## Lecture Notes:

Day 16, March 6, 1997
• CURVED SPACE:
The fundamental nature of our world is the curvature of our space-time according to Einstein. Since its so important, it is important and may eve be fun to learn some more features of curved space. The shortest distance between two points is always a geodesic, which is a great circle on a sphere.. Triangles are constructed by connecting three points by three geodesics.

• The three types of curvature are flat, positive, and negative. Each can be distinguished by a number of techniques such as the sum of the interior angles of triangles, the number of line that can pass through a point and not intersect with an adjacent line, the circumference of a circle divided by its radius:

 Sum of Angles # of lines Circumference/radius Flat 180 1 twice pi Positive > 180 none less than twice pi Negative < 180 infinite greater than twice pi

• The sum of the interior angles of a triangle drawn upon a sphere depends upon the amount of space of the sphere covered by the triangle. If the area of the triangle is << (much less) than the surface area of the sphere, the sum is the same as flat space.

• If the triangle covers of a hemisphere, there are three 90 degree angles and hence a sum of 270 degrees

• If the triangle covers an entire hemisphere there are two 90 degree angles and one 360 degree angle and hence a total of 540 degrees.

• If a black hole were to have a triangle drawn on its, the interior angles would reach their maximum sum. What can that be??

• A circle drawn on a sphere also changes its geometric rules depending upon its size. If it covers half of a sphere, the circumference/radius is equal to 4. When the circle includes the entire sphere the circumference is zero. The extraordinary consequence is that for such a circle there is no boundary to the outside. There is no way to get into it or out of it because its boundary has disappeared. This is another feature of a black hole; a structure that is so curved that its boundary has vanished.

• Human kind first realized that the natural world might be composed of curved space after the eclipse observation of the curvature of star light around the sun. Recently the Hubble Space Telescope has photographed incredible double images of galaxies in which light has been bent above and below a dark mass to produce two identical images: a very dramatic demonstration of space curvature.

• CURVED TIME:
We experience time running slowly when time is curved. The greatest curvature of time, namely that near a black hole, results in time stopping entirely. The effect of the slowing of time is obvious in its influence on light. Light is red shifted in the vicinity of massive object and the curvature of time.

Light is produced by the oscillation of an electron which generates an electromagnetic wave. The wavelength of the wave is determined by the rapidity of oscillation of the electron. In a hot gas when the electron is oscillating rapidly, the wavelength is short and the radiation is bluish; when it is a cool gas and the electrons oscillate slowly, the wavelength is long and the radiation is toward the red.

Very simply:
HOT GAS, FAST ELECTRONS, SHORT WAVELENGTH, BLUE : Rigel
COOL GAS, SLOW ELECTRONS, LONG WAVELENGTH, RED : Betelgeuse

• A second way of changing the color of light is due to motion of the source of light. When the source of light is moving toward you, the waves are squeezed together, the wavelength is short and the light is shifted to the blue. When the source of light is moving away, the wave are pulled apart, the wavelength are long, and the light is shifted to the red. See SNOW 121. This effect is known as the Doppler Effect. The equation is z = v/c where z is delta lambda divided by lambda and v is the speed of the source.

• The third way of changing the color of light is by slowing time such as occurs in the vicinity of a mass. For small mass, the effect is z = 0.5*(v/c)^2. Where in this equation v is the escape velocity from the object that is producing the light. In the case of the earth for which v = 11 km/sec, the gravitational red shift is 6.7*10^-10. Very small but recently measured, verifying Einstein's ideas. Gravitational red shifts from various objects can be easily determined once you know the escape velocity, such as 618 km/sec for the sun; 7000 km/sec for Sirius B.

• SIRIUS B.
The white dwarf companion of Sirius revolves with a period of 50 years at a distance of 20 au. From this one can calculate the sum of the mass of the two stars: M + m = a^3/P^2 = 3.2 solar masses. The white dwarf, Sirius B, has a mass of about that of the sun, but it has a radius of 5800 km, slightly smaller than the earth. Its density is enormous such that a teaspoon weighs more than a T. rex or Brontosaurus, or maybe even two!