Visualizing Geodesics in Space

A previous presentation displayed the geodesics of two-dimensional metrics as surfaces embedded in three-dimensional space. With a third variable in the metric, the resulting ‘surfaces’ would need four dimensions for complete visualization. This is physically difficult for mere mortals.

This presentation will instead display such geodesics in three dimensions to explore whether one can intuit the structure of the higher-dimensional ‘surfaces’ from within our own physical space. Since the visualizations become complicated quickly, only a few compact metrics will be considered, all diagonal in form for simplicity of the mathematics.

In the previous presentation, it was pointed out that one can derive geodesic equations straightforwardly from a Lagrangian describing a given metric. Define this function for the present setting as

$L = \frac{1}{2} [E {\overset{\cdot}{u}}^{2} + G {\overset{\cdot}{v}}^{2} + J {\overset{\cdot}{t}}^{2}]$

where the dot is differentiation with respect to invariant interval. The notation of the first two variables and their corresponding metric functions is kept consistent with the previous presentation. The Lagrange equations are

$\frac{d}{d s} \frac{\partial L}{\partial {\overset{\cdot}{q}}_{i}} = \frac{\partial L}{\partial q_{i}}$

which under explicit evaluation become

$\begin{array}{r} \frac{d}{d s} [E \overset{\cdot}{u}] = \frac{1}{2} [E_{1} {\overset{\cdot}{u}}^{2} + G_{1} {\overset{\cdot}{v}}^{2} + J_{1} {\overset{\cdot}{t}}^{2}] \\ \frac{d}{d s} [G \overset{\cdot}{v}] = \frac{1}{2} [E_{2} {\overset{\cdot}{u}}^{2} + G_{2} {\overset{\cdot}{v}}^{2} + J_{2} {\overset{\cdot}{t}}^{2}] \\ \frac{d}{d s} [J \overset{\cdot}{t}] = \frac{1}{2} [E_{3} {\overset{\cdot}{u}}^{2} + G_{3} {\overset{\cdot}{v}}^{2} + J_{3} {\overset{\cdot}{t}}^{2}] \end{array}$

Expanding left-hand sides and rearranging, the second derivatives of the independent variables are

$\begin{array}{l} \overset{\cdot \cdot}{u} = \frac{1}{2 E} [- E_{1} {\overset{\cdot}{u}}^{2} + G_{1} {\overset{\cdot}{v}}^{2} + J_{1} {\overset{\cdot}{t}}^{2} - 2 E_{2} \overset{\cdot}{u} \overset{\cdot}{v} - 2 E_{3} \overset{\cdot}{u} \overset{\cdot}{t}] \\ \overset{\cdot \cdot}{v} = \frac{1}{2 G} [E_{2} {\overset{\cdot}{u}}^{2} - G_{2} {\overset{\cdot}{v}}^{2} + J_{2} {\overset{\cdot}{t}}^{2} - 2 G_{1} \overset{\cdot}{u} \overset{\cdot}{v} - 2 G_{3} \overset{\cdot}{v} \overset{\cdot}{t}] \\ \overset{\cdot \cdot}{t} = \frac{1}{2 J} [E_{3} {\overset{\cdot}{u}}^{2} + G_{3} {\overset{\cdot}{v}}^{2} - J_{3} {\overset{\cdot}{t}}^{2} - 2 J_{1} \overset{\cdot}{u} \overset{\cdot}{t} - 2 J_{2} \overset{\cdot}{v} \overset{\cdot}{t}] \end{array}$

As remarked in the previous presentation, these equations are generally nonlinear and can be left in this form for the required numerical integration. Note also with $t = 0$ and ignoring derivatives with respect to this vanishing variable, the equations are equivalent to those of the previous presentation with $F = 0$ .

As a first and simplest example, consider the metric of a 3-sphere. To keep the notation as consistent with the lower dimensional case as possible, choose the parametrization

$\begin{array}{l} z = a cos u \\ x = a sin u cos v \\ y = a sin u sin v cos t \\ w = a sin u sin v sin t \end{array} 0 \leq u \leq π, 0 \leq v \leq 2 π, 0 \leq t \leq 2 π$

When extending a three-dimensional parametrization to one higher dimension, there are clearly multiple slots to add the new variable. For simplicity and consistency of the presentation, the additional variable will extend the last coordinate as written. That means $t = 0$ implies $w \equiv 0$ , reducing this metric to a 2-sphere in three dimensions.

