# Orthogonal - a model of the world with an alternative theory of relativity

In 2011-2013 Australian writer Greg Egan has published the Orthogonal trilogy (The Clockwork Rocket, The Ethernal Flame, The Arrows of Time). The books describe an amazing world in which there are no fluids and electric charges, four-eyed intelligent creatures live, able to change shape and multiply by fission, using air not for chemical reactions, but for cooling their body, and light - for transmitting nerve impulses. The speed of light in this world is variable: violet photons move much faster than red ones. Therefore, stars do not look like white dots, but like rainbow stripes

Even in the first book, the heroes found out that the reason for this behavior of light lies in the properties of the space-time of their universe: unlike our world, which is Minkowski space, their spatial and temporal coordinates are completely equal. Any body moves along its path in four-dimensional space-time with constant speed, uniform motion there looks like a straight line, and accelerated - like an arc. For example, a flight of a spaceship to another star and vice versa can be represented as follows:

We consider the thrust of the ship during acceleration and braking to be constant, and in this case the trajectory of its movement in space-time will be an arc of a circle. In a finite time, the ship will reach infinite (by the clock of a stationary observer) speed, and the main part of the flight will take place in zero time. At the same time, the time for the passengers of the ship will go as usual, and it can be measured along the length of the trajectory in the figure. When the ship returns to the starting point, it turns out that only a few years have passed on the home planet, while centuries could have passed for the passengers of the ship. Moreover, if the acceleration / deceleration phases last a little longer, the ship may return at the same moment when it started, or maybe even earlier:

True, the Universe will have to somehow solve the paradoxes that arise in this case, and these decisions may turn out to be unexpected for the inhabitants of the planet.
Photons obey the same laws as other bodies. They differ in their own vibrations. All photons are the same, and in their own frame of reference they have the same frequency (and zero wavelength). But when the observer sees photons moving at different speeds, then their frequencies seem different to him. For red light (the slowest), the frequency and wavelength is minimal, and for violet it is maximum:

Here, the red line is the trajectory of the red photon, and the violet is the trajectory of the violet, respectively. The eye sees photons whose trajectories are between these lines.
The heroes of the book see light in a speed range from 76/144 to 192/144 of the speed of blue light (blue photons are those that fly in spacetime at an angle of 45 degrees to the observer, that is, their apparent speed in space is equal to any speed reference systems in space-time). Thus, the observer sees only those photons whose trajectory lies between the two cones:
Half the angle at the apex of the inner (red) cone is 27 degrees, and the outer (purple) is 54 degrees. If the trajectory of a star intersects this space, then the star can be seen:

Here, a slowly moving star was considered. If the star’s speed becomes greater, then the trajectory will consist of two parts:

Soon after the start of the trilogy, strange stars began to appear in the sky - Hartlers (Hurtler - a pest?). They arose as violet dots, from which iridescent stripes quickly diverged in two directions:

Heroes suggested (and correctly) that hartlers are stars whose trajectory is perpendicular to the trajectory of their world. That is, each such star exists only at one moment in time, but at the same time occupies its entire trajectory. In space-time, it could look like this:

But if you look closely at this picture, it turns out that half of the trajectory directed towards the hartler’s movement is formed by photons that fly backward in time in the hartler’s frame of reference! Judging by the contents of the third book, stars do not emit such photons, so in reality the picture should look like this:

