# Note on the asymptotic structure of Kerr-Schild form

Pujian Mao and Weicheng Zhao

Note on the asymptotic structure of Kerr-Schild form

Pujian Mao and Weicheng Zhao

Center for Joint Quantum Studies and Department of Physics,

School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China

Abstract. The Kerr-Schild form provides a natural way of realizing the
classical double copy that relates exact solutions in general relativity to
exact solutions in gauge theory. In this paper, we examine the asymptotic
structure of Kerr-Schild form. In Newman-Unti gauge, we find a generic
solution space satisfying the Kerr-Schild form in series expansion around null
infinity. The news function in the solution space is chiral and can not lead
to a mass loss formula. A class of asymptotically flat complex pp-wave
solutions in closed form is obtained from the solution space.

##  1 Introduction

The double copy structure [1, 2] reveals a remarkable relation between gravity
and gauge theory in the context of scattering amplitudes. Apart from the
fruitful results and applications (see, e.g., [3] for a review and references
therein), it is still somewhat mysterious that what is the interpretation of
double copy from the field theory side, in particular at the classical level.
In a pioneering paper [4], an exact classical double copy was demonstrated for
a particular class of solutions in gravity that admits a Kerr-Schild form.
Shortly, this type of classical double copy has been extensively investigated
and developed [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43].

The success of Kerr-Schild double copy is temporarily subject to algebraically
special solutions. One of the remarkable features of the Kerr-Schild metric
(see, e.g., [44] for detailed introduction) is the fact that the null vector
and the scalar field in the decomposition can be interpreted as perturbations
on the flat background. Such a property is very crucial in connecting
solutions of gravitational theory to gauge theory. Given the fact that the
numbers of known exact solutions of gravity are indeed limited, in particular
the ones admitting a Kerr-Schild form in four dimensional Einstein gravity,
the connection between gravity and gauge theory in the Kerr-Schild double copy
formalism is very restricted. However, if one considers solutions of Einstein
equation in a perturbative way, namely, given in series expansion, there can
be a solution space including infinite amount of solutions. In the seminal
work by Bondi et. al. [45], it is shown that when expanding the metric fields
in inverse powers of a radius coordinate in asymptotic region, the Einstein
equation is formulated as a characteristic initial value problem. In this
framework, the gravitational radiation is characterized by the news functions.
Once reasonable news functions and initial data are given, a solution of
Einstein equation is determined. Naturally one would wonder that if the
asymptotic framework can allow us to extend the application of classical
double copy.

Indeed, such idea has already been implemented in literature recently. In
[42], the authors consider the rotating STU supergravity black holes [46],
which are only algebraically special asymptotically. The classical double copy
is confirmed at the leading order in the asymptotic expansion. In [43], a
precise prescription is given to connect asymptomatically flat gauge and
gravitational fields. In this paper, we start from an alternative point of
view. We adopt the Kerr-Schild form in the asymptotic framework and study the
solution space admitting Kerr-Schild form in asymptotic expansion around null
infinity in four dimensional spacetime. This computation is also motivated by
the triangle relation [47] where the scattering amplitudes in the low energy
limit, i.e., soft theorems, are connected to asymptotic symmetries and memory
effects. Clearly the soft theorems obey a double copy structure that the soft
graviton theorem is double the soft photon one. While the study of asymptotic
symmetry and memories are mostly done in the region around null infinity. The
Newman-Unti (NU) gauge [48] is adopted to derive the solution space.

Because of the constraint from the Kerr-Schild form, the solutions in NU gauge
satisfying the real condition must be spherically symmetric. Then Birkhoff’s
theorem implies that the only vacuum solution is Schwarzschild. Nevertheless,
we continue to derive a solution space by removing the real condition. There
are many reasons to explore complex solutions of Einstein equation, in
particular in the context of classical double copy. The complex solutions in
the present work are real solutions in the split metric signature $(+,+,-,-)$.
The self-dual sector in light-cone coordinates with split signature has very
important applications in classical double copy [4, 13, 31, 33]. Other
applications of complex solutions can be found, e.g., in [49, 50, 51]. In the
stereographic coordinates, we find a self-consistent solution space. The news
function and the initial data are constrained from the Kerr-Schild form. It is
difficult to give a generic solution to the Kerr-Schild constraint.
Alternatively, we provide two simple solutions for the news function and
leading initial data. Those two cases correspond to two classes of solutions.
It is worthwhile to emphasize that any solution of the news function and the
initial data to the Kerr-Schild constraint will give a solution of vacuum
Einstein equation. We also find an interesting solution in closed form. It is
pp-wave solution and is a linear superposition of self-dual and anti-self-dual
solutions. In the complex case, the news function is chiral. Consequently, it
can not lead to a mass loss formula which is not surprising in the sense that
the radiation taking away the energy from the system should satisfy real
condition.

The organization of this paper is as follows: In the next section, we give a
brief review of the NU gauge. In section 3, we combine the NU gauge with the
Kerr-Schild form to fix the gauge conditions for the present work. Section 4
is devoted to the deviation of the solution space satisfying the gauge
conditions. Exact solutions in the anti-self-dual or the self-dual sectors are
also provided in this section. We close in the last section with some remarks.

Note added: When writing up this paper, we became aware of [42, 43] with
overlapping motivations with ours.

##  2 The Newman-Unti gauge

The retarded coordinates $(u,r,z,\bar{z})$ is the most convenient choice of
the coordinates system to work with physical fields at future null infinity
where $(z,\bar{z})$ is the complex stereographic coordinates that is related
to the usual angular variables $(\theta,\phi)$ by
$z=\cot\frac{\theta}{2}e^{i\phi}$. The NU gauge was originally introduced in
the Newman-Penrose formalism [52] to study the asymptotic behavior of Weyl
tensor and metric tensor. Adapted to metric formalism, the metric and inverse
metric have the form [48]

$g_{\mu\nu}=\begin{pmatrix}g_{uu}&-1&g_{uz}&g_{u\bar{z}}\\\ -1&0&0&0\\\
g_{uz}&0&g_{zz}&g_{z\bar{z}}\\\
g_{u\bar{z}}&0&g_{z\bar{z}}&g_{\bar{z}\bar{z}}\end{pmatrix}\ ,\quad
g^{\mu\nu}=\begin{pmatrix}0&-1&0&0\\\ -1&g^{rr}&g^{rz}&g^{r\bar{z}}\\\
0&g^{rz}&g^{zz}&g^{z\bar{z}}\\\
0&g^{r\bar{z}}&g^{z\bar{z}}&g^{\bar{z}\bar{z}}\end{pmatrix}\ .$ (1)

For computational simplicity, we map the celestial sphere at null infinity to
a 2d plane. The connection between those two choices can be found, for
instance, in [53, 54, 55, 56]. The line element of the Minkowski spacetime now
is

$\text{d}s^{2}=-2\text{d}u\text{d}r+2r^{2}\text{d}z\text{d}\bar{z}.$ (2)

The asymptotic behaviors of the non-vanishing metric components preserving
asymptotic flatness are required as

$\begin{split}&g_{uu}={\cal O}{(r^{-1})}\,,\quad g_{uz}={\cal O}(1)\,,\quad
g_{u\bar{z}}={\cal O}(1)\,,\\\ &g_{zz}={\cal O}(r)\,,\quad
g_{\bar{z}\bar{z}}={\cal O}(r)\,,\quad g_{z\bar{z}}=r^{2}+{\cal
O}(1)\,.\end{split}$ (3)

To solve the Einstein equation, it is very useful to arrange all components as
follows:

  * •

Four hypersurface equations:

$G_{r\mu}=0\ .$ (4)

  * •

Two standard equations:

$G_{zz}=G_{\bar{z}\bar{z}}=0\ .$ (5)

  * •

One trivial equation:

$G_{z\bar{z}}=0\ .$ (6)

