A 3+1 formulation of the $1/c$ expansion of General Relativity

Mahmut Elbistan1, Efe Hamamcı1,
Dieter Van den Bleeken1,2 and Utku Zorba1

1) Physics Department, Boğaziçi University

34342 Bebek / Istanbul, Turkey

2) Secondary address:

Institute for Theoretical Physics, KU Leuven

3001 Leuven, Belgium

mahmut.elbistan, efe.hamamci, dieter.van, utku.zorba @boun.edu.tr

Abstract

Expanding General Relativity in the inverse speed of light, $1/c$, leads to a
nonrelativistic gravitational theory that extends the Post-Newtonian expansion
by the inclusion of additional strong gravitational potentials. This theory
has a fully covariant formulation in the language of Newton-Cartan geometry
but we revisit it here in a 3+1 formulation. The appropriate 3+1 formulation
of General Relativity is one first described by Kol and Smolkin (KS), rather
than the better known Arnowitt-Deser-Misner (ADM) formalism. As we review, the
KS formulation is dual to the ADM formulation in that the role of tangent and
co-tangent spaces get interchanged. In this 3+1 formulation the $1/c$
expansion can be performed in a more systematic and efficient fashion,
something we use to extend the computation of the effective Lagrangian beyond
what was previously achieved and to make a number of new all order
observations.

###### Contents

  1. 1 Introduction
  2. 2 Two dual 3+1 formulations of GR
    1. 2.1 Choice of coordinates
    2. 2.2 Decomposition of the metric
    3. 2.3 Decomposition of the connection
    4. 2.4 Decomposition of the Einstein-Hilbert action
  3. 3 Dynamics and symmetries in the KS formulation
    1. 3.1 Einstein equations
    2. 3.2 Diffeomorphism symmetries and Noether identities
    3. 3.3 Conformal redefinition and the relativistic Poisson equation
    4. 3.4 Making time derivatives explicit
  4. 4 The $1/c$ expansion in KS formalism
    1. 4.1 Structure of the expansion
    2. 4.2 Explicit expansion up to NNLO
  5. 5 The $1/c^{2}$ expansion in KS formalism
    1. 5.1 Structure of the expansion
    2. 5.2 Explicit expansion up to NNLO
  6. 6 Discussion
  7. A Additional technical arguments
    1. A.1 Expanded action and equations of motion
    2. A.2 Gauge fixing the traces: details
  8. B Extra trace terms
  9. C Equations of motion

##  1 Introduction

The geometric formulation of nonrelativistic physics has experienced a
resurgence of interest in the last few years, see e.g. [1, 2, 3, 4] for a
recent overview of the relevant geometries, symmetries and some applications.
Formulating the nonrelativistic expansion of general relativity in a geometric
fashion has the advantage that it keeps, by construction, general coordinate
invariance manifest. Building on [5] this idea was applied to the Post-
Newtonian (PN) expansion in [6], reformulating it in terms of Newton-Cartan
geometry. The PN expansion, see e.g. [7] for an introduction, is an expansion
around flat space-time and is for this reason a double expansion:
nonrelativistic, i.e. an expansion in the inverse speed of light $1/c$, as
well as weakly coupled, i.e. an expansion in Newton’s constant
$G_{\mathrm{N}}$. From the geometric perspective the assumption of weak
coupling was implemented in [5, 6] as an assumption on the nonrelativistic
connection. In [8] this assumption was relaxed, which extends the
nonrelativistic theory to so called twistless torsional Newton-Cartan
geometry. Physically speaking, this extended theory adds to Newtonian gravity
an additional gravitational potential that describes strong gravitational time
dilation effects. Rather than an expansion around flat space-time, it is an
expansion, in powers of $1/c^{2}$, around an arbitrary static space-time [9]
and as such extends the PN expansion. One of the advantages of the $1/c^{2}$
expansion is that an expansion of the Einstein equations goes hand in hand
with an expansion of the Einstein-Hilbert Lagrangian [10, 11]. This makes it
possible to formulate the expansion – truncated up to a given order – as a
self-consistent nonrelativistic theory with a fully geometric/covariant action
and associated variational principle.

Despite the elegance of the geometrically formulated $1/c^{2}$ expansion [8,
11], manifest space-time coordinate invariance comes with a cost, in that it
introduces a number of fields which are pure gauge, and this increases the
complexity of the expansion. Some work has been done to streamline this [12,
13, 14] or to organize it based on symmetry considerations [10, 15, 16, 17],
but this has so far not led to higher order results.

In this paper we revisit the $1/c$ and $1/c^{2}$ expansions of general
relativity in a 3+1 formulation, where an explicit choice of time coordinate
is made. Although the theory loses some of its elegance in such formalism, it
keeps some key features: manifest invariance under spatial coordinate
transformations as well as an action with well defined variational principle.
Furthermore, it has the advantage that it makes the physical degrees of
freedom more explicit and the expansion more transparent. This allowed us
first of all to carry out the expansion to higher order than before. It also
reveals that the $1/c$ expansion is equivalently an expansion in time
derivatives around a quasi-stationary relativistic background, something which
was pointed out at leading order in [18]. Also the relation between the $1/c$
expansion and the $1/c^{2}$ expansion, a consistent truncation to even orders,
becomes more clear so that we could use it to devise an algorithm that
constructs the $1/c^{2}$ expansion to power $c^{-2N}$ from the $1/c$ expansion
to power $c^{-N}$.

A 3+1 formulation of Newton-Cartan theory was recently considered in [19, 20],
for use in Newtonian cosmology. Rather than introducing the 3+1 formulation at
the nonrelativistic level, we will start from General Relativity (GR) in 3+1
form and then expand. The most familiar 3+1 formulation of GR is that of
Arnowitt, Deser and Misner (ADM) [21], see e.g. [22] for a detailed
introduction. In ADM form the 4d relativistic metric is

$ds^{2}=-c^{2}N^{2}dt^{2}+h_{ij}(dx^{i}+cN^{i}dt)(dx^{j}+cN^{j}dt)$ (1.1)

where the fields $(N,N^{i},h_{ij})$ depend on time $t$ as well as the spatial
coordinates $x^{i}$. Mathematically speaking $h_{ij}$ is the induced metric on
a spatial hypersurface with tangent vectors $\partial_{i}$. The basis in
tangent space is completed by the vector
$u=N^{-1}(c^{-1}\partial_{t}-N^{i}\partial_{i})$, which is normal to the
hypersurface. The factors of the speed of light $c$ are naturally there for
dimensional reasons. Now observe that the ADM decomposition becomes singular
in the $c\rightarrow\infty$ limit, since $u$ is then no longer linearly
independent of the $\partial_{i}$. This implies the ADM decomposition is not a
suitable starting point for a $1/c$ expansion. (The same feature makes
it111Remark that the introduction of $c$ on dimensional grounds, as in (1.1),
is not the unique appearance of powers of $c$ such that the frame
$\\{u,\partial_{i}\\}$ remains non-degenerate in the $c\rightarrow 0$ limit.
One could for example consider replacing $N^{i}\rightarrow c^{-1}N^{i}$ in
(1.1) and the frame would remain non-degenerate as $c$ approaches 0. a good
starting point for the ultra relativistic expansion in small $c$, see e.g.
[23, 24, 25]). A 3+1 decomposition suitable for the $1/c$ expansion was
introduced by Kol and Smolkin (KS) in [26]. In KS form the 4d relativistic
metric reads

$ds^{2}=-M^{2}(cdt+C_{i}dx^{i})^{2}+h_{ij}dx^{i}dx^{j}$ (1.2)

where again the fields $(M,C_{i},h_{ij})$ depend on $t$ and $x^{i}$. This
decomposition is based on a preferred basis for the cotangent space. The basis
$dx^{i}$ for the spatial subspace gets extended to a basis of the total 4
dimensional cotangent space by the one-form $n=M(cdt+C_{i}dx^{i})$. One sees
that this basis remains generating in the limit $c\rightarrow\infty$ since $n$
remains linearly independent of the $dx^{i}$. This indicates that the KS
formulation is the appropriate 3+1 decomposition for the nonrelativistic
expansion. Kol and Smolkin originally introduced the decomposition (1.2) in
[27] in the context of the PN expansion. In that case, where one assumes $M$
to be of the weak field form $M=1-\frac{2\varphi}{c^{2}}+{\cal O}(c^{-4})$,
the KS ansatz, also known as the Kaluza-Klein decomposition, greatly
simplifies (part of) the PN expansion, see e.g. [28] for a review.

In this work, where we consider the $1/c$ expansion and make no weak field
assumption, we will allow the fields $(M,C_{i},h_{ij})$ to be arbitrary
analytic functions in $1/c$, which in particular means that
$M={\overset{\,\scalebox{0.5}{($0$)}}{M}}{}+{\overset{\,\scalebox{0.5}{($1$)}}{M}}{}c^{-1}+{\overset{\,\scalebox{0.5}{($2$)}}{M}}{}c^{-2}+{\cal
O}(c^{-4})$, with ${\overset{\,\scalebox{0.5}{($k$)}}{M}}{}$ arbitrary
functions of $t,x^{i}$ (only constrained by the expanded Einstein equations).
While we can relate ${\overset{\,\scalebox{0.5}{($2$)}}{M}}{}$ to the
Newtonian potential $\varphi$, ${\overset{\,\scalebox{0.5}{($0$)}}{M}}{}$ and
${\overset{\,\scalebox{0.5}{($1$)}}{M}}{}$ are additional potentials not
present in the PN expansion, capturing strong nonrelativistic gravitational
effects. A similar observation holds for the expansion of $C_{i}$ and
$h_{ij}$. The precise relation to the PN expansion (up to 1PN order) is worked
out in section 5.2, see also table 2.

Our paper is organized as follows. In section 2 we review both the ADM and KS
decomposition of GR, we do so in a way that allows to treat both
decompositions simultaneously and which emphasizes their dual nature. In
section 3 we focus on the KS formulation and discuss the Einstein equations as
well as the diffeomorphism symmetry and the associated Noether/Bianchi
identities from the point of view of this 3+1 decomposition. At the end of the
section, in subsections 3.3 and 3.4 we motivate and perform the convenient
field redefinition $M=e^{\frac{\psi}{2}}$, $h_{ij}=e^{-\psi}\gamma_{ij}$ and
make factors of $c$ and time derivatives explicit. In section 4 we then come
to the $1/c$ expansion. The new formalism allows us to make a number of new
observations: we show how all subleading equations take the form of a sourced
linear second order PDE and compute the linear differential operator that is
the same at all orders. We also discuss how by a convenient choice of gauge
all the subleading coefficients in the expansion of $\gamma_{ij}$ can be made
trace-less. We then compute the Lagrangian of the expansion up to NNLO, i.e.
up to order $c^{-2}$. Previously only the leading order, i.e. order $c^{0}$,
had been computed [18]. In section 5 we recall how the $1/c^{2}$ expansion is
a consistent truncation of the $1/c$ expansion to even orders. Here the use of
the 3+1 formulation makes the connection to the $1/c$ expansion more
transparent and allows us to derive an algorithm, that we call the shuffling
algorithm, with which the $1/c^{2}$ expansion up to order $c^{-2N}$ can be
constructed from the $1/c$ expansion up to order $c^{-N}$. At NLO, i.e. order
$c^{-2}$, we recover the results of [8, 11] and spell out the precise
relation. The shuffling algorithm then allows us to compute the Lagrangian up
to NNLO, i.e. order $c^{-4}$, for the first time. This is the order where the
1PN correction to Newtonian gravity finds itself and we show how indeed, upon
setting to zero a number of additional potentials, the $1/c^{2}$ expansion
reproduces the well-known 1PN equations. We conclude in section 6 where we
also point towards possible future directions. Finally there are three
appendices. In appendix A we spell out two technical derivations of results
used in the main text, in appendix B we provide some extra terms present
outside the traceless gauge and the (un-expanded) equations of motion with
powers of $c$ and time derivatives made explicit are written down in appendix
C.

##  2 Two dual 3+1 formulations of GR

Let ${\cal M}$ be a $d+1$ dimensional manifold equipped with a Lorentzian
metric $g_{\mu\nu}$. We’ll denote the associated Levi-Civita connection with
$\nabla$.

###  2.1 Choice of coordinates

Intuitively a (local) $d+1$ formulation would amount to a (local) choice of
coordinates $(x^{\mu})=(t,x^{i})$ with $t$ a ’time coordinate’ and the $x^{i}$
’spatial coordinates’. We could try to make this more precise by requiring
$\partial_{t}$ to be a time-like vector and $\partial_{i}$ to be space-like
vectors. Although imposing both these conditions is always possible, e.g. in
Gaussian normal (GN) coordinates, this would restrict the allowed set of
coordinates very much. A weaker choice, which we’ll call Arnowitt-Deser-Misner
(ADM) coordinates [21], only requires the $\partial_{i}$ to be spatial. Let us
however point out that this automatically implies that the one-form $dt$ will
be time-like. So in this case we see that it is a mix of the coordinate basis
of tangent vectors and that of co-tangent vectors which is required to be
space/time-like respectively. This then suggests a naturally dual alternative,
which is to choose coordinates such that the $dx^{i}$ are space-like which
then implies that $\partial_{t}$ is time-like. We’ll call such coordinates
Kol-Smolkin (KS) coordinates [26]. For a summary of the definitions see table
1.

| space-like | time-like
---|---|---
GN | $\partial_{i}\,,\ dx^{i}$ | $\partial_{t}\,,\ dt$
ADM | $\partial_{i}$ | $dt$
KS | $dx^{i}$ | $\partial_{t}$
Table 1: We’ll refer to a coordinate system $(t,x^{i})$ as a space-time split
or $d+1$ decomposition if it falls into either the ADM or KS class (the GN
class is the intersection of both). They are defined by the properties of the
coordinate bases of tangent and co-tangent spaces, as listed in this table.

##### Elementary example

Before discussing these classes in full generality in the next subsection, let
us illustrate them in a simple example. Consider the Minkowski metric in
inertial coordinates

$ds^{2}=-dt^{2}+dx^{i}dx^{i}\,.$ (2.1)

One easily verifies that $(t,x^{i})$ satisfy our definition of GN coordinates.

If we define new coordinates $(\hat{t},\hat{x}^{i})$, with $\hat{t}=t$ and
$\hat{x}^{i}=x^{i}-v^{i}t$ for some constants $v^{i}$, then the metric takes
the form

$ds^{2}=-d\hat{t}^{2}+(d\hat{x}^{i}+v^{i}d\hat{t})(d\hat{x}^{i}+v^{i}d\hat{t})\,.$
(2.2)

While $\hat{\partial}_{i}$ remains spatial, $\hat{\partial}_{t}$ is no longer
time-like when $v^{i}v^{i}\geq 1$. But since $\hat{g}^{tt}=-1$ independent of
$v^{i}$, it follows that $d\hat{t}$ remains time-like. The coordinates
$(\hat{t},\hat{x}^{i})$ are thus an example of ADM coordinates.

Alternatively we can work in coordinates $(\tilde{t},\tilde{x}^{i})$, with
$\tilde{t}=t-w_{i}x^{i}$ and $\tilde{x}^{i}=x^{i}$ for some constants $w_{i}$.
In these coordinates the metric reads

$ds^{2}=-(d\tilde{t}+w_{i}d\tilde{x}^{i})^{2}+d\tilde{x}^{i}d\tilde{x}^{i}\,.$
(2.3)

One verifies that $\tilde{\partial}_{t}$ is always time-like, while
$\tilde{\partial}_{i}$ is not space-like when $w_{i}^{2}\geq 1$. But the
$d\tilde{x}^{i}$ are always space-like and so $(\tilde{t},\tilde{x}^{i})$ are
an example of KS coordinates for Minkowski space, independent of the values of
$w_{i}$.

Note that indeed the ADM and KS classes are not mutually exclusive, their
intersection is exactly the GN class.

###  2.2 Decomposition of the metric

Although our aim is to decompose the space-time metric $g_{\mu\nu}$ in either
ADM or KS coordinates, it will be interesting to introduce a formalism that
can treat both at the same time. This will then also allow us to decompose the
Einstein-Hilbert action simultaneously for both cases in section 2.4. The
decomposition of the metric we’ll perform here will also shed light on the
definitions of the previous subsection, which might have appeared a bit ad-hoc
there.

We start by choosing222Mathematically speaking a Lorentzian metric introduces
an O(1, $d$)-structure on ${\cal M}$. The addition of the vector field
$u^{\mu}$ refines this to an O($d$)-structure, also known as an Aristotelian
structure, which is a combination of compatible Galilean and Carrollian
structures, see e.g. [29]. The tensors defining this Aristotelian structure
are $u^{\mu},n_{\mu}=g_{\mu\nu}u^{\nu}$,
$h^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$ and
$h_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu}$.  a time-like vector field $u^{\mu}$,
which without loss of generality we can assume to be normalized:
$g_{\mu\nu}u^{\mu}u^{\nu}=-1$. Of course this is equivalent to the choice of a
time-like one-form $n_{\mu}=g_{\mu\nu}u^{\nu}$, which is again normalized.

We can complete the time-like vector into a tangent frame $(u,e_{i})$, with
each of the $e_{i}$ orthogonal to $u$: $g_{\mu\nu}u^{\mu}e_{i}^{\nu}=0$. The
relations $e^{i}_{\mu}e_{j}^{\mu}=\delta_{j}^{i}$, $u^{\mu}e_{\mu}^{i}=0$ then
provide a dual frame $(n,e^{i})$. In summary this amounts to

$-u^{\mu}n_{\nu}+e^{\mu}_{i}e_{\nu}^{i}=\delta^{\mu}_{\nu},\qquad
e_{i}^{\mu}e_{\mu}^{j}=\delta_{i}^{j}\,.$ (2.4)

Note that we do not require the frame $e_{i}$ to be orthonormal, instead we
define

$h_{ij}=g_{\mu\nu}e^{\mu}_{i}e^{\nu}_{j}\,.$ (2.5)

It follows that $h^{ij}=g^{\mu\nu}e_{\mu}^{i}e_{\nu}^{j}$ is the inverse of
$h_{ij}$.

In terms of this frame and co-frame the metric and its inverse decompose as

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-n^{2}+h_{ij}e^{i}e^{j}\,,\qquad
g^{\mu\nu}\partial_{\mu}\partial_{\nu}=-u^{2}+h^{ij}e_{i}e_{j}\,.$ (2.6)

As we will now point out, there is a natural choice of frames of the form
above associated to both ADM and KS coordinates.

####  2.2.1 ADM decomposition

In ADM coordinates, where $\partial_{i}$ is spatial and $dt$ time-like, a
natural choice of frames satisfying (2.4) is

$\displaystyle e_{i}=\partial_{i}\,,$ $\displaystyle n=-Ndt\,,$ (2.7)
$\displaystyle e^{i}=dx^{i}+N^{i}dt\,,$ $\displaystyle
u=N^{-1}(\partial_{t}-N^{i}\partial_{i})\,.$ (2.8)

Then, the metric (2.6) takes the well-known ADM form

$ds^{2}=-N^{2}dt^{2}+h_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt)\,.$ (2.9)

Note that the inverse metric takes the form

$g^{\mu\nu}\partial_{\mu}\partial_{\nu}=-N^{-2}(\partial_{t}-N^{i}\partial_{i})^{2}+h^{ij}\partial_{i}\partial_{j}\,.$
(2.10)

####  2.2.2 KS decomposition

In KS coordinates it is $dx^{i}$ which is spatial while $\partial_{t}$ is
time-like, so in this case the natural choice of frames satisfying (2.4) is

$\displaystyle e^{i}=dx^{i}\,,$ $\displaystyle u=M^{-1}\partial_{t}\,,$ (2.11)
$\displaystyle e_{i}=\partial_{i}-C_{i}\partial_{t}\,,$ $\displaystyle
n=-M(dt+C_{i}dx^{i})\,.$ (2.12)

So the metric (2.6) takes the following form in KS coordinates:

$ds^{2}=-M^{2}(dt+C_{i}dx^{i})^{2}+h_{ij}dx^{i}dx^{j}\,,$ (2.13)

while the inverse metric is

$g^{\mu\nu}\partial_{\mu}\partial_{\nu}=-M^{-2}\partial_{t}^{2}+h^{ij}(\partial_{i}-C_{i}\partial_{t})(\partial_{j}-C_{j}\partial_{t})\,.$
(2.14)

Comparing the forms (2.9, 2.10) with (2.13, 2.14) one explicitly sees how they
are dual, in the sense that the role of metric and inverse metric – i.e.
tangent space and co-tangent space – get interchanged.

