Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A fresh look at boundary terms in Einstein-Hilbert gravity via an initial value variational principle

Songmin Ha, Alexander Rothkopf

Abstract

A key tenet of general relativity is the dynamical nature of space-time, ideally represented as an initial value problem. Here we explore the variational formulation of classical Einstein-Hilbert gravity as initial value problem by constructing its Schwinger-Keldysh-Galley (SKG) action, including a careful treatment of boundary terms. The construction is based on a doubling of degrees of freedom and independent of a foliation. The action naturally decomposes into a bulk term furnishing Einstein's equations and a boundary term, which is related to conserved quantities, such as the Komar mass. We find that since only trivial connecting conditions must be specified on boundaries, the variational action principle for gravity as an initial value problem is rendered well-posed without the need to add additional boundary terms. The SKG approach to gravity offers a novel and complementary avenue to solve for the metric of spacetime directly from the action, bypassing the governing equations.

A fresh look at boundary terms in Einstein-Hilbert gravity via an initial value variational principle

Abstract

A key tenet of general relativity is the dynamical nature of space-time, ideally represented as an initial value problem. Here we explore the variational formulation of classical Einstein-Hilbert gravity as initial value problem by constructing its Schwinger-Keldysh-Galley (SKG) action, including a careful treatment of boundary terms. The construction is based on a doubling of degrees of freedom and independent of a foliation. The action naturally decomposes into a bulk term furnishing Einstein's equations and a boundary term, which is related to conserved quantities, such as the Komar mass. We find that since only trivial connecting conditions must be specified on boundaries, the variational action principle for gravity as an initial value problem is rendered well-posed without the need to add additional boundary terms. The SKG approach to gravity offers a novel and complementary avenue to solve for the metric of spacetime directly from the action, bypassing the governing equations.
Paper Structure (12 sections, 49 equations, 1 figure)

This paper contains 12 sections, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hamilton's principle for Einstein-Hilbert gravity
  3. Schwinger--Keldysh--Galley action principle
  4. Point particle mechanics
  5. Scalar field theory
  6. Initial-value action principle for gravity
  7. Constructing the SKG Einstein-Hilbert action
  8. Variation and boundary term analysis
  9. SKG action and conserved quantities
  10. Noether currents
  11. Relation to the Komar mass
  12. Conclusion

Figures (1)

  • Figure 1: (left) Hamilton's boundary value action principle for field theory requires specification of field data at initial and final time (green). After imposition of these Dirichlet temporal boundary values the classical field configuration (blue) emerges from the extremum of the action $S_{\rm BVP}$ under arbitrary variations (gray). (right) The SKG action principle asks us to specify only initial data for the physical degrees of freedom $\phi_+(t_i,x)$ and $\dot \phi_+(t_i,x)$ (green). The advanced components $\phi_-(x)$ and their temporal derivatives are set to zero on the final time slice (white). The extremum of the action under imposition of initial and connecting conditions produces the classical field configuration for the retarded components $\phi_{\rm cl}(x)=\phi_+(x)=(\phi_1(x)+\phi_2(x))/2$ for which the solution on the forward and backward branch agrees such that $\phi_-(x)=\phi_1(x)-\phi_2(x)=0$.