  * •

Three supplementary equations:

$G_{uz}=G_{u\bar{z}}=G_{uu}=0\ .$ (7)

The Bianchi identities can be written as

$\nabla_{\mu}{G^{\mu}}_{\nu}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}{G^{\mu}}_{\nu}\right)+\frac{1}{2}G_{\mu\rho}\partial_{\nu}g^{\mu\rho}=0\
.$ (8)

When the hypersurface equations and standard equations are satisfied, the
Bianchi identity $\nabla_{\mu}{G^{\mu}}_{r}=0$ yields

$G_{z\bar{z}}\partial_{r}g^{z\bar{z}}=0\ .$ (9)

Hence $G_{z\bar{z}}=0$ because of the boundary condition for $g_{z\bar{z}}$,
which means that the trivial equation is satisfied automatically. Then one
obtains that $R=0$. The Bianchi identity $\nabla_{\mu}{G^{\mu}}_{z}=0$ reduces
to $\partial_{r}\left(\sqrt{-g}{G}_{uz}\right)=0$, noticing that the
hypersurface equations and standard equations are satisfied. Similarly, the
Bianchi identity $\nabla_{\mu}{G^{\mu}}_{\bar{z}}=0$ yields
$\partial_{r}\left(\sqrt{-g}{G}_{u\bar{z}}\right)=0$. The last Bianchi
identity $\nabla_{\mu}{G^{\mu}}_{u}=0$ leads to
$\partial_{r}\left(\sqrt{-g}{G}_{uu}\right)=0$ when the rest equations are
satisfied. Hence three supplementary equations are only left with only one
order to solve in the asymptotic expansion.

##  3 The Kerr-Schild form

A metric of Kerr-Schild form is obtained from a deformation of a fixed
background metric $\bar{g}_{\mu\nu}$. In the retarded coordinates
$(u,r,z,\bar{z})$, we choose the background metric as the Minkowski spacetime
(2). The deformation is made by a null vector field $k_{\mu}$ and a scalar
function $\phi$ in the form

$g_{\mu\nu}=\eta_{\mu\nu}+\phi k_{\mu}k_{\nu}\ .$ (10)

One can show that $k^{\mu}$ is null when contracted with both the full metric
$g_{\mu\nu}$ and the Minkowski one $\eta_{\mu\nu}$. Therefore, the inverse
metric is just

$g^{\mu\nu}=\eta^{\mu\nu}-\phi k^{\mu}k^{\nu}\ .$ (11)

Hence we can raise the index of $k$ using both $g^{\mu\nu}$ and
$\eta^{\mu\nu}$. In the NU gauge, it is required that

$g_{rr}=g_{rz}=g_{r\bar{z}}=0\ ,\quad g_{ur}=-1\ .$ (12)

Combining these conditions with the Kerr-Schild ansatz (10), one should
require that $k_{r}=0$. Because $k_{\mu}$ is a null vector respect to
$\eta^{\mu\nu}$, the solution of the remaining components is either $k_{z}=0$
or $k_{\bar{z}}=0$. It is important to point out that such solution will not
lead to a real metric. In other works, the only solution in the usual real
angular variables $(\theta,\phi)$ is the trivial one $k_{\theta}=k_{\phi}=0$.
However, it is still meaningful to continue the computation in the complex
coordinates with the chiral solution. On the one hand, the self-dual metric
fields in the retarded coordinates $(u,r,z,\bar{z})$ are exactly in this form.
The self-dual gravity is a very important realization of the classical double
copy [4, 13, 31, 33]. Other important applications of self-dual gravity can be
found, e.g., in [57, 58, 59] and references therein. On the other hand, this
situation is somehow similar to the gluon three point amplitude at tree level.
As the smallest or starting point amplitude in the BCFW recursion relation
[60], it is vanishing for real null momenta. Alternatively, one works with
complex momenta where the helicity spinor $\lambda_{i}$ and
$\tilde{\lambda}_{i}$ are independent and the three-gluon amplitudes are
derived from either left-handed spinors or right-handed spionrs, see, e.g.,
[61, 62] for a textbook treatment. We choose $k_{\bar{z}}=0$ in the present
work. Substituting this solution back to the metric, one obtains

$g_{u\bar{z}}=0=g_{\bar{z}\bar{z}}\ .$ (13)

The remaining components of the metric have the form

$g_{uu}=\phi k_{u}k_{u}\ ,\quad g_{zz}=\phi k_{z}k_{z}\ ,\quad g_{uz}=\phi
k_{u}k_{z}\ .$ (14)

Thus they should satisfy the relation as follows

$g_{uu}g_{zz}=g^{2}_{uz}\ .$ (15)

The aim of the present work is to check the constraints from the Kerr-Schild
form (13) and (15) on the solution space in NU gauge. Before doing so, we
would like to point out that a double Kerr-Schild form

$g_{\mu\nu}=\eta_{\mu\nu}+\phi k_{\mu}k_{\nu}+\Psi l_{\mu}l_{\nu}\ ,$ (16)

where $k^{2}=l^{2}=k\cdot l=0$, can not bring the other half copy of the
metric. The NU gauge condition will fix both $k_{r}=0$ and $l_{r}=0$. The null
conditions of $k_{\mu}$ and $l_{\mu}$ bring the same constraint as single
Kerr-Schild form. Let us choose $k_{\bar{z}}=0$ and $l_{z}=0$ to keep both
components $g_{zz}$ and $g_{\bar{z}\bar{z}}$ in the metric. But the last
condition that $k\cdot l=0$ eventually leads to $k_{z}l_{\bar{z}}=0$. Hence we
are back to the single Kerr-Schild form.

##  4 The solution space

Once the gauge condition from the Kerr-Schild is clarified, the solution of
Einstein equation in the retarded coordinates can be derived in a
straightforward way following the computation in [45, 63, 64, 65, 66]. We
start with the hypersurface equations which determine the radial dependence of
the metric from the initial data. The standard equations control the time
evolutions of the initial data. Then the time evolution of the integration
constants from the hypersurface equations are fixed by the supplementary
equations. In this work, there is one more constraint from the Kerr-Schild
form.

###  4.1 Hypersurface equations

The hypersurface equations $G_{rr}=0$ and $G_{r\bar{z}}=0$ are satisfied once
the Kerr-Schild form is applied by imposing (13). The next one $G_{rz}=0$
yields

$\partial_{r}\left(r^{4}\partial_{r}(\frac{g_{uz}}{r^{2}})\right)=r^{2}\partial_{r}(\frac{\partial_{\bar{z}}g_{zz}}{r^{2}})\
.$ (17)

Hence $g_{uz}$ can be solved out from $g_{zz}$ as

$g_{uz}=\frac{N(u,z,\bar{z})}{r}+r^{2}\int^{\infty}_{r}\text{d}r^{\prime}\frac{1}{{r^{\prime}}^{4}}\int^{\infty}_{r^{\prime}}\text{d}r^{\prime\prime}{r^{\prime\prime}}^{2}\partial_{r^{\prime\prime}}(\frac{\partial_{\bar{z}}g_{zz}}{{r^{\prime\prime}}^{2}})\
,$ (18)

where function $N(u,z,\bar{z})$ is an integration constant in $r$. We continue
with $G_{ur}=0$, from which one obtains

$\partial_{r}(rg_{uu})=\frac{1}{2r^{2}}\left[\partial_{r}(r^{2}\partial_{\bar{z}}g_{uz})-\partial_{\bar{z}}^{2}g_{zz}\right]\
.$ (19)

Thus $g_{uu}$ is fixed as

$g_{uu}=\frac{M(u,z,\bar{z})}{r}-\frac{1}{r}\int^{\infty}_{r}\text{d}r^{\prime}\frac{1}{2{r^{\prime}}^{2}}\left[\partial_{r^{\prime}}({r^{\prime}}^{2}\partial_{\bar{z}}g_{uz})-\partial_{\bar{z}}^{2}g_{zz}\right]\
,$ (20)

where $M(u,z,\bar{z})$ is another integration constant.

