READ: Chapter 8 and Section 12.4 (Binaries with Black Holes)

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1 HW The Nature of Space and Time-General Relativity it Black Holes READ: Chapter 8 and Section 12.4 (Binaries with Black Holes) Chapter 8: Questions (HW#28)1, (HW#29)3 Problems (HW#30)2, (HW#31) 3, (HW#32) 6, (HW#33)7, (HW#34)8, (HW#35) 10 Chapter 12 (HW#36) Problem 12.9(note, inclination is the angle between the plane of the orbit of the two objects and a perpendicular p plane to our line of sight..see Fig 5.7)

2 Curved Space time Shortest distance between two points is called the geodesic (light path in GR?) We normally think in terms of Flat or Euclidean space, geodesic = a straight line. Consider sum of angles of a triangle =? o But consider spherical space: geodesic = great circles, hence sum of angles of triangle =? Ex? 90 o What curves Space time Einstein said MASS! 90 o 90 o

3 General Relativity A new description of gravity Postulate: Equivalence Principle: Observers can not distinguish locally between inertial forces due to acceleration and uniform gravitational ti forces due to the presence of massive bodies.

4 Another Thought Experiment Imagine a light source on board a rapidly accelerated space ship: Light source a Time a a Time a g As seen by a tti stationary observer As seen by an observer on board the space ship

5 Thought Experiment (2) For the accelerated observer, the light ray appears to bend downward! d! Now, we can t distinguish between this inertial effect and the effect of gravitational forces Thus, a gravitational force equivalent to the inertial force must also be able to bend light!

6 Thought Experiment This bending of light by the gravitation of massive bodies has indeed been observed: During total solar eclipses: The positions of stars apparently close to the sun are shifted away from the position of the sun. New description of gravity as curvature of space-time!

7 Principle of Equivalence A uniform gravitational field in some direction is indistinguishable from a uniform acceleration in the opposite direction. Note: if you are in an accelerating reference frame pseudo-forces appear in the Direction opposite to the true acceleration. Example 1. Car breaks ( a true force (friction accelerates the car backwards) but you accelerate forwards and think a force (pseudo-force) pushes you forward. You can accelerate out the windshield but no true force is on you! Example 2. To travel in a circle you need an acceleration toward a center of a Circle (Centrifugal force) but you feel a force (pseudo-force) outward. Exs. Amusement park rides, pail of water in a circle, Centrifugal devices to separate Out blood serum or heavy elements in solution, David and Goliath Example 3. Person in elevator on a scale see Fig 8.4 in text Reminder Newtons second law F=ma Case I: elevator is not accelerating -mg+f s =0 Or F s =mg! Third law-> force of scale on person is Same of person on scale=reading of scale! mg is called the Weight or W=mg F s mg s g

8 Equivalence principle illustrated Example 3 case II Elevator is in deep space..no gravity. The elevator with the Person on the scale is accelerating at a! (Rocket engine) Set a=g What does the scale read? Scale has to push person to g acceleration or F s =ma=mg reaction on the scale Is mg or scale reads F s =mg! Person does not know whether they are on earth Or in deep space accelerating at g! Hence equivalence! Case III. Elevator cable breaks and is in free fall. What is F s =? Free fall also when in orbit

9 A DEEPER MEANING! Inertial mass, m, ie a=f/m or m acts like a resistance to F. Inertial mass Gravitational mass Formally, Inertial Mass = Gravitational Mass! DEEPER!

10 Inertial Mass = Gravitational Mass! This is the reason all bodies near the surface of the Earth have the same Acceleration! Given a mass, m, near the Earths surface. Gravitaional Force on the mass m from the Earth M E is F G =GmM E /R E 2 here m is the gravitational mass Since F=ma with m the inertial mass. And Inertial Mass = Gravitational Mass! the acceleration experienced by mass m by gravity in this case Is a =F G /m =GM E /R 2 E or all bodies of any mass m have same a Near the surface of the Earth as found by Galileo and proved by Newton As above. It is not obvious that the inertial mass and the Gravitational mass are the same There must be something special about Gravity!

11 Other Effects of General Relativity Perihelion advance (in particular, of Mercury) Closer to the mass creating the curvature the greater the Effect! Hence Mercury! Since Curvature changes this causes the elliptical orbit not to close but to advance as follows sun Advance is 5600 sec as follows but all can Be accounted for but 43 sec of arc/century

12 Einsteins results Were 43 sec/century From GR! See text about Challenges to this! GR implies! Planets follow The curvature of Space-time Due to the mass of the Sun( or star) Try rolling marbles In a gently sloping Salad bowl? Record your Observations for EC!

