mathematics

Regge calculus

Regge calculus is a mathematics 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 Regge calculus rather than just read about it. In short: In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in 1961.

Key takeaways

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

Reference excerpt

In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in 1961.

Overview The starting point for Regge's work is the fact that every four dimensional time orientable Lorentzian manifold admits a triangulation into simplices. Furthermore, the spacetime curvature can be expressed in terms of deficit angles associated with 2-faces where arrangements of 4-simplices meet. These 2-faces play the same role as the vertices where arrangements of triangles meet in a triangulation of a 2-manifold, which is easier to visualize. Here a vertex with a positive angular deficit represents a concentration of positive Gaussian curvature, whereas a vertex with a negative angular deficit represents a concentration of negative Gaussian curvature. The deficit angles can be computed directly from the various edge lengths in the triangulation, which is equivalent to saying that the Riemann curvature tensor can be computed from the metric tensor of a Lorentzian manifold. Regge showed that the vacuum field equations can be reformulated as a restriction on these deficit angles. He then showed how this can be applied to evolve an initial spacelike hyperslice according to the vacuum field equation. The result is that, starting with a triangulation of some spacelike hyperslice (which must itself satisfy a certain constraint equation), one can eventually obtain a simplicial approximation to a vacuum solution. This can be applied to difficult problems in numerical relativity such as simulating the collision of two black holes. The elegant idea behind Regge calculus has motivated the construction of further generalizations of this idea. In particular, Regge calculus has been adapted to study quantum gravity.

See also

Notes

References John Archibald Wheeler (1965). "Geometrodynamics and the Issue of the Final State, in "Relativity Groups and Topology"". Les Houches Lecture Notes 1963, Gordon and Breach. {{cite journal}}: Cite journal requires |journal= (help) Misner, Charles W. Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 978-0-7167-0344-0.{{cite book}}: CS1 maint: multiple names: authors list (link) See chapter 42. Herbert W. Hamber (2009). Hamber, Herbert W (ed.). Quantum Gravitation - The Feynman Path Integral Approach. Springer Publishing. doi:10.1007/978-3-540-85293-3. ISBN 978-3-540-85292-6. Chapters 4 and 6. [1] [2] James B. Hartle (1985). "Simplicial MiniSuperSpace I. General Discussion". Journal of Mathematical Physics. 26 (4): 804–812. Bibcode:1985JMP....26..804H. doi:10.1063/1.526571. Ruth M. Williams & Philip A. Tuckey (1992). "Regge calculus: a brief review and bibliography". Class. Quantum Grav. 9 (5): 1409–1422. Bibcode:1992CQGra...9.1409W. doi:10.1088/0264-9381/9/5/021. S2CID 250776873. Available (subscribers only) at "Classical and Quantum Gravity". Tullio E. Regge and Ruth M. Williams (2000). "Discrete Structures in Gravity". Journal of Mathematical Physics. 41 (6): 3964–3984. arXiv:gr-qc/0012035. Bibcode:2000JMP....41.3964R. doi:10.1063/1.533333. S2CID 118957627. Available at [3]. Herbert W. Hamber (1984). "Simplicial Quantum Gravity, in the Les Houches Summer School on Critical Phenomena, Random Systems and Gauge Theories, Session XLIII". North Holland Elsevier: 375–439. {{cite journal}}: Cite journal requires |journal= (help) [4] Adrian P. Gentle (2002). "Regge calculus: a unique tool for numerical relativity". Gen. Rel. Grav. 34 (10): 1701–1718. doi:10.1023/A:1020128425143. S2CID 119090423. eprint Renate Loll (1998). "Discrete approaches to quantum gravity in four dimensions". Living Rev. Relativ. 1 (1): 13. arXiv:gr-qc/9805049. Bibcode:1998LRR.....1...13L. doi:10.12942/lrr-1998-13. PMC 5253799. PMID 28191826. Available at "Living Reviews of Relativity". See section 3. J. W. Barrett (1987). "The geometry of classical Regge calculus". Class. Quantum Grav. 4 (6): 1565–1576. Bibcode:1987CQGra...4.1565B. doi:10.1088/0264-9381/4/6/015. S2CID 250783980. Available (subscribers only) at "Classical and Quantum Gravity".

External links Regge calculus on ScienceWorld

Worked examples

Example 1 — a first encounter with Regge calculus

Start with the simplest possible case. Write down what Regge calculus claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Regge calculus 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 Regge calculus 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 Regge calculus

In research: Regge calculus appears in mathematics 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 Regge calculus 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: Regge calculus is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematics of general relativity, Numerical analysis, Simplicial sets, so understanding it makes those chapters shorter.
In everyday life: Look for Regge calculus 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 Regge calculus in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Regge calculus 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 Regge calculus out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Regge calculus in simple terms?

Why does Regge calculus matter?

Because it connects several mathematics 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 Regge calculus?

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 Regge calculus.