CosmologyHubble’s Law

Equivalence PrincipleCurvature of Spacetime

Einstein’s Field EquationsRed Shift

Black Holes

“

Wheaton HS Principal to Edwin Hubble at 1906 graduation

“Edwin Hubble, I have watched for four years and I have never seen you study for ten minutes." He then paused for what was an awful moment for Edwin, and continued, "Here is a scholarship to the University of Chicago.”

Edwin Hubble

1889-1953

astronomer

discoverer of the expanding universe

all around guyFingerprints on everything done in astronomy

ex-boxer, ex-lawyer, ex-basketball player, ex-high school teacher, ex-Rhodes scholar

• graduate of Wheaton, IL high school, star of the Big 10 Champion ship (1910) & National Champion (1909) University of Chicago basketball team

• Rhodes scholar, returned with Law degree and practiced briefly in Kentucky, resigned to teach high school and then returned to graduate school in astronomy

front-line Lt in WWIspent brief post-war time in Cambridge, but Eddington was busy (later)mustered out of military as Major…directly to the new Mt. Wilson Observatory: 60” and 100” optical telescopehe wanted to work on nebulae: “He was sure of himself - of what he wanted to do, and how he wanted to do it.”He was the discoverer of one of the most remarkable regularities in the universe.

1909 National Championship BB, University of Chicago

basically: spherical, elliptical, bar, and irregular

major accomplishments1922-1926: Hubble classification scheme for galaxies

there were others, but it’s acknowledged that Hubble was a writer of significant capability and that his papers carried a power of persuasion just through their English and reasoning.

Measurement of distances is Complicated in Astronomy

In 1920’s Henrietta Leavitt discovered the Cepheid variable stars, whose brightness pulsates with periods which are directly related to their absolute luminosities

Knowing the absolute luminosities, astronomers can determine the distance from 1/r2

http://hyperphysics.phy-astr.gsu.edu/hbase/astro/cepheid.html#c2

they are the yardsticks for deep-space measurements - distances to galaxies containing Cepheids can be determined

Hubble Space Telescope (HST)

Observation of variables in M100…a member of the Virgo cluster of galaxies - too faint to have been observed from the ground

Hubble’s Cepheids were everywhere

M31, Andromeda

2900 thousand light years

M33, Triangulum

3000 thousand light years

NGC 6822, Barnard’s Galaxy

1700 thousand light years

from nice web Messier Object site

Where were “nebulae”? In our galaxy?

Nope. In 1925 or so, Hubble found Cepheids in these nebulae - and established their distances from us

…finalized the speculation about whether “nebulae” were in our galaxy…indeed, that there were other galaxies beyond ours.

distance units:

lightyear (ly)= distance light travels in a year, 9.46 x 1017cm

Astronomical unit (AU) = average earth orbit, 1.496 x 1013 cm

parsec (pc) = 3.26 ly, 3.086 x 018cm; also kiloparsec (kpc) and Mpc (distance at which 1AU subtends one second of arc.

a little doppler’ll do ‘ya

you’ve all had the experience of listening to the sound of a moving object change pitch

vdthe motion toward the left means that R is seeing more peaks in a given time than L

vs

If not sound, but light and if the relative speeds are relativisitc, there is a Relativistic Doppler Shift:

instead of frequency, use wavelengths:

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λd =λ'1+β1−β

http://cas.sdss.org/dr4/en/tools/explore/obj.asp?id=588848899898278062

star spectraStars can be very colorful

they have colors due to the atomic and molecular transitions and the absorptive media between the stars and us...

Here is an arbitrary star from the Sloan Digital Sky Survey...and the spectrum that was produced by that telescope

can identify elements based on their wavelength patterns

Hubble’s lawHubble had 2 tools: spectroscopy and the Cepheid distance measure

with his yardstick, he could make better measurements…and he did

with his persuasive description, his landmark work was believed almost immediately

Using spectra of stars in galaxies, he could identify their chemical elements

and he got a surprise.

http://www.astro.ucla.edu/~wright/doppler.htm

Ca H I Mg I Na I

eg, Doppler shifts at work:

Wavelengths shifted to longer - “redshifted”

meaning objects moving away from observer (Hubble)

His remarkable conclusion:

All galaxies are moving away from us!

Note, there is a correction/reinterpretation to this that is based on General Relativity...more in a minute

Hubble’s law states that the velocity of recession is proportional to the distance of separation:

v = H0 d, where H0 is called the Hubble Constant…a measurable of some importance

currently, it is determined to be H0-1 = 1.5 x 1010 y

The farthest objects appear to have enormous velocities, which astronomers characterize by their “redshift”, Z.

