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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 they meet in an attempt to mesh them into one. In the presence of gravity, space-time is flat only locally but must be curved globally.

Any theory of gravity that fulfils EEP is called a "metric" theory (from the geometrical, curved-space-time view of gravity). Because the equivalence principle is a crucial foundation for this view, it has been well tested. Versions of the Eotvos experiment performed in Princeton in 1964 and in Moscow in 1971 verified EEP to 1 part in 1012. The first Eotvos experiment was performed by gathering gravitational red shift measurements using gamma rays climbing a tower on the Harvard University campus in 1965. Other Eotvos experiments include observing wavelengths of light emitted from the surface of the Sun and using atomic clocks flown in aircraft and rockets to verify the EEP effect to a precision of better than 1 percent.

General Relativity

Einstein's Field Equations and Brans-Dicke Metric Theory

The principle of equivalence and its experimental confirmation reveal that space-time is curved by the presence of matter, but they do not indicate how much space-time curvature matter actually produces. To determine this curvature requires a specific metric theory of gravity, such as general relativity, which provides a set of equations that allow computation of the space-time curvature from a given distribution of matter. These are called field equations. Einstein's aim was to find the simplest field equations that could be constructed in terms of the space-time curvature and that would have the matter distribution as source. The result was a set of 10 equations, which are unfortunately extremely difficult to comprehend. This is not, however, the only possible metric theory. In 1960, C. H. Brans and Robert Dicke developed a metric theory that proposed, in addition to field equations for curvature, equations for an additional gravitational field whose role was to mediate and augment the way in which matter generated curvature. Between 1960 and 1976 it became a serious competitor to general relativity. Many other metric theories have also been invented since 1916.

An important issue, therefore, is whether general relativity is indeed the correct theory of gravity. The only way to answer this question is by means of experiment. In the past scientists customarily spoke of the three classical tests proposed by Einstein: gravitational red shift, light deflection, and the perihelion shift of Mercury. The red shift, however, is a test of the equivalence principle, not of general relativity itself, and two new important tests have been discovered since



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