text:
\section{Introduction: file preparation and submission}

The \verb"iopart" \LaTeXe\ article class file is provided to help authors prepare articles for submission to IOP Publishing journals.
  This document gives advice on preparing your submission, and specific instructions on how to use \verb"iopart.cls" to follow this advice.  You
  do not have to use \verb"iopart.cls"; articles prepared using any other common class and style files can also be submitted.
    It is not necessary to mimic the appearance of a published article.

The advice
on \LaTeX\ file preparation in this document applies to
the journals listed in table~\ref{jlab1}.  If your journal is not listed please go to the journal website via \verb"http://iopscience.iop.org/journals" for specific
submission instructions.

\begin{table}
\caption{\label{jlab1}Journals to which this document applies, and macros for the abbreviated journal names in {\tt iopart.cls}. Macros for other journal titles are listed in appendix\,A.}
\footnotesize
\begin{tabular}{@{}llll}
\br
Short form of journal title&Macro name&Short form of journal title&Macro name\\
\mr
2D Mater.&\verb"\TDM"&Mater. Res. Express&\verb"\MRE"\\
Biofabrication&\verb"\BF"&Meas. Sci. Technol.$^c$&\verb"\MST"\\
Bioinspir. Biomim.&\verb"\BB"&Methods Appl. Fluoresc.&\verb"\MAF"\\
Biomed. Mater.&\verb"\BMM"&Modelling Simul. Mater. Sci. Eng.&\verb"\MSMSE"\\
Class. Quantum Grav.&\verb"\CQG"&Nucl. Fusion&\verb"\NF"\\
Comput. Sci. Disc.&\verb"\CSD"&New J. Phys.&\verb"\NJP"\\
Environ. Res. Lett.&\verb"\ERL"&Nonlinearity$^{a,b}$&\verb"\NL"\\
Eur. J. Phys.&\verb"\EJP"&Nanotechnology&\verb"\NT"\\
Inverse Problems&\verb"\IP"&Phys. Biol.$^c$&\verb"\PB"\\
J. Breath Res.&\verb"\JBR"&Phys. Educ.$^a$&\verb"\PED"\\
J. Geophys. Eng.$^d$&\verb"\JGE"&Physiol. Meas.$^{c,d,e}$&\verb"\PM"\\
J. Micromech. Microeng.&\verb"\JMM"&Phys. Med. Biol.$^{c,d,e}$&\verb"\PMB"\\
J. Neural Eng.$^c$&\verb"\JNE"&Plasma Phys. Control. Fusion&\verb"\PPCF"\\
J. Opt.&\verb"\JOPT"&Phys. Scr.&\verb"\PS"\\
J. Phys. A: Math. Theor.&\verb"\jpa"&Plasma Sources Sci. Technol.&\verb"\PSST"\\
J. Phys. B: At. Mol. Opt. Phys.&\verb"\jpb"&Rep. Prog. Phys.$^{e}$&\verb"\RPP"\\
J. Phys: Condens. Matter&\verb"\JPCM"&Semicond. Sci. Technol.&\verb"\SST"\\
J. Phys. D: Appl. Phys.&\verb"\JPD"&Smart Mater. Struct.&\verb"\SMS"\\
J. Phys. G: Nucl. Part. Phys.&\verb"\jpg"&Supercond. Sci. Technol.&\verb"\SUST"\\
J. Radiol. Prot.$^a$&\verb"\JRP"&Surf. Topogr.: Metrol. Prop.&\verb"\STMP"\\
Metrologia&\verb"\MET"&Transl. Mater. Res.&\verb"\TMR"\\
\br
\end{tabular}\\
$^{a}$UK spelling is required; $^{b}$MSC classification numbers are required; $^{c}$titles of articles are required in journal references; $^{d}$Harvard-style references must be used (see section \ref{except}); $^{e}$final page numbers of articles are required in journal references.

\end{table}
\normalsize

Any special submission requirements for the journals are indicated with footnotes in table~\ref{jlab1}.
Journals which require references in a particular format will need special care if you are using BibTeX, and you might need to use a \verb".bst" file
that gives slightly non-standard output in order to supply any extra information required.  It is not
necessary to give references in the exact style of references used in published articles, as long as all of
the required information is present.

Also note that there is an incompatibility
between \verb"amsmath.sty" and \verb"iopart.cls" which cannot be completely worked around.  If your article relies
on commands in \verb"amsmath.sty" that are not available in \verb"iopart.cls", you may wish to consider using a different
class file.

