Black Hole Gravitational Lensing

Core Physics Concepts

This visualization demonstrates key principles of General Relativity:

Spacetime Curvature: Massive objects like black holes curve the spacetime around them.
Gravitational Lensing: Light rays, which follow the shortest path through spacetime, appear to bend when passing near a massive object.
Dependence on Mass (M) and Impact Parameter (b): The amount of bending is proportional to the black hole's mass (M) and inversely proportional to the light ray's closest distance (b) to the black hole.

Theory and Explanation:

Gravitational lensing occurs when light rays bend around a massive object like a black hole.

The bending angle $\alpha$ can be approximated by:
$$ \alpha \approx \frac{4GM}{c^2 b} $$
where:
- G: gravitational constant
- M: mass of black hole
- c: speed of light
- b: impact parameter (closest distance of the light ray to black hole)

In this visualization, the black hole is at the origin.
Multiple light rays start from the left with different offsets.
Rays bend near the black hole with bending angle inversely proportional to b.

Use the slider to adjust the mass M and see how the bending changes.

Problem for Students

A light ray passes by a black hole with a mass M. The ray is observed to bend with an angle of $\alpha = 0.5$ radians. Use the formula provided in the visualization to solve the following problems. Note: For this problem, assume the normalized units from the simulation (G = 1, c = 1).

If the light ray's impact parameter is $b = 2$, what is the mass (M) of the black hole?
If the black hole's mass is $M = 1.5$, what is the impact parameter (b) of a light ray that bends by $\alpha = 0.5$ radians?

Show Answer

Given Formula:

$\alpha = \frac{4GM}{c^2 b}$

Given Values:

$\alpha = 0.5$
$G = 1$
$c = 1$

Solution to Problem 1:

Find M when b = 2

Substitute the given values into the formula:

$$0.5 = \frac{4 \cdot 1 \cdot M}{1^2 \cdot 2}$$ $$0.5 = 2M$$ $$M = \frac{0.5}{2}$$

Answer: M = 0.25

Solution to Problem 2:

Find b when M = 1.5

First, rearrange the formula to solve for b:

$$b = \frac{4GM}{c^2 \alpha}$$

Substitute the given values:

$$b = \frac{4 \cdot 1 \cdot 1.5}{1^2 \cdot 0.5}$$ $$b = \frac{6}{0.5}$$

Answer: b = 12