---
title: Generalized Wave Map Gauge
url: https://www.emergentmind.com/topics/generalized-wave-map-gauge
type: topic
---

# Generalized Wave Map Gauge

Generalized wave map gauge is a covariant gauge fixing for Einstein’s equations in which the identity map \(\mathrm{id}:(\mathcal M,g)\to(\mathcal M,\bar g)\) is required to be a wave map, possibly with prescribed gauge source functions. In one standard form, the gauge constraint is
\[
C^\mu \;:=\; g^{\alpha\beta}\Big(\Gamma^{\mu}{}_{\alpha\beta}(g)-\bar\Gamma^{\mu}{}_{\alpha\beta}(\bar g)\Big)-H^\mu,
\]
and the gauge condition is \(C^\mu=0\). In generalized harmonic form one writes \(C_\mu=\Gamma_\mu-H_\mu=0\), with \(\Gamma_\mu\) the contracted Christoffel symbols of \(g\). Choosing \(H^\mu=g^{\alpha\beta}\bar\Gamma^\mu_{\alpha\beta}(\bar g)\) identifies the two descriptions. This gauge converts the Einstein equations into quasilinear wave systems, provides a natural setting for constraint propagation and hyperbolicity, and has been used both in asymptotic analyses near null infinity and in singular or numerical formulations of general relativity [2205.09405] [1602.03018] [1303.4783] [0904.4873].

## 1. Covariant definition and equivalence with generalized harmonic gauge

In the generalized wave map gauge, the unknown Lorentzian metric \(g\) is compared to a fixed background metric \(\bar g\), and the coordinate condition is imposed by requiring the identity map to be a wave map up to gauge source functions. The gauge constraint vector is
\[
C^\mu(x;g,\partial g)\;:=\;g^{\alpha\beta}\Big(\Gamma^{\mu}{}_{\alpha\beta}(g)-\bar\Gamma^{\mu}{}_{\alpha\beta}(\bar g)\Big)-H^\mu(x,g,\partial g),
\]
with gauge condition \(C^\mu=0\). A widely used specialization is generalized harmonic gauge, where one drops the background term and sets \(C^\mu=\Gamma^\mu-H^\mu\), and harmonic gauge is the special case \(H^\mu=0\) [1602.03018] [1303.4783].

The same structure appears in the notation \(D_i=F_i-T_i=0\), with \(T_i:=g^{k\ell}\Gamma_{k\ell i}=\Gamma_i\). In this notation the coordinate functions satisfy the inhomogeneous wave equations
\[
\square_g x^i = g^{k\ell}\Gamma^{i}{}_{k\ell}(g)=F^i,
\]
which is why the construction is termed a generalized wave, or wave map, gauge [1602.03018].

The equivalence between generalized harmonic gauge and generalized wave map gauge is particularly explicit in the null-asymptotic formulation of “Peeling in Generalized Harmonic Gauge” [2205.09405]. There one uses three covariant derivatives: \(\nabla\), the Levi–Civita connection of \(g\); \(\mathring{\nabla}\), the flat Cartesian covariant derivative; and \(\mathbullet{\nabla}\), the flat “shell” covariant derivative. The gauge source functions are chosen as
\[
F^{a}=\mathring{F}^{a}:=g^{bc}\,\Gamma[\mathring{\nabla},\mathbullet{\nabla}]^{a}{}_{bc},
\]
so that \(Z^a:=\Gamma^a+F^a\) satisfies
\[
g^{bc}\,\Gamma[\nabla,\mathring{\nabla}]_{b}{}^{a}{}_{c}=0.
\]
This is the wave map gauge with reference connection \(\mathring{\nabla}\), and in index notation
\[
H^\mu = g^{\alpha\beta}\,\bar\Gamma^\mu_{\alpha\beta}(\bar g), \qquad \bar\Gamma=\Gamma[\mathring{\nabla},\mathbullet{\nabla}],
\]
so generalized harmonic and generalized wave map gauges coincide in that formulation [2205.09405].