Whatever journal you are submitting to, please look at recent published articles (preferably
articles in your subject area) to familiarize yourself with the features of the journal.  We do not demand
that your \LaTeX\ file closely resembles a published article---a generic `preprint' appearance of the sort
commonly seen on \verb"arXiv.org" is fine---but your submission should be presented
in a way that makes it easy for the referees to form an opinion of whether it is suitable for the journal.
The generic advice in this document---on what to include in an abstract, how best to present complicated
mathematical expressions, and so on---applies whatever class file you are using.

\subsection{What you will need to supply}
Submissions to our journals are handled via the ScholarOne web-based submission system.  When you submit
a new article to us you need only submit a PDF of your article.  When you submit a revised version,
we ask you to submit the source files as well.  Upon acceptance for publication we will use the source files to produce a proof of your article in the journal style.

\subsubsection{Text.}When you send us the source files for a revised version of your submission,
you should send us the \LaTeX\ source code of your paper with all figures read in by
the source code (see section \ref{figinc}).  Articles can be prepared using almost any version of \TeX\ or \LaTeX{},
not just \LaTeX\ with the class file \verb"iopart.cls".  You may split your \LaTeX\ file into several parts, but please show
which is the `master' \LaTeX\ file that reads in all of the other ones by naming it appropriately.  The `master'
\LaTeX\ file must read in all other \LaTeX\ and figure files from the current directory.  {\it Do not read in files from a different directory, e.g. \verb"\includegraphics{/figures/figure1.eps}" or
\verb"\include{../usr/home/smith/myfiles/macros.tex}"---we store submitted files
all together in a single directory with no subdirectories}.
\begin{itemize}
\item {\bf Using \LaTeX\ packages.} Most \LaTeXe\ packages can be used if they are
available in common distributions of \LaTeXe; however, if it is essential to use
a non-standard package then any extra files needed to process the article must
also be supplied.  Try to avoid using any packages that manipulate or change the standard
\LaTeX\ fonts: published articles use fonts in the Times family, but we prefer that you
use \LaTeX\ default Computer Modern fonts in your submission.  The use of \LaTeX\ 2.09, and of plain
\TeX\ and variants such as AMSTeX is acceptable, but a complete PDF of your submission should be supplied in these cases.
\end{itemize}
\subsubsection{Figures.} Figures should ideally be included in an article as encapsulated PostScript files
(see section \ref{figinc}) or created using standard \LaTeX\ drawing commands.
 Please name all figure files using the guidelines in section \ref{fname}.
We accept submissions that use pdf\TeX\ to include
PDF or bitmap figures, but please ensure that you send us a PDF that uses PDF version 1.4 or lower
(to avoid problems in the ScholarOne system).
You can do this by putting \verb"\pdfminorversion=4" at the very start of your TeX file.

\label{fig1}All figures should be included within the body of the text
at an appropriate point or grouped together with their captions at the end of the article. A standard graphics inclusion package such as \verb"graphicx" should be used for figure inclusion, and the package should be declared in the usual
way, for example with \verb"\usepackage{graphicx}", after the \verb"\documentclass" command.
Authors should avoid using special effects generated by including verbatim
PostScript code in the submitted \LaTeX\ file. Wherever possible, please try to use standard \LaTeX\ tools
and packages.

\subsubsection{References.\label{bibby}}
You can produce your bibliography in the standard \LaTeX\ way using the \verb"\bibitem" command. Alternatively
you can use BibTeX: our preferred  \verb".bst" styles are:

\begin{itemize}
\item For the numerical (Vancouver) reference style we recommend that authors use
 \verb"unsrt.bst"; this does not quite follow the style of published articles in our
 journals but this is not a problem.  Alternatively \verb"iopart-num.bst" created by Mark A Caprio
 produces a reference style that closely matches that in published articles.  The file is available from
\verb"http://ctan.org/tex-archive/biblio/bibtex/contrib/iopart-num/" .
\item For alphabetical (Harvard) style references we recommend that authors use the \verb"harvard.sty"
in conjunction with the \verb"jphysicsB.bst" BibTeX style file.  These, and accompanying documentation, can be downloaded
from \penalty-10000 \verb"http://www.ctan.org/tex-archive/macros/latex/contrib/harvard/".
Note that the \verb"jphysicsB.bst" bibliography style does not include article titles
in references to journal articles.
To include the titles of journal articles you can use the style \verb"dcu.bst" which is included
in the \verb"harvard.sty" package.  The output differs a little from the final journal reference
style, but all of the necessary information is present and the reference list will be formatted
into journal house style as part of the production process if your article is accepted for publication.
\end{itemize}

\noindent Please make sure that you include your \verb".bib" bibliographic database file(s) and any
\verb".bst" style file(s) you have used.

