Special relativity shader in GLSL

Question

I'm trying to implement a GLSL shader which helps understanding special relativity Lorentz Transformation.

Let's take two axis-aligned inertial observer O and O' . The observer O' is in motion w.r.t observer O with velocity v=(v_x,0,0).

When described in terms of O' coordinates, an event P' = (x',y',z',ct') has transformed coordinates (x,y,z,ct)= L (x',y',z',ct')

where L is a 4x4 matrix called Lorentz transformation which helps us writing the coordinates of event P' in O coordinates.

(for details look http://en.wikipedia.org/wiki/Lorentz_transformation#Boost_in_the_x-direction)

I've wrote down a first preliminary vertex shader that apply the Lorentz transformation given the velocity to every vertex, but I can't get the transformation to work correctly.

vec3 beta= vec3(0.5,0.0,0.0);
float b2 = (beta.x*beta.x + beta.y*beta.y + beta.z*beta.z )+1E-12; 
float g=1.0/(sqrt(abs(1.0-b2))+1E-12); // Lorentz factor (boost)
float q=(g-1.0)/b2;

//http://en.wikipedia.org/wiki/Lorentz_transformation#Matrix_forms
vec3 tmpVertex = (gl_ModelViewMatrix*gl_Vertex).xyz;
float w = gl_Vertex.w;

mat4  lorentzTransformation =
        mat4(
            1.0+beta.x*beta.x*q ,   beta.x*beta.y*q ,   beta.x*beta.z*q , beta.x*g ,
            beta.y*beta.x*q , 1.0+beta.y*beta.y*q ,   beta.y*beta.z*q , beta.y*g ,
            beta.z*beta.x*q ,   beta.z*beta.y*q , 1.0+beta.z*beta.z*q , beta.z*g ,
            beta.x*g , beta.y*g , beta.z*g , g
            );
vec4 vertex2 = (lorentzTransformation)*vec4(tmpVertex,1.0);


gl_Position = gl_ProjectionMatrix*(vec4(vertex2.xyz,1.0) );

This shader should apply to every vertex and perform the non-linear Lorentz transformation, but the transformation it performs is clearly different from what I'd expect (in this case a length-contraction on x-axis).

Has somebody already worked on special relativity shader for 3D videogame?

Answer 1

To implement Lorentz contraction, your best bet is probably just to explicitly scale the object by 1/gamma along the direction of motion.

The trouble is that the Lorentz transformation displaces vertices in the time direction as well as in space, so by itself it will not give you what a moving object looks like at a specific moment in time. To do that, you would have to first transform the whole object then take a "slice" through it parallel to the space axes, like in this diagram:

To calculate this for real, you'd effectively have to raytrace in 4D, intersecting the world-line of the vertex with the 3D hyperplane of the current moment of time in the observer's reference frame. I believe the result of doing this is the same as simply scaling by 1/gamma.

(For extra credit, take into account the fact that an observer wouldn't actually see the whole object at a single moment in time: they'd see it using light rays. So you'd need to intersect the world-line of the vertex with the past light cone of the observer. This actually changes the results significantly: an object moving away from you will look shortened, but an object moving toward you will appear elongated and an object moving sidways will be rotated - see Penrose-Terrell rotation for more.)