Hyperspace

For an account of the concept of "hyperspace" used in science fiction, largely unrelated to the topic of this article, see hyperspace (science fiction).

In physics, hyperspace is a theoretical entity. The theory consists of the idea that our own universe is connected to other universes through wormholes, and all of the universes are found within "hyperspace".

The geometry of space-time in special relativity with hyperspace added in SR uses a 'flat' 4-dimensional Minkowski space, usually referred to as space-time. This space, however, is very similar to the standard 3-dimensional Euclidean space, and fortunately by that fact, very easy to work with.

The differential of distance (ds) in Cartesian 3D space is defined as:

[itex]ds^2 = dx_1^2 + dx_2^2 + dx_3^2\,[itex]

where

[itex](dx_1,dx_2,dx_3)\,[itex]

are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension, time, is added, with units of c, so that the equation for the differential of distance becomes:

[itex]ds^2 = dx_1^2 + dx_2^2 + dx_3^2 - c^2\, dt^2.[itex]

In many situations it may be convenient to treat time as imaginary (e.g. it may simplify equations), in which case t in the above equation is replaced by i.t', and the metric becomes

[itex]ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + c^2(dt')^2.\,[itex]

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space,

[itex]ds^2 = dx_1^2 + dx_2^2 - c^2 dt^2\,[itex]

We see that the null geodesics lie along a dual-cone: ￼defined by the equation

[itex]ds^2 = 0 = dx_1^2 + dx_2^2 - c^2 dt^2,\,[itex]

or

[itex]dx_1^2 + dx_2^2 = c^2 dt^2,\,[itex]

which is the equation of a circle with r = c dt.

If we extend this to three spatial dimensions, the null geodesics are continuous concentric spheres, with radius = distance = c·([itex]\pm[itex])time.

[itex]ds^2 = 0 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2 dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2\,[itex]

This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old.", we are looking down this line of sight: a null geodesic. We are looking at an event

[itex]d = \sqrt{x_1^2+x_2^2+x_3^2}\,[itex]

meters away and d/c seconds in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".) ￼The cone in the −t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.

The biggest difference with adding in extra dimensional hyperspace is that the cone spreads out and shortens in height for the hyperspace frame almost becoming an instanton where far larger distances are encompassed in relation to our normal space-time frame. However, when you try and compare events from one frame to another with the limits our frame imposes you find that while event rates in hyperspace yield a superluminal path that path in relation to our frame moves into the future. Thus, while for anyone using such a superluminal path (see Fernando Loups works on Cern) their journey will appear to be faster than light in our normal space-time they have simply journeyed into our far future. What has happened is the null geodesics which are continuous concentric spheres, with radius = distance = c·([itex]\pm[itex])time, has in the hyperspace frame, our normal concentric sphere a vastly shrunken version. Yet, in our frame of reference the hyperspace sphere may be many spheres removed from our own.

When a careful comparison of frame to frame is done one finds that unless one cares to limit this future time travel through hyperspace to C or less paths that such travel does not actually get around some of the current problems we face when it comes to traveling to the stars.

• Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons (Oxford University Press) by Clifford A. Pickover
• The Fabric of the Cosmos: Space, Time, and the Texture of Reality (Knopf) by Brian Greene
• Hyperspace A Vanishing Act by P. Hoiland (http://doc.cern.ch//archive/electronic/other/ext/ext-2004-109.pdf)

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