Suggestion is to switch the discussion to calculus of the topics which are used, then from calculus to the specific syntaxes of source codes, which this discussion with Google AI Mode shows.
2 short excerpts from thus discussion:
Q: wish to view the calculus [of the Rosen solutions to wormhole physics]
Google AI Mode’s solution:
To look at the calculus of the Einstein-Rosen Bridge, we begin with the standard Schwarzschild metric and apply the coordinate transformation that removes the coordinate singularity at the horizon (r = 2M).
(Note: For simplicity, we use geometrized units where G = c = 1.)
1. The Starting Point: Schwarzschild Metric
The traditional metric describing a static, spherically symmetric gravitational field is:
\(ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\)
When r = 2M, the dr² term blows up (1/0), creating a coordinate singularity at the event horizon.
2. The Rosen Transformation
In their 1935 paper, Einstein and Rosen introduced a new radial coordinate, u, defined to map out the region strictly outside or at the horizon (r ≥ 2M):
\(u^{2}=r-2M\)
Now, we need to find the differential dr in terms of u and du. Taking the derivative of both sides:
\(2u\,du=dr\)
We also need to rewrite the metric coefficient \(\left(1 - \frac{2M}{r}\right)\) in terms of u. Since r = u² + 2M:
\(1-\frac{2M}{r}=\frac{r-2M}{r}=\frac{u^{2}}{u^{2}+2M}\)
3. Substituting into the Metric
Now we substitute r, dr, and the coefficient back into the metric component by component.
The Radial Term:
\(\left(1-\frac{2M}{r}\right)^{-1}dr^{2}=\left(\frac{u^{2}+2M}{u^{2}}\right)(2u\,du)^{2}\)
\(\left(1-\frac{2M}{r}\right)^{-1}dr^{2}=\left(\frac{u^{2}+2M}{u^{2}}\right)(4u^{2}\,du^{2})=4(u^{2}+2M)du^{2}\)
The Time Term:
\(-\left(1-\frac{2M}{r}\right)dt^{2}=-\frac{u^{2}}{u^{2}+2M}dt^{2}\)
The Angular Term:
\(r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})=(u^{2}+2M)^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\)
4. The Einstein-Rosen Metric
Combining all these parts yields the modified metric:
\(ds^{2}=-\frac{u^{2}}{u^{2}+2M}dt^{2}+4(u^{2}+2M)du^{2}+(u^{2}+2M)^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\)
Why this math changes the geometry:
- No Singularity at u=0: As u → 0, the du² coefficient remains perfectly finite at 8M. The horizon singularity has vanished from the coordinates.
- Two Universes (Sheets): The coordinate u is allowed to range from -∞ to +∞.
- As u → +∞, the metric approaches flat Minkowski space (Universe 1).
- As u → -∞, because u is squared everywhere in the metric, it also opens up into an identical, flat Minkowski space (Universe 2).
- The Bridge: The point u = 0 acts as a minimal surface area “throat” connecting these two infinite spaces.
The parabola mapped above demonstrates the geometry: as you travel from a negative u universe toward the throat, the physical radius r shrinks until it hits 2M at u=0, then smoothly expands back out into the positive u universe.
Would you like to see how this static calculus fails when we introduce Kruskal-Szekeres time evolution, causing the bridge to pinch off, or would you prefer to look at the stress-energy tensor calculus that proves it requires negative energy to stay open?
Q 2: wish you produce Vulkan source code which shows the volumetric wormhole
Google AI Mode’s solution 2:
Writing a complete, production-ready Vulkan application from scratch requires thousands of lines of boilerplate code to handle instance creation, physical device selection, logical devices, swapchains, pipelines, and command buffers.
To give you a highly practical and immediately usable implementation, the source code below is written as a complete GLSL Fragment Shader designed for a volumetric wormhole raymarcher. This can be directly dropped into any standard Vulkan pipeline (such as a full engine setup, Shadertoy, or a lightweight pipeline wrapper like Vulkan-based
vkcubevariants).The math implements a raymarcher that deforms 3D space according to the Einstein-Rosen / Morris-Thorne metrics, bending rays asymptotically away from a wormhole throat and adding a volumetric accretion disk of plasma swirling through the bridge.