We should emphasize that the spatial metric $h_{ij}$ in the KS form of the
metric (2.13) is different from the spatial metric $h_{ij}$ in the ADM form of
the metric (2.9). Indeed, in case the coordinates $(t,x^{i})$ are in the GN
class, i.e. of both ADM and KS type, they are related as

$h_{ij}^{\mathrm{ADM}}=h_{ij}^{\mathrm{KS}}-M^{2}C_{i}C_{j}\quad\mbox{and}\quad
h^{ij}_{\mathrm{KS}}=h^{ij}_{\mathrm{ADM}}-N^{2}N^{i}N^{j}\,.$ (2.15)

It is $h_{ij}^{\mathrm{ADM}}$ which has the geometrical interpretation of the
pull-back of the Lorentzian metric $g_{\mu\nu}$ on the constant $t$ hyper-
surfaces. Although $h_{ij}^{\mathrm{KS}}$ does not have this natural geometric
interpretation it is, by construction, a well-defined Riemannian metric on the
constant $t$ hyper-surfaces when $(t,x^{i})$ are in the KS class. Only if
$(t,x^{i})$ are in the ADM class will the constant $t$ hypersurfaces be
spatial so that then $h_{ij}^{\mathrm{ADM}}$ is Riemannian.

###  2.3 Decomposition of the connection

Given a choice of frame and dual frame that satisfy (2.4) one has the
following identities

$\displaystyle n_{\mu}\nabla_{\nu}u^{\mu}$
$\displaystyle=u^{\mu}\nabla_{\nu}n_{\mu}=0\,,$ (2.16) $\displaystyle
e^{\mu}_{i}\nabla_{\nu}e_{\mu}^{j}$
$\displaystyle=-e_{\mu}^{j}\nabla_{\nu}e^{\mu}_{i}\,,$ (2.17) $\displaystyle
e^{\mu}_{i}\nabla_{\nu}n_{\mu}$
$\displaystyle=-n_{\mu}\nabla_{\nu}e^{\mu}_{i}\,,$ (2.18) $\displaystyle
e^{i}_{\mu}\nabla_{\nu}u^{\mu}$
$\displaystyle=-u^{\mu}\nabla_{\nu}e^{i}_{\mu}\,.$ (2.19)

Furthermore in either the ADM or KS case the natural frames (2.7, 2.8) or
(2.11, 2.12) satisfy the further identities

$e_{\mu}^{i}\partial_{\nu}e^{\mu}_{j}=-e^{\mu}_{j}\partial_{\nu}e_{\mu}^{i}=0,$
(2.20)

Our results below hold for more general frames than (2.7, 2.8) or (2.11,
2.12), as long as we assume them to satisfy the additional condition (2.20) in
addition to (2.4).

The identities listed above are then sufficient to decompose the Levi-Civita
connection of $g_{\mu\nu}$ in the independent components333Note that when
combined with (2.17) the assumption (2.20) results in
$-e^{\mu}_{i}\nabla_{\nu}e_{\mu}^{j}=e_{\mu}^{j}\nabla_{\nu}e^{\mu}_{i}=e_{\mu}^{j}\Gamma_{\rho\nu}^{\mu}e^{\rho}_{i}$.
This in turn guarantees that $\hat{\Gamma}_{ij}^{k}$ is symmetric in $ij$
since it can be written as
$\hat{\Gamma}_{ij}^{k}=e_{i}^{\mu}e_{j}^{\nu}e_{\rho}^{k}\Gamma_{\mu\nu}^{\rho}$.

$\displaystyle K_{ij}$
$\displaystyle=e_{i}^{\mu}e_{j}^{\nu}\nabla_{\mu}n_{\nu}\,,$ (2.21)
$\displaystyle\Omega_{i}$
$\displaystyle=u^{\mu}e_{i}^{\nu}\nabla_{\mu}n_{\nu}\,,$ (2.22)
$\displaystyle\Delta_{i}^{j}$
$\displaystyle=u^{\mu}e^{j}_{\nu}\nabla_{\mu}e^{\nu}_{i}-K_{i}{}^{j}\,,$
(2.23) $\displaystyle\hat{\Gamma}_{ij}^{k}$
$\displaystyle=e_{\mu}^{k}e_{i}^{\nu}\nabla_{\nu}e_{j}^{\mu}\,.$ (2.24)

The key point of introducing these objects is that, via (2.16-2.19), they are
sufficient to decompose the covariant derivatives of the frames. In turn that
allows to decompose covariant derivatives of tensors as

$\displaystyle\nabla_{\mu}(e^{\nu}_{i}V^{i})$
$\displaystyle=-n_{\mu}u^{\nu}\Omega_{i}V^{i}-n_{\mu}e^{\nu}_{i}(\hat{D}_{t}V^{i}+V^{j}K_{j}{}^{i})+e_{\mu}^{i}u^{\nu}K_{ij}V^{j}+e_{\mu}^{i}e^{\nu}_{j}\hat{D}_{i}V^{j}\,,$
(2.25) $\displaystyle\nabla_{\mu}(e_{\nu}^{i}V_{i})$
$\displaystyle=-n_{\mu}n_{\nu}\Omega^{i}V_{i}-n_{\mu}e_{\nu}^{i}(\hat{D}_{t}V_{i}-K_{i}{}^{j}V_{j})+e_{\mu}^{i}n_{\nu}K_{i}{}^{j}V_{j}+e_{\mu}^{i}e_{\nu}^{j}\hat{D}_{i}V_{j}\,.$
(2.26)

It is a bit tedious but rather obvious to see how this generalizes to tensors
with any number of upper and lower ’spatial’ indices, e.g.
$\nabla_{\mu}(e_{\nu}^{i}e^{\rho}_{j}V_{i}{}^{j})$, so we will not write out
these expressions explicitly.

In (2.25, 2.26) we introduced a modified covariant derivative $\hat{D}_{t}$
along $t$ and a modified covariant derivative $\hat{D}_{i}$ along the $x^{i}$.
Their precise definitions on a ’spatial’ tensor $T^{i_{1}\ldots
i_{m}}_{j_{1}\ldots j_{n}}(t,x)$ are

$\displaystyle\hat{D}_{t}T^{j_{1}\ldots j_{m}}_{k_{1}\ldots k_{n}}$
$\displaystyle=\hat{\partial}_{t}T^{j_{1}\ldots j_{m}}_{k_{1}\ldots
k_{n}}+\Delta^{j_{1}}_{i}T^{ij_{2}\ldots j_{m}}_{k_{1}\ldots
k_{n}}+\ldots-\Delta_{k_{1}}^{i}T^{j_{1}\ldots j_{m}}_{ik_{2}\ldots
k_{n}}-\ldots\,,$ (2.27) $\displaystyle\hat{D}_{i}T^{j_{1}\ldots
j_{m}}_{k_{1}\ldots k_{n}}$ $\displaystyle=\hat{\partial}_{i}T^{j_{1}\ldots
j_{m}}_{k_{1}\ldots k_{n}}+\hat{\Gamma}_{il}^{j_{1}}T^{lj_{2}\ldots
j_{m}}_{k_{1}\ldots k_{n}}+\ldots-\hat{\Gamma}_{ik_{1}}^{l}T^{j_{1}\ldots
j_{m}}_{lk_{2}\ldots k_{n}}-\ldots\,.$ (2.28)

where

$\hat{\partial}_{t}=u^{\mu}\partial_{\mu}\qquad\mbox{and}\qquad\hat{\partial}_{i}=e_{i}^{\mu}\partial_{\mu}\,.$
(2.29)

Note that $\hat{D}_{i}$ differs from the standard covariant derivative $D_{i}$
with respect to the Levi-Civita connection of $h_{ij}$, i.e.

$\displaystyle D_{i}T^{j_{1}\ldots j_{m}}_{k_{1}\ldots k_{n}}$
$\displaystyle=\partial_{i}T^{j_{1}\ldots j_{m}}_{k_{1}\ldots
k_{n}}+\Gamma^{j_{1}}_{il}T^{lj_{2}\ldots j_{m}}_{k_{1}\ldots
k_{n}}+\ldots-\Gamma^{l}_{ik_{1}}T^{j_{1}\ldots j_{m}}_{lk_{2}\ldots
k_{n}}-\ldots\,,$ (2.30) $\displaystyle\Gamma_{ij}^{k}$
$\displaystyle=\frac{1}{2}h^{kl}\left(\partial_{i}h_{jl}+\partial_{j}h_{il}-\partial_{l}h_{ij}\right).$
(2.31)

The difference is two-fold, firstly in $\hat{D}_{i}$ the standard partial
derivative $\partial_{i}$ is replaced by $e_{i}^{\mu}\partial_{\mu}$ and
secondly the ’spatial connection’ $\hat{\Gamma}_{ij}^{k}$ is used instead of
$\Gamma_{ij}^{k}$. Both these differences vanish in the ADM case, but not in
the KS case – as we’ll see below. It is important to point out that the
modified covariant derivative (2.28) remains compatible with $h_{ij}$:

$\hat{D}_{i}h_{jk}=0$ (2.32)

This is not so for the time-like covariant derivative, but one finds the
elegant expression

$\hat{D}_{t}h_{ij}=2K_{(ij)}\,.$ (2.33)

Let us remark that similarly to the decomposition (2.25) one computes that

$\nabla_{\mu}(Vu^{\mu}+V^{i}e_{i}^{\mu})=\hat{D}_{t}V+KV+\hat{D}_{i}V^{i}+\Omega_{i}V^{i}\,,$
(2.34)

and finally we also point out the relations

$\displaystyle\hat{\partial}_{t}\hat{\partial}_{i}-\hat{\partial}_{i}\hat{\partial}_{t}$
$\displaystyle=\Omega_{i}\hat{\partial}_{t}+\Delta_{i}^{j}\hat{\partial}_{j}\,,$
(2.35)
$\displaystyle\hat{\partial}_{i}\hat{\partial}_{j}-\hat{\partial}_{j}\hat{\partial}_{i}$
$\displaystyle=K_{[ij]}\hat{\partial}_{t}\,.$ (2.36)

####  2.3.1 The ADM case

When choosing the frames to be of the ADM type (2.7, 2.8) the hatted partial
derivatives take the form

$\hat{\partial}_{t}=N^{-1}(\partial_{t}-N_{i}\partial_{i})\,,\qquad\hat{\partial}_{i}=\partial_{i}\,.$
(2.37)

The hatted time derivative, which in this case is hypersurface orthogonal,
includes the well known shift by the vector field $N^{i}$ (see e.g. [22]),
while the hatted spatial derivatives are simply partial derivatives.

The connection components (2.21-2.24) become in this case

$\displaystyle K_{ij}$
$\displaystyle=N^{-1}\left(\frac{1}{2}\partial_{t}h_{ij}-D_{(i}N_{j)}\right)\,,$
(2.38) $\displaystyle\Omega_{i}$ $\displaystyle=N^{-1}\partial_{i}N\,,$ (2.39)
$\displaystyle\Delta_{i}^{j}$ $\displaystyle=N^{-1}\partial_{i}N^{j}\,,$
(2.40) $\displaystyle\hat{\Gamma}_{ij}^{k}$ $\displaystyle=\Gamma_{ij}^{k}\,.$
(2.41)

First one recognizes in $K_{ij}$ the extrinsic curvature of the induced metric
$h_{ij}$, while $\Omega_{i}$ is the so called Eulerian acceleration vector
(see e.g. [22]). Secondly one sees that the connection $\hat{\Gamma}_{ij}^{k}$
in this case equals the Levi-Civita connection (2.31), so that
$\hat{D}_{i}=D_{i}$. Finally the object $\Delta_{i}^{j}$ plays a role as a
’connection’ in the time-like covariant derivative (2.27). Writing that out
explicitly one gets the expression

$\hat{D}_{t}T^{j_{1}\ldots j_{m}}_{k_{1}\ldots
k_{n}}=N^{-1}(\partial_{t}-L_{N})T^{j_{1}\ldots j_{m}}_{k_{1}\ldots k_{n}}\,,$
(2.42)

where $L_{N}$ is the Lie derivative with respect to the vector field $N^{i}$.
Remark that this also allows one to rewrite (2.38) as

$K_{ij}=\frac{1}{2}\hat{D}_{t}h_{ij}\,.$ (2.43)

####  2.3.2 The KS case

Now we assume the frames to take the KS form (2.11, 2.12). The hatted partial
derivatives now are

$\hat{\partial}_{t}=M^{-1}\partial_{t}\,,\qquad\hat{\partial}_{i}=\partial_{i}-C_{i}\partial_{t}\,.$
(2.44)

The situation is exactly opposite – or dual – to the ADM case (2.37), in that
the hatted time derivative is simply (up to a prefactor) the partial
derivative, while it is the spatial hatted derivatives that get a shift, this
time by the one-form $C_{i}$.

When writing out the objects (2.21-2.24) one finds they are naturally
expressed in terms of these hatted partial derivatives444This is also true in
the ADM case, as there $\partial_{i}=\hat{\partial}_{i}$ and (2.38) can be re-
expressed as (2.43).:

$\displaystyle K_{ij}$
$\displaystyle=\frac{1}{2}\hat{\partial}_{t}h_{ij}-M\hat{\partial}_{[i}C_{j]}\,,$
(2.45) $\displaystyle\Omega_{i}$
$\displaystyle=M^{-1}\hat{\partial}_{i}M-M\hat{\partial}_{t}C_{i}\,,$ (2.46)
$\displaystyle\Delta_{i}^{j}$ $\displaystyle=0\,,$ (2.47)
$\displaystyle\hat{\Gamma}_{ij}^{k}$
$\displaystyle=\frac{1}{2}h^{kl}\left(\hat{\partial}_{i}h_{jl}+\hat{\partial}_{j}h_{il}-\hat{\partial}_{l}h_{ij}\right)\,.$
(2.48)

It might appear as if the introduction of the hatted partial derivatives is
simply a notation to simplify the expressions (2.45-2.48). But, as we will
discuss in section 3, these hatted derivatives are the natural ones invariant
under redefinitions of the time coordinate, which is why all invariant
quantities involving derivatives can be expressed in terms of them.

Let us now come to the expressions (2.45-2.48) themselves. While the symmetric
part of $K_{ij}$ is analogous to the ADM expression (2.43) (see also (2.49)),
it no longer has an interpretation as extrinsic curvature. More importantly,
$K_{ij}$ now also has a non-vanishing anti-symmetric part (absent in the ADM
case), which takes the form of a (generalized) curvature for the 1-form
potential $C_{i}$. The vector $\Omega_{i}$ can also in this KS case be
interpreted as an ’acceleration vector’, in that it provides a relativistic
version of Newton’s gravitational field vector (see sections 3.3). Since
$\Delta^{i}_{j}$ vanishes we can conclude that

$\hat{D}_{t}=\hat{\partial}_{t}\,.$ (2.49)

On the contrary, $\hat{\Gamma}_{ij}^{k}$ is in this case not the Levi-Civita
connection, rather

$\hat{\Gamma}_{ij}^{k}=\Gamma_{ij}^{k}+\Delta_{ij}^{k},\qquad\Delta_{ij}^{k}=-\frac{1}{2}h^{kl}\left(C_{i}\partial_{t}h_{jl}+C_{j}\partial_{t}h_{il}-C_{l}\partial_{t}h_{ij}\right)\,.$
(2.50)

As mentioned before, this implies that in the KS case, the spatial covariant
derivative (2.28) is not the standard one, in that $\hat{\partial}_{i}$ has a
shift proportional to $\partial_{t}$, see (2.44), and the connection
coefficients have the extra contribution $\Delta_{ij}^{k}$. When discussing
general relativity in KS form the expression (2.28), together with (2.44) and
(2.48) is most convenient to work with.

###  2.4 Decomposition of the Einstein-Hilbert action

To rewrite the Einstein-Hilbert action in $d$+1 form one needs to decompose
the Ricci scalar. This can be done by first expressing the Ricci tensor as
$R_{\mu\nu}=-2(\nabla_{[\rho}\nabla_{\nu]}u^{\rho})n_{\mu}+2(\nabla_{[\rho}\nabla_{\nu]}e^{\rho}_{i})e^{i}_{\mu}$
and re-expressing the covariant derivatives in terms of (2.21-2.24). Upon
contraction with $g^{\mu\nu}$ one finds555Note that there a priory is an extra
term proportional to $h^{ij}K_{[kj]}\Delta_{i}^{k}$, but this vanishes in both
the ADM and KS case, see (2.39, 2.40) and (2.46, 2.47) respectively.

$R[g]=\hat{R}+K_{ij}K^{ij}-K^{2}+2\left(\hat{D}_{t}K+K^{2}-\hat{D}_{i}\Omega^{i}-\Omega_{i}\Omega^{i}\right)\,,$
(2.51)

where

$K=h^{ij}K_{ij}\,,\quad\hat{R}=h^{ij}\hat{R}_{ij}\quad\mbox{and}\quad\hat{R}_{ij}=\hat{\partial}_{k}\hat{\Gamma}_{ij}^{k}-\hat{\partial}_{j}\hat{\Gamma}_{ik}^{k}+\hat{\Gamma}_{lk}^{l}\hat{\Gamma}_{ij}^{k}-\hat{\Gamma}_{ik}^{l}\hat{\Gamma}_{lj}^{k}\,.$
(2.52)

Via (2.34) one recognizes the term in between brackets in (2.51) as a total
derivative so that666We assume boundary conditions so that the total
derivative term can be ignored. Analyzing that more thoroughly could be
interesting.

$S_{\mathrm{EH}}=\int
R[g]\,\sqrt{-g}\,d^{d+1}x=\int(\hat{R}+K_{ij}K^{ij}-K^{2})\,n_{t}dt\,\sqrt{h}\,d^{d}x\,.$
(2.53)

It is gratifying to see that the Einstein-Hilbert action takes a simple and
universal form for both ADM and KS type decompositions.

####  2.4.1 The ADM Lagrangian

The ADM Lagrangian [21] is well-known:

${\cal L}_{\mathrm{ADM}}=N\sqrt{h}\left(R+K_{ij}K^{ij}-K^{2}\right)\,.$ (2.54)

Indeed it equals (2.53) when one takes the frames of the ADM form (2.7, 2.8),
since then $n_{t}=N$ and – as discussed in the previous subsection –
$\hat{\partial}_{i}=\partial_{i},\hat{\Gamma}_{ij}^{k}=\Gamma_{ij}^{k}$, so
that $\hat{R}=R$, the Ricci scalar of the Levi-Civita connection of $h_{ij}$.
Furthermore $K_{ij}$ has the standard ADM form (2.38).

####  2.4.2 The KS Lagrangian

Since for the KS frames (2.11, 2.12) $n_{t}=M$ we recover from (2.53) the KS
Lagrangian [26]:

${\cal L}_{\mathrm{KS}}=M\sqrt{h}\left(\hat{R}+K_{ij}K^{ij}-K^{2}\right)\,.$
(2.55)

Although this expression might appear almost identical to the ADM Lagrangian
(2.54), it is rather different. This is since now, via (2.44) and (2.50),
$\hat{R}$ has a number of additional contributions apart from $R$, the Ricci
tensor of the Levi-Civita connection of $h_{ij}$. Secondly the tensor $K_{ij}$
is also different from its ADM analog, see (2.45) versus (2.38), in particular
it contains a non-trivial anti-symmetric part. When we write out the time
derivatives explicitly in section 3.4, the Lagrangian (2.55) becomes (3.45,
3.57) and one sees some of the complexity that is elegantly packaged in the
form (2.55).