## 2. Reduced Einstein equations and wave-system structure

The principal utility of generalized wave map gauge is that it reduces the Einstein equations to quasilinear wave equations for the metric. In a standard covariant presentation, one introduces the reduced operator
\[
E_{\mu\nu}(g)\;:=\;R_{\mu\nu}(g)-\nabla_{(\mu}C_{\nu)}+\text{lower-order terms}(g,\partial g,C),
\]
and solves \(E_{\mu\nu}(g)=0\). If \(C^\mu=0\), then the reduced system implies the vacuum Einstein equations \(R_{\mu\nu}(g)=0\). The principal part is the covariant wave operator acting on \(g_{\mu\nu}\), yielding a second-order wave system and corresponding first-order hyperbolic reductions [1602.03018] [1303.4783].

In the asymptotic null formulation of [2205.09405], one writes
\[
\mathcal{R}_{ab}:=R_{ab}-\nabla_{(a}Z_{b)}+W_{ab}, \qquad Z^a:=\Gamma^a+F^a,
\]
where \(W_{ab}\) is a constraint addition homogeneous in \(Z^a\), so that \(Z^a=0\) implies \(W_{ab}=0\). Vacuum Einstein equations are imposed through the reduced system \(\mathcal{R}_{ab}=0\). The paper also defines
\[
\tilde{\mathcal R}_{ab}:=\mathcal R_{ab}+\nabla_{(a}F_{b)}-W_{ab}.
\]
With null vectors adapted to shell coordinates \((T,R,\theta^A)\), the inverse metric is decomposed into ten variables
\[
\varphi,\quad \mathcal C_\pm^R,\quad \mathcal C_A^\pm,\quad \epsilon,\quad h_+,\quad h_\times,
\]
where \(h_+\) and \(h_\times\) parameterize the two gravitational-wave polarizations. The resulting field equations form a closed system of ten coupled quasilinear wave equations driven by \(\mathbullet{\square}\), the wave operator associated with \(\mathbullet{\nabla}\) [2205.09405].

The same reduction mechanism appears in symmetry-reduced singular problems. For Gowdy \(T^3\) spacetimes, the generalized wave gauge reduction yields a 6-component quasilinear wave system for
\[
u(t,x)=(g_{00},\,g_{11},\,g_{01},\,R,\,E,\,Q)^T,
\]
equivalently a first-order symmetric hyperbolic system
\[
S^0\,D U + S^1\,t\,\partial_x U + N\,U = f[U],
\]
with \(D=t\,\partial_t\). In that setting the source combines gauge couplings through \(F_0,F_1\), lower-order quadratic Ricci terms, and terms proportional to the gauge violations \(D_i=F_i-\Gamma_i\), while the Bianchi identities furnish a hyperbolic propagation equation for the gauge constraints [1602.03018].

## 3. Hyperbolicity, pure gauge subsystems, and 3+1 parametrizations

A central structural result is that, from the free-evolution point of view, only wave-like pure gauges can be coupled to the field equations to obtain a properly defined wave-like formulation. In the constrained Hamiltonian model analyzed in “Hyperbolicity of Physical Theories with Application to General Relativity” [1303.4783], the total principal symbol is block triangular, with diagonal pure-gauge, constraint, and physical blocks. Strong hyperbolicity of the full formulation therefore requires strong hyperbolicity of each diagonal block; in particular, a pure gauge can be used to form a strongly hyperbolic formulation if and only if the pure gauge subsystem is itself strongly hyperbolic [1303.4783].