Bending of EM radiation y Radius of sun=b A Classical argument for the bend to get a Feel for the GR solution Consider a photon whose E=h is Equivalent to a mass =m 2 ph c

and the distance y=ct With t equal to the time the m ph is under No sun The acceleration a.

2 Which predicts an effect 8 times larger than a classical approach! Even so the angle with b as the solar radius is largest at only 1.

13 Bending of EM radiation y Radius of sun=b A Classical argument for the bend to get a Feel for the GR solution Consider a photon whose E=h is Equivalent to a mass =m 2 ph c m ph grazing the sun will experience a Gravitational force =GM s m ph /b 2 =m ph a Or a=gm s /b 2 b=radius The displacement d under acceleration a d Is given by d=at 2 /2 and the distance y=ct With t equal to the time the m ph is under No sun The acceleration a. path Tan d/y =GM s /b 2 t 2 /2 /ct =1/2tGM s /b 2 c An estimate of time is order of t=b/c So GM s /bc 2 GR via Einsteins curved space time formulation predicts =4GM s /bc 2 Which predicts an effect 8 times larger than a classical approach! Even so the angle with b as the solar radius is largest at only 1.74 of arc A difficult measurement which has been repeated (Eddington 1914,1919) 1919) and Extended to Radio time delays via interplanetary satellites in the extra path around the sun confirmed GR effect to 0.1%! The bending due to galaxies curving distance Other galaxies causes a lens!

14 Another manifestation of bending of light: Gravitational lenses A massive galaxy cluster is bending and g y g focusing the light from a background object. See also fig 8.1 in your text!

15 Gravitational red shift: Light from sources near massive bodies seems shifted towards longer wavelengths (red). Remember a photon has Energy E=h and since c c/ so E=hc/ photon has its energy change as it moves away from a massive body. That is, it looses some energy as the gravitational field does work on it. This means that must change. Since E goes down gets bigger or we have a RED shift!

16 GR Redshift From A static and spherically symmetric mass distribution From a solutions from the Schwarzchild solution to Einstein s Space-time field equations: This solution takes a mathematical form called a metric that describes the Geometry of space-time around a mass. One can derive the red shift of a Photon escaping a mass starting at r 1 =R and at a distance r 2 =r: similar to Eq. 8.4 in Kutner from the Schwarzchild metric. (Very complicated Tensor Mathematics!) r R (1-2GM/rc 2 ) (1-2GM/Rc 2 ) [ ] 1/2 What happens to if (1-2GM/Rc 2 )->0!? What is the value of R that does this? R=? Solve now! This R is called the Schwarzchild radius: Welcome to a BLACK HOLE!

17 Gravitational red shift: a classical approach to get a feel for GR solution When a particle, mass m, is in a gravitational field it has Gravitational potential energy = -GMm/r We again can consider the photon has an effective mass Ie E ph =h = hc = m eff c 2 or m eff =h/c r 1 r 2 E total = E ph GMm eff /r=hc/ GMh/rc Total Energy =photon energy +gravitational Potential energy for m eff At each point r 1 and r 2 the total energy is the same(conservation of Energy) using the last expression at each point we get hc/ 1 GMh/r 1 c 1 =hc/ 2 GMh/r 2 c 2 -> 1 1 (1-GM/r 1 c 2) = 1/ 2 (1-GM/r 2 c 2 ) or the nonrigorous classical result-> 2 / 1 = (1-GM/r 2 c 2 )/ (1-GM/r 1 c 2 ) eq 8.3 GR result-> = (1-2GM/r 2 (1-2GM/r 2 1/2 2 1 ( 2 c )/ 1 c ) ) eq 8.4 For small shifts (1-x) 1/2 ~= 1-x/2 8.4-> 8.3 as other R expressions

18 Gravitational red shift: for photon reaching us r 2 -> infinity from an object radius r 2 GR-> =( (1-2GM/r 2 (1-2GM/r 2 1/ c )/ 1 c ) ) r 2 -> infinity with r 1 =r we get 2 / 1 = (1/(1-2GM/rc 2 ) 1/2 = (1-2GM/rc 2 ) -1/2 Expanding as before 2 / 1 =1+GM/rc 2 or GM/rc 2 eq 8.6 Kutner for photon leaving surface of radius r! When is the GR redshift the most for values of r? How big are these shifts? be sure to do Example 8.1 in the text very small Sun not measurable but White dwarfs ok Best 3 x 10-4 Sirius B and =6 x Eridani