###  4.2 Standard equations

Equation $G_{\bar{z}\bar{z}}=0$ is satisfied automatically because of the
Kerr-Schild form. The other standard equation leads to

$\partial_{u}\partial_{r}(\frac{g_{zz}}{r})=\frac{1}{2r^{4}}\bigg{[}4rg^{2}_{uz}+2g_{zz}\partial_{\bar{z}}g_{uz}+4r^{2}g_{zz}\partial_{r}g_{uu}+r\partial_{\bar{z}}g_{zz}\partial_{r}g_{uz}\\\
+r^{3}(\partial_{r}g_{uz})^{2}-r\partial_{\bar{z}}g_{uz}\partial_{r}g_{zz}-r^{3}\partial_{r}g_{uu}\partial_{r}g_{zz}\\\
-2rg_{zz}\partial_{r}\partial_{\bar{z}}g_{uz}-g_{uz}(4\partial_{\bar{z}}g_{zz}+4r^{2}\partial_{r}g_{uz}-2r\partial_{r}\partial_{\bar{z}}g_{zz})\\\
+2r^{3}\partial_{r}\partial_{z}g_{uz}+r^{3}g_{zz}\partial_{r}^{2}g_{uu}-rg_{uu}(2g_{zz}-2r\partial_{r}g_{zz}+r^{2}\partial_{r}^{2}g_{zz})\bigg{]}\
.$ (21)

Clearly, there is no constraint at the order ${\cal O}(r)$ of $g_{zz}$ from
the standard equation. The field at this order is related to the news function
in the system which indicates propagating degree of freedom.

###  4.3 Solution space in series expansion

Suppose that $g_{zz}$ is given as initial data as111The absent of
$\bar{z}$-dependence in $f_{0}(u,z)$ is to avoid logarithmic terms which will
involve a polyhomogeneous solution space.

$g_{zz}=c(u,z,\bar{z})r+f_{0}(u,z)+\sum\limits^{\infty}_{i=1}\frac{f_{i}(u,z,\bar{z})}{r^{i}}\
.$ (22)

Inserting the initial data into (18) and (20), one obtains

$g_{uz}=\frac{1}{2}\partial_{z}c+\frac{N}{r}-\sum\limits^{\infty}_{i=1}\frac{i+2}{i(i+3)}\frac{\partial_{\bar{z}}f_{i}}{r^{i+1}}\
,$ (23)

and

$g_{uu}=\frac{M}{r}-\frac{\partial_{\bar{z}}N}{2r^{2}}+\sum\limits^{\infty}_{i=1}\frac{1}{i(i+3)}\frac{\partial_{\bar{z}}^{2}f_{i}}{r^{i+2}}\
.$ (24)

###  4.4 Supplementary equations

With the series expansion, one can continue to solve the supplementary
equations. From $G_{uz}=0$, we obtain

$\partial_{u}N=\frac{1}{3}\partial_{z}M-\frac{1}{6}\partial_{z}\partial_{\bar{z}}^{2}c\
.$ (25)

Next, $G_{u\bar{z}}=0$ yields

$2\partial_{\bar{z}}M+\partial_{\bar{z}}^{3}c=0\ .$ (26)

The last supplementary equation $G_{uu}=0$ leads to

$\partial_{u}M=\frac{1}{2}\partial_{u}\partial_{\bar{z}}^{2}c\ .$ (27)

It is of convenience to define

$M_{0}=M-\frac{1}{2}\partial_{\bar{z}}^{2}c\ .$ (28)

Then (27) and (25) reduce to

$\partial_{u}M_{0}=0\ ,\quad\partial_{u}N=\frac{1}{3}\partial_{z}M_{0}\ .$
(29)

Hence $M_{0}$ is independence of $u$ and
$N=N_{0}(z,\bar{z})+\frac{1}{3}u\partial_{z}M_{0}$. So we do not have a mass
loss formula in this system as $M_{0}$ is not changing. Moreover, (26) becomes

$\partial_{\bar{z}}M_{0}+\partial_{\bar{z}}^{3}c=0\ .$ (30)

Since $M_{0}$ is independence of $u$, one can have that
$\partial_{u}\partial_{\bar{z}}^{3}c=0$. Therefore, a generic solution to $c$
should have the form

$c={\tilde{c}}(z,\bar{z})+c_{0}(u,z)+c_{1}(u,z)\bar{z}+c_{2}(u,z)\bar{z}^{2}\
.$ (31)

Hence

$M_{0}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$M$\kern
0.0pt}}}(z)-\partial_{\bar{z}}^{2}\tilde{c}\ ,\quad M=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$M$\kern
0.0pt}}}-\frac{1}{2}\partial_{\bar{z}}^{2}\tilde{c}+c_{2}\ ,\quad
N=N_{0}+\frac{1}{3}u(\partial_{z}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$M$\kern
0.0pt}}}-\partial_{z}\partial_{\bar{z}}^{2}\tilde{c})\ .$ (32)

###  4.5 The Kerr-Shcild constraint

The first two constraints in (13) from the Kerr-Schild form have been
implemented when solving the Einstein equation. The last one in (15) can be
considered as a constraint on the initial data. In the series expansion (22),
the fields from each order are constrained from both the standard equation
(21) and the Kerr-Schild form (15). For the first order, the news aspect $c$
is only constrained from the Kerr-Schild form as

$cM=\frac{1}{4}(\partial_{\bar{z}}c)^{2}\ ,$ (33)

For the second order, one obtains

$\displaystyle\partial_{u}f_{0}=0\ ,$ (34) $\displaystyle
f_{0}M=N\partial_{\bar{z}}c+\frac{1}{2}c\partial_{\bar{z}}N\ .$ (35)

Clearly, the first equation shows that $f_{0}$ has only $z$ dependence. Hence
the second equation will lead to another constraint on $c$. Though it is not
easy to give a generic solution for $c$, one can easily find some non trivial
solutions, e.g.,

$f_{0}=M_{0}=N_{0}=\tilde{c}=c_{0}=c_{1}=0\ ,\quad M=c_{2}(u,z)\ ,\quad
c=c_{2}(u,z)\bar{z}^{2}\ .$ (36)

and

$M=\tilde{c}=c_{1}=c_{2}=0\ ,\quad N=N_{0}(z)\ ,\quad c=c_{0}(u,z)\ .$ (37)

We can continue to trace higher orders. For every $f_{i}(i\geq 1)$, there are
two differential equations. Considering $f_{1}$ as example, one has

$\displaystyle
2f_{0}\partial_{\bar{z}}N-3\partial_{\bar{z}}f_{1}\partial_{\bar{z}}c-c\partial_{\bar{z}}^{2}f_{1}-4f_{1}M+4N^{2}=0\
,$ (38)
$\displaystyle\partial_{u}f_{1}=\frac{1}{8}c(2M-\partial_{\bar{z}}^{2}c)+\frac{1}{2}\partial_{z}N\
.$ (39)

Again, a generic solution for $f_{1}$ is very to find once the other fields in
the equations are known. One can show that there is non trivial solution when
the other fields are given as (36). A simple one is
$f_{1}=\frac{\tilde{f}_{1}(z)}{\bar{z}}$ where $\tilde{f}_{1}(z)$ is an
arbitrary function of $z$. Hence, one can expect that there should not be more
constraint on the news aspect $c$ from the equations at higher orders. Since
the dynamics is only encoded in the news aspect $c$, the equations from higher
orders will not bring any new information.