\subsection{\label{copyright}Copyrighted material and ethical policy} If you wish to make use of previously published material for which you do not own the copyright then you must seek permission from the copyright holder, usually both the author and the publisher.  It is your responsibility to obtain copyright permissions and this should be done prior to submitting your article. If you have obtained permission, please provide full details of the permission granted---for example, copies of the text of any e-mails or a copy of any letters you may have received. Figure captions must include an acknowledgment of the original source of the material even when permission to reuse has been obtained.  Please read our ethical policy before writing your article.

\subsection{Naming your files}
\subsubsection{General.}
Please name all your files, both figures and text, as follows:
\begin{itemize}
\item Use only characters from the set a to z, A to Z, 0 to 9 and underscore (\_).
\item Do not use spaces or punctuation characters in file names.
\item Do not use any accented characters such as
\'a, \^e, \~n, \"o.
\item Include an extension to indicate the file type (e.g., \verb".tex", \verb".eps", \verb".txt", etc).
\item Use consistent upper and lower case in filenames and in your \LaTeX\ file.
If your \LaTeX\ file contains the line \verb"\includegraphics{fig1.eps}" the figure file must be called
\verb"fig1.eps" and not \verb"Fig1.eps" or \verb"fig1.EPS".  If you are on a Unix system, please ensure that
there are no pairs of figures whose names differ only in capitalization, such as \verb"fig_2a.eps" and \verb"fig_2A.eps",
as Windows systems will be unable to keep the two files in the same directory.
\end{itemize}
When you submit your article files, they are manipulated
and copied many times across multiple databases and file systems. Including non-standard
characters in your filenames will cause problems when processing your article.
\subsubsection{\label{fname}Naming your figure files.} In addition to the above points, please give each figure file a name which indicates the number of the figure it contains; for example, \verb"figure1.eps", \verb"figure2a.eps", etc. If the figure file contains a figure with multiple parts, for example figure 2(a) to 2(e), give it a name such as \verb"figure2a_2e.eps", and so forth.
\subsection{How to send your files}
Please send your submission via the ScholarOne submission system.  Go to the journal home
page, and use the `Submit an article' link on the right-hand side.

\section{Preparing your article}

\subsection{Sample coding for the start of an article}
\label{startsample}
The code for the start of a title page of a typical paper in the \verb"iopart.cls" style might read:
\small\begin{verbatim}
\documentclass[12pt]{iopart}
\begin{document}
\title[The anomalous magnetic moment of the
neutrino]{The anomalous magnetic moment of the
neutrino and its relation to the solar neutrino problem}

\author{P J Smith$^1$, T M Collins$^2$,
R J Jones$^3$\footnote{Present address:
Department of Physics, University of Bristol, Tyndalls Park Road,
Bristol BS8 1TS, UK.} and Janet Williams$^3$}

\address{$^1$ Mathematics Faculty, Open University,
Milton Keynes MK7~6AA, UK}
\address{$^2$ Department of Mathematics,
Imperial College, Prince Consort Road, London SW7~2BZ, UK}
\address{$^3$ Department of Computer Science,
University College London, Gower Street, London WC1E~6BT, UK}
\ead{williams@ucl.ac.uk}

\begin{abstract}
...
\end{abstract}
\keywords{magnetic moment, solar neutrinos, astrophysics}
\submitto{\jpg}
\maketitle
\end{verbatim}
\normalsize

At the start of the \LaTeX\ source code please include
commented material to identify the journal, author, and (if you are sending a revised
version or a resubmission) the reference number that the journal
has given to the submission. The first non-commented line should be
\verb"\documentclass[12pt]{iopart}"  to load the preprint class
file.  The normal text will be in the Computer Modern 12pt font.
It is possible to specify 10pt font size by passing the option \verb"[10pt]" to the class file.
Although it is possible to choose a font other than Computer Modern by loading external packages, this is not recommended.

The article text begins after \verb"\begin{document}".
Authors of very long articles may find it convenient to separate
their article into a series of \LaTeX\ files each containing one section, and each of which is called
in turn by the primary file.  The files for each section should be read in from the current directory;
please name the primary file clearly so that we know to run \LaTeX\ on this file.