##  3 Dynamics and symmetries in the KS formulation

General relativity in the ADM formulation has been widely discussed. This is
much less so for the KS formulation, so in this section we shortly discuss the
Einstein equations in the KS formulation, as well as the diffeomorphism
invariance of the theory. This means that in this section all definitions are
those of the KS subsections of the previous section, i.e. the key definitions
used here are (2.11-2.13), (2.44-2.48) and (2.55).

###  3.1 Einstein equations

The Einstein equations in KS form can be derived in two parallel and
equivalent ways. The first would be to decompose the $d+1$-covariant Einstein
equations using the frames $(\ref{basexp2},\ref{basexp2a})$, the second to
vary the KS Lagrangian with respect to the fields $h_{ij},C_{i},M$. Let us
make the link between these two approaches explicit. The discussion in section
2.4 showed that

${\cal L}_{\mathrm{EH}}={\cal L}_{\mathrm{KS}}+\partial_{\mu}\Theta^{\mu}\,.$
(3.1)

Via the KS decomposition of the metric (2.13) and the definition of the KS
frames $(\ref{basexp2},\ref{basexp2a})$ it follows that

$\delta g_{\mu\nu}=-2M^{-1}\delta Mn_{\mu}n_{\nu}+2M\delta
C_{i}n_{(\mu}e_{\nu)}^{i}+e_{\mu}^{i}e_{\nu}^{j}\delta h_{ij}\,.$ (3.2)

This implies that if we define

$\delta{\cal L}_{\mathrm{EH}}=\delta{\cal
L}_{\mathrm{KS}}+\partial_{\mu}\delta\Theta^{\mu}=\sqrt{h}M\left(2G^{0}M^{-1}\delta
M-2G^{i}M\delta C_{i}-G^{ij}\delta
h_{ij}\right)+\partial_{\mu}\theta^{\mu}\,,$ (3.3)

one gets the relations

$G^{0}=G^{\mu\nu}n_{\mu}n_{\nu}\,,\qquad
G^{i}=G^{\mu\nu}n_{(\mu}e_{\nu)}^{i}\,,\qquad
G^{ij}=G^{\mu\nu}e_{\mu}^{i}e_{\nu}^{j}\,.$ (3.4)

Here $G^{\mu\nu}=-\frac{1}{\sqrt{-g}}\frac{\delta{\cal
L}_{\mathrm{EH}}}{\delta g_{\mu\nu}}$ is the Einstein tensor of the metric
$g_{\mu\nu}$. In summary we see that indeed there are two ways to compute the
$G$’s: either via (3.4), or via (3.3) by a variation of ${\cal
L}_{\mathrm{KS}}$. One checks that both calculations match, with the result

$\displaystyle G^{0}=$
$\displaystyle\,\frac{1}{2}\left(\hat{R}+3K_{[ij]}K^{ij}-K_{(ij)}K^{ij}+K^{2}\right)\,,$
(3.5) $\displaystyle G^{i}=$
$\displaystyle\,\hat{D}_{j}(K^{ij}-h^{ij}K)+2K^{[ij]}\Omega_{j}\,,$ (3.6)
$\displaystyle G^{ij}=$
$\displaystyle\,\hat{G}^{ij}-2K^{k(i}K^{j)}{}_{k}+KK^{(ij)}+h^{ik}h^{jl}\hat{\partial}_{t}K_{(kl)}-\hat{D}^{(i}\Omega^{j)}-\Omega^{i}\Omega^{j}$
$\displaystyle+\frac{1}{2}h^{ij}\left(3K_{(kl)}K^{kl}-K_{[ij]}K^{ij}-K^{2}-2h^{kl}\hat{\partial}_{t}K_{kl}+2\hat{D}_{i}\Omega^{i}+2\Omega_{i}\Omega^{i}\right)\,.$
(3.7)

In the above $\hat{G}^{ij}=\hat{R}^{ij}-\frac{1}{2}h^{ij}\hat{R}$. The vacuum
Einstein equations are thus equivalent to putting (3.5-3.7) to zero. It is
interesting to point out that $M$ appears algebraically in ${\cal
L}_{\mathrm{KS}}$ (similarly to $N$ in the ADM formulation), so that $G^{0}=0$
has the interpretation of a constraint. Another consequence is that the
equations do not contain a second time derivative of $M$ (while the second
time derivative of both $h_{ij}$ and $C_{i}$ does appear). One obtains an
equation for the second spatial derivatives of $M$, by taking the trace of
$G^{ij}$, we will discuss that equation a bit more in section 3.3.

One can couple matter to the theory by introducing a matter Lagrangian ${\cal
L}_{\mathrm{mat}}$. This then allows to define the KS energy-momentum tensors
$T$:

$\delta{\cal L}_{\mathrm{mat}}=\sqrt{h}M\left(-2T^{0}M^{-1}\delta
M+2T^{i}M\delta C_{i}+T^{ij}\delta h_{ij}\right)\,.$ (3.8)

The Einstein equations then take the form

$G^{0}=T^{0}\,,\qquad G^{i}=T^{i}\,,\qquad G^{ij}=T^{ij}\,.$ (3.9)

Apart from computing the $T$’s by a variation of the matter Lagrangian wrt
$M,C_{i}$ and $h_{ij}$, one can also obtain them via a decomposition of the
usual energy momentum tensor $T^{\mu\nu}=\frac{1}{\sqrt{-g}}\frac{\delta{\cal
L}_{\mathrm{mat}}}{\delta g_{\mu\nu}}$. Via (3.2) one finds

$T^{0}=T^{\mu\nu}n_{\mu}n_{\nu}\,,\qquad
T^{i}=T^{\mu\nu}n_{(\mu}e_{\nu)}^{i}\,,\qquad
T^{ij}=T^{\mu\nu}e_{\mu}^{i}e_{\nu}^{j}\,.$ (3.10)

By comparing (3.10) to (3.4) the equations (3.9) then follow immediately. One
advantage of the form (3.10) is that it can also be used in cases where the
matter has no Lagrangian description and is only defined in terms of an energy
momentum tensor $T^{\mu\nu}$. Clearly the conservation of energy momentum
$\nabla_{\mu}T^{\mu\nu}=0$ will become equivalent to some equations for the
$T$’s. These equations, and their origin in diffeomorphism invariance, will be
discussed in the next subsection.

###  3.2 Diffeomorphism symmetries and Noether identities

Because a choice of KS coordinates amounts to a choice of preferred time
coordinate, it breaks manifest diffeomorphism invariance. Since they leave the
time coordinate invariant, spatial diffeomorphisms, i.e.
$\tilde{x}^{i}(x^{j})$, do remain manifest. But more general coordinate
transformations map one choice of KS coordinates into another; this will leave
the theory invariant, but in a less manifest way. One reason to have a closer
look at diffeomorphism invariance in the KS formulation is that via the
associated Noether identities it allows to find the equivalent of the Bianchi
identity and energy-momentum conservation of the standard covariant
formulation. Of course these equations can also be obtained in more direct
fashion via the decompositions (3.4, 3.10). A second reason is that invariance
under these symmetries will explain much of the structure of the Lagrangian
and the modified derivatives appearing.

It will be useful to split the $d+1$ dimensional diffeomorphisms into two
classes, which we will refer to as time redefinitions and time-dependent
spatial diffeomorphisms. This split originates in the split of an
infinitesimal diffeomorphism $\xi^{\mu}$ into $\Lambda=\xi^{\mu}n_{\mu}$ and
$\xi^{i}=\xi^{\mu}e_{\mu}^{i}$, via the KS frames (2.11, 2.12).

Although it is customary to discuss diffeomorphisms in the passive
formulation, where one keeps the coordinates fixed and the action on tensors
is via the Lie derivative, we find in this case the active formulation to be
more insightful. For convenience we remind the reader of the relation between
infinitesimal passive and active transformations and their action on an
arbitrary tensor $S$:

$\displaystyle\\!\\!\\!\\!\\!\delta_{\mathrm{A}}x^{\mu}$
$\displaystyle=\tilde{x}^{\mu}-x^{\mu}=-\xi^{\mu}$
$\displaystyle\delta_{\mathrm{A}}S^{\mu}_{\nu}$
$\displaystyle=\tilde{S}^{\mu}_{\nu}(\tilde{x})-S^{\mu}_{\nu}(x)=-\partial_{\lambda}\xi^{\mu}S^{\lambda}_{\nu}+\partial_{\nu}\xi^{\lambda}S^{\mu}_{\lambda}\,,$
(3.11) $\displaystyle\\!\\!\\!\delta_{\mathrm{P}}x^{\mu}$ $\displaystyle=0$
$\displaystyle\delta_{\mathrm{P}}S^{\mu}_{\nu}$
$\displaystyle=\tilde{S}^{\mu}_{\nu}(x)-S^{\mu}_{\nu}(x)=\delta_{A}S^{\mu}_{\nu}-\partial_{\lambda}S^{\mu}_{\nu}\delta_{\mathrm{A}}x^{\lambda}=L_{\xi}S^{\mu}_{\nu}.$
(3.12)

It might be relevant to note how derivatives transform differently, i.e.
$\delta_{\mathrm{P}}\partial_{\mu}=0$, so that
$\delta_{\mathrm{P}}\partial_{\lambda}S^{\mu}_{\nu}=\partial_{\lambda}\delta_{\mathrm{P}}S^{\mu}_{\nu}$,
but $\delta_{\mathrm{A}}\partial_{\mu}=\partial_{\mu}\xi^{\nu}\partial_{\nu}$
so that

$\delta_{\mathrm{A}}\partial_{\lambda}S^{\mu}_{\nu}=\partial_{\lambda}\delta_{\mathrm{A}}S^{\mu}_{\nu}+\partial_{\lambda}\xi^{\rho}\partial_{\rho}S^{\mu}_{\nu}\,.$
(3.13)

The above generalizes straightforwardly to tensors with an arbitrary number of
upper and lower indices. In the remainder of this section all infinitesimal
transformations will be active and we drop the A subscript.

####  3.2.1 Time redefinitions

First we consider redefinitions of the coordinate $t$, i.e. diffeomorphisms of
the form $\xi^{\mu}n_{\mu}=-M\Lambda$, $\xi^{\mu}e_{\mu}^{i}=0$. The
coordinates transform as

$\displaystyle\delta_{\Lambda}t$ $\displaystyle=-\Lambda(x,t)\,,$ (3.14)
$\displaystyle\delta_{\Lambda}x^{i}$ $\displaystyle=0\,.$ (3.15)

Demanding the line element $ds^{2}$ – see (2.13) – to be form invariant, leads
to the transformations

$\displaystyle\delta_{\Lambda}(-M^{-1})$
$\displaystyle=\hat{\partial}_{t}\Lambda\,,$ (3.16)
$\displaystyle\delta_{\Lambda}C_{i}$
$\displaystyle=\hat{\partial}_{i}\Lambda\,,$ (3.17)
$\displaystyle\delta_{\Lambda}h_{ij}$ $\displaystyle=0\,.$ (3.18)

Note the strong similarity to the U(1) gauge transformations of Maxwell
theory, as this will provide some intuition in the discussion of invariants
below. We should stress however that this is but a similarity rather than a
real equality, since contrary to standard U(1) gauge transformations here also
the coordinates transform, the transformations are not abelian and it is the
hatted rather than standard partial derivatives that appear in (3.16-3.18).

A short computation reveals that the following objects are invariant under
these time redefinitions, i.e. $\delta_{\Lambda}X=0$, for $X$ any of

$h_{ij}\,,\ \hat{\partial}_{t}\,,\ \hat{\partial}_{i}\,,\ K_{[ij]}\,,\
K_{(ij)}\,,\ \Omega_{i}\,,\ \hat{\Gamma}_{ij}^{k}\,,\ \hat{R}_{ij}\,,\
Mdtd^{d}x\,.$ (3.19)

First of all, one sees that although the partial derivatives are not
invariant777Remember we are discussing active transformations., their hatted
versions are. So we see that the appearance of these hatted partial
derivatives is no coincidence, but fully dictated by the invariance under time
redefinitions. Since $\hat{\Gamma}_{ij}^{k}$ is built out of $h_{ij}$ and
$\hat{\partial}_{i}$, it also follows that it is invariant, which makes it the
natural connection to use. Similarly it is $\hat{R}_{ij}$, the modified Ricci
tensor which is invariant rather than the standard Ricci tensor $R_{ij}$. The
symmetric part $K_{(ij)}=\frac{1}{2}\hat{\partial}_{t}h_{ij}$ is manifestly
invariant while the anti-symmetric part takes a form similar to a U(1)
curvature tensor $K_{[ij]}=-M\hat{\partial}_{[i}C_{j]}$.

Given the list of invariants (3.19), the KS Lagrangian (2.55) is manifestly
invariant. The same is true for the equations of motion by inspection of
(3.5-3.7). More generally speaking, time redefinition invariance guarantees
that $M$, $C_{i}$ and derivatives can only appear through the objects (3.19).

####  3.2.2 Time dependent spatial diffeomorphisms

A second set of diffeomorphisms are those for which the generating vector
field satisfies $\xi^{\mu}n_{\mu}=0$. The remaining components,
$\xi^{i}=e^{i}_{\mu}\xi^{\mu}$, define a spatial coordinate transformation,
but one that is time dependent. Explicitly

$\displaystyle\delta_{\xi}t$ $\displaystyle=0\,,$ (3.20)
$\displaystyle\delta_{\xi}x^{i}$ $\displaystyle=-\xi^{i}(x,t)\,.$ (3.21)

It will be useful to introduce a separate notation for the time derivative of
this vector field:

$\hat{f}_{i}=h_{ij}\hat{\partial}_{t}\xi^{j}\,.\\\ $ (3.22)

Invariance of $ds^{2}$ then defines the transformations of the fields:

$\displaystyle\delta_{\xi}(-M)^{-1}$ $\displaystyle=C_{i}\hat{f}^{i}\,,$
(3.23) $\displaystyle\delta_{\xi}C_{i}$ $\displaystyle={\cal
C}_{\xi}C_{i}-M^{-1}\hat{f}_{i}\,,$ (3.24) $\displaystyle\delta_{\xi}h_{ij}$
$\displaystyle={\cal C}_{\xi}h_{ij}\,.$ (3.25)

Here we introduced the following linear operation on tensor components:

$\displaystyle{\cal C}_{\xi}S^{i_{1}\ldots i_{n}}_{j_{1}\ldots j_{m}}=$
$\displaystyle-S^{ki_{2}\ldots i_{n}}_{j_{1}\ldots
j_{m}}\hat{\partial}_{k}\xi^{i_{1}}-\ldots-S^{i_{1}\ldots
i_{n-1}k}_{j_{1}\ldots j_{m}}\hat{\partial}_{k}\xi^{i_{n}}$
$\displaystyle+S^{i_{1}\ldots i_{n}}_{kj_{2}\ldots
j_{m}}\hat{\partial}_{j_{1}}\xi^{k}+\ldots+S^{i_{1}\ldots i_{n}}_{j_{1}\ldots
j_{m-1}k}\hat{\partial}_{j_{m}}\xi^{k}\,.$ (3.26)

The introduction of ${\cal C}_{\xi}$ is useful when performing variations of
objects build out of the basic fields, since this operation is nicely
compatible with tensor multiplication and contraction. E.g.

${\cal C}_{\xi}(T^{i_{1}\ldots i_{n}}_{j_{1}\ldots j_{m}}S^{j_{1}\ldots
j_{m}}_{k_{1}\ldots k_{p}})={\cal C}_{\xi}(T^{i_{1}\ldots i_{n}}_{j_{1}\ldots
j_{m}}){\cal C}_{\xi}(S^{j_{1}\ldots j_{m}}_{k_{1}\ldots k_{p}})\,.$ (3.27)

The KS action (2.55) is not manifestly invariant under the transformations
(3.23-3.25), a somewhat tedious calculation reveals that

$\delta_{\xi}\left[Mdt\sqrt{h}d^{d}x\left(\hat{R}+K_{ij}K^{ij}-K^{2}\right)\right]=4Mdt\sqrt{h}d^{d}x\big{(}\hat{D}_{i}+\Omega_{i}\big{)}\left(\hat{f}_{j}K^{ji}-\hat{f}^{i}K\right).$
(3.28)

It is then important to observe that

$(\hat{D}_{i}+\Omega_{i})S^{[ij_{1}\ldots
j_{n}]}=\frac{1}{M\sqrt{h}}\partial_{i}\left(M\sqrt{h}S^{[ij_{1}\ldots
j_{n}]}\right)-\frac{1}{M\sqrt{h}}\partial_{t}\left(M\sqrt{h}\,C_{i}S^{[ij_{1}\ldots
j_{n}]}\right)$ (3.29)

which implies

$\delta_{\xi}S_{\mathrm{KS}}=\int
dtd^{d}x\left[\partial_{i}\left(M\sqrt{h}(\hat{f}_{j}K^{ji}-\hat{f}^{i}K)\right)-\partial_{t}\left(M\sqrt{h}C_{i}(\hat{f}_{j}K^{ji}-\hat{f}^{i}K)\right)\right]=0$
(3.30)

This explicitly confirms that indeed general relativity in KS form is
invariant also under time-dependent spatial diffeomorphisms, albeit not
manifestly so. We stress that this is nothing but a consistency check, since
the KS action is identical to the Einstein-Hilbert action and the time-
dependent spatial diffeomorphisms are nothing but a subclass of $d+1$
dimensional diffeomorphisms, so invariance is guaranteed by construction.

####  3.2.3 Noether identities

In any theory with a gauge or local symmetry the equations of motion are not
all independent and the (differential) relations between them go under the
name of Noether identities (sometimes also called classical Ward identities),
see e.g. [30] for a pedagogic introduction. In the case of general relativity
the Noether identities amount to the Bianchi identity for the Einstein tensor
and the conservation equation for the energy-momentum tensor.

Given a local symmetry of the form888Also here we take the active point of
view

$\delta
x^{\mu}=H^{\mu}\lambda\,,\qquad\delta\phi=F(\phi,\partial\phi)\lambda+F^{\mu}(\phi)\partial_{\mu}\lambda\,,$
(3.31)

the associated Noether identity is

$(F-H^{\mu}\partial_{\mu}\phi){\cal E}=\partial_{\mu}(F^{\mu}{\cal E})\,.$
(3.32)

Here ${\cal E}$ are the Euler-Lagrange equations, defined through a generic
variation of the Lagrangian as

$\delta{\cal L}=-{\cal E}\delta\phi+\partial_{\mu}\theta^{\mu}\,.$ (3.33)

We can then apply this to the Lagrangian ${\cal L}_{\mathrm{KS}}(M,C,h)$
(2.55) both for the time redefinitions and time dependent spatial
diffeomorphisms discussed above. The resulting Noether identities are
respectively equivalent to

$\displaystyle\hat{D}_{i}G^{i}+2\Omega_{i}G^{i}-\hat{\partial}_{t}G^{0}-KG^{0}-K_{ij}G^{ij}$
$\displaystyle=0\,,$ (3.34)
$\displaystyle\hat{D}_{j}G^{ji}+\Omega_{j}G^{ji}-\hat{\partial}_{t}G^{i}-2G^{j}K_{j}{}^{i}-KG^{i}+\Omega^{i}G^{0}$
$\displaystyle=0\,.$ (3.35)

Of course the same equations can also be obtained by decomposing the Bianchi
identity $\nabla_{\mu}G^{\mu\nu}=0$ via the methods of the previous section.
Through the Einstein equations in the form (3.9) one then directly obtains the
energy-momentum conservation equations:

$\displaystyle\hat{D}_{i}T^{i}+2\Omega_{i}T^{i}-\hat{\partial}_{t}T^{0}-KT^{0}-K_{ij}T^{ij}$
$\displaystyle=0\,,$ (3.36)
$\displaystyle\hat{D}_{j}T^{ji}+\Omega_{j}T^{ji}+\Omega^{i}T^{0}-\hat{\partial}_{t}T^{i}-2T^{j}K_{j}{}^{i}-KT^{i}$
$\displaystyle=0\,.$ (3.37)

###  3.3 Conformal redefinition and the relativistic Poisson equation

The field content of GR in KS formulation, as we introduced it above, is the
triple $(M,C_{i},h_{ij})$. But since $M$ appears algebraically in the
Lagrangian its equation of motion, eqn (3.5), is a constraint and does not
involve any derivatives on $M$. Second derivatives on $M$ do appear in eqn
(3.7) but this is a tensorial equation instead of a scalar equation. This
suggests that $M$ is not a relativistic analog of the Newtonian potential, nor
is (3.5) the analog of the Poisson equation. Indeed in a non-relativistic
expansion, see section 5 or [26, 11], it is the combination
$G^{0}+G^{ij}h_{ij}$ that becomes the Newtonian Poisson equation. This
particular linear combination appears naturally as the equation of motion
$E^{\psi}$ of a scalar $\psi$ introduced through the field redefinition999Let
us point out that from here onward in the paper we set the number of spatial
dimensions $d=3$.