The corresponding three-dimensional metric is

$d s^{2} = d x^{2} + d y^{2} + d z^{2} + d w^{2} = a^{2} (d u^{2} + {sin}^{2} u d v^{2} + {sin}^{2} u {sin}^{2} v d t^{2})$

The geodesic equations for this metric are

$\begin{array}{l} \overset{\cdot \cdot}{u} = \frac{1}{2 E} [G_{1} {\overset{\cdot}{v}}^{2} + J_{1} {\overset{\cdot}{t}}^{2}] = sin u cos u ({\overset{\cdot}{v}}^{2} + {sin}^{2} v {\overset{\cdot}{t}}^{2}) \\ \overset{\cdot \cdot}{v} = \frac{1}{2 G} [J_{2} {\overset{\cdot}{t}}^{2} - 2 G_{1} \overset{\cdot}{u} \overset{\cdot}{v}] = sin v cos v {\overset{\cdot}{t}}^{2} - 2 cot u \overset{\cdot}{u} \overset{\cdot}{v} \\ \overset{\cdot \cdot}{t} = - \frac{1}{J} [J_{1} \overset{\cdot}{u} \overset{\cdot}{t} + J_{2} \overset{\cdot}{v} \overset{\cdot}{t}] = - 2 cot u \overset{\cdot}{u} \overset{\cdot}{t} - 2 cot v \overset{\cdot}{v} \overset{\cdot}{t} \end{array}$

which reduce as expected to those of the 2-sphere for $t = 0$ .

Since one cannot of course visualize four dimensions directly, a projection of some sort to a lower dimension must be used, and stereographic projection appears most appropriate. This procedure has been discussed in general form in another presentation, but the necessary formula can be derived rather simply.

Consider a point on a sphere and a line through it from the top of the sphere to an intersecting plane through the origin. In cross section this looks like

Evaluating the line through the two points and solving for the projected value gives

$w = a - \frac{x}{X} \to X = \frac{x}{a - w}$

This simple formula will be used for projections in this presentation, where the extension to a higher dimension will employ a spherical variable.

Before visualizing the projected geodesics of a 3-sphere, an application first to a 2-sphere is most informative. Here is an interactive graphic from the previous presentation with a few modifications, illustrating the projection from three dimensions down to two:

Note that the projection being used produces a stable visual result regardless of the parameter a. This is because the parameter multiplies all variables involved, and so factors out of the projection formula. The parameter can thus be set equal to unity in what follows, not being useful in practice for higher-dimensional graphics.

A spray of geodesics in two dimensions is set in the initial conditions from a single angular variable, and so can be colored by translating this value to a hue for the line. In three dimensions one needs two angular variables for the spray, and assigning unique colors to each line takes somewhat more effort. Fortunately this has been developed in another presentation, whose final color component functions have been copied into the source of this document.

The resulting sprays are visualized in three dimensions as

These sprays represent directions in the three-dimensional angular space, and so can be used to set initial angular velocities around an initial point in angular space.

And now for the the real meat of this presentation! The projection of geodesics on a 3-sphere down one dimension looks like this:

These projected geodesics have the general form of a sphere, as should be expected. By extending the interval of integration one can see how they wrap around the 3-sphere to a location opposite the initial point.

Setting the initial angular velocities from the spray of geodesics around the initial point encounters the same curiosity of code as that in the two-dimensional case. For the 2-sphere the differential of v is multiplied by $sin u$ in the metric, so its initial velocity was divided by this quantity to achieve parity with the variable u when assigning equally spaced values. In this three-dimensional case, the differential of v has the same factor of $sin u$ , while the differential of t has the factor $sin u sin v$ . For both of these variables the initial velocities are divided by their respective factors for the same reason.

Further, the regions around $u = 0$ and $u = π$ , as well as those around $v = 0$ and $v = 2 π$ are avoided due to singularities in the second and third geodesic equations.

As a next example consider the metric of a 3-ellipsoid of rotation with one longer axis along the topmost variable. The coordinate parametrization is

$\begin{array}{l} z = c cos u \\ x = a sin u cos v \\ y = a sin u sin v cos t \\ w = a sin u sin v sin t \end{array} 0 \leq u \leq π, 0 \leq v \leq 2 π, 0 \leq t \leq 2 π$

The corresponding three-dimensional metric is

$\begin{array}{l} d s^{2} = d x^{2} + d y^{2} + d z^{2} + d w^{2} \\ = (a^{2} {cos}^{2} u + c^{2} {sin}^{2} u) d u^{2} \\ + a^{2} ({sin}^{2} u d v^{2} + {sin}^{2} u {sin}^{2} v d t^{2}) \end{array}$

