1. Black Holes and General Relativity
The General Theory of Relativity
Intervals and Geodesics
Black Holes

2. The General Theory of Relativity
Newton
Einstein
Simple
Works (extremely) well
Describes how things fall on earth
Describes motion of solar system objects…predicted existence of Neptune
Unable to account for …
for 43”/century of observed 574”/century shift of mercury perihelion…hmmm?
Elegant
Works even better
Alters basic view of Universe
Mass  space-time
Accounts for …
additional 43”/century of observed 574”/century shift of mercury perihelion…
Shifting of star positions during eclipse…

3. Advance of the Perihelion of Mercury
Perihelion-nearest approach point of Mercury’s orbit advances due to effects from planets,etc…
Newtonian Gravitation can not account for total observed shift of 574”/century….Planet Vulcan???
Einstein’s General Theory of relativity could fully account for the observed perihelion advance
Need to account for trivial precession of equinoxes 1.5 degree/century

4. Advance of the Perihelion of Mercury

5. Curvature of Space-Time?
Salvador Dali's "Soft Watch at Moment of First Explosion"

6. Curvature of Space-Time
The General Theory of Relativity is fundamentally a geometric description of how distances (intervals) are measured in the presence of mass.
Relativity deals with a unified space-time.
Distances between points in the space surrounding a massive object are altered in a way that can be interpreted as space becoming curved through a fourth spatial dimension
Rubber sheet analogy:
Closer to ball more curvature
Distance between two points increases more
Mass acts on Spacetime, telling it how to curve
Curved space-time acts on mass telling it how to move

7. Curvature of Space-Time
Curved space….can see that….kinda
Curved time?
The rate of flow of time is determined by the strength of the gravitational field where it is measured…

8. Curvature of Space-Time Space and Time
Gravity curves space ,eh?
Looks like balls follow different curves to me!!...
The picture below shows two balls being shot across a room by a machine located at the lower right side of the room. The fast ball shoots across the room with very little curvature in the 3-D space. The higher ball is slower and therefore it must take a higher path to guarantee that it reaches the other side. This curved path action is recorded by the experimenter inside the room. He captures the side view of both particles. It is obvious that space curves the path of these two balls quite differently.
The experimenter to the left stands outside the room and records the vertical action of the two balls as seen through the room's end window. The view that she captures is the most relevant for gravity since gravity only acts in the vertical direction. The side view obtained in the picture above is too complicated for our purposes. We are not interested in the right-side to left-side motion. It has no connection to gravity.

9. Curvature of Space-Time Space and Time
The higher ball's vertical motion is shown in the top strip. The height of the ball is being compared to the top of the small table that is located in the center of the room. The horizontal direction represents the time direction. The left frames are earlier than the right frames. The bottom strip represents the vertical motion sequences of the fast lower ball. There are less frames for the faster ball because the faster ball took less time to cross the room. These time sequenced pictures show a remarkable thing. If we focus our attention on the curved path through time we can construct a connecting curve for all of the balls. After doing this we notice that the curvature of the path through time and space (i.e., spacetime) in the bottom three frames is exactly equal to the curvature of the path through spacetime of the central 3 frames in the ball in the top strip. The Earth's gravity effect is the same for both balls.  The spacetime curvature around the Earth acts on fast or slow balls in exactly the same way. It tells these balls how to move.
Need to look at space and time….In space-time the balls trajectory do indeed have the same curvature…

10. Curvature of Space-Time Space and Time
Bullet and Ball initially take same paths in space. Why don’t they continue following same path if gravity is curving space?
Bullet and Ball take different paths in space-time
 experience different curvature

11. Curvature of Space-Time Space and Time
Bullet and Ball initially take same paths in space. Why don’t they continue following same path if gravity is curving space?
Bullet and Ball take different paths in space-time
 experience different curvature

12. Curvature of Space-Time
Since spacetime itself is curved
even the trajectory of massless photons deviates from a “straight” line

13. The Principle of Equivalence
We’re in a “no gravity” state?

14. The Principle of Equivalence
In any free-float situation, such as that in a freely falling spaceship, the paths of objects will never bend in any direction when they are given a certain speed. These objects will move in completely straight lines.

15. The Principle of Equivalence
When the Earth (or a rocket) pushes on the spaceship, the tracks curve relative to the spaceship.

16. The Principle of Equivalence
The ball appears to move in the familiar curved path which we have come to view (since the time of Newton) as the effect of a gravitational force directed towards the center of the Earth.
With the platform severed from its attachment to the Earth, the small house is in a free-fall situation. This time the ball will move in a straight line, unaffected by any so-called 'gravity' force.

17. The Principle of Equivalence

18. The Principle of Equivalence
Weak Equivalence principle
mg/mi is a constant
The Principle of Equivalence: All local, freely falling, nonrotating laboratories are fully equivalent for the performance of all physical experiments

19. The Bending of light

20. The Bending of light

21. The Bending of light

22. Gravitational Redshift and Time Dilation
An outside observer, not in free-fall inside of the lab, would measure only the gravitational redshift (blueshift if the photon were going downward)

23. Gravitational Redshift and Time Dilation
Light pulse generated at instant cable is released. In the time for the photon to cross the elevator cabin, the meter has attained a speed v=gt=gh/c.
Doppler blueshift should change meter’s frequency measurement by
In fact no frequency shift would be observed in accordance with principle of equivalence. The gravitational redshift exactly compensates by
An outside observer, not in free-fall inside of the lab, would measure only the gravitational redshift (blueshift if the photon were going downward)

