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• Test next week (all about stars)• Second project is due in three weeks

So many things I would have done,

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--Joni Mitchell

Black Holes

10 November 2006

Today:

• Special relativity theory (for high-speed motions)

• General relativity: a new theory of gravity

• Black holes

• How to detect black holes

Special Relativity Theory

• A revision of Newton’s laws of motion

• Especially important when objects move at close to the speed of light

• Proposed by Einstein in 1905

• Resolved apparent paradoxes in electromagnetic theory Albert Einstein,

1879 - 1955

Special Relativity Theory• The “principle of relativity” (Galileo, Descartes) says that motion is

relative: There’s no way to tell who’s really moving, so long as the motion is straight and uniform.

• But electromagnetic waves are predicted (and observed) to travel at a fixed speed: 300,000 km/s.

• Before Einstein, physicists thought this meant that the principle of relativity doesn’t apply to electromagnetic waves--just measure the speed of light relative to you, and you’ll learn how fast you’re really moving.

• Einstein proposed that the speed of light is fixed, regardless of the motion of the source or the observer!

• Implications: Time itself behaves in bizarre ways, which become noticeable only when objects move fast relative to each other.

Implications of Special Relativity

• The time interval measured between two events depends on exactly how it is measured.

Alpha Cent., 4 l.y. away

Amy Beth

Speed = 4/5 speed of light

Amy stays home on earth, while Beth travels to Alpha Centauri and back at 4/5 the speed of light.

Amy measures the time between Beth’s departure and return to be 10 years (5 years each way), as we would expect.

Beth measures the time to be only 6 years! ( 10 x 1–(4/5)2 )

Implications of Special Relativity

• When you “add” speeds, the result is less than you thought!

Amy

Beth

1/2 speed of light w.r.t. earth

Amy measures the cannon ball’s speed to be only 4/5 the speed of light!

1/2 speed of light w.r.t. Beth

1

2

1

2+

1 +1

2

1

2.

4

5=

Implications of Special Relativity

• The speed of light (300,000 km/s) is a “Cosmic Speed Limit”

Combining two speeds less than the speed of light always results in a speed that is also less;

Photons and some other signals move at the speed of light, regardless of the motion of the source;

It would take an infinite amount of energy to accelerate a physical object up to the speed of light;

If a signal could travel faster than light with respect to Amy, then it could travel backwards in time with respect to Beth (who’s moving w.r.t. Amy).

Special Relativity

The mathematics of special relativity isn’t advanced--just basic algebra. There are many good books on the subject, or you can learn more by taking Physics 2220 (Physics for Scientists and Engineers).

What about Gravity?

Newton: Force on the moon points toward where earth is now.

Force

Einstein: That’s inconsistent with the Cosmic Speed Limit! If the earth suddenly moves, the force on the moon can’t change until 1.3 seconds later.

We need a new theory of gravity…

General Relativity Theory

• Einstein’s theory of gravity

• Especially important for fast-moving objects and for very dense objects

• Proposed by Einstein in 1915

General Relativity: The Basic Idea

The graph appears curved because space-time itself is curved (distorted by the earth’s mass).

Po

sitio

n

Time

Why does the apple fall?

Aristotle: It’s seeking its natural place toward the center of the universe.

Newton: The earth exerts a force, pulling the apple downward.

Einstein: It’s moving along the straightest possible path through curved space-time.

Why do golf balls and bowling balls fall at the same rate?

Because the trajectory is a property of space-time itself, not a property of the object.

Consequences of General Relativity(Advanced mathematics required!)

• Inverse-square force law isn’t exact; minor corrections to solar system trajectories

Mercury’s elliptical orbit gradually precesses, mostly due to the gravitational pull of other planets. But a small part of the precession (43 arc-seconds per century) is due to relativistic corrections to Newtonian gravity.

Consequences of General Relativity

• Gravitational deflection of light (twice the amount that Newton would have predicted)

Arthur Eddington (tested prediction in 1919)

Consequences of General Relativity

• Gravitational waves: ripples in space-time curvature that propagate outward from accelerating massive objects (e.g., supernova explosions, close binary systems)

“LIGO” (Hanford, WA site)

Binary pulsar (in Aquila)

Consequences of General Relativity

• Black holes: Objects so dense that not even light can escape.

(2-D analogy of 3-D curvature)

Black Holes in Brief

• A spherical “horizon” surrounds the region from which nothing can escape

• The radius of the horizon is proportional to the black hole’s mass (surface area is 4πR2)

• For a solar-mass black hole, the horizon radius is 3 km (only a few times smaller than a neutron star)

• Neutron stars, like white dwarfs, shrink with increasing mass; maximum mass is 2-3 solar masses

• So a collapsing stellar core more massive than this should form a black hole!

How to detect a black hole?• Only through its gravitational pull on other objects!

• An isolated black hole would be virtually impossible to detect

• In a close binary system, a black hole can suck gas off the companion star…as the gas spirals inward it becomes extremely hot and emits x-rays

• Most famous example: Cygnus X-1. Binary system (8000 ly away) of a blue supergiant star and a compact object, both 20-30 solar masses--too massive for a neutron star.

Dust ring (800 ly wide) swirling into center of galaxy NGC 4261

More Black Hole Physics• What happens if you fall into one? Nothing

special when you cross the horizon (unless you try to turn around). But from then on, you’re doomed. The equations say that when you reach the center, your time stops. (The equations don’t take quantum effects into account.)

• Because black holes are perfect absorbers of radiation, they also emit thermal radiation from their horizons! Smaller black holes are hotter. But for a solar-mass black hole, T is only about 10–6 kelvin, so the radiation is negligible.

Stephen Hawking