![]() at the same time, affect different properties of a black hole 62,63,64,65,66,67. He was stationed on the Russian front as an artillery officer when he wrote his two papers on general relativity in the following year-the very year Einstein published the theory. In this work, test particle dynamics around a static regular Bardeen. Schwarzschild had volunteered for the military at the outbreak of the war in 1914. Credit: Bildarchiv Preussischer, Kulturbesitz, BerlinĪn interesting historical aspect of Schwarzschild’s discovery is that it was made during his service in the German army during World War I. He was the first person to solve Einstein’s equations for a spherically symmetric system. Figure 11.2: Karl Schwarzschild was a German astrophysicist. It applies to any spherically symmetric gravitating system, including non-spinning black holes-we will have a look at spinning black holes later. The interval describes the Schwarzschild geometry, discovered by the German astronomer Karl Schwarzschild (1873–1916) in 1915. ![]() That is what is meant by spherically symmetric. This is clear from the lack of modifications to the angular term in the expression for the interval. The curvature has no dependence on the direction considered. Those are the only terms where we find a difference from the flat space interval, the one used for special relativity. The spacetime interval may look complicated, but the curvature depends only on the distance from the origin, d, and time, t. The angular part is given by the last term, with which we do not have to concern ourselves it is the same as for flat space and does not contribute to spacetime curvature. Gravitational Radiation from a Particle Falling Radially into a Schwarzschild Black Hole Marc Davis, Remo Ruffini, William H. Here G is the universal gravitational constant, c is the speed of light, M is the mass of the object, d is the distance from the origin, and t is the time. Using the solution (0.2) in this equation, show that - og(-7/ t(T)2M log (0.\right) (c\Delta t)^2 r^2 \Delta \Omega^2 \nonumber \] (b) Show that the equation of motion for t(T) for observer A is dt (1-2M/r). Note that T is negative while A is outside r = 2M. Substitute the ansatz r(T) 2M -T O(72) (0.2) 1 into the equation of motion and find the number y. We 11 only need to solve for the motion of the observer near r = 2M, and for this purpose we can use a Taylor expansion. ![]() ![]() Without loss of generality take T = 0 as the observer crosses r 2M. (a) Show that the equation for the path r(7) of observer A is (dr/dr)2 2M/r. In this problem you will show that as A approaches the horizon r 2M, B sees A's broadcasts become enormously redshifted, with the observed frequency wB varying with B's proper time te as tB (0.1) Wв х ехp 4M Note that we have chosen units where G = 1 in this problem. (This is similar to the thought experiment we did in lecture, but now the infalling observer is in free-fall, not hovering at a fixed value of r.) Another observer B is stationary at r R with R M, and monitors A's broadcasts. While falling, A broadcasts a description of what he or she experiences using radio waves. The black hole’s extreme gravity alters the paths of light coming from different parts of the disk, producing the warped image where we see the disk behind the black hole as if it is simultaneously on the top and the bottom of the black hole. ![]() An observer A falls radially in the Schwarzschild metric, starting from rest at r = o0. Seen nearly edgewise, the turbulent disk of gas churning around a black hole takes on a bizarre double-humped appearance. ![]()
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