Введение
Термин в общей теории относительности
In general relativity, 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 is closed, i. e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary , 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
where is the Einstein–Hilbert action, is the Gibbons–Hawking–York boundary term, is the induced metric (see section below on definitions) on the boundary, its determinant, is the trace of the second fundamental form, is equal to where the normal to is spacelike and where the normal to is timelike, and are the coordinates on the boundary. Varying the action with respect to the metric , subject to the condition
gives the Einstein equations; the addition of the boundary term means that in performing the variation, the geometry of the boundary encoded in the transverse metric is fixed (see section below). There remains ambiguity in the action up to an arbitrary functional of the induced metric
That a boundary term is needed in the gravitational case is because , the gravitational Lagrangian density, contains second derivatives of the metric tensor. This is a non typical feature of field theories, which are usually formulated in terms of Lagrangians that involve first derivatives of fields to be varied over only. The GHY term is desirable, as it possesses a number of other key features. When passing to the Hamiltonian formalism, it is necessary to include the GHY term in order to reproduce the correct Arnowitt–Deser–Misner energy (ADM energy). 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. This term has had more recent applications in loop quantum gravity in calculating transition amplitudes and background independent scattering amplitudes. In order to determine a finite value for the action, one may have to subtract off a surface term for flat spacetime:
where is the extrinsic curvature of the boundary imbedded flat spacetime. As is invariant under variations of , this addition term does not affect the field equations; as such, this is referred to as the non dynamical term.
В общей теории относительности термин границы Гиббонса — Хокинга — Йорка — это член, который необходимо добавить к действию Эйнштейна — Гильберта, когда лежащее в основе пространственно-временное многообразие имеет границу. Действие Эйнштейна — Гильберта является основой для самого элементарного вариационного принципа, из которого можно определить уравнения поля общей теории относительности. Однако использование действия Эйнштейна — Гильберта уместно только тогда, когда лежащее в основе пространственно-временное многообразие замкнуто, то есть является компактным и не имеет границы. В случае, если многообразие имеет границу, действие следует дополнить граничным членом, чтобы вариационный принцип был корректно определен. Необходимость такого граничного члена была впервые осознана Джеймсом У. Йорком, а затем незначительно уточнена Гари Гиббонсом и Стивеном Хокингом. Для многообразия, которое не замкнуто, соответствующее действие имеет вид:
In general relativity, 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 is closed, i. e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary , 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
where is the Einstein–Hilbert action, is the Gibbons–Hawking–York boundary term, is the induced metric (see section below on definitions) on the boundary, its determinant, is the trace of the second fundamental form, is equal to where the normal to is spacelike and where the normal to is timelike, and are the coordinates on the boundary. Varying the action with respect to the metric , subject to the condition
gives the Einstein equations; the addition of the boundary term means that in performing the variation, the geometry of the boundary encoded in the transverse metric is fixed (see section below). There remains ambiguity in the action up to an arbitrary functional of the induced metric
That a boundary term is needed in the gravitational case is because , the gravitational Lagrangian density, contains second derivatives of the metric tensor. This is a non typical feature of field theories, which are usually formulated in terms of Lagrangians that involve first derivatives of fields to be varied over only. The GHY term is desirable, as it possesses a number of other key features. When passing to the Hamiltonian formalism, it is necessary to include the GHY term in order to reproduce the correct Arnowitt–Deser–Misner energy (ADM energy). 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. This term has had more recent applications in loop quantum gravity in calculating transition amplitudes and background independent scattering amplitudes. In order to determine a finite value for the action, one may have to subtract off a surface term for flat spacetime:
where is the extrinsic curvature of the boundary imbedded flat spacetime. As is invariant under variations of , this addition term does not affect the field equations; as such, this is referred to as the non dynamical term.
где — действие Эйнштейна — Гильберта, — граничный член Гиббонса — Хокинга — Йорка, — индуцированная метрика (см. раздел ниже об определениях) на границе, — её определитель, — след второй фундаментальной формы, — равен , где нормаль к пространственноподобна, и — равен , где нормаль к временоподобна, а — координаты на границе. Варирование действия по метрике , при условии
In general relativity, 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 is closed, i. e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary , 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
where is the Einstein–Hilbert action, is the Gibbons–Hawking–York boundary term, is the induced metric (see section below on definitions) on the boundary, its determinant, is the trace of the second fundamental form, is equal to where the normal to is spacelike and where the normal to is timelike, and are the coordinates on the boundary. Varying the action with respect to the metric , subject to the condition
gives the Einstein equations; the addition of the boundary term means that in performing the variation, the geometry of the boundary encoded in the transverse metric is fixed (see section below). There remains ambiguity in the action up to an arbitrary functional of the induced metric
That a boundary term is needed in the gravitational case is because , the gravitational Lagrangian density, contains second derivatives of the metric tensor. This is a non typical feature of field theories, which are usually formulated in terms of Lagrangians that involve first derivatives of fields to be varied over only. The GHY term is desirable, as it possesses a number of other key features. When passing to the Hamiltonian formalism, it is necessary to include the GHY term in order to reproduce the correct Arnowitt–Deser–Misner energy (ADM energy). 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. This term has had more recent applications in loop quantum gravity in calculating transition amplitudes and background independent scattering amplitudes. In order to determine a finite value for the action, one may have to subtract off a surface term for flat spacetime:
where is the extrinsic curvature of the boundary imbedded flat spacetime. As is invariant under variations of , this addition term does not affect the field equations; as such, this is referred to as the non dynamical term.
