Schwarzschild Radius
The German scientist Albert Einstein discovered the theory of general relativity and its novel vision of gravity, then the German physicist Karl Schwarzschild took on this question in 1916. His work published the limit at which gravity succeeded over the other physical forces, creating a black hole. Hence the theory is called Schwarzschild radius. This theory was discovered in the early 20th century. Besides, there is more to the black hole story, which Thentarts in the late 1700s with a little-known scientist named John Michell. Michell devised the torque balance, a piece of equipment that prepares the strength of forces to be computed quite accurately.
The Schwarzschild radius definition is defined as, it is a physical parameter that occurs in the Schwarzschild solution to Einstein's field equations, proportional to the radius defining the event horizon of a Schwarzschild black hole. Schwarzschild radius is also called a gravitational radius. It is a characteristic radius related to any quantity of mass. Karl Schwarzschild calculated this solution for the theory of general relativity in 1916.
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The minimum radius beyond which a body's gravitational interaction between particles would cause irreversible gravitational collapse. This phenomenon is thought to be the destiny of the more massive stars (see black hole). Simply, the black hole becomes, when a star's mass is so large that nothing can stop its own gravity from compressing all the materials that make up the star. When this happens, the mass density and gravitational force inside the black hole is very strong. In addition, the laws of physics cannot explain what happens there anymore. The gravitational field inside a black hole is so strong that not even light can escape from it. Hence the black hole is black. The separation between the region where we know how things work and the region where we don't work hence is called the event horizon, and in a black hole, it is known as the Schwarzschild radius. In the Spanish language, the Schwarzschild radius is called radio de Schwarzschild.
The Schwarzschild radius is denoted by Rg and is an object of mass M is given by the following formula, The Schwarzschild radius formula is given as,
\[R_{g} = 2\frac{GM}{c^{2}}\]
Where,
G is the universal gravitational constant
c is the speed of light.
Let derive the expression of the Schwarzschild radius equation,
Here g is a acceleration due to gravity is given as,
\[g= 2\frac{GM}{Rg^{2}}\]
The escape velocity is equal to speed of light hence kinetic energy is equal to potential energy.
P= .E = K.E
In general, the P.E is equal to mgh and K.E is equal to \[\frac {1}{2}\] mv2
P.E = K.E
mgh = \[\frac {1}{2} mv^2\]
gh = \[\frac {1}{2} v^2 gh\]
= v2 = 2gh
= \[v^{2}=2\frac{GM}{R_{g}^{2}}\] where h is equal to the radius of the black hole.
= \[v^{2}=2\frac{GM}{R_{g}}\]
= \[R_{g}=2\frac{GM}{v^{2}}\]
In a general situation, calculate the gravitational acceleration by simply using the gravitational force calculator and mount one of the masses to 1 kg. This is a logical procedure to calculate the gravity from a normal object or the black hole gravity at a point away from the surface of the black hole. However, there is a point of unique interest for black holes called the event horizon or Schwarzschild radius formula which exactly calculates the black hole gravity.
Let's see how to use the black hole Schwarzschild radius calculator.
What is the Event Horizon Schwarzschild Radius of a Black Hole?
The event horizon is the most important concept when talking about black holes. In this Schwarzschild radius calculator, we can simply compute where it is located. The event horizon Schwarzschild radius is the point of a collection of points in space that divide two areas that cannot be transmitted back and forth.In the case of a black hole, it is the point where the escape velocity is the speed of light. The event horizon of a black hole separates the points from the area of space where light can also escape velocity due to the lure of such a large object. Nothing, not even light, can withstand the black hole's pull. This means that effectively falling inside the event horizon is "lost forever" and we can never recover it.
Because of this effect, the event horizon is generally considered informally as the surface of a black hole.
Schwarzschild showed the same situation for anybody but the mass could become a black hole if that mass were compressed into a sufficiently small sphere and the sphere with a radius R, which we now call the Schwarzschild radius. To calculate the Schwarzschild radius of any object, planet, galaxy, even an apple all you need to know is the mass to be compressed.
Let's Discuss the Schwarzschild Radius of Earth.