$M=e^{\frac{\psi}{2}}\,,\qquad h_{ij}=e^{-\psi}\gamma_{ij}\,.$ (3.38)

Indeed, defining101010Throughout the remainder of the paper we will use the
notation ${\cal L}=\sqrt{\gamma}L$ and ${\cal E}=\sqrt{\gamma}E$ to
distinguish between tensorial quantities and the associated densities.

$\delta{\cal
L}_{\mathrm{KS}}=\sqrt{\gamma}\left(E^{\psi}\delta\psi-2E^{i}\delta
C_{i}-E^{ij}\delta\gamma_{ij}\right)+\partial_{\mu}\tilde{\theta}^{\mu}\,,$
(3.39)

a comparison to (3.3) reveals that

$E^{\psi}=e^{-\psi}(G^{0}+h_{ij}G^{ij}),\qquad E^{i}=e^{-\psi/2}G^{i},\qquad
E^{ij}=e^{-2\psi}G^{ij}\,.$ (3.40)

This suggests $\psi$ is a natural relativistic analog of the Newtonian
potential, and indeed this change of variables has proven very fruitful in the
nonrelativistic expansion; this was the key insight of [27] that motivated the
KS formulation of [26] and it also appeared in the $1/c$ expansion in [11,
18].

Expressed in terms of the alternative fields (3.38), and upon dropping some
total derivative terms, the KS action (2.55) becomes

$S_{\mathrm{KS}}=\int
dtd^{3}x\sqrt{\gamma}\Big{(}\hat{R}+e^{-2\psi}(\tilde{K}_{(ij)}\tilde{K}^{ij}-\tilde{K}^{2})+e^{2\psi}\tilde{K}_{[ij]}\tilde{K}^{ij}-\frac{1}{2}\hat{\partial}_{i}\psi\hat{\partial}^{i}\psi-2\tilde{\Omega}^{i}\hat{\partial}_{i}\psi\Big{)}\,,$
(3.41)

where now

$\hat{R}=\hat{R}[\gamma]=\gamma^{ij}\hat{R}_{ij}[\gamma]\quad\mbox{and}\quad\hat{R}_{ij}[\gamma]=\hat{\partial}_{k}\hat{\Gamma}_{ij}^{k}[\gamma]-\hat{\partial}_{j}\hat{\Gamma}_{ik}^{k}[\gamma]+\hat{\Gamma}_{lk}^{l}[\gamma]\hat{\Gamma}_{ij}^{k}[\gamma]-\hat{\Gamma}_{ik}^{l}[\gamma]\hat{\Gamma}_{lj}^{k}[\gamma]\,,$
(3.42)

with $\Gamma_{ij}^{k}[\gamma]$ the Levi-Civita connection of $\gamma_{ij}$.
Furthermore

$\tilde{K}_{ij}=e^{\psi}K_{ij}=\frac{1}{2}(\partial_{t}\gamma_{ij}-\gamma_{ij}\partial_{t}\psi)-\hat{\partial}_{[i}C_{j]}\,,\qquad\tilde{\Omega}_{i}=-\partial_{t}C_{i}\,.$
(3.43)

Note that if one interprets $\psi$ as the relativistic analog of the Newtonian
potential then
$\Omega^{i}=\frac{1}{2}\hat{\partial}^{i}\psi+\tilde{\Omega}^{i}$ is a natural
analog of the gravitational force vector.

###  3.4 Making time derivatives explicit

General relativity in the KS formulation appears – at least to us – most
elegant in the form (2.55), which is simple and very similar to the ADM form.
In the previous subsection we made a change of variables that is more suitable
for comparison to Newtonian gravity. This introduces a few extra terms in the
Lagrangian, see (3.41), but still it retains some of its elegance due to the
use of the quantities $\hat{R}$, $\tilde{K}$ and the hatted partial derivative
$\hat{\partial}_{i}$ (see (2.44)). These quantities each contain ’hidden’ time
derivatives, which we would now like to make explicit. This is motivated by
our aim, in sections 4 and 5, to make an expansion in (inverse) powers of $c$,
which naturally accompany each time derivative. The key point in this
subsection is that in particular the time dependence in the term
$\sqrt{\gamma}\hat{R}$ can be simplified quite a lot by discarding a total
derivative term.

We start by introducing the speed of light $c$ in the original relativistic
metric (2.13), which together with the redefinition (3.38) is then of the form

$ds^{2}=-e^{\psi}(c\,dt+C_{i}dx^{i})^{2}+e^{-\psi}\gamma_{ij}dx^{i}dx^{j}\,.$
(3.44)

Since apart from an overall factor in front of the action this is the only
place $c$ appears in the purely metric sector of the theory it implies all
factors of $c$ can always be re-absorbed by simply rescaling $t$. Equivalently
this implies that each time derivative has to be accompanied by a power
$c^{-1}$. Since the Lagrangian is maximally second order in derivatives it
implies we can write

${\cal L}=\sqrt{\gamma}\left(L_{0}+c^{-1}L_{1}+c^{-2}L_{2}\right)\,,$ (3.45)

with $L_{0}$ containing no time derivatives, $L_{1}$ being first order in time
derivatives and $L_{2}$ second order. We will discard total derivatives such
that $L_{1}$ and $L_{2}$ will be quadratic in derivatives rather than contain
second derivatives. $L_{0}$ will still contain second spatial derivatives. One
can think of $L_{2}$ as the kinetic term, of $L_{1}$ as a Lorentz coupling to
velocity and of $L_{0}$ as the potential term.

To simplify notation we will introduce a dot notation for time derivatives,
i.e. $\dot{{}}=\partial_{t}$ . We make the important conventional choice that
the time derivative will always act on tensors with lower indices. This
implies that if one sees an object with upper indices and a dot, one should
first lower the indices before interpreting the dot as a time derivative! For
example:

$\dot{C}_{i}=\partial_{t}C_{i}\ \mbox{ and }\
\dot{C}^{i}=\gamma^{ij}\dot{C}_{j}=\gamma^{ij}\partial_{t}C_{j}\,,\
\mbox{while }\
\partial_{t}C^{i}=(\partial_{t}\gamma^{ij})C_{j}+\dot{C}^{i}\neq\dot{C}^{i}\,,$
(3.46)

or

$\dot{\gamma}_{ij}=\partial_{t}\gamma_{ij}\ \mbox{ and }\
\dot{\gamma}^{ij}=\gamma^{ik}\gamma^{kl}\dot{\gamma}_{ij}=\gamma^{ik}\gamma^{kl}\partial_{t}\gamma_{ij}\,,\mbox{
while }\ \partial_{t}\gamma^{ij}=-\dot{\gamma}^{ij}\neq\dot{\gamma}^{ij}\,.$
(3.47)

Note that under our convention we also have

$\dot{\gamma}=\gamma^{ij}\dot{\gamma}_{ij}=\gamma^{ij}\partial_{t}\gamma_{ij}=\partial_{t}\log\det(\gamma_{ij})\,,$
(3.48)

One could proceed by simply inserting (3.44) into the Einstein Hilbert action,
but we can use the results of the previous subsections to start from (3.41)
instead. First we introduce the appropriate factors of $c$ and split all the
objects appearing in (3.41) in parts containing equal powers of $c$:

$\displaystyle\tilde{K}_{(ij)}$
$\displaystyle=\frac{c^{-1}}{2}(\dot{\gamma}_{ij}-\gamma_{ij}\dot{\psi})\,,$
(3.49) $\displaystyle\tilde{K}_{[ij]}$
$\displaystyle=-\frac{1}{2}C_{ij}+c^{-1}C_{[i}\dot{C}_{j]},\qquad\quad
C_{ij}=\partial_{i}C_{j}-\partial_{j}C_{i}\,,$ (3.50)
$\displaystyle\tilde{\Omega}_{i}$ $\displaystyle=-c^{-1}\dot{C}_{i}\,,$ (3.51)
$\displaystyle\hat{\partial}_{i}\psi$
$\displaystyle=\partial_{i}\psi-c^{-1}C_{i}\dot{\psi}\,.$ (3.52)

In rewriting $\hat{R}[\gamma]$ it is useful to isolate a total derivative (the
second line below), which will drop out of the action. One calculates that

$\displaystyle\hat{R}=$
$\displaystyle\,R+c^{-1}{\Gamma}^{ij\,kl}\dot{\gamma}_{ij}D_{k}C_{l}+c^{-2}\left({\Gamma}^{ij\,kl}\dot{C}_{i}C_{j}\dot{\gamma}_{kl}-\frac{1}{4}{\Gamma}^{ij\,kl\,mn}C_{i}C_{j}\dot{\gamma}_{kl}\dot{\gamma}_{mn}\right)$
$\displaystyle-2c^{-1}D_{i}\left({\Gamma}^{ij\,kl}C_{j}\dot{\gamma}_{kl}\right)+c^{-2}D_{t}\left({\Gamma}^{ij\,kl}C_{i}C_{j}\dot{\gamma}_{kl}\right)\,,$
(3.53)

where $D_{i}$ and $R$ are defined with respect to the Levi-Civita connection
of $\gamma_{ij}$ and we additionally introduced

$\displaystyle D_{t}\cdot$
$\displaystyle=\frac{1}{\sqrt{\gamma}}\partial_{t}(\sqrt{\gamma}\,\cdot)=\partial_{t}\cdot+\frac{1}{2}\dot{\gamma}\cdot\,,$
(3.54) $\displaystyle{\Gamma}^{ij\,kl}$
$\displaystyle=-\gamma^{ij}\gamma^{kl}+\gamma^{il}\gamma^{kj}\,,$ (3.55)
$\displaystyle{\Gamma}^{ij\,kl\,mn}$
$\displaystyle=\gamma^{ij}(\gamma^{kl}\gamma^{mn}-\gamma^{kn}\gamma^{ml})+2\gamma^{in}(\gamma^{kj}\gamma^{ml}-\gamma^{kl}\gamma^{mj})\,.$
(3.56)

Inserting (3.49-3.52) and (3.53) into (3.41) leads, upon dropping total
derivatives, to

$\displaystyle L_{0}=$
$\displaystyle\,R+\frac{1}{4}e^{2\psi}C_{ij}C^{ij}-\frac{1}{2}\partial_{i}\psi\partial^{i}\psi\,,$
$\displaystyle L_{1}=$
$\displaystyle\,{\Gamma}^{ij\,kl}D_{i}C_{j}\dot{\gamma}_{kl}+(2\partial^{j}\psi-e^{2\psi}C^{ij}C_{i})\dot{C}_{j}+C^{i}\partial_{i}\psi\dot{\psi}\,,$
$\displaystyle L_{2}=$
$\displaystyle\,-{\Gamma}^{ij\,kl}C_{i}\dot{C}_{j}\dot{\gamma}_{kl}+e^{-2\psi}\dot{\gamma}\dot{\psi}-2C^{i}\dot{C}_{i}\dot{\psi}$
(3.57)
$\displaystyle+\frac{1}{4}(e^{-2\psi}{\Gamma}^{kl\,mn}+{\Gamma}^{ij\,kl\,mn}C_{i}C_{j})\dot{\gamma}_{kl}\dot{\gamma}_{mn}-\frac{1}{2}e^{2\psi}{\Gamma}^{ij\,kl}C_{i}C_{j}\dot{C}_{k}\dot{C}_{l}$
$\displaystyle-\frac{1}{2}(3e^{-2\psi}+C_{i}C^{i})\dot{\psi}^{2}\,.$

Let us stress again that the Lagrangian (3.45, 3.57) equals the Einstein-
Hilbert Lagrangian up to a total derivative. We refer to Appendix C for the
equations of motion in a form with explicit time derivatives. These can be
obtained by varying (3.45, 3.57) or equivalently by rewriting (3.5-3.7).

We will refrain from an extensive discussion of the symmetries in this form,
as we already discussed them in some detail in a more compact form in section
3. The discussion there also implies the transformations (3.58) and (3.59) do
indeed leave the action for (3.45, 3.57) invariant. Still let us shortly
mention a few key points and formulas. For use in the next sections we find it
convenient to revert in this and the next sections to the passive formulation.
In the passive formulation, using (3.11,3.12), and upon (3.38), the time
redefinition symmetry (3.16-3.18) takes the form

$\displaystyle\delta_{\Lambda}\psi$
$\displaystyle=c^{-1}(\Lambda\dot{\psi}+2\dot{\Lambda})\,,$
$\displaystyle\delta_{\Lambda}C_{i}$
$\displaystyle=\partial_{i}\Lambda+c^{-1}(\Lambda\dot{C}_{i}-\dot{\Lambda}C_{i})\,,$
(3.58) $\displaystyle\delta_{\Lambda}\gamma_{ij}$
$\displaystyle=c^{-1}(\Lambda\dot{\gamma}_{ij}+2\dot{\Lambda}\gamma_{ij})\,.$

while the time-dependent spatial diffeomorphisms (3.23-3.25) become

$\displaystyle\delta_{\xi}\psi$
$\displaystyle=L_{\xi}\psi+2c^{-1}C_{i}\partial_{t}\xi^{i}\,,$
$\displaystyle\delta_{\xi}C_{i}$
$\displaystyle=L_{\xi}C_{i}-c^{-1}(e^{-2\psi}\gamma_{ij}+C_{i}C_{j})\partial_{t}\xi^{j}\,,$
(3.59) $\displaystyle\delta_{\xi}\gamma_{ij}$
$\displaystyle=L_{\xi}\gamma_{ij}+2c^{-1}(C_{k}\gamma_{ij}-C_{(i}\gamma_{j)k})\partial_{t}\xi^{k}\,.$

One point of importance below is that the time-redefinition symmetry (3.58)
implies a simpler scaling symmetry. Consider the special case
$\Lambda=c(t-t_{0})\alpha$, with $t_{0}$ an arbitrary constant and $\alpha$ a
real parameter. Then observe that (3.38) implies that, when evaluated at
$t=t_{0}$, the transformations of the fields and their time derivatives take
the scaling forms $\delta_{\alpha}X=[X]X$,
$\delta_{\alpha}\dot{X}=\alpha([X]+1)\dot{X}$, where the scaling weights are

$\\!\\!\\!\\!\\!\\!\\!\\![\partial_{i}]=[\partial_{i}\psi]=0\,,\,\,[\partial_{t}]=[\partial_{t}\psi]=-[C_{i}]=1\,,\,\,[e^{\psi}]=-[e^{-\psi}]=[\gamma_{ij}]=-[\gamma^{ij}]=2\,.$
(3.60)

From this it follows that $[\sqrt{\gamma}dtd^{3}x]=2$, which in turn implies
that invariance of the action requires $[L]=-2$. One can indeed check that
with the weights (3.60) all the terms in (3.57) carry weight $-2$. Note also
the simple fact that since only $\partial_{t}$ and $C_{i}$ have an odd weight,
every term with a single time derivative should carry an odd number of $C_{i}$
factors, something which indeed can also be verified by inspection of (3.57).
This feature will play a small but key role in the shuffling algorithm
introduced in section 5.1.4 and it is thus interesting to point out it has its
origin in the time redefinition symmetry of the theory.

The time redefinition symmetry (3.58) will also play a key role in gauge
fixing the trace of the subleading metric coefficients, as discussed in
section 4.1.2. In section 5.2 we will shortly discuss how the symmetries
(3.58, 3.59) upon expansion lead to the non-relativistic symmetries of [8, 10,
11].

##  4 The $1/c$ expansion in KS formalism

For a pedagogic summary and overview of earlier work on the $1/c$ expansion
see [31, 4]. All of this work has been in a form making $4$-dimensional
coordinate invariance manifest, and for that reason the analysis was based on
Newton-Cartan geometry. Here we will revisit this expansion, but now in a 3+1
formulation, i.e. one that only keeps $3$-dimensional coordinate invariance
manifest and uses an explicit choice of time. This is not unnatural for two
related reasons. First of all in the non-relativistic expansion there is a
time direction that all non-relativistic observers agree upon. Secondly, by
dimensional reasons, all time derivatives are accompanied by a power of $1/c$
and so the $1/c$ expansion is actually an expansion in time derivatives. As we
argued in section 1, it is the KS formalism that provides the natural 3+1
split to use for the $1/c$ expansion. In this 3+1 form the general structure
of the expansion will be more transparent, which allows us to make some new
all order observations as well as push the expansion to higher order than was
previously done. In section 5 we discuss the truncation of the $1/c$ expansion
to the $1/c^{2}$ expansion in detail.

###  4.1 Structure of the expansion

The starting point is the relativistic KS Lagrangian in the form (3.45, 3.57).
To perform the expansion we assume111111A priori one could consider a Laurent
expansion in $1/c$, a transseries including terms of the form
$e^{-\frac{A}{c}}$ or even more generic $c$ dependence in the fields. Such
more general ansatze remain largely unexplored and fall outside the scope of
this paper. the fields to be analytic in $1/c$ :

$\psi=\sum_{n=0}^{\infty}{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}c^{-n}\qquad
C_{i}=\sum_{n=0}^{\infty}{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{i}c^{-n}\qquad\gamma_{ij}=\sum_{n=0}^{\infty}{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}c^{-n}\,.$
(4.1)

Note that the Lagrangian (3.45, 3.57) has some explicit dependence on $1/c$ as
well. In combination with the $1/c$ dependence of the fields (4.1) one gets an
expansion

${\cal
L}[\psi,C_{i},\gamma_{ij};c]=\sum_{n=0}^{\infty}{\overset{\,\scalebox{0.5}{($n$)}}{{\cal
L}}}{}[{\overset{\,\scalebox{0.5}{($0$)}}{\psi}}{},{\overset{\,\scalebox{0.5}{($0$)}}{C}}{}_{i},{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij},\ldots,{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{},{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{i},{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}]c^{-n}\,.$
(4.2)

In the following subsections we will discuss some general features of this
expansion. It will often be useful to collect all fields and their expansion
coefficients in a single ’master field’:

$\Phi=(\psi,C_{i},\gamma_{ij})\,,\qquad\Phi=\sum_{n=0}^{\infty}{\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}c^{-n}\,.$
(4.3)

####  4.1.1 Actions and equations of motion

Note that one could either expand the Lagrangian ${\cal L}$ and then vary the
expansion coefficients ${\overset{\,\scalebox{0.5}{($n$)}}{{\cal L}}}{}$ with
respect to the various expanded fields to obtain equations of motion, or
alternatively one could expand the original equations of motion associated to
the unexpanded Lagrangian. In [11] the interplay and compatibility between
these two approaches was explained, for completeness we shortly revisit this
in appendix A.1. The key relation derived there is

$\frac{\delta{\overset{\,\scalebox{0.5}{($n+m$)}}{S}}{}}{\delta{\overset{\,\scalebox{0.5}{($m$)}}{\Phi}}{}}={\overset{\,\scalebox{0.5}{($n$)}}{{\cal
E}}}{}\,.$ (4.4)

In words: the $n$’th order equations of motion can be obtained by varying the
$(n+m)$’th order action with respect to the $m$’th order fields. The relations
(4.4) can be interpreted in two ways. One point of view is that to work up to
$n$’th order we can simply use the action
${\overset{\,\scalebox{0.5}{($n$)}}{S}}{}[{\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{},\ldots,{\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}]$,
which can reproduce all equations of motion up to $n$’th order by variation
with respect to the fields ${\overset{\,\scalebox{0.5}{($k$)}}{\Phi}}{}$,
$k=0,\ldots,n$. I.e. one could expand the action up to the preferred order $n$
and then vary it with respect to all fields up to that order, to obtain all
the relevant equations of motion:

$\displaystyle{\overset{\,\scalebox{0.5}{($k$)}}{{\cal
E}}}{}=\frac{\delta{\overset{\,\scalebox{0.5}{($n$)}}{S}}{}}{\delta{\overset{\,\scalebox{0.5}{($n-k$)}}{\Phi}}{}}\qquad
k\leq n\,.$ (4.5)

In practice this approach is however a rather contrived way to find, say,
${\overset{\,\scalebox{0.5}{($1$)}}{{\cal E}}}{}$, which one could also obtain
by varying ${\overset{\,\scalebox{0.5}{($1$)}}{S}}{}$ with respect to
${\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{}$, a much shorter calculation.
Indeed from a computational point of view it is more natural to compute order
by order:

${\overset{\,\scalebox{0.5}{($k$)}}{{\cal
E}}}{}=\frac{\delta{\overset{\,\scalebox{0.5}{($k$)}}{S}}{}}{\delta{\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{}}\qquad
k\leq n\,.$ (4.6)

I.e. one can obtain all equations of motion up to a given order $n$, by
expanding the action up to order $n$ and then varying all of these actions
only with respect to the leading order fields
${\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{}$. As we will discuss in the next
subsection, section 4.1.2, this approach has the additional advantage that if
we restrict only to variations with respect to the leading order fields
${\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{}$ then the action
${\overset{\,\scalebox{0.5}{($k$)}}{S}}{}$ can be replaced by an equivalent
but simpler action ${\overset{\,\scalebox{0.5}{($k$)}}{\bar{S}}}{}$.

