\section{Introduction}\label{sec:intro}
Deep Neural Networks (DNNs) have revolutionized Computer-Assisted Interventions (CAI) with applications ranging from instrument tracking to quality control \cite{MaierHein.2017,Vercauteren.2020}.
However, the design of these neural architectures is a time-consuming and complex optimization task requiring extensive hyper-parameter testing.
Consequently, CAI researchers often adopt established neural architectures designed for other vision tasks such as large-scale image classification \cite{Twinanda.2017,Hajj.2018,Miao.2016, Unberath.2019}.
But CAI problems requiring regression instead of classification or segmentation on scarcely annotated and small datasets differ from these tasks on a fundamental level.
This CAI-centric challenge is featured in instrument pose estimation for minimally-invasive temporal bone surgery \cite{Schipper.2004}:
Training on synthetic data is necessary, because hard-to-acquire real-world datasets with high-quality annotation are reserved for evaluation only.
The state-of-the-art method (i3PosNet \cite{Kugler.2020}) relies on an architecture optimized for classification.
It disregards the specialization potential as described by the no-free-lunch-theorem and as realized by DNNs for registration \cite{Balakrishnan.2019} demanding a method to automatically improve the architecture.

Optimizing neural architectures for a specific problem is challenging on its own due to the following requirements:
1)~an \textit{Efficient Environment} to test candidate performance,
2)~a \textit{Succinct Representation} to describe the architecture, and
3)~an \textit{Efficient Search Algorithm} to improve candidates quickly with limited hardware.
Automatic Neural Architecture Search (NAS) strategies were initially introduced in computer vision classification.
Previous work \cite{Elsken.2019} can be classified into two groups: discrete and continuous.
The discrete strategy (e.g. NASNet) \cite{Zoph.2017,Luo.2018} iteratively proposes, tests and improves blocks.
These blocks consist of multiple “NAS units” and are themselves combined to form full architectures.
Despite being widely used in various publications, these units are not particularly efficient in terms of both optimization and functional redundancy.
However, the iterative improvement scales well for distributed computing with massive computational effort (>=200 GPU days).
The continuous strategy (e.g. DARTS \cite{Liu.2019}) stacks all layer options together and calculates a weighted sum, motivating the name continuous.
All architectures are trained at the same time and weights are shared. This approach is more computationally efficient (4 GPU days), but very VRAM-demanding because of “stacks of layers”.
AutoSNAP combines the flexibility of NASNet with the speed of DARTS by introducing an intuitive yet succinct representation (instead of NAS units) and improving the efficient search and optimization strategy.
The medical imaging community has recently confirmed the potential of NAS methods to segmentation \cite{Dong.2019b,Zhu.2019,Yu.20191220,Weng.2019} with adaptations for scalable \cite{Kim.2019} and resource-constrained \cite{Bae.2019} environments.
We are not aware of any application of NAS to CAI.

We introduce problem-dependent learning and optimization of neural architectures to instrument pose estimation. \footnote{We will publish our code at \url{https://github.com/MECLabTUDA/AutoSNAP}.}
AutoSNAP implements problem-specific and limited-resources NAS for CAI with three major contributions:
1)~the integration of a CAI-framework as an \emph{efficient testing environment} for performance analysis (\Cref{fig:overview:a}),
2)~an extensible, \emph{succinct representation} termed Symbolic Neural Architecture Pattern (SNAP, \Cref{fig:overview:b}) to describe architecture blocks, and
3)~an \emph{efficient search algorithm} guided in ``Optimization Space'' (auto-encoder latent space) to explore and discover ``new architectures'' (\Cref{fig:overview:c}).
By integrating these factors, AutoSNAP links architecture and performance allowing for end-to-end optimization and search.
We jointly train AutoSNAP's auto-encoder (\Cref{fig:overview:c}) using a multi-component loss.
In addition to reconstruction, this loss also uses on-the-fly performance metrics from the testing environment to predict the performance of a SNAP-based architecture.
In consequence, we enable the substitution of the optimization on SNAPs by the optimization in a traversable ``Optimization Space''.
We show experimentally, that our automated approach produces improved architecture designs significantly outperforming the non-specialized state-of-the-art design.
Additionally, AutoSNAP outperforms our reimplementation of the state-of-the-art NAS method DARTS \cite{Liu.2019} for pose estimation of surgical instruments (DARTS$^*$).
\section{Methods}
\label{sec:methods}
{
\setlength{\belowcaptionskip}{0pt}%
\setlength{\abovecaptionskip}{0pt}%
\setlength{\textfloatsep}{6pt}%
\setlength{\intextsep}{0pt}%
\begin{figure}[t!]
	\centering
	\begin{subfigure}{\linewidth}
		\centering
		\includegraphics[trim=0 -30 0 0,clip,width=\textwidth]{Fig1a.png
		\caption{\emph{Testing Environment} from CAI: we search for a performant architecture (SNAPNet, green architecture) to estimate the pose of a surgical instrument from X-ray images.\vspace{10pt}}\label{fig:overview:a}
	\end{subfigure}

	\begin{subfigure}{\linewidth}
		\centering
		\includegraphics[trim=0 0 0 0,clip,width=\textwidth]{Fig1b.png}
		\caption{\emph{Succinct Representation}: left side: a SNAP (top) defines a corresponding neural block (bottom); right: we build the architecture (SNAPNet) by repeating this block.\vspace{10pt}}\label{fig:overview:b}
	\end{subfigure}
	\begin{subfigure}{\linewidth}
		\centering
		\includegraphics[trim=0 0 0 0,clip,width=0.8\textwidth]{Fig1c.png}
		\caption{\emph{Efficient Search}: the transformation of SNAPs into a unified latent-space (auto-encoder) accelerates the search by gradient ascent on the value estimator surface.}\label{fig:overview:c}
	\end{subfigure}

	\caption{Overview of AutoSNAP components}
	\label{fig:overview}
\end{figure}%
}%
Here, we present the details of pose estimation (environment), SNAP (succinct representation), the auto-encoder and the optimization scheme (efficient search).
\subsection{Problem definition of pose estimation}
\label{ssec:problemdefinition}
To guide surgeons and robotic instruments in image-guided temporal bone surgery, instrument poses need to be estimated with high-precision.
Since the direct prediction of poses from full images is difficult, the state-of-the-art modular framework i3PosNet \cite{Kugler.2020} implements ``CROP'' and ``POSE'' operations (see \Cref{fig:overview:a}).
These simplifications significantly stabilize the learning problem by converting it to a patch-based prediction of ``virtual landmarks''.
``CROP'' uses a rough initial pose estimate to extract the region of interest,
``POSE'' geometrically reconstructs the 3D pose of surgical instruments from six ``virtual landmarks''.
\Cref{fig:neuralproblem} shows a patch from a real X-ray image and predicted landmarks.
i3PosNet then iterates these operations using earlier prediction as improved estimates.

Framing this prediction task as our \emph{environment}, we search for a neural architecture (green network in \Cref{fig:overview}) that minimizes the Mean-Squared Error of the point regression task (regMSE).
i3PosNet, on the other hand, only adapts a non-specialized VGG-based architecture for this task.
Our implementation parallelizes training and evaluation on a validation dataset on multiple machines.

\begin{figure}[!t]
	\centering
	\parbox{58mm}{%
		\includegraphics[trim=0 0 0 0, clip,width=56mm]{task_viz}
	}\begin{minipage}{63mm}%
		\centering%
		\begin{tabular}{p{28mm}p{28mm}}
			\textbf{layer} & \textbf{topology}\\
			\textbf{symbols} & \textbf{symbols}\\
			\midrule
			\texttt{Conv 1x1} & \texttt{branch}\\
			\texttt{Conv 3x3} & \texttt{merge\,(add)} \\
			\texttt{DW-Conv 3x3} & \texttt{switch}\\
			\texttt{DWS-Conv 3x3} \\
			\multicolumn{2}{l}{\texttt{Max-Pool 3x3} (stride 1)}\\
			\bottomrule
		\end{tabular}

	\end{minipage}

	\parbox{53mm}{%
		\caption{\label{fig:neuralproblem}X-ray image patch of a screw with ``virtual landmarks''.}
	}\hspace{5mm}~\begin{minipage}{60mm}%
		\captionof{table}{\label{tab:symbols}SNAP symbols: \texttt{Conv}: Convolution, \texttt{DW}: Depthwise, \texttt{DWS}: DW-Separable}
	\end{minipage}
\end{figure}

\subsection{Symbolic Neural Architecture Patterns (SNAPs)}
\label{ssec:snap}
With many layer types and the design of connections in DNNs, it is currently impractical to optimize the topology and layer choice of the full architecture.
As a result, our full architecture (SNAPNet) repeats a block representing multiple operations as illustrated in \Cref{fig:overview:b}.
The topology and layer choice of this block are defined by a SNAP sequence.

To automatically generate trainable models, we introduce a language to define blocks using 8 SNAP symbols (see \Cref{tab:symbols}).
Each symbol corresponds to a modification of a stack of tensors which is used to build the model.
Five symbols specify (trainable) layers (\Cref{tab:symbols}).
Results replace the top tensor using the previous value as input.
Convolutions are always preceeded with BatchNormalization and ReLU activation.
The three topology symbols realize modification of the stack size and order for example enabling skip connections (see \Cref{fig:overview:b}, left).
\texttt{branch} duplicates the top element on the stack, \texttt{switch} swaps the top two elements and \texttt{merge\,(add)} pops the top two inputs, applies \texttt{concat} + \texttt{Conv 1x1} and pushes the result.
The stack is initialized by the output of the last two blocks and
SNAPs end with an implicit \texttt{merge\,(add)} across all tensors on the stack (ignored in \Cref{fig:overview:b} for simplicity, but \Cref{fig:SNAP} includes these).

\subsection{AutoSNAP's auto-encoder}
\label{ssec:controller}
We introduce an auto-encoder architecture to transform the SNAP sequence into a 16-dimensional vector (latent space), since continuous vectors have favorable properties for optimization.
In addition to the Encoder and Decoder, a Value Estimator predicts the architecture performance (\Cref{fig:overview:c}).

The \textbf{Encoder} ($E$) and the \textbf{Decoder} ($D$) use a Recurrent Neural Networks with mirrored architectures of two bi-directional LSTMs and two fully connected layers\footnote{We provide additional diagrams of the architectures in the Supplementary Materials.}.
Since the last Encoder-layer uses tanh-activation, the latent space is bound to the interval of [-1,1] in each dimension.
As a conditional language model, the decoder generates a sequence of symbol probabilities from a latent vector.
Finally, the \textbf{Value Estimator} ($V$) is a linear regression layer with no activation function.
Since we are interested in both high accuracy and resolution for well-performing architectures (i.e. at very low regMSE values),
we estimate $-\log_{10}(\text{regMSE})$ of the candidate architecture on the validation dataset.
This value metric improves the resolution and gradients of the value estimator.

We train the auto-encoder on three sets of input and target data:
1)~Sequence Reconstruction, 2)~Latent-space consistency and 3)~Value Regression.
1)~A Cross-Entropy (CE) loss enforces successful \emph{reconstruction} on randomly generated SNAP sequences ($\hat{X}$).
2)~The \emph{consistency of the latent space} is further supported by mapping uniform random latent vectors ($\hat{z}$) to a sequence of symbol probabilities, back to the latent space under a Mean Squared Error (MSE) loss.
3)~Value estimator and encoder are jointly trained to estimate the value criterion ($Y$) of known SNAPs ($X$) via MSE loss.
All three loss functions are minimized simultaneously via Gradient Descent
\begin{equation}
\mathcal{L} = \mathcal{L}_{\mathrm{CE}}(D(E(\hat{X})),\hat{X})+\mathcal{L}_{\mathrm{MSE}}(V(E(X)),Y)+\mathcal{L}_{\mathrm{MSE}}(E(D(\hat{z})),\hat{z})
\label{eqn:loss}
\end{equation}
Three failure modes motivate this design:
1)~A disentanglement between value estimate and architecture description occurs, rendering optimization within the latent space futile.
2)~The encoder only projects to a limited region of the latent space, thus some latent vectors have an unconstrained value estimate and no corresponding architecture (see~a).
3)~The decoder overfits to known architectures, which leaves a strong dependence on initial samples.
The Latent-space consistency enforces a bijective mapping between latent and symbol spaces (cycle consistency loss) addressing issues 1 and 2.
We mitigate problem 3 by only training the auto-encoder on randomly generated sequences.

