1. Extragalactic Astronomy & Cosmology Lecture GR*
07/16/96
[4246] Physics 316
Extragalactic Astronomy & Cosmology Lecture GR
Jane Turner
Joint Center for Astrophysics
UMBC & NASA/GSFC
2003 Spring * ##

2. A Note on the Mid-Term Exam*
A Note on the Mid-Term Exam
07/16/96
Excludes Copernicus and anything before that…
Revision might start with Keplers Laws and Newtons version of Keplers laws and his Universal Law of Gravitation
Hubbles Law
What are SR, GR about, Worldlines in Spacetime diagrams
Galaxies & the history of discovering they were external to the Milky Way rather than nebulae * ##

3. *
General Relativity
07/16/96
The Universe is filled with masses - we need a theory which accommodates inertial & non-inertial frames & which can describe the effects of gravity * ##

4. GR in a nutshell
General Relativity is essentially a geometrical theory concerning
the curvature of Spacetime.
For this course, the two most important aspects of GR are needed:
Gravity is the manifestation of the curvature of Spacetime
Gravity is no longer described by a gravitational "field" /”force”
but is a manifestation of the distortion of spacetime.
Matter curves spacetime; the geometry of spacetime
determines how matter moves.
Energy and Mass are equivalent
Any object with energy is affected by the curvature of spacetime.

5. The Equivalence Principle
the effects of gravity are exactly equivalent to the effects of acceleration
thus you cannot tell the difference between being in a closed room on Earth and one accelerating through space at 1g
any experiments performed (dropping balls of different weights etc) would produce the same results in both cases

6. Back to Spacetime Accelerated Observer Inertial Observers
Consider a person standing on the Earth versus an astronaut accelerating through space … gravity and acceleration sure look different!
However, GR says in order to understand things properly you have to see the whole picture, i.e.
consider spacetime
Recall our spacetime diagrams
Accelerated Observer
Inertial Observers

7. Spacetime Curvature
We have considered flat spacetime diagrams, however spacetime can be curved and then different rules of geometry apply
consider how there is no straight line on the surface of the Earth, the shortest distance between 2 pts is a Great Circle-whose center is the center of the Earth

8. Rules of Geometry - Euclidean Space
Space has a flat geometry if these rules apply

9. Rules of Spherical Geometry
Geometric rules for the surface of a sphere

10. Rules of Hyperbolic Geometry

11. Rules of Hyperbolic Geometry
Cannot be visualized, although a saddle exhibits some of its properties - sometimes called a Saddle Shape Geometry

12. Summary of Geometries These three forms of curvature
the "closed" sphere
the "flat" case
the "open" hyperboloid
Einstein's SR is limited to
("flat") Euclidean spacetime.

13. Geometries
Why have we described three apparently arbitrary sets of geometries when there are an infinite number possible???
These three geometries have the properties of making space homogeneous and isotropic
-as is the observed universe (later)
so these three are the subset which are possible geometries for space in the universe

14. Reminder: Homogeneity/Isotropy
homogeneous - same properties everywhere
isotropic - no special direction
homogeneous but not isotropic
isotropic but not homogeneous

15. “Straight Lines” in Curved Spacetime*
“Straight Lines” in Curved Spacetime
07/16/96
Key to understanding spacetime is to be able to tell whether an object is following the straightest possible path between 2 pts in spacetime
Equivalence Principle provides the answers - can attribute a feeling of weight either to experiencing a grav field or an acceleration
Similarly can attribute weightlessness to being in free-fall or at const velocity far from any grav field
Traveling at const velocity means traveling in a straight line… * ##

16. “Straight Lines” in Curved Spacetime*
“Straight Lines” in Curved Spacetime
07/16/96
Traveling at const velocity means traveling in a straight line…
So, Einstein reasoned that weightlessness was a state of traveling in a straight line - leading to the conclusion:
If you are floating freely your worldline is
following the straightest possible path through
spacetime. If you feel weight then you are not on
the straightest possible path
This provides us a remarkable way to examine the geometry of spacetime, by looking at the shapes and speeds related to orbits * ##

17. “Straight Lines” in Curved Spacetime*
“Straight Lines” in Curved Spacetime
07/16/96
This provides us a remarkable way to examine the geometry of spacetime, by looking at the shapes and speeds related to orbits
e.g. changes the concept of Earths motion around the Sun, its not under the force of gravity, it is following the straightest possible path and spacetime is curved around the Sun due to its large mass
What we perceive as gravity arises from the curvature of spacetime due to the presence of massive bodies * ##

18. Note of Interest: Machs Principle
Newton’s contemporary and rival Gottfried Leibniz first voiced the idea that space and matter must be interlinked in some way
Ernst Mach first made a statement of this