####  4.1.2 Gauge fixing the trace

The non-relativistic theory obtained through the $1/c$ expansion contains a
tower ${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}$ of symmetric tensor
fields. It is natural to identify
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$ as a metric defining the
geometry underlying the expanded theory. Using this metric we can split the
subleading tensor fields into a trace and traceless part121212While
${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}$ indicates the trace of
${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}$ for $n\geq 1$, we will
use ${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}$ to indicate the
determinant of ${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$. Although
this might at first appear confusing this will not lead to any clash of
notation as the trace of ${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$
will never appear (since
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}^{ij}{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}=3$)
and neither will the determinants of
${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}$ (since the expansion of
$\det\gamma_{ij}$ can be expressed in terms of
$\det{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$ and traces of
polynomials in ${\overset{\,\scalebox{0.5}{($k$)}}{\gamma}}{}_{ij}$). While
the distinctive interpretation of
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}$ versus
${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}$, $n\geq 1$, puts extra load on
the notation, it prevents the proliferation of symbols, which is already quite
large.:

${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}={\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}^{ij}{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}\,,\qquad{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}_{ij}={\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}-\frac{1}{3}{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}\,,\qquad
n\geq 1\,.$ (4.7)

The trace appears frequently in the expansion, both in the equations of motion
and the Lagrangian, for example through the expansion of the determinant of
$\gamma_{ij}$. As we discuss in detail in appendix A.2, one can always go to a
coordinate gauge where all traces are zero, i.e.
${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}=0,\forall n\geq 1$. In such a
gauge the equations of motion will simplify accordingly. Concerning the action
one has to be a little more careful, since gauge fixing at the level of the
action can lead to a loss of equations of motion131313These are the so called
constraints, such as the well known Gauss constraint of electro-magnetism that
needs to be supplemented to the Euler-Lagrange equations obtained from the
Lagrangian in $A_{0}=0$ gauge.. This can however be circumvented in the $1/c$
expansion, in the following sense. Let us define

${\overset{\,\scalebox{0.5}{($n$)}}{\bar{S}}}{}={\overset{\,\scalebox{0.5}{($n$)}}{S}}{}\big{|}_{\begin{subarray}{c}{\overset{\,\scalebox{0.5}{($m$)}}{\gamma}}{}=0\\\
m\geq 1\end{subarray}}\,,\qquad{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\cal
E}}}{}=\frac{\delta{\overset{\,\scalebox{0.5}{($n$)}}{\bar{S}}}{}}{\delta{\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{}}\,,$
(4.8)

then it is shown in appendix A.2 that

$\left.\left\\{{\overset{\,\scalebox{0.5}{($0$)}}{{\cal
E}}}{}=0,\ldots,{\overset{\,\scalebox{0.5}{($n$)}}{{\cal
E}}}{}=0\right\\}\right|_{\begin{subarray}{c}{\overset{\,\scalebox{0.5}{($m$)}}{\gamma}}{}=0\\\
m\geq
1\end{subarray}}\quad\Leftrightarrow\quad\left\\{{\overset{\,\scalebox{0.5}{($0$)}}{\bar{\cal
E}}}{}=0,\ldots,{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\cal
E}}}{}=0\right\\}\,.$ (4.9)

With the above we mean that the gauge fixed collection of equations of motion
is equivalent to the collection of equations of motion obtained from the
action which has been gauge fixed.

We should point out that the precise interpretation of (4.8) is somewhat
subtle when the variation with respect to
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$ is considered. One can
treat ${\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}_{ij}$ as an
independent field that does not change under variation of
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$, which is consistent if
one does not forget to replace all expressions of the form
$\bar{\gamma}_{ij}F^{ij}$ by $\bar{\gamma}_{ij}\bar{F}^{ij}$ before varying.
Alternatively one can consider
${\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}_{ij}={\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}-\frac{1}{3}{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}^{kl}{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{kl}$,
keep ${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}$ fixed while varying
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$ and only after varying
impose ${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}=0$. Both approaches are
equivalent, as we spell out in appendix A.2.

####  4.1.3 The leading order as the stationary sector of GR

We have chosen notation and conventions in such a way that both the Lagrangian
and the fields start at order $c^{0}$, see (3.45) and (4.1, 4.2). It follows
that the leading order of the $1/c$ expansion is

${\overset{\,\scalebox{0.5}{($0$)}}{{\cal
L}}}{}=\sqrt{{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}}{\overset{\,\scalebox{0.5}{($0$)}}{L}}{}_{0}\,,$
(4.10)

where via (3.57)

${\overset{\,\scalebox{0.5}{($0$)}}{L}}{}_{0}=R[{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}]+\frac{1}{4}e^{2{\overset{\,\scalebox{0.5}{($0$)}}{\psi}}{}}{\overset{\,\scalebox{0.5}{($0$)}}{C}}{}_{ij}{\overset{\,\scalebox{0.5}{($0$)}}{C}}{}^{ij}-\frac{1}{2}\partial_{i}{\overset{\,\scalebox{0.5}{($0$)}}{\psi}}{}\partial^{i}{\overset{\,\scalebox{0.5}{($0$)}}{\psi}}{}\,.$
(4.11)

In the above $R[{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}]$ is the Ricci
tensor of the metric ${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$ and
${\overset{\,\scalebox{0.5}{($0$)}}{C}}{}_{ij}=\partial_{i}{\overset{\,\scalebox{0.5}{($0$)}}{C}}{}_{j}-\partial_{j}{\overset{\,\scalebox{0.5}{($0$)}}{C}}{}_{i}$,
note that furthermore all indices are raised with
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}^{ij}$, which is the inverse of
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$.

Since the leading order fields
${\overset{\,\scalebox{0.5}{($0$)}}{\psi}}{},{\overset{\,\scalebox{0.5}{($0$)}}{C}}{}_{i},{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$
will appear at all lower orders as well, it will be useful to introduce
notation removing their superscript. By abuse of notation we will refer to the
leading order fields in the expansion by exactly the same symbol as the un-
expanded fields. This should not lead to confusion as it should be clear from
the context whether we are discussing the $1/c$ expansion or not. I.e. from
here onwards:

${\overset{\,\scalebox{0.5}{($0$)}}{\psi}}{}\rightarrow\psi\,,\qquad{\overset{\,\scalebox{0.5}{($0$)}}{C}}{}_{i}\rightarrow
C_{i}\,,\qquad{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}\rightarrow\gamma_{ij}\,.$
(4.12)

Similarly we will simply write $\sqrt{\gamma}$, $R$ and $C_{ij}$ for
$\sqrt{{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}}$,
$R[{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}]$ and
${\overset{\,\scalebox{0.5}{($0$)}}{C}}{}_{ij}$ respectively. Additionally
indices will be raised at all orders of the expansion with $\gamma^{ij}$,
again abusive notation for
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}^{ij}$. Similarly we will simply
use $\Gamma_{ij}^{k}$ and $D_{i}$ for the Levi-Civita connection
$\Gamma_{ij}^{k}[{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}]$ and covariant
derivative $D_{i}[{\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}]$.

In this new notation then, we can summarize the Lagrangian at leading order as

${\overset{\,\scalebox{0.5}{($0$)}}{{\cal
L}}}{}=\sqrt{\gamma}\left(R+\frac{1}{4}e^{2\psi}C_{ij}C^{ij}-\frac{1}{2}\partial_{i}\psi\partial^{i}\psi\right)\,.$
(4.13)

This form immediately reveals that this leading order Lagrangian (4.13) is
nothing but the full KS Lagrangian (3.45, 3.57) with all time derivatives
removed. In other words, for time independent field configurations the leading
order $1/c$ expansion is exact. If one furthermore recalls the form of the
metric (3.44) then one sees that such time independent fields correspond
exactly to a 4 dimensional stationary (lorentzian) metric. We can thus
conclude that the leading order of the $1/c$ expansion captures the full non-
linear dynamics of the stationary sector of general relativity. The same
conclusion was reached in [18], but the approach taken in this paper has the
advantage that it is much more straightforward to arrive at the leading order
Lagrangian (4.13) starting from the KS Lagrangian (3.45), instead of using the
fully covariant approach of [18].

Note that stationary solutions to GR, in addition to solving the dynamics
described by (4.13), will also not source any higher order corrections. One
can however consider quasi-stationary solutions to GR, i.e. stationary
solutions in which one makes the integration constants time dependent. Then
such metrics will still solve the dynamics of (4.13), since this does not
contain any time derivatives, but the time dependent integration constants
will lead to non-zero time derivatives which in turn will source subleading
corrections. We discuss the structure of these subleading corrections in the
following subsection.

####  4.1.4 The universal linear part

As we discussed in the previous subsection, the leading order equations are
non-linear equations for the leading order fields
${\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{}$. But once one considers the
subleading equations, they are, as we will discuss in this subsection, linear
equations determining the subleading fields
${\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{},n\geq 1$. Furthermore these
equations take the schematic form

$\mathbb{D}^{2}{\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}={\overset{\,\scalebox{0.5}{($n$)}}{\mathbb{S}}}{}[{\overset{\,\scalebox{0.5}{($n-1$)}}{\Phi}}{},\ldots,{\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{},\partial_{t}\\!\\!{\overset{\,\scalebox{0.5}{($n-1$)}}{\Phi}}{},\ldots,\partial_{t}{\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{},\partial_{t}^{2}\\!{\overset{\,\scalebox{0.5}{($n-2$)}}{\Phi}}{},\ldots,\partial_{t}^{2}{\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{}]\,.$
(4.14)

Here $\mathbb{D}^{2}$ is a second order linear differential operator that is
universal, i.e. it is the same at each order $n\geq 1$. The right hand side
${\overset{\,\scalebox{0.5}{($n$)}}{\mathbb{S}}}{}$ is a source term, built
out of the fields of order lower than $n$, and their time derivatives. This
gives the equations of motion in the $1/c$ expansion a hierarchic structure:
at each order one determines ${\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}$ by
solving (4.14), and this field and its time derivatives then enter the source
term in the equation for the higher order fields.

To arrive at the equations in the form (4.14) we first recall that the
expansion of the Lagrangian takes the form:

${\cal L}=\sum_{n=0}^{\infty}{\overset{\,\scalebox{0.5}{($n$)}}{{\cal
L}}}{}c^{-n}\,,$ (4.15)

where, via (3.45),

${\overset{\,\scalebox{0.5}{($n$)}}{{\cal
L}}}{}={\overset{\,\scalebox{0.5}{($n$)}}{{\cal
L}}}{}_{0}+{\overset{\,\scalebox{0.5}{($n-1$)}}{{\cal
L}}}{}\\!\\!\\!{}_{1}+{\overset{\,\scalebox{0.5}{($n-2$)}}{{\cal
L}}}{}\\!\\!\\!{}_{2}\,.$ (4.16)

Now remark that the $n$’th order fields
${\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}$ do not appear into the last two
terms, but only in the first. Furthermore, since the fields
${\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}$ by definition enter with a power
$c^{-n}$ they will appear linearly in this first term, as
${\overset{\,\scalebox{0.5}{($n$)}}{{\cal L}}}{}$ comes with that same power.
This first of all suggests to decompose

${\overset{\,\scalebox{0.5}{($n$)}}{{\cal
L}}}{}={\overset{\,\scalebox{0.5}{($n$)}}{{\cal
L}}}{}_{\mathrm{lin}}+{\overset{\,\scalebox{0.5}{($n$)}}{{\cal
L}}}{}_{\mathrm{source}}\,,$ (4.17)

where

${\overset{\,\scalebox{0.5}{($n$)}}{{\cal
L}}}{}_{\mathrm{source}}=\left.{\overset{\,\scalebox{0.5}{($n$)}}{{\cal
L}}}{}_{0}\right|_{{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}=0,\,{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{i}=0,\,{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}=0}+{\overset{\,\scalebox{0.5}{($n-1$)}}{{\cal
L}}}{}\\!\\!{}_{1}+{\overset{\,\scalebox{0.5}{($n-2$)}}{{\cal
L}}}{}\\!\\!{}_{2}\,.$ (4.18)

The second observation is then that the linear part is universal. One computes
from (3.57) that141414We remind the reader that we have now started using
notation where $\psi={\overset{\,\scalebox{0.5}{($0$)}}{\psi}}{}$, etc. see
section 4.1.3. Furthermore we have gauge fixed the traces
${\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}$ to zero, see section 4.1.2,
replacing ${\overset{\,\scalebox{0.5}{($n$)}}{{\cal L}}}{}$ by
${\overset{\,\scalebox{0.5}{($n$)}}{\bar{{\cal L}}}}{}$. For those readers
interested in the results outside this gauge we have provided the additional
terms proportional to the trace in appendix B.

${\overset{\,\scalebox{0.5}{($n$)}}{\bar{\cal
L}}}{}_{\mathrm{lin}}=\sqrt{\gamma}\Bigg{(}-\partial_{i}\psi\partial^{i}{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}+\frac{{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}e^{2\psi}}{2}C_{ij}C^{ij}+\frac{e^{2\psi}}{2}C^{ij}{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{ij}-{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{ij}(R_{ij}+\frac{e^{2\psi}}{2}C_{ik}C_{j}{}^{k}-\frac{1}{2}\partial_{i}\psi\partial_{j}\psi)\Bigg{)}\,.$
(4.19)

The equations of motion then take the form (4.14) with

$\overline{\mathbb{D}}^{2}{\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}=\frac{1}{\sqrt{\gamma}}\frac{\delta{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\cal
L}}}{}_{\mathrm{lin}}}{\delta{\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{}}\qquad\qquad{\overset{\,\scalebox{0.5}{($n$)}}{\overline{\mathbb{S}}}}{}=-\frac{1}{\sqrt{\gamma}}\frac{\delta{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\cal
L}}}{}_{\mathrm{source}}}{\delta{\overset{\,\scalebox{0.5}{($0$)}}{\Phi}}{}}\,.$
(4.20)

Since ${\overset{\,\scalebox{0.5}{($n$)}}{\bar{\cal L}}}{}_{\mathrm{lin}}$ has
the same form $\eqref{linLag}$ at each order,
$\overline{\mathbb{D}}^{2}{\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}$ needs
to be computed only once. Doing so leads to

$\displaystyle\overline{\mathbb{D}}^{2}{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}=$
$\displaystyle\,D_{i}\partial^{i}{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}+{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}e^{2\psi}C_{ij}C^{ij}+e^{2\psi}C^{ij}{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{ij}+e^{2\psi}{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{ij}C_{ik}C_{j}{}^{k}+\partial_{i}\psi
D_{j}{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{ij}\,,$
$\displaystyle\overline{\mathbb{D}}^{2}{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}^{j}=$
$\displaystyle\,-D_{i}\left(e^{2\psi}({\overset{\,\scalebox{0.5}{($n$)}}{C}}{}^{ij}+2{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}C^{ij}+{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{ik}C_{k}{}^{j}-{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{jk}C_{k}{}^{i})\right)\,,$
$\displaystyle\overline{\mathbb{D}}^{2}{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{ij}=$
$\displaystyle\,\frac{1}{2}D^{2}{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{ij}+\partial^{(i}\psi\partial^{j)}{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}-{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}e^{2\psi}C^{i}{}_{k}C^{jk}-e^{2\psi}C^{(i}{}_{k}{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}^{j)k}$
(4.21)
$\displaystyle+\frac{1}{2}{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{kl}e^{2\psi}C_{k}{}^{i}C_{l}{}^{j}-\frac{1}{2}\gamma^{ij}\bigg{(}\partial_{k}\psi\partial^{k}{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}-\frac{{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}}{2}e^{2\psi}C_{kl}C^{kl}-\frac{e^{2\psi}}{2}C^{kl}{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{kl}\bigg{)}$
$\displaystyle+({\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{ik}\gamma^{jl}+{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{jk}\gamma^{il}-\frac{1}{2}\gamma^{ij}{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{kl})(R_{kl}+\frac{e^{2\psi}}{2}C_{km}C_{l}{}^{m}-\frac{1}{2}\partial_{k}\psi\partial_{l}\psi)$
$\displaystyle-\frac{1}{3}{\overset{\,\scalebox{0.5}{($n$)}}{\bar{\gamma}}}{}^{ij}(R+\frac{e^{2\psi}}{2}C_{kl}C^{kl}-\frac{1}{2}\partial_{k}\psi\partial^{k}\psi)\,.$

where

$\displaystyle
D^{2}{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}^{ij}=D_{k}D^{k}{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}^{ij}-D_{k}D^{i}{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}^{jk}-D_{k}D^{j}{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}^{ik}+\gamma^{ij}D_{k}D_{l}{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}^{kl}\,.$
(4.22)

Contrary to $\overline{\mathbb{D}}^{2}$, the source term
${\overset{\,\scalebox{0.5}{($n$)}}{\overline{\mathbb{S}}}}{}$ is different at
each order – indeed it becomes more intricate at each successive order – and
thus it needs to be computed for each order separately via (4.20). We provide
the results for ${\overset{\,\scalebox{0.5}{($1$)}}{\bar{\cal
L}}}{}_{\mathrm{source}}$ and ${\overset{\,\scalebox{0.5}{($2$)}}{\bar{\cal
L}}}{}_{\mathrm{source}}$ in section 4.2.

###  4.2 Explicit expansion up to NNLO

We discussed the leading order Lagrangian in section 4.1.3 and the all order
linear part of the Lagrangian and equations of motion in section 4.1.4. Now we
will present the complete Lagrangians at the first two subleading orders, in
the traceless gauge – see section 4.1.2. The additional terms that appear
without this gauge choice can be found in appendix B.

