Extragalactic Astronomy & Cosmology Lecture GR

[4246] Physics 316 Extragalactic Astronomy & CosmologyLecture GR 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 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

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 propertiesof 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 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

“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-affected 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 rc=c2/g = 9.17x1017 cm for Earth for larger masses, g is larger and rc smaller

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

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 @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 Einstein’s 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 geometry=matter + 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 Eddington’s 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

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

Extragalactic Astronomy & Cosmology Lecture GR