physics

Initial value formulation (general relativity)

Initial value formulation (general relativity) 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 Initial value formulation (general relativity) rather than just read about it. In short: The initial value formulation of general relativity is a reformulation of Albert Einstein's theory of general relativity that describes a universe evolving over time. Each solution of the Einstein field equations encompasses the whole history of a universe – it is not just some snapshot of how things are, but a whole spacetime: a statement encompassing the state of matter and geometry everywhere and at every moment…

Key takeaways

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

Reference excerpt

The initial value formulation of general relativity is a reformulation of Albert Einstein's theory of general relativity that describes a universe evolving over time. Each solution of the Einstein field equations encompasses the whole history of a universe – it is not just some snapshot of how things are, but a whole spacetime: a statement encompassing the state of matter and geometry everywhere and at every moment in that particular universe. By this token, Einstein's theory appears to be different from most other physical theories, which specify evolution equations for physical systems; if the system is in a given state at some given moment, the laws of physics allow you to extrapolate its past or future. For Einstein's equations, there appear to be subtle differences compared with other fields: they are self-interacting (that is, non-linear even in the absence of other fields); they are diffeomorphism invariant, so to obtain a unique solution, a fixed background metric and gauge conditions need to be introduced; finally, the metric determines the spacetime structure, and thus the domain of dependence for any set of initial data, so the region on which a specific solution will be defined is not, a priori, defined. There is, however, a way to re-formulate Einstein's equations that overcomes these problems. First of all, there are ways of rewriting spacetime as the evolution of "space" in time; an earlier version of this is due to Paul Dirac, while a simpler way is known after its inventors Richard Arnowitt, Stanley Deser and Charles Misner as ADM formalism. In these formulations, also known as "3+1" approaches, spacetime is split into a three-dimensional hypersurface with interior metric and an embedding into spacetime with exterior curvature; these two quantities are the dynamical variables in a Hamiltonian formulation tracing the hypersurface's evolution over time. With such a split, it is possible to state the initial value formulation of general relativity. It involves initial data which cannot be specified arbitrarily but needs to satisfy specific constraint equations, and which is defined on some suitably smooth three-manifold Σ {\displaystyle \Sigma } ; just as for other differential equations, it is then possible to prove existence and uniqueness theorems, namely that there exists a unique spacetime which is a solution of Einstein equations, which is globally hyperbolic, for which Σ {\displaystyle \Sigma } is a Cauchy surface (i.e. all past events influence what happens on Σ {\displaystyle \Sigma } , and all future events are influenced by what happens on it), and has the specified internal metric and extrinsic curvature; all spacetimes that satisfy these conditions are related by isometries. The initial value formulation with its 3+1 split is the basis of numerical relativity; attempts to simulate the evolution of relativistic spacetimes (notably merging black holes or gravitational collapse) using computers. However, there are significant differences to the simulation of other physical evolution equations which make numerical relativity especially challenging, notably the fact that the dynamical objects that are evolving include space and time itself (so there is no fixed background against which to evaluate, for instance, perturbations representing gravitational waves) and the occurrence of singularities (which, when they are allowed to occur within the simulated portion of spacetime, lead to arbitrarily large numbers that would have to be represented in the computer model).

Worked examples

Example 1 — a first encounter with Initial value formulation (general relativity)

Start with the simplest possible case. Write down what Initial value formulation (general relativity) 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 Initial value formulation (general relativity) 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 Initial value formulation (general relativity) 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 Initial value formulation (general relativity)

In research: Initial value formulation (general relativity) 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 Initial value formulation (general relativity) 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: Initial value formulation (general relativity) is common in secondary-school and first-year university syllabi. It links to neighbouring topics General relativity, so understanding it makes those chapters shorter.
In everyday life: Look for Initial value formulation (general relativity) 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 Initial value formulation (general relativity) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Initial value formulation (general relativity) 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 Initial value formulation (general relativity) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Initial value formulation (general relativity) in simple terms?

Why does Initial value formulation (general relativity) 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 Initial value formulation (general relativity)?

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 Initial value formulation (general relativity).