For general relativity in \(3+1\) form, this perspective yields a five-parameter generalization of harmonic gauge. With lapse \(\alpha\), shift \(\beta^i\), spatial metric \(\gamma_{ij}\), extrinsic curvature \(K_{ij}\), and Z4-type auxiliary constraints \(\Theta\) and \(Z_i\), the gauge conditions are
\[
\partial_t \alpha = - g_1 \alpha^2 K + g_2 (\partial_i \alpha)\beta^i + \beta^j \partial_j \alpha + 2 c_1 \alpha^2 \Theta,
\]
\[
\partial_t \beta^i = \alpha^2 [ g_3 \gamma^{ij}\gamma^{kl} + g_4 \gamma^{il}\gamma^{jk} ] \partial_l \gamma_{jk} - g_5 \alpha \partial_i \alpha + \beta^j \partial_j \beta^i + 2 \alpha^2 c_2 Z_i.
\]
The pure-gauge characteristic speeds are \(\pm\sqrt{g_3}\) and \(\pm v_\pm\), where
\[
2 v_\pm^2 = g_1 + \bar g_3 - g_2 g_5 \pm \sqrt{(g_1 + \bar g_3 - g_2 g_5)^2 - 4 (g_1-g_2)\bar g_3},
\qquad \bar g_3 := 2(g_3+g_4).
\]
Strong hyperbolicity holds if \(g_3>0\) and either \(0\neq g_2<g_1\) with the stated inequality for \(g_2g_5\), or \(g_2=0\) with \(\bar g_3\neq g_1\), or \(g_2=0\), \(\bar g_3=g_1\), and \(g_5=1\) [1303.4783].

The constraint subsystem can be decoupled at principal level by choosing
\[
c_1=g_1,\quad c_2=g_3,\quad c_3=c_5=c_6=0,\quad c_4=2,\quad 2c_7=1,\quad c_8=1,\quad c_9=1,\quad c_{10}=2\Big(1+\frac{g_4}{g_3}\Big).
\]
Then the constraint block has characteristic speeds \(\pm\sqrt{c_{10}}\) and \(\pm1\) with multiplicity three, while the physical block has characteristic speeds \(\pm1\). The harmonic or wave-map principal part is recovered by
\[
g_2=0,\qquad g_1=1,\qquad g_3=1,\qquad g_4=-\frac12,\qquad g_5=1,
\]
with the same constraint couplings. In this sense the five-parameter family is a \(3+1\) parametrization of generalized wave-map or generalized harmonic gauge at principal level [1303.4783].

## 4. Null asymptotics, polyhomogeneity, and peeling

Near null infinity, the reduced Einstein equations in generalized harmonic or generalized wave map gauge admit a stratification into “good,” “bad,” and “ugly” fields. In its basic form, the model system is
\[
\mathring{\square}\,g = \mathcal N_g,\qquad
\mathring{\square}\,b = (\nabla_T g)^2 + \mathcal N_b,\qquad
\mathring{\square}\,u = \frac{2}{R}\,\nabla_T u + \mathcal N_u,
\]
where \(\mathring{\square}\) is the Cartesian wave operator. Good fields satisfy pure wave equations and do not generate logarithmic terms at leading orders; bad fields are sourced quadratically by time derivatives of good fields and can generate logarithms already at low orders; ugly fields satisfy wave equations with singular transport-type terms \((2/R)\nabla_T\) and generically produce logarithmic contributions at higher orders. The leading null expansion
\[
\mathring{\square}\,\phi \simeq -2\,\nabla_{\psi}\nabla_T \phi - \frac{2}{R}\,\nabla_T \phi
\]
exhibits the weak null structure behind this classification [2205.09405].

For coupled systems with many good, bad, and ugly fields, formal polyhomogeneous expansions exist under asymptotic flatness assumptions:
\[
g \sim \frac{G_{1,0}(\psi^*)}{R} + \sum_{n\ge 2}\sum_{k=0}^{N_n^g} \frac{(\log R)^k\,G_{n,k}(\psi^*)}{R^n},
\]
\[
b \sim \frac{B_{1,0}(\psi^*) + (\log R)\,B_{1,1}(\psi^*)}{R} + \sum_{n\ge 2}\sum_{k=0}^{N_n^b} \frac{(\log R)^k\,B_{n,k}(\psi^*)}{R^n},
\]
\[
u \sim \frac{m_{u,1}(\theta^A)}{R} + \sum_{n\ge 2}\sum_{k=0}^{N_n^u} \frac{(\log R)^k\,U_{n,k}(\psi^*)}{R^n}.
\]
A key remark is that, in a system with only goods and uglies, and with all uglies having the same \(p\), there are no logarithms up to order \(n=p\); the first logs can appear at \(n=p+1\), provided none are inherited via coupling [2205.09405].

