In modern physics theories, forces are carried by particles. An example is the photon, a massless particle that carries the force of electromagnetism. (This is discussed more in chapter 13.) The graviton is the name given to the particle that would carry the force of gravity. It has never been detected, but is required if gravity is to be understood in the same way as the other fundamental forces. This way of thinking about the nature of gravity is different from the geometrical description in Einstein's general relativity. The two points of view are not necessarily incompatible, but reconciling them (in detail) will require a more complete theory of gravity (see Chapter 16).

A space warp is a term coined in science fiction, and as such has no clear meaning. However, one could take it to mean a non-Euclidean geometry for space. Then ``warped space'' would be any nonflat space. Why have science fiction writers latched onto this? They are trying to get around the limitation of the speed of light for traveling through the universe. If the geometry of space is more complicated than just flat geometry, there might be ``shortcuts'' between different points in space (see discussion of wormholes in Chapter 9). Or maybe you could get from this point to that point by going through ``hyperspace'' which would be some higher dimensional space in which our three dimensional space is embedded. Sort of like going from New York to Sydney through the Earth instead of around on its surface; you save some distance that way. Do you suppose that in another century people will think today's science fiction is as corny as we think last century's science fiction is (e.g. traveling to the moon in a balloon, or in a projectile fired from a cannon)?

Light is energy, and in general relativity energy is affected by gravity just as is mass.

What we are looking for when we are trying to do time travel, is a timelike worldline through space-time that moves into the future of some event A and ends up in the past of event A without ever exceeding the speed of light. This is called a ``closed timelike curve.'' Remarkably, there are solutions of Einstein's equations that contain such curves. One example, discovered recently, involves two infinitely long ``cosmic strings'' which are line-like mass-energy sources. Another is the so-called Goedel solution for a rotating universe. (This is discussed in Chapter 16.) So far all such solutions have one thing in common: the closed timelike curves are built in from the start. Nobody has ever found a solution in which time travel becomes possible (after not being possible) due to the evolution of spacetime. Some think that solutions to Einstein's equations that admit closed timelike curves are forbidden, that is, while mathematically they are solutions to the equations, physically they are impossible. Nobody knows how nature would accomplish this, though.

This is the problem with taking the analogy of the sphere too far. The
surface of your basic sphere is two-dimensional. Traveling through the
sphere means going through a third dimension. This seems reasonable
since we are 3D creatures, but it wouldn't work for 2D creatures living
on the surface of the sphere. So if the universe were spherical (in
3D) "going through the sphere" would mean traveling through some higher
dimension. Science fiction writers often assume the existence of
so-called hyperspaces, because it provides a way to get from point A to
point B rapidly. But: there is no reason why there should be any
higher dimensions. Curvature is an * intrinsic * property of a
geometry. A 3D spherical geometry doesn't require some higher
dimensional flat geometry to "curve in."

I am not sure whether one twin is supposed to live on the floor of the ocean or not. The maximum would presumably be a twin on the Earth and the other removed out into space. This has a relative boost factor of 1.0000000008, corresponding to about 0.024 seconds per year, giving a net difference over 75 years of 1.8 seconds.

You spin it up using jets along the rim of the space station. The jets all fire tangentially to the rim in the same sense (e.g. clockwise) and the whole station begins to spin. You want to make sure that the jets fire evenly so there isn't any wobble.

Special relativity is a subset of general relativity. (Note: if there is a cosmological constant - see Chapters 10 and 11 - then, strictly, the equations of GR do not reduce precisely to SR, although locally the Lambda term will be completely negligible.) SR is useful because it is much easier to work with, and most of the time you can use Minkowski space to do your calculations of (for instance) particle physics, with negligible error (due to ignoring gravitational fields).

A metric is the "measure" of the distance between points in a geometry. The distance between two points on a geometry such as a surface is certainly going to depend on how that surface is shaped. The metric is a mathematical function that takes such effects into account when calculating distances between points.

*
What kind of geometry can be non-Riemannian, i.e., not locally flat?
*

An example would include any geometry that has a cusp in it, or a point (like the apex of a cone). Those special locations don't become flat no matter how small a scale one considers.

*
Do you think Einstein's concept of general relativity will be found to
be an inadequate theory?
*

Certainly, it will have to be superseded by some theory of quantum gravity.

We are not certain of the true nature of gravity because it must be quantizable, but we have no theory of quantum gravity. Why does gravity still appear to be so geometrical? A major question. See discussion on page 459 of the text.

Major areas of research include deducing what Einstein's theory predicts (although we have the fundamental form of the equations, we still don't know all the solutions) and trying to determine how to combine GR with quantum mechanics to develop a theory of quantum gravity. As to practical applications of these theories, none immediately come to mind. (A civilization could use a spinning black hole to generate scads of power in principle, but it doesn't seem feasible in practice.) On the other hand, Maxwell probably didn't anticipate all the applications of his equations, so perhaps some practical application may be found someday.

"Gravity thrusters" are a popular way to propel ships in science fiction. It would seem to require some sort of antigravity (which is impossible.) Slightly more sophisticated sci-fi posits ships that can locally affect space-time curvature and use that to propel themselves along. One effect that does work in real life and has been exploited is to dive down into some gravitational field and fire the engines, leaving the fuel mass deep in the gravitational field and thus increasing the boost by the gravitational potential energy of the fuel. If you found a rotating black hole you could also tap into the hole's spin energy.

No, otherwise the Sun would also be blue. The blue color comes from scattering of light off gas molecules in the sky. The amount of scattering (Rayleigh scattering, for those who want to know more) is strongly dependent on the wavelength--specifically, it is proportional to the inverse fourth power of wavelength. Blue light is scattered much more than red light. We see the sky lit up by the blue light scattered toward us from all over. At sunset the sky is red because the blue light is strongly scattered out of our line of sight to the Sun due to the thick layer of atmosphere that the light must traverse near the horizon.

Well, they are tricky concepts to understand, and most people have no personal experience with weightlessness, and know only that things fall down. People making movies often care less about whether something is correct than whether the audience will like it or accept it. Most people know there is no sound in space, for example, but they still expect to hear big booms when spaceships blow up. The factual media has less of an excuse for screwing up. Ultimately the problem is you can only teach people who are willing to learn. (I would be happy to help George Lucas get the science right in his next movie.)

Not quite sure what you mean. But gravity is our name for the force (acceleration) we experience because we are not moving along a freefalling (inertial) trajectory. In other contexts such forces (accelerations) are often called "fictitious" (e.g. centrifugal force).

They are reciprocal in SR because all inertial frames are equivalent. (By the relativity principle.) All accelerating frames are not equivalent. Accelerations vary and are produced by physical effects (forces) proportional to their magnitude.

No. Distance can be defined only in terms of a spatial relationship between one object and another.

The graviton.

Yes, certainly. The geometries we are discussing are the complete range of isotropic and homogeneous geometries. If you throw out those requirements many other geometries are possible.

The overall geometry of the universe is the overall gravitational field of the universe. Obviously the individual things in the universe have their own gravity too. Think of the gravity of a galaxy like a dimple in the overall flat, spherical or hyperbolic geometry.

Your impression is correct. Space is bendable; massive bodies do it all the time. But space is very hard to bend (it is very stiff if you will). All the mass of the sun produces only a very modest curvature in space. I doubt any human contrivance could accomplish much in the way of space warping.

It changes its geometry. That is, matter determines what constitutes freefalling (inertial) worldlines in spacetime.

Return to Chapter 8 | Table of Contents

Copyright © 1998 John F. Hawley