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PHY390, the Kerr Metric and Black Holes

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PHY390, the Kerr Metric and Black Holes PHY390, The Kerr Metric and Black Holes James Lattimer Department of Physics & Astronomy 449 ESS Bldg. Stony Brook University April 1, 2021 Black Holes, Neutron Stars and Gravitational Radiation [email protected] James Lattimer PHY390, The Kerr Metric and Black Holes What Exactly is a Black Hole? Standard definition: A region of space from which nothing, not even light, can escape. I Where does the escape velocity equal the speed of light? r 2GMBH vesc = = c RSch This defines the Schwarzschild radius RSch to be 2GMBH M RSch = 2 ' 3 km c M I The event horizon marks the point of no return for any object. I A black hole is black because it absorbs everything incident on the event horizon and reflects nothing. I Black holes are hypothesized to form in three ways: I Gravitational collapse of a star I A high energy collision I Density fluctuations in the early universe I In general relativity, the black hole's mass is concentrated at the center in a singularity of infinite density. James Lattimer PHY390, The Kerr Metric and Black Holes John Michell and Black Holes The first reference is by the Anglican priest, John Michell (1724-1793), in a letter written to Henry Cavendish, of the Royal Society, in 1783. He reasoned, from observations of radiation pressure, that light, like mass, has inertia. If gravity affects light like its mass equivalent, light would be weakened. He argued that a Sun with 500 times its radius and the same density would be so massive that it's escape velocity would exceed light speed. He proposed using a prism to measure the gravitational weakening of starlight due to the surface gravity of the star. He predicted one day they could be observed if some were in binary stars, by their gravitational effect on their companions. Michell, around 1783, designed the experiment now attributed to Cavendish which first accurately measured the force of gravity between masses. This resulted in the first accurate values for G and M⊕. He invented the torsion balance for the experiment but didn't live to put it to use. His device was willed to his friend Henry Cavendish, who performed the experiment in 1797-8. James Lattimer PHY390, The Kerr Metric and Black Holes Michell's Apparatus { The Torsion Balance κθ = LF = LGmM=r 2 r r I mL2 T = 2π = 2π κ 2κ 2π2Lr 2 G = θ MT 2 Harvard Lecture Demonstration Cavendish 1798, Phil. Trans. Roy. Soc. Lon., 469 2 M⊕ = gR⊕=G James Lattimer PHY390, The Kerr Metric and Black Holes More About John Michell He discovered that double stars are more common than statistically probable, which must be a product of their mutual gravitation. Michell tried to measure the radiation pressure of light, but when he focused sunlight onto a compass needle, it melted. He first proposed the accepted explanation for the twinkling of starlight. He proposed that parallax, although too small at present to be observed, could in the future determine a star's distance, which together with its apparent brightness, could be used to measure it's true luminosity. He sold a 30" telescope to William Herschel, and introduced him to astronomy. In geology, he proposed that earthquakes were experienced as seismic waves of elastic compression travelling through the Earth, and determined the epicenter of the 1755 Lisbon earthquake. He also first suggested that tsunamis were caused by earthquakes. He laid the foundations of English stratigraphy, especially concerning the Mesozoic strata, without any knowledge of fossils. He first explained how to manufacture articial magnets much stronger than natural ones, and discovered the inverse square law of magnetism. James Lattimer PHY390, The Kerr Metric and Black Holes Other Early Work on Black Holes I The mathematician Pierre-Simon Laplace made similar arguments in 1796 in the first two editions of his book Exposition du syst´eme du Monde, although the discussion was removed from later editions. I Einstein's paper describing the advance of the perihelion of Mercury was published on Nov. 25th, 1915. It had an approximate metric, in Cartesian coordinates, valid in the weak-field limit. I Karl Schwarzschild within two weeks found the exact analytical solution for the spherical symmetry metric, valid for arbitary field strength, despite the fact that we was at the Russian front in WWI, and was ill with an autoimmune disease. He mailed it to Einstein, who read it on Dec. 15th. This solution was published in Jan. 1916. I A few months later, Johannes Droste, Lorentz's student, independently derived the solution to the metric. I The metric