25. Gravitational Redshift Calculation for a beam that escapes to infinity

26. Gravitational Redshift and Time Dilation
Gravitational time dilation: Time passes more slowly as the surrounding space-time becomes more curved

28. Intervals and Geodesics
General Relativity allows one to relate events (x,y,z,t) in spacetime in the presence of mass that results in the fabric of spacetime being curved!!!
T is the stress-energy tensor which evaluates the effect of a given distribution of mass and energy on the curvature of spacetime.
G is the Einstein Tensor for Gravity that mathematically describes the curvature of space-time.
Note that the Gravitational constant G and the speed of light play a role in the gravitational field equation.
Field Equation: for calculating the geometry of space-time produced by a given distribution of mass and energy

29. Worldlines and Light Cones
Worldline: The path followed by an object as it moves through spacetime.
Light Cone: worldline of photon originating at event A. The speed of light is taken to be 1.
What is the worldline for a freely falling object in response to the local curvature of spacetime?

30. Spacetime Intervals, Proper Time and Proper Distance
What is a “distance” in spacetime?
Spatial distance between two points (x1,y1,z1) and (x2,y2,z2) in flat space
Spacetime interval: between two events (xA,yA,zA,tA) & (xB,yB,zB,tB) in flat spacetime
s)2>0: timelike. Light has more than enough time to travel between the events A and B
s)2=0. Lightlike separation.
s)2<0: spacelike. Light does not have enough time to travel between the events
Proper time: the time between two events that occurs at the same location.
s is invariant under Lorentz transformations. An observer in another inertial frame S’ would measure the same interval between events A and B
Proper Distance: The distance measured between two events A and B in a reference frame for which they occur simultaneously (tA=tB).

31. The Metric for Flat Spacetime
The interval measured along any timelike interval is the proper time multiplied by c.
The proper time along any lightlike worldline is zero
The proper time along any spacelike worldline is undefined
In flat spacetime, the interval measured along a straight timelike worldline is a maximum!!!
Metric for flat 3-dimensional space
Path length for arbitrary path
Straightest possible line between points minimizes length
Metric for flat spacetime
Total interval along worldline

32. The Metric for Flat Spacetime

33. Curved Spacetime and the Schwarzschild Metric
Flat spacetime metric in polar coordinates
Schwarzschild metric in space time curved by a spherical body of mass M where r>R
Mass acts on spacetime, telling it how to curve
Spacetime in turn acts on mass, telling it how to move
Any freely falling particle (including a photon) follows the straightest possible worldline, a geodesic, through spacetime. For a massive particle the geodesic has a maximum or a minimum interval, while for light , the geodesic has a null interval.
Geodesic: “Straightest” possible worldline. In a flat spacetime a geodesic is a straight worldline. In a curved spacetime a geodesic will be curved.

34. Shortest distance between two points may not be a straight line

35. The Schwarzschild metric
Consider a sphere of radius R and mass M placed at the origin of the coordinate system. The coordinate r does NOT represent distance from the origin. A concentric sphere whose surface is at r would have a surface area 4r2.
A Flamm paraboloid helps “visualize” this curvature. Remember that “you” would be contained in the curved spacetime and you cannot directly “view” the curvature into the 4th dimension….
Proper distance along a radial line
Proper Time at radial coordinate r

36. The Orbit of a Satellite
Starting with Schwarzschild metric with dr=0, d=0 and d=dt where v/r

37. Tests of General Relativity
Classical Tests
Perihelion Precession of Mercury
Deflection of light by the Sun
Gravitational Redshift of Light
Modern Tests
Gravitational Lensing
Light travel time delay testing
Equivalence Principle tests
Gravitational redshift
Lunar Ranging…
Frame dragging tests
Strong Field Tests (Neutron stars,Black holes)
Gravitational Wave Detectors
Cosmological Tests

38. Gravitational Lens

39. Gravitational Lens “Einstein Cross”

40. Gravitational Lens Studies at the U

41. Gravitational Lens Studies at the U

42. Black Holes
In 1783 John Mitchell pondered that the escape velocity from the surface of a star 500 times larger than the sun with the same average density would equal the speed of light.
Light would not be able to escape from such a star!!!!
Naïve solution of Newtonian escape velocity equation for c gives a radius of R=2GM/c2 for a star whose escape velocity equals the speed of light.
R=2.95(M/M) km….kinda small!!!
In 1939 J. Robert Oppenheimer and Hartland Snyder described the ultimate gravitational collapse of a massive star that has exhausted its sources of nuclear fusion. They pondered what happened to the cores of stars whose mass exceeded the limit of neutron stars..
In 1967 the term “black hole” was coined
By John Archibald Wheeler

43. The Schwarzschild Radius
Consider the Schwarzschild metric
When the radial coordinate of the star’s surface has collapsed to RS=2GM/c2 the square roots in the metric go to zero. RS is known as the Schwarzschild Radius.
At r=RS the behavior of space and time is “remarkable”….
The proper time measured by a clock here is d=0.
Time has slowed to a complete stop! As measured from a vantage point far away. From this viewpoint nothing ever happens at the Scwarzschild radius
Does this mean that even light is frozen in time???
The speed of light by an observer suspended above the star must always be c. But from far away we can determine that light is delayed as it moves through curved spacetime…
The apparent speed of light, the rate at which the spatial coordinates of a photon change, is called the coordinate speed of light. For light ds=0.
For dd=0, we have
In flat spacetime dr/dt~c, however at r=RS dr/dt=0
Light does appear frozen in time at the Schwarzschild radius!!!