Re-cap from last time (General Relativity):

• Gravitation & Acceleration are equivalent

• Straightest path is path taken by light and in general it is curved

• Time runs slowly and lengths are contracted in a high gravity environment

• Light escaping from a high gravity environment is redshifted

• Light falling into a high gravity environment is blueshifted

• Is gravity really a force as Newton envisoned it?

Outline - Feb. 18, 2010

• Is gravity a classical force (like Newton said)?

• Warped “spacetime”

• Tests of Special and General Relativity (Ch. S3, 449-465)

• Black Hole basics (pg. 595-601)

• Should you believe in black holes? How would you go about finding one?

Gravity: Newton vs. Einstein

“Spooky” Action at Distance (Newton):

Sun “tugs” on the planets and pulls them around in their orbits, like a string tied to a whirling kitty toy. But how?? Where’s the string??

If the sun disappears, the planets should instantly “fly off” into space on straight line trajectories. But information can’t travel faster than c, so how can they know “instantly” that the sun is gone?

General Relativity (Einstein):

Matter/mass tells space how to CURVE

Curvature of space tells things how to MOVE

Information about changes comes from gravity waves (ripples of curvature in spacetime) that travel at the speed of light

2-dimensional, rubber sheet analogy to General Relativity(Note, in reality this is 4-dimensional in the universe.)

If you start a marble rolling across the rubber sheet in a straight line, what happens?

Einstein’s view of orbits

Planets move along their natural curves in space, caused by the mass of the sun “warping” space. Now what happens if you pluck the sun out of the center of the solar system?

Who’s right? Einstein or Newton?

Classic tests of General Relativity:

1. Precession of the perihelion of Mercury

2. Gravitational lensing

Einstein’s theory of gravity gives the same answers as Newton’s theory in the limit of extremely weak gravity. They only differ where gravity is particularly strong (e.g., nearby very massive objects such as stars, centers of galaxies, galaxy clusters)

Precession of the Perihelion of Mercury

This effect is noticeable for the innermost planets.

Every century the total change in the location of “periheilon” for Mercury is 43 arcseconds (= 0.012 degrees). For Venus the change is 8.6 arcseconds (= 0.002 degrees), and for Earth the change is 3.8 arcseconds (= 0.001 degrees)

Gravitational Lensing

Light has to follow the curved path of space around a massive object (like the sun).

The closer the light passes to the sun, the more it is “deflected” by the curved path.

Gravitational lensing by the sun first detected in 1919, validating General Relativity over Netwonian gravity.

Gravitation lensing can create multiple images of the same object

We actually see this in Nature!!

“QUASAR” named QSO 0957+561

Two images of the same object, discovered in 1979

Einstein “Cross”; 4 images of the same object

Gravitational lenses make REALLY bad eyeglasses

The images are highly DISTORTED!

Black Hole Basics:Warped Spacetime

Black holes are so “compressed” that spacetime nearby a black hole is warped to the point of being cut off from the rest of the universe.

Black Hole Basics:Escape Speed

Black hole is a region of space where gravity is so strong that not even light can escape.

Minimum speed of escape from the gravitational pull of an object with mass M is:

Vesc = ( 2GM / R )1/2

Imagine compressing the earth.

How would the escape speed from the surface of the earth change?

Radius Vesc

R = Rearth = 6500km 11 km/s (Saturn V rocket)

R = 0.25 Rearth = 1600 km 22 km/s

R = 1 km 890 km/s

R = 1 cm 300,000 km/s = c

Black Hole Basics:Escape Speed, II

Compress the earth to a radius smaller than a grape and Vesc > c, so not even light could escape off the surface of the planet!

Note: black holes are not necessarily infinitely dense with the mass contained within a physical radius of zero size (a “singularity”), they just need to be sufficiently dense that Vesc > c.

Note: the FARTHER you are from a black hole (the larger “R” is in the equation), the LOWER is Vesc.

How close can you get to a black hole and still escape?

Black Hole Basics:Schwarzschild Radius and Event Horizon

black hole

Schwarzschild radius is the distance from the black hole at which Vesc = c. If R < RSch, Vesc > c. If R > RSch, Vesc < c.

Event Horizon is the effective “surface” of a black hole. It is a sphere with radius equal to RSch that defines the region of space within which no event can be seen, heard, or known by an outside observer.RSch = 3 x ( Mbh / Msun ) km

Black Hole Basics:Don’t be frightened…

Black holes are NOT cosmic vacuum cleaners!

(but they are cosmic “heaters”…)

What happens as you travel toward a black hole?

Release a space probe near a black hole and watch it fall in.

The space probe has a clock on board and it sends out a radio signal to a stationary observer.

As distance between probe and black hole decreases:

• gravitational pull increases, spacetime become more warped

• as seen by an outside observer, the clock on probe ticks more and more slowly

• wavelength at which and outside observer receives the signal gets longer (have to tune to lower and lower frequencies)

Travel to a Black Hole, II

As seen by an outside observer:

• as probe nears event horizon, time on probe comes to a stop (takes an infinite amount of time to pass between events on board the probe)

• everything on probe is frozen in time and it takes an infinite amount of time for the probe to cross the event horizon

• black holes are the ultimate in length contraction, time dilation, and gravitational redshift

Travel to a Black Hole, III

Suppose somebody was crazy enough to go for a ride on the space probe…

Very close to the event horizon, the occupant of the space probe:

• would notice nothing strange as far as length and time on the probe (physics appears normal within the probe as you approach the black hole)

• would say time is running faster and faster outside the probe, and all light is blueshifted (falling toward the black hole)

• can only see in the direction from which the probe came, not in the direction of travel (the direction of the black hole)

Do Black Holes Always Chew their Food?

