1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
|
# Mathematical Theory of Black Hole Geodesics
### Einstein’s Field Equations
General Relativity describes gravity not as a force, but as the curvature of spacetime caused by mass and energy. This curvature is captured by **Einstein’s field equations**:
$$
G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}
$$
Here:
* $G_{\mu\nu}$ is the **Einstein tensor**, describing spacetime curvature.
* $T_{\mu\nu}$ is the **stress-energy tensor**, representing matter and energy.
* $G$ is Newton’s gravitational constant, and $c$ is the speed of light.
### Spacetime Metric
Distances in spacetime are described using the **metric tensor** $g_{\mu\nu}$:
$$
ds^2 = g_{\mu\nu} dx^\mu dx^\nu
$$
where $ds^2$ is the spacetime interval between two events.
## Schwarzschild Metric
For a **spherically symmetric, non-rotating mass** (like a static black hole), the Schwarzschild solution gives the spacetime geometry:
$$
ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2)
$$
* $M$ = mass of the black hole
* $r, \theta, \phi$ = spherical coordinates
* $t$ = time coordinate
The **Schwarzschild radius** $r_s$ marks the event horizon:
$$
r_s = \frac{2GM}{c^2}
$$
Inside $r_s$, not even light can escape.
We often write the metric using the **lapse function** $f(r)$:
$$
ds^2 = -f(r) dt^2 + f(r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2), \quad f(r) = 1 - \frac{r_s}{r}
$$
## Geodesics: Paths of Free-Falling Particles and Light
A **geodesic** is the path that a particle follows when moving under gravity alone. For light rays, $ds^2 = 0$ (null geodesics).
### Lagrangian Formulation
We can derive the geodesic equations from a Lagrangian:
$$
L = \frac{1}{2} g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu, \quad \dot{x}^\mu = \frac{dx^\mu}{d\lambda}
$$
where $\lambda$ is an affine parameter along the geodesic.
### Conserved Quantities
Because the Schwarzschild metric is **time-independent** and **spherically symmetric**, we have two key conserved quantities:
1. **Energy** (from time translation symmetry):
$$
E = - g_{tt} \frac{dt}{d\lambda} = f(r) \frac{dt}{d\lambda}
$$
2. **Angular Momentum** (from rotational symmetry):
$$
L = g_{\phi\phi} \frac{d\phi}{d\lambda} = r^2 \sin^2 \theta \frac{d\phi}{d\lambda}
$$
### Derivation of the Geodesic Equations
Geodesics satisfy the **Euler-Lagrange equations**:
$$
\frac{d}{d\lambda} \left(\frac{\partial L}{\partial \dot{x}^\mu}\right) - \frac{\partial L}{\partial x^\mu} = 0
$$
#### 1. Radial Motion
For the Schwarzschild metric:
$$
L = \frac{1}{2} \left[-f(r) \dot{t}^2 + f(r)^{-1} \dot{r}^2 + r^2 (\dot{\theta}^2 + \sin^2\theta \, \dot{\phi}^2)\right]
$$
The radial Euler-Lagrange equation becomes:
$$
\ddot{r} = -\frac{GM}{r^2} (\dot{t})^2 + \frac{GM}{r^2 f(r)} (\dot{r})^2 + r f(r) \left(\dot{\theta}^2 + \sin^2\theta \, \dot{\phi}^2\right)
$$
#### 2. Angular Motion
$$
\ddot{\theta} = -\frac{2}{r} \dot{r} \dot{\theta} + \sin\theta \cos\theta \, \dot{\phi}^2
$$
$$
\ddot{\phi} = -\frac{2}{r} \dot{r} \dot{\phi} - 2 \cot\theta \, \dot{\theta} \dot{\phi}
$$
Here, $\dot{}$ denotes derivative with respect to $\lambda$.
These equations fully describe how light or particles move around a Schwarzschild black hole.
### Numerical Implementation
In a shader or simulation, we integrate these equations using:
```glsl
void GeodesicRHS(Ray ray, out vec3 d1, out vec3 d2)
{
float r = ray.r;
float theta = ray.theta;
float dr = ray.dr;
float dtheta = ray.dtheta;
float dphi = ray.dphi;
float f = 1.0 - SagA_rs / r;
float dt_dL = ray.E / f;
d1 = vec3(dr, dtheta, dphi);
d2.x = - (SagA_rs / (2.0 * r*r)) * f * dt_dL * dt_dL
+ (SagA_rs / (2.0 * r*r * f)) * dr * dr
+ r * (dtheta*dtheta + sin(theta)*sin(theta)*dphi*dphi);
d2.y = -2.0*dr*dtheta/r + sin(theta)*cos(theta)*dphi*dphi;
d2.z = -2.0*dr*dphi/r - 2.0*cos(theta)/(sin(theta)) * dtheta * dphi;
}
```
### Conserved Quantities in Code
```glsl
ray.E = f * dt_dL; // Energy
ray.L = ray.r * ray.r * sin(ray.theta) * ray.dphi; // Angular momentum
```
### Effective Potential
The **radial motion** can be described using an effective potential:
$$
V_\text{eff}(r) = \left(1 - \frac{r_s}{r}\right) \frac{L^2}{r^2}
$$
This potential defines the possible orbits of light or particles.
### Relativistic Effects Around Black Holes
* **Gravitational Lensing:** Light bends around the black hole, producing Einstein rings, multiple images, or distorted images.
* **Event Horizon:** Located at $r = r_s$, where nothing escapes.
* **Photon Sphere:** At $r = 1.5 r_s$, light can orbit in unstable circular paths.
### Numerical Considerations
* **Event Horizon:** Integration becomes singular at $r = r_s$. Use adaptive step sizes or terminate integration near the horizon.
* **Coordinate Poles:** Spherical coordinates have singularities at $\theta = 0, \pi$. Avoid direct integration through these points or use transformations.
|
