Question

If it is fundamental to nature that a given mass has a critical radius at which something extraordinary happens (i.e., a black hole forms), we might guess that this radius should depend only on the mass and fundamental constants of nature. Assuming that \(r_{\text {critical }}\) depends only' on \(M, G\), and \(c\), show that dimensional analysis gives the equation for the Schwarzschild radius to within a multiplicative constant.

Short answer

The Schwarzschild radius, up to a constant factor, is \(MG/c^2\).

Step-by-step solution

Understanding the variables involved

It is given that the critical radius \(r_{critical}\) depends on the mass \(M\), the gravitational constant \(G\), and the speed of light \(c\). First, recognize the dimensions of each of these quantities. Mass \(M\) has a dimension of [M] (mass), the gravitational constant \(G\) has dimension of [M\(^{-1}\)L\(^3\)T\(^{-2}\)] and the speed of light \(c\) has a dimension of [LT\(^{-1}\)], where L is length and T is time.

Set up the dimensional analysis

We want to find \(r_{critical}\) which has dimension of [L] using \(M\), \(G\), and \(c\). According to the principle of dimensional analysis, dimensions on both sides of an equation must match. We can write the dependence of \(r_{critical}\) on \(M\), \(G\), and \(c\) as follows: \[r_{critical} =M^aG^bC^c\] where \(a\), \(b\), and \(c\) are powers to be determined.

Compare the dimensions on both sides

The critical radius, \(r_{critical}\), has a dimension of [L]. Therefore, the dimensions on the right side of the equation also need to add up to [L]. Equate the dimensions on both sides of the equation to get the following equation: \[[L]=[M]^a[M\(^{-1}\)L\(^3\)T\(^{-2}\)]^b[LT\(^{-1}\)]^c\] which simplifies to \[[L]=[ML\(^2\)T\(^{-1}\)]^a[ML\(^3\)T\(^{-2}\)]^b[LT\(^{-1}\)]^c\] comparing the coefficients of M,L and T gives the following equations: \[a-b=0\] \[2a+3b+c=1\] \[-b-c=0\]

Solve the equations and find the Schwarzschild radius

Solving the above equations, we get \(a=b=1\) and \(c=-1\). Substituting \(a\), \(b\), and \(c\) back into the equation we obtained in Step 2, we find \(r_{critical}=MG/c^2\), up to a constant factor. So the Schwarzschild radius, up to a constant factor, is given by the product of the mass and the gravitational constant, divided by the square of the speed of light.

Key concepts

Dimensional Analysis

Dimensional analysis is a tool used in physics and engineering to understand the relationship between different physical quantities. It involves analyzing the dimensions (mass, length, time, etc.) of the quantities involved in a physical situation to ensure that both sides of an equation are consistent in terms of their units.

For instance, in the exercise you have, dimensional analysis helps us deduce the Schwarzschild radius formula by equating the dimensions of quantities on both sides of an equation. Since the Schwarzschild radius has the dimension of length [L], the goal is to combine the dimensions of mass [M], the gravitational constant [M^-1L³T^-2], and the speed of light [LT^-1] in a way that results in the dimension of length [L]. The outcome confirms that it is possible to derive the Schwarzschild radius using only these fundamental constants and mass, reinforcing the power of dimensional analysis in physics.

Gravitational Constant

The gravitational constant (denoted as G) is a fundamental constant in physics that appears in Newton's law of universal gravitation. It describes the strength of the gravitational force between two masses.

The dimension of the gravitational constant is [M^-1L³T^-2], reflecting the relationship between mass (M), length (L), and time (T). In the exercise, G's role is pivotal since it relates mass and the fabric of spacetime to form a black hole at the Schwarzschild radius. The gravitational constant thus bridges the gap between massive objects and their gravitational effects on the surrounding space-time continuum.

Speed of Light in Physics

The speed of light, commonly represented by the symbol c, is a fundamental physical constant that denotes the speed at which light travels in a vacuum. It has a value of approximately 299,792,458 meters per second.

In physics, the speed of light is crucial not only as a speed measurement but also as a constant that relates energy to mass (as per Einstein's E=mc²) and appears in the calculation of the Schwarzschild radius. It highlights that light's speed is a limit for the propagation of causality and information in the universe. It also defines the scale for the maximum speed at which gravity itself can propagate according to general relativity.

Mass in Physics

Mass is a fundamental property of physical objects which quantitatively measures the amount of matter within them. In the international system of units (SI), mass is measured in kilograms (kg).

In the context of the Schwarzschild radius, mass represents one of the central factors in black hole formation. The mass of an object is critical in determining whether or not it will collapse under its own gravity to form a black hole once it is within its Schwarzschild radius. This exercise demonstrates that a black hole's boundary is intrinsically tied to the mass of the object, revealing the profound link between an object's mass and the curvature of space-time it can produce.