Changes

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.)

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.

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 coordinate 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.

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.