Flux Conservative Scalar Wave Equation

Why?

Writing the scalar wave equation in a flux conservative form facilitates some tasks related to it's numerical evolution, namely:

1. Showing that the system is strongly hyperbolic thus well posed.
2. Computing the system's characteristic variables/equations.
3. Determining equations for boundary condition impositions.
4. Eliminating the necessity of explicitly computing Christoffel symbols and second spatial derivatives.

The system

On an arbitrary background spacetime, a scalar field of mass $m$ propagates according to the Klein-Gordon (KG)

$$ \nabla_\mu \nabla^mu \Phi - m^2 \Phi = 0 \quad (1) $$

Using the well-know formula

$$ \nabla_\mu T^\mu = \frac{1}{\sqrt{g}} \partial_\mu \left( \sqrt{-g} T^\mu \right), \quad (2) $$

we can write

$$ \nabla_\mu \nabla^\mu \Phi = \nabla_\mu \left( g^{\mu\nu} \nabla_\nu \Phi \right) = \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g} g^{\mu\nu} \nabla_\nu \Phi \right), \quad (3) $$

and using the fact that $\nabla_\mu \Phi = \partial_\mu \Phi$, the KG equation becomes

$$ \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g} g^{\mu \nu} \partial_\nu \Phi \right) - m^2 \Phi = 0. \quad (4) $$

Flux conservative form

To get the flux conservative form, we first write out the time and spatial derivatives explicitly:

$$ \partial_t \left( \sqrt{-g} g^{t\mu} \partial_\nu \Phi \right) + \partial_i\left( \sqrt{-g} g^{i\nu} \partial_\nu \Phi \right) - \sqrt{-g} m^2 \Phi = 0, \quad (5) $$

where spatial indices label spatial dimensions. By defining

$$ \Psi_i \partial_i \Phi \quad (6) $$

we can define a new function $\Pi$ as

$$ \Pi \equiv \sqrt{-g} g^{t\mu}\partial_\nu\Phi = \sqrt{-g} \left( g^{tt} \partial_t \Phi + g^{ti} \Psi_i \right) \quad (7) $$

which we can immediately invert to yield a time evolution equation for \Phi

$$ \partial_t \Phi = \frac{1}{g^{tt}}\left( \frac{\Pi}{\sqrt{-g}} - g^{ti} \Psi_i \right). \quad (8) $$

By defining the flux vector $F^i$ as

$$ F^i \equiv \sqrt{-g} g^{i\nu} \partial_\nu \Phi = \sqrt{-g} \left\{ g^{ti} \left[ \frac{\Pi}{\sqrt{-g}} - g^{tj}\Psi_j \right] + g^{ij}\Psi_j \right\} \quad (9) $$

the original KG equation becomes

$$ \partial_t \Pi + \partial_i F^i - \sqrt{-g} m^2 \Phi = 0, \quad (10) $$

which immediately yields a time evolution equation for $\Pi$.

In order to complete the system, we need to find the time evolution equations for the $\Psi_i$ variables, which can be easily done using Eq. (6) and assuming that the $\Psi_i$ functions are such that their second order partial derivatives commute:

$$ \partial_t\Psi_i = \partial_t \left( \partial_i\Phi \right) = \partial_i \left( \partial_t \Phi \right) = \partial_i \left[ \frac{1}{g^{tt}} \left( \frac{\Pi}{\sqrt{-g}} - g^{tj}\Psi_j \right) \right]. \quad (11) $$

Using ADM variables

Often in numerical codes the spacetime metric and curvature is presented in the form of ADM objects, namely the lapse ( $\alpha$ ), the shift vector ( $\beta^i$ ) the spatial metric ( $\gamma_{ij}$ ) and the extrinsic curvature ( $K_{ij}$ ). By noting that

$$ g^{tt} = -\alpha^{-2} \quad (12) $$

$$ g^{ti} = \alpha^{-2}\beta^i \quad (13) $$

$$ g^{ij} = \gamma^{ij} - \alpha^{-2} \beta^i \beta{j} \quad (14) $$

and

$$ \sqrt{-g} = \alpha \sqrt{\gamma} \quad (15) $$

the full evolution system becomes

$$ \partial_t \Pi = \alpha \sqrt{\gamma} m^2 \Phi - \partial_i F^i \quad (16) $$

$$ \partial_t \Psi_i = \partial_i \left( \beta^j\Psi_j - \frac{\alpha \Pi}{\sqrt{\gamma}} \right) \quad (17) $$

$$ \partial_t \Phi = \left( \beta^i\Psi_i - \frac{\alpha \Pi}{\sqrt{\gamma}} \right) \quad (18) $$

and

$$ F^i = \alpha \sqrt{\gamma} \gamma^{ij} \Psi_j - \beta^i \Pi \quad (19) $$