Authors may use any common \LaTeX\ \verb".sty" files.
Authors may also define their own macros and definitions either in the main article \LaTeX\ file
or in a separate \verb".tex" or \verb".sty" file that is read in by the
main file, provided they do not overwrite existing definitions.
It is helpful to the production staff if complicated author-defined macros are explained in a \LaTeX\ comment.
The article class \verb"iopart.cls" can be used with other package files such
as those loading the AMS extension fonts
\verb"msam" and \verb"msbm", which provide the
blackboard bold alphabet and various extra maths symbols as well as symbols useful in figure
captions.  An extra style file \verb"iopams.sty" is provided to load these
packages and provide extra definitions for bold Greek letters.

\subsection{\label{dblcol}Double-column layout}
The \verb"iopart.cls" class file produces single-column output by default, but a two-column layout can be obtained by
using \verb"\documentclass[10pt]" at the start of the file and \verb"\ioptwocol" after the \verb"\maketitle" command.  Two-column output will begin
on a new page (unlike in published double-column articles, where the two-column material
starts on the same page as the abstract).

In general we prefer to receive submissions in single-column format even for journals
published in double-column style; however, the \verb"\ioptwocol" option may be useful to test figure sizes
and equation breaks for these journals.  When setting material
in two columns you can use the asterisked versions of \LaTeX\ commands such as \verb"\begin{figure*} ... \end{figure*}"
to set figures and tables across two columns.  If you have any problems or any queries about producing two-column output, please contact us at \verb"submissions@iop.org".

\section{The title and abstract page}
If you use \verb"iopart.cls", the code for setting the title page information is slightly different from
the normal default in \LaTeX.  If you are using a different class file, you do not need to mimic the appearance of
an \verb"iopart.cls" title page, but please ensure that all of the necessary information is present.

\subsection{Titles and article types}
The title is set using the command
\verb"\title{#1}", where \verb"#1" is the title of the article. The
first letter
of the title should be capitalized with the rest in lower case.
The title appears in bold case, but mathematical expressions within the title may be left in light-face type.

If the title is too long to use as a running head at the top of each page (apart from the
first) a short
form can be provided as an optional argument (in square brackets)
before the full title, i.e.\ \verb"\title[Short title]{Full title}".

For article types other than papers, \verb"iopart.cls"
has a generic heading \verb"\article[Short title]{TYPE}{Full title}"
and some specific definitions given in table~\ref{arttype}. In each case (apart from Letters
to the Editor and Fast Track Communications) an
optional argument can be used immediately after the control sequence name
to specify the short title; where no short title is given, the full title
will be used as the running head.  Not every article type has its own macro---use \verb"\article" for
any not listed.  A full list of the types of articles published by a journal is given
in the submission information available via the journal home page.
The generic heading could be used for
articles such as those presented at a conference or workshop, e.g.
\small\begin{verbatim}
\article[Short title]{Workshop on High-Energy Physics}{Title}
\end{verbatim}\normalsize
Footnotes to titles may be given by using \verb"\footnote{Text of footnote.}" immediately after the title.
Acknowledgment of funding should be included in the acknowledgments section rather than in a footnote.

\begin{table}
\caption{\label{arttype}Types of article defined in the {\tt iopart.cls}
class file.}
\footnotesize\rm
\begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus12pt}}l}}
\br
Command& Article type\\
\mr
\verb"\title{#1}"&Paper (no surtitle on first page)\\
\verb"\ftc{#1}"&Fast Track Communication\\
\verb"\review{#1}"&Review\\
\verb"\topical{#1}"&Topical Review\\
\verb"\comment{#1}"&Comment\\
\verb"\note{#1}"&Note\\
\verb"\paper{#1}"&Paper (no surtitle on first page)\\
\verb"\prelim{#1}"&Preliminary Communication\\
\verb"\rapid{#1}"&Rapid Communication\\
\verb"\letter{#1}"&Letter to the Editor\\
\verb"\article{#1}{#2}"&Other articles\\\ & (use this for any other type of article; surtitle is whatever is entered as {\tt
\#1})\\
\br
\end{tabular*}
\end{table}

\subsection{Authors' names and addresses}
For the authors' names type \verb"\author{#1}",
where \verb"#1" is the
list of all authors' names. Western-style names should be written as initials then
family name, with a comma after all but the last
two names, which are separated by `and'. Initials should {\it not} be followed by full stops. First (given) names may be used if
desired.  Names in Chinese, Japanese and Korean styles should be written as you want them to appear in the published article. Authors in all IOP Publishing journals have the option to include their names in Chinese, Japanese or Korean characters in addition to the English name: see appendix B for details.