###  4.6 Exact solution

There is a simple but very interesting solution in closed form in the solution
space. The metric is given by

$\text{d}s^{2}=-2\text{d}u\text{d}r+2r^{2}\text{d}z\text{d}\bar{z}+r[c(u,z)+f(r,z)]\text{d}z^{2}\
.$ (40)

This solution is non-trivial in the sense that it has non vanishing components
of Riemann tensor which are

$R_{uzuz}=-\frac{1}{2}r\partial_{u}^{2}c\ ,\quad
R_{rzrz}=-\frac{1}{2}r\partial_{r}^{2}f\ .$ (41)

The square of the Riemann tensor $R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$
is zero. The vector $k=\frac{\partial}{\partial\bar{z}}$ is null and tangent
to an expansion-free, shear-free and twist-free null geodesic congruence,
i.e., satisfying $\nabla_{\mu}k_{\nu}=0$. So this solution belongs to the
class of pp-wave solutions following the definition in [67, 68].
Interestingly, this pp-wave solution is asymptotically flat when the
asymptotic behavior of $f(r,z)$ is ${\cal O}(\frac{1}{r})$ which is a direct
consequence of the NU setup in section 2 with a special boundary choice (2),
i.e., a 2d plane. Coordinate $z$ labels the null wave surfaces and coordinates
$u$ and $r$ span the wave surfaces. There is another null direction
$l=\frac{\partial}{\partial r}$ that has expansion and shear. With the
following change of coordinates

$\bar{z}=\rho\ ,\quad z=-v\ ,\quad r=\frac{1}{\zeta}\ ,\quad u=\bar{\zeta}\ ,$
(42)

the line element (40) becomes

$\text{d}s^{2}=\frac{1}{\zeta^{2}}\left[-2\text{d}v\text{d}\rho-2H(v,\zeta,\bar{\zeta})\text{d}v^{2}+2\text{d}\zeta\text{d}\bar{\zeta}\right],$
(43)

where $H(v,\zeta,\bar{\zeta})=-\frac{1}{2}\zeta[c(v,\bar{\zeta})+f(v,\zeta)]$.
This metric is conformal to the Brinkmann form [68] with a complex conformal
factor. This solution might be considered as a generalization of the Brinkmann
form to the complex case. Writing the line element (40) in a more generic form
as

$\text{d}s^{2}=-2\text{d}u\text{d}r+2r^{2}\text{d}z\text{d}\bar{z}+rH(u,r,z)\text{d}z^{2}\
,$ (44)

the vacuum Einstein yields

$\partial_{u}\partial_{r}H(u,r,z)=0\ .$ (45)

The precise choice in (40) is just the general solution to this equation.

By choosing four null basis as

$e^{1}_{\mu}=(0,1,0,0)\ ,e^{2}_{\mu}=(1,0,0,0)\ ,e^{3}_{\mu}=(0,0,r,0)\
,e^{4}_{\mu}=(0,0,\frac{H}{2},r)\ ,$ (46)

the solution can be described in a Newman-Penrose formalism. The non-vanishing
Newman-Penrose variables are

$\begin{split}&\Psi_{0}=\frac{\partial_{r}^{2}H}{2r}\ ,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{4}=\frac{\partial_{u}^{2}H}{2r}\ ,\\\ &\rho=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\rho$\kern 0.0pt}}}=\frac{1}{r}\
,\quad\sigma=-\frac{H-r\partial_{r}H}{2r^{2}}\ ,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}=-\frac{\partial_{u}H}{2r}\ .\end{split}$ (47)

Since (45) is a linear equation, a new solution can be obtained simply by
superposing distinct solutions with different expressions for $H$ just like
the case of pp-wave in Brinkmann form. An interesting feature of the solution
(40) is the fact that it can be considered as a linear superposition of self-
dual and anti-self-dual solutions. The case $c=0$ and $f\neq 0$ represents a
self-dual solution while the other case $c\neq 0$ and $f=0$ represents an
anti-self-dual solution. To see that, we define the spinorial tetrad as

${\sigma_{\mu}}_{A\dot{A}}=\sigma^{a}_{A\dot{A}}e^{b}_{\mu}\eta_{ab}\ ,$ (48)

where

$\eta_{ab}=\eta^{ab}=\begin{pmatrix}0&-1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}\
,\quad\sigma^{a}=\frac{1}{2}(\sigma^{2}+i\sigma^{3},\sigma^{2}-i\sigma^{3},\mathbb{1}-\sigma^{1},\mathbb{1}+\sigma^{1})\
,$ (49)

$\sigma^{k}(k=1,2,3)$ are the Pauli matrices and the tetrad is chosen as (46).
In our convention, $\varepsilon^{12}=1$. One can easily verify that

${\sigma_{\mu}}_{A\dot{A}}{\sigma_{\nu}}_{B\dot{B}}\varepsilon^{AB}\varepsilon^{\dot{A}\dot{B}}=g_{\mu\nu}\
.$ (50)

The spinoral form of the Weyl tensor is

$W_{A\dot{A}B\dot{B}C\dot{C}D\dot{D}}=C_{ABCD}\varepsilon_{\dot{A}\dot{B}}\varepsilon_{\dot{C}\dot{D}}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$C$\kern
0.0pt}}}_{\dot{A}\dot{B}\dot{C}\dot{D}}\varepsilon_{AB}\varepsilon_{CD}\ .$
(51)

$C_{ABCD}$ and $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$C$\kern 0.0pt}}}_{\dot{A}\dot{B}\dot{C}\dot{D}}$ are totally symmetric
and they represent the anti-self-dual and self-dual parts of the curvature
respectively. For a real metric, they are complex conjugation of each other.
Direct computation regarding to the solution (40) gives

$C_{1111}=C_{1122}=C_{2222}=\frac{1}{8r}\partial_{u}^{2}c\ ,\quad
C_{1112}=C_{1222}=-\frac{1}{8r}\partial_{u}^{2}c\ ,$ (52)

and

$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$C$\kern
0.0pt}}}_{1111}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$C$\kern 0.0pt}}}_{1122}=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$C$\kern
0.0pt}}}_{2222}=\frac{1}{8r}\partial_{r}^{2}f\ ,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$C$\kern
0.0pt}}}_{1112}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$C$\kern 0.0pt}}}_{1222}=-\frac{1}{8r}\partial_{r}^{2}f\ .$ (53)

##  5 Discussion

We obtain a generic solution space in series expansion near null infinity in
the NU gauge satisfying the Kerr-Schild form. The solutions only exist in
complex coordinates which should correspond to a complex classical double
copy. Interesting exact solutions in the anti-self-dual and the self-dual
sectors can be read out from the solution space. As the single copy or zeroth
copy version of the solution space, the null vector $k_{\mu}$ and the scalar
field $\phi$ in the Kerr-Schild form can be easily derived from (14).

The solution space is self-contained. But the result itself somehow indicates
the incompatibility between the NU gauge and Kerr-Schild form. The only non
trivial real solution in NU gauge satisfying the Kerr-Schild form is the
spherically symmetric one. With such shortcoming, applications of the present
result in the triangle relation are now less clear. For instance, what is a
memory effect in the Kerr-Schild form and what is the interpretation of the
memory effect from the point of view of classical double copy structure. We do
not yet have a clear answer to those questions and leave them for future
investigation.

## Acknowledgments

The authors thank Zhengwen Liu, Hong Lü, Jun-Bao Wu, and Xiaoning Wu for
useful discussions. This work is supported in part by the National Natural
Science Foundation of China under Grant No. 11905156 and No. 11935009.

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# TrAME: Trajectory-Anchored Multi-View Editing for Text-Guided 3D Gaussian
Splatting Manipulation

Chaofan Luo1,2, Donglin Di 1, Yongjia Ma1, Zhou Xue1,
Chen Wei1, Xun Yang∗ 2, Yebin Liu3
1Space AI, Li Auto; 2University of Science and Technology of China; 3Tsinghua
University
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>Corresponding authors.