\subsection{Exploration \& Optimization}
\label{ssec:optimization}
Our approach introduces the continuous optimization of an architecture block by gradient ascent inside the latent space of the auto-encoder (\Cref{fig:overview:c}).
The gradient for this optimization is provided by the value estimator which predicts the performance of the architecture pattern vector of a corresponding block (i.e. SNAP).
By the design of the auto-encoder, the latent space is learned to represent a performance-informed vector representation.
The thereby improved convexity of the latent space w.r.t. performance is the central intuition of the search.

The architecture search\label{chap:methods:opt}
consists of three iterative steps:
1)~Retrain the auto-encoder on all previously evaluated SNAPs (initially this is  a random sample of evaluated architectures).
2)~Transform a batch of best known SNAPs into the latent space,
optimize them via gradient ascent on value estimator gradients until the decoded latent vector no longer maps to a SNAP with known performance.
After a limit of 50 gradient steps (i.e. in vicinity of the SNAP all architectures are evaluated), a new latent vector is sampled randomly (uniform distribution) from the latent space and gradient ascent resumed.
3)~Evaluate SNAPs found in step 2) using the test environment and add them to the set of known architectures.
Finally, repeat the iteration at step 1).
\section{Experiments}
\label{sec:experiments}
We evaluate our method on synthetic and real radiographs of surgical screws in the context of temporal bone surgery \cite{Schipper.2004}.
In this section, we summarize data generation, the parameters of the AutoSNAP optimization and settings specific to pose estimation as well as the evaluation metrics.

\subsection{Training and Evaluation Datasets}
\label{ssec:dataset}
We use the publicly available i3PosNet Dataset \cite{Kugler.2020} assuming its naming conventions. Dataset A features synthetic and Dataset C real radiographs.
We train networks exclusively on Dataset A, while evaluating on synthetic and real images.
\ \\
\textbf{Dataset A: Synthetic Images}:
The dataset consists of 10,000 digital images (Subjects 1 and 2) for training and 1,000 unseen images (Subject 3) for evaluation with geometrically calculated annotations.
Training images are statically augmented 20-fold by random shifts and rotations to ensure similarity between training runs.
For architecture search, we split the training dataset by 70/10/20 (training/online validation/testing and model selection) to identify the performance of candidate models without over-fitting to the evaluation dataset.
\ \\
\textbf{Dataset C: Real Images}: Real X-ray images of medical screws on a phantom head are captured with a Ziehm c-arm machine (totaling 540 images).
Poses are manually annotated with a custom tool.

\subsection{Details of Optimization}
\label{ssec:controllersettings}
We randomly choose 100 SNAP architectures for the initial training of the autoencoder and the value-estimator.
During the search phase, 100 additional models are tested by training the models for 20 epochs and evaluation on the validation set.
We stop the search after 1500 tested models.
Our small-scale test environment uses 4 blocks with a pooling layer in the center.
Convolutions have 24/48 features before/after the pooling layer.
In total, the search takes 100 GPU hours (efficiently parallelized on two NVIDIA GeForce GTX 1080 Ti for \SI{50}{\hour}) and requires no human interaction.

\subsection{DARTS architecture search}
\label{ssec:darts}
We compare our results with DARTS${}^*$, our reimplementation of DARTS \cite{Liu.2019} where the $^*$ indicates our application to CAI.
DARTS is an efficient, state-of-the-art NAS approach for classification from computer vision (CIFAR-10).
We ported the DARTS framework to tensorflow implementing all operations as documented by the authors.
This process yields a large ``continuous model'' with weights for the contribution of individual layers.
For the evaluation and comparison with SNAPNet, we discretize and retrain the continuous model of DARTS$^*$ in analogy to the DARTS transfer from CIFAR-10 to ImageNet.
Similar to DARTS on ImageNet, our DARTS${}^*$ search took approximately 4 days on one GPU.
Inherently, DARTS cannot efficiently be parallelized across multiple GPUs or machines because all updates are applied to the same continuous model.

\begin{table*}[t]
\begin{adjustbox}{center}
\begin{tabular}{ll|rr||rr}\toprule
	& & \multicolumn{2}{c||}{Dataset A: synthetic images} & \multicolumn{2}{c}{Dataset C: real images} \\%& \textbf{weight}\\
&\textbf{Model} & \textbf{Position [mm]} & \textbf{Angle\,[deg.]} &\textbf{Position [mm]} & \textbf{Angle [deg.]}
\\ \midrule
\parbox[t]{4mm}{\multirow{4}{*}{\rotatebox[origin=c]{90}{3 iterations}}}
&i3PosNet \cite{Kugler.2020} & $0.024 \pm 0.016$ & $0.92 \pm 1.22$ & $1.072 \pm 1.481$ & $9.37 \pm 16.54$
\\
&DARTS${}^*$ \cite{Liu.2019} & $0.046 \pm 0.105$ & $1.84 \pm 6.00$ & $1.138 \pm 1.199$ & $9.76 \pm 18.60$
\\  \cline{2-6}
&SNAPNet-A (Ours)& $0.017 \pm 0.012$ & $0.52 \pm 0.88$ & $0.670 \pm 1.047$ & $7.55 \pm 14.22$
\\
&\textbf{SNAPNet-B (Ours)} & $\mathbf{0.016} \pm 0.011$ & $\mathbf{0.49} \pm 0.84$ & $\mathbf{0.461} \pm 0.669$ & $\mathbf{5.02} \pm 9.28$
\\ \midrule[1pt]
\parbox[t]{2mm}{\multirow{4}{*}{\rotatebox[origin=c]{90}{1 iteration}}}
&i3PosNet \cite{Kugler.2020} & $0.050 \pm 0.139$ & $1.14 \pm 1.50$ & $0.746 \pm 0.818$ & $6.59 \pm 10.36$
\\
&DARTS${}^*$ \cite{Liu.2019} & $0.062 \pm 0.146$ & $1.81 \pm 4.20$ & $0.810 \pm 0.770$ & $7.68 \pm 12.70$
\\  \cline{2-6}
&SNAPNet-A (Ours) & $0.026 \pm 0.029$ & $0.72 \pm 1.19$ & $0.517 \pm 0.678$ & $5.32 \pm 8.85$
\\
&\textbf{SNAPNet-B (Ours)} & $\mathbf{0.025} \pm 0.028$ & $\mathbf{0.65} \pm 1.06$ & $\mathbf{0.419} \pm 0.486$ & $\mathbf{4.36} \pm 6.88$
\\\bottomrule
\end{tabular}
\end{adjustbox}
\caption{Pose Evaluation (lower is better) for Datasets A and C (synthetic and real X-ray images). Evaluation for one and three iterations of the i3PosNet-scheme. Mean Absolute Error $\pm$ Standard Deviation of the absolute error.}
\label{tab:results}
\end{table*}
\subsection{Full-scale Retraining}
We retrain the full final architecture on the common training data of synthetic images from Dataset A \cite{Kugler.2020}.
Since efficiency is not a constraining factor for full training, we increase the number of blocks to a total of eight, four before and after the central pooling layer (see \cref{fig:overview:b}).
While SNAPNet-A uses 24/48 feature channels (same as the test environment), we increase the number of features to 56/112 (before/after the pooling layer) for SNAPNet-B.
In consequence, the number of weights approximately quadruple from SNAPNet-A and the discrete DARTS$^*$ model to SNAPNet-B and again to i3PosNet.
Like i3PosNet, we train models for 80 epochs, however using RMSProp instead of Adam.
Following the spirit of automatic machine learning, we obtained hyperparameters for these models using bayesian optimization.

\subsection{Evaluation Metrics}
To maximize comparability and reproducibility, we follow the evaluation protocol introduced by i3PosNet \cite{Kugler.2020}.
Similar to i3PosNet, we report mean and standard deviation of the absolute error for position and forward angle.
These are calculated w.r.t. the projection direction ignoring depth.
The forward angle is the angle between the image's x-axis and the screw axis projected into the image (\Cref{fig:overview:a}).
The architecture optimization performance and effectiveness is reported by the value metric ($-\log_{10}(\text{regMSE})$, see \Cref{ssec:controllersettings}).

\section{Results}
\label{sec:results}
\begin{figure}[t]
	\centering
	\parbox{68mm}{%
		\includegraphics[trim=28 5 50 30, clip,width=68mm]{search-progress-with-random}
	}
	~
	\begin{minipage}{47mm}%
		\centering%
		\includegraphics[trim=0 0 0 0, clip,width=47mm]{best-snap}
	\end{minipage}

	\parbox{68mm}{%
		\caption{\label{fig:searchefficiency}Comparison of search efficiency for \mbox{AutoSNAP} (red) and Random Search (black) for the value metric. No significant events occur after 2 GPU days (800 models).}
	}
	~
	\begin{minipage}{47mm}%
		\caption{\label{fig:SNAP}Best SNAP of Auto\-SNAP. Implicit merge of skip-connections, outputs of the previous 2 blocks are used as input.}
	\end{minipage}
\end{figure}
We compare our final architecture (SNAPNet) with two state-of-the-art methods:
1)~the manually designed i3PosNet \cite{Kugler.2020}, and
2)~an automatically identified architecture using the DARTS$^*$ \cite{Liu.2019} NAS approach.

Both AutoSNAP-based architectures outperform both reference methods by a considerable margin approximately doubling the pose estimation performance.
DARTS-based results do not even reach i3PosNet levels and show the potential of AutoSNAP for CAI applications.
For synthetic images, SNAPNet consistently outperforms position and angle estimates of all other methods including a substantial increase in performance when using i3PosNet's iterative scheme.
For difficult real X-ray images, on the other hand, SNAPNet can significantly reduce the instability of the iterative scheme resulting in a significant reduction of 90\% and 95\% confidence intervals.
In general, \emph{performance gains are slightly more pronounced for real images than for synthetic images}.

The AutoSNAP search strategy is extremely effective, discovering this best performing architecture after less than 10 GPU hours.
We illustrate the top-1 SNAP and neural block used for SNAPNet-A/B in \Cref{fig:SNAP}.
\Cref{fig:searchefficiency} compares the convergence of AutoSNAP to random search (a common NAS baseline), which samples architectures randomly from the search space.
While additional well-performing architectures are discovered after this block confirming flexibility of the approach,
even the 25th-best SNAPNet architecture outperforms the best architecture produced by random search on the validation set.

\section{Conclusion}
\label{sec:conclusion}
We propose AutoSNAP, a novel approach targeting efficient search for high-quality neural architectures for instrument pose estimation.
While the application of neural architecture search to CAI is already a novelty, our contribution also introduces SNAPs (to represent block architectures) and an auto-encoder-powered optimization scheme (efficient search algorithm).
This optimization operates on a continuous representation of the architecture in latent space.
We show more than 33 \% error reduction compared to two state-of-the-art methods: the hand-engineered i3PosNet and DARTS, a neural architecture search method.

The application of NAS to CAI is generally limited by a scaling of the  search cost with more operations (\Cref{tab:symbols}) and limited to block optimization, e.g. no macro-architecture optimization.
With respect to learning, AutoSNAP requires stable task evaluations either by good reproducibility or by experiment repetition.
Especially, the identification and exploration of unexplored architectures remains a challenge.

Methods like AutoSNAP enable efficient development and improvement of neural architectures.
It promises to help researchers in finding well-performing architectures with little effort.
In this manner, researchers can focus on the integration of deep neural networks into CAI problems.
While originally designed with instrument pose estimation in CAI in mind, in the future, we will expand AutoSNAP to other CAI and medical imaging problems.