The record holder is a quasar with an amazing Z = 4.9.

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Z ≡λd

λ−1=

1+β1−β

Using this object, we can estimate (nonrelativistically) the radius of the universe now:

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RU = vH0

=vc

cH0

=0.94⋅(3×108m/s)(1.5×1010y)

≅1.3×1026m

v/c

z

boom

That everything is streaming away from everything

Suggests that:• the universe started from a single point in time and

space• which we now call the Big Bang

Hubble’s law is not the only evidence• however, his idea, and subsequent confirmation and

extension suggest for the first time that the universe is not static and that it had a beginning

This confused Einstein for a whileand caused him to adjust his original General Relativity Theory

• to his later dismayBut he never would have believed the late-20C conclusions on this score

stay tuned.

the Beginning.

Suppose a car is moving away from you at 50mph

T = x/v, right?• at 50 miles away, how long has it been traveling?50 mi/50 mi/h = 1 h• at 100 miles away, how long has it been traveling?100 mi/50 mi/h = 2 h

Or...v = x(1/T) ...for constant v.• Let’s call H = 1/T, then

v = H x

That’s the Hubble relation:so, 1/H is a measure of the time that the universe has been “traveling”: H0

-1 = 1.5 x 1010 y• the age of the Universe.

This is the beginning of quantitative Cosmology.

the general theory of relativityWhat’s the “special” in “special” relativity?

the physics of inertial framesnot for accelerating frames or gravitation

Non-inertial frames are maybe more common…Frames in which an observer is rotating

• a rotating frame – very different impressions for observers inside and outside of such a frame: a passenger in a car going around a curve

what does the passenger feel? a force, pushing her against the doorwhat does an outside observer see? no force

until she hits the door…she’s just moving according to Newton’s 1st law

seat-back-end friction provide the necessary centripetal force in the car

We tend to treat the effects of gravity differently from the effects of ‘everyday’ or inertial forces

With gravity, we tend to think that we’re in an inertial frame and something outside reaches in and grabs us…by virtue of our mass

• Einstein challenged that point of view…again, with simple questions

Equivalence Principle

Einstein’s happy ideaIn a local sense, it is not possible to

distinguish between a frame “at rest” in the presence of a gravitational field and a

frame being uniformly accelerated in empty space

this actually makes the gravitational force somewhat different from other forces

gooooiiiing up! (or down)The principle of equivalence means that if it happens for gravity, it happens with inertial forces - and visa versa

the idea that Einstein called the “happiest thought of my life”• reputedly after interviewing a house painter about his trip

after he fell from a substantial heighttwo closed rooms…

one on Earth & one accelerating up at g far away from Earth

gThe observers both make measurements of the ball and find that they both accelerate to the floor with 9.8m/s2

What’s the difference? So, by thinking about simple things, Einstein has come up with another relativity statement:

There is no measurement that can be performed to distinguish a non-intertial, accelerating frame from an inertial frame embedded a gravitational field

sooo… Remember that from the roadside, the passenger doesn’t appear to experience a force. Is there a frame in which gravity can be then similarly “transformed” away?

• If WE watch the ball from a frame in which the RH elevator appears to be going up at 9.8m/s2…we’ll see the ball stationary

• If WE observe the LH ball from a frame which is in free-fall relative to earth, we’ll see the ball stationary

We can “transform” the earth’s gravity away and that’s what NASA does in roller-coaster airplanes to train astronauts on how to heroically throw up.

the Real Equivalence Principle

Einstein generalized:All of the laws of physics are

• the same in a gravitational field as in an inertial frame undergoing uniform acceleration.

that means electromagnetism too: no optical experiment can distinguish gravity from inertial acceleration

So, if you can think of something that ought to happen in one circumstance, then it must happen in the other one.

you notice that for arguably one of the most technically challenging pieces of physics ever, the basics come from only logic and words

• that’s Einstein’s (& Feynman’s) gift…thinking simply and therefore thinking big.

a real benderBack to our upwardly accelerating guys… now, observing a flashlight burst through a hole in the side

from the “lab” (the rest frame of the flashlight), we see that the beam of light travels horizontally

but from inside the elevator frame, the beam curves to the floor, following a parabolic path…

So, the equivalence principle demands that the same thing happen in a gravitational field

and that it be possible to transform it away into a freely falling frame…

gg

g

Given our realization that mass and energy are equivalent, perhaps this isn’t as much of a surprise as it might have been?

