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Black Holes

Black Holes. Newtonian. Universal Mutual Gravitation Isaac Newton, in his Principia , formulated the Law of Universal Mutual Gravitation: Gravity is an Attractive force: Works to bring massive objects closer together. Gravity is a Universal force: Works everywhere in the Universe.

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Black Holes

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  1. Black Holes

  2. Newtonian • Universal Mutual Gravitation Isaac Newton, in his Principia, formulated the Law of Universal Mutual Gravitation: • Gravity is an Attractive force: • Works to bring massive objects closer together. • Gravity is a Universal force: • Works everywhere in the Universe. • Gravity is a Mutual force: • Works between pairs of massive objects

  3. Gravitational Force Force of gravity between any two objects depends only upon: • The masses of the two objects: • More massive objects feel a stronger force. • The distance between them: • Objects closer together feel a stronger force. • It does not depend at all on the shapes, colors, or compositions of the two objects.

  4. The Law of Universal Gravitation The force of gravitational attraction between any two massive bodies is proportional to their masses and inversely proportional to the square of the distance between their centers. • The Force of Gravity is an example of an "Inverse Square Law Force"

  5. Stated Mathematically: • Where: • F = force due to gravity. • M1 = mass of the first object • M2 = mass of the second object • d = distance between their centers. • G = "Gravitational Force Constant"

  6. The Flaw • Doesn’t work when you are talking about intense gravitational force of black holes and neutron stars • Thankfully Einstein created the principles of relativity

  7. In Newtonian gravitation, an orbit is always an ellipse. • As the gravitating body becomes more massive and the test particle orbits it more closely, the speed of the particle in its orbit increases without bound, always balancing the gravitational force. For a black hole, Newton's theory predicts orbital velocities greater than the speed of light,

  8. 1st Flaw • It gave the wrong prediction for the precession of the perihelion of Mercury's orbit. • Mercury's orbit is elliptical, as predicted by Newton's theory of gravity, but the ellipse doesn't stay in precisely the same place all the time. • It precesses, which is to say that as Mercury orbits the sun, the entire ellipse rotates about the focal point (i.e. the sun) as shown in the in the picture

  9. 2nd Flaw • It did not explain why the gravitational force on an object was proportional to its inertial mass. • In other words it did not explain why gravitational acceleration is independent of the mass or composition of an object.

  10. 3rd Flaw • It was inconsistent with the Special Theory of Relativity. That is, if an instantaneous force of gravitational attraction exists between two objects then information about the location of one object would be transmitted to another object instantaneously by changes in the gravitational force. Thus it would be possible to send information faster than the speed of light.

  11. The special theory of relativity changes our conceptions of space and time • This theory, published by Einstein in 1905, is based on the notion that there is no such thing as absolute space or time • Space and time are not wholly independent of each other, but are aspects of a single entity called spacetime

  12. Special Relativity • General Relativity developed from Special Relativity • universal speed limit = speed of light = c = 300,000 km/sec • Example A: Nolan Ryan on a train: • train moves East at vtrain= 30 m/sec (~70 mph) • Nolan, who is on the train, throws his fastball at vball= 40 m/sec (90 mph) • Nolan sees the ball move at vball= 40 m/sec (90 mph) • We see the ball move at vtrain+vball= 70 m/sec (~160 mph) from the ground

  13. Example B: Nolan switches on a flashlight • Nolan turns on his flashlight pointing East • Nolan sees the light move at c = 300,000 km/sec • Do we see the light move at vtrain+c = 300,000.03 km/sec? NO!! • We also see the light move at exactly c = 300,000 km/sec! • Even if the train moved at vtrain= 200,000 km/sec, we'd still see the light move at velocity c!

