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# 3: Spacetime and General Relativity

• 3.1: Minkowski Metric
Hermann Minkowski discovered that if the temporal (dt) and spatial (dx, dy, dz) separation between two events are combined appropriately,  the resulting quantity, the spacetime interval, is the same for all observers. This result is the metric of the four-dimensional flat spacetime that obeys Special Relativity. This metric is referred to as the Minkowski metric.
• 3.2: Schwarzchild Metric
In General Relativity, the flatspace Minkowski metric cannot be used to describe spacetime. In fact, the metric depends (in a very complicated way) on the exact distribution of mass and energy in its vicinity. This metric is referred to as the Schwarzchild metric, and describes the shape of space near a spherical mass such as (approximately) the earth or the sun, as well as the space surrounding a black hole.
• 3.3: Activities
• 3.4: Global Positioning System (Project)
The timing accuracy required by the GPS system is so great that general relativistic effects are central to its performance. First, clocks run at different rates when they are at different distances from a center of gravitational attraction. Second, both satellite motion and Earth rotation must be taken into account; neither the moving satellite nor Earth's surface corresponds to a stationary frame of reference. In this project you will investigate these effects.
• 3.5: Falling into a Black Hole - Easy Version (Project)
Consider falling feet first into a black hole of mass M . Although you may think that this would be a rather unpleasant experience, the truth may surprise you. Sure, you will be increasingly stretched until you are torn apart near the end of your journey, but for most of your journey you feel quite pleasant. You are freely floating through space, enjoying the ride.
• 3.6: Falling into a Black Hole - Hard Version (Project)
The text describes how the Schwarzchild metric can be used to directly compare time and length intervals measured by observers at rest at different locations in spacetime. However, measurements made by, and of, moving observers are slightly more difficult to compare. In order to determine how the observations of a person falling into a black hole compare to observers at rest, additional mathematical machinery is necessary.