In the Cartesian harmonic specialization of [2205.09405], after rescaling \(\hat{\mathcal C}_A^\pm:=R\,\mathcal C_A^\pm\), the first-order asymptotic system separates as follows.

| Class | Variables | Leading asymptotic equation |
|---|---|---|
| Good | \(\varphi,\ h_+,\ h_\times\) | \(\mathring{\square}\phi=\mathcal N_\phi\) |
| Bad | \(\mathcal C_-^R\) | \(\mathring{\square}\mathcal C_-^R=-\tfrac12(\nabla_T h_+)^2-\tfrac12(\nabla_T h_\times)^2+\mathcal N_{\mathcal C_-^R}\) |
| Ugly | \(\mathcal C_+^R,\ \hat{\mathcal C}_A^+,\ \epsilon\) | \(\mathring{\square}\phi=\tfrac{2}{R}\nabla_T\phi+\mathcal N_\phi\) |
| Ugly | \(\hat{\mathcal C}_A^-\) | \(\mathring{\square}\hat{\mathcal C}_A^-=\tfrac{4}{R}\nabla_T\hat{\mathcal C}_A^-+\mathcal N_{\hat{\mathcal C}_A^-}\) |

This classification determines the maximal logarithmic powers \(N_n^\phi\) order by order. For example, at first order \(N_1^{\mathcal C_-^R}=1\) while the other listed variables have \(N_1^\phi=0\); at second order \(N_2^{\mathcal C_-^R}=N_2^{\mathcal C_A^-}=2\), whereas \(N_2^{\varphi}=N_2^{\mathcal C_+^R}=N_2^{\mathcal C_A^+}=N_2^\epsilon=N_2^{h_+}=N_2^{h_\times}=1\) [2205.09405].

These expansions feed directly into the peeling analysis of the Weyl tensor. Using Newman–Penrose scalars, peeling means
\[
\Psi_N \sim O(r^{N-5}),\qquad N=0,\dots,4.
\]
In Cartesian harmonic gauge, [2205.09405] finds
\[
\Psi_4 = -\frac{\ddot{h}_{+(1,0)} + i\,\ddot{h}_{\times(1,0)}}{R} + o(R^{-1}),
\]
\[
\Psi_3 = \frac{1}{4R^2}\,\big(\text{derivatives of }\dot{h}_{\pm(1,0)}, \dot{\varepsilon}_{(1,0)}, \dot{\varphi}_{(1,0)}, \ddot{\mathcal C}^+_{A(1,0)}\big)+o(R^{-2}),
\]
which are consistent with peeling at those orders. For \(\Psi_2\), however, logarithmic terms appear at order \(R^{-3}\):
\[
\Psi_2' = \frac{\Psi_{2(3,0)}}{12\,R^3} + \frac{\log R}{4\,R^3}\,\Psi_{2(3,1)} + o(R^{-3}),
\]
with
\[
\Psi_{2(3,1)} = -\mathcal C^R_{-(1,1)} + 2\,\dot{\mathcal C}^R_{+(2,1)} - 2\,\dot{\varepsilon}_{(2,1)}.
\]
Since these coefficients are nonzero for generic data in that gauge, classical peeling is violated in pure Cartesian harmonic gauge. The obstruction is traced to the bad field \(\mathcal C_-^R\) and the ugly fields carrying transport terms proportional to \((2/R)\nabla_T\) [2205.09405].