For notational convenience we have decided to remove the superscripts
indicating the order and instead use different characters to label the
different coefficients151515As before, see section 4.1.3, we indicate the
leading order field with the same symbol as the complete $c$ dependent field.
:

$\displaystyle\psi$ $\displaystyle=\psi+c^{-1}\chi+c^{-2}\phi+\ldots\,,$
$\displaystyle C_{i}$ $\displaystyle=C_{i}+c^{-1}B_{i}+c^{-2}A_{i}+\ldots\,,$
(4.23) $\displaystyle\gamma_{ij}$
$\displaystyle=\gamma_{ij}+c^{-1}\alpha_{ij}+c^{-2}\beta_{ij}+\ldots\,.$

Furthermore

$C_{ij}=\partial_{i}C_{j}-\partial_{j}C_{i}\,,\qquad
A_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}\,,\qquad
B_{ij}=\partial_{i}B_{j}-\partial_{j}B_{i}\,.$ (4.24)

As before we split the Lagrangians into a linear part and a source part, and
additionally we organize the source part by the number of time derivatives:

$\displaystyle{\overset{\,\scalebox{0.5}{($1$)}}{\bar{\cal L}}}{}$
$\displaystyle=\sqrt{\gamma}\left({\overset{\,\scalebox{0.5}{($1$)}}{\bar{L}}}{}_{\mathrm{lin}}+{\overset{\,\scalebox{0.5}{($1$)}}{\bar{L}}}{}_{1\mathrm{td}}\right)\,,$
(4.25) $\displaystyle{\overset{\,\scalebox{0.5}{($2$)}}{\bar{\cal L}}}{}$
$\displaystyle=\sqrt{\gamma}\left({\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}_{\mathrm{lin}}+{\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}_{0\mathrm{td}}+{\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}_{1\mathrm{td}}+{\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}_{2\mathrm{td}}\right)\,.$
(4.26)

Although the linear part is the same at all orders and was already computed
above in (4.19) we will add it here as well, to provide a complete reference.

####  4.2.1 NLO

From the split of the relativistic Lagrangian (3.45, 3.57) it follows that

${\overset{\,\scalebox{0.5}{($1$)}}{\bar{\cal
L}}}{}_{1\mathrm{td}}={\overset{\,\scalebox{0.5}{($0$)}}{\bar{{\cal
L}}}}{}_{1}\,.$ (4.27)

An explicit calculation then yields

$\displaystyle{\overset{\,\scalebox{0.5}{($1$)}}{\bar{L}}}{}_{\mathrm{lin}}=$
$\displaystyle\,-\partial_{i}\psi\partial^{i}\chi+\frac{\chi}{2}e^{2\psi}C_{ij}C^{ij}+\frac{1}{2}e^{2\psi}C^{ij}B_{ij}-\bar{\alpha}^{ij}(R_{ij}+\frac{e^{2\psi}}{2}C_{ik}C_{j}{}^{k}-\frac{1}{2}\partial_{i}\psi\partial_{j}\psi)\,,$
$\displaystyle{\overset{\,\scalebox{0.5}{($1$)}}{\bar{L}}}{}_{1\mathrm{td}}=$
$\displaystyle\,\Gamma^{ij\,kl}D_{i}C_{j}\dot{\gamma}_{kl}+(2\partial^{j}\psi-e^{2\psi}C^{ij}C_{i})\dot{C}_{j}+C^{i}\partial_{i}\psi\dot{\psi}\,.$
(4.28)

####  4.2.2 NNLO

At the next order the split of the relativistic Lagrangian (3.45, 3.57)
implies

${\overset{\,\scalebox{0.5}{($2$)}}{\bar{\cal
L}}}{}_{0\mathrm{td}}={\overset{\,\scalebox{0.5}{($2$)}}{\bar{\cal
L}}}{}_{0}-{\overset{\,\scalebox{0.5}{($2$)}}{\bar{\cal
L}}}{}_{\mathrm{lin}}\,,\qquad{\overset{\,\scalebox{0.5}{($2$)}}{\bar{\cal
L}}}{}_{1\mathrm{td}}={\overset{\,\scalebox{0.5}{($1$)}}{\bar{\cal
L}}}{}_{1}\,,\qquad{\overset{\,\scalebox{0.5}{($2$)}}{\bar{\cal
L}}}{}_{2\mathrm{td}}={\overset{\,\scalebox{0.5}{($0$)}}{\bar{\cal
L}}}{}_{2}\,.$ (4.29)

After an explicit calculation one finds

$\displaystyle{\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}_{\mathrm{lin}}=$
$\displaystyle-\partial_{i}\psi\partial^{i}\phi+\frac{\phi}{2}e^{2\psi}C_{ij}C^{ij}+\frac{1}{2}e^{2\psi}C^{ij}A_{ij}-\bar{\beta}^{ij}(R_{ij}+\frac{e^{2\psi}}{2}C_{ik}C_{j}{}^{k}-\frac{1}{2}\partial_{i}\psi\partial_{j}\psi)\,,$
$\displaystyle{\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}_{0\mathrm{td}}=$
$\displaystyle-\frac{1}{4}\bar{\alpha}^{kl}\bar{\alpha}_{kl}(R-\frac{1}{2}\partial_{i}\psi\partial^{i}\psi+\frac{e^{2\psi}}{4}C_{ij}C^{ij})+\frac{1}{4}\Gamma^{ij\,kl\,mn}D_{i}\bar{\alpha}_{kl}D_{j}\bar{\alpha}_{mn}-\frac{1}{2}\partial_{i}\chi\partial^{i}\chi$
$\displaystyle+\bar{\alpha}^{ik}\bar{\alpha}_{k}{}^{j}(R_{ij}-\frac{1}{2}\partial_{i}\psi\partial_{j}\psi+\frac{e^{2\psi}}{2}C_{i}{}^{l}C_{jl})+\bar{\alpha}^{ik}\bar{\alpha}^{jl}\frac{e^{2\psi}}{4}C_{ij}C_{kl}+\frac{1}{2}\chi^{2}e^{2\psi}C_{ij}C^{ij}$
$\displaystyle+\bar{\alpha}^{ij}(\partial_{i}\psi\partial_{j}\chi-\chi
e^{2\psi}C_{ik}C_{j}{}^{k}-e^{2\psi}B_{ik}C_{j}{}^{k})+\frac{1}{4}e^{2\psi}B^{ij}B_{ij}+\chi
e^{2\psi}C^{ij}B_{ij}\,,$
$\displaystyle{\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}_{1\mathrm{td}}=$
$\displaystyle\,\,C_{i}\Big{(}\dot{\gamma}D_{j}\bar{\alpha}^{ij}+\dot{\gamma}_{jk}(\frac{1}{2}D^{i}\bar{\alpha}^{jk}-D^{k}\bar{\alpha}^{ij})-\dot{\psi}\bar{\alpha}^{ij}\partial_{j}\psi+\dot{\psi}\partial^{i}\chi+\dot{\chi}\partial^{i}\psi\Big{)}$
$\displaystyle-\dot{\gamma}\Big{(}D^{i}B_{i}-\bar{\alpha}^{ij}D_{i}C_{j}\Big{)}+\dot{\gamma}^{ij}\Big{(}D_{i}B_{j}-\bar{\alpha}_{ik}D^{k}C_{j}-\bar{\alpha}_{jk}D_{i}C^{k}+\bar{\alpha}_{ij}D_{k}C^{k}\Big{)}$
$\displaystyle+B_{i}(\dot{\psi}\partial^{i}\psi-e^{2\psi}\dot{C}_{j}C^{ij})+2\dot{C}_{i}(\partial^{i}\chi-\bar{\alpha}^{ij}\partial_{j}\psi)+\dot{\bar{\alpha}}^{ij}D_{i}C_{j}-\dot{\bar{\alpha}}D^{i}C_{i}+2\dot{B}_{i}\partial^{i}\psi$
$\displaystyle+C_{i}e^{2\psi}\left[\dot{C}_{j}\Big{(}-2\chi
C^{ij}-B^{ij}+C_{kl}(\gamma^{ik}\bar{\alpha}^{jl}-\gamma^{jk}\bar{\alpha}^{il})\Big{)}-\dot{B}_{j}C^{ij}\right]\,,$
$\displaystyle{\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}_{2\mathrm{td}}=$
$\displaystyle-\Gamma^{ij\,kl}C_{i}\dot{C}_{j}\dot{\gamma}_{kl}+e^{-2\psi}\dot{\gamma}\dot{\psi}-2C^{i}\dot{C}_{i}\dot{\psi}+\frac{1}{4}(e^{-2\psi}\Gamma^{kl\,mn}+\Gamma^{ij\,kl\,mn}C_{i}C_{j})\dot{\gamma}_{kl}\dot{\gamma}_{mn}$
$\displaystyle-\frac{1}{2}e^{2\psi}\Gamma^{ij\,kl}C_{i}C_{j}\dot{C}_{k}\dot{C}_{l}-\frac{1}{2}(3e^{-2\psi}+C_{i}C^{i})\dot{\psi}^{2}\,.$
(4.30)

Previously only the leading order of an expansion of GR in $1/c$ including odd
powers was computed [18], the results in sections 4.2.1 and 4.2.2 are new.
They show the computational advantage of the 3+1 formulation since the already
quite involved expressions we obtained here would become quite a bit more
involved to derive in their fully covariant Newton-Cartan form.

##  5 The $1/c^{2}$ expansion in KS formalism

In this section we review how the $1/c$ expansion contains the $1/c^{2}$
expansion as a self-consistent sub-theory. In addition we spell out how the
$1/c^{2}$ expansion up to order $c^{-2n}$ can be obtained from the $1/c$
expansion up to order $c^{-n}$ by a reshuffling of the terms. We use this to
compute up to order $c^{-4}$ and compare to the results in the literature: in
section 5.2.1 we rederive the results of [8, 11] while in section 5.2.2 we
show how a further truncation of our result reproduces the 1PN order of the PN
expansion.

###  5.1 Structure of the expansion

####  5.1.1 The even power ansatz and truncation

The explicit form of the relativistic metric (3.44) reveals that $c$ naturally
appears with an odd power. This can be circumvented by defining

$C_{i}=c^{-1}B_{i}\,,$ (5.1)

so that the metric takes the form

$ds^{2}=-c^{2}e^{\psi}(dt+c^{-2}B_{i}dx^{i})^{2}+e^{-\psi}\gamma_{ij}dx^{i}dx^{j}\,.$
(5.2)

It is important to stress that this is much more than a simple redefinition
when combined with the assumption, crucial for an expansion in $1/c$, that
$C_{i}$ is analytic in $1/c$. Then (5.1) implies that $B_{i}$ is analytic, but
more importantly also that $C_{i}$ is ’subleading’. I.e. the relation (5.1)
should be interpreted as the non-trivial assumption that
$\lim_{c\rightarrow\infty}C_{i}=0$.

Inserting the ansatz (5.1) into the KS action in the form (3.45, 3.57) one
gets

${\cal
L}^{\mathrm{e}}=\sqrt{\gamma}\left(L_{0}^{\mathrm{e}}+c^{-2}L_{2}^{\mathrm{e}}+c^{-4}L_{4}^{\mathrm{e}}+c^{-6}L_{6}^{\mathrm{e}}\right)\,,$
(5.3)

where now

$\displaystyle L_{0}^{\mathrm{e}}=$
$\displaystyle\,R-\frac{1}{2}\partial_{i}\psi\partial^{i}\psi\,,$
$\displaystyle L_{2}^{\mathrm{e}}=$
$\displaystyle\,\frac{1}{4}e^{2\psi}B_{ij}B^{ij}+{\Gamma}^{ij\,kl}D_{i}B_{j}\dot{\gamma}_{kl}+2\partial^{j}\psi\dot{B}_{j}+B^{i}\partial_{i}\psi\dot{\psi}$
$\displaystyle+e^{-2\psi}\left(\frac{1}{4}{\Gamma}^{kl\,mn}\dot{\gamma}_{kl}\dot{\gamma}_{mn}+\dot{\gamma}\dot{\psi}-\frac{3}{2}\dot{\psi}^{2}\right)\,,$
(5.4) $\displaystyle L_{4}^{\mathrm{e}}=$
$\displaystyle-e^{2\psi}B^{ij}B_{i}\dot{B}_{j}-{\Gamma}^{ij\,kl}B_{i}\dot{B}_{j}\dot{\gamma}_{kl}-2B^{i}\dot{B}_{i}\dot{\psi}$
$\displaystyle+\frac{1}{4}{\Gamma}^{ij\,kl\,mn}B_{i}B_{j}\dot{\gamma}_{kl}\dot{\gamma}_{mn}-\frac{1}{2}B_{i}B^{i}\dot{\psi}^{2}\,,$
$\displaystyle L_{6}^{\mathrm{e}}=$
$\displaystyle-\frac{1}{2}e^{2\psi}{\Gamma}^{ij\,kl}B_{i}B_{j}\dot{B}_{k}\dot{B}_{l}\,.$

In summary, after the redefinition (5.1) both the metric (5.2) and Lagrangian
(5.3) contain only even powers of $c$. This in turn implies that if we assume
the fields to be analytic in $1/c^{2}$, i.e.

$\psi=\sum_{n=0}^{\infty}{\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{}c^{-2n}\,,\qquad
B_{i}=\sum_{n=0}^{\infty}{\overset{\,\scalebox{0.5}{($2n$)}}{B}}{}_{i}c^{-2n}\,,\qquad\gamma_{ij}=\sum_{n=0}^{\infty}{\overset{\,\scalebox{0.5}{($2n$)}}{\gamma}}{}_{ij}c^{-2n}\,,$
(5.5)

then we get a consistent expansion, that we’ll refer to as the $1/c^{2}$
expansion. Furthermore this expansion is, via (5.1), nothing but a truncation
of the $1/c$ expansion:

$(\mathrm{ev.trunc.})\,\qquad{\overset{\,\scalebox{0.5}{($2m+1$)}}{\psi}}{}\\!\\!\\!=0,\quad{\overset{\,\scalebox{0.5}{($2m$)}}{C}}{}_{i}=0,\quad{\overset{\,\scalebox{0.5}{($2m+1$)}}{C}}{}\\!\\!\\!{}_{i}={\overset{\,\scalebox{0.5}{($2m$)}}{B}}{}_{i},\quad{\overset{\,\scalebox{0.5}{($2m+1$)}}{\gamma}}{}\\!\\!\\!{}_{ij}=0\qquad\forall
m\geq 0\,.$ (5.6)

In particular, for the expansion coefficients of the action this implies

${\overset{\,\scalebox{0.5}{($2n$)}}{S}}{}^{\mathrm{e}}=\left.{\overset{\,\scalebox{0.5}{($2n$)}}{S}}{}\right|_{\mathrm{ev.trunc.}}\qquad
0=\\!\\!\\!{\overset{\,\scalebox{0.5}{($2n+1$)}}{S}}{}\Big{|}_{\mathrm{ev.trunc.}}\,.$
(5.7)

At the level of the equations of motion observe that

$\displaystyle{\overset{\,\scalebox{0.5}{($2n$)}}{{\cal
E}}}{}\\!{}_{\psi}^{\mathrm{e}}=\frac{\delta\\!\\!{\overset{\,\scalebox{0.5}{($2n+2k$)}}{S}}{}\\!\\!\\!\\!{}^{\mathrm{e}}}{\delta{\overset{\,\scalebox{0.5}{($2k$)}}{\psi}}{}}=\frac{\delta\Big{(}{\overset{\,\scalebox{0.5}{($2n+2k$)}}{S}}{}\Big{|}_{\mathrm{ev.trunc.}}\Big{)}}{\delta{\overset{\,\scalebox{0.5}{($2k$)}}{\psi}}{}}=\left.\frac{\delta{\overset{\,\scalebox{0.5}{($2n+2k$)}}{S}}{}}{\delta{\overset{\,\scalebox{0.5}{($2k$)}}{\psi}}{}}\right|_{\mathrm{ev.trunc.}}=\left.{\overset{\,\scalebox{0.5}{($2n$)}}{{\cal
E}}}{}\\!{}_{\psi}\right|_{\mathrm{ev.trunc.}}\,,$ (5.8)

The same observation holds for ${\overset{\,\scalebox{0.5}{($2n$)}}{{\cal
E}}}{}\\!{}_{\mathrm{e}}^{ij}$,

$\displaystyle{\overset{\,\scalebox{0.5}{($2n$)}}{{\cal
E}}}{}{}_{\mathrm{e}}^{ij}=\frac{\delta\\!\\!{\overset{\,\scalebox{0.5}{($2n+2k$)}}{S}}{}\\!\\!\\!\\!{}^{\mathrm{e}}}{\delta{\overset{\,\scalebox{0.5}{($2k$)}}{\gamma}}{}\\!{}_{ij}}=\left.{\overset{\,\scalebox{0.5}{($2n$)}}{{\cal
E}}}{}\\!{}^{ij}\right|_{\mathrm{ev.trunc.}}\,,$ (5.9)

but gets a little twist in the case of
${\overset{\,\scalebox{0.5}{($2n$)}}{{\cal E}}}{}\\!{}_{\mathrm{e}}^{i}$:

$\displaystyle{\overset{\,\scalebox{0.5}{($2n$)}}{{\cal
E}}}{}{}_{\mathrm{e}}^{i}=\frac{\delta\\!\\!{\overset{\,\scalebox{0.5}{($2n+2k$)}}{S}}{}\\!\\!\\!\\!{}^{\mathrm{e}}}{\delta{\overset{\,\scalebox{0.5}{($2k$)}}{B}}{}{}_{i}}=\frac{\delta\Big{(}{\overset{\,\scalebox{0.5}{($2n+2k$)}}{S}}{}\Big{|}_{\mathrm{ev.trunc.}}\Big{)}}{\delta{\overset{\,\scalebox{0.5}{($2k$)}}{B}}{}_{i}}=\left.\frac{\delta{\overset{\,\scalebox{0.5}{($2n+2k$)}}{S}}{}}{\delta{\overset{\,\scalebox{0.5}{($2k+1$)}}{C}}{}\\!\\!\\!{}_{i}}\right|_{\mathrm{ev.trunc.}}=\left.{\overset{\,\scalebox{0.5}{($2n-1$)}}{{\cal
E}}}{}^{i}\right|_{\mathrm{ev.trunc.}}\,.$ (5.10)

Finally note that

$0=\left.{\overset{\,\scalebox{0.5}{($2n+1$)}}{{\cal
E}}}{}\\!\\!\\!{}_{\psi}\right|_{\mathrm{ev.trunc.}}\qquad
0=\left.{\overset{\,\scalebox{0.5}{($2n+1$)}}{{\cal
E}}}{}\\!\\!\\!{}^{ij}\right|_{\mathrm{ev.trunc.}}\qquad
0=\left.{\overset{\,\scalebox{0.5}{($2n$)}}{{\cal
E}}}{}^{i}\right|_{\mathrm{ev.trunc.}}\,.$ (5.11)

This follows from the fact that the action as expressed in terms of
$\psi,\gamma_{ij}$ and $B_{i}$ contains only even powers, see (5.3, 5.4), and
the same is thus true for the equations of motion ${\cal
E}_{\psi}=\frac{\delta S}{\delta\psi}$ and ${\cal E}^{ij}=\frac{\delta
S}{\delta\gamma_{ij}}$ when expressed in terms of the fields
$\psi,\gamma_{ij}$ and $B_{i}$. Note that via (5.1) ${\cal E}^{i}=\frac{\delta
S}{\delta C_{i}}=c\frac{\delta S}{\delta B_{i}}$ and so this equation will
only contain odd powers when expressed in terms of $\psi,\gamma_{ij}$ and
$B_{i}$.

The discussion above shows in detail that the truncation (5.6) from the $1/c$
expansion to the $1/c^{2}$ expansion is a consistent truncation. This means it
can be performed at the level of the action and that variation of the
truncated action will reproduce the truncated equations of motion.
Equivalently it also shows that the even coefficients do not source the odd
coefficients, in case all of those are set to zero (the reverse is not true).
We should point out that our discussion of the even power truncation is
restricted to the pure gravitational or vacuum sector. In the presence of non-
trivial energy momentum one needs to perform a similar analysis of that sector
as well.

