/newtonian_black_holes

Raytracing black holes using (and abusing) Newtonian mechanics.

Primary LanguageCGNU General Public License v3.0GPL-3.0

Table of Contents

  1. Newtonian Black Holes
  2. Benchmarks
  3. Benchmarks (w/Parallelization)
  4. License

Newtonian Black Holes

This is a very small project to experiment with ray tracing and black holes using (and slightly abusing) Newtonian mechanics.

Newtonian mechanics predicts the existence of black holes. Take the escape velocity formula and apply it to the speed of light. The result is the Schwarzschild radius, which is the same formula one derives if they were to perform to full general relativistic calculation.

$$R=\frac{2GM}{c^{2}}$$

where $M$ is the mass of the object, $c$ is the speed of light, and $G$ is the universal gravitational constant. Next we abuse Newtonian mechanics slightly. Newton's second law says

$$\mathbf{F}(t)=m\mathbf{a}(t)=m\ddot{\mathbf{r}}(t)$$

where $\mathbf{F}$ is the force on an object of mass $m$, $\mathbf{a}$ is the acceleration vector, $\mathbf{r}$ is the position vector, and double-dots denote the second derivative with respect to time. For gravity we have

$$m\ddot{\mathbf{r}}(t)=\frac{GMm}{||\mathbf{r}(t)||^{3}}\mathbf{r}(t)$$

where $||\mathbf{r}(t)||$ denotes the norm or Euclidean length of the vector $\mathbf{r}(t)$ and $M$ is the mass of the gravitating object. Dividing by $m$ we obtain the following equation of motion:

$$\ddot{\mathbf{r}}(t)=\frac{GM}{||\mathbf{r}(t)||^{3}}\mathbf{r}(t)$$

This is all fine and dandy so long as $m\ne{0}$, otherwise we have an illegal division. Let us suppose that this final equation is true even for massless particles (like light). Or pretend that light has a mass $m$ that is so stupidly small (but positive) that it will never be measurable (The reciprocal of Graham's number, if you'd like, in kilograms). We may then take this equation and use it to approximate how light may behave under the force of gravity of some object (such as a black hole). We could then apply raytracing methods and draw a nice picture. This directory contains files to do exactly this.

This is not physically realistic, obviously, but the resulting images do look like what one might expect a black hole to be.

Newtonian Black Hole

Benchmarks

These benchmarks used a Ryzen 9 7950x on Debian 12.

Language Implementation Time (s) Version
C gcc 11.396 gcc (Debian 12.2.0-14) 12.2.0
C++ g++ 11.840 g++ (Debian 12.2.0-14) 12.2.0
D gdc 12.186 gdc (Debian 12.2.0-14) 12.2.0
C clang 12.199 Debian clang version 14.0.6
D ldc 12.204 LLVM D Compiler (1.30.0)
Swift swiftc 12.216 Swift version 5.9 (swift-5.9-RELEASE)
C++ clang++ 12.361 Debian clang version 14.0.6
Fortran gfortran 12.633 GNU Fortran (Debian 12.2.0-14) 12.2.0
Rust rustc 12.681 rustc 1.72.0 (5680fa18f 2023-08-23)
Java openJDK 12.910 openjdk 17.0.8 2023-07-18
Python Pypy 15.481 PyPy 7.3.11 with GCC 12.2.0
Go golang 18.118 go 1.19.8
Pascal fpc 18.755 Free Pascal Compiler 3.2.2 20230330
Go gccgo 18.932 gccgo (Debian 12.2.0-14) 12.2.0
D dmd 35.832 DMD64 D Compiler v2.107.0
C pcc 62.625 Portable C Compiler 1.2.0.DEVEL 20220331
C tcc 70.031 tcc version 0.9.27
Python CPython 1876.293 Python 3.11.2
IDL GDL 6677.710 GNU Data Language, Version 1.0.1

Benchmarks (w/Parallelization)

I haven't made parellel versions in every language, but here are the benchmarks for those that have been done. Ryzen 9 7950x (16-core) on Debian 12.

Language Implementation Time (s) Version
C gcc 0.665 gcc (Debian 12.2.0-14) 12.2.0
C++ g++ 0.666 g++ (Debian 12.2.0-14) 12.2.0
C++ clang++ 0.961 Debian clang version 14.0.6
C clang 0.971 Debian clang version 14.0.6
Go golang 1.334 go 1.19.8
Go gccgo 1.684 gccgo (Debian 12.2.0-14) 12.2.0

License

newtonian_black_holes is free software: you can redistribute it and/or
modify it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

newtonian_black_holes is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License along
with newtonian_black_holes.  If not, see <https://www.gnu.org/licenses/>.