{
"cells": [
{
"cell_type": "markdown",
"id": "3f0290f9-403f-4070-9626-a1acfcc93d20",
"metadata": {},
"source": [
"# A false trapped surface in Minkowski spacetime\n",
"\n",
"This Jupyter/SageMath notebook is relative to the lectures\n",
"[Geometry and physics of black holes](https://relativite.obspm.fr/blackholes/)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "8b7488fd-bbdf-45fb-b6d2-ebbef4d78732",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'SageMath version 10.0, Release Date: 2023-05-20'"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"version()"
]
},
{
"cell_type": "markdown",
"id": "26fca8a0-5657-4e98-870c-d47d1ed011a5",
"metadata": {},
"source": [
"The two intersecting light cones:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "4935752c-640c-46e8-a962-d04361bb5098",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t, ph = var('t ph')\n",
"c1 = parametric_plot3d([(1 - t)*cos(ph), (1 - t)*sin(ph), t],\n",
" (t, 0, 1), (ph, 0, 2*pi), color='palegreen')\n",
"\n",
"c2 = parametric_plot3d([1 + (1 - t)*cos(ph), (1 - t)*sin(ph), t],\n",
" (t, 0, 1), (ph, 0, 2*pi), color='palegreen')\n",
"c1 + c2"
]
},
{
"cell_type": "markdown",
"id": "4d4e5f8f-4dae-4c16-905e-c2346c8ea035",
"metadata": {},
"source": [
"The 2-surface $\\mathscr{S}$:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "87947f7d-c277-4ae1-8963-85614221d6ae",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(t) = 1 - sqrt(t^2 + 1/4)\n",
"s = parametric_plot3d([1/2, t, f(t) + 0.01], (t, -sqrt(3)/2, sqrt(3)/2),\n",
" color='red', thickness=4)\n",
"c1 + c2 + s"
]
},
{
"cell_type": "markdown",
"id": "64305109-11aa-4cfa-9a32-e4e4bd97a150",
"metadata": {},
"source": [
"Adding some light rays:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "f09f30f6-b107-4fad-8d65-8d46ed320d83",
"metadata": {},
"outputs": [],
"source": [
"l1 = (line([(1/2, 1/3, f(1/3)), (0,0,1)], color='green', thickness=3)\n",
" + line([(1/2, 1/2, f(1/2)), (0,0,1)], color='green', thickness=3) \n",
" + line([(1/2, 2/3, f(2/3)), (0,0,1)], color='green', thickness=3))\n",
"l2 = (line([(1/2, 1/3, f(1/3)), (1,0,1)], color='green', thickness=3)\n",
" + line([(1/2, 1/2, f(1/2)), (1,0,1)], color='green', thickness=3) \n",
" + line([(1/2, 2/3, f(2/3)), (1,0,1)], color='green', thickness=3))\n",
"graph = c1 + c2 + s + l1 + l2"
]
},
{
"cell_type": "markdown",
"id": "851a8496-3fcd-4421-8473-66b0ad91d080",
"metadata": {},
"source": [
"and some labels:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "0e65695f-f496-468c-9133-a7c3d0693b9f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"