The same paper also shows that this obstruction is gauge-sensitive rather than unavoidable. With modified gauge source functions and constraint additions
\[
F^a=\mathring F^a+\check F^a,
\]
\[
W_{\psi\psi}=-\tfrac12(Z^\sigma)^2+\tfrac{p_1}{R}Z^\sigma,\qquad
\slashed W=-\tfrac{p_2}{R}Z^{\sigmab},\qquad
W_{\psi A}=\tfrac{3-p_A}{R}Z^A,
\]
and by choosing \(p:=p_1=p_2=p_A\) together with \(\check F^a\) so that all non-radiative variables satisfy ugly equations with the same transport coefficient \(2p/R\), the first logarithms are pushed to arbitrarily high order. A gauge-driver field \(\check F_1^{\sigmab}\) is introduced to maintain hyperbolicity. Choosing \(p\) sufficiently large, for example \(p\ge 7\), removes logarithms from the orders contributing to the leading Newman–Penrose scalars and restores peeling at leading order, with \(\Psi_2\sim O(R^{-3})\), \(\Psi_1\sim O(R^{-4})\), and \(\Psi_0\sim O(R^{-5})\) [2205.09405].

## 5. Singular initial value problems and AVTD behavior in generalized wave gauges

Generalized wave map gauge is also a framework for singular analyses near spacelike singularities. For vacuum Gowdy spacetimes with \(T^3\)-spatial topology, [1602.03018] proves the existence of smooth solutions that are asymptotically velocity term dominated (AVTD) in an infinite-dimensional family of generalized wave gauges. The formulation uses a symmetry-preserving metric ansatz
\[
g = g_{00}(t,x)\,dt^2 + 2g_{01}(t,x)\,dt\,dx + g_{11}(t,x)\,dx^2 + R(t,x)\,E(t,x)\,(dy+Q(t,x)\,dz)^2 + \frac{R(t,x)}{E(t,x)}\,dz^2,
\]
with \(g_{02}=g_{03}=g_{12}=g_{13}=0\). Areal coordinates are recovered as the special case \(R(t,x)=t\), \(g_{01}\equiv0\), and \(g_{00}=-g_{11}\) [1602.03018].

Motivated by Kasner asymptotics, the gauge source functions are chosen in the form
\[
F_0(t,x)=-\frac{1}{t}+F_0(t,x),\qquad F_1(t,x)=F_{10}(x)+F_1(t,x),\qquad F_2=F_3=0,
\]
where \(F_0,F_1\) belong to weighted function spaces encoding decay near \(t=0\). More generally, \(F_0\) and \(F_1\) may depend on the metric via rational function operators in the authors’ weighted Sobolev framework. This is the source of the infinite-dimensional gauge freedom in which AVTD is established [1602.03018].

The leading asymptotics are parameterized by a smooth velocity function \(k(x)\in(0,3/4)\) and free asymptotic data \((g_{00}^*,g_{11}^*,R_+,E_+,Q_*,Q_+^*)\). The main theorem gives
\[
g_{00}(t,x)=g_{00}^*(x)\,t^{(k^2(x)-1)/2} + w_{00}(t,x),\qquad
g_{11}(t,x)=g_{11}^*(x)\,t^{(k^2(x)-1)/2} + w_{11}(t,x),
\]
\[
g_{01}(t,x)=w_{01}(t,x),\qquad
R(t,x)=R_+(x)\,t+w_R(t,x),
\]
\[
E(t,x)=E_+(x)\,t^{-k(x)} + w_E(t,x),\qquad
Q(t,x)=Q_+^*(x)\,t^{2k(x)} + w_Q(t,x),
\]
with remainders in explicit weighted spaces and with all time derivatives controlled in the same classes [1602.03018].

The proof proceeds through a singular initial value problem for a first-order symmetric hyperbolic Fuchsian system. The analytical ingredients are weighted spaces
\[
\|w\|_{\delta,\mu,q}:=\sup_{t\in(0,\delta]}\|R[\mu]\,w(t,\cdot)\|_{H^q(T^n)},
\]
rational function operators, block diagonality with respect to the exponent vector, and the eigenvalue condition
\[
p(x)\ge \mathrm{Re}\,A(x)
\]
for \(A=(S^0)^{-1}N\). Under these conditions, the singular initial value problem admits a unique remainder \(W\in X_{\delta,\kappa+p,\infty}\), and the gauge constraints propagate to zero once suitable asymptotic constraints on \(F_{10}\) and the asymptotic data are imposed [1602.03018].