###### Abstract

Despite significant strides in the field of 3D scene editing, current methods
encounter substantial challenge, particularly in preserving 3D consistency in
multi-view editing process. To tackle this challenge, we propose a progressive
3D editing strategy that ensures multi-view consistency via a Trajectory-
Anchored Scheme (TAS) with a dual-branch editing mechanism. Specifically, TAS
facilitates a tightly coupled iterative process between 2D view editing and 3D
updating, preventing error accumulation yielded from text-to-image process.
Additionally, we explore the relationship between optimization-based methods
and reconstruction-based methods, offering a unified perspective for selecting
superior design choice, supporting the rationale behind the designed TAS. We
further present a tuning-free View-Consistent Attention Control (VCAC) module
that leverages cross-view semantic and geometric reference from the source
branch to yield aligned views from the target branch during the editing of 2D
views. To validate the effectiveness of our method, we analyze 2D examples to
demonstrate the improved consistency with the VCAC module. Further extensive
quantitative and qualitative results in text-guided 3D scene editing indicate
that our method achieves superior editing quality compared to state-of-the-art
methods. We will make the complete codebase publicly available following the
conclusion of the double-blind review process.

##  1 Introduction

3D technologies are pivotal in various fields, such as virtual reality (VR),
the film and gaming industries, and 3D design [40, 11, 22, 12]. Recent
advances in Neural Radiance Fields (NeRF) and 3D Gaussian Splatting (3DGS)
have pushed the boundary of 3D representation, enabling versatile 3D
applications [21, 37, 23, 31, 7, 6, 17, 35, 33, 3]. Leveraging the powerful
generative capabilities of text-to-image diffusion models [2, 26], text-guided
3D editing now allows for intricate adjustments in shape, style, texture, and
lighting, marking a significant advancement in the flexibility of 3D scene
manipulation [9, 16, 11, 22, 12, 18].

One key challenge in 3D editing is maintaining multi-view consistency when
applied to real scenes [29, 32], which is essential for preventing visual
artifacts and inconsistent appearances when observed from different view
angles. Current studies of 3D editing can be divided into two categories,
namely optimization-based and reconstruction-based methods. The optimization-
based 3D editing methods [19, 27, 20, 41, 16] usually adopt the Score
Distillation Sampling (SDS) loss [25] or its modified versions [13, 24]. They
often suffer from sub-optimal editing quality such as over-saturation, over-
smoothing, and a lack of diversity due to the misalignment between the
randomly sampled timestep $t$ in the SDS optimization process and the
corresponding timestep $t$ in the diffusion sampling process [15, 40]. The
second category of methods, reconstruction-based 3D editing methods [9, 5, 32,
29, 18], employ readily available 2D editing diffusion models to edit 3D
scene. However, 2D diffusion models struggle to achieve multi-view consistency
because of their inherent limitation of independently processing each view.

In this paper, to mitigate the above-mentioned issues, we first theoretically
unveil the relation between these two branches of methods, demonstrating that
optimization-based methods are special cases of reconstruction-based methods.
Specifically, we uncover an intrinsic equivalence between the optimization
process of SDS [25] and an iterative reconstruction process that aims to match
pseudo-ground-truths obtained from the Denoising Diffusion Consistent Model
(DDCM) [34]. This analysis unites these two branches of 3D editing methods,
offering a unified perspective for making superior design choices. Based on
the analysis, we introduce a progressive 3D editing framework named
“Trajectory-Anchored Multi-View Editing (TrAME)”. The core idea of this
framework is the Trajectory-Anchored Scheme (TAS), a unified paradigm designed
to gradually consolidate incremental 2D view edits onto 3D scenes. Inspired by
an empirical practice of utilizing feedback from rendered images of the
updated 3DGS to correct errors that arise during the 2D editing process[4,
42], we design TAS to enhance the interplay between the image editing process
and the 3DGS update process, allowing for active rectification of
inconsistencies within 2D multi-view edits through the 3D constrained
rendering process (_i.e._ , rendering in the loop). Utilizing a modified DDCM
editing trajectory [34], we incrementally generate pseudo-ground-truths for
progressive 3D Gaussian editing. Furthermore, to enhance 3D-consistency during
the 2D view editing process, we design a VCAC module with a dual-branch
editing strategy. The edited views maintain structural multi-view consistency
via self-attention queries injected from the source branch into the target
branch during the early stages of diffusion. To enhance semantic multi-view
consistency, we facilitate self-attention Key-Value (KV) propagation and KV
reference mechanisms, propagating the self-attention keys and values of
keyframes within the same context and inflating the self-attention keys and
values across different keyframes for a mutual reference.

We conduct extensive experiments to demonstrate the effectiveness of our
proposed TAS strategy and VCAC module. Both qualitative and quantitative
results of 3D editing highlight the superiority of our approach in comparison
to state-of-the-art methods. Additionally, the qualitative results from
ablation studies demonstrate enhanced consistency with our proposed TAS and
VCAC module. Our main contributions are summarized as follows:

$\bullet$ Our theoretical analysis bridges the gap between optimization-based
and reconstruction-based editing methods, offering a unified perspective for
selecting superior design choices.

$\bullet$ We propose a progressive 3D editing strategy that integrates a
Trajectory-Anchored Scheme (TAS). This design ensures multi-view consistency
by tightly coupling the iterative processes of 2D view editing and 3D scene
updating, thereby preventing error accumulation during text-to-image process.

$\bullet$ We present a tuning-free VCAC module that enhances the 3D
consistency of editing results. The VCAC leverages cross-view semantic and
geometric references from the source branch to yield aligned views in the
target branch via query injection, KV propagation and KV reference mechanisms.

$\bullet$ Extensive experimental results in text-guided 3D scene editing
demonstrate that our method achieves improved multi-view consistency editing
results compared with state-of-the-art methods.

##  2 Related Work

In this section, we provide a concise overview of two primary branches of
methodologies employed in the task of 3D scene editing: optimization-based
methods and reconstruction-based methods.

Regarding optimization-based 3D editing, significant advancements have been
made, building upon the foundational SDS loss introduced by DreamFusion [25].
This has catalyzed the development of various innovative methods [19, 27, 20,
41, 16, 24] aimed at refining 3D model editing via optimizing the SDS loss.
Notably, Vox-E [27] and DreamEditor [41] have harnessed explicit 3D
representations, including voxels and meshes. These approaches utilize cross-
attention mechanisms to facilitate precise manipulations within specific
regions of 3D models. Furthermore, RePaint-NeRF [39] has advanced the
application of SDS in 3D editing by integrating a semantic mask to guide and
constrain modifications within the background elements. In a similar vein, ED-
NeRF [24] has introduced an enhanced loss function specifically designed for
3D editing tasks. It extends the Delta Denoising Score (DDS) loss [13] into
the three-dimensional space, offering a more refined approach to editing.
Despite these advancements, a common challenge still persists among methods
that utilize randomly sampled timesteps $t$ within the SDS framework: This
approach tends to lead to deviations from the intended diffusion sampling
trajectories, which can result in compromised quality of the 3D edits.

For reconstruction-based 3D editing, InstructN2N [9] alongside subsequent
studies [5, 32, 29, 18], have leveraged advancements in 2D diffusion editing
techniques, notably InstructP2P [2] and ControlNet [38], for enhancing scene
updates through Iterative Dataset Update [9]. GaussianEditor [5] and GaussCtrl
[32] introduce methods for precise control in partial 3D editing.
GaussianEditor utilizes semantic tracing to segment the 3D Gaussians to be
edited. GaussCtrl proposes depth-guided editing and attention-based latent
alignment. However, these methods often neglect inter-view correlations,
resulting in a gap in maintaining semantic and structural consistency across
multiple views, ultimately leading to error accumulation. Our work bridges
this gap by integrating Trajectory-Anchored Scheme to minimize error
accumulation and introducing attention manipulation mechanisms to enforce
structural and semantic consistency across views in 3D editing through our
VCAC module.

