Science 3210 001 : Introduction to Astronomy

Lecture 10 : Relativity, Black Holes

Robert Fisher

Items

! Homework and first observational project due next week.

! Second midterm will be on April 11th. Covers the material in

between the last midterm and the end of the next lecture.

! We plan to have a guest speaker sometime shortly after the

second midterm. Lunch in the loop (on me) with the guest

speaker following the lecture at Frontera Fresco for anyone who

wants to join us.

! Short proposals (1 paragraph) for final projects will be due on

April 18th. We will have a peer review session of the proposals

that day.

! Final projects due on the last day of class, along with a short (5

minute) presentation that day.

Final Project

! Your final project is to construct a creative interpretation a scientifictheme we encountered during the class. You will present your work in afive minute presentation in front of the entire class on May 11.

! The project must have both a scientific component and a creative one.

! For instance, a Jackson Pollock-lookalike painting would fly, but ONLY ifyou said that it was your interpretation of the big bang cosmologicalmodel AND you could also demonstrate mastery of the basicastrophysics of the big bang while presenting your work.

! Be prepared to be grilled!

! Ideas :

! Mount your camera on a tripod and shoot star trails.

! Create a “harmony of the worlds” soundtrack for the Upsilon Andromedasystem.

! Paint the night sky as viewed from an observer about to fall behind thehorizon of a black hole.

! Write a short science fiction story about the discovery of intelligent life in theuniverse.

Review of Three Weeks Ago

! Extrasolar planets

! 51b Peg

! HD209458b

Review of Two Weeks Ago

! Interstellar Medium and Star Formation

! Binary Stars

! Star Clusters

! HR (Hertzsprung-Russell) Diagram of Stars

Review of Last Week

! Stellar Structure

! Stellar Evolution

! Evolution of a low-mass star

! Evolution of a high-mass star

! Supernovae

Today -- Relativity and Black Holes

! Special Relativity

! Michelson-Morley Experiment

! Introduction to Spacetime Physics

! Relativity of Simultaneity

! General Relativity

! Black Holes

The Aether

! Late 19th century scientists attempting to make sense of the wavelikebehavior of light argued that light must be a wave like other wavesknown at that time -- water waves, sound waves, seismic waves, and soon.

! The common opinion developed was that waves are the result of amechanical disturbance in a physical medium -- for instance, waterwaves oscillate once a rock is dropped in a pond.

! By analogy, light must be the result of a disturbance in an undetectedmedium known as the aether (sometimes ether or luminiferous aether).

! If the aether did exist, it must carry physical properties like mass andmomentum, just like a pond. If it has physical properties, it must bedetectable.

! If that is all true, then where was all of the evidence for the existence ofthe aether ??

Michelson-Morley Experiment

! In 1887, physicists Michelson and Morley devised a brilliant

method to detect the aether.

! To understand how their experiment worked, consider Alexis,

who is standing on the shore watching Bettie, moving on a boat

moving at a fixed speed through a river.

! When Bettie is moving downstream, the boat moves with a speed

relative to Alexis which is the sum of the boat speed and the

water speed.

Bettie

Michelson-Morley Experiment

! When Bettie is moving upstream, the boat moves with a speed

relative to Alexis which is the difference of the boat speed and the

water speed.

! Even if Alexis observed only the motion of the boat in both

directions, she could easily infer both the direction and speed of

the water current.

! By analogy, Michelson and Morley hoped to measure the

difference in the speed of light as it moved relative to the aether,

and from that knowledge, both establish the existence of the

aether and also its direction of motion and speed.

Bettie

Michelson-Morley Experiment

! Believing that the speed of light relative to the Earth must vary as

the Earth moves through the aether, physicists Michelson and

Morley planned a highly-sensitive experiment to measure this

effect.

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Michelson-Morley Video

Classical Space and Classical Time

! Classical physics prior to Einstein considered motion taking place

upon a fixed space and a universal time.

