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Penrose Diagram Hoodie

ALL OF FOREVER, ON ONE PAGE | Pixelated Physics

\[ds^2 = -\left(1-\frac{2GM}{c^2r}\right)c^2dt^2 + \frac{dr^2}{1-\frac{2GM}{c^2r}} + r^2d\Omega^2\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

A Penrose diagram squeezes all of an infinite spacetime onto one page, while keeping light rays at 45°. The maximally extended Schwarzschild black hole, compactified: two exteriors, a horizon cross and a singularity on top. Infinity fits. 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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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

A Penrose diagram squeezes an infinite spacetime onto a finite page while keeping light rays at 45 degrees. For a Schwarzschild black hole it shows the outside universe, the horizon, the singularity, and regions that can never send signals to us.

A Penrose diagram squeezes all of an infinite spacetime onto one page, while keeping light rays at 45°.

This one is the full Schwarzschild black hole. Start from Kruskal coordinates U and V, in which the horizon is harmless, then take the arctangent of each. Infinity lands on the edges of the diamond.

The right diamond is our universe. The left is a second exterior that the exact solution also contains. The top triangle is the black hole interior, capped by the singularity r = 0, drawn as a zigzag; the bottom one is the time-reversed white hole. The cross through the middle is the horizon.

Once inside, every future-pointing path, at 45° or steeper, hits the zigzag. The lines in the art are surfaces of constant r and constant t.

All of forever, on one page.

Equations

\[ds^2 = -\left(1-\frac{2GM}{c^2r}\right)c^2dt^2 + \frac{dr^2}{1-\frac{2GM}{c^2r}} + r^2d\Omega^2\]
\[u = \arctan U,\qquad v = \arctan V\]
\[UV = \left(1-\frac{r}{2M}\right)e^{r/2M}\quad (G=c=1)\]
SymbolMeaningUnit
\(ds\) spacetime interval \(\mathrm{m}\)
\(t\) Schwarzschild time \(\mathrm{s}\)
\(r\) areal radius \(\mathrm{m}\)
\(M\) black hole mass \(\mathrm{kg}\)
\(d\Omega\) element of solid angle \(\mathrm{sr}\)
\(U, V\) Kruskal–Szekeres null coordinates \(\mathrm{1}\)
\(u, v\) compactified null coordinates \(\mathrm{rad}\)

Sources

  1. Schwarzschild (1916) Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsber. Preuss. Akad. Wiss., 189
  2. Kruskal (1960) Maximal Extension of Schwarzschild Metric, Phys. Rev. 119, 1743
  3. Penrose (1964) Conformal treatment of infinity, in Relativity, Groups and Topology (Les Houches)
  4. Hawking & Ellis, The Large Scale Structure of Space-Time (1973), Sec. 5.5

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

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