##  3 Preliminaries

In this section, we first briefly review Score Distillation Sampling (SDS) and
Denoising Diffusion Consistent Model (DDCM) before delving into the analysis
of the relation between optimization-based editing and reconstruction-based
editing.

###  3.1 Score Distillation Sampling

Score Distillation Sampling (SDS) [25], also known as Score Jacobian Chaining
(SJC) [30], is a commonly adopted technique in 3D asset generation and
editing. It performs sampling in parameter space by optimizing a diffusion
distillation loss function. Given a camera pose $c$ and the parameters of a 3D
representation $\theta$, noisy latents ${\bm{z}}_{t}$ are obtained by first
rendering different views and then adding noise to the latents of the rendered
images. The gradient with respect to the parameter of the 3D representation
$\theta$ is defined as in Eq. 1, using a pretrained diffusion model
${\bm{\epsilon}}_{\phi}$ to predict the noise estimate conditioned by $y$ and
match it against the ground truth noise ${\bm{\epsilon}}$. This gradient is
backpropagated through a differentiable render function ${\bm{g}}(\cdot,c)$,
scaled by a weighting schedule $\omega(t)$.

$\nabla_{\theta}{\mathcal{L}}_{\mbox{\tiny
SDS}}(\theta)\coloneq\mathbb{E}_{t,{\bm{\epsilon}},c}\,\left[\omega(t)({\bm{\epsilon}}_{\phi}({\bm{z}}_{t},t,y)-{\bm{\epsilon}})\frac{\partial{\bm{g}}(\theta,c)}{\partial\theta}\right].$
(1)

Additionally, variant loss functions, such as Delta Denoising Score, are often
used as a substitute for the vanilla SDS loss.

###  3.2 Denoising Diffusion Consistent Model

Denoising Diffusion Implicit Models (DDIMs) [28] characterize a family of
generative models with non-Markovian forward processes. Given a noisy latent
${\bm{z}}_{t}$ with condition $y$ at timestep $t$, the sampling procedure of
DDIM is formulated as:

${\bm{z}}_{t-1}=\sqrt{\alpha_{t-1}}\underbrace{\left(\frac{{\bm{z}}_{t}-\sqrt{1-\alpha_{t}}{\bm{\epsilon}}_{\phi}({\bm{z}}_{t},t,y)}{\sqrt{\alpha_{t}}}\right)}_{\text{
predicted
}\hat{{\bm{z}}}_{0}^{(t)}}+\underbrace{\sqrt{1-\alpha_{t-1}-\sigma_{t}^{2}}\cdot{\bm{\epsilon}}_{\phi}({\bm{z}}_{t},t,y)}_{\text{direction
pointing to
}{\bm{z}}_{t}}+\underbrace{\sigma_{t}{\bm{\epsilon}}_{t}}_{\text{random
noise}},$ (2)

where ${\bm{\epsilon}}_{\phi}$ is the noise predictor parameterized by
parameter $\phi$ and $\alpha_{t}$ corresponds to diffusion schedule. Different
instances of $\sigma_{t}$ in Eq. 2 yield different generative processes [28].
Consider a special generative process where
$\sigma_{t}\coloneq\sqrt{1-\alpha_{t-1}}$. The second term on the right-hand
side of Eq. 2 is canceled out, leaving only $\hat{{\bm{z}}}_{0}^{(t)}$ and the
newly added noise $\sigma_{t}{\bm{\epsilon}}_{t}$:

${\bm{z}}_{t-1}=\sqrt{\alpha_{t-1}}\left(\frac{{\bm{z}}_{t}-\sqrt{1-\alpha_{t}}{\bm{\epsilon}}_{\phi}({\bm{z}}_{t},t,y)}{\sqrt{\alpha_{t}}}\right)+\sqrt{1-\alpha_{t-1}}{\bm{\epsilon}}_{t}\;,$
(3)

which is formulated as Denoising Diffusion Consistent Model (DDCM) [34].

##  4 Methodology

Figure 1: Illustration of the proposed method, Trajectory-Anchored Multi-View
Editing for 3D Gaussian Splatting Manipulation (TrAME). Our method comprises a
Trajectory Anchored Scheme (TAS) as well as a View-Consistent Attention
Control (VCAC) module. Given a source prompt, a target prompt and the original
3DGS $\theta^{(0)}$ as input, the VCAC module can yield 3D-consistent and
progressively edited views with a single-step inference to update 3DGS.
Conversely, the views rendered from the updated 3DGS correct minor
inconsistencies from previous view edits and serve as inputs for subsequent
steps, thereby preventing error accumulation from the 2D editing process. This
process alternatively update the 2D views and 3DGS in a synchronized and
progressive manner, producing the final edited 3DGS $\theta^{(T)}$.

We first analyze the relation between optimization-based and reconstruction-
based 3D scene editing methods in detail. Then, we elaborate on the proposed
progressive Trajectory-Anchored Scheme (TAS) and the design of View-Consistent
Attention Control (VCAC). The overall framework is illustrated in Figure 1.

###  4.1 Analysis on Optimization-based Editing

We analyze the optimization process of score distillation sampling, and
generalize it to its variants. The gradient of the SDS loss function, as
defined in Eq. 1, is equivalent to the gradient obtained by optimizing the
rendered views to match a pseudo-ground-truth $\hat{{\bm{z}}}_{0}^{(t)}$:

$\displaystyle\nabla_{\theta}{\mathcal{L}}_{\mbox{\tiny SDS}}(\theta)$
$\displaystyle\coloneq\mathbb{E}_{t,{\bm{\epsilon}},c}\,\left[\omega(t)\left({\bm{\epsilon}}_{\phi}({\bm{z}}_{t},t,y)-{\bm{\epsilon}}\right)\frac{\partial{\bm{g}}(\theta,c)}{\partial\theta}\right]$
(4)
$\displaystyle=\mathbb{E}_{t,{\bm{\epsilon}},c}\,\left[\omega(t)\frac{\sqrt{\alpha_{t}}}{\sqrt{1-\alpha_{t}}}\left({\bm{z}}_{\pi}-\hat{{\bm{z}}}_{0}^{(t)}\right)\frac{\partial{\bm{g}}(\theta,c)}{\partial\theta}\right]\;,$

where ${\bm{z}}_{\pi}$ is the rendered image’s latent.

Prior works have highlighted the drawbacks of employing a uniform timestep
schedule in the optimization process of SDS [8, 36, 15]. The optimization
process of SDS presents a dilemma: a large noise scale is necessary in the
early stages to tackle the out-of-domain issue of the rendered images.
However, this large noise scale also corrupts the information of the original
images, making the optimization process unstable. Therefore, it is reasonable
to adopt an annealing timestep schedule so that coarse-grained structures are
generated at the early stage with a large noise level, while intricate details
are refined at a later stage with a lower noise level.

