Even the supermassive black holes now thought to inhabit the centers of most galaxies, which weigh in at millions or billions of our sun’s mass and in some cases have diameters larger than our solar system, are so far away from Earth that they subtend incredibly tiny angles on the sky. Notably we have to contend with the black hole’s tiny size when viewed from Earth. There is, of course, a catch: developing a telescope that can resolve a black hole horizon poses several challenges. If we could observe a black hole with a telescope with enough magnifying power to resolve the event horizon, we could follow matter as it spirals down toward the point of no return and see whether it behaves as general relativity says it should. This causes the infalling matter to reach temperatures of billions of degrees-which, ironically, makes the vicinity immediately surrounding a black hole one of the brightest spots in the cosmos. Near the black hole, the crushing force of gravity compresses inflowing matter, known as the accretion flow, into ever smaller volumes. The interior of a black hole is unobservable, but the gravitational field surrounding these objects causes matter close to the horizon to produce huge amounts of electromagnetic radiation that telescopes can detect. And nowhere in the universe today is gravity stronger than at the edge of a black hole-at the event horizon, the boundary beyond which gravity is so overwhelming that light and matter that pass through can never escape. To put general relativity to its greatest test, we need to see whether it holds up where gravity is extremely strong. Every assessment to date has been conducted in rather weak gravitational fields. So far, however, Einstein’s theory has had it easy. Calculate the duration of this unpleasant time interval! Does the size of the black hole matter? (4 p) (C) Suppose that x = 100 (m/s)/m (that is, 10 g per meter).Scientists have been trying unsuccessfully to poke holes in Albert Einstein’s general theory of relativity for a full century. Given this value x, at what radius does it start to hurt? (1 p) (d) So pain sets in at the radius in (c), and you are certainly dead at r=0. Suppose that the tidal stretching becomes painful as the tidal acceleration per meter exceeds a certain value x. (a) What is the tidal acceleration per meter along your body that you will experience the moment you pass the horizon of the black hole? Is it more or less painful if the black hole is large? (1 p) (b) What is the value in (a) for a black hole with the mass of the sun in standard units)? Would you be alive as you pass the event horizon of such a black hole? (1 p) (c) Now, return to the case with general black hole mass, and geometrized units. (21.31)) that the tidal stretching in the radial direction is 2M d7² where x is the radial component of the deviation vector as measured in your freely falling frame (defined by eqns. From the geodesic deviation equation with the relevant component of the Riemann tensor inserted, Hartle shows (eqn. You are diving into the black hole, head first, so that your body, from head to feet, is in the radial direction. Suppose that you are falling radially into a Schwarzschild black hole, with line element ds? = 2M Idi² + -11-20) (1-2 M 2M) *4r? + rºd = You start from rest far away from the black hole (for the calculations we can assume that you start infinitely far away), and then just fall freely.
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