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General theory of relativity (view source)
Revision as of 18:55, November 15, 2009
, 18:55, November 15, 2009First stirrings of coordinate system curvature ==> fictitious forces. Need 3 pics.
==Coordinate Systems and Spacetime Diagrams==
==Coordinate Systems and Spacetime Diagrams==
Figure 1 shows a "spacetime diagram", with my house, my neighbor's house, and my neighbor walking from his house to mine.
:This is the same kind of diagram that is used in explanations of special relativity. The "spacetime" is sometimes called "Minkowski space". Spacetime is actually four-dimensional, but we can only show two dimensions, so we leave out y and z. The single x spatial coordinate is good enough for our purposes, so the diagram has x going from left to right, and t (time) going upward. For the purposes of this explanation, don't worry about the considerations of special relativity such as the speed of light, the Lorentz transform, or light cones. None of that is important just now.
The diagram shows the calibration, in space (that is, x) and time. These measurements are made with respect to my (stationary) frame of reference. My house is at x=0, and my neighbor's house is at x=1250 (feet). My neighbor walks at 250 feet per minute.
The diagram shows some "events"—my house, now; my house, 5 minutes from now; and my neighbor's house now and 5 minutes from now. The diagonal line depicts my neighbor walking from his house to mine, arriving 5 minutes from now. That line is called his ''world line''. The line going straight up in my house is my own world line (I'm sitting at home.)
A car is driving down the street, from left to right. Figure 2 shows the same four events and two world lines, but with different calibration—the car's own corrdinate system. The car is driving 500 feet per minute, but in the opposite direction. The event of my neighbor's arrival at my house is now at x=-2500. It's way behind the car, though the car was directly in front of my house at t=0.
Because the car's frame of reference is in motion, the calibration lines in figure 2 are not perpendicular. The formerly vertical lines are now slanted. But there is something very important to notice about the two coordinate systems: They are ''flat''<ref>The words "flat" and "curved" used in this article are the same terms used by differential topology experts.</ref>. The flatness comes from the fact that the calibration lines are straight and parallel. The boxes created by the lines are parallelograms. But note that the lines don't have to be perpendicular, and the boxes don't have to be rectangles. Straight parallel lines and parallelograms are all that is required.
These two flat coordinate systems have a very important physical property: Neither I, sitting at home, nor a passenger in the car, experiences any "[[fictitious force]]s". That is, people in the car don't feel any recoil from acceleration, or centrifugal force, or Coriolis force. These frames of reference are said to be ''inertial''. This leads to an important principle of geometrical physics:
::*''Inertial frames of reference have flat coordinate systems. Flat coordinate systems lead to an absence of fictitious forces.''
Now consider figure 3. The coordinate system is once again that of the car, but the car is accelerating, starting at a standstill in front of my house at t=0. Its world line is curved. Once again, it crosses paths with my neighbor. This case is very different from the other two. The calibration lines are curved, and the boxes that they create are not parallelograms. This coordinate system is ''curved''. Another thing to notice is that people in the car will feel a fictitious force—a "recoil" force agains the back of the seat. This frame of reference is not inertial.
::*''Accelerating frames of reference have curved coordinate systems. Curved coordinate systems lead to fictitious forces.''
::::: .... !!!! We need these three diagrams, of course. I'll do them, but they will take a lot of work. If anyone else has the tools and expertise to do this, and more skill than I, feel free to make them, or to communicate with me (PatrickD).
... In progress ...
... In progress ...