19. Mach's Principle (restated)
Ernst Mach's principle (1893) states that
“the inertial effects of mass are not an innate property of the body,
rather the result of the effect of all the other matter in the universe”
(local behavior of matter is influenced by the global properties of the universe)
More specifically
“It is not absolute acceleration, but acceleration relative to the center of
mass of the universe that determine the inertial properties”
It is incorrect… it is incompatible with GR
- there is no casual relation between the distant
universe & a “local” inertial frame
- “local” properties are determined by “local” spacetime
However, Mach's Principle was "popularized" by Albert Einstein, and undoubtedly played
some role as Einstein formulated his GR. Indeed Einstein spent at least some effort (in vain)
to incorporate the theory into GR

20. “Straight Lines” in Curved Spacetime*
“Straight Lines” in Curved Spacetime
07/16/96
What we perceive as gravity arises from the curvature of spacetime
Things can approximate to different geometries on different size scales. The Earth’s surface seems flat to us, but when we consider large scales we know the Earth is a sphere.
Geometry of spacetime depends locally on mass
When we expand our consideration to a general geometry the 4-dimensional universe must have some geometry determined by the total mass in it * ##

21. “Straight Lines” in Curved Spacetime*
“Straight Lines” in Curved Spacetime
07/16/96
When we expand our consideration to a general geometry the 4-dimensional universe must have some geometry determined by the total mass in it
As noted earlier, our 3
geometries are possibilities * ##

22. “Straight Lines” in Curved Spacetime*
“Straight Lines” in Curved Spacetime
07/16/96
Our 3 geometries are possibilities as they fit the properties of homogeneity/isotropy
Spacetime would be infinite in
the flat or hyperbolic cases with no center or edges
Spherical case is finite, but the surface of sphere has no center or edges * ##

23. Mass Curves spacetimee
The greater the mass, the greater the distortion of spacetime and thus the stronger gravity

24. *
General Relativity
07/16/96
Compare an acceleration of a gravitationally-affected frame vs an inertial frame - light apparently bent by gravity/accln is light following the shortest path * ##

25. Radius of Curvature Radius of the circle fitting the curvature*
Radius of Curvature
07/16/96
Radius of the circle fitting the curvature
rc=c2/g = 9.17x1017 cm for Earth
for larger masses, g is larger and rc smaller * ##

26. *
Curvature of Space
07/16/96
The rubber-sheet analogy can’t show the time dimension
Of course, objects cannot return to the same point in spacetime because they always move forward in time
Even orbits which
bring earth back to
the same point in space
(relative to the Sun)
move along the time axis * ##

27. GR - Gravitational Redshift*
07/16/96
GR - Gravitational Redshift
Thought Experiment:
Shine light from bottom of tower to top, has energy Estart
When light gets to top, convert its energy to mass m= Estart /c2
Drop mass, it accelerates due to g
At bottom, convert back to energy
Eend = Estart+ Egrav
(From Chris Reynolds
Web
Cannot have created energy! * ##

28. GR - Gravitational Redshift*
GR - Gravitational Redshift
07/16/96
The light travelling upwards must have lost energy due to gravity!
At start, bottom of tower,
high frequency wave, high energy
Upon reaching the top of the tower,
low frequency wave, lower energy
Gravity affects the frequency of light * ##

29. GR - Gravitational Time Dilation*
GR - Gravitational Time Dilation
07/16/96
Consider a clock where 1 tick is time for a certain number of waves of light to pass, gravity slows down the waves and thus the clock.
Clocks run slow in gravitational fields
This is why clocks run slow near a black hole * ##

30. GR - Gravitational Redshift*
GR - Gravitational Redshift
07/16/96
From the Equivalence Principle, the same effect occurs in an accelerating frame
The stronger the gravity and thus the greater the curvature of spacetime the larger the time- dilation factor
Time runs slower on the surface of the Sun than the Earth
-extreme case, a Black Hole ! * ##

31. General Relativity -Tidal forces*
General Relativity -Tidal forces
07/16/96
Consider a giant elevator in free-fall. We have two balls, one released above the other.
Bottom ball is closer to Earth (thus stronger gravitational force)
Bottom ball accelerates faster than top ball. Balls drift apart.
Tidal forces are clues to space-time curvature, gradients of curvature are extreme near v. massive objects, and todal forces there are very destructive * ##

32. The Metric Equation How about some sort of metric then….
A metric is the "measure" of the distance between points in a geometry
The distance between two points on a geometry such as a surface is certainly going to depend on how that surface is shaped
The metric is a mathematical function that takes such effects into account when calculating distances between points

33. The Metric Equation In Euclidean space the distance between points is
r2= x2+ y2
In general geometries the distance between points is
r2= fx2+ 2g x y + hy2 - metric equation
f,g,h depend on the geometry - metric coefficients
-valid for points close together
-a metric eqn is a differential distance formula, integrate it to get the total distance along a path
For 2 arbitrary points we also need to know the path along which we want to measure the distance

34. The Metric Equation For close points
r2= fx2+ 2g x y + hy2 - metric equation
so for any 2 points sum the small steps along the path- integrate!
A general spacetime metric is
s2= ac2t2 -bctx-gx for coordinate x
a, b, g depend on the geometry

