# NYU Black Renaissance Noire NYU Black Renaissance Noire V. 16.1 - Page 17

When massive clouds of interstellar dust coalesce, condense, and begin to radiate, a star is born. Within a few billion years from the time of their birth, all stars age and eventually die. But they have a very interesting afterlife. After a life of burning, stars use up their fuel and cool, and with the lack of outward radiation pressure, they eventually collapse under their own inward gravitational pull. In 1931, Nobel Prize-winning Indian physicist Subrahmanyan Chandrasekhar showed that when all the mass of a dying star collapsed to within a s mall enough volume, it formed an enchanting object called a white dwarf — a quiet remnant of the former star with the pressure of its own constituent electrons holding it up against gravity. One day, our sun will become a white dwarf, shrinking roughly to the size of Earth. In 1939, Robert Oppenheimer and George Volkoff, with the work of Richard Tolman, showed that for stars more massive than the sun, even just one and a half times bigger, their gravity would be too great for their constituent electrons to hold them up. These stellar remnants collapse further until finally their neutrons take up the slack, pushing back against gravity. The result? Neutron stars. For stars greater yet, three times or more massive than the sun, even neutrons can’t fight gravity. The nuclei collapse — and then our theories teeter on the edge of our understanding. In step black holes. BLACK RENAISSANCE NOIRE The great power of mathematical symmetry is that it can reduce the complexity of the equations. Imagine there are two separate equations that describe the oscillation of two particles, particle X and particle Y. One example of a “symmetric” situation would be if the behavior of X was exactly the same as Y. The two differential equations could thus be reduced to one, and once a solution for either X or Y was found, the solution of the other would follow. Sometimes, nature actually provides these serendipitous situations of high symmetry, and physicists can delight in discovering the solutions. In the case of Einstein’s equations, spherical symmetry was a good place to start. Spheres could model the structure of stars, like our sun. The geometry of spheres allowed gravity to be reduced to a radially uniform field around a compact central source. It was such a natural and simple idea that, within a few months of Einstein developing his theory, Karl Schwarzschild, German physicist and astronomer, found a spherically symmetric solution to the equations. But there was a glitch. As smaller and smaller radii were considered, a radius was reached, now known as the Schwarzschild radius, where the equations revealed something called a singularity — mathematically the sort of thing you get if you divide by zero. Physicists don’t like singularities. They usually imply regions of infinite energy or force. Really, most singularities tell us that something is wrong with our theory in the regions where they show their face. But this singularity was pointing to something new and downright awe inspiring about our spherical friends, stars. 15 To really understand the magic behind Einstein’s ten coupled differential equations, it is useful to begin by considering a solution to them. But given their complexity, it is no easy task to dream up a physical space-time configuration that satisfies them. It’s no longer a case of studying a graph and guessing the form of the function, as we did with Newton’s equations. Even today, with the help of powerful computers, we still cannot find exact solutions of the gravitational field for interesting astrophysical systems. Nevertheless, just after Einstein developed his theory, physicists were buzzing with curiosity about his new space-time concept and eager to find solutions. For starters, they armed themselves with Dirac’s trusty method: using the power of symmetry.