####  5.1.2 The leading order as the static sector of GR

The leading order of the $1/c^{2}$ expansion can be obtained directly by
truncating the leading order of the $1/c$ expansion, which we discussed in
section 4.1.3. This simply amounts to removing the field
${\overset{\,\scalebox{0.5}{($0$)}}{C}}{}_{i}=0$ while keeping
${\overset{\,\scalebox{0.5}{($0$)}}{\psi}}{}$,
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$. From now on, as we did
previously, we will simply denote
${\overset{\,\scalebox{0.5}{($0$)}}{\psi}}{}$,
${\overset{\,\scalebox{0.5}{($0$)}}{\gamma}}{}_{ij}$ as $\psi$ and
$\gamma_{ij}$ and leave it up to context to determine if they are to be
interpreted as the relativistic fields or simply the leading coefficients in
their expansion. The truncated Lagrangian (4.13) takes the form

${\overset{\,\scalebox{0.5}{($0$)}}{{\cal
L}}}{}^{\mathrm{e}}=\sqrt{\gamma}\left(R-\frac{1}{2}\partial_{i}\psi\partial^{i}\psi\right)\,.$
(5.12)

This Lagrangian coincides with the fully relativistic one (3.45, 3.57) where
all time derivatives and the field $C_{i}$ have been put to zero. Comparing to
the form of the relativistic metric (3.44) we see that extrema
$(\psi,\gamma_{ij})$ of (5.12) coincide with quasi-static solutions of GR,
i.e. static solutions with time dependent integration constants. This
identification of the leading order of the $1/c^{2}$ expansion with the static
sector of GR was previously made in [9].

####  5.1.3 The universal linear part and partial decoupling

Let us recall from section 4.1.4 that the $n$’th order equations of motion
${\overset{\,\scalebox{0.5}{($n$)}}{E}}{}$ are equations of motion determining
the $n$’th order fields ${\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}$. They
take the form
$\mathbb{D}^{2}{\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}={\overset{\,\scalebox{0.5}{($n$)}}{\mathbb{S}}}{}$.
Here $\mathbb{D}^{2}$ is a second order linear differential operator that is
universal, i.e. the same at all orders $n\geq 1$. The source term
${\overset{\,\scalebox{0.5}{($n$)}}{\mathbb{S}}}{}$ is a function of the lower
order fields ${\overset{\,\scalebox{0.5}{($m$)}}{\Phi}}{}$, $m<n$, and their
time derivatives, only.

Since, as we discussed in detail above, the $1/c^{2}$ expansion is a
truncation of the $1/c$ expansion, this structure carries over to the
equations of motion
${\overset{\,\scalebox{0.5}{($2n$)}}{E}}{}\\!{}_{\mathrm{e}}$ in this
expansion. But due to the redefinition (5.1) we will need to treat
${\overset{\,\scalebox{0.5}{($2n$)}}{E}}{}^{i}\\!\\!\\!{}_{\mathrm{e}}$
separately from
${\overset{\,\scalebox{0.5}{($2n$)}}{E}}{}^{\mathrm{e}}\\!\\!\\!{}_{\psi}$ and
${\overset{\,\scalebox{0.5}{($2n$)}}{E}}{}^{ij}\\!\\!\\!\\!{}_{\mathrm{e}}{}$.
Defining
${\overset{\,\scalebox{0.5}{($2n$)}}{\varPhi}}{}=({\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{},{\overset{\,\scalebox{0.5}{($2n$)}}{\gamma}}{}_{ij})$
the equations of motion in the $1/c^{2}$ expansion will take the form

${\overset{\,\scalebox{0.5}{($2n$)}}{E}}{}_{\mathrm{e}}=\mathbb{D}^{2}_{\mathrm{e}}{\overset{\,\scalebox{0.5}{($2n$)}}{\varPhi}}{}-{\overset{\,\scalebox{0.5}{($2n$)}}{\mathbb{S}}}{}_{\mathrm{e}}\qquad{\overset{\,\scalebox{0.5}{($2n$)}}{E}}{}_{\mathrm{e}}^{i}={\cal
D}^{2}_{\mathrm{e}}{\overset{\,\scalebox{0.5}{($2n-2$)}}{B}}{}^{i}-{\overset{\,\scalebox{0.5}{($2n-2$)}}{\mathbb{S}}}{}_{\mathrm{e}}^{i}\qquad
n\geq 1\,.$ (5.13)

By combining (5.8-5.10) with (4.21) one finds the explicit form of universal
the linear operators161616Note that here, for simplicity, we again present the
result in the traceless gauge (hence the bars), see section 4.1.2. The extra
terms appearing without this choice of gauge can be found in appendix B.:

$\displaystyle\overline{\mathbb{D}}^{2}_{\mathrm{e}}{\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{}=$
$\displaystyle\,D_{i}\partial^{i}{\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{}+\partial_{i}\psi
D_{j}{\overset{\,\scalebox{0.5}{($2n$)}}{\bar{\gamma}}}{}^{ij}\,,$
$\displaystyle\overline{\mathbb{D}}^{2}_{\mathrm{e}}{\overset{\,\scalebox{0.5}{($2n$)}}{\bar{\gamma}}}{}^{ij}=$
$\displaystyle\,\frac{1}{2}D^{2}{\overset{\,\scalebox{0.5}{($2n$)}}{\bar{\gamma}}}{}^{ij}+\partial^{(i}\psi\partial^{j)}{\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{}-\frac{1}{2}\gamma^{ij}\partial_{k}\psi\partial^{k}{\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{}$
$\displaystyle+({\overset{\,\scalebox{0.5}{($2n$)}}{\bar{\gamma}}}{}^{ik}\gamma^{jl}+{\overset{\,\scalebox{0.5}{($2n$)}}{\bar{\gamma}}}{}^{jk}\gamma^{il}-\frac{1}{2}\gamma^{ij}{\overset{\,\scalebox{0.5}{($2n$)}}{\bar{\gamma}}}{}^{kl})(R_{kl}-\frac{1}{2}\partial_{k}\psi\partial_{l}\psi)$
(5.14)
$\displaystyle-\frac{1}{3}{\overset{\,\scalebox{0.5}{($2n$)}}{\bar{\gamma}}}{}^{ij}(R-\frac{1}{2}\partial_{k}\psi\partial^{k}\psi)\,,$
$\displaystyle\bar{\cal
D}^{2}_{\mathrm{e}}{\overset{\,\scalebox{0.5}{($2n+2$)}}{B}}{}\\!\\!{}^{j}=$
$\displaystyle\,-D_{i}(e^{2\psi}{\overset{\,\scalebox{0.5}{($2n+2$)}}{B}}{}\\!\\!{}^{ij})\,.$

where $D^{2}{\overset{\,\scalebox{0.5}{($2n$)}}{\bar{\gamma}}}{}^{ij}$ is
defined in (4.22).

Note now that apart from a great simplification with respect to (4.21), the
linear operator (5.14) is also block diagonal, so that the field
${\overset{\,\scalebox{0.5}{($2n$)}}{B}}{}_{i}$ decouples from the fields
${\overset{\,\scalebox{0.5}{($2n$)}}{\varPhi}}{}=({\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{},{\overset{\,\scalebox{0.5}{($2n$)}}{\gamma}}{}_{ij})$.
The operator $\mathbb{D}^{2}$ is fully off-diagonal, i.e. it mixes all
components of
${\overset{\,\scalebox{0.5}{($n$)}}{\Phi}}{}=({\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{},{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{i},{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij})$.
This has as a consequence that, unless one diagonalizes this operator, one has
to solve for all
$({\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{},{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{i},{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij})$
simultaneously at a given order $n$ in the $1/c$ expansion. On the contrary,
at order $2n$ in the $1/c^{2}$ expansion, one can solve for
${\overset{\,\scalebox{0.5}{($2n$)}}{B}}{}_{i}$ independently of
$({\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{},{\overset{\,\scalebox{0.5}{($2n$)}}{\gamma}}{}_{ij})$
and vice versa. Schematically this result can be written as

$\left.\mathbb{D}^{2}\right|_{\mathrm{ev.trunc.}}=\begin{pmatrix}\mathbb{D}^{2}_{\mathrm{e}}&0\\\
0&{\cal D}^{2}_{\mathrm{e}}\end{pmatrix}\,.$ (5.15)

####  5.1.4 The $1/c^{2}$ expansion as a shuffled $1/c$ expansion.

The $1/c^{2}$ expansion is a truncation of the $1/c$ expansion such that if
one were to compute the $1/c$ expansion up to order $c^{-2N}$ and put all odd
power coefficients to zero then one recovers the $1/c^{2}$ expansion up to
order $c^{-2N}$. This is however not the most efficient way to relate the
$1/c$ expansion to the $1/c^{2}$ expansion. If we compare the expansion
ansatze (5.5) and (4.1) then we see they are equivalent under the replacement
$c\rightarrow c^{2}$ and

${\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}\rightarrow{\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{}\,,\qquad{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{i}\rightarrow{\overset{\,\scalebox{0.5}{($2n$)}}{B}}{}_{i}\,,\qquad{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}\rightarrow{\overset{\,\scalebox{0.5}{($2n$)}}{\gamma}}{}_{ij}\,.$
(5.16)

This rather straightforward observation implies that the $1/c^{2}$ expansion
will contain exactly the same terms as the $1/c$ expansion with the
replacement (5.16). The key difference the replacement (5.16) makes is that it
changes the order of the various terms. As we’ll now discuss, this change of
order has a little twist, leading to a shuffling of terms among orders. But
keeping track of the order and the shuffling is not too hard.

We start by associating a weight171717The weight is the power of $c^{-1}$ with
which this object appears. The only factors of $c$ appearing are those
associated to the weights as listed in (5.17, 5.18). Please be aware that this
notion of weight as we use it in this section is unrelated to the weight as
defined in (3.60) to the relevant objects in each expansion:

$\displaystyle 1/c\mbox{ expansion:}$
$\displaystyle[{\overset{\,\scalebox{0.5}{($n$)}}{\gamma}}{}_{ij}]=[{\overset{\,\scalebox{0.5}{($n$)}}{\psi}}{}]=[{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{i}]=n\,,\qquad[\partial_{t}]=1\,.$
(5.17) $\displaystyle 1/c^{2}\mbox{ expansion:}$
$\displaystyle[{\overset{\,\scalebox{0.5}{($2n$)}}{\gamma}}{}\\!{}_{ij}]=[{\overset{\,\scalebox{0.5}{($2n$)}}{\psi}}{}]=2n\,,\quad[{\overset{\,\scalebox{0.5}{($2n$)}}{B}}{}_{i}]=2n+1\,,\quad[\partial_{t}]=1\,.$
(5.18)

Let us now consider terms in the $1/c^{2}$ expansion and trace back their
origin in the $1/c$ expansion through the replacement (5.16). Every term will
be polynomial181818To be precise: polynomial in the coefficients of weight
$\geq 1$. The zero weight fields can appear non-polynomially but this is
irrelevant for our counting argument and so we can ignore that. in the
expansion coefficients. We can group $\Psi=(\psi,\gamma)$ since they have the
same weight. Since spatial derivatives, as well as indices, have weight zero
we can ignore them. We should keep track of time derivatives, but it is not
relevant on which coefficient they act, so we will simply indicate the number
of time derivatives at the beginning of the expression. So in this schematic
fashion a term in the $1/c^{2}$ expansion has the form

$T\sim\partial_{t}^{\lambda}{\overset{\,\scalebox{0.5}{($2k_{1}$)}}{\Psi}}{}^{\nu_{1}}\ldots{\overset{\,\scalebox{0.5}{($2k_{p}$)}}{\Psi}}{}^{\nu_{p}}{\overset{\,\scalebox{0.5}{($2l_{1}$)}}{B}}{}^{\mu_{1}}\ldots{\overset{\,\scalebox{0.5}{($2l_{q}$)}}{B}}{}^{\mu_{q}}\,.$
(5.19)

Please note that $\nu_{a}$ and $\mu_{a}$, as well as $\lambda\leq 2$, indicate
positive integer powers. Via (5.18) it follows that the weight of this term is

$[T]=\lambda+2\sum_{a=1}^{p}k_{a}\nu_{a}+\sum_{a=1}^{q}(2l_{a}+1)\mu_{a}\,.$
(5.20)

Viewed through the replacement (5.16), the term $T$ originated from a term
$\tilde{T}$ in the $1/c$ expansion, with

$\tilde{T}\sim\partial_{t}^{\lambda}{\overset{\,\scalebox{0.5}{($k_{1}$)}}{\Psi}}{}^{\nu_{1}}\ldots{\overset{\,\scalebox{0.5}{($k_{p}$)}}{\Psi}}{}^{\nu_{p}}{\overset{\,\scalebox{0.5}{($l_{1}$)}}{C}}{}^{\mu_{1}}\ldots{\overset{\,\scalebox{0.5}{($l_{q}$)}}{C}}{}^{\mu_{q}}\,,$
(5.21)

which, via (5.17), has weight

$[\tilde{T}]=\lambda+\sum_{a=1}^{p}k_{a}\nu_{a}+\sum_{a=1}^{q}l_{a}\mu_{a}\,.$
(5.22)

Let us denote the weight of $T$ as $[T]=2N$ and the total number of $B$ (or
$C$) coefficients as $M=\sum_{a}\mu_{a}$. One then finds that

$[\tilde{T}]=N+\frac{\lambda-M}{2}\,.$ (5.23)

First of all, as a sanity check, remark that $\lambda-M$ is always even,
guaranteeing $[\tilde{T}]$ is integer as it should. This follows by inspection
of (5.4) or (3.57) and observing that the terms with an even number of time
derivatives come with even powers of $B$, respectively $C$, while the terms
with one time derivative come with an odd power of $B$, respectively $C$. This
is a consequence of invariance under the time redefinition symmetry (3.58), as
discussed at the end of section 3.4.

Taking this into account, together with the fact that $M$ is positive by
definition, we can conclude

$[\tilde{T}_{0\mathrm{td}}]\leq N\,,\qquad[\tilde{T}_{1\mathrm{td}}]\leq
N\,,\qquad[\tilde{T}_{\mathrm{2td},\,C}]\leq
N\,,\qquad[\tilde{T}_{\mathrm{2td},\,\mathrm{no}\,C}]=N+1\,,$ (5.24)

where $\tilde{T}_{\sharp\mathrm{td}}$ denotes terms with $\sharp$ time
derivatives and we split the case with two time derivatives in those that
contain coefficients of $C$ and those that do not.

Formula (5.23) shows that the presence of the
${\overset{\,\scalebox{0.5}{($2n$)}}{B}}{}_{i}/{\overset{\,\scalebox{0.5}{($n$)}}{C}}{}_{i}$
and time derivatives shuffles orders under the replacement (5.16) rather than
simply relating them by a factor of two. The upshot of our discussion is
however (5.24) which states that all terms up to order $c^{-2N}$ in the
$1/c^{2}$ expansion originate from a term up to order $c^{-N}$ in the $1/c$
expansion, the only exception being those terms containing two time
derivatives and no coefficients
${\overset{\,\scalebox{0.5}{($2k$)}}{B}}{}_{i}$, which originate from terms
with two time derivatives and no
${\overset{\,\scalebox{0.5}{($k$)}}{C}}{}_{i}$ at order $c^{-N-1}$ in the
$1/c$ expansion.

We can transform the above conclusion to the following algorithm, which we’ll
refer to as shuffling, to obtain the $1/c^{2}$ expansion up to a given order
$c^{-2N}$.

##### Shuffling algorithm

Compute all terms in the $1/c$ expansion up to order $c^{-N}$. Make the
replacement (5.16). Collect the resulting terms order by order by using the
rule (5.18) to determine the order. Compute the $c^{-N-1}$’th order in the
$1/c$ expansion of the two time-derivative terms without $C_{i}$ factors in
(3.57). Make the replacement (5.16). Add the result to the collection of terms
at order $c^{-2N}$.

Note that under this procedure some terms will get a weight greater than $2N$,
these can be discarded as they do not appear in the $1/c^{2}$ expansion up to
order $c^{-2N}$. For this reason the $1/c^{2}$ expansion up to order $c^{-2N}$
is simpler than the $1/c$ expansion up to order $c^{-N}$. The shuffling
algorithm can be applied to the Lagrangian as well as the equations of motion.

Let us compare the three ways to compute the $1/c^{2}$ expansion:

  * •

Direct approach
Insert the ansatz (5.5) into the relativistic Lagrangian in the form (5.3,
5.4). Expand up to order $c^{-2N}$.

  * •

Shuffling
Obtain the $1/c$ expansion up to order $c^{-N}$, use the shuffling algorithm
outlined above to get the $1/c^{2}$ expansion up to order $c^{-2N}$.

  * •

Truncation
Obtain the $1/c$ expansion up to order $c^{-2N}$, put all odd power
coefficients to zero to get the $1/c^{2}$ expansion up to order $c^{-2N}$.

In the absence of any results on the $1/c$ expansion the direct approach is
the most efficient, since the $1/c^{2}$ expansion contains less terms than the
$1/c$ expansion, and so it is easier to directly expand in $1/c^{2}$ up to
order $c^{-2N}$ than it is to first expand in $1/c$ up to order $c^{-N}$ and
then to shuffle191919Note that just like the $1/c^{2}$ expansion, also the PN
expansion is a truncation of the $1/c$ expansion. Similar to the $1/c^{2}$
expansion, a direct computation in the PN expansion will always be more
efficient computationally than one via the $1/c$ expansion combined with
shuffling and truncation.. In case there is a result of the $1/c$ expansion up
to some order $c^{-M}$ available, then the most efficient way to obtain the
$1/c^{2}$ expansion up to order $c^{-2\lfloor\frac{M}{2}\rfloor}$ is of course
to simply truncate. But shuffling is more powerful in that case, since it
directly allows to reproduce the $1/c^{2}$ expansion all the way up to order
$c^{-2(M-1)}$, and with the relatively little extra work of expanding the two
time-derivative, no $C_{i}$, part of the Lagrangian (3.57) to order $c^{-M-1}$
it provides the $1/c^{2}$ expansion up to order $c^{-2M}$.

We will use this in practice in section 5.2. Since we computed the $1/c$
expansion of the Lagrangian up to order $c^{-2}$ in section 4 we can use the
shuffling algorithm to easily find the $1/c^{2}$ expansion of the Lagrangian
up to order $c^{-4}$, which goes beyond earlier results in the literature. At
the next to leading order, i.e. order $c^{-2}$, one can explicitly see that
the result obtained by shuffling matches with that obtained by truncation, as
well as with results in the literature obtained by the direct approach.

###  5.2 Explicit expansion up to NNLO

We will now perform the $1/c^{2}$ expansion explicitly up to NNLO, i.e. order
$c^{-4}$. For notational convenience we indicate the various expansion
coefficients with different symbols rather than with their superscript, as we
did for the $1/c$ expansion. Furthermore we continue to use the same (abuse
of) notation introduced in section 4.1.3, indicating the leading order fields
with the same symbol as the full $c$ dependent field. More precisely our
expansion ansatz is

$\displaystyle\psi$ $\displaystyle=\psi+c^{-2}\phi+c^{-4}\tau+\ldots\,,$
$\displaystyle B_{i}$ $\displaystyle=B_{i}+c^{-2}Z_{i}+\ldots\,,$ (5.25)
$\displaystyle\gamma_{ij}$
$\displaystyle=\gamma_{ij}+c^{-2}\beta_{ij}+c^{-4}\epsilon_{ij}+\ldots\,.$

With this notation the replacement (5.16), used in the shuffling algorithm,
becomes

$\chi\rightarrow\phi\,,\quad\phi\rightarrow\tau\,,\quad C_{i}\rightarrow
B_{i}\,,\quad A_{i}\rightarrow
Z_{i}\,,\quad\alpha_{ij}\rightarrow\beta_{ij}\,,\quad\beta_{ij}\rightarrow\epsilon_{ij}\,.$
(5.26)

We present the results for the expansion of the Lagrangian (5.3) as

${\cal
L}^{\mathrm{e}}=\sqrt{\gamma}({\overset{\,\scalebox{0.5}{($0$)}}{L}}{}^{\mathrm{e}}+c^{-2}{\overset{\,\scalebox{0.5}{($2$)}}{L}}{}^{\mathrm{e}}+c^{-4}{\overset{\,\scalebox{0.5}{($4$)}}{L}}{}^{\mathrm{e}}+{\cal
O}(c^{-6}))\,.$ (5.27)

Note that we will present these results in the traceless gauge, see section
4.1.2, which we indicate by putting a bar on the relevant expressions. The
extra terms appearing outside this gauge can be found in appendix B.

