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Hyperspace and Warped Space Help (page 2)

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Time-space

You’ve seen time lines in history books. You’ve seen them in graphs of various quantities, such as temperature, barometric pressure, or the Dow Jones industrial average plotted as functions of time. Isaac Newton, one of the most renowned mathematicians in the history of the Western world, imagined time as flowing smoothly and unalterably. Time, according to Newtonian physics, does not depend on space, nor space on time.

Wherever you are, however fast or slow you travel, and no matter what else you do, the cosmic clock, according to Newtonian (or classical) physics, keeps ticking at the same absolute rate. In most practical scenarios, this model works quite well; its imperfections are not evident. It makes the time line a perfect candidate for a “fourth perpendicular axis.” Nowadays we know that Newton’s model represents an oversimplification; some folks might say it is conceptually flawed. But it is a good approximation of reality under most everyday circumstances.

Mathematically, we can envision a time line passing through 3D space, perpendicular to all three spatial axes such as the intersections between two walls and the floor of a room. The time axis passes through three-space at some chosen origin point, such as the point where two walls meet the floor in a room, or the center of the earth, or the center of the sun, or the center of the Milky Way galaxy.

In four-dimensional (4D) Cartesian time-space (or simply time-space) , each point follows its own time line. Assuming none of the points is in motion with respect to the origin, all the points follow time lines parallel to all the other time lines, and they are all constantly perpendicular to three-space. Dimensionally reduced, this sort of situation can be portrayed as shown in Fig. 11-2.

 

Hyperspace and Warped Space Cartesian n-Space Time-space

Fig. 11-2 . Time as a fourth dimension. Three-space is shown dimensionally reduced, as a plane. Each stationary point in space follows a time line perpendicular to 3D space and parallel to the time axis.

Position Vs Motion

Things get more interesting when we consider the paths of moving points in time-space. Suppose, for example, that we choose the center of the sun as the origin point for a Cartesian three-space coordinate system.

Imagine that the x and y axes lie in the plane of the earth’s orbit around the sun. Suppose the positive x axis runs from the sun through the earth’s position in space on March 21, and thence onward into deep space (roughly towards the constellation Virgo for you astronomy buffs). Then the negative x axis runs from the sun through the earth’s position on September 21 (roughly through Pisces), the positive y axis runs from the sun through the earth’s position on June 21 (roughly toward the constellation Sagittarius), and the negative y axis runs from the sun through the earth’s position on December 21 (roughly toward Gemini). The positive z axis runs from the sun toward the north celestial pole (in the direction of Polaris, the North Star), and the negative z axis runs from the sun toward the south celestial pole. Let each division on the coordinate axes represent one-quarter of an astronomical unit (AU), where 1 AU is defined as the mean distance of the earth from the sun (about 150,000,000 kilometers). Figure 11-3A shows this coordinate system, with the earth on the positive x axis, at a distance of 1 AU. The coordinates of the earth at this time are (1,0,0) in the xyz -space we have defined.

Of course, the earth doesn’t remain fixed. It orbits the sun. Let’s take away the z axis in Fig. 11-3A and replace it with a time axis called t . What will the earth’s path look like in xyt -space, if we let each increment on the t axis represent exactly one-quarter of a year? The earth’s path through this dimensionally-reduced time-space is not a straight line, but instead is a helix as shown in Fig. 11-3B. The earth’s distance from the t axis remains nearly constant, although it varies a little because the earth’s orbit around the sun is not a perfect circle. Every quarter of a year, the earth advances 90° around the helix.

 

Hyperspace and Warped Space Some Hyper Objects Time As Displacement

Hyperspace and Warped Space Some Hyper Objects Time As Displacement

Fig. 11-3 . (A) A Cartesian coordinate system for the position of the earth in 3D space. (B) A dimensionally reduced Cartesian system for rendering the path of the earth through 4D time-space.

Practice problems for these concepts can be found at:  Hyperspace And Warped Space Practice Test.

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