🌌 torch-relativistic

Relativistic visual effects and physics simulation toolkit for PyTorch

GPU-accelerated Lorentz transforms, Terrell-Penrose distortion, relativistic aberration and Doppler shift — differentiable and ready for rendering, simulation, and astrophysics.

The Terrell-Penrose effect reproduced computationally: a cube appears rotated, not contracted, at relativistic speeds — validated against the analytical formula arcsin(v/c) to machine precision. Same physics as Schattschneider et al. (2025).

What this is

torch-relativistic provides differentiable PyTorch implementations of special-relativistic transformations that operate on 3D geometry, point clouds, meshes, and spacetime coordinates. Everything runs on GPU and supports autograd.

Core capabilities

Lorentz boost — transform 4D spacetime coordinates between reference frames moving at relative velocity v (LorentzBoost)
Terrell-Penrose distortion — compute the apparent visual distortion of objects at relativistic speeds: the rotation, not the contraction (TerrellPenroseTransform, terrell_rotation_angle)
Relativistic aberration — how observation angles shift when the observer is in motion
Doppler shift — frequency / colour shift for approaching and receding sources, including the transverse (time-dilation) component
Lorentz factor (gamma), time dilation, length contraction, relativistic velocity addition — all as differentiable tensor ops
Minkowski metric, Levi-Civita tensor, spherical harmonics — for building custom relativistic computations

Use cases

Domain	Example
Relativistic rendering	What does a spaceship / star / accretion disk look like at 0.9c? Physically correct visual distortion for games, VR, educational tools.
Astrophysics visualization	Relativistic jets, binary pulsars, cosmological simulations with real particle velocities.
Science education	Interactive demos for special relativity courses — the same kind of demonstration Schattschneider et al. built in the lab, but as a software tool.
Shader / post-processing	The Terrell-Penrose distortion field as a velocity-parameterized render pass, applicable to any 3D scene.
Physics simulation	Differentiable relativistic transforms for optimization, inverse problems, or learnable physics.

Installation

uv add torch-relativistic

Development install:

git clone https://github.com/synapticore-io/torch-relativistic.git
cd torch-relativistic
uv sync --dev

Requirements: Python ≥ 3.11, PyTorch ≥ 2.0.

Quick Start

Terrell-Penrose distortion of a 3D object

import torch, math
from torch_relativistic.utils import terrell_rotation_angle

# At what angle does a cube appear rotated at 80% light speed?
v = torch.tensor(0.8)
angle = terrell_rotation_angle(v)
print(f"Apparent rotation: {math.degrees(angle):.1f}°")  # 53.1°

# This matches the analytical formula: arcsin(v/c)
assert abs(angle.item() - math.asin(0.8)) < 1e-6

Lorentz boost on spacetime coordinates

import torch
from torch_relativistic.transforms import LorentzBoost

# 4D spacetime events: (t, x, y, z)
events = torch.randn(100, 8)  # batch of 100 events, 8-dim features

boost = LorentzBoost(feature_dim=8, time_dim=0, max_velocity=0.9)
boosted = boost(events)  # transformed to a moving reference frame

Relativistic Doppler shift

import torch
from torch_relativistic.utils import relativistic_doppler_factor, calculate_gamma

v = torch.tensor(0.5)  # 50% light speed
doppler = relativistic_doppler_factor(v)
print(f"Doppler factor (head-on): {doppler:.3f}")  # blueshift for approach

# Transverse Doppler = pure time dilation
import math
gamma = calculate_gamma(v)
doppler_transverse = relativistic_doppler_factor(v, torch.tensor(math.pi / 2))
# doppler_transverse ≈ gamma (the transverse Doppler effect)

Interactive 3D visualization

# Generate an interactive demo of the Terrell-Penrose effect
uv run python examples/terrell_penrose_demo.py
# → opens examples/terrell_penrose_demo.html in your browser

Physics reference

The Terrell-Penrose effect

In 1959, James Terrell and Roger Penrose independently showed that a rapidly moving object does not appear Lorentz-contracted to an observer — instead it appears rotated by an angle

$$\theta = \arcsin(v/c)$$

This counter-intuitive result arises because photons from the far side of the object were emitted earlier (when the object was in a different position) and arrive at the same time as photons from the near side.

After 66 years as a purely theoretical prediction, the effect was first observed in the lab in May 2025 by Schattschneider et al. at TU Wien using high-speed cameras and laser pulses — the direct inspiration for this library.

Validated against analytical formulas

The terrell_rotation_angle() function matches arcsin(v/c) to machine precision at all tested velocities (see examples/terrell_penrose_demo.py):

v/c   | arcsin (deg) | torch-relativistic | match
0.50  |      30.0000 |           30.0000  | yes
0.80  |      53.1301 |           53.1301  | yes
0.95  |      71.8051 |           71.8051  | yes

API overview