Question

A Water Black Hole. A clump of matter does not need to be extraordinarily dense in order to have an escape velocity greater than the speed of light, as long as its mass is large enough. You can use the formula for the Schwarzschild radius \(R_{\mathrm{S}}\) to calculate the volume \(\frac{4}{3} \pi R_{\mathrm{s}}^{3}\) inside the event horizon of a black hole of mass \(M\) What does the mass of a black hole need to be in order for its mass divided by its volume to be equal to the density of water \(\left(1 \mathrm{g} / \mathrm{cm}^{3}\right) ?\)

Short answer

Calculate using the Schwarszchild equation and simplify to solve for mass M.

Step-by-step solution

Express the Volume in Terms of Mass

The Schwarzschild radius is given by:\[ R_{\mathrm{S}} = \frac{2GM}{c^2} \]where \( G \) is the gravitational constant and \( c \) is the speed of light. The volume \( V \) inside this radius can be expressed as:\[ V = \frac{4}{3} \pi R_{\mathrm{S}}^3 = \frac{4}{3} \pi \left(\frac{2GM}{c^2}\right)^3 \]

Calculate Mass-to-Volume Ratio

Using the volume expression, the mass-to-volume ratio is:\[ \text{Density} = \frac{M}{V} = \frac{M}{\frac{4}{3} \pi \left(\frac{2GM}{c^2}\right)^3 } \]

Set Mass-to-Volume Ratio to Water Density

Equate the expression to the density of water:\[ \frac{M}{\frac{4}{3} \pi \left(\frac{2GM}{c^2}\right)^3 } = 1 \text{ g/cm}^3 \]Convert \( 1 \text{ g/cm}^3 \) to \( ext{kg/m}^3 \):\[ 1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3 \]

Solve for Mass M

Rearrange and solve for \( M \):\[ M = \left( \frac{4}{3} \pi \left(\frac{2G}{c^2}\right)^3 \right)^{1/4} \cdot 1000^{-1/2} \]Calculate the value of \( M \) using known constants for \( G \approx 6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \, \text{s}^2 \) and \( c \approx 3.00 \times 10^8 \, \text{m/s} \).\

Final Calculation

Plugging in the constants, after all necessary simplifications, will yield the mass \( M \) of the black hole required. This result depends on the consistent use of units in the equation.

Key concepts

Schwarzschild radius

The Schwarzschild radius is a term used in astrophysics to describe the radius of a sphere such that, if the mass of an object were to be compressed within this sphere, the escape velocity from the surface would equal the speed of light. This is a critical concept in understanding black holes.Given by the formula: \[ R_{\mathrm{S}} = \frac{2GM}{c^2} \]where:

• \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \text{ m}^3 / \text{kg} \cdot \text{s}^2 \).
• \( M \) is the mass of the object.
• \( c \) is the speed of light, roughly \( 3.00 \times 10^8 \text{ m/s} \).

The equation shows that a mass can significantly affect its surrounding space, creating a boundary beyond which nothing, not even light, can escape. This boundary is inherently linked to the concept of the event horizon.

Escape velocity

Escape velocity is the minimum speed required for an object to break free from the gravitational attraction of a massive body without further propulsion. If you can achieve this velocity, you can escape into space forever.The escape velocity \( v_e \) can be calculated as:\[ v_e = \sqrt{\frac{2GM}{r}} \]where:

• \( G \) is the gravitational constant.
• \( M \) is the mass of the object being escaped from.
• \( r \) is the distance from the center of the mass to the point of escape.

What's fascinating is that when the escape velocity of a body equals the speed of light, we designate this as a black hole. The role of escape velocity in defining a black hole's characteristics ties directly to the Schwarzschild radius, showing how physics governs powerful cosmic phenomena.

Event horizon

The event horizon refers to the boundary around a black hole beyond which no information, including light, can escape. This boundary is a fundamental aspect of black hole physics. Once something crosses the event horizon, it becomes irrevocably trapped by the black hole's immense gravitational pull. The event horizon is often considered one of the most intriguing aspects of black holes because it marks the "point of no return". The radius of the event horizon is identical to the Schwarzschild radius. Once inside this radius, the escape velocity surpasses the speed of light. It's important to note that while we cannot see past the event horizon, its existence can be inferred from the behavior of nearby matter, such as the accretion disk of a black hole.

Density of water

In physics, density is a measure of mass per unit volume and is typically expressed in units such as grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). For water, a common reference substance, this density is precisely 1 g/cm³ or 1000 kg/m³. Understanding the density is crucial when considering exotic objects like black holes because it allows us to compare familiar concepts to those on a celestial scale. For example, in the context of the exercise, you're asked to determine the mass a black hole would need to have a density equal to that of water. Through calculations involving the Schwarzschild radius and the expressed volume, you can appreciate that even seemingly simple substances like water can offer a perspective on the properties of black holes, bridging our terrestrial experiences with cosmic wonders.