physics

Hoop conjecture

Hoop conjecture is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hoop conjecture rather than just read about it. In short: The hoop conjecture, proposed by Kip Thorne in 1972, states that an imploding object forms a black hole when, and only when, a circular hoop with a specific critical circumference could be placed around the object and rotated about its diameter. In simpler terms, the entirety of the object's mass must be compressed to the point that it resides in a perfect sphere whose radius is equal to that object's Schwarzschild…

Key takeaways

Hoop conjecture belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hoop conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hoop conjecture from memory before moving on to harder problems.

Reference excerpt

The hoop conjecture, proposed by Kip Thorne in 1972, states that an imploding object forms a black hole when, and only when, a circular hoop with a specific critical circumference could be placed around the object and rotated about its diameter. In simpler terms, the entirety of the object's mass must be compressed to the point that it resides in a perfect sphere whose radius is equal to that object's Schwarzschild radius; if this requirement is not met, then a black hole will not be formed. The critical circumference required for the imaginary hoop is given by the following equation listed below.

c = 2 π r s , {\displaystyle c=2\pi r_{\text{s}},}

where

c {\displaystyle c} is the critical circumference;

r s {\displaystyle r_{\text{s}}} is the object's Schwarzschild radius. Thorne calculated the effects of gravitation on objects of different shapes (spheres, and cylinders that are infinite in one direction), and concluded that the object needed to be compressed in all three directions before gravity led to the formation of a black hole. With cylinders, the event horizon was formed when the object could fit inside the hoop described above. The mathematics to prove the same for objects of all shapes was too difficult for him at that time, but he formulated his hypothesis as the hoop conjecture. By Penrose singularity theorem of 1964 it is known that if there is trapped null surface (and some other conditions) then a singularity must form, in 1983 Richard Schoen and Shing-Tung Yau proved how much matter must be crammed into a given volume to create a closed trapped surface, sometimes referred as the Schoen–Yau black hole existence theorem and more recently in 2023 using Mikhael Gromov's cube inequality some tori inequalities used in the results of 1983 have been generalized to cube ones that are more akin to Thorne's circular hoops.

See also General relativity Bounding sphere Black hole stability conjecture

References Thorne, Kip (January 1, 1995), Black Holes and Time Warps: Einstein's Outrageous Legacy (Reprint ed.), W. W. Norton & Company, ISBN 0-393-31276-3

Worked examples

Example 1 — a first encounter with Hoop conjecture

Start with the simplest possible case. Write down what Hoop conjecture claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hoop conjecture before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hoop conjecture ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hoop conjecture

In research: Hoop conjecture appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hoop conjecture in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hoop conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conjectures, General relativity, Relativity stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Hoop conjecture outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hoop conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hoop conjecture means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hoop conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hoop conjecture in simple terms?

Why does Hoop conjecture matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hoop conjecture?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hoop conjecture.