Extragalactic Astronomy & Cosmology

Lecture GRJane Turner

Joint Center for AstrophysicsUMBC & NASA/GSFC

2003 Spring

[4246] Physics 316

Jane Turner [4246] PHY 316 (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

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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.

Jane Turner [4246] PHY 316 (2003 Spring)

The Equivalence Principlethe 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

Jane Turner [4246] PHY 316 (2003 Spring)

Back to SpacetimeConsider 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

Jane Turner [4246] PHY 316 (2003 Spring)

Spacetime CurvatureWe 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

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

Rules of Spherical Geometry

Geometric rules for the surface of a sphere

Jane Turner [4246] PHY 316 (2003 Spring)

Rules of Hyperbolic Geometry

Jane Turner [4246] PHY 316 (2003 Spring)

Rules of Hyperbolic Geometry

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

Jane Turner [4246] PHY 316 (2003 Spring)

Summary of Geometries

These three forms of curvature the "closed" spherethe "flat" casethe "open" hyperboloid

Einstein's SR is limited to ("flat") Euclidean spacetime.

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

Reminder: Homogeneity/Isotropy

homogeneous - same properties everywhereisotropic - no special direction

homogeneous but not isotropic

isotropic but not homogeneous

Jane Turner [4246] PHY 316 (2003 Spring)

“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…

Jane Turner [4246] PHY 316 (2003 Spring)

“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 isfollowing 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

Jane Turner [4246] PHY 316 (2003 Spring)

“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

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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 playedsome role as Einstein formulated his GR. Indeed Einstein spent at least some effort (in vain) to incorporate the theory into GR

Jane Turner [4246] PHY 316 (2003 Spring)

“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 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

Jane Turner [4246] PHY 316 (2003 Spring)

“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

Jane Turner [4246] PHY 316 (2003 Spring)

“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

Jane Turner [4246] PHY 316 (2003 Spring)

Mass Curves spacetimee

The greater the mass, the greater the distortion of spacetime and thus the stronger gravity

Jane Turner [4246] PHY 316 (2003 Spring)

General Relativity

Compare an acceleration of a gravitationally-affected frame vs an inertial frame - light apparently bent by gravity/accln is light following the shortest path

Jane Turner [4246] PHY 316 (2003 Spring)

Radius of Curvature

Radius of the circle fitting the curvature

rc=c2/g = 9.17x1017 cm for Earthfor larger masses, g is larger and rc smaller

Jane Turner [4246] PHY 316 (2003 Spring)

Curvature of Space

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

Jane Turner [4246] PHY 316 (2003 Spring)

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 energyEend = Estart+ Egrav

(From Chris Reynolds

Web site @UMCP)

Cannot have created energy!

Jane Turner [4246] PHY 316 (2003 Spring)

GR - Gravitational Redshift

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

The light travelling upwards must have lost energy due to gravity!

Jane Turner [4246] PHY 316 (2003 Spring)

GR - Gravitational Time Dilation

This is why clocks run slow near a black hole

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

Jane Turner [4246] PHY 316 (2003 Spring)

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 !

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

The Metric EquationHow 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

Jane Turner [4246] PHY 316 (2003 Spring)

The Metric EquationIn Euclidean space the distance between points is r2= x2+ y2

In general geometries the distance between points is r2= fx2+ 2g x y + hy2 - metric equationf,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

Jane Turner [4246] PHY 316 (2003 Spring)

The Metric EquationFor close points r2= fx2+ 2g x y + hy2 - metric equation

so for any 2 points sum the small steps along the path- integrate!

A general spacetime metric is

s2= c2t2 -ctx-x2 for coordinate x, , depend on the geometry

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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 Einstein’s constraints for making the physical model, GR

Jane Turner [4246] PHY 316 (2003 Spring)

One-line description of the Universe

G=8GT

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

led to…

Jane Turner [4246] PHY 316 (2003 Spring)

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…

Jane Turner [4246] PHY 316 (2003 Spring)

Tests of GR - Light Bending

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

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

Jane Turner [4246] PHY 316 (2003 Spring)

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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: WeakPropagate at the speed of lightShould compress & expand objects they pass by

Can we look for more direct proof these exist?

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

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 starsbinary BHsstars falling onto BHs

Jane Turner [4246] PHY 316 (2003 Spring)

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

Jane Turner [4246] PHY 316 (2003 Spring)

Signatures of material spiraling onto a black hole

Jane Turner [4246] PHY 316 (2003 Spring)

Determine whether the black hole is spinning...