\renewcommand{\bibsection}{\section*{References}}
\bibliographystyle{splncsnat}
\begingroup
  \small


\section{Introduction}

Dealing with the full machinery of Einstein's gravity in astrophysical situations is often a formidable task both conceptually and practically, especially in the numerical study. It is desirable to have approximation methods if the situation allows. One such a well known approximation is the post-Newtonian (PN) method (Chandrasekhar 1965; Chandrasekhar \& Nutku 1969; Chandrasekhar \& Esposito 1970; Weinberg 1972; Blanchet, Damour \& Sch\"afer 1990; Shibata \& Asada 1995; Asada, Shibata \& Futamase 1996; Asada \& Futamase 1997; Takada \& Futamase 1999; Hwang, Noh \& Puetzfeld 2008, HNP hereafter). The PN approximation takes into account of the effect of Einstein's gravity as correction terms in Newtonian equations, thus weakly relativistic but fully nonlinear. In this work, we will present the first PN (1PN) hydrodynamic equations in the {\it general} temporal gauge (hypersurface or slicing) condition, Section \ref{sec:PN}.

Besides the 1PN approximation, our main objective in this work is to present another approximation which is the special relativistic hydrodynamics in the presence of weak self-gravity with the action-at-a-distance limit. As we take into account the fully relativistic pressure, anisotropic stress and velocity, this approximation is complementary to the 1PN method and full nonlinear as well; compared with the 1PN approximation our gravity and the action-at-a-distance conditions are weaker. Our equations will reveal how the relativistic pressure, anisotropic stress and velocity gravitate, Section \ref{sec:Summary}. The equations will be derived consistently from Einstein's equation, Section \ref{sec:Derivation}.

Einstein's gravity causes the result to depend on the temporal gauge choice. Previously we studied it in the zero-shear gauge (ZSG) (Hwang \& Noh 2013c, HNc hereafter; Hwang et al.\ 2016). Here, we take the maximal slicing (MS) which sets the trace of extrinsic curvature equal to be zero (Smarr \& York 1978). The MS causes a difference {\it only} in the Poisson's equation which becomes much simpler than the previous one in the ZSG and suitable for numerical study. Here we include the anisotropic stress.

We use the terms temporal gauge, hypersurface and slicing condition interchangeably. In cosmology the MS corresponds to the uniform-expansion gauge taking the perturbed part of the expansion scalar in the normal frame (which is the same as the perturbed part of trace of extrinsic curvature with a minus sign) to be zero. The gauge degrees of freedom are completely fixed in this slicing (as well as the ZSG) together with our (spatial) gauge conditions imposed in the spatial part of the metric, and each variable in this gauge has a unique corresponding gauge-invariant combination to the fully nonlinear order (Bardeen 1988; Noh \& Hwang 2004; Hwang \& Noh 2013a, HNa hereafter).

Previously we presented a fully nonlinear and exact cosmological perturbation theory in the absence of spatially transverse-tracefree perturbation (HNa). Although the formulation was made in the cosmological context it is applicable to the non-expanding background by setting the scale factor equal to one, and letting the background Robertson-Walker metric to be the Minkowsky one. We have shown that in the $c$-goes-to-infinity limit our perturbation formulation of Einstein's gravity properly reproduce the Newtonian hydrodynamics in two gauge conditions (Hwang \& Noh 2013b, HNb hereafter): these are the ZSG and the uniform-expansion gauge (the MS in Minkowsky background). The 1PN equations are also properly recovered without fixing the temporal gauge condition (Noh \& Hwang 2013).

As we have a curious difference of the presence/absence of the $3p/c^2$ term in the Poisson's equation depending on the gauge [compare Equations (\ref{Poisson-MS}) and (\ref{Poisson-ZSG})], we pay attention to it in the corresponding PN formulation (Section \ref{sec:PN}) as well as in the gauge transformation to the linear order (Section \ref{sec:GT}).

In the Appendix A we present the special relativistic hydrodynamics in the Minkowsky spacetime now including the anisotropic stress; as far as we are aware the case with the stress is not known in the literature. In the Appendix B we present the fully nonlinear and exact perturbation equations in a gauge ready (i.e., without taking the temporal gauge) form ignoring the transverse-tracefree type spatial metric in nonexpanding background; this is also new in the literature.

Section \ref{sec:Summary} is a summary of the newly proposed special relativistic hydrodynamic formulation with gravity. Section \ref{sec:Derivation} derive the formulation from full Einstein's gravity in the two gauge conditions. Section\ \ref{sec:PN} presents the complementary 1PN equations without taking the slicing condition.

For readers interested in numerical implementation, here we summarize the fundamental sets of equations derived in this work. Equations (\ref{continuity-practice})-(\ref{Poisson-MS}) are the complete set of equations for special relativistic hydrodynamic with gravity in the MS; for more generally valid form see Equations (\ref{continuity})-(\ref{M-conservation}).  An alternative expression is Equations (\ref{continuity-SR-practice})-(\ref{M-conservation-SR-practice}) and (\ref{Poisson-MS-E}). In the post-Newtonian approximation Equations (\ref{1PN-0})-(\ref{1PN-5}) are the complete set with the general gauge condition in Equation (\ref{1PN-gauge}).

\begin{widetext}
\section{Special relativistic hydrodynamics with gravitation: summary}
                                             \label{sec:Summary}

The special relativistic extension of the mass conservation, the energy conservation, the momentum conservation, and the Poisson's equations, respectively, in the MS is
\bea
   & &
       {d \overline \varrho \over dt}
       + \overline \varrho
       \left( \nabla \cdot {\bf v} + {d \over dt} \ln{\gamma} \right)
       = 0,
   \label{continuity-practice} \\
   & &
       {d \varrho \over dt}
       + \left( \varrho + {p \over c^2} \right)
       \left( \nabla \cdot {\bf v} + {d \over dt} \ln{\gamma} \right)
       = - {1 \over c^2} \Pi_i^j \nabla_j v^i
       - {1 \over c^4} \Pi_{ij} v^i \dot v^j,
   \label{E-conservation-practice} \\
   & &
       {d {\bf v} \over dt}
       =
       \nabla \Phi
       - {1 \over \varrho + p/c^2} {1 \over \gamma^2}
       \left( \nabla p
       + {1 \over c^2} {\bf v} \dot {p} \right)
   \nonumber \\
   & & \qquad
       + {1 \over \varrho + p/c^2} {1 \over \gamma^2} \left\{
       - \Pi^j_{i,j}
       + {1 \over c^2} \left[ {\bf v} \left( \Pi_j^k v^j \right)_{,k}
       - {1 \over \gamma^2} \left( \Pi_{ij} v^j \right)^{\displaystyle\cdot} \right]
       + {1 \over c^4} {\bf v}
       \left( \Pi_{jk} v^j v^k \right)^{\displaystyle\cdot} \right\},
   \label{M-conservation-practice} \\
   & &
       \Delta \Phi
       + 4 \pi G \left( \varrho + 3 {p \over c^2} \right)
       = - {8 \pi G \over c^2} \left[
       \left( \varrho + {p \over c^2} \right) \gamma^2 v^2
       + \Pi^i_i \right],
   \label{Poisson-MS}
\eea
with
\bea
   & &
       {d \over dt} \equiv {\partial \over \partial t}
       + {\bf v} \cdot \nabla, \quad
       \gamma = {1 \over \sqrt{ 1 - {v^2/ c^2}}},
   \label{gamma}
\eea
where $\varrho \equiv \overline \varrho (1 + \Pi/c^2)$, $p$ and ${\bf v}$ are the density, pressure and velocity, respectively, and $\gamma$ is the Lorentz factor with $v^2 \equiv v^k v_k$; $\overline \varrho$ is the rest mass density and $\overline \varrho \Pi$ is the internal energy density; $\Pi_{ij}$ is the anisotropic stress, see Equation (\ref{Pi_ij}).

These are complete (closed) system; the $p$ and $\Pi_{ij}$ are provided by the equations of state depending on the medium we are considering. Equation for $\ln{\gamma}$ follows from Equation (\ref{M-conservation-practice}), and is presented in Equation  (\ref{M-conservation-3}). Notice that the effect of gravity on the dynamics in Equation (\ref{M-conservation-practice}) is rather conventional. Except for the gravity Equations (\ref{continuity-practice})-(\ref{M-conservation-practice}) are the same as in the special relativistic hydrodynamics (see the Appendix A); For conventional forms often used in the numerical studies see Equations (\ref{continuity-SR})-(\ref{M-conservation-SR}). Equations (\ref{continuity-practice}) and (\ref{E-conservation-practice}) give
\bea
   & &
       \overline \varrho {d \Pi \over dt}
       + p \left( \nabla \cdot {\bf v} + {d \over dt} \ln{\gamma} \right)
       = - \Pi_i^j \nabla_j v^i
       - {1 \over c^2} \Pi_{ij} v^i \dot v^j.
   \label{Pi-equation-practice}
\eea

Our metric convention is
\bea
   & &
       ds^2 = - \left( 1 - {2 \Phi \over c^2} \right) c^2 d t^2
       - 2 \chi_i c dt d x^i
       + \left( 1 + {2 \Psi \over c^2} \right) \delta_{ij} d x^i d x^j.
   \label{metric}
\eea
The index of $\chi_i$ is raised and lowered by $\delta_{ij}$ as the metric. Keeping $\chi^i$ in the metric is essentially important in the MS analysis; see below Equation (\ref{Phi-Psi}). We {\it ignored} the transverse-tracefree perturbation in the spatial metric, and imposed spatial gauge conditions so that the spatial part of the metrics becomes simple as in Equation (\ref{metric}) to the fully nonlinear orders (Bardeen 1988; Noh \& Hwang 2004; HNa); in the spatial part of the metric tensor
\bea
   & &
       g_{ij} = \left( 1 + {2 \Psi \over c^2} \right) \delta_{ij}
       + 2 \overline \gamma_{,ij}
       + C^{(v)}_{i,j} + C^{(v)}_{j,i}
       + 2 C^{(t)}_{ij},
\eea
with $C^{(v)i}_{\;\;\;\;\;\;,i} \equiv 0$ and $C^{(t)j}_{\;\;\;\;\;i,j} \equiv 0 \equiv C^{(t)i}_{\;\;\;\;\;i}$, we imposed the spatial gauge conditions as
\bea
   & &
       \overline \gamma \equiv 0 \equiv C^{(v)}_i,
   \label{spatial-gauge}
\eea
and set $C^{(t)}_{ij} = 0$ as we ignore the transverse-tracefree part of the metric. We take the same spatial gauge in the 1PN approximation, see Equation (\ref{metric-1PN}) (HNP). Although we ignore the transverse-tracefree part of the metric we keep the corresponding part of the anisotropic stress. Thus we are considering situations where the tensor part (gravitational waves) of the metric is not significantly excited; for example, in the PN approximation the $C^{(t)}_{ij}$ part appears from the 2.5PN order (Chandrasekhar \& Esposito 1970).

The $\Psi$ does not affect the fluid motions and is determined by
\bea
   & &
       \Delta \Psi + 4 \pi G \varrho
       = - {4 \pi G \over c^2} \left[ \left(
       \varrho + {p \over c^2} \right) \gamma^2 v^2
       + \Pi^i_i \right].
   \label{Poisson-Psi}
\eea
Thus, with relativistic pressure, anisotropic stress or velocity, $\Psi$ in general differs from $\Phi$. Notice how the relativistic pressure, anisotropic stress and velocity gravitate. We may call Equations (\ref{Poisson-MS}) and (\ref{Poisson-Psi}) the extended Newtonian and the PN Poisson's equations, respectively, where $\Phi$ and $\Psi$ may be termed as the Newtonian and the PN gravitational potentials, respectively; in the Newtonian limit $\Psi$ does not have role in Newtonian dynamics, but we have $\Psi = \Phi$.

In the case of the ZSG which sets the longitudinal part of $\chi_i$ equal to zero, the only difference appears in the Newtonian Poisson's equation. Instead of Equation (\ref{Poisson-MS}) we have
\bea
   & &
       \Delta \Phi
       + 4 \pi G \varrho
       = - {8 \pi G \over c^2} \left[
       \left( \varrho + {p \over c^2} \right)
       \gamma^2 v^2 + \Pi^i_i \right]
       + {12 \pi G \over c^2}
       \Delta^{-1} \nabla_i \nabla_j
       \left[ \left( \varrho + {p \over c^2} \right)
       \gamma^2 {v^i v^j} + \Pi^{ij} \right],
   \label{Poisson-ZSG}
\eea
and Equations (\ref{continuity-practice})-(\ref{M-conservation-practice}) and (\ref{Poisson-Psi}) remain the same (Hwang et al.\ 2016). Notice the complication involving the inverse Laplacian operator $\Delta^{-1}$ and the absence of $3 p/c^2$ term in Equation (\ref{Poisson-ZSG}) compared with Equation (\ref{Poisson-MS}) in the MS.

