IndexRelatividade e CosmosSticker sheet

On Cosmic Time Sticker Sheet

Muonaut, Orbitick, Relicta & Primordix | 9 Pixel Art s

\[\begin{gathered} \Delta t = \gamma\,\tau, \qquad L = \frac{L_0}{\gamma} \\ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \end{gathered}\]
$19.99 USD
$19.99 USD
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The On Cosmic Time sticker sheet gathers several Pixelated Physics characters on one page, each built on a real piece of physics. Nine kiss-cut pixel-art stickers on one A5 sheet: Muonaut (time dilation), Orbitick (GPS clocks), Relicta (CMB), Primordix (Big Bang helium), plus props. Peel them off one at a time and stick them wherever the physics belongs.

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Details

What you getA5 sheet · 9 stickers

One A5 kiss-cut sheet (5.83″ × 8.27″) on white sticker paper with 9 stickers, made to order.

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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.

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The physics

Stickers about cosmic time: time dilation of fast muons, the relativistic clock correction built into GPS, the cooling of the cosmic microwave background, and the helium made in the first minutes after the Big Bang.

Nine stickers on one sheet: two characters who keep relativistic time, and two who remember the early universe.

Muonaut’s line is In My Frame, the Sky Is Short. Cosmic-ray muons are made 10–15 km up and live 2.2 μs at rest, long enough for only about 660 m at light speed. Many still reach the ground: from the ground their clocks run slow, and in their own frame the atmosphere is contracted. That is the falling μ and the 15 KM tag.

Orbitick’s line is Wrong on Purpose. Right in Orbit. A GPS satellite clock gains about 45.7 μs a day from weaker gravity (general relativity) and loses about 7.1 μs a day from its speed (special relativity), a net of about 38.6 μs a day. So the clocks are set slightly slow before launch, to 10.22999999543 MHz. That is the satellite marked GR+SR.

Relicta’s line is I Was 3000 K Once. Now I’m 2.7 K. The cosmic microwave background was released when the universe became transparent, at about 3000 K, and expansion has since stretched it to a near-perfect blackbody at 2.72548 K. Arno Penzias and Robert Wilson found it in 1965. That is the blackbody curve labelled 2.7 K.

Primordix’s line is Three Minutes. Neutrons Won’t Wait. In the first minutes, neutrons froze out at about one per six protons and kept decaying until fusion could lock them into helium-4 at about one per seven. Nearly all of them ended up there, so helium is about a quarter of ordinary matter by mass, as observed. That is the four-nucleon helium nucleus and the ≈1/4 tag.

Equations

\[\begin{gathered} \Delta t = \gamma\,\tau, \qquad L = \frac{L_0}{\gamma} \\ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \end{gathered}\]
\[\begin{gathered} \frac{\Delta f}{f} \approx \frac{\Delta\Phi}{c^2} - \frac{v^2}{2c^2} \end{gathered}\]
\[\begin{gathered} T(z) = T_0\,(1+z) \\ T_0 = 2.72548\ \text{K} \end{gathered}\]
\[\begin{gathered} Y_p \simeq \frac{2\,(n/p)}{1 + n/p} \\ \begin{gathered} \frac{n}{p} \approx \frac{1}{7} \;\Rightarrow \\ Y_p \approx \frac{2/7}{8/7} = 0.25 \end{gathered} \end{gathered}\]
SymbolMeaningUnit
\(\Delta t\) muon lifetime as timed by clocks on the ground \(\mathrm{s}\)
\(\tau\) mean lifetime in the muon's own rest frame (PDG 2024: 2.1969811 μs; the half-life is shorter, 1.52 μs) \(\mathrm{s}\)
\(\gamma\) Lorentz factor \(\mathrm{1}\)
\(L_0\) thickness of the atmosphere in the ground frame \(\mathrm{m}\)
\(L\) the same thickness measured in the muon's frame \(\mathrm{m}\)
\(v\) speed of the muon (in γ), or orbital speed of the GPS satellite (in Δf/f) \(\mathrm{m/s}\)
\(c\) speed of light in vacuum \(\mathrm{m/s}\)
\(\Delta f/f\) fractional rate of a GPS satellite clock relative to a clock on the geoid (positive: it runs fast) \(\mathrm{1}\)
\(\Delta\Phi\) gravitational potential at the satellite minus the effective potential on the geoid, which includes Earth's rotation term \(\mathrm{J/kg}\)
\(T(z)\) temperature of the background radiation at redshift z \(\mathrm{K}\)
\(z\) redshift (last scattering: about 1090) \(\mathrm{1}\)
\(T_0\) temperature today (Fixsen 2009: 2.72548 ± 0.00057 K) \(\mathrm{K}\)
\(Y_p\) primordial helium-4 fraction by mass (observed 0.245 ± 0.003) \(\mathrm{1}\)
\(n/p\) neutron-to-proton ratio \(\mathrm{1}\)

Sources

  1. Rossi & Hall (1941) Variation of the Rate of Decay of Mesotrons with Momentum, Phys. Rev. 59, 223
  2. Frisch & Smith (1963) Measurement of the Relativistic Time Dilation Using μ-Mesons, Am. J. Phys. 31, 342
  3. Ashby (2003) Relativity in the Global Positioning System, Living Rev. Relativ. 6, 1
  4. Ashby (2003) Relativity in the Global Positioning System — open full text (PubMed Central)
  5. Penzias & Wilson (1965) A Measurement of Excess Antenna Temperature at 4080 Mc/s, ApJ 142, 419
  6. NobelPrize.org — The Nobel Prize in Physics 1978 (Penzias and Wilson, cosmic microwave background radiation)
  7. Alpher, Bethe & Gamow (1948) The Origin of Chemical Elements, Phys. Rev. 73, 803 (history; not the modern calculation)
  8. Fields, Molaro & Sarkar — Big-Bang Nucleosynthesis, Particle Data Group Review of Particle Physics (2024)

Source Rossi, B.; Hall, D. B. (1941). Variation of the Rate of Decay of Mesotrons with Momentum. Physical Review 59, 223–228.doi.org/10.1103/PhysRev.59.223

Field notes

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