Will the space probe actually survive crossing the event horizon? Maybe, maybe not.

We have to consider tidal gravitational forces acting across the probe. Gravitational pull near the black hole increases so rapidly that the nosecone may experience a greater pull than the booster rockets and could be turn apart.

Properly, it’s the curvature of space (caused by the black hole) compared to the size of the space probe that truly matters.

Surviving the initial fall into a black hole…

Two extreme scenarios:

• Curvature of space at the event horizon is large compared to the size of the probe (BH with M = 1 Msun has RSch = 3 km, so a probe that is a few meters long is BIG compared to the effective radius of the BH)

• Curvature of space at the event horizon is small compared to the size of the probe (BH with M = 106 Msun has RSch = 3x106 km, so a probe that is a few meters long is SMALL compared to the effective radius of the BH)

Case 2: probe (and any occupants) pass through the event horizon and don’t notice anything strange occurring. Like a boat pulled along by current, they fall toward center (won’t see mass at center because all light is pulled inward). Eventually will be torn apart as go deeper into the black hole.

Should you believe in black holes?Is this science or fiction?

Suppose you wanted to find a black hole. What would you look for?

Remember, a black hole itself emits no light!!

But, material near to the black hole and outside the event horizon may emit light that you could observe.

Key properties to black hole observations: large, “invisible” amount of mass crammed into a very tiny space

Black Hole “Binary” System

A star is observed to be in orbit (by Doppler shift of spectrum) around an “invisible” object.

If star is sufficiently large and is sufficiently close to its unseen companion, matter from the star may transfer over and build up in an “accretion disk” around the black hole.

Gas spirals toward BH, is accelerated up to high speeds by gravity, suffers violent collisions and heats up (millions of degrees = X-ray emission).

Note: stars are not strong X-ray sources

Artist’s conception of black hole binary system Cygnus X-1

Optical image of Cygnus X-1

Constraining the Size of the Region that Contains the Invisible Mass

If the X-ray light flickers (on/off) very rapidly, this places a direct constraint on the size of the accretion disk (just outside the event horizon).

Time scale over which you observe the light to be flickering must be smaller than the time it takes for light to travel across the accretion

disk, or you won’t notice the flickering - it will be smeared out!!

Example: sound waves (time delay of arrival of sound due to its distance; e.g. thunder vs. lightning)

World’s Longest, Loudest Marching Band

All band members play one short, staccato note.

What do you hear?

Speed of sound = 343 m/s, so you don’t hear the back row of the band until 10 seconds after the single

note is played

If band plays 1 staccato note every half second you would hear continuous sound (no “quiet” or “off” time)

If band plays staccato notes more than 10 seconds apart, then you will notice breaks in the sound

Time to traverse the length of the band has to be shorter than the time between which the notes occur in order for you to experience “off” time (same goes

for light)

Band is 37.5 times the length of a football field…

Constraining the Size, II

Diameter of the emitting region has to be less than the distance light could travel over a time equal to the time scale for flickering (t)

D < c t

If t < 1 second, D < 300,000 km (i.e., 20% of the diameter of the sun)

So, if you see flickering on a time scale less than about 4 or 5 seconds, the size of the emitting region (the accretion disk) is smaller than a star, so the companion cannot possibly be a star!

Back to Cygnus X-1

Cygnus X-1 consists of a bright star with mass = 18 Msun and an unseen companion with mass = 10 Msun

Rapidly flickering of X-rays says companion is much too small to be a star

Most theoretically conservative conclusion: companion is a black hole

Many such X-ray binary systems exist in our Galaxy, with black holes that have masses between 4 Msun and 10 Msun

These black holes were formed when an extremely massive star died in a supernova explosion

“Supermassive” Black HolesMBH > 106 Msun

If something (star, disk of gas) is orbiting about a black hole, the speed of rotation should decrease with distance from the black hole:

V = (G MBH / R)1/2

If you can measure V and R, you can deduce MBH

Look for rapidly rotating disks at the very centers of big galaxies, motions of stars near the very center of our own galaxy (Milky Way).

What do you find?

“Supermassive” Black HolesMBH > 106 Msun

Rapidly rotating disk within only 16 light years of the center of giant elliptical galaxy M87 gives MBH = 3 x 109 Msun

“Supermassive” Black HolesMBH > 106 Msun

Rapidly rotating disk within only 0.64 light years of the center of spiral galaxy NGC 4258 gives MBH = 4 x 107 Msun

“Supermassive” Black HolesMBH > 106 Msun

Over course of about 20 years astronomers have followed the motions of stars at the very center of the Milky Way, and have determined their orbits

with very high accuracy. From orbital speeds of stars within 0.03 light years of the center of the Milky Way, MBH = 106 Msun

Last word on Black Holes

• Black holes really do exit

• Black holes with mass MBH < 10 Msun probably result from death of massive star in a supernova explosion

• Probably all large galaxies (galaxies at least as big as our own) harbor “supermassive” black holes at their centers (formation mechanism not yet understood)

• You have nothing to fear from black holes, you just want to stay far enough away that the maximum speed of your space ship exceeds the local escape speed.