! Space is the stage where all action takes place. Everyone always

agrees upon distances measured on the stage -- it is absolute

and unchanging.

! Time is a universal concept as well. Everyone’s clock always

precisely agrees with everyone else’s clock.

Spacetime

! One of the key ideas in relativity theory is that space and time aredramatically different from the classical viewpoint.

! The concepts which Einstein hit upon are radically different thanboth the classical point of view, and our own everydayexperience.

! Because relativity is so radically different from our everydayexperience, Einstein had proceed using razor-sharp logic, startingfrom basic axioms.

! This reasoning was often applied to extraordinary situationsknown as thought experiments (sometimes gedankenexperimentfrom the German).

A Short Note on Historical Attributions

! The popular conception is that the

theory of relativity was nearly single-

handedly created by Einstein. While

largely true, it is far from the entire

story.

! Early ideas remarkably similar to

Einstein’s were espoused by Karl

Friedrich Gauss and Behrnard

Riemann.

! Key contributions to the theory were

made by several other scientists,

including George Francis Fitzgerald,

Hendrik Lorentz, and Henre

Poincare.

Einstein & Lorentz

A Short Note on Historical Attributions

! “Whoever speaks of absolute space usesa word devoid of meaning. This is a truththat has been long proclaimed by all whohave reflected on the question, but onewhich we are too often inclined to forget…have shown elsewhere what are theconsequences of these facts from thepoint of view of the idea that we shouldconstruct non-Euclidean and other

analogous geometries.” -- HenriPoincare, Science and Method, 1897

! Recently more controversialsuggestions have been made thatEinstein’s first wife Mileva Mariccontributed substantially to therelativity, and even that otherscientists came upon E = m c2

independently.

! What remains true is that the wholeof relativity theory owes more to onesingle individual more than anyother major theory in modernphysics.

Henri Poincare

Einstein

! A large part of the Einstein myth

are the circumstances in which

his first papers were published.

! In 1905, while publishing his

“miracle year” papers on

relativity and other subjects,

Einstein was employed as a

clerk (third class) at the Swiss

patent office in Zurich.

! He remained a clerk in the office

well afterwards -- until he was

appointed “Extraordinary

Professor of Physics” at the

University of Zurich -- in 1909.

Spacetime Preliminaries

! In building a conception of how space and time work, it is first

crucial to define what we mean by such basic concepts as

‘space’, ‘time’, ‘event’, and ‘simultaneity’.

! The elementary building block in this framework is the event.

! An event defines a single point in space and time.

! For our thought experiments, we can imagine that events are

defined by flashes of light which move spherically outwards from

their sources -- for instance, as set off by an electronic light

source.

The Building Block of Spacetime -- The Event

! An event has no duration or spatial extent -- it is a single point in

space and in time.

! A distant observer will note the event when the light from the

event first reaches him or her.

! It is important to note that the light flash itself at the source and

the event of detection are two distinct events.

Spacetime Preliminaries -- Measuring Time

! Fundamental to this picture is that spacetime is filled with

hypothetical observers who can conduct observations and

measurements on their own.

! Each observer carries with him or her a clock (which we will

describe in detail later) to measure elapsed time.

! Using the pulses of light from events, and his or her clock, each

observer can measure time intervals between events.

Spacetime Preliminaries -- Measuring Distance

! Using events, light pulses, and clocks, observers can also

measure distances between spacetime events.

! Consider, for instance, measuring the distance between yourself

and the wall of a room. You send a light pulse out, which defines

event A. A mirror hanging on the wall reflects the light pulse,

which returns to you at event B.

! The distance between you and the wall is easily determined from

d = c t -- the speed of light times the elapsed time, divided by two

(to account for there and back again).

A

B

Mirror

An Important Word of Caution About Spacetime

Misconceptions

! Most students have common hangups when first learning relativity.

! In one hangup, some students do not see any immediate flaw in thelogic, and so accept the basic logic and conclusions of the theory.