####  5.2.1 NLO

The next to leading order, i.e. order $c^{-2}$, of the Lagrangian in the
$1/c^{2}$ expansion can be most easily obtained by simply truncating
${\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}$ as obtained in the $1/c$
expansion, see (4.26, 4.30). Alternatively one can also find it by performing
the shuffling algorithm of section 5.1.4 to
${\overset{\,\scalebox{0.5}{($1$)}}{\bar{L}}}{}$, which is given in (4.25,
4.28). Both methods give the same result, which is

$\displaystyle{\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}^{\mathrm{e}}=$
$\displaystyle\,\frac{1}{4}e^{2\psi}B_{ij}B^{ij}-\partial_{i}\psi\partial^{i}\phi-\bar{\beta}^{ij}(R_{ij}-\frac{1}{2}\partial_{i}\psi\partial_{j}\psi)$
$\displaystyle+\Gamma^{ij\,kl}D_{i}B_{j}\dot{\gamma}_{kl}+2\partial^{j}\psi\dot{B}_{j}+B^{i}\partial_{i}\psi\dot{\psi}$
(5.28)
$\displaystyle+e^{-2\psi}\left(\frac{1}{4}{\Gamma}^{kl\,mn}\dot{\gamma}_{kl}\dot{\gamma}_{mn}+\dot{\gamma}\dot{\psi}-\frac{3}{2}\dot{\psi}^{2}\right)\,.$

The $1/c^{2}$ expansion up to this order was first discussed at the level of
the equations of motion in [8] and later at the level of the action in [10,
11]. The Lagrangian ${\cal L}_{\mathrm{NRG}}$ as given in (3.29) in [11]
equals our ${\overset{\,\scalebox{0.5}{($2$)}}{{\cal L}}}{}^{\mathrm{e}}$ up
to (irrelevant) total derivatives:

${\cal L}_{\mathrm{NRG}}={\overset{\,\scalebox{0.5}{($2$)}}{{\cal
L}}}{}^{\mathrm{e}}+\sqrt{\gamma}D_{i}\left(\phi\partial^{i}\psi-\Gamma^{ijkl}B_{j}(\dot{\gamma}_{kl}-\gamma_{kl}\,\dot{\psi})+2B^{i}\dot{\psi}\right)-2\partial_{t}(\sqrt{\gamma}B^{i}\partial_{i}\psi)$
(5.29)

Here ${\overset{\,\scalebox{0.5}{($2$)}}{{\cal L}}}{}^{\mathrm{e}}$ is the
Lagrangian density outside the traceless gauge, which via (B.14) is related to
(5.28) as

${\overset{\,\scalebox{0.5}{($2$)}}{{\cal
L}}}{}^{\mathrm{e}}=\sqrt{\gamma}\left({\overset{\,\scalebox{0.5}{($2$)}}{\bar{L}}}{}^{\mathrm{e}}\big{|}_{\bar{\beta}_{ij}\rightarrow\beta_{ij}}+\frac{\beta}{2}(R-\frac{1}{2}\partial_{i}\psi\partial^{i}\psi)\right)\,.$
(5.30)

One can verify the equality (5.29) via the identification of our variables
with those of [11]:

$\displaystyle\tau_{\mu}$ $\displaystyle=e^{\psi/2}\ \delta_{\mu}^{t}\,,$
$\displaystyle h_{\mu\nu}$ $\displaystyle=e^{-\psi}\
\delta_{\mu}^{i}\delta_{\nu}^{j}\ \gamma_{ij}\,,$ $\displaystyle m_{\mu}$
$\displaystyle=e^{\psi/2}\left(\frac{1}{2}\delta_{\mu}^{t}\phi+\delta_{\mu}^{i}B_{i}\right)\,,$
(5.31) $\displaystyle\Phi_{\mu\nu}$ $\displaystyle=e^{-\psi}\
\delta_{\mu}^{i}\delta_{\nu}^{j}\ \left(\beta_{ij}-\phi\gamma_{ij}\right)\,.$

One of the interesting insights provided in [11], is that apart from obtaining
the Lagrangian by expansion of the Einstein Hilbert action, it can also be
constructed purely from symmetry considerations. Let us thus shortly comment
on the relation between the symmetries as discussed in [8, 11] and their shape
in the KS formalism.

The gauge parameters $\Lambda$ and $\xi^{\mu}$ in the transformations (3.58,
3.59) are a priori themselves functions of $c^{-1}$ and should thus be
expanded. Compatibility with the expansion ansatz (5.25) requires202020As
previously we indicate the leading order of the vectorfield $\xi^{i}$ with the
same symbol as the $c$ depedent vector field.

$\displaystyle\Lambda$
$\displaystyle=cf+c^{-1}\lambda+\mathcal{O}(c^{-3})\,,\qquad\partial_{i}f=0\,,$
(5.32) $\displaystyle\xi^{i}$ $\displaystyle=\xi^{i}+c^{-2}\zeta^{i}+{\cal
O}(c^{-4})\,.$ (5.33)

Ignoring $\delta_{f}$, we have three type of transformations of the fields. We
can start with the leading order diffeomorphisms $\xi^{i}$:

$\displaystyle\delta_{\xi}\psi$ $\displaystyle=L_{\xi}\psi\,,$
$\displaystyle\quad\delta_{\xi}\gamma_{ij}$
$\displaystyle=L_{\xi}\gamma_{ij}\,,\quad\delta_{\xi}B_{i}=L_{\xi}B_{i}+e^{-2\psi}\gamma_{ij}\partial_{t}\xi^{j}\,,$
(5.34) $\displaystyle\delta_{\xi}\phi$
$\displaystyle=L_{\xi}\phi+2B_{i}\partial_{t}\xi^{i}\,,$
$\displaystyle\delta_{\xi}\beta_{ij}$
$\displaystyle=L_{\xi}\beta_{ij}+2(B_{k}\gamma_{ij}+B_{(i}\gamma_{j)k})\partial_{t}\xi^{k}\,.$
(5.35)

These correspond to spatial diffeomorphisms also present in [11], but note
that the extra time derivative terms in the transformation of the subleading
fields correspond to an additional Milne boost in the language of [8, 11]. The
precise boost parameter in the conventions of [11] (HHO) is

$\lambda_{\mu}^{\mathrm{HHO}}=\delta_{\mu}^{i}e^{-3\psi/2}\gamma_{ij}\partial_{t}\xi^{j}\,.$
(5.36)

For the $\lambda$ transformation (5.32) one finds

$\delta_{\lambda}\psi=0\,,\,\,\,\delta_{\lambda}\gamma_{ij}=0\,,\,\,\,\delta_{\lambda}B_{i}=\partial_{i}\lambda\,,\,\,\,\delta_{\lambda}\phi=\lambda\dot{\psi}+2\dot{\lambda}\,,\,\,\,\delta_{\lambda}\gamma_{ij}=\lambda\dot{\gamma}_{ij}+2\dot{\lambda}\gamma_{ij}\,.$
(5.37)

while the subleading diffeomorphisms parameterized by $\zeta^{i}$ lead to

$\displaystyle\delta_{\zeta}\psi=0\,,\quad\delta_{\zeta}\gamma_{ij}=0\,,\quad\delta_{\zeta}B_{i}=0\,,\quad\delta_{\zeta}\phi=L_{\zeta}\psi\,,\quad\delta_{\zeta}\beta_{ij}=L_{\zeta}\gamma_{ij}\,.$
(5.38)

The transformations (5.37, 5.38) are related to those of e.g. [11] via the
field redefinition (5.31) and the following relation between the parameters

$-\Lambda_{\mathrm{HHO}}v^{\mu}+h^{\mu\nu}\zeta_{\nu}^{\mathrm{HHO}}=\lambda\delta^{\mu}_{t}+e^{-\psi}\zeta^{i}\delta^{\mu}_{i}\,.$
(5.39)

####  5.2.2 NNLO

We now proceed to the next order, i.e. order $c^{-4}$. As far as we are aware
the gravitational action has not been previously expanded to this order,
keeping all even power potentials as we do. As we will discuss below, upon
further truncation of most of these potentials our result reproduces the post-
Newtonian expansion at 1PN order. Given our computation of
${\overset{\,\scalebox{0.5}{($1$)}}{L}}{}$ and
${\overset{\,\scalebox{0.5}{($2$)}}{L}}{}$, in the $1/c$ expansion, see
sections 4.2.1, 4.2.2, it is a surprisingly short calculation to obtain
${\overset{\,\scalebox{0.5}{($4$)}}{L}}{}^{\mathrm{e}}$ by the shuffling
algorithm of section 5.1.4. The result is:

$\displaystyle\\!\\!\\!{\overset{\,\scalebox{0.5}{($4$)}}{\bar{L}}}{}^{\mathrm{e}}=$
$\displaystyle\,\,\frac{\phi}{2}e^{2\psi}B_{ij}B^{ij}+\frac{1}{2}e^{2\psi}B^{ij}Z_{ij}-\bar{\beta}^{ij}\frac{e^{2\psi}}{2}B_{ik}B_{j}{}^{k}-e^{2\psi}B^{ij}B_{i}\dot{B}_{j}-\partial_{i}\psi\partial^{i}\tau$
$\displaystyle-\bar{\epsilon}^{ij}(R_{ij}-\frac{1}{2}\partial_{i}\psi\partial_{j}\psi)-\frac{1}{4}\bar{\beta}^{kl}\bar{\beta}_{kl}(R-\frac{1}{2}\partial_{i}\psi\partial^{i}\psi)+\frac{1}{4}\Gamma^{ij\,kl\,mn}D_{i}\bar{\beta}_{kl}D_{j}\bar{\beta}_{mn}$
$\displaystyle-\frac{1}{2}\partial_{i}\phi\partial^{i}\phi+\bar{\beta}^{ik}\bar{\beta}_{k}{}^{j}(R_{ij}-\frac{1}{2}\partial_{i}\psi\partial_{j}\psi)+\bar{\beta}^{ij}\partial_{i}\psi\partial_{j}\phi-\dot{\gamma}\Big{(}D^{i}Z_{i}-\bar{\beta}^{ij}D_{i}B_{j}\Big{)}$
$\displaystyle+B_{i}\Big{(}\dot{\gamma}D_{j}\bar{\beta}^{ij}+\dot{\gamma}_{jk}(\frac{1}{2}D^{i}\bar{\beta}^{jk}-D^{k}\bar{\beta}^{ij})-\dot{\psi}\bar{\beta}^{ij}\partial_{j}\psi+\dot{\psi}\partial^{i}\phi+\dot{\phi}\partial^{i}\psi\Big{)}+\dot{\bar{\beta}}^{ij}D_{i}B_{j}$
$\displaystyle+\dot{\gamma}^{ij}\Big{(}D_{i}Z_{j}-\bar{\beta}_{ik}D^{k}B_{j}-\bar{\beta}_{jk}D_{i}B^{k}+\bar{\beta}_{ij}D_{k}B^{k}\Big{)}+Z_{i}\dot{\psi}\partial^{i}\psi+2\dot{B}_{i}(\partial^{i}\phi-\bar{\beta}^{ij}\partial_{j}\psi)$
$\displaystyle-\dot{\bar{\beta}}D^{i}B_{i}+2\dot{Z}_{i}\partial^{i}\psi-\Gamma^{ij\,kl}B_{i}\dot{B}_{j}\dot{\gamma}_{kl}-2B^{i}\dot{B}_{i}\dot{\psi}-\frac{1}{2}B_{i}B^{i}\dot{\psi}^{2}$
(5.40)
$\displaystyle+e^{-2\psi}\left(\frac{1}{2}\Gamma^{kl\,mn}\dot{\gamma}_{kl}\dot{\bar{\beta}}_{mn}+\frac{\bar{\beta}^{ij}}{2}(\dot{\gamma}_{ij}\dot{\gamma}-\dot{\gamma}_{i}{}^{k}\dot{\gamma}_{kj})+\dot{\bar{\beta}}\dot{\psi}-\bar{\beta}^{ij}\dot{\gamma}_{ij}\dot{\psi}+\dot{\gamma}\dot{\phi}-3\dot{\psi}\dot{\phi}\right)$
$\displaystyle-2\phi
e^{-2\psi}\left(\frac{1}{4}\Gamma^{kl\,mn}\dot{\gamma}_{kl}\dot{\gamma}_{mn}+\dot{\gamma}\dot{\psi}-\frac{3}{2}\dot{\psi}^{2}\right)+\frac{1}{4}\Gamma^{ij\,kl\,mn}B_{i}B_{j}\dot{\gamma}_{kl}\dot{\gamma}_{mn}\,.$

As before we presented the Lagrangian in trace-fixed form, see section 4.1.2.
The additional terms present outside this gauge choice are given in appendix
B.

Let us now discuss how the above Lagrangian describes an extension of the
post-Newtonian expansion up to 1PN order. The metric in the 1PN approximation,
see e.g. [7], is

$\displaystyle ds^{2}=$
$\displaystyle-c^{2}(1-2Uc^{-2}-2(\Psi-U^{2})c^{-4})dt^{2}-8U_{i}c^{-2}dx^{i}dt$
$\displaystyle+(1+2Uc^{-2})dx^{i}dx^{i}+{\cal O}(c^{-4})\,.$ (5.41)

The Ricci tensor of this metric is

$\displaystyle R_{tt}$
$\displaystyle=-\partial_{i}\partial^{i}U+\left(-\partial_{i}\partial^{i}\Psi+4U\partial_{i}\partial^{i}U-4\partial_{i}\dot{U}^{i}-3\ddot{U}\right)c^{-2}+{\cal
O}(c^{-4})\,,$ (5.42) $\displaystyle R_{ti}$
$\displaystyle=\left(2\partial_{j}U^{j}{}_{i}-2\partial_{i}\dot{U}\right)c^{-2}+{\cal
O}(c^{-4})\,,\qquad U_{ij}=\partial_{i}U_{j}-\partial_{j}U_{i}\,,$ (5.43)
$\displaystyle R_{ij}$
$\displaystyle=-\partial_{k}\partial^{k}U\delta_{ij}c^{-2}+{\cal
O}(c^{-4})\,.$ (5.44)

and the coefficients of the various powers of $c^{-2}$ provide the vacuum
gravitational equations up to 1PN order. If we compare (5.41) to our
relativistic metric (5.2) and expansion ansatz (5.25) one finds the
identification

$\displaystyle\psi=0\,,\,\,\,\gamma_{ij}=\delta_{ij}\,,\,\,\,B_{i}=0\,,\,\,\,\phi=-2U\,,\,\,\,\beta_{ij}=0\,,\,\,\,Z_{i}=4U_{i}\,,\,\,\,\tau=-2\Psi\,.$
(5.45)

while the field $\epsilon_{ij}$ remains undetermined by the 1PN metric ansatz
(5.41).

One sees that most of the fields that a priori can be non-trivial in the
$1/c^{2}$ expansion are assumed to be zero in the 1PN expansion. This is
because, by definition, the Post-Newtonian expansion is a non-relativistic
expansion around flat space, making it a weak gravity expansion as well. The
$1/c^{2}$ expansion is however a non-relativistic expansion around an
arbitrary quasi-static metric and includes certain non-linear, strong
gravitational effects exactly.

Although it would still be quite an effort to derive the full equations of
motion ${\overset{\,\scalebox{0.5}{($4$)}}{E}}{}$ from (5.40), it becomes
rather easy under the assumption (5.45) where most fields are set to trivial
values. From $\eqref{evnnlo}$ together with (5.12) and (5.28) we can find all
equations of motion up to order $c^{-4}$, the result is212121It is rather
straightforward to verify that the trace terms do not contribute under the
ansatz (5.45), so one can work with the trace fixed Lagrangians without loss
of generality.

$\displaystyle{\overset{\,\scalebox{0.5}{($0$)}}{E}}{}^{\mathrm{e}}_{\psi}$
$\displaystyle=0\,,\qquad\qquad{\overset{\,\scalebox{0.5}{($0$)}}{E}}{}_{\mathrm{e}}^{ij}=0\,,\qquad\qquad{\overset{\,\scalebox{0.5}{($0$)}}{E}}{}_{\mathrm{e}}^{i}=0\,.$
(5.46)
$\displaystyle{\overset{\,\scalebox{0.5}{($2$)}}{E}}{}^{\mathrm{e}}_{\psi}$
$\displaystyle=\partial_{i}\partial^{i}\phi=-2\partial_{i}\partial^{i}U\,,\quad\qquad{\overset{\,\scalebox{0.5}{($2$)}}{E}}{}_{\mathrm{e}}^{i}=0\,,\quad\qquad{\overset{\,\scalebox{0.5}{($2$)}}{E}}{}_{\mathrm{e}}^{ij}=0\,.$
(5.47)
$\displaystyle{\overset{\,\scalebox{0.5}{($4$)}}{E}}{}^{\mathrm{e}}_{\psi}$
$\displaystyle=\partial_{i}\partial^{i}\tau-2\partial_{i}\dot{Z}^{i}+3\ddot{\phi}=-2\partial_{i}\partial^{i}\Psi-8\partial_{i}\dot{U^{i}}-6\ddot{U}\,,$
(5.48)
$\displaystyle{\overset{\,\scalebox{0.5}{($4$)}}{E}}{}_{\mathrm{e}}^{i}$
$\displaystyle=-\partial_{j}Z^{ji}-2\partial^{i}\dot{\phi}=-4\partial_{j}U^{ji}+4\partial^{i}\dot{U}\,.$
(5.49)

Here we see explicitly how the equations of motion in the $1/c^{2}$ expansion
up to order $c^{-4}$ reproduce (a non-degenerate linear combination of) the
1PN equations (5.42-5.44). Note that we did not include the equation of motion
${\overset{\,\scalebox{0.5}{($4$)}}{E}}{}_{\mathrm{e}}^{ij}$, since it
involves the field $\epsilon_{ij}$, which is of higher than 1PN order.

##  6 Discussion

In this paper we revisited the $1/c$ expansion starting from the KS 3+1
formulation of GR, a lesser known dual version of the better known ADM
decomposition. Although this 3+1 formulation renders space-time diffeomorphism
invariance non-manifest, it preserves manifest spatial diffeomorphism
invariance and keeps the variational principle intact. It has the advantage it
makes the degrees of freedom more explicit, with the 4 dimensional Lorentzian
metric being parameterized by a scalar $\psi$, a vector field $C_{i}$ and a
metric $\gamma_{ij}$, with the last two carrying spatial indices only. These
fields are assumed to be analytic in the inverse speed of light $1/c$ and the
coefficients in their series expansion form the effective fields of the $1/c$
expansion. All effective fields up to order $c^{-4}$ are listed at the top of
table 2. The 3+1 formulation makes the structure of the expansion more
transparent and that allowed us to compute the effective Lagrangians to higher
order than before. We extended the computation of the Lagrangians from leading
order [18] to next to next to leading order in case of the $1/c$ expansion
including odd terms, which is described by the fields listed in the second
table from the left in table 2. In addition we made some all order
observations as well, with a computation of the universal linear part of the
expanded equations of motion and an all order gauge fixing of the trace of the
metric coefficients. The 3+1 formulation also clarifies the relation between