STANDING QWhere systems break — boundary tests, edge-case audits, the failure modes that teach.
🌲 OPATHORLOKAN UNIVERSITY
Lab 4.9.8 — Edge Cases · v0.0
BOTH AT ONCEholding togetherγ = 1.00
PUSH THE VELOCITY SLIDER. PUSH THE DISTANCE SLIDER. NOTICE WHAT YOUR EYE EXPECTS.
A body at rest in flat space. Move it toward the speed of light, and
it contracts along its direction of motion. Move it toward
a black hole, and it stretches radially — the tidal force pulls its
far end faster than its near end. Open both effects at once and the body is being
shrunk and pulled apart on the same axes simultaneously.
Two Limits
The two boundaries physics found in the 20th century
In 1905, working alone in a patent office in Bern, Albert Einstein
wrote four papers in one year, any one of which would have made his career. The
one we want here is the third: special relativity. It said something almost too
simple to believe. The speed of light is the same for everyone, no matter how
they are moving. From that one premise, everything else followed. Moving clocks
run slow. Moving rulers shrink along the direction they're moving. The classical
idea of "now" stopped being the same for everyone. Length contraction is the
ruler shrinking: a body moving at velocity v through your frame is
shorter, along its direction of motion, by a factor of √(1 − v²/c²).
Not because it has been squeezed. Because that's what length is when
relative motion is in play.
Ten years later, the same physicist finished a much harder problem. General
relativity said gravity wasn't a force; it was the geometry of spacetime
itself. Mass curves space. Objects roll along the curves. In 1915, Einstein
published the field equations. In 1916, Karl Schwarzschild —
writing from a German trench during the First World War, where he would die a
few months later of pemphigus — solved the equations for a spherically
symmetric mass and found something nobody wanted. Past a certain radius, the
equations broke. Light couldn't escape. The radius bore his name.
The body of a person falling toward a black hole feels tidal forces. The
gravitational pull on their feet is stronger than the pull on their head, because
gravity falls off with distance, and feet are closer to the center than head.
The body stretches. Get close enough and the stretching becomes the dominant
force in your life. In 1972, Kip Thorne's graduate students gave
this its working name: spaghettification. It went into the journals
under that word.
This lab asks the question that physics undergraduates always ask the first
time they meet both effects: what happens if you do them at the same time?
Why combining them is hard
Length contraction lives in special relativity, where spacetime is flat. Tidal
stretching lives in general relativity, where spacetime is curved. Computing
what happens when both act on the same body, rigorously, requires solving
Einstein's field equations for the trajectory through curved spacetime. Most
working physicists never do this for a falling person. They solve simpler test
problems and trust the qualitative picture. The qualitative picture is what
this lab shows.
Why It Matters
The shape of the answer is the lesson
Two limits matter in physics, and only two: the smallest (Planck scale) and the
fastest (speed of light), plus the densest, which is a black hole. This lab puts
a body at the intersection of two of those limits at once. The numerical answer
isn't the lesson — getting the digits right requires general relativity, and
this lab uses cartoon math. The shape of the answer is the lesson.
What each mode teaches
LORENTZ ONLY. The body shrinks along its direction of motion. Slide v
from 0.5c to 0.9c to 0.99c and the rate of change accelerates — small
pushes near the end of the slider produce huge contractions, because gamma is a
reciprocal of a square root that's collapsing to zero. The body never has zero
length in any frame where it's measured at all; it just gets shorter without
bound.
TIDAL ONLY. The body stretches in the direction of the gravity well.
Slide r from 10 rₛ toward 1 rₛ and the stretch grows because tidal
force scales like 1/r³. Long before you reach the horizon, the stretch is
strong enough to pull a human body apart. For a stellar-mass black hole, that
happens hundreds of kilometers above the horizon. For a supermassive black hole,
you can cross the horizon without noticing the stretching at all.
BOTH AT ONCE — radial motion. The body moves toward the black hole.
Length contraction shrinks it along the motion axis; tidal stretch pulls it
along the same axis. The two effects fight on the same axis.
Push both sliders to the limit and you have an undefined product: zero length
from contraction, infinite length from tidal stretch. The cartoon shows you the
ratio of those infinities at any given pair of slider values. The honest answer
is "general relativity is needed to choose between them."
BOTH AT ONCE — orbital motion. The body moves perpendicular to the
black hole direction. Length contraction shrinks it perpendicular to the radial
axis; tidal stretch pulls it along the radial axis. The two effects
compound on perpendicular axes — one squeezes width, the other
pulls height. The body becomes a thin tall noodle.
Where you'll meet this again
GPS satellites. They run fast clocks (because they're far
from Earth, less time dilation from gravity) and slow clocks (because they're
moving relative to the ground). The two effects don't cancel; they have to be
calculated separately and corrected. Your phone's location services run on
this calculation every second.
Particle accelerators. A proton at the LHC moves at 0.999999991c.
Its lifetime as measured in the lab is dilated by gamma = 7,500. Particles
that should decay in nanoseconds travel kilometers. This is engineering, not
philosophy.
Gravitational-wave detection. When LIGO catches two black holes
merging, the waveform encodes a body experiencing both extreme relativistic
motion and extreme spacetime curvature. The data analysis is exactly this
lab's math, done in earnest.
Cosmology. The early universe was both very dense (gravity
effects huge) and expanding very fast (relativistic velocities everywhere).
The two limits both apply in the first microsecond after the Big Bang.
About & Sources
What this is and isn't
A browser visualization of two relativistic limit cases acting on the same body.
Length contraction is computed with the exact special-relativity formula
L = L₀ · √(1 − v²/c²). Tidal stretching is rendered with a
cartoon formula that grows like 1 / (r/rₛ − 1)1.5, clamped
to a maximum factor of 18× for visual sanity. The cartoon captures the
qualitative scaling of tidal forces near the horizon — that the stretch
grows without bound as you approach rₛ — but is not the
full general-relativistic answer.
Computing what really happens when both effects act on a body falling into a
black hole requires solving the geodesic equation in Schwarzschild spacetime
with a finite-size test body, and the answer differs between the body's proper
frame and a distant observer's frame in fundamentally incompatible ways. This
lab shows the cartoon. The cartoon is enough to see the shape of the question.
The full answer is two semesters of graduate GR.
Section 4.9.8 · Standalone lab, sibling to the OPA Browser Physics Lab Suite
(4.9.4) but not part of the six-lab waves+optics roster. Filed under Building 9
(Stephens Science Center), College IX (Science), Opathorlokan University, 900
Arkadelphia Road, Birmingham, Alabama 35254.
Built by Travis Jenkins (User Zero) with Claude. The lab exists to give a
curious mind one moment of seeing both relativity limits act on the same body
at the same time — not because the answer is in the digits, but because
the question is in the shape.