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Extragalactic Astronomy Cosmology

Lecture GR

4246 Physics 316

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

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

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

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.

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

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

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

Rules of Geometry - Euclidean Space

Space has a flat geometry if these rules apply

Rules of Spherical Geometry

Geometric rules for the surface of a sphere

Rules of Hyperbolic Geometry

Rules of Hyperbolic Geometry

Cannot be visualized, although a saddle exhibits

some of its properties - sometimes called a

Saddle Shape Geometry

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.

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

Reminder Homogeneity/Isotropy

homogeneous - same properties everywhere isotropic

- no special direction

homogeneous but not isotropic

isotropic but not homogeneous

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

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

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

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

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

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

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

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

Mass Curves spacetimee

The greater the mass, the greater the distortion

of spacetime and thus the stronger gravity

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

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

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

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!

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

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

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 !

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

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

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

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

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

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

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

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

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

Tests of GR - Light Bending

Light going close to a massive object falls in

the gravitational field and travels through

curved spacetime

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

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

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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

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

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

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

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?

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

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

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

Signatures of material spiraling onto a black

hole

Determine whether the black hole is spinning...