Lecture 10 : Relativity, Black Holes

Science 3210 001 : Introduction to Astronomy

Lecture 10 : Relativity, Black Holes

Robert Fisher

Items

Nathan Hearn guest lecture on dark matter on April 20th. Lunch in the loop (on me) with Nathan following the lecture at Frontera Fresco for anyone who wants to join us.

Second midterm next week. Covers the material in between the last midterm and the end of today’s lecture.

Adler Planetarium field trip on May 4th - $16/person. Sign up today!!

Final projects due May 11th, along with a short (5 minute) presentation that day.

Final Project

Your final project is to construct a creative interpretation a scientific theme we encountered during the class. You will present your work in a five 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 if you said that it was your interpretation of the big bang cosmological model AND you could also demonstrate mastery of the basic astrophysics 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 Andromeda system. Paint the night sky as viewed from an observer about to fall behind the horizon of a

black hole. Write a short science fiction story about the discovery of intelligent life in the universe.

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 wavelike behavior of light argued that light must be a wave like other waves known at that time -- water waves, sound waves, seismic waves, and so on.

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

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

If the aether did exist, it must carry physical properties like mass and momentum, just like a pond. If it has physical properties, it must be detectable.

If that is all true, then where was all of the evidence for the existence of the 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


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


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

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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 are dramatically different from the classical viewpoint.

The concepts which Einstein hit upon are radically different than both the classical point of view, and our own everyday experience.

Because relativity is so radically different from our everyday experience, Einstein had proceed using razor-sharp logic, starting from basic axioms.

This reasoning was often applied to extraordinary situations known as thought experiments (sometimes gedankenexperiment from 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 uses a word devoid of meaning. This is a truth that has been long proclaimed by all who have reflected on the question, but one which we are too often inclined to forget… have shown elsewhere what are the consequences of these facts from the point of view of the idea that we should construct non-Euclidean and other analogous geometries.” -- Henri Poincare, Science and Method, 1897

Recently more controversial suggestions have been made that Einstein’s first wife Mileva Maric contributed substantially to the relativity, and even that other scientists came upon E = m c2 independently.

What remains true is that the whole of relativity theory owes more to one single individual more than any other major theory in modern physics.

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

BMirror

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 the logic, and so accept the basic logic and conclusions of the theory.

However, the conclusions are simply too “weird” to fully accept, so they come to believe that because the conclusions are based on measurements made by observers, relativity is actually an illusionary trick played on their instruments. The “real world” behaves differently.

This misses one of the key logical premises of the theory -- that we know of space and time only through our measurements. Any presumed “real world” outside of our measurements cannot be verified by any experiment and so does not exist.

This viewpoint is further refuted by the fact that relativity has real, observable consequences -- sometimes startling. We will discuss some of 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


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


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


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.

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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 past one another at constant speed in distant interstellar space, far away from any other objects.

From Albert’s viewpoint, he is at rest, and Heindrik is moving relative to him.

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

A

H

Albert’s Frame ofReference

Question

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

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H


A

H

Heindrik’s Frame ofReference

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

AAlbert’s Frame ofReference


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.

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A


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.

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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.

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A



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



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



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?

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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.


H H

Relativity of Time

Albert measures that the light ray in Heindrik’s clock moves for a further distance than Heindrik.



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.



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 gravity into his work.

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

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

Einstein’s vision was to create a theory of gravity which accounted for this “action at a distance” mystery, while also limiting to his previous work on relativity as a “special” case. Hence, “Special Relativity” and “General Relativity”.

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 Relativity may seem astonishing, though it is important to realize that for relatively weak gravitational fields like the Earth’s, the effect is incredibly tiny -- though still measurable.

In 1959, Pound and Rebka constructed an nuclear experiment in the physics building at Harvard using a radioactive source (57Fe) in the basement, and a moveable detector 22 meters above it.

Their measurements confirmed that the gamma rays emitted by the iron were indeed redshifted, and consistent with the gravitational redshift effect predicted by Einstein.

Relativity and GPS

GPS satellites require extraordinarily precise timing -- to within nanoseconds per day -- in order to obtain accurate measurements of locations on the Earth.

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

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

40 microseconds per day may not seem like a large effect, but without relativity, it would amount to an error of nearly 10 km per day -- which would 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.

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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, the size of a galaxy. Bettie stays nearby and sends a radio message to her. Alexis radios a message back to her.

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

Conversely, Alexis sees Bettie’s signal become more and more blueshifted -- 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 the entire universe moves in fast forward.

Black Hole Physics

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

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

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

Radio waves

Beyond the Horizon

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

What happens behind the horizon, and in particular what happens as one approaches the center of the black hole is a matter of intense speculation, but is not understood in the current framework of physics.

According to General Relativity, all of the mass of the black hole is concentrated in a single point of infinite density -- the singularity. This is in fact a breakdown of the theory itself, and so General Relativity cannot be used to understand what goes on at the 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?

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Lecture 10 : Relativity, Black Holes

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