The legomenon blog

July 13, 2008

Why three space dimensions are necessary for sight

Filed under: Philosophy — legomenon @ 6:26 pm

John Wheeler is reported to have said:

Time is what prevents everything from happening at once.
Space is what prevents everything from happening to me.

I won’t address the dimensionality of time here, I’ll save that for a later post maybe. Let me just briefly remark that I think the observation that time is 1-dimensional is a consequence of the fact that only a 1-dimensional time makes it possible for different observers to share an objective reality where causality is possible. So let’s stick with one single time dimension, and consider space and how it’s possible to see things at a distance.

In order to see things clearly at a distance, it must be possible to transport light signals across space with little or no distortion. Since light is a type of electromagnetic wave, the propagation of waves must allow such transporting of signals. The wave equation in n-dimensional space looks like this:

\displaystyle \frac{\partial^{2}\psi}{\partial x_1^{2}} + \frac{\partial^{2}\psi}{\partial x_2^{2}} + \dotsb + \frac{\partial^{2}\psi}{\partial x_n^{2}} = \frac{1}{c^2}\frac{\partial^{2}\psi}{\partial t^2}

where c is the propagation speed of the waves, and \psi represents the stuff that is “waving”, whatever it is. More precisely, \psi is a function of space and time coordinates, and the value of that function can be any mathematical object for which partial derivatives are defined: for example a simple scalar, a complex number or quaternion, a vector, a matrix, and a lot of other things.

Now imagine a wave that propagates from a point-like source, symmetrically in all directions. The symmetry allows us to write the wave equation in terms of only r and t, where r is the distance from the source. This spherically symmetric wave equation looks like this:

\displaystyle \frac{\partial^{2}\psi}{\partial r^{2}} + \left(\frac{n-1}{r}\right)\frac{\partial\psi}{\partial r} = \frac{1}{c^2}\frac{\partial^{2}\psi}{\partial t^2}

In order to transport an arbitrary signal, the wave function must hold for all solutions of the form

\displaystyle \psi = f(r-ct) \gamma(r)

where f is an arbitrary function that represents the shape of the source signal, and \gamma is some fixed function that is independent of f. The factor \gamma(r) attenuates the signal as it spreads outward from the source. Since the wave equation must hold for any f, it must in particular hold for the case f \equiv 1, which gives the condition:

\displaystyle \frac{\partial^{2}\gamma}{\partial r^{2}} + \left(\frac{n-1}{r}\right)\frac{\partial\gamma}{\partial r} = 0

which, ignoring irrelevant constants and the pathological case n=1, has the solution

\displaystyle \gamma(r) = \frac{1}{r^{n-2}}

which gives us \displaystyle \psi = \frac{f(r-ct)}{r^{n-2}} for the general case, since \gamma is independent of f.

Inserting this into the spherically symmetric wave equation produces, after simplifications,

\displaystyle \frac{f''}{r^{n-2}} - 2\left(\frac{n-2}{r^{n-1}}\right)f' + \left(\frac{n-1}{r^{n-1}}\right)f' = \frac{f''}{r^{n-2}}

where f' and f'' are the first and second derivates of the function f with respect to its parameter. After further simplification the final result is simply n=3. Voila! So faithful propagation of arbitrary signals via spherically symmetric waves is only possible in 3 space dimensions.

Notes:

• I only bothered with so-called “retarded” waves. Since the wave equation only uses the second derivative with respect to t, any linear combination with a solution that uses -c instead of c is also a solution. Solutions of the form f(r+ct) are called “advanced” waves and are the time-inverse of the normal retarded solutions.

• Spherical symmetry is an idealization for electromagnetic waves, since the electric and magnetic fields (or the Faraday tensor if you prefer) have directions in space which are transverse to the direction of wave propagation. See The Hairy Ball Theorem. This doesn’t affect our conclusion, however.

• There are some speculative exotic theories (string theory etc) that postulate several extra “compact” space dimensions in addition to the familiar ones that we already know about. This does not affect our conclusion either, since such hypothetical dimensions do not contribute to any degrees of freedom in a macroscopical wave equation.

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