Question

Calculate the Schwarzschild radius of a black hole with the mass of a) the Sun. b) a proton. How does this result compare with the size scale of \(10^{-15} \mathrm{~m}\) usually associated with a proton?

Short answer

In this exercise, we calculated the Schwarzschild radius for a black hole with the mass of the Sun and a proton. The Schwarzschild radius for a black hole with the mass of the Sun is approximately 2.954 kilometers. Whereas the Schwarzschild radius for a black hole with the mass of a proton is approximately \(2.490 \times 10^{-54}\) meters, which is much smaller than the characteristic size of a proton (\(10^{-15} \mathrm{~m}\)).

Step-by-step solution

Part a) Calculate the Schwarzschild radius of a black hole with the mass of the Sun.

First, we need to determine the mass of the Sun. The mass of the Sun is given by \(M_{Sun} = 1.989 \times 10^30 \mathrm{~kg}\). Now we will plug the values into the Schwarzschild radius formula: \(R_{Sun} = \frac{2G M_{Sun}}{c^2}\), where \(G = 6.674 \times 10^{-11} \mathrm{m^3 kg^{-1} s^{-2}}\), and \(c = 2.998 \times 10^8 \mathrm{m~s^{-1}}\) \(R_{Sun} = \frac{2 (6.674 \times 10^{-11}) (1.989 \times 10^30)}{(2.998 \times 10^8)^2}\). By performing the calculations, we find that: \(R_{Sun} \approx 2.954 \times 10^3 \mathrm{~m}\). So, the Schwarzschild radius of a black hole with the mass of the Sun is approximately 2.954 kilometers.

Part b) Calculate the Schwarzschild radius of a black hole with the mass of a proton and compare it to the typical size scale of a proton.

First, we need to determine the mass of a proton. The mass of a proton is given by \(M_p = 1.673 \times 10^{-27} \mathrm{~kg}\). Now we will plug the values into the Schwarzschild radius formula: \(R_p = \frac{2G M_p}{c^2}\), where \(G = 6.674 \times 10^{-11} \mathrm{m^3 kg^{-1} s^{-2}}\), and \(c = 2.998 \times 10^8 \mathrm{m~s^{-1}}\) \(R_p = \frac{2 (6.674 \times 10^{-11}) (1.673 \times 10^{-27})}{(2.998 \times 10^8)^2}\). By performing the calculations, we find that: \(R_p \approx 2.490 \times 10^{-54} \mathrm{~m}\). So, the Schwarzschild radius of a black hole with the mass of a proton is approximately \(2.490 \times 10^{-54}\) meters. Now, let's compare this value to the typical size scale of a proton, which is \(10^{-15} \mathrm{~m}\). The Schwarzschild radius of a black hole with the mass of a proton is much smaller than the size scale of a proton. In conclusion, we found the Schwarzschild radius of a black hole with the mass of the Sun to be approximately 2.954 kilometers, and with the mass of a proton to be approximately \(2.490 \times 10^{-54}\) meters, which is much smaller than the characteristic size of a proton.