The Schwarzschild radius of earth is nearly equal to one inch, meaning that you could compress the entire mass of the Earth into a sphere the size of a basketball and still not have a black hole, light emitted from that mass can still escape the high gravitational attraction. However, if you compress the mass of the Earth into a sphere the size of a ping-pong ball, it becomes a black hole.
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Let’s Discuss Some Examples of Schwarzschild Radius
The Schwarzschild radius of the universe is observable when the universe's mass is approximately 13.7 billion light-years.
The Schwarzschild radius of a human is 10-23cm. If the mass is compressed as small as possible then the human being, the Schwarzschild radius is of the order of 10-23 cm, much smaller than the nucleus of an atom.
The Schwarzschild radius of the typical star Sun is about 3 km (2 miles).
The Schwarzschild radius of a proton, and approximately the quarks in a proton, are about 1000-100,000 times more massive, still more than approx.
FAQs on Schwarzschild Radius
1. What is the Schwarzschild radius?
The Schwarzschild radius is the critical radius to which a celestial body of a given mass must be compressed to become a black hole. It represents the boundary, known as the event horizon for a non-rotating black hole, from within which nothing, not even light, can escape the object's gravitational pull. It was calculated by Karl Schwarzschild in 1916 using Einstein's theory of general relativity.
2. How is the Schwarzschild radius of an object calculated?
The Schwarzschild radius (Rs) is calculated using the formula: Rs = 2GM/c². In this equation:
- G is the universal gravitational constant.
- M is the mass of the object.
- c is the speed of light in a vacuum.
This formula shows that the Schwarzschild radius is directly proportional to the mass of the object.
3. What is the difference between an object's physical radius and its Schwarzschild radius?
An object's physical radius is its actual, measurable size from its centre to its surface. In contrast, the Schwarzschild radius is a calculated, theoretical radius based purely on the object's mass. For almost all objects, like planets and stars, their physical radius is vastly larger than their Schwarzschild radius. A black hole is formed only when an object is compressed to the point where its physical radius becomes smaller than its Schwarzschild radius.
4. Are the Schwarzschild radius and the event horizon of a black hole always the same thing?
For a simple, non-rotating, and uncharged black hole (a Schwarzschild black hole), the Schwarzschild radius is precisely the location of the event horizon. However, for rotating black holes (Kerr black holes), the structure is more complex. In the context of the CBSE/NCERT syllabus for the 2025-26 session, the terms are often used interchangeably to describe the point of no return for a non-rotating black hole.
5. What would be the Schwarzschild radius for common celestial objects like the Earth or the Sun?
The Schwarzschild radius gives a fascinating perspective on the mass of objects. For example:
- The Sun, despite its enormous size, has a Schwarzschild radius of only about 3 kilometres.
- The Earth has a Schwarzschild radius of just 9 millimetres, roughly the size of a small marble.
Since the actual physical radii of the Sun and Earth are much larger than these values, they are not black holes.
6. Why doesn't every object collapse into a black hole if everything has a Schwarzschild radius?
While every object with mass has a calculated Schwarzschild radius, collapse into a black hole only occurs if the object's mass is concentrated within that radius. For ordinary objects like planets and stars, internal forces (like electromagnetic forces and quantum pressure) are far stronger than gravity, preventing them from collapsing. Only in extremely massive stars do gravitational forces become strong enough to overcome these opposing forces and trigger a gravitational collapse.
7. What is the fundamental condition for an object to become a black hole?
For an object to become a black hole, its entire mass must be compressed into a volume so small that its physical radius becomes less than or equal to its Schwarzschild radius. When this happens, the gravitational pull at its surface becomes so intense that the escape velocity exceeds the speed of light, trapping everything inside its event horizon.
8. What would happen if you could compress a common object, like a car, to its Schwarzschild radius?
If you could hypothetically overcome the immense internal forces and compress a car to its Schwarzschild radius, it would indeed become a micro black hole. However, its Schwarzschild radius would be incredibly small, on the order of 10⁻²⁴ metres, which is far smaller than a single proton. This example highlights that an object's density (mass per unit volume), not just its mass, is the critical factor in forming a black hole.