! However, the conclusions are simply too “weird” to fully accept, so theycome to believe that because the conclusions are based onmeasurements made by observers, relativity is actually an illusionarytrick played on their instruments. The “real world” behaves differently.

! This misses one of the key logical premises of the theory -- that we knowof space and time only through our measurements. Any presumed “realworld” outside of our measurements cannot be verified by anyexperiment and so does not exist.

! This viewpoint is further refuted by the fact that relativity has real,observable consequences -- sometimes startling. We will discuss someof these later.

Spacetime Diagram

! A key tool used to understand how spacetime works is the

spacetime diagram.

! In this diagram, only one spatial dimension is plotted along one

axis. The other two spatial dimensions are suppressed.

! Along the second axis, time is plotted.

time

space

Question

! Which of the following figures represents the spacetime motion of

a body (shown in red) at rest?

time

space

time

space

Question

! Which of the following figures represents the spacetime motion of

a body (shown in red) moving at constant speed?

time

space

time

space

Spacetime Diagram of a Pulse of Light

! Imagine that an observer sets off a pulse of light at the origin of

our spacetime diagram, O. This defines an event.

time

spaceO

Spacetime Diagram of a Pulse of Light

! Light, traveling at a constant speed, moves outward from the

origin.

! On the spacetime diagram, this is represented by the two rays

shown below.

time

spaceO

Spacetime Diagram of a Pulse of Light

! Note that in the full three dimensions of space, the region

encompassed by the expanding pulse is of course, spherical, like

the rings on the surface when a rock is dropped into a pond.

time

spaceO

From Above Spacetime Diagram

Spacetime Diagram of a Pulse of Light

! The region encompassed by the expanding light front is known as

the future light cone.

! Because nothing can travel faster than light, only those events

lying in the future light cone of an event are in causal contact with

it.

time

spaceO

Light cone

Question

! Which of the events shown below are out of causal contact with

event O?

time

spaceO

Light cone

A CB

Inertial Observers

! Imagine Albert and Heindrik, aboard two rocket ships gliding pastone another at constant speed in distant interstellar space, faraway from any other objects.

! From Albert’s viewpoint, he is at rest, and Heindrik is movingrelative to him.

! From Heindrik’s viewpoint, he is at rest, and it is Albert who ismoving relative to him.

A

H

Albert’s Frame of

Reference

Question

! How might we be able to settle the debate as to who is moving --Heindrik or Albert?

A

H

Albert’s Frame of

Reference

A

H

Heindrik’s Frame of

Reference

The Axioms of Special Relativity Theory

! In a famous paper written in 1905, “On the Electrodynamics of

Moving Bodies,” Einstein posited the following two basic

assumptions :

! The speed of light is constant for every observer.

! The laws of physics are identical for every inertial observer -- every

observer moving at a constant speed.

The Relativity of Simultaneity

! These two seemingly straightforward assumptions turn out to

have profound implications for physics, and for the structure of

spacetime itself.

! Consider two inertial observers, Hendrik and Albert, moving

relative to one another on identical moving trains. Suppose that

each train has marked off an identical, fixed distance, with two

fixed two stationary light detectors there.

H

A

Albert’s Frame of

Reference

The Relativity of Simultaneity

! Each observer is stationary in his own frame of reference, and

sees the other observer moving toward him.

! So far, without the introduction of light, there is nothing new here -

- this understanding of the relative properties of motion was

known all the way back to Galileo.

H

A

Heindrik’s Frame of

Reference

Relativity of Simultaneity

! The critical new ingredient in relativity is the introduction of light,

which is presumed to move at the same speed to all inertial

observers.

! Consider what happens when an observer sets off a flash of light

from the center of the train, halfway between the two light

detectors.

! The speed of light is constant in every frame, so both detectors

are set off simultaneously.

A

Albert’s Frame of

Reference

The Relativity of Simultaneity

! Now, imagine that at the instantaneous event when Heindrik and

Albert pass one another, a light flash is set off at their precise

location.

