General Relativity and Black Holes

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1 General Relativity and Black Holes Lecture 19 1

2 Lecture Topics General Relativity The Principal of Equivalence Consequences of General Relativity slowing of clocks curvature of space-time Tests of GR Escape Velocity Black Holes 2

3 General Relativity A Theory of Gravity Albert Einstein 1916 Incorporates accelerated motions into Special Relativity Albert Einstein ( ) 3

4 When you re really famous 4

5 Einstein s Insight Newton s law of gravity worked very well in predicting planetary motions. But Einstein wondered how gravity could be made consistent with Special Relativity. Einstein s insight was the Principal of Equivalence He realized that a gravitational field would bend light rays. He also realized that Euclidean geometry would not apply. 5

6 Principle of Equivalence Gravity and acceleration due to a force are indistinguishable. In a small local environment (must be a small enough box ) This is the foundation of General Relativity. 6

7 Historical Background Proved a relationship between symmetries in physics and conservation principles (1915 or so). This information was used by Einstein and is used in many areas of physics. Noether was going to be a language teacher but became interested in mathematics. David Hilbert and Felix Klein fought to get her on the faculty at University of Göttingen. The battle took four years, but she was finally appointed in She remained there until 1932 when the Nazis caused her dismissal because she was Jewish. She accepted a visiting professorship at Bryn Mawr College and also lectured at the Institute for Advanced Study at Princeton. Emmy Noether ( ) 7

8 Accelerating Rocket ship Gravity Imagine yourself in a closed room. By the principle of equivalence you could not tell if you were on earth or in space in an accelerating rocket. 8

9 Falling under gravity Floating in space Now imagine yourself falling in a closed room. By the principle of equivalence you could not tell if you were falling towards earth or floating in space. 9

10 Gravity and Time Accelerating Rocket ship A B Imagine two clocks in an accelerating rocket. Clock A is in the front. Clock B is in the back. Clock A emits pulses 1 second apart. How far apart are they at Clock B? 10

11 A emits flash B receives flash Clocks in GR NO ACCELERATION A B A B d 2 d 1 A B A B Flash emitted from A To reach B it travels a distance d 1. Since the rocket is not accelerating, we have for the next flash d 2 = d 1 Flashes arrive one second apart. 11

12 A emits flash B receives flash Clocks in GR A ACCELERATING Flash emitted from A A B A d 2 B A To reach B it travels a distance d 1. Accelerating rocket rocket is traveling faster for next flash d 2 < d 1 B d 1 B Flashes arrive less than 1 second apart. 12

13 Clocks in GR Clock A runs faster than clock B. The equivalence principle states Gravity and Acceleration are the same. Therefore, the same thing happens in a gravitational field! A clock on a mountain top will run faster than a clock at sea level. 13

14 Consequences of GR GR 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. 14

15 Curvature of space-time Sun Empty space is flat space-time. Space with matter is curved space-time. 15

16 Bending of light Sun Near a massive object, GR predicts that light will be deflected. GR predicts 1.75 for light grazing the Sun. Measurements of stars during a solar eclipse verified this to within 1%. (Eddington ). 16

17 Time delay of light Viking At Earth Sun Near a massive object, GR predicts that light will travel a longer path due to curved space-time. Verified by timing signals from Viking spacecraft passing by the Sun. 17

18 Binary Pulsar Two neutron stars orbiting one another Work done at Arecibo Observatory Orbit period = 8 hr, Orbit speed = 0.1c!! Serves as a test of General Relativity Precession (movement of orbit) on sky. Decay of orbit due to Gravitational Radiation. (New type of radiation!!) General relativity has been proven over and over to be correct. 18

19 Gravitational Redshift 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). 19

20 Escape velocity 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. The escape velocity depends on mass & radius 20

21 Escape velocity Escape velocity is the speed an object would need to escape from a celestial body. The escape velocity depends on mass. Examples: Earth: Moon: 1 km asteroid: Sun: 11.2 km/sec (25,000 mph) 2.4 km/sec 1.3 m/sec (you could jump off it!) 618 km/sec White Dwarf: 6000 km/sec!! How high can the escape velocity get? 21

22 Dark Stars Rev. 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. 22

23 Making a Dark Star Suppose the escape velocity of an object was equal to the speed of light. R s = Schwarzchild radius Putting in numbers: Rs = 3M R s in km M in solar masses 23

24 Warped Space Time

25 How big are black holes? Object Mass (M sun ) R s Star km Star 3 9 km Sun 1 3 km Earth 3 x mm 25

26 The Event Horizon The event horizon is located at R s. Anything inside the event horizon is gone from sight forever (nothing can escape). R s 26

27 Event Horizon

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