A significant implication is that AVTD is not an artifact of areal coordinates. The areal gauge is included as a special case, but the theorem applies to a much larger family of generalized wave gauges. The paper also studies coordinate transformations mapping areal solutions to wave-gauge representations and concludes that large subfamilies are connected in this way, while full equivalence remains an open question [1602.03018].

## 6. Gauge drivers, boundary conditions, and numerical realizations

In numerical relativity, generalized wave map gauge is often implemented through gauge drivers that evolve the gauge source vector rather than prescribing it algebraically. In the generalized harmonic formulation of [0904.4873], the coordinates satisfy
\[
H^a=\nabla^c\nabla_c x^a,
\]
and the gauge constraint is
\[
\mathcal C_a \equiv H_a + \Gamma_a = 0,
\]
so exact generalized harmonic gauge corresponds to \(H_a=-\Gamma_a\). To enforce a target gauge \(H_a^{(\mathrm{target})}=F_a(x,\psi,\partial\psi)\) while maintaining hyperbolicity, the paper promotes \(H_a\) to an independent field and introduces the first-order driver
\[
\partial_t H_a - N^k \partial_k H_a = -\mu\,(H_a-F_a) + \theta_a,
\]
\[
\partial_t \theta_a + \eta\,\theta_a = -\eta\,N^k \partial_k H_a,
\]
with \(\mu>0\) and \(\eta>0\) [0904.4873].

Coupled to the first-order generalized harmonic Einstein system, this driver yields a strongly and symmetric hyperbolic formulation. The characteristic fields include
\[
u^{\hat 3}_a = H_a,\qquad v=-n_k N^k,
\]
\[
u^{\hat 4}_a = \theta_a + \eta H_a,\qquad v=0
\]
in the single-frame case, together with the standard generalized harmonic metric characteristic fields. A positive-definite symmetrizer exists, and the evolution of the constraints is unchanged from the pure generalized harmonic system [0904.4873].

The same framework supports boundary conditions for both the gauge source and the incoming metric fields. For \(u^{\hat 3}_a=H_a\), one may enforce either \(\mathcal C_a=0\) or \(H_a=F_a\) at the boundary, with the latter reported as generally more effective. The paper also introduces a new boundary condition for the “gauge” projection of the incoming metric characteristic field \(u^{\hat 1-}_{ab}\), distinct from the constraint and physical projections. This separates gauge, constraint, and radiative control at the boundary in a way compatible with the generalized harmonic evolution system [0904.4873].

The numerical tests of [0904.4873] use a damped-wave target
\[
F_a = \mu_L\,\log\!\left(\frac{g^{\,p}}{N}\right) t_a \;-\; \mu_S\, N^{-1} g_{ai} N^i,
\]
with \(p=1/2\) and \(\mu_L=\mu_S=0.1\). In single-black-hole evolutions, the relative \(L^2\) mismatch \(\|H-F\|^2/\|F\|^2\) peaks at approximately \(2\times10^{-2}\) and then decays to very small values, while the apparent-horizon coordinate radius grows from \(0.86M\) to \(1.28M\). The first-order driver remains stable in both single-frame and dual-frame evolutions, unlike previous second-order drivers, and robustly controls \(\|H-F\|\) over a wide parameter range \(1\lesssim \mu,\eta \lesssim \text{few}\times10^2\) [0904.4873].

The broader significance is that generalized wave map gauge is not merely a coordinate condition in covariant form. It is also a hyperbolic reduction principle, a tool for asymptotic classification near null infinity, a framework for Fuchsian singular analysis, and a computational interface through which gauge targets such as damped-wave, Bona–Massó-type slicing, and \(\Gamma\)-driver-like shifts can be implemented while preserving well-posedness [1303.4783] [0904.4873].

Source: https://www.emergentmind.com/topics/generalized-wave-map-gauge