I was wondering what such a universe would look like if you navigate it in a very maneuverable ship. To do this, I decided to write a game with the simplest plot - there is the Universe, there are several stars in it that need to be visited and redeemed (just by flying nearby).
The first question was what form of space-time to choose. The heroes of the trilogy quickly came to the conclusion that the Universe should be finite, but for a long time doubted which one. As a result, they came to the conclusion that this should be a four-dimensional sphere (i.e. a sphere in 5-dimensional space). True, for some reason they needed to have regions with negative curvature in it (otherwise there would be some problems with entropy), that is, the sphere should be distorted in shape. But for simplicity, I took the usual homogeneous sphere.
The position and speed of the ship are described by a pair of perpendicular vectors. Vector P determines the current position in time space, V is the drift direction (drift speed is always the same). In addition, we need three vectors X, Y, Z, which determine the orientation of the ship in space (and the image on the screen). All vectors are taken in 5-dimensional space, have a length of 1 and are perpendicular to each other. Thus, the ship is described by the 5 * 5 orthogonal matrix.
It turns out that all the movements and maneuvers of the ship in this representation are just rotations of the matrix in coordinate planes. General drift - rotation in the (P, V) plane (vectors X, Y, Z remain unchanged), acceleration and braking in 3D - rotations in the (V, Z) plane, lateral accelerations - rotations in (V, X) and in ( V, Y), change of orientation of the ship - rotation in (X, Z) and (Y, Z). Rotation speeds are determined by the general parameters of the game, and they can be changed on the control panel.
The star trajectory is also a pair of perpendicular vectors (P0, V0). At any time T (according to the clock of the star itself), its position will be P1 = P0 * cos (T) + V0 * sin (T), and the drift velocity will be V1 = V0 * cos (T) -P0 * sin (T). To get an image of a star, we need to determine the parameters of a photon emitted from the point (P1, V1) and reaching our point (P, V): what color it will be and from which side it will arrive. To do this, it is enough for us to connect the points P and P1 with an arc of a large circle and see which way it goes from point P and which side it enters into P1.
For simplicity, we assume that a photon cannot fly more than a quarter of a circle. In fact, the book says nothing about the image of a star visible from the night side of the planet, or about phantom stars from the opposite side of the universe, the light from which was focused in the vicinity of the world of heroes of the trilogy. This means that it suffices to consider the case when the angle between the vectors P and P1 is sharp, i.e. (P, P1)> 0. It turns out that, firstly, the condition (P, V1)> 0 must be fulfilled - otherwise the star would have to emit a photon into the past, and secondly, (P1, V)> 0 - otherwise the photon will fly to us from the future, and without special means we can’t see him.
After that, it suffices to project the vector P1 onto the space (V, X, Y, Z) (tangent to the sphere at the point P). Let the vector S = (v, x, y, z) be obtained. Then the length L of the vector S corresponds to the distance traveled by the photon (more precisely, equal to its sine), the quantity v / L is the cosine of the angle between the path of the photon and our path in space-time, which determines the color of the photon, and (x, y, z) - the direction from which the photon arrived - and we can depict it by the usual methods.
It turns out that catching stars is far from easy. The simplest case is when the star is close, and our speeds do not differ very much (as was the first picture with cones). With the help of outboard engines, we can easily eliminate lateral speeds. The star on the screen from the rainbow strip will turn into a point, our trajectories in space-time will be in the same plane, and we will fly to the star () or from it - it's hard to tell in advance).

A natural desire is to aim at a star and begin to accelerate. But what will happen?

It can be seen that the portion of the trajectory that we see is getting farther - in fact, we are starting to see the star’s increasingly distant past. In addition, the visible star moves away and becomes smaller and dimmer. And if we have at least a slight lateral displacement, then we will see that the red part of the trajectory contracts, stopping on the green, and then on the blue part of the spectrum.
At this point, you can slow down a little and wait until the star becomes larger in size. And then start to catch her. But what happens if we miss?

From the side where we looked, the purple part of the track suddenly changes to red, and we will see that it is moving away. And the star itself will be on the opposite side from us. Therefore, we need to deploy the ship and move to the star again.

But catching a close star is not very difficult. Problems begin when the star is far away and we see a long and thin rainbow trail. Where to look for a star, where to fly is completely impossible to understand. Sometimes I succeed. More often than not.

If you want to experiment with this space, you can get the exe file here (for .NET 3.5), and the source here .
A more detailed description of the laws of physics of this world (up to the quantum level) is on the site of the author of the trilogy: www.gregegan.net