дает уравнения Эйнштейна; добавление граничного члена означает, что при выполнении вариации геометрия границы, закодированная в поперечной метрике , фиксируется (см. раздел ниже). Остается неоднозначность в действии до произвольной функциональности индуцированной метрики.
In general relativity, 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 is closed, i. e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary , 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
where is the Einstein–Hilbert action, is the Gibbons–Hawking–York boundary term, is the induced metric (see section below on definitions) on the boundary, its determinant, is the trace of the second fundamental form, is equal to where the normal to is spacelike and where the normal to is timelike, and are the coordinates on the boundary. Varying the action with respect to the metric , subject to the condition
gives the Einstein equations; the addition of the boundary term means that in performing the variation, the geometry of the boundary encoded in the transverse metric is fixed (see section below). There remains ambiguity in the action up to an arbitrary functional of the induced metric
That a boundary term is needed in the gravitational case is because , the gravitational Lagrangian density, contains second derivatives of the metric tensor. This is a non typical feature of field theories, which are usually formulated in terms of Lagrangians that involve first derivatives of fields to be varied over only. The GHY term is desirable, as it possesses a number of other key features. When passing to the Hamiltonian formalism, it is necessary to include the GHY term in order to reproduce the correct Arnowitt–Deser–Misner energy (ADM energy). 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. This term has had more recent applications in loop quantum gravity in calculating transition amplitudes and background independent scattering amplitudes. In order to determine a finite value for the action, one may have to subtract off a surface term for flat spacetime:
where is the extrinsic curvature of the boundary imbedded flat spacetime. As is invariant under variations of , this addition term does not affect the field equations; as such, this is referred to as the non dynamical term.
Тот факт, что в гравитационном случае необходим граничный член, объясняется тем, что гравитационная плотность лагранжиана содержит вторые производные метрического тензора. Это нетипичная особенность теорий поля, которые обычно формулируются в терминах лагранжианов, включающих только первые производные варьируемых полей. Термин GHY желателен, поскольку обладает рядом других ключевых свойств. При переходе к гамильтонову формализму необходимо включить член GHY для воспроизведения корректной энергии Арновитта — Дезера — Миснера (энергии ADM). Этот член необходим для обеспечения того, чтобы интеграл по траекториям (в духе Хокинга) для квантовой гравитации имел правильные свойства композиции. При вычислении энтропии чёрной дыры с использованием полуклассического евклидова подхода весь вклад исходит из члена GHY. Этот член нашел более поздние применения в петлевой квантовой гравитации при вычислении амплитуд перехода и амплитуд рассеяния, независимых от фона. Для определения конечного значения действия может потребоваться вычесть поверхностный член для плоского пространства-времени:
In general relativity, 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 is closed, i. e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary , 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
where is the Einstein–Hilbert action, is the Gibbons–Hawking–York boundary term, is the induced metric (see section below on definitions) on the boundary, its determinant, is the trace of the second fundamental form, is equal to where the normal to is spacelike and where the normal to is timelike, and are the coordinates on the boundary. Varying the action with respect to the metric , subject to the condition
gives the Einstein equations; the addition of the boundary term means that in performing the variation, the geometry of the boundary encoded in the transverse metric is fixed (see section below). There remains ambiguity in the action up to an arbitrary functional of the induced metric
That a boundary term is needed in the gravitational case is because , the gravitational Lagrangian density, contains second derivatives of the metric tensor. This is a non typical feature of field theories, which are usually formulated in terms of Lagrangians that involve first derivatives of fields to be varied over only. The GHY term is desirable, as it possesses a number of other key features. When passing to the Hamiltonian formalism, it is necessary to include the GHY term in order to reproduce the correct Arnowitt–Deser–Misner energy (ADM energy). 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. This term has had more recent applications in loop quantum gravity in calculating transition amplitudes and background independent scattering amplitudes. In order to determine a finite value for the action, one may have to subtract off a surface term for flat spacetime:
where is the extrinsic curvature of the boundary imbedded flat spacetime. As is invariant under variations of , this addition term does not affect the field equations; as such, this is referred to as the non dynamical term.