35. General Relativity -Curved Space*
General Relativity -Curved Space
07/16/96
What do we have so far?
-Masses define trajectories
-Geometries other than Euclidean may describe the universe
Now need formulae to describe how mass determines geometry and how geometry determines inertial trajectories - General Relativity * ##

36. Riemannian Geometries*
Riemannian Geometries
07/16/96
We know on small scales spacetime reduces to Special Relativistic case of Minkowskian spacetime (flat)
Only a few special geometries have the property of local flatness-called Riemannian geometries * ##

37. Riemannian Geometries*
Riemannian Geometries
07/16/96
Only a few special geometries have the property of local flatness-called Riemannian geometries
Also know an extended body suffers tidal forces due to gravity (paths in curved space do not keep two points a fixed dist. apart!)
OK, homogeneity, isotropy, local flatness, tidal forces & reduction to Newtonian physics for small gravitational force & velocity provided Einstein’s constraints for making the physical model, GR * ##

38. One-line description of the Universe*
One-line description of the Universe
07/16/96
led to…
G=8GT  c4
G, T are tensors describing curvature of spacetime & distribution of mass/energy, respectively G is the constant of gravitation
 are labels for the space & time components of these
This one form represents ten eqns! generally of the basic form geometry=matter + energy * ##

39. Tests of GR - Light Bending*
07/16/96
Tests of GR - Light Bending
Differences between the Newtonian view of the universe and GR are most pronounced for the strongest fields, ie. around the most massive objects.
Black holes provide a good test case and they will be discussed in the next lecture.
Everyday life offers few measurable deviations from Newtonian physics, so are there suitable ways to test GR?
Bending of light by Sun is twice as great in GR as in
Newtonian physics so eclipses offer a chance… * ##

40. Tests of GR - Light Bending*
07/16/96
Tests of GR - Light Bending
Light going close to a massive object falls in the gravitational field and travels through curved spacetime * ##

41. *
07/16/96
GR-Light Bending
Eddington’s measurements of star positions during eclipse of 1919 were found to agree with GR, Einstein rose to the status of a celebrity * ##

42. *
07/16/96
GR-Light Bending
Light bending can be most dramatic when a distant galaxy lies behind a very massive object (another galaxy, cluster, or BH)
Spacetime curvature from the intervening object can alter different light paths so they in fact converge at Earth - grossly distorting the appearance of the background object * ##

45. Tests of GR-Gravitational Lenses
Depending on the mass distribution for the lensing object, we may see multiple images of the background object, magnification, or just distortion

46. Measurements of the precise orbit of Mercury*
07/16/96
Measurements of the precise orbit of Mercury
GR also predicts the orbits of planets to be slightly different to Newtonian physics
The orbit of Mercury was a good test case, closest to the Sun it was likely to show deviations between the two theories most strongly
In fact it had long been know there was a deviation of 43” century of the actual orbit vs Newtonian-predicted case - Einstein was delighted to find GR exactly accounted for this discrepancy * ##

47. Measurements of the precise orbit of planets*
07/16/96
Measurements of the precise orbit of planets
Modern day radar measurements have helped determine planetary orbits to high degrees of accuracy, strengthening the agreements with GR over Newtonian physics * ##

48. GR-Gravitational Waves*
07/16/96
GR-Gravitational Waves
Changes in mass distribution can cause ripples of spacetime curvature which propagate like ripples after dropping a stone into a pond
A Supernova explosion may cause them
Also, moving masses like a binary system of two massive objects, can generate waves of curvature-like a blade turning in water
A gravitational field which changes with time produces waves in spacetime-gravitational waves * ##

49. GR - Gravitational Waves*
GR - Gravitational Waves
07/16/96
So, GR predicts compact/massive objects orbiting each other will give off gravitational waves, thus lose energy resulting in orbital decay. Such orbital decays detected, Taylor & Hulse in 1993 (Noble Prize) -indirect support of GR
Characteristics of gravitational waves:
Weak
Propagate at the speed of light
Should compress & expand objects they pass by
Can we look for more direct proof these exist? * ##

50. GR - Gravitational Waves*
GR - Gravitational Waves
07/16/96
A Laser Interferometer
-can detect compression/expansions of curvature in spacetime by splitting a light beam & sending round two perpendicular paths, if spacetime is distorted in either direction due to gravitational waves, then recombining the beam would produce interference * ##

51. GR - Gravitational Waves*
GR - Gravitational Waves
07/16/96
Laser Interferometer Gravitational Wave Observatory -will soon become operational (Louisiana/Washington)
Laser Interferometer Space Antenna - Space-based version of LIGO
These experiments will look for
binary stars
binary BHs
stars falling onto BHs * ##

52. GR - Gravitational Redshift/Time dilation*
GR - Gravitational Redshift/Time dilation
07/16/96
Gravitational redshift produces a shift of photons to lower energies, we see some evidence for this close to supermassive BHs in the centers of galaxies * ##

53. Signatures
of material spiraling onto
a black hole

54. Determine whether
the black hole is spinning...