In order to derive the above equations we {\it assume} the weak gravity and the action-at-a-distance conditions
\bea
   & &
       {\Phi \over c^2} \ll 1, \quad
       {\Psi \over c^2} \ll 1, \quad
       \gamma^2 {t_\ell^2 \over t_g^2} \ll 1,
   \label{assumptions}
\eea
where $t_g \sim 1/\sqrt{G \varrho}$ and $t_\ell \sim \ell/c \sim 2 \pi /(kc)$ are gravitational time scale and the light propagating time scale of a characteristic length scale $\ell$, respectively; $k$ is the wave number with $\Delta = - k^2$. The action-at-a-distance condition implies that we keep the action-at-a-distance nature of Newtonian theory. In our approximation presented above we {\it ignore} these dimensionless quantities compared with order unity.

We have
\bea
   & &
       \gamma^2 {t_\ell^2 \over t_g^2}
       \sim \gamma^2 {G \varrho \ell^2 \over c^2}
       \sim \gamma^2 {G M \over \ell c^2}
       \sim {\Phi \over c^2}
       \sim {\Psi \over c^2},
\eea
where in the last two steps we used our special relativistic Poisson's equations in Equations (\ref{Poisson-MS}), (\ref{Poisson-Psi}) and (\ref{Poisson-ZSG}); $M$ is the characteristic mass scale. This estimate shows that all the conditions in Equation (\ref{assumptions}) are of the same order. Thus, even for ultra-relativistic velocity ($\gamma \gg 1$) the action-at-a-distance condition is comparable to the weak gravity conditions due to the relativistic boost corrections in the Poisson's equations. Still, in such an ultra-relativistic situation we can make an argument that unless we have steep gradient in the gravitational potential the gravity term is negligible in the hydrodynamic equations, see below Equation (\ref{M-conservation-4}); without the steep gradient in $\Phi$, $\nabla \Phi$ term in Equation (\ref{M-conservation-practice}) is negligible compared with the convective term in the left-hand-side due to the weak gravity condition. This is true despite the fact that in Equations (\ref{Poisson-MS}), (\ref{Poisson-Psi}) and (\ref{Poisson-ZSG}) the Lorentz boosted density and pressure in the right-side-sides of these equations dominate the gravity. In the presence of steep gradients of $\Phi$ and $\Psi$ we have a more general extended hydrodynamic conservation equations derived in Equations (\ref{continuity})-(\ref{M-conservation}), while the Poisson's equations in Equations (\ref{Poisson-MS}), (\ref{Poisson-Psi}) and (\ref{Poisson-ZSG}) remain the same.
The gravity might have non-negligible roles in situations with mildly relativistic velocity (caused by non-gravitational means) with $\gamma \sim 1$.

\subsection{Newtonian limit}

By taking $c$-goes-to-infinity limit we recover the Newtonian hydrodynamic equations in both gauges (HNb). We have
\bea
   & &
       {\partial \overline \varrho \over \partial t}
       + \nabla \cdot \left( \overline \varrho {\bf v} \right)
       = 0,
   \\
   & &
       {\partial {\bf v} \over \partial t}
       + {\bf v} \cdot \nabla {\bf v}
       = \nabla \Phi
       - {1 \over \varrho} \left( \nabla p
       + \nabla_j \Pi^j_i \right),
   \\
   & &
       \Delta \Phi + 4 \pi G \varrho = 0.
\eea
Even in the Newtonian context, the internal energy equation  (\ref{Pi-equation-practice}) in the c-infinity limit becomes
\bea
   & &
       \overline \varrho {d \Pi \over dt}
       + p \nabla \cdot {\bf v}
       = - \Pi_i^j \nabla_j v^i.
\eea
Including the internal energy the energy conservation equation (\ref{E-conservation-practice}) becomes
\bea
   & &
       {d \varrho \over dt}
       + \left( \varrho + {p \over c^2} \right)
       \nabla \cdot {\bf v}
       = - {1 \over c^2} \Pi_i^j \nabla_j v^i.
\eea

\subsection{Slow-motion and Newtonian stress limit}

Now, for non-relativistic velocity ($v^2/c^2 \rightarrow 0$) and Newtonian anisotropic stress, but still with fully relativistic pressure, Equations (\ref{continuity-practice})-(\ref{Poisson-MS}) become
\bea
   & &
       {d \overline \varrho \over dt}
       + \overline \varrho \nabla \cdot {\bf v}
       = {\overline \varrho \over \varrho + p/c^2}
       {1 \over c^2} {\bf v} \cdot \nabla p,
   \label{continuity-NR} \\
   & &
       {d \varrho \over dt}
       + \left( \varrho + {p \over c^2} \right)
       \nabla \cdot {\bf v}
       = {1 \over c^2} {\bf v} \cdot \nabla p
       - {1 \over c^2} \Pi_i^j \nabla_j v^i,
   \label{E-conservation-NR} \\
   & &
       {d {\bf v} \over dt}
       = \nabla \Phi
       - {1 \over \varrho + p/c^2}
       \left( \nabla p
       + {1 \over c^2} {\bf v} \dot p
       + \nabla_j \Pi_i^j \right),
   \label{M-conservation-NR} \\
   & &
       \Delta \Phi
       + 4 \pi G \left( \varrho + 3 {p \over c^2} \right)
       = 0,
   \label{Poisson-MS-NR}
\eea
and Equation (\ref{Poisson-Psi}) becomes
\bea
   & &
       \Delta \Psi + 4 \pi G \varrho = 0.
   \label{Poisson-Psi-NR}
\eea
In the ZSG, instead of Equation (\ref{Poisson-MS-NR}) we have (HNc)
\bea
   & &
       \Delta \Phi + 4 \pi G \varrho
       = 0.
   \label{Poisson-ZSG-NR}
\eea
The only difference between the two gauges is the presence or absence of $3p/c^2$ term in the Poisson's equation. Equations (\ref{continuity-NR}) and (\ref{E-conservation-NR}) give
\bea
   & &
       \overline \varrho {d \Pi \over dt}
       + p \nabla \cdot {\bf v}
       = {p/c^2 \over \varrho + p/c^2} {\bf v} \cdot \nabla p
       - \Pi_i^j \nabla_j v^i.
\eea

\subsection{Static limit}

In a static equilibrium situation with all time derivatives vanishing, Equation (\ref{M-conservation-practice}) leads to
\bea
   & &
       \nabla_i \Phi = {1 \over \varrho + p/c^2}
       \left( \nabla_i p + \nabla_j \Pi_i^j \right).
\eea
Equations (\ref{Poisson-MS}) and (\ref{Poisson-Psi}) become
\bea
   & &
       \Delta \Phi = - 4 \pi G \left( \varrho + 3 {p \over c^2} \right),
   \label{Poisson-MS-static} \\
   & &
       \Delta \Psi = - 4 \pi G \varrho,
\eea
where we used Equation (\ref{Pi_ii}).

Further assuming a spherical symmetry, we have
\bea
   & &
       {dp \over dr}
       = - {4 \pi G \over r^2} \left( \varrho + {p \over c^2} \right)
       \int_0^r \left( \varrho + {3 p \over c^2} \right) r^2 dr,
   \label{static-case}
\eea
which is the well known Tolman-Oppenheimer-Volkoff relation in the weak gravity limit (Tolman 1939, Oppenheimer \& Volkoff 1939). The presence of $3p/c^2$-terms in the slow-motion limit of Equations (\ref{Poisson-MS}) and (\ref{Poisson-MS-NR}) are consistent with the historically known results (Whittaker 1935, McCrea 1951, Harrison 1965).

It is curious to notice, however, that this $3 p/ c^2$-term is absent in the ZSG: see Equations (\ref{Poisson-ZSG}) and (\ref{Poisson-ZSG-NR}).  In the ZSG, instead of Equation (\ref{Poisson-MS-static}), we have
\bea
   & &
       \Delta \Phi = - 4 \pi G \left( \varrho
       + {3 \over c^2} \Delta^{-1} \nabla_i \nabla_j \Pi^{ij} \right).
\eea
This makes the ZSG result inconsistent with the Tolman-Oppenheimer-Volkoff relation by missing $3p/c^2$ term in Equation (\ref{static-case}) which can be regarded as a serious disqualification in the spherically symmetric situation. At the moment the meaning of why we have such an inconsistent result in the ZSG is unclear to us; derivations in the ZSG in HNc and Hwang et al.\ (2016) are consistent with full Einstein's equations just as in the case for the MS performed in this work; the ZSG case is also derived in parallel below. This has motivated us to pay special attention to this difference (presence or absence of $3p/c^2$-term in the Poisson's equation depending on hypersurface) by analyzing the 1PN approximation and gauge transformation in perturbation theory: the difference is real, see paragraphs below Equation (\ref{1PN-gauge}) and Section \ref{sec:GT}.

\section{Derivation}
                                             \label{sec:Derivation}

Here, we derive Equations (\ref{continuity-practice})-(\ref{Poisson-MS}), (\ref{Poisson-Psi}) and (\ref{Poisson-ZSG}) from full Einstein's equations assuming Equation (\ref{assumptions}). There is no {\it a priori} reason for a formulation based on such assumptions should be possible. The proof based on full Einstein's equation will guarantee the consistency as a result. We will use the fully nonlinear and exact perturbation formulation of the perturbation equations summarized in the Appendix B.

In our derivation of the conservation equations, we apply the weak gravity condition, but {\it kept} ${\bf v} \cdot \nabla \Phi/c^2$ term compared with $\nabla \cdot {\bf v}$, and ${\bf v} {\bf v} \cdot \nabla \Phi/c^2$ term compared with ${\bf v} \cdot \nabla {\bf v}$, etc. (not in the combination with the anisotropic stress though), although these terms are practically negligible in our weak gravity condition, see below Equation (\ref{M-conservation-4}). We ignore $\dot \Psi /c^2$ term compared with unity. Thus, instead of Equations (\ref{continuity-practice})-(\ref{M-conservation-practice}), we will derive the following more general forms
\bea
   & &
       {d \overline \varrho \over dt}
       + \overline \varrho
       \left[ \nabla \cdot {\bf v} + {d \over dt} \ln{\gamma} + {1 \over c^2} {\bf v} \cdot \nabla \left( \Psi - \Phi \right) \right]
       = 0,
   \label{continuity} \\
   & &
       {d \varrho \over dt}
       + \left( \varrho + {p \over c^2} \right)
       \left[ \nabla \cdot {\bf v} + {d \over dt} \ln{\gamma} + {1 \over c^2} {\bf v} \cdot \nabla \left( \Psi - \Phi \right) \right]
       = - {1 \over c^2} \Pi^i_j v^j_{\;\;,i}
       - {1 \over c^4} \Pi_{ij} v^i \dot v^j,
   \label{E-conservation} \\
   & &
       {d {\bf v} \over dt}
       =
       \nabla \Phi
       - {1 \over c^2} {\bf v} {\bf v} \cdot \nabla \Phi
       + {v^2 \over c^2} \nabla \Psi
       - {1 \over \varrho + p/c^2} {1 \over \gamma^2}
       \left( \nabla p
       + {1 \over c^2} {\bf v} \dot {p} \right)
   \nonumber \\
   & & \qquad
       + {1 \over \varrho + p/c^2} {1 \over \gamma^2} \left\{
       - \Pi^j_{i,j}
       + {1 \over c^2} \left[ {\bf v} \left( \Pi_j^k v^j \right)_{,k}
       - {1 \over \gamma^2} \left( \Pi_{ij} v^j \right)^{\displaystyle\cdot} \right]
       + {1 \over c^4} {\bf v}
       \left( \Pi_{jk} v^j v^k \right)^{\displaystyle\cdot} \right\}.
   \label{M-conservation}
\eea
Equations (\ref{continuity}) and (\ref{E-conservation}) give
\bea
   & &
       \overline \varrho {d \Pi \over dt}
       + p \left[ \nabla \cdot {\bf v} + {d \over dt} \ln{\gamma}
       + {1 \over c^2} {\bf v} \cdot \nabla \left( \Psi - \Phi \right) \right]
       = - \Pi^i_j v^j_{\;\;,i}
       - {1 \over c^2} \Pi_{ij} v^i \dot v^j.
   \label{Pi-equation}
\eea
In this more general forms, the relativistic pressure, anisotropic stress and velocity in general demand the presence of two gravitational potentials, $\Phi$ and $\Psi$. Although these will be derived consistently from Einstein's gravity, we will argue that these additional terms are, unless we have steep gradients in $\Psi$ and $\Psi$, practically negligible, see below Equation (\ref{M-conservation-4}). By ignoring these terms, thus strictly applying the weak gravity conditions in Equation (\ref{assumptions}), we have Equations (\ref{continuity-practice})-(\ref{M-conservation-practice}).

