A Qualitative Outline of General Relativity and Space-Time Curvature

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Outlining General Relativity and Space Time Curvature

In the real world, smooth, uniform motion is more an exception than a rule. Technically, any change in speed or direction is called acceleration (or deceleration), which can thus mean slowing down as well as speeding up, or simply a redirection. Ordinarily, an observer in an accelerating frame of reference can perceive its motion. Passengers in a car, for example, fell themselves pressed backward if the car starts suddenly from a dead stop. Their awareness seems to imply that acceleration is absolute, not relative; they need not refer to anything outside their frame of reference to detect their own motion. But if accelerated motion is absolute, it would have to be subject to a different set of natural laws from those that apply to uniform motion – a proposition that Einstein found highly objectionable. He thus set out to conjure a more general theory that would apply to motion of all sorts. In the process, he developed a new theory of gravity.

The starting point was Galileo's finding that falling objects accelerate at the same rate despite differences in their mass: if dropped from the same height in a vacuum, a cannonball and a feather would hit the ground at the same time, due to lack of air resistance. Einstein was sceptical of Newton's explanation that the force of gravitational attraction precisely equalled an object's inertial mass. Einstein rejected the notion that this uncanny coincidence was merely an accident of nature.

The Principle of Equivalence and the Weak Equivalence Principle (WEP) The exact Minkowski space-time of special relativity is incompatible with the existence of gravity, because of gravity's physical quality of ‘warping' space-time. A frame chosen to be inertial for a particle far from the Earth where the gravitational field is negligible will not be inertial for a particle near the Earth. An approximate compatibility between the two, however, can be achieved through a remarkable property of gravitation called the weak equivalence principle (WEP): all modest-sized bodies fall in a given external gravitational field with the same acceleration regardless of their mass, composition, or structure. Baron Roland von Eotvos (after such experiments have been named) has checked experimentally by Galileo, Newton, and Friedrich Bessel and in the early 20th century the principle's validity. If an observer were to ride in an elevator falling freely in a gravitational field, then all bodies inside the elevator, because they are falling at the same rate, would consequently move uniformly in straight lines as if gravity had vanished. On the contrary, in an accelerated elevator in free space, bodies would fall with the same acceleration (because of their inertia), just as if there were a gravitational field. The Einstein Equivalence Principle (EEP)

Einstein's great insight was to postulate that this "vanishing" of gravity in free-fall applied not only to mechanical motion but to all the laws of physics, such as electromagnetism. In any freely falling frame, therefore, the laws of physics should (at least locally) take on their special relativistic forms. This postulate is called the Einstein equivalence principle (EEP). One consequence is the gravitational red shift, a shift in frequency ‘f' for a light ray that climbs through a height ‘h' in a gravitational field, given by where ‘g' is the gravitational acceleration. (If the light ray descends, it is blue-shifted.) Equivalently, this effect can be viewed as a relative shift in the rates of identical clocks at two heights. A second consequence of EEP is that space-time must be curved. Although this is a highly technical issue, consider the example of two frames falling freely, but on opposite sides of the Earth. According to EEP, Minkowski space-time is valid locally in each frame; however, because the frames are accelerating toward each other, the two Minkowski space-times cannot be extended until...
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