The Einstein–Hilbert action is the basis for the most elementary variational principle from which the field equations of general relativity can be 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.
The term is required to ensure the path integral (a la Hawking) for quantum gravity has the correct composition properties.
When calculating black hole entropy using the Euclidean semiclassical approach, the entire contribution comes from the GHY term.
In order to determine a finite value for the action, one may have to subtract off a surface term for flat spacetime: where
In a four-dimensional spacetime manifold, a hypersurface is a three-dimensional submanifold that can be either timelike, spacelike, or null.
, it is easy to check where Note that variation subject to the condition implies that
So finally we have Gathering the results we obtain We next show that the above boundary term will be cancelled by the variation of
The total variation of the gravitational action is: This produces the correct left-hand side of the Einstein equations.
This result was generalised to fourth-order theories of gravity on manifolds with boundaries in 1983[2] and published in 1985.
gives zero and so does not effect the field equations, its purpose is to change the numerical value of the action.
The numerical value of the gravitational action is then where we are ignoring the non-dynamical term for the moment.
On the three cylinder, in coordinates intrinsic to the hyper-surface, the line element is meaning the induced metric is so that
There are many theories which attempt to modify General Relativity in different ways, for example f(R) gravity replaces R, the Ricci scalar in the Einstein–Hilbert action with a function f(R).
[4] They found that the "modified action in the metric formalism of f(R) gravity plus a Gibbons–York–Hawking like boundary term must be written as:" where
By using the ADM decomposition and introducing extra auxiliary fields, in 2009 Deruelle et al. found a method to find the boundary term for "gravity theories whose Lagrangian is an arbitrary function of the Riemann tensor.
"[5] This method can be used to find the GHY boundary terms for Infinite derivative gravity.
[6] As mentioned at the beginning, the GHY term is required to ensure the path integral (a la Hawking et al.) for quantum gravity has the correct composition properties.
This older approach to path-integral quantum gravity had a number of difficulties and unsolved problems.
The starting point in this approach is Feynman's idea that one can represent the amplitude to go from the state with metric
Taking the implications of this into account, it can then be shown that the composition rule will hold if and only if we include the GHY boundary term.
In the quantum theory, the object that corresponds to the Hamilton's principal function is the transition amplitude.
Consider gravity defined on a compact region of spacetime, with the topology of a four dimensional ball.
In pure gravity without cosmological constant, since the Ricci scalar vanishes on solutions of Einstein's equations, the bulk action vanishes and the Hamilton's principal function is given entirely in terms of the boundary term, where
No spacetime is assumed a priori, but rather it is built up by the states of theory themselves – however scattering amplitudes are derived from
The relation between the background-independent formalism and the conventional formalism of quantum field theory on a given spacetime is far from obvious, and it is far from obvious how to recover low-energy quantities from the full background-independent theory.
-point functions of the theory from the background-independent formalism, in order to compare them with the standard perturbative expansion of quantum general relativity and therefore check that loop quantum gravity yields the correct low-energy limit.
A strategy for addressing this problem has been suggested;[8] the idea is to study the boundary amplitude, or transition amplitude of a compact region of spacetime, namely a path integral over a finite space-time region, seen as a function of the boundary value of the field.
-point function is determined by the state of the gravitational field on the boundary of the spacetime region considered.
In other words, the boundary formulation realizes very elegantly in the quantum context the complete identification between spacetime geometry and dynamical fields.