## 4-this, 4-that, 4-vectors

Special relativity merges space and time into a single, 4-dimensional unity, space-time. In doing so, concepts that had, in classical physics, been separated into distinct spatial and temporal aspects, become unified into 4-dimensional aggregates. The heart of the matter begins with the contravariant 4-vector for the position of an event in space-time, relative to some coordinate system, . We could write this as , or more formally as or as just . Our next step is to consider the invariant distance between two events as This can also be written as where is the metric tensor on a Euclidean...

Read More## Energy and momentum

When Einstein began to develop the theory that became general relativity, there were a number of elements that were going to be essential to the result. The basic papers on special relativity were published in 1905 and general relativity was not made public until 1915. Special relativity had already dealt with coordinate transformations between inertial frames of reference; general relativity was to incorporate accelerated frames of reference. Einstein would also have had in mind the equivalence principle; namely, that a gravitational field was equivalent to an accelerated frame of...

Read More## The heart of special relativity

Often, a theory is built upon a very simple observation. For example, Galileo rolled balls of different sizes and masses down an inclined plane and discovered that they increased their speed at a constant rate, independent of their size or mass. From this, he could work out the acceleration due to gravity and that this acceleration applied to all bodies uniformly. You could repeat Galileo’s experiment at home yourself. Here is another simple observation, based upon the Michelson-Morley experiment: the speed of light is constant to all observers. I’ve had the advantage of being...

Read More## Geodesics and general relativity

“What is a geodesic?”, you ask. The answer is simple, it is the straightest possible line that you, or anything else for that matter, can travel on through space. If space were flat (Euclidean), then the geodesic is just a straight line. On the surface of a sphere, a geodesic is a Great Circle. If you’ve ever looked at one of those maps in the back of the airline flight magazine while flying from point A to point B, you will have seen that the Great Circle routes that airlines like to fly, especially on a Mercator projection map of the Earth. The arcs curve to the north in...

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