To further unveil the connections between SDS optimization and DDCM sampling,
we make a reasonable assumption that the timestep $t$ should follow an
annealing schedule, as suggested by extensive prior research. We can write the
pseudo-ground-truth as a function of the rendered image’s latent
${\bm{z}}_{\pi}$, timestep $t$, condition $y$ and added noise
${\bm{\epsilon}}$:

$\displaystyle\hat{{\bm{z}}}_{0}^{(t)}\coloneq{\bm{f}}_{\phi}({\bm{z}}_{\pi},{\bm{\epsilon}},t,y)$
$\displaystyle=\frac{({\bm{z}}_{t}-\sqrt{1-\alpha_{t}}{\bm{\epsilon}}_{\phi}({\bm{z}}_{t},t,y))}{\sqrt{\alpha_{t}}}$
(5)
$\displaystyle=\frac{\left(\sqrt{\alpha_{t}}{\bm{z}}_{\pi}+\sqrt{1-\alpha_{t}}{\bm{\epsilon}}-\sqrt{1-\alpha_{t}}{\bm{\epsilon}}_{\phi}({\bm{z}}_{t},t,y)\right)}{\sqrt{\alpha_{t}}}$
$\displaystyle={\bm{z}}_{\pi}+\frac{\sqrt{1-\alpha_{t}}}{\sqrt{\alpha_{t}}}\left({\bm{\epsilon}}-{\bm{\epsilon}}_{\phi}(\sqrt{\alpha_{t}}{\bm{z}}_{\pi}+\sqrt{1-\alpha_{t}}{\bm{\epsilon}},t,y)\right)\;.$

Note that Eq. 3 in DDCM describes the dynamics of ${\bm{z}}_{t}$. We
reparameterize it to model the dynamics of $\hat{{\bm{z}}}_{0}^{(t)}$ instead,
formulated as:

$\displaystyle\hat{{\bm{z}}}_{0}^{(t)}$
$\displaystyle=\frac{{\bm{z}}_{t}-\sqrt{1-\alpha_{t}}{\bm{\epsilon}}_{\phi}}{\sqrt{\alpha_{t}}}=\frac{\sqrt{\alpha_{t}}\hat{{\bm{z}}}_{0}^{(t+1)}+\sqrt{1-\alpha_{t}}{\bm{\epsilon}}_{t+1}-\sqrt{1-\alpha_{t}}{\bm{\epsilon}}_{\phi}}{\sqrt{\alpha_{t}}}$
(6)
$\displaystyle=\hat{{\bm{z}}}_{0}^{(t+1)}+\frac{\sqrt{1-\alpha_{t}}}{\sqrt{\alpha_{t}}}\left({\bm{\epsilon}}_{t+1}-{\bm{\epsilon}}_{\phi}\right)\;.$

We observe that the dynamics described by Eq. 6 closely resembles the
optimization of SDS Eq. 5. With the assumption that SDS following an annealing
timestep schedule, the SDS optimization is equivalent to an iterative
reconstruction procedure: 1) sample next-step pseudo-ground-truth using noisy
latents produced from previous rendered images with DDCM; 2) optimize a
reconstruction loss to match the view with pseudo-ground-truth. This finding
allows us to consider the design of 3D editing methods from a more general
iterative reconstruction perspective. We expand this formulation to include
variants of SDS, summarizing their pseudo-ground-truth parameterizations in
Table LABEL:table:f-variants of the Appendix.

###  4.2 Trajectory-Anchored Scheme for Progressive 3D Gaussian Editing

Our previous analysis in Sec. 4.1 bridges the gap between optimization-based
and reconstruction-based 3D Gaussian editing methods, demonstrating that
optimization-based approaches are special cases of reconstruction-based
methods. Without loss of generality, we consider the design of 3D editing
methods from a broader reconstruction-based perspective. The key questions for
designing improved 3D Gaussian editing methods become:  (1) which appropriate
reconstruction pseudo-ground truths to use, and (2) how to schedule the
reconstruction process of 3D Gaussians in a progressive manner for 3D editing.

With this in mind, we propose a trajectory-anchored progressive 3D Gaussian
editing scheme, as shown in Figure 1. This approach more tightly couples the
image editing process with the rendering process, enabling 3D rendering to
actively rectify the inconsistencies in the progressive 2D multi-view edits.
Specifically, we use images generated by a modified DDCM editing trajectory,
similar to InfEdit [34], as suitable pseudo-ground-truths for 3D Gaussian
editing. The detailed sampling process for generating pseudo-ground-truths is
described in Algorithm 1 in the Appendix. These pseudo-ground-truths enable a
smooth transition from the source image to the target image (as illustrated in
Figure 2), making them ideal for the progressive 2D editing and 3D updating
scheme. Progressive edits on 2D views can be promptly applied to 3D Gaussians.
Conversely, minor view inconsistencies arising from 2D editing can be promptly
rectified through 3D constrained rendering.

Figure 2: The editing trajectory exhibits a smooth and incremental transition
from the original image to the final edited image.

In line with GaussianEditor [5], we adopt a combination of reconstruction
$\mathcal{L}_{1}$ loss, perceptual loss $\mathcal{L}_{\text{LPIPS}}$, and
anchor loss $\mathcal{L}_{\text{anchor}}$ as our loss function $\mathcal{L}$:

$\mathcal{L}=\mathcal{L}_{1}+\lambda_{\text{LPIPS}}\cdot\mathcal{L}_{\text{LPIPS}}+\lambda_{\text{anchor}}\cdot\mathcal{L}_{\text{anchor}}\;,$
(7)

where $\lambda_{\text{LPIPS}}$ and $\lambda_{\text{anchor}}$ respectively
denote the weight coefficient of perceptual loss and anchor loss.

###  4.3 View-Consistent Attention Control

Ensuring consistent edits across multiple views is crucial for effective
optimization of 3D Gaussians, as inconsistent edits among different views may
fail to consolidate in 3D scenes [10]. To address this, we proposed a dual-
branch editing scheme that utilizes structural information from the original
views as 3D view prior, imposing structural constraints on the spatial layouts
of the target views. Furthermore, we employed KV propagation and KV reference
mechanisms to enhance semantic consistencies across views.

We utilize a source branch and a target branch to achieve 3D consistent
editing. The source branch takes in unedited original views, acting as a
reference for the target branch, providing structural and semantic guidance
for generating edited views. Previous studies have noted that the structural
layout of an image is formed in the early stages of diffusion and is closely
linked to self-attention queries. To ensure that the edited views produced by
the target branch maintain structural multi-view consistency, we opt to inject
self-attention queries from the source branch into the target branch during
the early stages of diffusion (mid-bottom part of Figure 1). Specifically,
denote the attention calculation as:

$\text{Attn}({\bm{Q}},{\bm{K}},{\bm{V}})=\mathrm{softmax}\left(\frac{{\bm{Q}}{{\bm{K}}}^{\top}}{\sqrt{d}}\right){\bm{V}}\;.$
(8)

The query injection is performed as:

$\text{QInj}({\bm{Q}}_{\text{src}},{\bm{Q}}_{\text{tgt}},{\bm{K}}_{\text{tgt}},{\bm{V}}_{\text{tgt}},t)=\left\\{\begin{array}[]{ll}\text{Attn}({\bm{Q}}_{\text{src}},{\bm{K}}_{\text{tgt}},{\bm{V}}_{\text{tgt}})&t\leq
t_{q}\\\
\text{Attn}({\bm{Q}}_{\text{tgt}},{\bm{K}}_{\text{tgt}},{\bm{V}}_{\text{tgt}})&t>t_{q}\end{array}\right.\;,$
(9)

where $t_{q}$ is the threshold that determines the number of steps for which
the injection applies.