где — внешняя кривизна границы, встроенной в плоское пространство-время. Поскольку инвариантен относительно вариаций , этот дополнительный член не влияет на уравнения поля; следовательно, он называется нединамическим членом.
In general relativity, 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 is closed, i. e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary , 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
where is the Einstein–Hilbert action, is the Gibbons–Hawking–York boundary term, is the induced metric (see section below on definitions) on the boundary, its determinant, is the trace of the second fundamental form, is equal to where the normal to is spacelike and where the normal to is timelike, and are the coordinates on the boundary. Varying the action with respect to the metric , subject to the condition
gives the Einstein equations; the addition of the boundary term means that in performing the variation, the geometry of the boundary encoded in the transverse metric is fixed (see section below). There remains ambiguity in the action up to an arbitrary functional of the induced metric
That a boundary term is needed in the gravitational case is because , the gravitational Lagrangian density, contains second derivatives of the metric tensor. This is a non typical feature of field theories, which are usually formulated in terms of Lagrangians that involve first derivatives of fields to be varied over only. The GHY term is desirable, as it possesses a number of other key features. When passing to the Hamiltonian formalism, it is necessary to include the GHY term in order to reproduce the correct Arnowitt–Deser–Misner energy (ADM energy). 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. This term has had more recent applications in loop quantum gravity in calculating transition amplitudes and background independent scattering amplitudes. In order to determine a finite value for the action, one may have to subtract off a surface term for flat spacetime:
where is the extrinsic curvature of the boundary imbedded flat spacetime. As is invariant under variations of , this addition term does not affect the field equations; as such, this is referred to as the non dynamical term.
Амплитуды перехода и основная функция Гамильтона
В квантовой теории объектом, соответствующим главной функции Гамильтона, является амплитуда перехода. Рассмотрим гравитацию, определенную на компактной области пространства-времени с топологией четырехмерного шара. Граница этой области представляет собой трехмерное пространство с топологией трехмерной сферы, которую мы будем называть. В чистой гравитации без космологической постоянной, поскольку скаляр Риччи обращается в ноль на решениях уравнений Эйнштейна, объемное действие обращается в ноль, а главная функция Гамильтона задается исключительно через граничный член:
где – внешняя кривизна границы, – трехмерная метрика, индуцированная на границе, и – координаты на границе. Функционал является весьма нетривиальным для вычисления, поскольку внешняя кривизна определяется решением в объеме, выделенным внутренней геометрией на границе. Таким образом, является нелокальным. Знание общей зависимости от эквивалентно знанию общего решения уравнений Эйнштейна.
Амплитуды рассеивания, не зависящие от фона
Квантовая гравитация с петлей сформулирована на языке, независимом от фона. Априорно пространство-время не предполагается, а скорее возникает из состояний самой теории, однако амплитуды рассеяния выводятся из точечных функций (корреляционной функции (квантовая теория поля)), которые, будучи сформулированными в обычной квантовой теории поля, являются функциями точек фонового пространства-времени. Связь между фоно-независимым формализмом и обычным формализмом квантовой теории поля на заданном пространстве-времени далеко не очевидна, и неясно, как получить низкоэнергетические величины из полной фоно-независимой теории. Желательно вывести точечные функции теории из фоно-независимого формализма, чтобы сравнить их со стандартным возмутительным разложением квантовой общей теории относительности и, таким образом, проверить, что квантовая гравитация с петлей дает правильный низкоэнергетический предел. Предложена стратегия решения этой проблемы: изучить амплитуду на границе или амплитуду перехода для компактной области пространства-времени, то есть интеграл по траекториям в конечной области пространства-времени, рассматриваемую как функция граничного значения поля. В обычной квантовой теории поля эта граничная амплитуда хорошо определена и кодирует физическую информацию теории; то же самое происходит и в квантовой гравитации, но в полностью фоно-независимой манере. Обычно ковариантное определение точечных функций может быть основано на идее, что расстояние между физическими точками, являющимися аргументами точечной функции, определяется состоянием гравитационного поля на границе рассматриваемой области пространства-времени. Ключевым наблюдением является то, что в гравитации граничные данные включают гравитационное поле, следовательно, геометрию границы, следовательно, все соответствующие относительные расстояния и временные интервалы. Иными словами, граничная формулировка очень элегантно реализует в квантовом контексте полное отождествление геометрии пространства-времени и динамических полей.