not just a spacey theorist…

Einstein knew how to predict a measurement

In 1911, he calculated that light would bend around our sun by 0.875’’ of arc at grazing incidence

apparent direction of star

actual positionlight trajectory

During a total eclipse, this could be measurable by comparing with 6 months before or after…but wasn’t doable experimentally at that time

That was lucky, as he’d not realized that he had one other thing to take into account to get the right deflection - a complication that would take him many years to figure out.

give it to me straight…what could be straighter than a beam of light? and yet, with some simple thought…followed by some hard thought:

• Either light can’t provide a ‘straightedge’ for regions where there is gravity

• ie, where there are massesor the notion of what’s “straight,” or the shortest distance between two points, needs modification

So, geometry (“straight”) and dynamics (“forces”) come to have much to do with one anotherthat’s what took Einstein 5 more years after his first “happy thought”he had to learn the entirety of the new “Riemannian Geometry,” and for that he relied on his college friend, Marcel Grossmann

a simple thought experiment suggests why geometry and gravity are connected Consider a rotating wheel… with S’ riding along the rim & S in an inertial frame in which the wheel

rotates

They both set about to measure the relationship between radius and circumference of the wheel with identical straight, rigid rulers laid end to end.

S dutifully measures r and C: he finds that C = 2πr

S’ measures it the same way. S notices that the rods placed end to end are Lorentz-contracted

so it takes more of them to get around the circle than for his measurement

so S’ concludes that C’ > 2πr ! NOT an expectation according to “regular” Geometry according to Euclid

She would conclude that she’s not in a non-Euclidean geometry

r

SS’

Home: a non-Euclidean Geometry where the shortest distance isn’t a straight line

When you fly from North America to Japan, what route do you take?

Straight across the Pacific along a constant latitude?

No, you fly on a “great circle” route…because it’s the shortest distance over the surface of a sphere

A clear example of how the shortest distance between two points is not a straight line

The distance between two points on the surface of the earth for “coordinates” θ and φ:

Δs2 = R2Δθ2 + R2 cos2θ Δφ2 …like Δs2 = Δx2 + Δy2 for flat space

notice that it mixes up the two coordinates in the second term.

Suppose you and a friend meet at the equator, turn back to back and walk along the equator, each for a thousand miles

then, each turns toward north and walks parallel to the other

you meet at the pole

say what? parallel?

On the surface of a sphere, parallel lines meet. Euclid’s 5th Axiom is not general

Suppose you lay out a large triangle…and measure the interior angles

you find them to be less than 180

say what?

All examples of non-Euclidean geometries…the sort that General Relativity proscribes for the universe

The degree of curvature is created by the mass-energy content in the universe

and it is the curvature that dictates how matter will move - it will follow a geodesic in spacetime

In General Relativity, the gravitational field is gone and replaced by the curvature of spacetime

don’t muck with it!

How curvature is generatedThe nitty gritty of Einstein’s “field equations”

• 8π TμνGμνHere, the equation says energy terms = geometry terms

• 10 non-linear differential equations to be solved simultaneously

They say that (energy/momentum) → curvature• free trajectories in space will follow the geometry• giving the shortest distances between two points...not

necessarily “straight” lines

This equation does not permit a stable universe - it has to expand. Einstein thought: “silly…”,

so he included an arbitrary term to stop that expansion: the “Cosmological Constant”

• then he removed it when Hubble’s work was announced.He called it the worst mistake of his life...but stay tuned. It’s back.

cute dimples

Curvature of spacetime was missing from his original light-bending calculation• when included for the curvature of spacetime around the sun, then he got twice the original predictionthis realization coincided with the end of WWI and so then an expedition was mounted by (Sir) Arthur Eddington in 1919 to take advantage of an eclipse due to occur in western Africa• the prediction was right on

I do not doubt any more the correctness of the whole system, whether the observation of the solar eclipse succeeds or not

eclipse announcement at scientific meeting, 11/07/19: instant celebrity,

11/08/19

caption: “A new great figure in world history: Albert Einstein, whose investigations signify a complete revision of our concepts of Nature, and are on a par with the insights of a Copernicus, a Kepler, and a Newton.”

cover of December 14, 1919 issue of Berliner Illustrirte

New York Times, November 10, 1919

now recovered from exhaustion and photogenic: 1920

New York Times, December 3, 1919

gravitational lensingToday, the dramatic effects of light bending are observed in the form of gravitational lensing

this is the bending of light around a very massive object, like a large galaxy

massive galaxy

remote object

apparent direction

apparent direction

“Einstein Cross” - 4 images of a quasar“Einstein Cross” - 4 images of a quasar

FlatlandSo…how do we think about this?