! Each observer sees himself as the center of the expanding front

of light.

H

A

Albert’s Frame of

Reference

The Relativity of Simultaneity

! This poses an immediate paradox -- how can two different inertial

observers at two different spatial locations both be at the center

of the same sphere -- the one determined by the expanding flash

of light??

H

A

Albert’s Frame of

Reference

The Relativity of Simultaneity

! This poses an immediate paradox -- how can two different inertial

observers at two different spatial locations both be at the center

of the same sphere -- the one determined by the expanding flash

of light??

H

A

Heindrik’s Frame of

Reference

The Relativity of Simultaneity

! According to Albert, the light pulse reaches both of his detectors

simultaneously -- at the same time. However, he sees the pulse

reach Heindrik’s detectors at two distinct times.

! Heindrik, of course, observes precisely the opposite -- the light

pulse reaches both of his detectors simultaneously, but Albert’s

detectors at two distinct times.

! How can these two evidently contradictory conclusions be

resolved?

H

A

Albert’s Frame of

Reference

Resolution of Relativity of Simultaneity --

Gott im Himmel !

! Einstein’s solution to this paradox was bold and radical. Rather

than suggesting a modification to the two basic axioms laid out,

he proposed that our understanding of space and time itself

had to be modified.

! In the case of Heindrik and Albert, it is clear that we must reject

the notion that two events which are viewed as simultaneous by

one inertial observer must also be simultaneous for all other

inertial observers.

! The very concept of simultaneity itself only makes sense

when we also specify WHO is making the measurement.

Relativity of Simultaneity

! Einstein’s resolution to the paradox also clarifies that indeed, both

observers are correct to state that they are the center of the

expanding sphere of light.

! The conflict arises when we inject our prior notions of a fixed

absolute space and a fixed absolute time, independent of all

observers, into the picture.

! Therefore, there can be no such absolute space, and no such

absolute time.

! Einstein’s theory elegantly describes how space and time can be

unified and understood as a single entity -- spacetime.

Relativity of Simultaneity Video

Relativity of Time

! Einstein also analyzed the flow of time. Consider Heindrik and

Albert again, and let us assume both carries a light clock.

! The light clock consists of two mirrors, which light bounces in

between.

! When viewed at rest, in the frame of reference of the clock, the

distance between the two mirrors is fixed, and the speed of light

is constant, so each cycle of the clock has a fixed time.

Relativity of Time

! Consider how Albert sees Heindrik’s moving clock.

Albert’s Frame of

Reference

H H

Relativity of Time

! Albert measures that the light ray in Heindrik’s clock moves for a

further distance than Heindrik.

Albert’s Frame of

Reference

Heindrik’s Frame of

Reference

Heindrik’s Clock

Relativity of Time

! Since the speed of light is the same in all reference frames, and

because Albert sees Heindrik’s light rays move further, Albert

measures the time between cycles on Heindrik’s clock as longer

than Albert’s.

! Consequently, according to Albert, Heindrik’s clock is slow

relative to Albert’s.

Albert’s Frame of

Reference

Heindrik’s Frame of

Reference

Heindrik’s Clock

Relativity of Time

! Framing the same issues from the standpoint of Heindrik instead

of Albert, it is clear that Heindrik also sees Albert’s clock move

more slowly relative to his own.

! This relativistic effect -- that an inertial observer measures a

moving clock as progressing more slowly than his own -- is

referred to as “time dilation”.

Discussion

! Albert sees Heindrik’s clock move slowly relative to his own.

Heindrik sees Albert’s clock move slowly relative to his own.

! Who do you think is correct? Albert or Heindrik? Why?

Putting Relativity to the Test --

Decaying Muons

! At first glance, the predictions of the theory of relativity may seem

so bizarre that they cannot possibly be true.

! The theory has been tested numerous times to extraordinary

accuracy, and each time, the experiments have proven to be in

complete agreement with the theory.