The geodesic equations for this metric are

$\begin{array}{l} \overset{\cdot \cdot}{u} = \frac{1}{2 E} [- E_{1} {\overset{\cdot}{u}}^{2} + G_{1} {\overset{\cdot}{v}}^{2} + J_{1} {\overset{\cdot}{t}}^{2}] \\ = \frac{sin u cos u}{a^{2} {cos}^{2} u + c^{2} {sin}^{2} u} [(a^{2} - c^{2}) {\overset{\cdot}{u}}^{2} + a^{2} {\overset{\cdot}{v}}^{2} + a^{2} {sin}^{2} v {\overset{\cdot}{t}}^{2}] \\ \overset{\cdot \cdot}{v} = \frac{1}{2 G} [J_{2} {\overset{\cdot}{t}}^{2} - 2 G_{1} \overset{\cdot}{u} \overset{\cdot}{v}] = sin v cos v {\overset{\cdot}{t}}^{2} - 2 cot u \overset{\cdot}{u} \overset{\cdot}{v} \\ \overset{\cdot \cdot}{t} = - \frac{1}{J} [J_{1} \overset{\cdot}{u} \overset{\cdot}{t} + J_{2} \overset{\cdot}{v} \overset{\cdot}{t}] = - 2 cot u \overset{\cdot}{u} \overset{\cdot}{t} - 2 cot v \overset{\cdot}{v} \overset{\cdot}{t} \end{array}$

which differ from the previous example by one additional term and a denominator in the first equation. Naturally for $c = a$ these equations are the same as those of the first example.

While the separate parameter a is as previously stated not useful, the geodesic equations do depend on the ratio $c / a$ which cannot be ignored. The same ratio will appear in the projection down to three dimensions but only on the topmost variable, where it does not factor out.

The projection of geodesics on a 3-ellipsoid down one dimension is thus

The projected geodesics here are similar to those of the 3-sphere, but can as expected appear oblate or prolate depending on whether $c / a < 1$ or $c / a > 1$ , respectively.

For the final example, consider a torus extended into a fourth dimension by an additional spherical variable. The coordinate parametrization to be used is

$\begin{array}{l} z = a cos u \\ x = (b + a sin u) cos v \\ y = (b + a sin u) sin v cos t \\ w = (b + a sin u) sin v sin t \end{array} 0 \leq u \leq 2 π, 0 \leq v \leq 2 π, 0 \leq t \leq 2 π$

This parametrization is identical to that of the 3-sphere when $b = 0$ , thereby indicating which of the two parameters can be set to unity in practice. The corresponding three-dimensional metric is

$\begin{array}{l} d s^{2} = d x^{2} + d y^{2} + d z^{2} + d w^{2} \\ = a^{2} d u^{2} + (b + a sin u)^{2} (d v^{2} + {sin}^{2} v d t^{2}) \end{array}$

The geodesic equations for this metric are

$\begin{array}{l} \overset{\cdot \cdot}{u} = \frac{1}{2 E} [G_{1} {\overset{\cdot}{v}}^{2} + J_{1} {\overset{\cdot}{t}}^{2}] = \frac{1}{a} (b + a sin u) cos u ({\overset{\cdot}{v}}^{2} + {sin}^{2} v {\overset{\cdot}{t}}^{2}) \\ \overset{\cdot \cdot}{v} = \frac{1}{2 G} [J_{2} {\overset{\cdot}{t}}^{2} - 2 G_{1} \overset{\cdot}{u} \overset{\cdot}{v}] = sin v cos v {\overset{\cdot}{t}}^{2} - \frac{2 a cos u}{b + a sin u} \overset{\cdot}{u} \overset{\cdot}{v} \\ \overset{\cdot \cdot}{t} = - \frac{1}{J} [J_{1} \overset{\cdot}{u} \overset{\cdot}{t} + J_{2} \overset{\cdot}{v} \overset{\cdot}{t}] = - \frac{2 a cos u}{b + a sin u} \overset{\cdot}{u} \overset{\cdot}{t} - 2 cot v \overset{\cdot}{v} \overset{\cdot}{t} \end{array}$

which reduce as expected to those of a torus in three-dimensional space for $t = 0$ .

The separate parameter a is again not useful, but like the previous example the geodesic equations depend on a ratio of parameters, in this case $b / a$ . This ratio will also appear in the downward projection in a manner somewhat more complicated than the previous example, but straightforward.

The projection of geodesics on this four-dimensional torus down one dimension looks like this:

The maximum range of this interactive graphic has been increased relative to the other two due to the extended nature of these projected geodesics. As expected the graphic looks like the first one for $b / a = 0$ but markedly different in general, although one can distinguish geodesics around the smaller radius of the torus.

This presentation is an experiment in identifying higher-dimensional behavior in our three-dimensional reality. These spatially four-dimensional objects are clearly mathematical abstractions, but there are real-world applications possible for this method: the general relativistic geodesics around a gravitating mass. Can one see a black hole simply from its geodesics? To be continued...