remember Hubble’s ideathink about our situation - we’re 3 dimensional beings trapped inside of an unsketchable 4 dimensional world

• we can make analogy with 2 dimensional beings trapped inside of a 3 dimensional world…then we can make drawings

• this was done in 1845 by Edwin Abbott in his curious novel Flatland

In this world, say on a sphere, the horizons of Flatlanders are limited

there is no concept of “up” or “down”

only back and forth, to and fro

Their world is finite in volume…and infinite in extent

if they start walking…um, sliding,…they never find an edge and indeed they might come back to where they started - limitless

Certainly their world is non-Euclidean

Suppose they measured that all spots on the surface got further away from one another in time…

They would have to conclude that the (unobservable) radius of their world is getting bigger in time

remember Vulcan?Remember that there was a long-standing problem with the orbit of Mercury

the issue was that the perihelion (the distance of closest approach to the sun) precessed...advanced by a tiny amount...the amount is only 574” of arc per century and most could be accounted for by the influence of the other planets...

However, there was a small chunk - 43” of arc per century - which could not - so, there must be another planet...Vulcan.

In 1916 Einstein set about to calculate it as an effect of the curvature of spacetime introduced by the sun

He found it to be precisely what was needed

and was reported to have been so excited by this that he had heart palpitations at the realization

This is why he KNEW that GR was correct...before the eclipse expeditions

Earth too has a precession which GR would predict to be about 10% of Mercury’s...measurements are consistent with what’s expected, but hard to do.

heavy dopplerSuppose a light source on the ceiling of the elevator emits light…gathered by our guy with a frequency meter

He’ll catch up to light with an increasing velocity and will observe a Doppler-shifted frequency which is

fd = f (1+ v/c) (remember, if v<<c)

Let’s suppose that it takes time T for the light to reach the meter’s position in an inertial frame...T = L/c

If the observer is accelerating at a...then after time T

a = vT and we get that

fd = f (1+ aL/c2)

So, by the principle of equivalence,replacing a with -g, the same situation on should exist on earth, we get that there should be a Gravitational Redshift (wavelength gets longer, frequency gets shorter ) for light going away from a mass

This means that two identical clocks, separated to different elevations in a gravitational field will lose their synchronicity - they will keep different times

a

L L

black holesWithin a year of Einstein’s General Relativity publication

various people solved it for a variety of scenarios• one of the more surprising solutions was for the

curvature around a non-rotating, massive sphereCalculated by Karl Schwartzchild in 1916The issue was: at what radius around a star is its escape velocity equal to c

normally, you ask a different question - namely, what is the escape velocity.

recall, it is

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vesc=2GM

Rfor vesc = c,

the corresponding radius is:

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c=2GMRS

so: RS =2GM

c2

The only circumstance in which one could conceive of such a thing must involve stars or galaxies (eg, for the earth, RS is less than a centimeter...for the sun, ~100km).

The conditions under which one could achieve such a condition, then, depends on a critical density of matter.

RS

falling for youA general solution to Einstein’s equations includes a relationship for the interval, which here is a short element of spacetime (recall the distance between two points on earth before):

Δs2 = g00c2Δt 2 − g11Δr2 − r2Δθ 2 − r2 sin2θΔφ2

as in special relativity, the space pieces are negative and the time piece is positive. The last two terms (Δθ & Δφ) refer to the “longitude” and “latitude” of the solution - not interesting here.

Schwartzchild’s case is interesting… suppose ds is measuring a patch of spacetime a distance r from a very massive stellar object. There he found that the components of Δs2 were:

€

g00=1−RS /r ; g11=−1

1−RS /r

RS is the Schwartzchild radius from before…

Notice some interesting things:

1. r is an arbitrary distance…RS depends on the body. So, RS is usually inside of most objects. Very massive stars can actually contract to the point where their size becomes smaller than RS - this is a black hole.

2. when r = RS, notice that g00 becomes zero - so the time part of the interval disappears - so, time has in effect stopped - clocks will run infinitely slowly, or anything leaving has an infinite redshift;

notice that g11 has blown up, so you can no longer use normal coordinate descriptions of this particular radius. This is called the “Event Horizon” - a place where light cannot get out

3. when r < RS…then the time and the space terms change sign - so, the time piece becomes space-like (negative) and the space piece becomes timelike (positive). This is very peculiar behavior

There are other odd things…at r = 3/2 RS, then light cannot escape, but goes into circular orbits - this is called the photon sphere. Beyond, that, light can escape.

