Foundations of Modern Cosmology

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Chapter 8: General Relativity

Chapter Summary Special relativity showed that the absolute space and time of Newtonian physics could be only an approximation to their true nature. However, the special theory of relativity is incapable of explaining gravity because SR assumes the existence of inertial frames; it does not explain how inertial frames are to be determined. Mach's principle, which states that the distribution of matter determines space and time, suggests that matter is related to the definition of inertial frames, but Mach never elucidated any means by which this might happen. General relativity attacks this problem and in so doing, discovers that gravity is related to geometry. The equivalence principle is the fundamental basis for the general theory of relativity. The strict equivalence between gravity and inertial acceleration means that freefalling frames are completely equivalent to inertial frames. In general relativity, it is spacetime geometry that determines freefalling (inertial, geodesic) worldlines, telling matter how to move. Matter, in turn, tells spacetime how to curve. Geometry is related to matter and energy through Einstein's equation. The metric equation provides a general formalism for the spacetime interval in general geometries, not just the Minkowski (flat) spacetime of special relativity. Matter and energy determine inertial frames, but within an inertial frame there is no influence by any outside matter. Thus Mach's principle is present more in spirit than in actuality in the general theory of relativity.

Tidal forces prevent a perfect equivalence of freefall and gravity. If the gravitational field diverges over the size of an object, the various parts of the object will be pulled by different amounts or in different directions. These differential effects are known as tidal forces. The equivalence principle requires only that the size of the inertial frame be sufficiently small that tidal forces are negligible.

General Relativity predicts the bending of light by gravity, gravitational time dilation and length contraction, gravitational redshifts and blueshifts, the precession of Mercury's orbit, and the existence of gravitational radiation. All these effects have been measured, although gravitational radiation has been observed only indirectly via the decay of the orbits of binary pulsars. The LIGO project is an attempt to build a giant Michelson-Morley type of interferometer to detect gravitational radiation directly. Two interferometers will be built, each one with perpendicular light-carrying vacuum pipes 5 kilometers long. Figure 8.11 is an image of an artist's conception of the LIGO interferometer.

Here is an updated photo showing the Hanford Gravitational Observatory site, with construction well along.

Points to Ponder
  • What is meant by a non-Euclidean geometry? What are the properties of spherical and hyperbolic geometries?
Potential Pitfalls

The equivalence principle seems to have some strange consequences. If you have difficulty visualizing light falling in a gravitational field, consider the freefalling elevator. Draw a picture showing the elevator at different points. Inside the elevator, the light will always strike the same point on the wall. Where is that point as a function of time, as seen from outside the elevator?

A consequence of the equivalence principle is that even though you may be sitting at rest in your chair, you are in an accelerated, and hence noninertial, reference frame merely by virtue of residing in a gravitational field. But if you are sitting at rest, should not the net force be zero? How can you be accelerated? The answer lies in the fact that in order to claim zero net force, we must include the "fictitious" gravitational force in the resultant. As long as we take gravity into account, we can apply Newton's laws within our noninertial frame, and we can regard an object in freefall as accelerating toward the Earth. In actuality, however, the object in freefall is not accelerated, while we in our noninertial frame are accelerated.

Questions & Answers

Questions and answers about general relativity.

Web World

Interested in learning more about special and general relativity? The numerical relativity group at the National Center for Supercomputing Applications has some nice relativity pages. Go here for an index . A couple of specific examples include a page on Einstein . There are a lot of interesting and informative links from there. To start at the beginning of their general relativity tour, go to the Spacetime Wrinkles Homepage.

Visit the LIGO home page. See for yourself how far construction has proceeded with the new gravity wave detectors!