19 Other GR considerations The Mossbauer effect is used to verify GR redshift on Earth Kutner p 146 Essentially, a crystal emits a well defined Gamma ray and identical crystal can absorb that gamma ray provided it is the same wavelength. Put a crystal emitter in the basement and a crystal to absorb on the roof. The gamma ray photon moving through earths gravitaional field under goes a red shift and cannot be absorbed on the roof. Moving the crystal together one changes the wavelength and can calculate the shift which matches GR very well. Time dilation In a gravitational field can also me derived and expression of change is similar to GR redshift t 2 /t 1 = ( (1-2GM/r 2 c 2 )/ (1-2GM/r 1 c 2 ) ) 1/2 Airplane and rocket tests with atomic clocks follow this formula after taking into account SR time dilation! GR PREDICTS GRAVITATIONAL RADIATION! =BRIEF DISTORTION OF SPACE-TIME!

20 OLD AGE Stellar Collapse: Review: The Remnants of Sun-Like Stars: White Dwarfs Sunlike stars build up a Carbon-Oxygen (C,O) core, which does not ignite Carbon fusion. He-burning shell keeps dumping C and O onto the core. C,O core collapses and the matter becomes degenerate. Formation of a White Dwarf

21 White Dwarfs Degenerate stellar remnant (C,O core) Extremely dense: 1 teaspoon of WD material: mass 16 tons!!! Chunk of WD material the size of a beach ball would outweigh an ocean liner! White Dwarfs: Mass ~ M sun Temp. ~ 25,000 K Luminosity ~ 0.01 L sun Core has high Pressure!

22 Degenerate Matter Pressure! Depends on the Pauli exclusion principle: no two electrons can be in the same state: Regular gas pressure cannot stop the collapse in this case. Electrons packed in tight have a higher energy than normal and exert a pressure higher than normal ideal gas. Hence, this degenerate situation stops the collapse: P ~ density) about 100 x P (ideal gas) Electron energy and momentum become relativistic as calculated by S. Chandraseker (nobel prize in 83 for stellar structure)

23 The Chandrasekhar Limit The more massive a white dwarf, the smaller it is. Pressure becomes larger, until electron degeneracy pressure can no longer hold up against gravity. The relativistic i electrons create a pressure that t has a maximum support! WDs with more than ~ 1.4 solar masses can not exist! If the core mass Is greater than 1.4 star collapses Further! We get A Neutron star-> PULSARS: Our next chapter!

24 Black Holes beyond Neutron stars Just like white dwarfs (Chandrasekhar limit: 1.4 M sun ), there is a mass limit it for neutron stars: Neutron stars can not exist with masses > 3 M sun We know of no mechanism to halt the collapse of a compact object with > 3 M sun. It will collapse into a single point a singularity: => A Black Hole!

Escape Velocity Velocity needed to escape Earth s gravity from

If you could compress Earth to a smaller radius If you could

25 Escape Velocity Velocity needed to escape Earth s gravity from the surface: v esc 11.6 km/s. Now, gravitational force decreases with distance (~ 1/d 2 ) => Starting out high above the surface => lower escape velocity. If you could compress Earth to a smaller radius If you could compress Earth to a smaller radius => higher escape velocity from the surface. v esc v esc v esc

26 Non-relativistic! =(2GM/R) 1/2

27 Examples V escape Earth F g =GM 1 M 2 /R 2 units 1 dyne=g g 2 /cm 2 But F=ma ->1 dyne =g cm/s 2 or units of G=kgm/s 2 /kg 2 /m 2 = cm 3 /s 2 g or G= x cm 3 /s 2 g v es =(2GM/R) 1/2 =(2 x 667x cm 3 /s 2 gx598x10 x5.98 x 27 g/6.37 x 10 8 cm) 1/2 =1.12 x 10 6 cm/s =11.2 km/s ~ 7mi/s A NEUTRON STAR V es = 2 x 10 8 cm/s ~ 2/3 c If a star hits the Schwartzchild Radius we solved before,namely R s =2GM/c 2 WHAT IS V es = for this case? Solve now!

28 The Schwarzschild Radius So we see => This is a limiting radius where the escape velocity reaches the speed of light, c: R s = 2GM c 2 V esc = c G = Universal const. of gravity M = Mass R s is called the Schwarzschild Radius of a Black Hole! And the Red shift is????

29 What is the values for the Schwarzchild Radius Black holes are determined by the Mass and Size (Angular Momentum and electric charge is also considered but we will ignore these effects) How big is the R S for an object of one solar mass in km?. M sun =2 x g C=3 x cm/s G=6.67 x 10-8 dyne cm 2 /g 2 Calculate NOW!! R S =? R s = 2GM/c 2 That was example 8.2!

