Lecture Notes:

Day 15, March 4, 1997
• The quiz on March 20 will cover material in Snow through page 162. Look at the thought questions #1, 4, 6, 7, 9 of Chapter 4 and #3, 4, 5, 7, 10 of Chapter 5

• Important issues to look at in Chapter 5 are contained in the sections on escape velocity, relativity (insight 5.2) and tidal "forces".

• It is important that you understand fully the nature of tides: namely that two high tides always exist, one toward and one away from the moon and the earth rotates under those tidal bulges. As a consequence we experience two high tides a day as we are carried through the bulges.

• The heights of the tides increase with the mass of the object responsible for raising them. The extreme case involves a black hole for which tidal effects will so stretch an object that it can destroy it before it crosses the event horizon. Humans are destroyed at the "tearing apart" radius of a black hole, for which with a 1 solar mass black hole occurs at approximately 100 km away.

• RESONANCE:
We demonstrated resonance with a swinging pendulum. Hitting it at will stop its swing. But hitting it once every swing in the same direction will build up the amplitude of the pendulum, like pushing a swing in a playground. If you push it once every swing, that corresponds to a 1:1 resonance; if you hit it every second time it is a 1:2 resonance.

• INERTIA:
The concept was first developed by Galileo (SNOW 70) as the property of an object that resists change such that an object in motion will continue in motion and that at rest will continue at rest. Newtons refinement of the concept of inertia is contained in his first law of motion: Free particles move along straight lines with constant speeds.

• EINSTEIN'S VIEW OF INERTIA:
Einstein modified Newton by simply replacing "straight line" by geodesic. The geodesic is the defines the shortest distance between to points, regardless of the surface. If the surface is flat, the geodesic is a straight line; on a sphere the geodesic is a great circle. The fundamental explanation of gravity by Einstein is contained in his proposition: Free particles move along geodesics with constant speeds.

• DEMONSTRATIONS:
In class we gave demonstrations of various forms of inertia:
1. Hitting a block of lead
2. Loose axe head
3. Bucket of water
4. Grinding wheel
5. Table cloth
6. Vertical cannon

• FICTITIOUS FORCES:

Inertial processes often give the appearance of forces; the baby flying out of a fast moving perambulator is an example of the working of inertia such that it appears that a force pushes the baby out of the buggy. But there is no force operating on the baby. It is a free object until it hits the pavement.

Centrifugal force, which by definition comes out from a center, is another fictitious force. When one is thrown out of a car when it goes around a corner, one might feel that a force is operating, but one is only continuing along a straight line at a constant speed and the car moves away. One can verify that there is no centrifugal force by inspecting the streaks of blood and noting that they do not proceed radially from the center of turning, but are continuations of the initial direction of motion.

Gravitational forces appear to arise when space is curved. When apples fall from trees or rats fall from high buildings, because of the curvature of geodesics downward to the earth. On a space time diagram, geodesics illustrate why object appear to be "pulled" to the earth; they are, of course, not being pulled by anything, but are simply free objects moving along curving geodesics, which always curve CONCAVE in the direction of the mass. AGAIN: There is no force of gravity; it was invented by Newton. The force didn't exist before he conceived it and doesn't exist now after Einstein showed us a better way to understand gravity.

• CURVED SPACE-TIME

Gravity results from curved space and time. We live in a world of four dimensions; three of space and one of time. In the three space dimensions we are free to move forward and backward and stop; in the time dimensions we do not have that freedom but are inexorably carried forward in time, never slowing down, stopping or reversing. The curvature of space is easy to understand and visualize. The extreme curvature of space around a black hole means that space is so curved in upon itself that there are no pathways outward from the black holes event horizon.

Curvature of time results in a slowing down of time. Wherever there is mass time advances slowly; the greater the experience of gravity the slower is time. At sea level in California, time advances more slowly than in Boulder, because we are further away from the center of the earth. Time advances fastest in empty space and most slowly at the event horizon of a black hole where it stops entirely. The stopping of time is another way of understanding how a black hole works. When time stops change also stops, and therefore nothing can change its position from inside the event horizon to outside the event horizon. Everything, even light is thus trapped inside the event horizon.

The slowing down of time near the event horizon means that light is reddened; slower time means slower oscillations of light and therefore longer wavelengths. As a watch falls into a black hole, it steadily ticks slower and slower compared to an outside watch, and its color gets redder and redder. Eventually, it gets so red that it becomes invisible.

• EVENT HORIZON OF A BLACK HOLE

The easiest way of determining the radius of the event horizon is to use the equation for escape velocity and set the velocity equal to c, the speed of light. The radius then turns out to be 3 km times the mass of the object divided by the mass of the sun. Thus an object with a mass of 10 solar masses has a radius of 30 km.