Which statement about Schwarzschild radii is true?
A.The Schwarzschild radius of a black hole can be defined only in binary systems.
B.The Schwarzschild radius of a black hole depends only on its mass.
C.The Schwarzschild radius is used to describe the size of neutron stars.
D.The more massive the black hole, the smaller the Schwarzschild radius.
The Correct Answer and Explanation is :
The correct answer is B. The Schwarzschild radius of a black hole depends only on its mass.
Explanation:
The Schwarzschild radius (( r_s )) is a critical concept in general relativity that defines the radius of the event horizon of a non-rotating black hole. The formula for the Schwarzschild radius is given by:
[
r_s = \frac{2GM}{c^2}
]
where ( G ) is the gravitational constant, ( M ) is the mass of the black hole, and ( c ) is the speed of light in a vacuum. This equation illustrates that the Schwarzschild radius is directly proportional to the mass of the black hole. As the mass increases, the Schwarzschild radius increases linearly.
Why the Other Statements Are Incorrect:
- A. The Schwarzschild radius of a black hole can be defined only in binary systems.
This statement is incorrect because the Schwarzschild radius applies to any non-rotating black hole, regardless of whether it is part of a binary system or not. It can be defined for any isolated black hole based solely on its mass. - C. The Schwarzschild radius is used to describe the size of neutron stars.
This statement is misleading. The Schwarzschild radius specifically applies to black holes and does not describe the physical size of neutron stars. Neutron stars are incredibly dense and have a different set of properties that do not fall under the same framework as black holes. - D. The more massive the black hole, the smaller the Schwarzschild radius.
This statement is false. In fact, the opposite is true: the more massive the black hole, the larger the Schwarzschild radius. For example, a black hole with ten times the mass of the Sun will have a Schwarzschild radius that is ten times larger than that of a black hole with the mass of the Sun.
In summary, the Schwarzschild radius is a fundamental characteristic of black holes that depends solely on their mass, distinguishing it from other astronomical objects and systems.