30 Schwarzschild Radii Since, R s is directly proportional to M One can see R s =3km (M/M sun ) Recall, density, Vol : Vol =4/3 R3 Calculate l for one solar mass Rs =? Equation 8.11 for black hole ~=10 17 g/cm 3 (M/M sun ) -2 Derive for extra credit!

31 Schwarzschild Radius and Event Horizon Event horizon No object can travel faster than the speed of light => nothing (not even light) can escape from inside the Schwarzschild radius We have no way of finding out what s happening inside the Schwarzschild radius.

Tides of the moon at The earth depends on The Force on a Mass (m) at a

32 Black Hole Tidal Effects I Tides depend on the Difference of the force Of gravity. Or the rate Of change of gravity. Tides of the moon at The earth depends on The Force on a Mass (m) at a distance r is Different at each point On Earth. Low Forces High Differentials High From center force Low

33 Black hole Tidal Effects on an unfortunate Astronaut Consider the force of Gravity on a mass m leads to an acceleration g, derived from F=mg =GMm/r 2 or g(r) () =GM/r 2 g at earths surface~ 10m/s 2 ~10 3 cm/s 2 The tidal acceleration is the differential in the gravitational acceleration. Or g =-2GM/r 3 r -> How did I get this? Example 8.3-What is the difference in the acceleration of gravity at the feet to the Head of on an astronaut whose height is 2 meters drifting into a black hole of 1 Solar mass ->R=3km and how does this compare With the gravitaional accelaration at earth s surface: Calculate now! A bl k h l t M i h t b t thi As black holes get more Massive what can we say about this Tidal effect???? Hint how does the Radius go!

34 An astronaut descending down towards the BH will be stretched vertically and squeezed laterally (tidal effects). This effect is called spaghettification

General Relativity Effects Near Black Holes :one Astronut!

35 General Relativity Effects Near Black Holes :one Astronut! Travels in to it! Time dilation Clocks starting at 12:00 at each point. After 3 hours (for an observer far away Clocks closer to the from the BH): BH run more slowly. Time dilation becomes infinite at the event horizon. Astronaut far away Never sees falling Event Horizon Astronaut reach the R s since the Time is infinite!

All wavelengths of emissions from near the event

Or as the astronut travels inward he emits a wave as he

36 General Relativity Effects Near Black Holes Gravitational Red Shift =GM/R s c 2 from the surface! All wavelengths of emissions from near the event horizon are stretched (red shifted). Or as the astronut travels inward he emits a wave as he get closer each wave received by the normal one gets longer! See text sec Event Horizon

37 Traveling toward R s Since space-time is sharply curved near the Black hole as Astronut approaches and sends light out the photons bend with the curvature Photons escape here Exit Cone Captured Gets smaller Nothing escapes Passed Event Horizon on SEE THE EFFECTS OF ROTATION Fig 8.13 At r=3/2r s photons Emitted horizontally Orbit in photon sphere! You can see The Back of your Head here!

38 Black Holes in Supernova Remnants Some supernova remnants with no pulsar / neutron star in the center may contain black holes.

Observing Stellar Black Holes No light can escape a

Material falling into a black hole emits x-rays If an

can pulsate then we can estimate its mass from the

39 Observing Stellar Black Holes No light can escape a black hole => Black holes can not be observed directly. Material falling into a black hole emits x-rays If an invisible compact object is part of a binary, x- rays can pulsate then we can estimate its mass from the orbital period and radial velocity. Mass > 3 M sun => Black hole!

40 O star =15 M sun Read sec 12.4 for details

41 Stellar black hole Supermassive black holes Are found At the core of galaxies Matter falling in accelerates and spirals out in two jets: The exact mechanism is still unknown and various models have Been proposed.

42 General Relativity :Visualizations At a distance, the gravitational fields of a black hole and a star of the same mass are virtually identical. At small distances, the much deeper gravitational potential will become noticeable.

43 World line = path of the photon

46 Movies to Visualize Space time Movies from the Edge of Spacetime

47 HW REMINDER! The Nature of Space and Time-General Relativity it Black Holes READ: Chapter 8 and Section 12.4 (Binaries with Black Holes) Chapter 8: Questions (HW#28)1, (HW#29)3 Problems (HW#30)2, (HW#31) 3, (HW#32) 6, (HW#33)7, (HW#34)8, (HW#35) 10 Chapter 12 (HW#36) Problem 12.9(note, inclination is the angle between the plane of the orbit of the two objects and a perpendicular p plane to our line of sight..see Fig 5.7)

Evolution of High Mass stars

Evolution of High Mass stars Neutron Stars A supernova explosion of a M > 8 M Sun star blows away its outer layers. The central core will collapse into a compact object of ~ a few M Sun. Pressure becomes