! One of the most amazing experimental tests of relativity makes

use of an exotic, more massive cousin of the electron -- the

muon. It has 207 times the mass of the electron.

! The muon is identical to the electron in all respects apart from its

mass and lifetime. It is essentially the fat cousin of the electron,

with a lifetime of about 2 microseconds.

Decay of Muons

! Muons are created in the upper atmosphere of the Earth during

bombardment by cosmic rays, and stream downwards towards

the surface of the Earth.

! In 1941, physicists measured found that the number of muons

present at 2 km altitude at the peak of a mountain was about 1.4

times that at its base, implying that the majority of muons made

the descent without decaying.

2 km

Muons

Decaying Muons

! Multiplying the speed of light by the lifetime of the muon, we

would infer that the muons could travel only about 0.6 km on

average before decaying into other particles. If this were true,

fewer than 1 in 10 particles could traverse 2 km.

! The resolution to this paradox is quite simple -- according to the

observer on the Earth, the muon’s clock is running more slowly

than the Earthbound clock.

! Consequently, fewer muons decay than one would have

expected based on on the elapsed interval on the Earth.

! Once the time dilation effect of the muons is taken into account,

relativity accounts for just the muon fluxes we observe.

Einstein’s General Theory of Relativity

! From 1905 to 1915, Einstein struggled with the problem of fitting gravityinto his work.

! From the time of Newton, the origin of gravity appeared to be a completemystery. How could it seem to act instantaneously over vast distances ofspace with nothing in between??

! “I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I donot feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; andhypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place inexperimental philosophy. In this philosophy particular propositions are inferred from the phenomena, andafterwards rendered general by induction.” -- Sir Isaac Newton

! Einstein’s vision was to create a theory of gravity which accounted forthis “action at a distance” mystery, while also limiting to his previous workon relativity as a “special” case. Hence, “Special Relativity” and “GeneralRelativity”.

The Principle of Equivalence

! The tremendous challenge to Einstein was to explain the effects

of gravity as a curvature in spacetime. The motion of falling

bodies provided a starting place.

! The special theory of relativity asserted as an axiom that the laws

of physics are the same for any uniformly-moving observer.

! Einstein had a flash of intuition when he realized that there is no

experiment that could distinguish between a stational laboratory

in the gravitational field of the Earth, and a rocket ship

accelerating at exactly 1 g in distant space.

The Principle of Equivalence

! The general theory begins with the assertion that the laws of physics are

the same for any freely-falling observer.

! This principle became known as the principle of equivalence.

! Because the motion of the rocket ship can be understood as a bending

of the trajectory of an inertial observer, it suggested to Einstein that our

understanding of gravity can be framed as a purely geometric problem.

Earth Rocket

gravity

Acceleration of rocket

Greene Video

General Theory of Relativity Summary

! The whole picture of the General Theory can be summarized by a

simple viewpoint.

! Matter tells spacetime how to curve.

! Curved spacetime tells matter how to move.

! The mathematics of computing how spacetime becomes curved,

and determining how a body moves in the curved spacetime is

quite complex, but the basic idea remains simple.

Predictions of The General Theory --

Bending of Starlight

! If the Special Theory of Relativity seems strange at first, the

General Theory may seem downright impossible.

! The first truly new prediction of the theory to be confirmed (during

the eclipse of 1919 by Eddington) was the bending of starlight.

Gravitational Lensing Due to A Cluster of

Galaxies

! Spectacular instances of extragalactic gravitational lensing have

been observed over the last 20 years.

Prediction of Relativity Theory -- Gravitational

Redshift

! Imagine that we send a laser

light beam inside the

accelerating rocket ship.

! A detector placed at the top of

the rocket ship will be moving

away from the source, and so

there will be a Doppler shift of

the light beam toward the red.

! By the equivalence principle, an

identical laer light ray sent

upwards in the laboratory in a

gravitational field will experience

a redshift -- a gravitational

redshift.

