General Relativity ASTR 2110 Sarazin. Gravitational Waves from Merging Black Holes

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1 General Relativity ASTR 2110 Sarazin Gravitational Waves from Merging Black Holes

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3 General Relativity ASTR 2110 Sarazin Gravitational Waves from Merging Black Holes

4 General Relativity Not related to Corporal Punishment!!

5 General Relativity Newtonian Gravity and Special Relativity are incompatible Need a relativistic theory of gravity for: Black Holes Cosmology Relativity must enter if either: Things orbit at v c Always true for light! Gravitational potential φ = GM/r c 2 (v esc c)

6 Special Relativity Some results: 1. Uniform motion is relative All v = constant systems are equivalent 2. c = maximum speed 3. E = m c 2 Energy mass (matter) 4. Simultaneity, time dilation, etc.

7 Special Relativity Key Idea: Canvas of the Universe = 4-dimension spacetime z x y Newton: space and time Einstein: spacetime Point (x,y,z,t) = event

8 Spacetime Point (x,y,z,t) = event

9 Speed of Light What so special about light? c = maximum speed Is light egotistical? No, just converts units of time units of space

10 Analogy - Height and Length Imagine that we measured horizontal distances (length) in inches, and vertical distances (height) in cm. Then, many equations would contain: 2.54 cm/inch H = (2.54 cm/inch) L H (cm) c just converts time (sec) space (cm) x = c t L (inch)

11 Gravity and Special Relativity Why are only v = constant motions relative? With acceleration, feel inertial effects Why?

12 Gravity and Special Relativity Why are only v = constant motions relative? With acceleration, feel inertial effects Why? Mach s Principle (Ernst Mach, ~1900) Inertial effects are due to acceleration relative to the average of all the material in the Universe In an empty Universe, inertia would not exist

13 Venus Analogy Imagine Earth were a cloud covered, rotating planet Rotation not known at first, but Laws of physics depend on location F = m a + m a cent + m a corr depend on latitude Build lab at pole, rotate once/day opposite of Earth Physics simple F = ma

14 Venus Analogy One day, the clouds clear, and stars appear Sky rotates, except at pole lab a cent due to motion relative to average of stars?

15 Mach s Principle Inertial effects are due to acceleration relative to the average of all the material in the Universe In an empty Universe, inertia would not exist What is mechanism? Only long range force is Gravity Generalizing Relativity to accelerations relativistic theory of Gravity(?)

16 Newtonian Gravity Violates 1. Newton: Gravity forces have infinite speed F = GMm r 2 Special Relativity ˆ e r m M Move M, change r force changes instantly across entire Universe

17 Newtonian Gravity Violates Special Relativity 1. Newton: Gravity forces have infinite speed Solution: Consider electromagnetism E field line charge charge Accelerating charges EM wave (light) Prediction: Gravitational waves at v = c!!

18 First Detection of Gravitational Waves

19 Gravity waves from 2 merging black holes

20 Gravity Waves Gravity = curvature (distortions) of spacetime Wave amplitude = distortion in space (or time) h = Δl / l, where l is length between two objects h "strain"

21 Quadrupole waves

22 LIGO

23 Gravity Waves Gravity = curvature (distortions) of spacetime Wave amplitude = distortion in space (or time) h = Δl / l, where l is length between two objects h "strain" LIGO detection h max Δl km cm < 10 3 nucleus

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25 Two black holes spiral together and merge 29 M and 36 M è 62 M 3 M into energy, gravitational waves 50 x the luminosity of all the stars in the universe together, for 20 millionths of a sec.

26 Newtonian Gravity Violates Special Relativity 2. Newtonian Gravity Violates Energy Mass Equivalence a) Passive effect: Energy affected by gravity i. Bending of light by gravity

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29 Newtonian Gravity Violates Special Relativity ii. Gravitational Redshift Drop mass m in gravity field m φ = GM r gravitational potential Initially, E i =mc 2 M, R Drop, PE KE E f = mc 2 + KE = mc 2 + m Δφ Turn matter into energy, convert all energy into a photon

30 Newtonian Gravity Violates Special Relativity ii. Gravitational Redshift E em = hν em = E f = mc 2 + mδφ Photon moves back to start If no effect of gravity on photon, E obs = hν obs = mc 2 + mδφ ΔE = mδφ m M, R Energy not conserved. Convert photon back to mass, have more than at start

31 Newtonian Gravity Violates Special Relativity ii. Gravitational Redshift Photon must do work, lose energy, just like matter Frequency must decrease E obs = hν obs = mc 2 ν em ν obs = λ obs λ em = (mc2 + mδφ)/h mc 2 /h In GR, ν em ν obs = λ obs λ em = 1 1 2Δφ c 2 =1+ Δφ c 2 = 1 1 2GM rc 2 m M, R if r obs

32 Principle of Equivalence How to build in the physics of gravity Principle of Equivalence: Gravity acceleration locally Freefall no gravity locally g a = g

33 Principle of Equivalence Automatically gives bending of light a = g

34 Principle of Equivalence Freefall no gravity locally g a = g Everything falls together = same as no gravity Or, gravity and acceleration cancel one another

35 Weightlessness

36 Weightlessness

37 Freefall

38 Principle of Equivalence 1907: While sitting at his desk in patent office in Bern, Einstein realizes that a freefalling person will not feel gravity. This lead him to the Principle of Equivalence, the happiest thought of my life (Einstein)

39 Principle of Equivalence Gravity = acceleration locally gives Mach s Principle (locally) Explains why motion is independent of mass F = ma = GMm r 2 How to see gravity? Nonlocally Gravity appears tidally (in changes of motion with position) mass

40 Accelerations in Spacetime Acceleration curved path in spacetime a = 0 straight motion at constant speed Centrifugal a curved path in space curved path in spacetime Linear accelerated motion x = (1/2) a t 2 Parabola curved path in spacetime ct x

41 Curvature of Spacetime 1. Principle of Equvalence: gravity acceleration locally 2. Acceleration curved path in spacetime In gravitational field, all particle paths are curved How is this possible? Gravity spacetime itself is curved!!