Maintaining structural multi-view consistency alone is insufficient; semantic
multi-view consistency is also crucial. To achieve this, we implemented KV
propagation and KV reference mechanisms (bottom-left of Figure 1). Given a
sequence of $N$ views $\\{{\bm{I}}^{n}\\}_{n=1}^{N}$ resulting from
consecutive camera motion, we divide the sequence into equal-partitioned
context segments $\\{{\bm{C}}^{i}\\}_{i=1}^{L}$. Each context ${\bm{C}}^{i}$
has a length of $N/L$ frames and contains a keyframe ${\bm{I}}_{1}^{i}$. Views
within the same context exhibit minimal movement or view shift, their
semantics are relatively consistent. For views within the same context, we
utilized an efficient KV propagation method, where the self-attention keys and
values of the keyframe $\\{{\bm{K}}_{1}^{i},{\bm{V}}_{1}^{i}\\}$ of $i$-th
context is directly propagated to the rest of the frames in the same context.
Denote the output of self-attention of $j$-th frame in context $i$ as
${\bm{O}}^{i}_{j}$, we can formulate this attention propagation as:

${\bm{O}}^{i}_{j}=\text{Attn}\left({\bm{Q}}_{j}^{i},{\bm{K}}_{1}^{i},{\bm{V}}_{1}^{i}\right)\;.$
(10)

Significant view shifts can occur across different contexts, leading to
noticeable semantic differences. Applying KV propagation on keyframes across
different contexts would accumulate significant distortion. To mitigate this,
we employ KV reference for keyframes across contexts. This involves inflating
the self-attention keys and values across keyframes of different contexts and
processing them jointly. The self-attention keys and values from other
keyframes serve as a mutual reference for the semantic information of the
edited entity. For self-attention keys $\\{{\bm{K}}^{i}\\}_{i=1}^{l}$ and
values $\\{{\bm{V}}^{i}\\}_{i=1}^{l}$ from $l$ different keyframes, we
concatenate them respectively to form cross-context KV pairs. The operation is
formulated as:

${\bm{O}}_{1}^{i}=\text{Attn}\left({\bm{Q}}_{1}^{i},\left[{\bm{K}}^{1},\dots,{\bm{K}}^{l}\right],\left[{\bm{V}}^{1},\dots,{\bm{V}}^{l}\right]\right)\;.$
(11)

To maintain fidelity with the original images in the non-edited areas, we
applied a cross-attention control (bottom-right of Figure 1) and local
blending strategy similar to Prompt-to-Prompt [14] and InfEdit [34]. Cross-
attention control is done by replacing the attention maps corresponding to
non-edited tokens in the target branch with those from the source branch. We
use the semantic mask ${\bm{M}}$ obtained from semantic tracing [5] to blend
noisy target latent with source latent, formulated as:

${\bm{z}}^{\text{tgt}}_{t}={\bm{M}}\odot{\bm{z}}^{\text{tgt}}_{t}+(1-{\bm{M}})\odot{\bm{z}}^{\text{src}}_{t}\;,$
(12)

where $\odot$ is the Hadamard product. Consequently, the 3D consistent view
edits consolidate onto the 3D Gaussians in a rectified and progressive manner.

Table 1:  Quantitative Results. Compared to DDS and GaussianEditor, our method outperforms across evaluated metrics. Methods | Reference | CLIP Score [2] | CLIP Directional Score [2] | User Study
---|---|---|---|---
DDS [13] | ICCV (2023) | 0.2242 | 0.1336 | 5.59%
GaussianEditor [5] | CVPR (2024) | 0.2343 | 0.1548 | 25.47%
TrAME (Ours) | Ours | 0.2550 | 0.2002 | 68.94%
Figure 3: Qualitative comparison. A comparative analysis of experimental
results for single-scene editing across State-of-the-Art methods and ours
under identical conditions.

##  5 Experiments

In this section, we provide a comprehensive overview of our experimental
setup, which encompasses the testing data, baseline methods and implementation
details. We examine the effectiveness of our method with quantitative and
qualitative results. Additionally, we conduct ablation studies on each
proposed component to dissect their individual contributions to the overall
performance.

###  5.1 Implementation Details

We implement our method based on a modified version of GaussianEditor [5].
Each editing typically takes about 5-15 minutes on average, depending on the
complexity of the scene. For scenes involving partial editing, we adhere to
the semantic tracing approach outlined in GaussianEditor. The context length
of the VCAC module is set to 4. A view angle difference threshold of 25
degrees is set to determine if the displacement between two views is too
large, thereby deciding whether to use the KV propagation or KV reference
approach in the VCAC module. We evaluate our method on Instruct-N2N [10] and
MipNeRF-360 [1] dataset. The experiments are conducted using a single Nvidia
A100 80G GPU, with VRAM consumption around 30GB.

###  5.2 Baseline Methods

We compare our method with two state-of-the-art 3DGS editing methods: (1)
Delta Distillation Sampling (DDS)[13], which represents the SDS optimization
strategy for editing and uses an implicit score function loss derived from a
diffusion model as guidance; (2) GaussianEditor [5], which represents an
editing method based on Instruct-P2P [2], guiding the editing of the 3D
Gaussian Splatting (3DGS) by iteratively updating the training set. Note that
other 3D scene editing methods either lack publicly available code or utilize
3D representations other than 3DGS. We conduct a quantitative comparison using
the CLIP Score and CLIP Directional Score [2] to evaluate the visual
resemblance between the edited 3DGS and the given prompt. Additionally, we
carry out a user study aimed at assessing the effectiveness of these editing
methods from a human perspective.

###  5.3 Quantitative Results

In our quantitative analysis, we employ the CLIP Score and CLIP Directional
Score [2] to assess the alignment of the edited 3D models with the target text
prompts on the rendered views of the edited 3DGS. These results indicate that
our 3D editing method is effective at consolidating user edits into 3D scenes.
To further validate our approach, we organized a user study involving 23
participants over 14 different cases, where users were asked to evaluate and
choose the best edits based on image quality, alignment with the target
prompt, and consistency across different views. The quantitative evaluation
results, presented in Table 1, compare our method against baseline methods
across various test scenes. Our method excels in CLIP Score, CLIP Directional
Score, and user study scores, confirming its superior editing outcomes.

###  5.4 Qualitative Results

As shown in Figure 3 and Figure 4, our qualitative analysis showcases that our
method ensures multi-view consistency and fine-grained details in 3D scene
editing. We display outcomes from various perspectives in different scenes,
underscoring the effectiveness of our method. Based on the results, we can
observe that DDS faces challenges in achieving detailed and faithful editing.
GaussianEditor struggles with maintaining multi-view consistency, as seen in
inconsistencies such as the hair color of the elf and the nose of the clown
when viewed from different angles, and in consolidating fine-grained details
onto 3D textures, such as accumulated snow on park benches and patterns on
marble tables (examples shown in the second and third columns of Figure 4).
This shortcoming of the compared methods becomes particularly apparent when
trying to capture the intricate patterns and fine details inherent to such
materials, highlighting a critical area where our approach excels in producing
view-consistent and detailed 3D scene edits.

Figure 4:  Qualitative comparison. A comparative analysis of experimental
results for various scenes encompassing both partial and global editing under
identical conditions.

###  5.5 Ablation Studies

Study on VCAC. The results presented in Figure 5 demonstrate that the VCAC
module effectively enhances cross-view structural and semantic consistency.
The views modified by our VCAC module align more accurately with the original
views in terms of structural elements like poses and facing directions, and
they also demonstrate better semantic coherence, exhibiting similar
appearances. In particular, the outputs generated by our VCAC module maintain
a uniform appearance across various views, with the facing direction of
Spiderman in the figures aligning cohesively with the pose of the man in the
original views.

Study on TAS. The ablated results obtained without TAS were derived by
applying 3D editing solely based on the final obtained edited views, without
feedback from the 3D Gaussians. The results depicted in Figure 6 show that TAS
produces more naturalistic and consistent appearances, while the ablated
results exhibit undesirable blurry artifacts and inconsistencies across views.
Specifically, the hair color around the clown’s forehead in the ablated
results varies across views, and the lips of the clown show ghosting shadows.

Figure 5: Ablation study on the VCAC module for 2D view editing. The results
display edited views with (w. VCAC) and without the VCAC module (w/o VCAC).
Figure 6: Ablation study on TAS for 3DGS editing. Comparative results are
shown with (w. TAS) and without the adoption of the TAS (w/o TAS).

##  6 Conclusion and Future Work

Our research introduces TrAME, a progressive 3DGS editing framework that
effectively improves multi-view consistency via Trajectory Anchored Scheme
(TAS) and View-Consistent Attention Control (VCAC) module. Our work bridges
the gap between optimization-based and reconstruction-based editing methods,
offering a unified perspective for selecting better design choices in 3D scene
editing methods. Our limitations reside in complex view-consistent edits with
large object deformations, which will be the focus of our future work.

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