Rocketlaser

laser Lab

Gravitational Redshift and

Gravitational Time Dilation

! Earlier we discussed one method of constructing a clock based

on two mirrors and a light beam.

! A second clock can be constructed by simply measuring the

frequency of a laser beam of light. Blue light has a higher

frequency than red light, and so oscillates more times per second.

! From the existence of a gravitational redshift effect, it follows that

for two observers at two altitudes in a gravitational field like that of

the Earth, the clock at a lower altitude moves more slowly

than that of an observer at a higher altitude.

Pound-Rebka Experiment

! This prediction of General Relativitymay seem astonishing, though it isimportant to realize that for relativelyweak gravitational fields like theEarth’s, the effect is incredibly tiny --though still measurable.

! In 1959, Pound and Rebkaconstructed an nuclear experiment inthe physics building at Harvard usinga radioactive source (57Fe) in thebasement, and a moveable detector22 meters above it.

! Their measurements confirmed thatthe gamma rays emitted by the ironwere indeed redshifted, andconsistent with the gravitationalredshift effect predicted by Einstein.

Relativity and GPS

! GPS satellites require extraordinarily precise timing -- to withinnanoseconds per day -- in order to obtain accurate measurements oflocations on the Earth.

! The time dilation effect of Special Relativity due to the moving satellite,AND the gravitational redshift effect of General Relativity are absolutelycritical to the GPS system.

! In fact, taking into account both the gravitational redshift effect and thetime dilational effect predicts clocks on the ground are slower than on thesatellite, by about 40 microseconds per day.

! 40 microseconds per day may not seem like a large effect, but withoutrelativity, it would amount to an error of nearly 10 km per day -- whichwould make the GPS system completely useless.

Black Holes

! If the gravitational field of a body becomes sufficiently strong, not

even light can escape from it.

! This body is known as a black hole. The boundary beyond which

there is no escape is known as the horizon.

Cygnus X-1

! The first strong case for the detection of a black hole was made in

the Cygnus X-1 x-ray emitting system in the 1970s.

Black Hole Physics

! Imagine what it might be like to make a trip to a black hole.

! Alexis uses a rocket ship to move close to a huge black hole, thesize of a galaxy. Bettie stays nearby and sends a radio messageto her. Alexis radios a message back to her.

! As Alexis nears the horizon, Bettie sees her signal become moreand more redshifted. Alexis’s clock slows down further and furtheras she approaches the horizon.

! Conversely, Alexis sees Bettie’s signal become more and moreblueshifted -- until it is not a radio wave at all, but infrared,visible… eventually become high-frequency gamma radiation.Alexis sees Bettie’s clock move faster and faster, and indeed theentire universe moves in fast forward.

Black Hole Physics

! In addition, as she nears the horizon, only those photons fromAlexis moving nearly vertically have a chance to escape; the onesmoving horizontally begin to fall into the black hole.

! This means that Bettie sees the signal from Alexis become moreand more highly-beamed as she moves further in.

! Alexis, on the other hand, sees the sky overhead begin to darkento absolute black apart from a narrow cone above her.

Radio waves

Beyond the Horizon

! While Bettie will never see Alexis move behind the horizon, Alexisactually falls behind the horizon in a finite time.

! What happens behind the horizon, and in particular whathappens as one approaches the center of the black hole is amatter of intense speculation, but is not understood in the currentframework of physics.

! According to General Relativity, all of the mass of the black holeis concentrated in a single point of infinite density -- thesingularity. This is in fact a breakdown of the theory itself, and soGeneral Relativity cannot be used to understand what goes on atthe location of the singularity.

Next Week : More Black Holes and Galaxies

! What would happen if two regions of spacetime were tied

together in a “wormhole”?

! What do we think happens at the very smallest scales in which

gravity and quantum effects both become important?

! And is there a black hole at the center of the Milky Way?