We consider a general fluid energy-momentum tensor
\bea
   & &
       T_{ab} = \left( \varrho c^2 + p \right) u_a u_b + p g_{ab}
       + \pi_{ab},
   \label{Tab}
\eea
with $\mu \equiv \varrho c^2$. The anisotropic stress is introduced as
\bea
   & &
       \pi_{ij} \equiv \Pi_{ij},
   \label{Pi_ij}
\eea
where the indices of $\Pi_{ij}$ is raised and lowered by $\delta_{ij}$ as the metric. From $\pi^c_c \equiv 0$ we have
\bea
      \Pi^i_i = \Pi_{ij} {v^i v^j \over c^2}.
   \label{Pi_ii}
\eea
The fluid four-vector is introduced as
\bea
   & &
       u_i \equiv \gamma {v_i \over c}, \quad
       \gamma \equiv - n_c u^c
       = {1 \over \sqrt{ 1 - {v^k v_k
       \over c^2 (1 + 2 \Psi/c^2) }}},
   \label{v_i}
\eea
where $v_i$ is the fluid three-velocity measured by the Eulerian observer (HNa); the index of $v_i$ is raised and lowered by $\delta_{ij}$ as the metric; $n_a$ is the normal-frame four-vector with $n_i \equiv 0$. In the weak gravity approximation the Lorentz factor $\gamma$ becomes the one in Equation (\ref{gamma}).

The ADM (Arnowitt-Deser-Misner) momentum constraint in Equation (\ref{eq3}) gives
\bea
   & &
       {2 \over 3} \kappa_{,i}
       + c \left( {1 \over 2} \Delta \chi_i
       + {1 \over 6} \chi^j_{\;\;,ij} \right)
       = - {8 \pi G \over c^2}
       \left[ \left( \varrho + {p \over c^2} \right)
       \gamma^2 v_i + {1 \over c^2} \Pi_{ij} v^j \right].
   \label{mom-constraint}
\eea
We decompose $\chi_i$ into the longitudinal and transverse parts as $\chi_i \equiv \chi_{,i} + \chi^{(v)}_i$ with $\chi^{(v)i}_{\;\;\;\;\;\;,i} \equiv 0$. The MS takes $\kappa \equiv 0$, and the ZSG takes $\chi \equiv 0$ as the slicing condition; $\kappa$ is the trace of extrinsic curvature, and the tracefree part of extrinsic curvature vanishes for $\chi_i = 0$, see Equation (B7) in HNa.
In the MS, thus $\kappa \equiv 0$, this equation can be decomposed as
\bea
   & &
       \chi = - {12 \pi G \over c^3} \Delta^{-2} \nabla^i
       \left[ \left( \varrho + {p \over c^2} \right)
       \gamma^2 v_i + {1 \over c^2} \Pi_{ij} v^j \right],
   \label{mom-constraint-chi} \\
   & &
       \Delta \chi^{(v)}_i
       = - {16 \pi G \over c^3}
       \left\{ \left( \varrho + {p \over c^2} \right)
       \gamma^2 v_i + {1 \over c^2} \Pi_{ij} v^j
       - \nabla_i \Delta^{-1} \nabla^j \left[
       \left( \varrho + {p \over c^2} \right)
       \gamma^2 v_j + {1 \over c^2} \Pi_{jk} v^k \right] \right\}.
   \label{mom-constraint-chi-v}
\eea
Similarly, in the ZSG, thus $\chi \equiv 0$, we have
\bea
   & &
       \kappa = - {12 \pi G \over c^2} \Delta^{-1} \nabla^i
       \left[ \left( \varrho + {p \over c^2} \right)
       \gamma^2 v_i + {1 \over c^2} \Pi_{ij} v^j \right],
   \label{mom-constraint-kappa}
\eea
and $\chi^{(v)}_i$ is the same as in Equation (\ref{mom-constraint-chi-v}). From these we can estimate
\bea
   & &
       {1 \over c} \Delta^{-1} \kappa_{,i}
       \sim \chi_{,i} \sim \chi^{(v)}_i
       \sim \left( \gamma^2 {t_\ell^2 \over t_g^2} \right)
       {v_i \over c},
   \label{chi-estimate}
\eea
where it is understood that the MS and the ZSG consider $\kappa \equiv 0$ and $\chi \equiv 0$, respectively. Thus, under the action-at-a-distance condition $\Delta^{-1} \kappa_{,i}/c$, $\chi_{,i}$ and $\chi^{(v)}_i$ are negligible compared with $v_i/c$.

\subsection{Conservation equations}

Using Equations (\ref{assumptions}) and (\ref{chi-estimate}), the covariant momentum conservation in Equation (\ref{eq7-cov}) gives
\bea
   & &
       {1 \over \gamma} {d \over dt} \left( \gamma {\bf v} \right)
       = \nabla \Phi + {v^2 \over c^2} \nabla \Psi
       - {1 \over \varrho + p/c^2} \left( {1 \over \gamma^2} \nabla p + {\bf v} {1 \over c^2} {d {p} \over dt} \right)
   \nonumber \\
   & & \qquad
       + {1 \over \varrho + p/c^2} \left\{
       - {1 \over \gamma^2} \Pi^j_{i,j}
       + {1 \over c^2} \left[ {\bf v} \Pi_j^k v^j_{\;\;,k}
       - {1 \over \gamma^2} \left( \Pi_{ij} v^j \right)^{\displaystyle\cdot} \right]
       + {1 \over c^4} {\bf v} \Pi_{jk} v^j \dot v^k \right\},
   \label{M-conservation-1}
\eea
with ${\bf v} = v_i$. This can be arranged to give Equation (\ref{M-conservation}) and
\bea
   & &
       {d \over dt} \ln{\gamma}
       = {1 \over c^2} {\bf v} \cdot \nabla \Phi
       - {1 \over \varrho + p/c^2}
       {1 \over c^2}
       \left( {d p \over dt}
       - {1 \over \gamma^2} \dot {p} \right)
   \nonumber \\
   & & \qquad
       + {1 \over \varrho + p/c^2} {1 \over c^2} \left\{
       - {1 \over \gamma^2} v^i \Pi^j_{i,j}
       + {1 \over c^2} \left[ v^2 \Pi_j^k v^j_{\;\;,k}
       - {1 \over \gamma^2} v^i \left( \Pi_{ij} v^j \right)^{\displaystyle\cdot} \right]
       + {1 \over c^4} v^2 \Pi_{jk} v^j \dot v^k \right\}.
   \label{M-conservation-3}
\eea

The continuity equation $(\overline \varrho u^c)_{;c} = 0$ can be written as $\widetilde {\dot {\overline \varrho}} + \overline \varrho \theta = 0$, and leads to Equation (\ref{continuity}). We used
\bea
   & &
       \widetilde {\dot \varrho} \equiv c \varrho_{,c} u^c = \gamma {d \varrho \over dt}, \quad
       \theta \equiv c u^c_{\;\;;c} = \gamma \left[ \nabla \cdot {\bf v} + {d \over dt} \ln{\gamma} + {1 \over c^2} {\bf v} \cdot \nabla \left( \Psi - \Phi \right) \right],
\eea
which follow from Equations (C2), (C3), (D10) and (D11) in HNa. Equation (\ref{eq0}) also gives Equation (\ref{continuity}) directly.

Using Equation (\ref{M-conservation-3}) we have
\bea
   & &
       {d \overline \varrho \over dt}
       + \overline \varrho \nabla \cdot {\bf v}
       =
       - \overline \varrho {1 \over c^2}
       {\bf v} \cdot \nabla \Psi
       + {\overline \varrho \over c^2}
       {1 \over \varrho + p/c^2}
       \left( {d p \over dt}
       - {1 \over \gamma^2} \dot {p} \right)
   \nonumber \\
   & & \qquad
       - {1 \over c^2}
       {\overline \varrho \over \varrho + p/c^2} \left\{
       - {1 \over \gamma^2} v^i \Pi^j_{i,j}
       + {1 \over c^2} \left[ v^2 \Pi_j^k v^j_{\;\;,k}
       - {1 \over \gamma^2} v^i \left( \Pi_{ij} v^j \right)^{\displaystyle\cdot} \right]
       + {1 \over c^4}  \Pi_{jk} v^j \dot v^k \right\}.
   \label{Continuity-1}
\eea

The covariant energy conservation in Equation (\ref{eq6-cov}) gives Equation (\ref{E-conservation}). Using Equation (\ref{M-conservation-3}) we have
\bea
   & &
       {d \varrho \over dt}
       + \left( \varrho + {p \over c^2} \right)
       \nabla \cdot {\bf v}
       =
       - \left( \varrho + {p \over c^2} \right) {1 \over c^2}
       {\bf v} \cdot \nabla \Psi
       + {1 \over c^2} \left( {d p \over dt}
       - {1 \over \gamma^2} \dot {p} \right)
   \nonumber \\
   & & \qquad
       + {1 \over c^2} \left[ - 2 \Pi^j_i v^i_{\;,j}
       + {1 \over \gamma^2} \left( \Pi^j_i v^i \right)_{,j} \right]
       + {1 \over c^4} \left[ - 2 \Pi_{ij} v^i \dot v^j
       + {1 \over \gamma^2} \left( \Pi_{ij} v^i v^j
       \right)^{\displaystyle\cdot} \right].
   \label{E-conservation-1}
\eea

For later use, we present alternative expressions of the conservation equations
\bea
   & &
       {\partial \over \partial t} \left[ \left(
       \varrho + {p \over c^2} \right) \gamma^2 \right]
       + \nabla \cdot \left[ \left(
       \varrho + {p \over c^2} \right)
       \gamma^2 {\bf v} \right]
   \nonumber \\
   & & \qquad
       = \left( \varrho + {p \over c^2} \right)
       {1 \over c^2} \gamma^2 {\bf v} \cdot
       \nabla \left( 2 \Phi - \Psi \right)
       + {1 \over c^2} \dot {p}
       - {1 \over c^2} \nabla_i \left( \Pi^i_j v^j \right)
       - {1 \over c^4} \left( \Pi_{ij} v^i v^j \right)^{\displaystyle\cdot},
   \label{E-conservation-2} \\
   & &
       {\partial \over \partial t} \left[
       \left( \varrho + {p \over c^2} \right) \gamma^2 {\bf v} \right]
       + \nabla_j \left[
       \left( \varrho + {p \over c^2} \right) \gamma^2 {\bf v} v^j \right]
   \nonumber \\
   & & \qquad
       = \left( \varrho + {p \over c^2} \right) \gamma^2
       \left[ \nabla \Phi
       + {v^2 \over c^2} \nabla \Psi
       + {1 \over c^2} {\bf v} {\bf v} \cdot \nabla
       \left( \Phi - \Psi \right) \right]
       - \nabla p
       - \nabla_j \Pi^j_i
       - {1 \over c^2} \left( \Pi_{ij} v^j \right)^{\displaystyle\cdot},
   \label{M-conservation-4}
\eea
which also follow from Equations (\ref{eq6-ADM}) and (\ref{eq7-ADM}), respectively. These conservation equations are valid in both gauges.

