Manifestations of General Relativity. Relativity and Astrophysics Lecture 32 Terry Herter

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1 Manifestations of General elativity elativity and Astrophysics Lecture 32 Terry Herter Outline Consequences of General elativity Tests of G Escape Velocity => Black holes Black holes Size, Event Horizon, Singularity Homework Problems 8-3, 8-25 Prelim Wednesday, Nov 18 Closed book and notes, will cover lectures But some of this material depends on earlier lectures Should know Will have both qualitative and quantitative questions Most equations will be provided (if needed) A Manifestations of G 2 A

2 Consequences of G G changes our concepts of space and time (gravity and geometry are linked). Einstein no longer thought of gravity as a force but a curvature of space-time. Space is curved by massive objects causing objects to fall toward them. A Manifestations of G 3 Curvature of space-time Sun Empty space is flat space-time. Space with matter is curved space-time. A Manifestations of G 4 A

3 Bending of light Sun Near a massive object, G predicts that light will be deflected. G predicts 1.75 for light grazing the Sun. Measurements of stars during a solar eclipse verified this to within 1%. (Eddington ). A Manifestations of G 5 Time delay of light Viking At Earth Sun Near a massive object, G predicts that light will travel a longer path due to curved space-time. Verified by timing signals from Viking spacecraft passing by the Sun. A Manifestations of G 6 A

4 Gravitational edshift Light from the surface of a massive object will be redshifted. The more massive and/or more compact an object, the greater the redshift. ~ 0.01 A for the Sun. ~ 1 A for a white dwarf. Gravitational redshift verified to 0.01% by hydrogen masers (one in space, the other on the ground). A Manifestations of G 7 Gravitational Potential Energy Suppose we drop a rock from very far out in space. How fast is it going when it hits? The potential energy of the ball is: GMm PE M, m = mass of Earth and rock = distance from center of Earth In general, the potential energy difference between two points surrounding a mass is GMm GMm PE 1 > When 2 -> infinity (very large distance), get back first equation A Manifestations of G 8 A

5 Potential => Kinetic Energy The kinetic energy of the rock when it hits is: KE mv conv This KE comes from the conversion of PE into KE (by gravity). KEimpact PE start All the PE the rock had when it started is converted to KE at impact. 1 2 mv conv 2 GMm 2GM v conv A Manifestations of G 9 The impact velocity Putting in some numbers M earth = kg earth = 6400 km = m G = N-m 2 /kg 2 (m 3 /kg/s 2 ) GM v conv v = 11,200 m/sec = 11.2 km/sec A Manifestations of G 10 A

6 The Escape Velocity everse the problem: What is the minimum speed upward the rock must have to escape the Earth Leaves the Earth unpowered It s the same as if you let if fall (only going the other way)! 2GM v escape A Manifestations of G 11 Escape velocity Escape velocity is the speed an object would need to escape from a celestial body. The escape velocity depends on mass and radius Examples: Earth: 11.2 km/sec (25,000 mph) Moon: 2.4 km/sec 1 km asteroid: 1.3 m/sec (you could jump off it!) Sun: 618 km/sec White Dwarf: 6000 km/sec!! How high can the escape velocity get? A Manifestations of G 12 A

7 Dark Stars ev. John Mitchell An object more massive than the Sun could have an escape velocity greater than the speed of light! Today we call this object a black hole. An object from which no light can escape Suppose the escape velocity of an object was equal to the speed of light 2GM v escape c 2GM s c 2 s = Schwarzchild adius A Manifestations of G 13 Schwarzchild adius s is the radius which an object must be to become a black hole. s is given by: s = 3M s in km M in solar masses Derived by Karl Schwarzchild using General elativity A Manifestations of G 14 A

8 How big are black holes? Object Mass (M sun ) s Star km Star 3 9 km Sun 1 3 km Earth mm A Manifestations of G 17 How dense are black holes? The average density of a black hole is: but M V 3M 3 4 s 1 s M 2 M More massive black holes are less dense For a black hole with M = M sun. = 2 x g/cm 3 For M = 10 M sun = 2 x g/cm 3 A Manifestations of G 19 A

9 Very Massive Black Holes Suppose we could make a black hole a big as the solar system, e.g s = 40 AU. Then M = 2 x10 9 M sun and = g/cm 3 (!) -A 2 x10 9 M sun black hole can not be formed by a single star but these do exist in the centers of active galaxies (quasars). A Manifestations of G 20 The Event Horizon The event horizon is located at s. Anything inside the event horizon is gone from sight forever (nothing can escape). s A Manifestations of G 21 A

10 Time Dilation & edshift Time Dilation ecall that clocks run slower on the surface of the earth than on a mountain top. Viewed from space clocks slow down as they approach the event horizon. At the event horizon, the clock stops! Gravitational edshift The gravitational redshift gets larger and larger as objects approach the event horizon. At the event horizon the redshift becomes infinite! A Manifestations of G 23 Falling into a black hole Black Hole What happens if you fall in? A Clock A falls in while B stays outside. B A Manifestations of G 25 A

11 Person A Falling into BH Person outside BH sees Photons from A redshifted. Clock A slow down. Person A stretched and ripped apart by tidal forces. Black Hole Person falling in sees Clock B getting faster. Photons coming from person B and the rest of the universe are blueshifted. Visible photons become X- rays and -rays! The tidal forces and the rain of high energy photons will be very bad for the person falling into the black hole. A Manifestations of G 26 Tides Tidal forces are due to the difference in the gravitational force across an object. Near a black hole gravity changes very rapidly with distance. neutron stars too! Tides pull on the object and stretch it in the direction of the star. A Manifestations of G 27 A

12 People lying in 12 ft. spaceship near the walls. Tides Tides pull the two people towards the outer walls. Black Hole Distance Tidal Force from BH ( g s ) 5000 km km km 1.4x km 1.8x10 7 A Manifestations of G 28 Are black holes dangerous? BHs don t go around scooping up people and stars. Only if you get very close to one is there a problem. eplacing the sun with a 1 M sun black hole would not change the orbits of the planets! But we d have a problem keeping warm. A Manifestations of G 30 A