Stranger than fiction? maybe…but it’s nonfiction.

rΔs

RS

€

c=2GMRS

so: RS =2GM

c2

yes, Virginia…there are black holes

There are thought to be at least 3 kinds of black holes:

1. stellar - ie, individual stars

2. miniature - speculation, no evidence…some thought of production in particle accelerators

Cygnus A…the brightest X-ray source, also the first accepted candidate for a black hole first positive evidence of a stellar black

hole as a binary…optical partner brightens due to gravitational (micro) lensing…when it passes behind the black hole

In some sense, black holes are like the dog that didn’t bark in Sherlock Holmes

can’t see them, so evidence is from something else nearby

Black holes have an appetite - for matter

So, should it have originally been a partner in a binary star system, then once it has contracted, it will begin to suck off material from its companion

the accretion of material into the hole will cause so much heating that there will be radiation (X-ray, radio) given off which is then visible…sometimes redshifted enough to be definitive.

QuickTime™ and aCinepak decompressor

are needed to see this picture.

Quasars, Active Galactic Nuclei (AGN)3. supermassive - masses of billions of stars, probably at the center of every galaxy where the

density is above critical due to the close packing of stars:

jet from M87 galactic center…evidence perhaps of a massive, spinning black hole. Charged particles are ejected and radiate.

evidence of the influx of matter into a black hole a the center of M84. The doppler shifting of matter coming towards us and away from us in the spectrum is shown.

Quasars (“QUAsiStellAR” objects) were initially discovered as hugely energetic radio frequency emissions - 1000 galaxies’ worth per second, but nearly point-sized

the speed of influx is so large, that the emission is many times more energy than the galaxy itself

Suggestive of being extremely far away

The mechanism now is thought to be a supermassive black hole at the center of these galaxies

Those with less- massive black holes are called Active Galaxies, and the black hole, an Active Galactic Nucleus

Our galaxy has not got enough gas to be captured by our black hole.

Chandra X-Ray Observatorythis year: eclipsed black hole, NGC 1365

extras...

a little Doppler do ‘yaSuppose you have a source of sound…and a detector

the locations of the peaks in the intensity are uniform and exactly what was played by the source, the same for each microphone

now, suppose that the RH microphone is moving toward the source at velocity vd…the speed of the wave is independent of vd - it’s the speed of sound in air, w

vdthe motion toward the left means that R is seeing more peaks in a given time than L

number of additional wavefronts that R sees in time t is:

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additional wavefronts=vdtλ

or, number per second vd / λ, so the frequency changes by that amount

€

fd =f +vd

λsince λ = w/f, then vd/ λ = vd(f/w), so

€

fd =fw±vd

w⎛ ⎝ ⎜

⎞ ⎠ ⎟ =f 1±

vd

w⎛ ⎝ ⎜

⎞ ⎠ ⎟

This changing of frequency due to relative motion between a source of sound and the observer is called the Doppler Effect…familiar with sirens, train whistles, etc.

(where the bottom sign is for the observer moving away, top sign, toward)

going and comingof course, if the picture is view from the perspective of the listener… then the source of sound appears to be moving and the same phenomenon is observed, but with slightly different results…they are different physical phenomena

let T = 1/f be the period of the wave: the time that it takes for one cycle

during this time, the source has moved by vsT

so, the wavelength of the sound arriving at R is not w/f but w/f - vs /f and the frequency heard is

vs

€

fd =wλ d

=w

w / f −vs / f= f

ww−vs

= fw

wmvs(where, again, the bottom sign is for the source moving away, top sign, toward)

So:

When the source is moving toward the observer, the frequency gets higher (wavelength shorter)

When the source is moving away from the observer, the frequency gets lower (wavelength longer)

The Doppler effect is used in all sorts of applications: police radar…

lightCrucial is the difference that having a medium makes-get two different results for two different frames.

What if the velocities of source/receiver are relativistic and light is the disturbance?

Using our old designation of S and S’ frames

Suppose the source ( S’) emits light of frequency f’ in S’.

The time between pulses in S’ is T’ = 1/f’

…to an observer (S) watching S’ move at velocity u, the frequency of these pulses is observed to be f and the time between them, T.

€

T =T'

1−u2 /c2

These two times are related by the Lorentz Transformation…

Going through a similar analysis as before, but now with the transformed period gives:

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fd =f 'c−uc+u

rewriting and modifying for wavelengths:

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λd =λ'1+β1−β