The presence of the post-Newtonian potential $\Psi$ in Equations (\ref{continuity})-(\ref{M-conservation}) demands two Poisson's equations. In Hwang et al.\ (2016), however, we have argued that although the $(v^2/c^2)\nabla\Psi$ term in Equation (\ref{M-conservation-1}) is comparable to the $\nabla\Phi$ term in our approximation, it is negligible compared with the convective term ${\bf v} \cdot \nabla {\bf v}$ in the left-hand-side due to the weak gravity condition.
[In the ultra-relativistic case ($v \simeq c$) the $(v^2/c^2)\nabla\Phi$ term is, unless we have steep gradients in the gravitational potentials $\Phi$ and $\Psi$, negligible compared with the convective term by the weak gravity condition; in this case the $\nabla\Psi$ term can be negligible as well. In the relativistic case (we keep $v^2/c^2$ order, but $v^2 \ll c^2$) we have the $(v^2/c^2)\nabla\Psi$-term much smaller than the $\nabla \Phi$-term, and the latter is again much smaller than the convective term: thus although we may have to keep the $\nabla\Phi$ term, the $(v^2/c^2)\nabla\Psi$ term can be ignored.]
This applies similarly to $\Psi$ and $c^{-2} {\bf v} {\bf v} \cdot \nabla \Phi$ terms in Equations (\ref{continuity})-(\ref{Pi-equation}), (\ref{M-conservation-3}) and (\ref{E-conservation-1})-(\ref{M-conservation-4}), and $\nabla ( \Phi - \Psi)$ terms in Equations (\ref{E-conservation-2}) and (\ref{M-conservation-4}), see Equation (\ref{Phi-Psi}).

Thus, the basic set of conservation equations effectively becomes Equations (\ref{continuity-practice})-(\ref{M-conservation-practice}). We have kept these in practice negligible terms because, first it is consistent to keep these terms in our derivation from Einstein's field equations, second these general equations might relevant in the presence of steep gradients of gravitational potentials, and third the results are also consistent with the 1PN expansion to be presented in the next section, see the last paragraph in Section \ref{sec:PN}.

Now we examine the remaining Einstein's field equations in the two gauge conditions separately, and show the full consistency of the derived equations.

\subsection{Einstein's equations in the MS}

In the MS, the ADM energy-constraint and the trace of ADM propagation in Equations (\ref{eq4}) and (\ref{eq2}) give Equations (\ref{Poisson-MS}) and (\ref{Poisson-Psi}), respectively. In order to show the consistency of these two equations we need to examine the tracefree ADM propagation in Equation (\ref{eq5}). This gives
\bea
   & &
       \left( \nabla^i \nabla_j - {1 \over 3} \delta^i_j \Delta \right)
       \left( \Phi - \Psi \right)
       - {8 \pi G \over c^2} \left[
       \left( \varrho + {p \over c^2} \right) \gamma^2 \left( v^i v_j
       - {1 \over 3} \delta^i_j v^2 \right)
       + \Pi^i_j - {1 \over 3} \delta^i_j \Pi^k_k \right]
   \nonumber \\
   & & \qquad
       = - c {\partial \over \partial t}
       \bigg[ \left( \nabla^i \nabla_j
       - {1 \over 3} \delta^i_j \Delta \right) \chi
       + {1 \over 2} \left( \chi^{(v)i}_{\;\;\;\;\;\;,j}
       + \chi^{(v),i}_j \right) \bigg]
   \nonumber \\
   & & \qquad
       = {4 \pi G \over c^2} \Delta^{-1} {\partial \over \partial t}
       \bigg\{ 2 \nabla_j \left[ \left( \varrho
       + {p \over c^2} \right) \gamma^2 v^i
       + {1 \over c^2} \Pi^i_k v^k \right]
       + 2 \nabla^i \left[ \left( \varrho
       + {p \over c^2} \right) \gamma^2 v_j
       + {1 \over c^2} \Pi_{jk} v^k \right]
   \nonumber \\
   & & \qquad \qquad
       - \left( \nabla^i \nabla_j \Delta^{-1} + \delta^i_j \right)
       \nabla_k \left[ \left( \varrho + {p \over c^2} \right)
       \gamma^2 v^k
       + {1 \over c^2} \Pi^k_\ell v^\ell \right] \bigg\}
   \nonumber \\
   & & \qquad
       = {4 \pi G \over c^2} \Delta^{-1} \bigg\{
       - 2 \nabla_j \nabla_k \left[ \left( \varrho
       + {p \over c^2} \right) \gamma^2 v^i v^k
       + \Pi^{ik} \right]
       - 2 \nabla^i \nabla_k \left[ \left( \varrho
       + {p \over c^2} \right) \gamma^2 v_j v^k
       + \Pi^k_j \right]
   \nonumber \\
   & & \qquad \qquad
       + \left( \nabla^i \nabla_j \Delta^{-1} + \delta^i_j \right)
       \nabla_k \nabla_\ell \left[
       \left( \varrho + {p \over c^2} \right) \gamma^2 v^k v^\ell
       + \Pi^{k \ell} \right]
       - 3 \left( \nabla^i \nabla_j
       - {1 \over 3} \delta^i_j \Delta \right) p
       \bigg\},
   \label{tracefree-ADM-propagation}
\eea
where we used Equations (\ref{mom-constraint-chi}), (\ref{mom-constraint-chi-v}) and (\ref{M-conservation-4}). Thus we have
\bea
   & &
       \Delta \left( \Phi - \Psi \right)
       = - {4 \pi G \over c^2} \left[ 3 p
       + \left( \varrho + {p \over c^2} \right)
       \gamma^2 v^2 + \Pi^i_i \right].
   \label{Phi-Psi}
\eea
Using Equation (\ref{Phi-Psi}), Equations (\ref{Poisson-MS}) and (\ref{Poisson-Psi}) become consistent. We note that $\chi_i$ has a nontrivial role only in Equations (\ref{tracefree-ADM-propagation}) and (\ref{Phi-Psi}); Equation (\ref{tracefree-ADM-propagation}) still has contribution from $\chi^{(v)}_i$ part, but the term has no role in deriving Equation (\ref{Phi-Psi}). In this sense, we have to keep $\chi^{(v)}_i$ in our metric although it has no role in deriving the final result.

One remaining equation to be checked for consistency is the trace of extrinsic curvature in Equation (\ref{eq1}). This gives
\bea
   & &
       c \Delta \chi = - 3 {\dot \Psi \over c^2}.
   \label{kappa-def}
\eea
We can show that this is consistent with Equations (\ref{Poisson-Psi}), (\ref{mom-constraint-chi}) and (\ref{E-conservation-2}). This completes the proof of full consistency of Equations (\ref{continuity})-(\ref{M-conservation}), (\ref{Poisson-MS}) and (\ref{Poisson-Psi}) in Einstein's gravity, under the assumptions of Equation (\ref{assumptions}).

\subsection{Einstein's equations in the ZSG}

The case of ZSG proceeds similarly. Equation (\ref{eq2}) gives Equation (\ref{Poisson-Psi}). Equation (\ref{eq2}) gives
\bea
   & &
       \Delta \Phi
       + 4 \pi G \left( \varrho + {3 p \over c^2} \right)
       = \dot \kappa
       - {8 \pi G \over c^2} \left[
       \left( \varrho + {p \over c^2} \right)
       \gamma^2 v^2 + \Pi^i_i \right].
\eea
From Equation (\ref{mom-constraint-kappa}), using Equation (\ref{M-conservation-4}), we can derive Equation (\ref{Poisson-ZSG}). In order to show the consistency between Equation (\ref{Poisson-Psi}) and Equation (\ref{Poisson-ZSG}) we need to examine Equation (\ref{eq5}). This gives
\bea
   & &
       \left( \nabla^i \nabla_j - {1 \over 3} \delta^i_j \Delta \right)
       \left( \Phi - \Psi \right)
       - {8 \pi G \over c^2} \left[
       \left( \varrho + {p \over c^2} \right) \gamma^2 \left( v^i v_j
       - {1 \over 3} \delta^i_j v^2 \right)
       + \Pi^i_j - {1 \over 3} \delta^i_j \Pi^k_k \right]
   \nonumber \\
   & & \qquad
       = - c {\partial \over \partial t}
       {1 \over 2} \left( \chi^{(v)i}_{\;\;\;\;\;\;,j}
       + \chi^{(v),i}_j \right)
   \nonumber \\
   & & \qquad
       = {8 \pi G \over c^2} \Delta^{-1} {\partial \over \partial t}
       \bigg\{ \nabla_j \left[ \left( \varrho
       + {p \over c^2} \right) \gamma^2 v^i
       + {1 \over c^2} \Pi^i_k v^k \right]
       + \nabla^i \left[ \left( \varrho
       + {p \over c^2} \right) \gamma^2 v_j
       + {1 \over c^2} \Pi_{jk} v^k \right]
   \nonumber \\
   & & \qquad \qquad
       - 2 \nabla^i \nabla_j \Delta^{-1}
       \nabla_k \left[ \left( \varrho + {p \over c^2} \right)
       \gamma^2 v^k
       + {1 \over c^2} \Pi^k_\ell v^\ell \right] \bigg\}
   \nonumber \\
   & & \qquad
       = {8 \pi G \over c^2} \Delta^{-1} \bigg\{
       - \nabla_j \nabla_k \left[ \left( \varrho
       + {p \over c^2} \right) \gamma^2 v^i v^k
       + \Pi^{ik} \right]
       - \nabla^i \nabla_k \left[ \left( \varrho
       + {p \over c^2} \right) \gamma^2 v_j v^k
       + \Pi^k_j \right]
   \nonumber \\
   & & \qquad \qquad
       + 2 \nabla^i \nabla_j \Delta^{-1}
       \nabla_k \nabla_\ell \left[
       \left( \varrho + {p \over c^2} \right) \gamma^2 v^k v^\ell
       + \Pi^{k \ell} \right]
       \bigg\},
   \label{tracefree-ADM-propagation-ZSG}
\eea
where we used Equations (\ref{mom-constraint-chi}), (\ref{mom-constraint-chi-v}) and (\ref{M-conservation-4}). Thus we have
\bea
   & &
       \Delta \left( \Phi - \Psi \right)
       = {12 \pi G \over c^2} \Delta^{-1} \nabla_i \nabla^j
       \left[ \left( \varrho + {p \over c^2} \right)
       \gamma^2 \left( v^i v_j - {1 \over 3} \delta^i_j v^2 \right)
       + \Pi^i_j - {1 \over 3} \delta^i_j \Pi^k_k \right].
   \label{Phi-Psi-ZSG}
\eea
Using Equation (\ref{Phi-Psi-ZSG}), we can show that Equations (\ref{Poisson-Psi}) and (\ref{Poisson-ZSG}) are consistent. We note that $\chi_i$ has a nontrivial role only in Equations (\ref{tracefree-ADM-propagation-ZSG}) and (\ref{Phi-Psi-ZSG}); Equation (\ref{tracefree-ADM-propagation-ZSG}) still has contribution from $\chi^{(v)}_i$ part, but the term has no role in deriving Equation (\ref{Phi-Psi}). In this sense, we have to keep $\chi^{(v)}_i$, thus $\chi_i$, in our metric although it has no role in deriving the final result.

One remaining equation to be checked for consistency is the trace of extrinsic curvature in Equation (\ref{eq1}). This gives
\bea
   & &
       \kappa = - 3 {\dot \Psi \over c^2}.
   \label{kappa-def-ZSG}
\eea
We can show that this is consistent with Equations (\ref{Poisson-Psi}), (\ref{mom-constraint-chi}) and (\ref{E-conservation-2}).

