## Lecture Notes:

Day 14 February 27, 1997 So far this semester the major topics we have covered are
1. Ancient astronomy in the American southwest, emphasizing observations of the cycles of the sun and moon: namely solstices, equinox, and standstill.
2. Prediction of eclipses: eclipse seasons, lunar/solar eclipses; major/minor standstills.
3. Impacts and extinction events produced by comets and asteroids: asteroid belt and resonances; Oort comet cloud. Nature of comets and asteroids. Evidence for impacts in the past.
4. NOW: Gravity and black holes.
Gravity started during the summer of 1667 when Newton was at his mother's farm and he realized that the same force that pulls the moon also pulls the apple to the earth. Newton was wrong about the idea of a force but very correct in making the experience of gravity a cosmic phenomenon, which included the earth and the most distant reaches of the universe.

He invented the equation for gravitational force: F= MmG/R squared The equation predicts behavior reasonably well in weak gravity and short distances, but fails in strong gravitation systems. The force of gravity is strange, mysterious, and counterintuitive, if you think about it. The gravitational strings that pull you down are invisible, can not be cut or blocked: there are no anti-gravity shields.

For 250 years from 1667 to 1917 the force of gravity as invented by Newton reigned supreme. Newtons universal law of gravity seemed absolutely TRUE.

Einstein developed two kinds of relativity: Special and General. Special Relativity deals with the changes of mass, time, and length when objects are moving at constant speeds and predicts that at a relative speed of light, object acquire infinite mass, their time stops, and their length parallel to their direction of motion shrinks to zero

General Relativity deals with curved space and time and gravity. Einstein showed that the force of gravity is an illusion, resulting entirely from the curvature of space and time. Space is curved near a massive object; the greater the mass the greater the curvature.

The first real test of Einstein's ideas about gravity came during the total solar eclipse of May 29 1919 when the curvature of light around the sun was verified. Stars were displaced outward at the eclipse compared to their positions when the sun was not between them and the earth.

Another test for the curvature of space is simply the sum of the interior angles of a triangle. In flat Euclidean space the sum is exactly 180 degrees. But in positively curved space, such as the surface of a sphere, the sum is always greater than 180. In the planetarium, we demonstrated a triangle drawn on the interior of the dome which had three points connected by three great circles ( also known as geodesics); each of the three interior angles was 90 degrees and the sum was therefore 270 degrees.

The curvature of space around the sun can be verified by three satellites that are in radio contact. The triangle connecting them has a sum of its interior angles greater than 180.

Because of the curvature of space, which is always curved in the direction of the mass, (concave toward the mass) it is difficult to escape from a mass: space is always curving you back toward it. The greater the mass, the harder it is to escape. Every mass has an escape velocity (SNOW: 95-98) which is the square root of the product of twice G times M divided by R. (sorry folks, writing equations is difficult). The escape velocity of the earth of 11 km/sec; that from Jupiter is 61; and from the sun, 618 km/sec is required to escape forever from its gravity.

Since only a larger mass is required to increase escape velocity (or a smaller radius) it is quite easy to imagine larger and larger escape velocities. Ultimately, an escape velocity of 300,000 km/sec can be required; such is the speed of light, which we represent by the symbol c. An object with an escape velocity equal to that of light is a black hole by definition.

If one sets the escape velocity equal to c, then one can calculate the fundamental equation for black holes, the radius of a black hole which is R= 2GM/c squared. This is the radius of the "event horizon" of a black hole and is also known as the Schwarzschild radius (SNOW 512). The sun has a radius of 700,000 km; if its mass stays the same and it shrinks to 3 km radius, it would be a black hole.

Tides on the earth are caused by the curvature of space; the changing curvature of space stretches objects. In the case of the earth, we are stretched such there are two equally high tides on opposite sides of the earth. (SNOW:98-100). There are enormous tides produced by a black hole because of its huge distortion and curvature of space. A person falling into a black hole with the mass of the sun would be stretched to death well before he/she crossed the event horizon. Larger black holes are not so painful. In fact, we may be living in a black hole right now.