was followed by its uniform density solution in Feb. 1916. I Schwarzschild died in May 1916 at age 41. I The metric had a bad behavior (a singularity) at the so-called Schwarzschild radius, but it was only later (1939) that Oppenheimer, Tolman and Volkov interpreted this radius as the boundary of a bubble where time stopped. This led to the idea of \frozen stars". James Lattimer PHY390, The Kerr Metric and Black Holes The Trouble With Singularities I In 1958, David Finkelstein identified the Schwarzschild radius with the event horizon, a perfect unidirectional membrane: causal influence can cross it in only one direction. I He and Martin Kruskal extended the Schwarzshild solution into the interior of the event horizon (so it could be applied to infalling observers) by means of a coordinate transformation, thereby showing it was not a true singularity. I The origin, however, contains a true singularity. I Although black holes seem exotic, a massive one has the interesting property that, within the event horizon, it has a low average density 3M 3c6 109M 2 ρ¯ = = ' g cm−3: 4πR3 32πG 3M2 M I But back to that singularity. James Lattimer PHY390, The Kerr Metric and Black Holes The Trouble With Singularities Coordinate singularities are certainly not confined to general relativity. Consider spherical coordinates at the poles. The north pole has θ = 0, but 0 ≤ φ ≤ 2π. So that point has infinitely multi-valued coordinates. Points separated by zero distance must be the same point, even if the coordinate distance between them does not vanish. This is a symptom of a bad choice of coordinates. In relativity, the situation is more subtle because of certain curves, called null curves, that have zero invariant distance between them. It wasn't until 1960 that the true nature of the Schwarz- schild event horizon became apparent. James Lattimer PHY390, The Kerr Metric and Black Holes Infalling Particles and Proper Time Let a particle fall inwards to r = 2M from R > r. How much proper time elapses (the time measured on a clock attached to the particle)? For radial infall, dr 2 2M dr = E~2 − 1 + ; dτ = − dτ r q E~2 − 1 + 2M=r where E~ = −p0=m = −U0 is the energy per unit mass of the particle. We choose the minus sign to represent infall. ~2 R 2M If E > 1 (unbound particle), R > 0. If E~ = 1 (particle at rest at 1), the integral is 4M r 3=2R τ(R) − τ(2M) = > 0: 3 2M 2M 2 2 If E~ < 1, the particle can't get closer than 1 − E~ = 2M=ri or 2 ri = 2M=(1 − E~ ) > 2M. Any particle reaching 2M gets there in a finite proper time. In fact, why stop at 2M? The proper time stays finite to any r. James Lattimer PHY390, The Kerr Metric and Black Holes Coordinate Time and Infalling Particles Now let's ask the same question about coordinate time. Using the fact that the time component of the four-velocity is U0 = dt=dτ, dt p 2M −1 U0 = = g 00U = g 00 0 = −g 00E~ = 1 − E~: dτ 0 m r Ed~ τ (1 − 2M=r)−1Edr~ dt = = − : 1 − 2M=r (E~2 − 1 + 2M=r)1=2 Define " = r − 2M, so that (" + 2M)3=2Ed~ " 2Md" dt = − −! − : ((2M + ")E~2 − ")1=2" "!0 " This diverges logarithimically for all E~, so an infinite coordinate time elapses. It is behaving badly. Nothing happens to the particle as it falls through the event horizon. Tidal forces don't suddenly become infinite there. James Lattimer PHY390, The Kerr Metric and Black Holes Inside 2M Now define " = 2M − r, so that the metric becomes " 2M − " ds2 = dt2 − d"2 + (2M − ")2dΩ2: 2M − " " On a trajectory for which t; θ and φ are constant, ds2 < 0, which means it is timelike. Thus ", and r, is timelike, while t is now spacelike. The infalling particle must follow a timelike world line, so r perpetually decreases. Eventually, r = 0 is reached, a point at which there is finally a true singularity. A particle sending an outwardly directed photon does so futiley, because the photon must also go forward in time as seen locally; it also has to follow a timelike path, which means to decreasing r. Nothing inside r = 2M gets out and is doomed to encounter the singularity at r = 0. Nothing inside the event horizon can be seen outside of it; hence the name horizon. James Lattimer PHY390, The Kerr Metric and Black Holes The Coordinate Singularity Consider the light cones of various events occuring outside the event horizon. These are cones showing radially ingoing and outgoing null lines emanating from the events. They are computed by solving ds2 = 0 for dθ = dφ = 0 (radial), dt 2M −1 = ± 1 − : dr r Far from the star, dt=dr = ±1, 45◦. For events close to r = 2M, dt=dr ! ±∞; the cones close up. Particles are confined to move within light cones, limited in velocity by c. As r ! 2M, they can't get to r = 2M until t ! 1.
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