Lecture 8 - General Relativity

Statement of Special Relativity

All bodies in reference frames K and K', which are moving with uniform linear motion relative to one another, are equivalent for the description of all natural phenomena in a Euclidian geometry.

Problems of Special Relativity

It only holds for inertial reference frames. That is reference frames that are moving a constant velocity with respect to each other with no rotation. For example, Newton's First Law states that all objects will remain at rest or travel along straight lines unless acted upon by some external force. But, this only applies for reference frames in linear motion. If a reference frame is accelerated, this doesn't work.

It says that nothing can move faster than the speed of light. This arguably goes against Newton's Third Law (gravitation). If one moves a mass from one place to another, Newton's law says that its influence on all other matter is instantaneous. This doesn't fit notion of light being fastest and finite.

Statement of General Relativity

All bodies in refernce frames K and K' are equivalent for the description of all natural phenomena no matter what their state of motion in a Gaussian (or Riemannian, non-Euclidian) geometry.

This theory will not work for Euclidean geometry. It does work for more general curvilinear coordinate systems called Gaussian or Riemannian geometries. These geometries can describe the universe and happen to be curved depending on wher mass in located.

In this statement, the effects we associate with gravity can be seen to be equivalent with the effects of uniform acceleration of our reference frame.

Mr. Thompkin's first visit to the Land of Relativity

(I - City Speed Limit)

Most of the activities which Mr. Thompkins (MT) sees during his first trip to the land of relativity can be explained by the Special Theory of Relativity. One sequence, however, must be explained by the General Theory of Relativity.

* At the train station MT sees a young man coming off the train meeting an old lady. It turns out the young man is the grandfather of the old lady. Man spends a lot of time travelling where time moves more slowly. Therefore he ages more slowly than people who live in town and do not travel much. [This can be explained by the Special Theory of Relativity, travel at relativistic speeds causes time to slow down. But, it can also be explained in terms of the General Theory of Relativity - see below.]

* Train engineer claims he is responsible for making people age more slowly on trains because he is brakeman. Suggests time slows down due to acceleration and decceleration. [ The Genral Theory of Relativity states that time slows down whenever acceleration (or decceleration) is occurring. So young man nay be young due to effects of either Special or General Theory of Relativity.

Twin Paradox

The General Theory of Relativity also provides a more realistic explanation of the Twin Paradox. When we first visited the Twin Paradox, we stated that the space ship went away from the Earth at constant linear relativisitic speed, then turned around and came back. We attributed difference in age of twins to time spent at constant relativiistic speed (slower time). But, realistically, the space ship had to accelerate and deccelerate twice to really make the trip. Age difference could also be due to this.

Perspectives on General Relativity

**Example **1:

The reasoning which led to the General Theory of Relativity (GTR) included simple examples of how things behave in systems under acceleration. For example, consider a space ship at rest (of in constant liear motion) somewhere far from sources of gravitation. Inside the spaceship a man (or any object) will float in the air and a beam of light shone from one location will travel in a straight line. If the ship starts to accelerate, the person or any object will move in adirection opposite the direction of acceleration until there is contact witht he ship. Then theperson/object will move with the ship but feel the force of acceleration. A person would interpret this as gravity! This establishes the equivalence of acceleration andgravity in GTR.

Under acceleration, light shone from one location in the ship will no longer travel in a straight line, but rather a curved arc (unless light direction is exactly parallel to direction of acceleration). This suggests light rays can be bent by sources of gravity. This effect is visible as light from distant stars goe close to, but past, the Sun.

**Example **2:

Another example (Mr. Thompkins, Chapter 4), is to imagine a merry-go-round which is walled off so that anyone on the merry-go-round can't see that they are rotating. Let the merry-go-round spin at a speed near the speed of light. (Remember, rotation even at constant velocity is a form of acceleration because the direction is always changing.) In this example, people of the merry-go-round will 'feel' a gravitation that is outward toward the edges ofthe merry-go-round. Speed in the merry-go-round will change from zero at the center to almost the speed of light at the edges.

Let a meter stick be used to measure the merry-go-round by laying the stick end-to-end from the center to the edge and then end-to-end around the circumference of the merry-go-round. In Euclidean space (inertial reference frames), the ratio of the radius to circumference is 2*pi, where pi is 3.14159... But, in the case of the merry-go-round, the ratio will be bigger. The reason is that the meter stick is always the same length as one progresses towar dthe edge ofthe merry-go-round because lengths only change in direction of motion (Lorentz transformations). But, masurement along the edge will align the meter stick with the direction of motion and it will change its length.

Also, if one tried to stretch a piece of string or rope from one place on the merry-go-round edge to another place, the string would prefer to 'bend' toard the middle. This is because the shortest distance between two points in an accelerated reference frame is no longer a straight line. Euclidean geometry doesn't hold! What we have is a curvilinear non-Euclidean (Riemanian ) geomtry. This is the geometry of our Universe!

Last but not least, a clock placed at the center of the merry-go-round will be faster than a clock at the edge due to efects of acceleration.

Key Experimental Tests of the General Theory of Relativity

1) Light travelling from distant stars ar bent around the Sun as the light rays go closely past it.

2) Precession of all planetary orbits around the Sun. This precession is most easily measured for Mercury.

Implications

The Universe is curvilinear in form and is a true space-time continuum.

The curvilinear nature can be attributed to location and size of masses in the Universe.

Planets can be viewed as moving along paths that are the shortest distance
along a curved surface near a strong source of mass (gravity), the Sun.