physics

Gibbons–Hawking–York boundary term

Gibbons–Hawking–York boundary term 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 Gibbons–Hawking–York boundary term rather than just read about it. In short: In general relativity and quantum gravity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary. The Einstein–Hilbert action is the basis for the most elementary variational principle from which the field equations of general relativity can be defined.

Key takeaways

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

Reference excerpt

In general relativity and quantum gravity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary. The Einstein–Hilbert action is the basis for the most elementary variational principle from which the field equations of general relativity can be defined. However, the use of the Einstein–Hilbert action is appropriate only when the underlying spacetime manifold M {\displaystyle {\mathcal {M}}} is closed, i.e. a manifold which is both compact and without boundary. In the event that the manifold has a boundary ∂ M {\displaystyle \partial {\mathcal {M}}} , the action should be supplemented by a boundary term so that the variational principle is well-defined. The necessity of such a boundary term was first realised by James W. York and later refined in a minor way by Gary Gibbons and Stephen Hawking. For a manifold that is not closed, the appropriate action is

S E H + S G H Y = 1 16 π ∫ M d 4 x − g R + 1 8 π ∫ ∂ M d 3 y ϵ h K , {\displaystyle {\mathcal {S}}_{\mathrm {EH} }+{\mathcal {S}}_{\mathrm {GHY} }={\frac {1}{16\pi }}\int _{\mathcal {M}}\mathrm {d} ^{4}x\,{\sqrt {-g}}R+{\frac {1}{8\pi }}\int _{\partial {\mathcal {M}}}\mathrm {d} ^{3}y\,\epsilon {\sqrt {h}}K,}

where S E H {\displaystyle {\mathcal {S}}_{\mathrm {EH} }} is the Einstein–Hilbert action, S G H Y {\displaystyle {\mathcal {S}}_{\mathrm {GHY} }} is the Gibbons–Hawking–York boundary term, h a b {\displaystyle h_{ab}} is the induced metric (see section below on definitions) on the boundary, h {\displaystyle h} its determinant, K {\displaystyle K} is the trace of the second fundamental form, ϵ {\displaystyle \epsilon } is equal to + 1 {\displaystyle +1} where the normal to ∂ M {\displaystyle \partial {\mathcal {M}}} is spacelike and − 1 {\displaystyle -1} where the normal to ∂ M {\displaystyle \partial {\mathcal {M}}} is timelike, and y a {\displaystyle y^{a}} are the coordinates on the boundary. Varying the action with respect to the metric g α β {\displaystyle g_{\alpha \beta }} , subject to the condition

δ g α β | ∂ M = 0 , {\displaystyle \delta g_{\alpha \beta }{\big |}_{\partial {\mathcal {M}}}=0,}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Gibbons–Hawking–York boundary term

Start with the simplest possible case. Write down what Gibbons–Hawking–York boundary term 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 Gibbons–Hawking–York boundary term 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 Gibbons–Hawking–York boundary term 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 Gibbons–Hawking–York boundary term

In research: Gibbons–Hawking–York boundary term 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 Gibbons–Hawking–York boundary term 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: Gibbons–Hawking–York boundary term is common in secondary-school and first-year university syllabi. It links to neighbouring topics General relativity, Lagrangian mechanics, Stephen Hawking, so understanding it makes those chapters shorter.
In everyday life: Look for Gibbons–Hawking–York boundary term 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Gibbons–Hawking–York boundary term” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Gibbons–Hawking–York boundary term in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Gibbons–Hawking–York boundary term 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 Gibbons–Hawking–York boundary term out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Gibbons–Hawking–York boundary term in simple terms?

Why does Gibbons–Hawking–York boundary term 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 Gibbons–Hawking–York boundary term?

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 Gibbons–Hawking–York boundary term.