# Phase Space GR 3+1 Form: Compact & Non-Compact

• shoehorn
In summary, the 3+1 formulation of general relativity (GR) includes variables such as the metric on a spatial surface, the momentum conjugate to the metric, and the shift vector and lapse function, which are both considered gauge variables. The phase space for GR is typically described as the space of pairs (metric, momentum) over a closed spatial manifold. However, if we extend this idea to include a non-zero boundary, the action for GR requires a boundary term and the phase space would need to be expanded to include additional variables such as a two-dimensional metric and its conjugate momentum. The dynamics of this extended phase space and its relationship to the dynamics of the original closed phase space are still unclear.
shoehorn
In the 3+1 formulation of GR we have the following basic variables:

$$g_{ij} = \textrm{metric on a spatial surface}$$
$$\pi^{ij} = \textrm{momentum conjugate to }g_{ij}$$
$$N^i = \textrm{shift vector}$$
$$N = \textrm{lapse function}$$

Both $N$ and $N^i$ are purely gauge variables, so are essentially unimportant. This means that the phase space for GR is essentially the space of pairs $(g_{ij},\pi^{ij})$ over a given spatial manifold $\Sigma$. (I know this isn't really the phase space since GR has constraints, but that's irrelevant for my current question.)

The point is that, in most treatments of which I am aware, the spatial manifold $\Sigma$ is taken to be closed, i.e., compact and without boundary. However, if we extend these ideas by setting $\partial\Sigma\ne0$ then we know that the action for GR also requires a boundary term. Does this then imply that the phase space for GR should be extended to the set

$$\{g_{ij},\pi^{ij},\gamma_{AB},p^{AB}\}$$

where $\gamma^{AB}$ is a two-dimensional metric on $\partial\Sigma$ and $p^{AB}$ is its conjugate momentum?

If so, then what would the dynamics of this extended phase space look like, and how would it be related to the dynamics of the original (closed) phase space?

Yes, extending the spatial manifold \Sigma to include a boundary \partial\Sigma would require the phase space for GR to be extended as well. This is because the boundary term in the action for GR is necessary to ensure that the variational principle is well-defined, and therefore the boundary variables \gamma^{AB} and p^{AB} must be included in the phase space.

Furthermore, the inclusion of the boundary variables also reflects the fact that the boundary conditions at \partial\Sigma play an important role in the dynamics of the system. In particular, the boundary conditions can affect the evolution of the system and the behavior of the gravitational fields.

In summary, while the phase space for GR is usually taken to be the space of pairs (g_{ij},\pi^{ij}) over a closed spatial manifold \Sigma, extending \Sigma to include a boundary \partial\Sigma would require the phase space to be extended to include the boundary variables \gamma^{AB} and p^{AB}. This is necessary to properly account for the dynamics and boundary conditions of the system.

## 1. What is Phase Space GR 3+1 Form?

Phase Space GR 3+1 Form is a mathematical formalism used in General Relativity (GR) to describe the dynamics of a system with three spatial dimensions and one time dimension. It represents the four-dimensional space-time as a set of points, with each point representing a possible state of the system.

## 2. What is the difference between compact and non-compact Phase Space GR 3+1 Form?

In compact Phase Space GR 3+1 Form, the space-time is finite and has boundaries, while in non-compact form, the space-time is infinite and has no boundaries. Compact form is often used to study systems with a finite number of degrees of freedom, while non-compact form is used for systems with an infinite number of degrees of freedom.

## 3. How is Phase Space GR 3+1 Form related to classical mechanics?

Phase Space GR 3+1 Form is an extension of Hamiltonian mechanics, a classical mechanics formalism that describes the dynamics of a system using position and momentum variables. In Phase Space GR 3+1 Form, the position and momentum are replaced by the space-time coordinates and their derivatives, respectively.

## 4. What are some applications of Phase Space GR 3+1 Form?

Phase Space GR 3+1 Form has applications in various fields such as cosmology, astrophysics, and high energy physics. It is used to study the evolution of the universe, black holes, and gravitational waves. It is also used in the development of theories beyond GR, such as string theory and loop quantum gravity.

## 5. Are there any limitations to using Phase Space GR 3+1 Form?

One limitation of Phase Space GR 3+1 Form is that it does not take into account the quantum nature of matter. It is a classical theory and cannot fully describe systems at the quantum level. Additionally, it is a highly mathematical formalism and can be challenging to apply to real-world situations without proper training and expertise.

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