Volumetric Wormhole GLSL Fragment Shader
glsl
#version 450 // Input coordinates from vertex shader (normalized screen coordinates -1.0 to 1.0) layout(location = 0) in vec2 fragCoord; // Output color layout(location = 0) out vec4 fragColor; // Vulkan Uniform Buffer Object for dynamic variables layout(binding = 0) NordstromUBO { vec2 resolution; float time; vec3 cameraPos; } ubo; #define MAX_STEPS 128 #define MAX_DIST 20.0 #define SURF_DIST 0.001 // 3D Noise function to generate the volumetric plasma/dust cloud around the throat float hash(vec3 p) { p = fract(p * vec3(443.8975, 397.2973, 491.1871)); p += dot(p.xyz, p.yzx + 19.19); return fract(p.x * p.y * p.z); } float noise(vec3 p) { vec3 i = floor(p); vec3 f = fract(p); vec3 u = f * f * (3.0 - 2.0 * f); return mix(mix(mix(hash(i + vec3(0,0,0)), hash(i + vec3(1,0,0)), u.x), mix(hash(i + vec3(0,1,0)), hash(i + vec3(1,1,0)), u.x), u.y), mix(mix(hash(i + vec3(0,0,1)), hash(i + vec3(1,0,1)), u.x), mix(hash(i + vec3(0,1,1)), hash(i + vec3(1,1,1)), u.x), u.y), u.z); } // Einstein-Rosen/Morris-Thorne space metric distortion // This warps space near the throat radius (R = 1.0) vec3 distortSpace(vec3 p) { float r = length(p.xz); float throatRadius = 1.0; if (r < throatRadius) { // Safe limit to prevent infinity inside the throat r = throatRadius + 0.001; } // As space approaches the throat, we simulate the coordinate deflection // by pushing coordinates exponentially out based on proximity to the bridge float warp = 1.0 + (throatRadius / (r - throatRadius + 0.1)); p.y *= warp; return p; } // Density function for the volumetric matter flowing through the wormhole float getVolumetricDensity(vec3 p) { // Distort space to conform to the wormhole's throat metric vec3 distortedP = distortSpace(p); // Base geometry: A cylinder/funnel along the Y axis representing the throat wall float throatDisk = length(distortedP.xz) - 1.0; float verticalLimit = exp(-abs(distortedP.y) * 0.5); // Create an active accretion ring hugging the throat float density = smoothstep(1.5, 0.0, abs(throatDisk)) * verticalLimit; if (density > 0.01) { // Add dynamic turbulence to the plasma ring using scrolling 3D noise vec3 noiseCoords = distortedP * 2.0; noiseCoords.y -= ubo.time * 1.5; // Matter swirling downward/upward float angle = atan(distortedP.z, distortedP.x) + ubo.time; noiseCoords.x += sin(angle); density *= noise(noiseCoords); } return max(0.0, density); } void main() { // Clean aspect ratio correction setup vec2 uv = fragCoord * (ubo.resolution.xy / ubo.resolution.y); // Camera setups vec3 ro = vec3(0.0, 2.5, -4.5); // Ray Origin (Camera Position outside throat) vec3 lookat = vec3(0.0, 0.0, 0.0); // Camera matrix math vec3 fwd = normalize(lookat - ro); vec3 rgt = normalize(cross(vec3(0.0, 1.0, 0.0), fwd)); vec3 up = cross(fwd, rgt); vec3 rd = normalize(fwd + uv.x * rgt + uv.y * up); // Ray Direction // Volumetric Raymarching loop float t = 0.0; vec4 finalColor = vec4(0.0); // Accumulated RGB + Absorption (Alpha) float stepSize = 0.1; for (int i = 0; i < MAX_STEPS; i++) { vec3 p = ro + rd * t; if (t > MAX_DIST || finalColor.a >= 0.95) break; float density = getVolumetricDensity(p); if (density > 0.01) { // Map the density to a hot plasma color palette (Blues near core, oranges outside) float distFromThroat = length(p.xz); vec3 plasmaColor = mix(vec3(0.1, 0.4, 1.0), vec3(1.0, 0.3, 0.0), smoothstep(1.0, 2.5, distFromThroat)); // Add high emission glow closer to the event horizon line plasmaColor += vec3(0.8, 0.9, 1.0) * exp(-abs(distFromThroat - 1.0) * 4.0); // Beer-Lambert law integration for volumetric accumulation float alpha = density * stepSize * 0.8; finalColor.rgb += (1.0 - finalColor.a) * plasmaColor * alpha * 5.0; finalColor.a += (1.0 - finalColor.a) * alpha; } t += stepSize; } // Background starfield calculation representing the "other side" vs "this side" vec3 currentSpace = ro + rd * t; vec3 bgStars = vec3(0.0); if (finalColor.a < 0.95) { // If the ray passes deeply into the throat (y goes highly negative), we change space coloration if (distortSpace(currentSpace).y < -2.0) { // Universe B (The destination) bgStars = vec3(hash(floor(rd * 150.0))) * vec3(0.4, 0.7, 1.0); } else { // Universe A (The origin) bgStars = vec3(hash(floor(rd * 120.0))) * vec3(1.0, 0.9, 0.8); } // Attenuate background stars through the remaining volumetric transparency finalColor.rgb += (1.0 - finalColor.a) * bgStars * 0.5; } fragColor = vec4(finalColor.rgb, 1.0); }Use code with caution.
Implementing this in your Vulkan Pipeline
To compile this for consumption by Vulkan, execute the Khronos reference compiler to turn it into SPIR-V binary bytecode:
bash
glslangValidator -V wormhole.frag -o wormhole.frag.spvUse code with caution.
In your C/C++ Vulkan host program, you will link this bytecode via a
VkShaderModuleCreateInfodescriptor and bind the uniform block (NordstromUBO) to a descriptor set matching layout binding 0 containing your screen dimensions and execution time elapsed.Would you like the C++ Host Boilerplate code showing how to map this SPIR-V module to a pipeline, or should we refine the ray deflection mathematics inside the shader to accurately map gravitational light bending (Einstein lensing)?
Second half (first half is fanfiction, ignore thus) of this discussion with Google AI Mode also includes some usable shader source code.
Notice: such synthetic neural tissues (such as Google AI Mode) have improved, but are still not suitable as substitution for human tutors.