass black hole. How does your answer compare with the size of our solar system (given by the diameter of Pluto's orbit)?

Understanding the concept of a black hole's mass is crucial in astrophysics and helps us comprehend the immense forces at play in our universe. A black hole's mass is at the center of its gravitational pull, with larger masses resulting in stronger gravitational attraction. In our exercise, the mass of a black hole is given in terms of solar masses, a common measurement in astrophysics. One solar mass is equivalent to the mass of our Sun, approximately 1.99 x \(10^{30}\) kilograms.

When we speak of a \(10^9\)-solar-mass black hole, we're referring to a black hole with a mass a billion times that of our Sun. To put it into a formula, the mass \( M \) of a black hole is \(10^9\) times the mass of the Sun, hence \( M = 10^9 \times 1.99 \times 10^{30} \) kg, which simplifies to approximately \(1.99 \times 10^{39}\) kg. This number is beyond everyday comprehension, showcasing the colossal scale of celestial objects. The black hole's mass is a fundamental input in calculating its Schwarzschild radius, the point at which nothing, not even light, can escape its gravitational influence.

The gravitational constant \( G \) is a key variable in the language of the cosmos, enabling us to measure the gravitational attraction between two objects. It is a cornerstone of Newton's law of universal gravitation and is also vital in the calculation of the Schwarzschild radius. The value of \( G \) is approximately \( 6.67430 \times 10^{-11} \) m³ kg^-1 s^-2. Despite its tiny numerical value, its impact on the universe is monumental.

The constant provides us with the proportionality factor needed to calculate the force of attraction between two masses. It is important to note that \( G \) is a universally constant value. This means that its value does not change regardless of the location, whether on Earth, near a black hole, or in any other part of the universe. In our exercise, we use \( G \) alongside the mass of the black hole to determine the Schwarzschild radius, which illustrates the boundary of the black hole's event horizon.

The speed of light, denoted as \( c \), is the fastest speed at which all energy, matter, and information in the universe can travel. It is one of the most significant constants in physics, with a value of exactly 299,792,458 meters per second (m/s). This incredible speed has far-reaching implications for the fabric of reality, including time dilation and the nature of causality.

In the realm of black holes, the speed of light plays a pivotal role in defining the event horizon, the 'point of no return'. The event horizon is closely related to the Schwarzschild radius, where the escape velocity equals the speed of light. In our calculations, we divide the product of the gravitational constant and the black hole's mass by the square of the speed of light. This accounts for the relativistic effects near such dense objects. Fascinatingly, within the Schwarzschild radius, the required escape velocity would exceed \( c \), making it physically impossible for light to escape, hence why we perceive black holes as 'black'.