\section{1PN approximation: complete equations without fixing slicing}
                                           \label{sec:PN}

Now, we switch our subject to the hydrodynamic 1PN approximation. The main purpose of the presentation is to show the consistency of our special-relativity-with-gravity approximation presented in the previous sections, but the general hydrodynamic set of 1PN equations {\it without} fixing the hypersurface condition presented in this section is also new in the literature; previously it was derived in the cosmological context (HNP). The complete set of 1PN equations without taking the hypersurface condition is
\bea
   & &
       \dot {\overline \varrho} + \nabla \cdot
       \left( \overline \varrho \overline {\bf v} \right)
       = - {1 \over c^2} \overline \varrho
       \left( {\partial \over \partial t}
       + \overline {\bf v} \cdot \nabla \right)
       \left( {1 \over 2} \overline v^2 + 3 U \right),
   \label{1PN-0} \\
   & &
       \dot {\overline \varrho} + \nabla \cdot
       \left( \overline \varrho \overline {\bf v} \right)
       = - {1 \over c^2} \left[ \overline \varrho
       \left( {\partial \over \partial t}
       + \overline {\bf v} \cdot \nabla \right)
       \left( {1 \over 2} \overline v^2 + 3 U + \Pi \right)
       + p \nabla \cdot \overline {\bf v}
       + \Pi_i^j \nabla_j v^i \right],
   \label{1PN-1} \\
   & &
       \dot {\overline {\bf v}}
       + \overline {\bf v} \cdot \nabla \overline {\bf v}
       - \nabla U + {1 \over \overline \varrho}
       \left( \nabla p
       + \nabla_j \Pi_i^j \right)
   \nonumber \\
   & & \qquad
       = {1 \over c^2} \bigg\{
       - 2 \nabla \left( U^2 - \widetilde \Phi \right)
       + \dot P_i + \overline v^j \left( P_{i,j} - P_{j,i} \right)
       - \overline {\bf v}
       \left( {\partial \over \partial t}
       + \overline {\bf v} \cdot \nabla \right)
       \left( {1 \over 2} \overline v^2 + 3 U \right)
       + \overline v^2 \nabla U
   \nonumber \\
   & & \qquad
       + {1 \over \overline \varrho}
       \left[
       \left( \overline v^2 + 4 U + \Pi + {p \over \overline \varrho} \right) \left( \nabla p + \nabla_j \Pi_i^j \right)
       - \overline {\bf v}
       \left( {\partial \over \partial t}
       + \overline {\bf v} \cdot \nabla \right) p
       - \left( \Pi_i^j \overline v^j \right)^{\displaystyle\cdot}
       - \overline {\bf v} \Pi_k^j \nabla_j v^k
       + 2 U \nabla_j \Pi_i^j
       \right] \bigg\},
   \label{1PN-2} \\
   & &
       \Delta U + 4 \pi G \overline \varrho
       = - {1 \over c^2} \bigg[ 3 \ddot U

       - 2 U \Delta U
       + 2 \Delta \widetilde \Phi
       + \dot P^i_{\;\;,i}
       + 8 \pi G \left( \overline \varrho \overline v^2 + {1 \over 2} \overline \varrho \Pi
       + {3 \over 2} p \right) \bigg],
   \label{1PN-3} \\
   & &
       0 = {1 \over 4} \left( P^j_{\;\;,ji} - \Delta P_i \right)
         + \nabla \dot U - 4 \pi G \overline \varrho \overline {\bf v},
   \label{1PN-4} \\
   & &
       0 = U - V.
   \label{1PN-5}
\eea
These follow from the continuity, the energy conservation, the momentum conservation, the trace of ADM propagation, the ADM momentum constraint, and the tracefree ADM propagation equations, respectively: see Equations (62), (114), (115), (119), (120) and (91), respectively, in HNP which were presented in the cosmological background\footnote{There are typos in the right-hand-side of Equation (115) of the paper. It should be read as
\bea
   & &
       = - {1 \over a \sigma} \Pi^j_{i|j}
       - {1 \over c^2} \dots
   \nonumber
\eea
};
in our non-cosmological background, we set $a \equiv 1$ and $\varrho_b \equiv 0 \equiv p_b$, etc. for the background quantities. As mentioned, these 1PN equations also follow from the fully nonlinear perturbation formulation (Noh \& Hwang 2013).

The left-hand and right-hand sides of Equations (\ref{1PN-0})-(\ref{1PN-5}) are the 0PN (Newtonian) and 1PN orders, respectively. From Equations (\ref{1PN-0}) and (\ref{1PN-1}) we have ${d\Pi/dt} + (p/\overline \varrho) \nabla \cdot {\bf v} = 0$. Our 1PN metric convention follows Chandrasekhar (1965)
\bea
   & &
       g_{00} = - \left[ 1 - {1 \over c^2} 2 U
       + {1 \over c^4} \left( 2 U^2
       - 4 \widetilde \Phi \right) \right], \quad
       g_{0i} = - {1 \over c^3} P_i, \quad
       g_{ij} = \left( 1 + {1 \over c^2} 2 V \right) \delta_{ij},
   \label{metric-1PN}
\eea
where the index of $P_i$ is raised and lowered by $\delta_{ij}$ as the metric. Compared with Equation (\ref{metric}) we have
\bea
   & &
       \Phi = U
       - {1 \over c^2} \left( U^2 - 2 \widetilde \Phi \right), \quad
       \chi_i = {1 \over c^3} P_i, \quad
       \Psi = V.
\eea
The velocity in the 1PN equations is defined as
\bea
   & &
       u^i \equiv u^0 {\overline v^i \over c},
\eea
where $\overline v^i$ is the fluid coordinate three-velocity (HNa); the index of $\overline v_i$ is raised and lowered by $\delta_{ij}$ as the metric. Compared with $v_i$ defined in Equation (\ref{v_i}) we have
\bea
   & &
       v_i = \overline v_i
       + {1 \over c^2} \left[ \left( U + 2 V \right) \overline v_i
       - P_i \right].
\eea

The above set of equations is still in a gauge-ready form, i.e., without taking the temporal gauge condition yet. The general 1PN hypersurface condition is (HNP)
\bea
   & &
       P^i_{\;\;,i} + n \dot U = 0.
   \label{1PN-gauge}
\eea
The choice of $n$ corresponds to taking different temporal gauges: notable ones are the harmonic gauge ($n \equiv 4$) in Weinberg (1972), the MS ($n \equiv 3$) in Chandrasekhar (1965) and the ZSG ($n \equiv 0$) (HNP).

Now, we show the gauge dependence of $3p/c^2$ term in the Poisson's equation (\ref{1PN-3}). To 0PN order Equation (\ref{1PN-3}) gives $\Delta U = - 4 \pi G \overline \varrho$. From this, using Equation (\ref{1PN-1}), we have $\Delta \dot U = 4 \pi G \nabla \cdot \left( \overline \varrho \overline {\bf v} \right)$ which also follows from Equation (\ref{1PN-4}). From this, using Equations (\ref{1PN-1}) and (\ref{1PN-2}), we can show
\bea
   & &
       \ddot U
       = - 4 \pi G p
       + 4 \pi G \Delta^{-1} \left[ \nabla \cdot
       \left( \overline \varrho \nabla U \right)
       - \nabla_i \nabla_j
       \left( \overline \varrho \overline v^i \overline v^j
       + \Pi^{ij}
       \right) \right].
   \label{ddot-U}
\eea
Imposing the general gauge condition in Equation (\ref{1PN-gauge}), thus still in a gauge-ready form, Equation (\ref{1PN-3}) gives
\bea
   & &
       \Delta U + 4 \pi G \varrho
       + {1 \over c^2} \big[
       - (n - 3) \ddot U + 12 \pi G p + \dots \big] = 0.
   \label{PN-Poisson-2}
\eea
Using Equation (\ref{ddot-U}), Equation (\ref{PN-Poisson-2}) can be arranged as
\bea
   & &
       \Delta U + 4 \pi G \varrho + {1 \over c^2} \big[
       4 \pi G n p + \dots \big]  = 0.
   \label{PN-Poisson-3}
\eea
From Equations (\ref{PN-Poisson-2}) and (\ref{PN-Poisson-3}) we notice that both the propagation speed of the gravitational potential and the presence of a pressure term in the 1PN Poisson's equation depend on the gauge condition, and these are related. Under the general gauge condition the propagation speed can be na\"ively read from Equation (\ref{PN-Poisson-2}) as $c/\sqrt{n-3}$ which becomes the speed of light only in the harmonic gauge with $n = 4$, and the pressure term can be read from Equation (\ref{PN-Poisson-3}) as $n p/c^2$.

The gauge dependence of the propagation speed of gravitational potential is not surprising because the potential is gauge dependent. Whereas the propagation speed of the (physical) Weyl tensor naturally does not depend on the gauge choice and is always $c$: see Section 7 of HNP. We have an exact analogy in the classical electromagnetism where the propagation speed of the field potential depends on the gauge choice, whereas the propagation speed of the (physical) field strength is always $c$ (Jackson 2002).

Likewise, the form of 1PN pressure term also depends on the gauge choice. It vanishes in the ZSG, and becomes $3 p/c^2$ and $4 p/c^2$ in the MS and the harmonic gauge, respectively. In the MS with $P^i_{\;\;,i} = - 3 \dot U$ the $3 \ddot U$ in Equation (\ref{1PN-3}) cancels by the $\dot P^i_{\;\;,i}$, thus leaving $12 \pi G p$. Similarly, in the ZSG with $ P^i_{\;\;,i} = 0$, the $12 \pi G p$ in Equation (\ref{1PN-3}) cancels by the $3 \ddot U$ using Equation (\ref{ddot-U}).

We can show that Equations (\ref{continuity})-(\ref{M-conservation}), (\ref{Poisson-MS}), (\ref{Poisson-Psi}) and (\ref{Poisson-ZSG}) in the two gauge conditions are {\it consistent} with the 1PN equations (\ref{1PN-0})-(\ref{1PN-5}), in the overlapping regime (i.e., for the weak gravity and the action-at-a-distance, and 1PN for the pressure, anisotropic stress and velocity) of the two approximations. In both gauges, according to the comment below Equation (\ref{assumptions}) we have ${\bf v} = \overline {\bf v}$, but $\overline {\bf v} \cdot \nabla \overline {\bf v} = {\bf v} \cdot \nabla {\bf v} - 3 {\bf v} {\bf v} \cdot \nabla \Phi/c^2$ and $\nabla \cdot (\overline \varrho \overline {\bf v}) = \nabla \cdot (\overline \varrho {\bf v}) - 3 \overline \varrho {\bf v} \cdot \nabla \Phi/c^2$. In the MS the Equations (\ref{1PN-0})-(\ref{1PN-5}) become
\bea
   & & \dot {\overline \varrho}
       + \nabla \cdot \left( {\overline \varrho} {\bf v} \right)
       = {1 \over c^2} \left(
       - {\overline \varrho} {\bf v} \cdot \nabla \Phi
       + {\bf v} \cdot \nabla p
       + v^i \nabla_j \Pi^j_i \right),
   \label{1PN-continuity} \\
   & & \dot {\overline \varrho}
       + \nabla \cdot \left( {\overline \varrho} {\bf v} \right)
       = {1 \over c^2} \left(
       - {\overline \varrho} {\bf v} \cdot \nabla \Phi
       - {\overline \varrho} {d \Pi \over dt}
       + {\bf v} \cdot \nabla p
       - p \nabla \cdot {\bf v}
       + v^i \nabla_j \Pi^j_i
       - \Pi^j_i \nabla_j v^i \right),
   \label{1PN-E-conservation} \\
   & & {d \over dt} {\bf v}
       - \nabla \Phi
       + {1 \over {\overline \varrho}} \left(
       \nabla p + \nabla_j \Pi_i^j \right)
   \nonumber \\
   & & \qquad
       = {1 \over c^2} \left\{
       v^2 \nabla \Phi
       - {\bf v} {\bf v} \cdot \nabla \Phi
       + {1 \over {\overline \varrho}}
       \left[
       - {\bf v} \dot p
       + \left( v^2 + \Pi + {p \over \overline \varrho} \right)
       \left( \nabla p
       + \nabla_j \Pi_i^j \right)
       + {\bf v} \nabla_j \left( \Pi^j_k v^k \right)
       - \left( \Pi_i^j v_j \right)^{\displaystyle\cdot}
       \right] \right\},
   \label{1PN-M-conservation} \\
   & & \Delta \Phi
       + 4 \pi G \left( \varrho + 3 {p \over c^2} \right)
       = - {8 \pi G \over c^2} \overline \varrho v^2,
   \label{1PN-Poisson-MS} \\
   & & 0
       = \Delta \Psi
       + 4 \pi G \overline \varrho,
   \label{1PN-Poisson-Psi} \\
   & & 0
       = \nabla_i \dot \Phi
       - 4 \pi G \overline \varrho v_i
       - {1 \over 4} c^3 \Delta \chi_i^{(v)}.
   \label{1PN-Phi-dot}
\eea
Equation (\ref{1PN-Phi-dot}) is consistent with Equations (\ref{kappa-def}), (\ref{kappa-def-ZSG}), (\ref{mom-constraint-chi}) and (\ref{mom-constraint-kappa}) valid to the 0PN order. In the ZSG instead of Equation (\ref{1PN-Poisson-MS}) we have
\bea
   \Delta \Phi + 4 \pi G \varrho
       = - {8 \pi G \over c^2} \overline \varrho v^2
       + {12 \pi G \over c^2}
       \Delta^{-1} \nabla_i \nabla_j \left( \overline \varrho v^i v^j
       + \Pi^{ij} \right),
   \label{1PN-Poisson-ZSG}
\eea
and the other equations remain the same.

