Quasi-Local Mass in General Relativity (for Numerical Relativists)

How would we, in general relativity, define conserved quantities such as energy for a finitely extended region of spacetime? We know how to handle mass and energy asymptotically, of course, but there's currently no agreed upon notion of quasi-local mass/energy (QLM/E)*

Such a quantity would be nice to have because statements in GR about gravitational collapse, such as the hoop conjecture, are concerned with mass content in a finite region of space. Likewise, in numerical relativity, where we work in a finite domain, we could compute such a quantity throughout a simulation for a highly dynamical binary system, and extract new physical information.

We thus look at Po-Ning Chen and Mu-Tau Wang's recent paper, which in part reviews several quasi-local mass quantities, and see which of these would be useful for numerical calculations.

First, we would like our definition of QLM/E to satisfy the following properties

1. Rigidity: QLM = 0 in Minkowski space
2. Positivity: QLE ≥ 0 under the dominant energy condition
3. Asymptotics: The QLM should reduce to the definitions of asymptotic (ADM, Bondi masses) and local masses (Bel-Robinson tensor, matter density), in the large sphere and small sphere limits, respectively. 
4. Monotonicity: The QLM should increase when we would expect it to increase (though it doesn't have to be strictly additive because of negative gravitational binding energy). 

These are mathematical properties that we would like to satisfy; an additional property for the  numerical relativists among us could be 

     0. Nice to compute in simulations

Presently, we don't have a quasi-local mass definition that is proven to satisfy all of the mathematical properties, but let's walk through the definitions that we do have, and talk about the (dis)advtanges of each.

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Hawking Mass

\[m_H(\Sigma) = \sqrt{\dfrac{|\Sigma|}{16\pi}}\left(1 - \dfrac{1}{16\pi} \int_{\Sigma} |H|^2 d\Sigma \right) \]

where $$\Sigma$$ is a spacelike 2-surface (the boundary of our region) and $$H$$ is the mean curvature (trace of the extrinsic curvature) vector of $$\Sigma$$ in the spacetime.

This has some proven monotonicity (for time-symmetric initial data (ID)), and asymptotics (for asymptotically flat ID, will approach ADM mass of the ID), but is proven to lack positivity under certain conditions, as well as rigidity for $\mathbb R^3$. As for easy of computation, it's composed of quantities we have access to throughout a simulation, and so we could feasibly compute it. However, it's too negative, as Wang put in a recent talk.

Bartnik Mass

It's defined in terms of an infimum among all admissible extensions - not easy to use for numerical computations.

Brown-York Mass

\[m_BY(\Sigma) = \dfrac{1}{8\pi} \int_{\Sigma} (H_0 - H) d\Sigma \]

where $$\Sigma$$ bounds a spacelike hypersurface $$\Omega$$, and $$H_0$$ is the mean curvature of the unique isometric embedding of $$\Sigma$$ into $$\mathbb R^3$$ and $$H$$ is the mean curvature of $$\Sigma$$ in $$\Omega$$.

This satisfies positivity in certain cases,  and some asymptotics, but it's also gauge dependent. One can, however, replace $\Omega$ with the entire spacetime to obtain a gauge independent quantity (the Liu-Yau mass). This, however, lacks the rigidity property. It's also too positive, as Wang put in a recent talk. As for ease of computation, since we work with spacelike hypersurfaces in NR, this would be a feasible, if not natural, mass to compute - the isometric embedding would involve some numerical use of Newton's method. 

Wang-Yau Energy

Warning, notation soup up ahead. For the Wang-Yau Energy, we compute the minimum over all choices of $$(X, T_0)$$ of

\[E(\Sigma, X, T_0) = \int_{\hat \Sigma} \hat H d \hat \Sigma - \int_\Sigma \left(\sqrt{1 + |\nabla \tau |^2}  \cosh \theta |H| - \nabla \tau \cdot \nabla \theta - \alpha_H (\nabla \tau) \right) d \Sigma \]

where $$\Sigma$$ is a spacelike 2-surface in the spacetime, with spacelike mean curvature vector $$H$$, $$\sigma$$ is the induced metric on $$\Sigma$$, $$\theta$$ is a function of $$|H|$$ and $$\tau$$, $$\Delta$$ and $$\nabla$$ are the gradient of Laplacian wrt $$\sigma$$, the induced 2-metric, $$\alpha_H$$ is the connection one form of the normal bundle wrt $$H$$. For an isometric embedding $$X: \Sigma \to \mathbb R^{3,1}$$, and a future timelike unit vector $$T_0$$, we can consider the projected embedding $$\hat X$$ into the orthogonal complement of $$T_0$$, from which we get the quantities with the hats. Finally, $$\tau = -X \cdot T_0$$ is the time function.

In order to find the isometric embedding that minimizes the quasi-local energy, we would have to solve a nonlinear elliptic equation - the Euler-Lagrange equation for the critical point of the energy as a function of $$\tau$$ (given in the paper, not reproduced here).

We do, however, have the numerical technology for this! And, unless I'm mistaken, we have access to all of these quantities in our simulations - though we have one more non-linear PDE to solve. Moreover, this definition of quasi-local mass satisfies the rigidity property, and certain positivity properties. 

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Thus, we have 3 quantities for the quasi-local mass that we can use for new physics computations. It would be cool to look at all of these (first for Schwarzschild and Kerr) for BH binary simulations :)


*Note that because of the equivalence principle and lack of symmetry in generic spacetimes, we can't formulate mass as a volume integral over mass density, as we would in Newtonian gravity. Note also that we can't take approaches meant for an isolated system, either - asymptotic masses are defined in regions where gravity is weak on the boundary, but for a quasi-local mass, gravity could be strong on the boundary.