Extragalactic Astronomy

1
Extragalactic Astronomy Cosmology
Lecture GR
4246 Physics 316

• Jane Turner
• Joint Center for Astrophysics
• UMBC NASA/GSFC
• 2003 Spring

2
A Note on the Mid-Term Exam
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
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
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
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
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
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
Newtons 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
What we perceive as gravity arises from the
curvature of spacetime Things can approximate to
different geometries on different size scales.
The Earths 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
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
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
Compare an acceleration of a gravitationally-affec
ted 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 rcc2/g 9.17x1017 cm for Earth for
larger masses, g is larger and rc smaller



26
Curvature of Space
The rubber-sheet analogy cant 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

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 site _at_UMCP)

Cannot have created energy!

28
GR - Gravitational Redshift

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

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

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
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?x2 for coordinate x a, b, g
depend on the geometry

35
General Relativity -Curved Space
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
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
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 Einsteins constraints for making the
physical model, GR



38
One-line description of the Universe
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
geometrymatter energy



39
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

Light going close to a massive object falls in
the gravitational field and travels through
curved spacetime



41
GR-Light Bending

Eddingtons measurements of star positions during
eclipse of 1919 were found to agree with GR,
Einstein rose to the status of a celebrity



42
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



43
(No Transcript)
44
(No Transcript)
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

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

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
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
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
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
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
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...