\section{Gauge transformation}
                                         \label{sec:GT}

Now, we can show that to the linear order in perturbation the presence or absence of $3p/c^2$ term in the extended Newtonian Poisson's equation in Equations (\ref{Poisson-MS}) or (\ref{Poisson-ZSG}) depends on the temporal gauge while Equations (\ref{continuity-practice})-(\ref{M-conservation-practice}) and (\ref{Poisson-Psi}) remain the same. Under the gauge transformation $\widehat x^c = x^c + \xi^t$, in a non-expanding background, the gauge transformation properties in Equation (252) of Noh \& Hwang (2004) give
\bea
   & &
       \widehat \Phi = \Phi + c \dot \xi^t, \quad
       \widehat \Psi = \Psi, \quad
       \widehat \chi = \chi - \xi^t, \quad
       \widehat \kappa = \kappa + c \Delta \xi^t, \quad
       \widehat \varrho = \varrho, \quad
       \widehat p = p, \quad
       \widehat v = v - c \xi^t,
   \label{GT-1}
\eea
where the spatial gauge transformation is already fixed by our spatial gauge condition (Bardeen 1988); similarly we can add $\widehat {\overline \varrho} = \overline \varrho$ for the rest mass density; we introduced $v_i \equiv v_{,i} + v_i^{(v)}$ with $v^{(v)i}_{\;\;\;\;\;\;,i} \equiv 0$. Thus, variables $\Psi$, $\overline \varrho$, $\varrho$ and $p$ are invariant under the gauge transformation. Lets consider $\widehat x^c$ and $x^c$ coordinates as the MS and the ZSG, respectively. Then using $\chi_{\rm ZSG} \equiv 0$ and $\kappa_{\rm MS} \equiv 0$, we have
\bea
   & &
       \Phi_{\rm MS} = \Phi_{\rm ZSG} + c \dot \xi^t, \quad
       \chi_{\rm MS} = - \xi^t, \quad
       \kappa_{\rm ZSG} = - c \Delta \xi^t, \quad
       v_{\rm MS} = v_{\rm ZSG} - c \xi^t,
   \label{GT-2}
\eea
and $\Psi_{\rm MS} = \Psi_{\rm ZSG}$ etc. Using these gauge transformation properties we can show the following. To the linear order Equation (\ref{Poisson-Psi}) is apparently gauge invariant. The forms of Equations (\ref{continuity-practice})-(\ref{M-conservation-practice}) to the linear order remain the same in both gauges in our approximation in Equation (\ref{assumptions}). From Equation (\ref{GT-2}) we have
\bea
   & &
       \Delta \Phi_{\rm MS} - \Delta \Phi_{\rm ZSG}
       = c \Delta \dot \xi^t
       = - c \Delta \dot \chi_{\rm MS}
       = - \dot \kappa_{\rm ZSG}
       = 3 {\ddot \Psi \over c^2}
       = - 12 \pi G {p \over c^2},
   \label{GT-Phi}
\eea
where we used Equations (\ref{kappa-def}), (\ref{kappa-def-ZSG}), (\ref{E-conservation-practice})-(\ref{Poisson-MS}) and (\ref{Poisson-ZSG}), and the approximation in Equation (\ref{assumptions}). Thus, we exactly recover the pressure difference.

\section{Another expressions}

In a conventional notation of special relativistic hydrodynamics suitable for numerical study (Wilson \& Mathews 2003; Ryu, Chattopadhyay \& Choi 2006), Equations (\ref{continuity})-(\ref{M-conservation}) can be arranged using Equations (\ref{E-conservation-2}) and (\ref{M-conservation-4}), as
\bea
   & &
       \dot D + \nabla_i \left( D v^i \right)
       = {1 \over c^2} D {\bf v} \cdot \nabla
       \left( \Phi - \Psi \right),
   \label{continuity-SR} \\
   & &
       \dot E
       + \nabla \cdot \left[ \left( E
       + p \right) {\bf v} \right]
       = {1 \over c^2} \left( E + p \right) {\bf v} \cdot
       \left[ \nabla \Phi
       + \nabla \left( \Phi - \Psi \right)
       + {v^2 \over c^2} \nabla \Psi \right]
       - \nabla_j \left( \Pi_i^j v^i \right)
       - {1 \over c^2} \left( \Pi_{ij} v^i v^j \right)^{\displaystyle\cdot},
   \label{E-conservation-SR} \\
   & &
       \dot M_i + \nabla_j \left( M_i v^j
       + p \delta^j_i \right)
       = {1 \over c^2} \left( E + p \right)
       \left[ \nabla_i \Phi
       + {1 \over c^2} v_i {\bf v} \cdot \nabla
       \left( \Phi - \Psi \right)
       + {v^2 \over c^2} \nabla_i \Psi \right]
       - \nabla_j \Pi_i^j
       - {1 \over c^2} \left( \Pi_{ij} v^j \right)^{\displaystyle\cdot},
   \label{M-conservation-SR}
\eea
where
\bea
   & &
       D \equiv \gamma \overline \varrho, \quad
       E \equiv \gamma^2 \overline \varrho c^2 h - p, \quad
       M_i \equiv \gamma^2 \overline \varrho h v_i,
\eea
are the coordinate (Eulerian) mass density, the ADM energy density and the coordinate momentum density, respectively, with $h \equiv 1 + \Pi/c^2 + p/(\overline \varrho c^2)$ the specific enthalpy; we have $E + p = \gamma^2 (\varrho c^2 + p)$. The gravitational potential $\Phi$ is determined by Equation (\ref{Poisson-MS}) in the MS and Equation (\ref{Poisson-ZSG}) in the ZSG, and $\Psi$ by Equation (\ref{Poisson-Psi}). In terms of the above notation we have
\bea
   & &
       \Delta \Phi
       + 4 \pi G \left[ {1 \over c^2} \left( E + 3 p \right)
       + {M^i M_i \over E + p} + {2 \over c^2} \Pi^i_i \right]
       = 0,
   \label{Poisson-MS-E} \\
   & &
       \Delta \Psi
       + {4 \pi G \over c^2} \left( E+ \Pi^i_i \right)
       = 0,
   \label{Poisson-Psi-E}
\eea
in the MS; in the case of ZSG, instead of Equation (\ref{Poisson-MS-E}) we have
\bea
   & &
       \Delta \Phi
       + 4 \pi G \left[
       {1 \over c^2} E
       + {M^i M_i \over E + p} + {2 \over c^2} \Pi^i_i
       - 3 \Delta^{-1} \nabla_i \nabla_j
       \left( {M^i M^j \over E + p} + \Pi^{ij} \right) \right]
       = 0.
   \label{Poisson-ZSG-E}
\eea

If we strictly apply the weak gravity limit in Equation (\ref{assumptions}), we can ignore the $\nabla (\Phi - \Psi)$ and $\nabla \Psi$ terms in Equations (\ref{continuity-SR})-(\ref{M-conservation-SR}). Although these terms are consistently derived in our special-relativity-with-gravity approximation, as we argued below Equation (\ref{M-conservation-4}), the strengths of these terms are in practice negligible compared with other terms in the equations. Thus, in practice, we have
\bea
   & &
       \dot D + \nabla_i \left( D v^i \right)
       = 0,
   \label{continuity-SR-practice} \\
   & &
       \dot E
       + \nabla_i \left[ \left( E
       + p \right) v^i \right]
       = {1 \over c^2} \left( E + p \right) {\bf v} \cdot
       \nabla \Phi
       - \nabla_j \left( \Pi_i^j v^i \right)
       - {1 \over c^2} \left( \Pi_{ij} v^i v^j \right)^{\displaystyle\cdot},
   \label{E-conservation-SR-practice} \\
   & &
       \dot M_i + \nabla_j \left( M_i v^j
       + p \delta^j_i \right)
       = {1 \over c^2} \left( E + p \right)
       \nabla_i \Phi
       - \nabla_j \Pi_i^j
       - {1 \over c^2} \left( \Pi_{ij} v^j \right)^{\displaystyle\cdot},
   \label{M-conservation-SR-practice}
\eea
which correspond to Equations (\ref{continuity-practice})-(\ref{M-conservation-practice}); $\Phi$ is determined by Equation (\ref{Poisson-MS-E}) in the MS, and by Equation (\ref{Poisson-ZSG-E}) in the ZSG. The gravity term in Equation (\ref{E-conservation-SR-practice}) is present even in Newtonian hydrodynamics, see Equation (2.32) in Shu (1992); this is because $E$ contains both $\overline \varrho c^2$ order term together with $\overline \varrho \Pi$ and $p$ terms\footnote{We thank professor Dongsu Ryu for pointing out this.
                       },
thus in order to derive this term we have to use Equation (\ref{continuity}) instead of Equation (\ref{continuity-practice}).

\section{Discussion}
                                             \label{sec:Discussion}

We presented two hydrodynamic approximations of Einstein's gravity. The first method, which is new, is the special relativistic hydrodynamics with fully relativistic pressure and velocity in the presence of self-gravity. In the MS Equations (\ref{continuity-practice})-(\ref{Poisson-MS}) are the complete set, and in the ZSG Equation (\ref{Poisson-ZSG}) replaces Equation (\ref{Poisson-MS}).

The second method is the 1PN approximation: our new contribution here is the gauge-ready expression in a non-expanding background.
Previously, the hydrodynamic 1PN approximation was studied by Chandrasekhar (1965) in the MS and by Weinberg (1972) in the harmonic gauge; the gauge-ready extension was made by us in the cosmological context (HNP). Equations (\ref{1PN-0})-(\ref{1PN-5}) are the complete set of 1PN equations with the general temporal-gauge condition in Equation (\ref{1PN-gauge}).

The fully general relativistic hydrodynamic equations in terms of $D$, $E$ and $M_i$ are known in the literature (Wilson \& Mathews 2003; Alcubierre 2008; Bona, Palenzuela-Luque \& Bona-Casas 2009; Baumgarte \& Shapiro 2010,; Gourgoulhon 2012; Rezzolla \& Zanotti 2013). The numerical relativity generally demands heavy computational resources with some conceptual difficulties compared with the Newtonian hydrodynamics. Our special relativistic hydrodynamics with self-gravity is formally quite similar to the Newtonian one. The special relativistic hydrodynamics with weak gravity was not known in the literature.

We can imagine many astrophysical situations where the special relativistic effect is important while the gravity is weak. Except near neutron stars or black holes, in most celestial bodies (planets, stars, star cluster, galaxies, galaxy clusters, interstellar and intergalactic media, and even including astrophysical regions some distances away from pulsar, quasars, jets, etc) the gravity is weak. The strength of the gravity is measured by $\Phi/c^2 \sim GM/(\ell c^2)$ with $M$ and $\ell$ the characteristic mass and size dimensions, respectively. In the celestial bodies mentioned above this dimensionless gravitational strength is of the order $\Phi/c^2 \sim 10^{-5} - 10^{-10}$ and even smaller. In such situations the gravity is indeed weak. We anticipate our two approximations may find wide applications in such astrophysical situations where the special relativistic activity is important.

In most realistic astrophysical situation where special relativity is important, however, often the magnetic activity is also involved. Extending our special-relativity-with-gravity approximation including special relativistic MHD (magnetohydrodynamics) is currently in progress.

\acknowledgments

We would like to thank the anonymous referee for clarifying and insightful comments which helped to much improve the manuscript. We thank professor Dongsu Ryu for useful suggestions and critical comments on special relativistic hydrodynamics. J.H.\ was supported by Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science, ICT and future Planning (No.\ 2013R1A2A2A01068519 and No.\ 2016R1A2B4007964). H.N.\ was supported by National Research Foundation of Korea funded by the Korean Government (No.\ 2015R1A2A2A01002791).