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Photon Sphere Hoodie

EVEN LIGHT GOES AROUND | Pixelated Physics

\[\frac{d^2u}{d\phi^2} + u = \frac{3GM}{c^2}\,u^2,\qquad u = \frac{1}{r}\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Far from a black hole, light runs straight. Close in, it bends. At one radius it can run in a circle. Exact light paths past a Schwarzschild black hole. At r = 3GM/c² a photon can circle, unstably; one nudge and it falls or escapes. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Near a black hole, gravity bends light so strongly that at one and a half times the event-horizon radius light can travel in a circle, the photon sphere. Rays that pass closer than a critical distance are captured, which sets the size of the black hole's shadow.

Far from a black hole, light runs straight. Close in, it bends. At one radius it can run in a circle.

That radius is r = 3GM/c², one and a half times the event horizon. Every ray in the art is an exact null geodesic of Karl Schwarzschild’s 1916 solution, integrated from the orbit equation on the back: u″ + u = (3GM/c²)u², with u = 1/r. Drop the right-hand side and you get a straight line; the u² term is general relativity.

Each ray is labelled by its impact parameter b, how far it would miss the hole if nothing bent it. Above b = 3√3 GM/c² it escapes. Below, it falls in. Aimed a hair above the critical value, it winds around the photon sphere several times and then leaves, which is the amber ray.

The orbit is unstable. A photon on it, nudged either way, spirals in or flies off. The edge of the black hole shadow imaged by the Event Horizon Telescope in 2019 is set by these same critical rays.

Light goes around. Not for long.

Equations

\[\frac{d^2u}{d\phi^2} + u = \frac{3GM}{c^2}\,u^2,\qquad u = \frac{1}{r}\]
\[r_{\mathrm{ph}} = \frac{3GM}{c^2} = \tfrac{3}{2}\,r_s\]
\[b_c = \frac{3\sqrt{3}\,GM}{c^2}\]
SymbolMeaningUnit
\(u\) inverse radial coordinate, 1/r \(\mathrm{m^{-1}}\)
\(\phi\) orbital angle in the equatorial plane \(\mathrm{rad}\)
\(r\) Schwarzschild radial coordinate \(\mathrm{m}\)
\(G\) gravitational constant \(\mathrm{m^3·kg^{-1}·s^{-2}}\)
\(M\) mass of the black hole \(\mathrm{kg}\)
\(c\) speed of light \(\mathrm{m/s}\)
\(r_s\) Schwarzschild radius, 2GM/c^2 \(\mathrm{m}\)
\(r_{\mathrm{ph}}\) radius of the photon sphere \(\mathrm{m}\)
\(b_c\) critical impact parameter: rays with smaller b are captured \(\mathrm{m}\)

Sources

  1. Schwarzschild (1916) Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsber. Preuss. Akad. Wiss., 189
  2. Misner, Thorne & Wheeler, Gravitation (1973), Ch. 25: Orbits in Schwarzschild geometry
  3. Event Horizon Telescope Collaboration (2019) First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, ApJL 875, L1

Source Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), 189–196.

Field notes

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