  14. Time reversed case: now lets throw a baseball up from the Earth (ignoring air friction) • I throw it at 20 m/sec - it goes up 20 m and falls back to Earth • Nolan Ryan throws it at 40 m/sec - it goes up to 80m at falls back to Earth • shoot it out of a cannon at 10 km/sec - it goes out beyond the communication satellites and then falls back to Earth • shoot it out of a cannon at 11 km/sec - and it goes up and slows down, but never comes back • This is the escape velocity

  15. The speed of light is the same to all observers, no matter how fast they are moving

  16. An observer will note a slowing of clocks and a shortening of rulers that are moving with respect to the observer This effect becomes significant only if the clock or ruler is moving at a substantial fraction of the speed of light

  17. The general theory of relativity is our most accurate description of gravitation • Published by Einstein in 1915, this is a theory of gravity • A massive object causes space to curve and time to slow down • These effects manifest themselves as a gravitational force • These distortions of space and time are most noticeable in the vicinity of large masses or compact objects

  18. The theory of relativity predicts a number of phenomena, including the bending of light by gravity and the gravitational redshift, whose existence has been confirmed by observation and experiment

  19. Escape Speed • Escape velocity is the speed an object would need to escape from a celestial body. • Gravity is low on an asteroid. You could throw a ball off it, or jump off it. • Thus, low escape velocity • The escape velocity depends on mass.

  20. The general theory of relativity predicts black holes

  21. If a stellar corpse has a mass greater than about 2 to 3 M, gravitational compression will overwhelm any and all forms of internal pressure • The stellar corpse will collapse to such a high density that its escape speed exceeds the speed of light

  22. Certain binary star systems probably containblack holes • Black holes have been detected using indirect methods • Some binary star systems contain a black hole • In such a system, gases captured from the companion star by the black hole emit detectable X rays

  23. Supermassive black holes exist at the centers of most galaxies These are detected by observing the motions of material around the black hole

  24. A nonrotating black hole has only a “center” and a “surface” • The entire mass of a black hole is concentrated in an infinitely dense singularity • The singularity is surrounded by a surface called the event horizon, where the escape speed equals the speed of light • Nothing—not even light—can escape from inside the event horizon

  25. Just 3 numbers completely describe the structure of a black hole • A black hole has only three physical properties: mass, electric charge, and angular momentum • A rotating black hole (one with angular momentum) has an ergoregion around the outside of the event horizon • In the ergoregion, space and time themselves are dragged along with the rotation of the black hole

  26. Falling into a black hole is an infinite voyage

  27. Could a black hole somehow be connected to another part of spacetime, or even some other universe? • General relativity predicts that such connections, called wormholes, can exist for rotating black holes

  28. Mass tells space how to curve • Space tells mass how to move

  29. Gravitational redshift: • light rays (i.e. photons) lose energy as they climb out of a gravitational field • So, they shift to larger wavelength, lower energy

  30. Gravitational Energy • Energy is conserved - i.e. the total energy does not change but it can be transferred into a different form • consider a baseball in outer space - very far from the Earth - we'll say infinitely far. • let it go from rest • it will reach a high velocity - and gain lots of energy of motion as it falls • energy is conserved - so where did the energy come from?

  31. Gravity - we assign a negative potential energy to an object in a gravitational field • so, the total energy is still the same as before • lots of energy of motion and • a negative gravitational energy that compensates for this to allow energy conservation

  32. Suppose the Earth was squeezed down to half its size, but kept the same mass • The escape velocity would be larger - 15 km/sec in this case • the baseball would slow down from 15 km/sec to 11 km/sec by the time it reached the current radius of the Earth

  33. Suppose the Earth was squeezed down to 1 cm • the escape velocity would be = c • any smaller and its a black hole - nothing can escape!

  34. Sample Escapes Velocities • Earth: 11.2 km/sec (25,000 mph) • Moon: 2.4 km/sec • 1 km asteroid: 1.3 m/sec • Sun: 618 km/sec • White Dwarf: 6000 km/sec !!

  35. black hole black hole evaporation equivalence principle ergoregion event horizon general theory of relativity gravitational radiation gravitational waves gravitational redshift Heisenberg uncertainty principle law of cosmic censorship length contraction Lorentz transformations mid-mass black hole no-hair theorem primordial black hole proper length (proper distance) proper time Schwarzschild radius (RSch) singularity spacetime special theory of relativity stellar-mass black hole supermassive black hole time dilation virtual pairs wormhole Key Words

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