Final answer:
The component g_rr in the Schwarzschild metric is the inverse 1/[1-(2GM)/(rc^2)] and is used to determine the Schwarzschild radius of a black hole. For Sgr A*, the black hole at the center of our galaxy, the Schwarzschild radius is calculated by substituting the known values of G, M, and c into the formula Rs = 2GM/c^2.
Step-by-step explanation:
The question pertains to the Schwarzschild metric, an exact solution to the Einstein field equations which describes the gravitational field outside a spherical mass. The component grr in the Schwarzschild metric is described by the inverse 1/[1-(2GM)/(rc2)], where c is the speed of light, G is the gravitational constant, and M is the mass of the black hole. To find the Schwarzschild radius for a black hole such as Sgr A* at the center of our galaxy, which has a mass of 4 million solar masses, you would substitute the values for G, M, and c into the formula Rs = 2GM/c2.
Using the values for G, 6.67 × 10−11 N·m²/kg², the mass of Sgr A* in kilograms (4 × 106 solar masses, with one solar mass equal to 1.99 × 1030 kg), and the speed of light c, which is 3.00 × 108 m/s, we obtain a Schwarzschild radius for this particular black hole.