Calculate the Schwarzschild radius using a semi-classical (Newtonian) gravitational theory, by calculating the minimum radius R for a sphere of mass M such that a photon can escape from the surface. (General Relativity gives
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- General Relativity Consider a spherical blackbody of constant temperature and mass M whose surface lies at radial coordinate r = R. An observer located at the surface of the sphere and a distant observer both measure the blackbody radiation given off by the sphere. If the observer at the surface of the sphere measures the luminosity of the blackbody to be L, use the gravitational time dilation formula, to show that the observer at infinity measures. 2GM L̟ = L] 1– Rcarrow_forwardThe Schwarzschild radius RBH for an object of mass M is defined as (See image.) where c is the speed of light and G is the universal gravitational constant. RBH gives the radius of the event horizon of a black hole with mass M. In other words, it gives the radius to which some amount of mass M would need to be compressed in order to form a black hole. 1. The mass of the Sun is about 1.99 × 1030 kg. What would be the radius of a black hole with this mass? 2. The mass of Mars is about 6.42 × 1023 kg. What would be the radius of a black hole with this mass? 3. Suppose you want to make a black hole that is roughly the size of an atom (take RBH = 1.10 x 10-10 m). What would be the mass M of such a black hole?arrow_forwardThe radius Rh of a black hole is the radius of a mathematicalsphere, called the event horizon, that is centered on the blackhole. Information from events inside the event horizon cannotreach the outside world. According to Einstein’s general theory ofrelativity, Rh = 2GM/c2, where M is the mass of the black hole andc is the speed of light.Suppose that you wish to study a black hole near it, at a radialdistance of 50Rh. However, you do not want the difference in gravitationalacceleration between your feet and your head to exceed10 m/s2 when you are feet down (or head down) toward the blackhole. (a) As a multiple of our Sun’s mass MS, approximately what isthe limit to the mass of the black hole you can tolerate at the givenradial distance? (You need to estimate your height.) (b) Is the limitan upper limit (you can tolerate smaller masses) or a lower limit(you can tolerate larger masses)?arrow_forward
- General Relativity Consider a spherical blackbody of constant temperature and mass M whose surface lies at radial coordinater = R. An observer located at the surface of the sphere and a distant observer both measure the blackbody radiation given off by the sphere.Both observers use the Stefan-Boltzmann law, to determine the radius of the spherical blackbody. Show that R R, = 1 – 2GM/Rc²arrow_forwardStephen Hawking has predicted the temperature of a black hole of mass M to be T = hc3/8πkGM, where k is Boltzmann’s constant. (a) Calculate the temperature of a black hole with the mass of the sun. Discuss the implications of the temperature you calculate. (b) Find the temperature of a supermassive black hole, which may exist at the center of some galaxies, with a mass 6.0x 109 times the sun’s massarrow_forwardCompact objects and black-holes 2. Consider three compact objects in the form of: a white dwarf of 0.5Mo; a neutron star of 1.4Mo and a black-hole of 50 Mo. The radii of the white dwarf and neutron star are: Rwp 5.5 106 m and and RNS 10 Km. (a) Determine the radii of curvature Re = c2/g (where c is the speed of light and g is the local gravitational acceleration) around cach objcct specifying which radius you assume for the BH.arrow_forward
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- Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage LearningPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning