Re-cap from last time (Special Relativity):

• “Motion” is a relative concept

• Intervals of time are relative, not absolute (time dilation)

• Measurements of distance are relative, not absolute (length contraction)

• Time travel into the future is possible if you’re going very close to the speed of light (and have an essentially limitless source of energy)

tb = tp db = dp mmove = mrest

Outline - Feb. 18, 2010

• Is any of this for real? Can we really demonstrate these effects?

• Basics of General Relativity

• Time dilation and length contraction due to gravity alone

• Gravitational Redshift

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

• Warped “spacetime”

• Tests of Special and General Relativity

• Black Hole basics

• Ch. S3, pages 449-465

General Relativity (1915)

A theory of gravity, much more general than Newton’s theory.

Newtonian gravity is a “special case”; applies when gravity is very weak.

ground

g = 10 m/s2

Describe the motion of the fruit RELATIVE TO the ground…

ground

a = 10 m/s2

Suppose gravity didn’t exist, and you accelerate the GROUND up to the fruit. Would the motion of the fruit RELATIVE TO THE GROUND look any different than before?

Einstein says you’ve got two possibilities…

If all objects are observed to accelerate similarly relative to a particular frame of reference then either:

1. Reference frame is inertial and gravity is present

**OR**

2. Gravity is not present but the reference frame is non-inertial (i.e., it is accelerating)

This is the Equivalence Principle of General Relativity.

Gravitation and (generic) acceleration are equivalent.

Consequence of Equivalence Principle for a Beam of Light

Return to our train car thought experiments, except this time the train is NOT in uniform motion. Now, we’re going to accelerate the train car!

Put a lamp in the floor of a train car, switch it on as the train car accelerates to your right.

What do YOU as a “bystander” say is the path taken by the light beam on its way to the top of the train car?

Equivalence Principle says gravitation and acceleration are equivalent, so the path of light must be curved by gravity!!

What more does the Principle of Equivalence have to say?

Consider rolling a “fast” object and a “slow” object past a very massive body (say, a huge star). What will the two paths look like?

Nothing can move faster than light, so light must take the “straightest” path (the minimum distance between any two points).

But, that path is always curved in the presence of gravity!

Geometry is no longer simple - not “plane” geometry in general - instead we have to worry about geometry on curved surfaces.

Generic Geometries

“Plane” geometry (flat surfaces, like table)

Spherical geometry (“positive” curvature)“Hyperbolic” geometry (“negative” curvature; saddle or Pringle’s chip)

Effect of Gravity on Time

high gravity planet

slow clock

fast clock

rotating centrifuge

fast clock

slow clock

Since gravitation and acceleration are equivalent, clocks at different altitudes will run at different rates!

Time Dilation and Length Contraction due to Gravity Alone

Clocks in a “high gravity” environment will run slow (i.e., they will tick more slowly) than clocks in a “low gravity” environment

Atomic clock kept by National Institute of Standards in Boulder, CO (altitude of about 1600m) runs faster than atomic clock at sea level close to Washington, DC!!

Because time dilation (slow running clock) occurs in “high gravity” environments, the equivalence principle will also require length contraction (foreshortened lengths) in high gravity environments. Clock on the centrifuge will look shortened in the direction of motion, which is the same as an acceleration since it is moving on a circle.

1971 Naval Observatory Experiment with 3 Atomic Clocks

Clock 1: left in DC (rotates with the earth, W to E)

Clock 2: flew W to E around the world (in direction of Earth’s rotation)

Clock 3: flew E to W around the world (opposite to direction of Earth’s rotation)

Special Relativity says:

Clock 2 is moving faster than Clock 1, so 2 runs slow compared to 1

Clock 1 is moving faster than Clock 3, so 3 runs fast compared to 1

General Relativity says:

Due to difference in altitude, clocks 2 and 3 should both run fast compared to clock 1

Net Effect compared to Clock 1:

Clock 2 should lose 40 billionths of a second

Clock 3 should gain 275 billionths of a second

Effect of Gravity on Wavelength/Frequency of Light

Put a source of light in the nosecone of a rocket.

Accelerate the rocket in the direction of the nosecone.

Accelerate the rocket away from the nosecone.

What do you observe about the wavelength/frequency of the light during the accelerations (compared to rest)?

Equivalence Principle says gravitation and acceleration are equivalent, so the same effect must hold true for light in the

presence of gravity. Light has to do work (and lose energy) to escape from the pull of gravity = gravitational redshift.

Why should you care about any of this?

1. Chances are good that some of it may show up on an exam

2. When you go home for Spring Break you can baffle your Aunt Martha when she asks you what you’ve been learning in your astronomy course

3. If you’re buried under snow after an avalanche, you want your emergency GPS locator to work so that the rescue team can dig you out before you suffocate

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.

Midterm Exam #1

Your score on this exam: 73.5 / 100

Your ranking in the class on this exam: 21 / 47

Approximate letter grade on this exam: B+

Curve boundaries for Midterm #1:

A > 83%

A- 78% to 83%

B+ 73% to 77%

B 69% to 72%

B- 66% to 68%

C+ 61% to 65%

C 59% to 60.5%

C- 55% to 58%

D 50% to 54%

F < 50%

Class letter grade average based on the curve is between B and B- (2.9 / 4.0)

This info is on the last page of your exam.