If the authors are at different addresses a superscripted number, e.g. $^1$, \verb"$^1$", should be used after each
name to reference the author to his/her address.
If an author has additional information to appear as a footnote, such as
a permanent address, a normal \LaTeX\ footnote command
should be given after the family name and address marker
with this extra information.

The authors' affiliations follow the list of authors.
Each address is set by using
\verb"\address{#1}" with the address as the single parameter in braces.
If there is more
than one address then the appropriate superscripted number, followed by a space, should come at the start of
the address.

E-mail addresses are added by inserting the
command \verb"\ead{#1}" after the postal address(es) where \verb"#1" is the e-mail address.
See section~\ref{startsample} for sample coding. For more than one e-mail address, please use the command
\verb"\eads{\mailto{#1}, \mailto{#2}}" with \verb"\mailto" surrounding each e-mail address.  Please ensure
that, at the very least, you state the e-mail address of the corresponding author.

\subsection{The abstract}
The abstract follows the addresses and
should give readers concise information about the content
of the article and indicate the main results obtained and conclusions
drawn. It should be self-contained---there should be no references to
figures, tables, equations, bibliographic references etc.  It should be enclosed between \verb"\begin{abstract}"
and \verb"\end{abstract}" commands.  The abstract should normally be restricted
to a single paragraph of around 200 words.

\subsection{Subject classification numbers}
We no longer ask authors to supply Physics and Astronomy Classification System (PACS)
classification numbers.  For submissions to {\it Nonlinearity}\/ we ask that you should
supply Mathematics Subject Classification (MSC) codes.  MSC numbers are included after the abstract
using \verb"\ams{#1}".

The command
\verb"\submitto{#1}" can be inserted, where \verb"#1" is the journal name written in full or the appropriate control sequence as
given in table~\ref{jlab1}. This command is not essential to the running of the file and can be omitted.

\subsection{Keywords}
Keywords are required for all submissions. Authors should supply a minimum of three (maximum seven) keywords appropriate to their article as a new paragraph starting \verb"\noindent{\it Keywords\/}:" after the end of the abstract.

\subsection{Making a separate title page}
To keep the header material on a separate page from the
body of the text insert \verb"\maketitle" (or \verb"\newpage") before the start of the text.
If \verb"\maketitle" is not included the text of the
article will start immediately after the abstract.

\section{The text}
\subsection{Sections, subsections and subsubsections}
The text of articles may be divided into sections, subsections and, where necessary,
subsubsections. To start a new section, end the previous paragraph and
then include \verb"\section" followed by the section heading within braces.
Numbering of sections is done {\it automatically} in the headings:
sections will be numbered 1, 2, 3, etc, subsections will be numbered
2.1, 2.2,  3.1, etc, and subsubsections will be numbered 2.3.1, 2.3.2,
etc.  Cross references to other sections in the text should, where
possible, be made using
labels (see section~\ref{xrefs}) but can also
be made manually. See section~\ref{eqnum} for information on the numbering of displayed equations. Subsections and subsubsections are
similar to sections but
the commands are \verb"\subsection" and \verb"\subsubsection" respectively.
Sections have a bold heading, subsections an italic heading and
subsubsections an italic heading with the text following on directly.
\small\begin{verbatim}
\section{This is the section title}
\subsection{This is the subsection title}
\end{verbatim}\normalsize

The first section is normally an introduction,  which should state clearly
the object of the work, its scope and the main advances reported, with
brief references to relevant results by other workers. In long papers it is
helpful to indicate the way in which the paper is arranged and the results
presented.

Footnotes should be avoided whenever possible and can often be included in the text as phrases or sentences in parentheses. If required, they should be used only for brief notes that do not fit conveniently into the text. The use of
displayed mathematics in footnotes should be avoided wherever possible and no equations within a footnote should be numbered.
The standard \LaTeX\ macro \verb"\footnote" should be used.  Note that in \verb"iopart.cls" the \verb"\footnote" command
produces footnotes indexed by a variety of different symbols,
whereas in published articles we use numbered footnotes.  This
is not a problem: we will convert symbol-indexed footnotes to numbered ones during the production process.

\subsection{Acknowledgments}
Authors wishing to acknowledge assistance or encouragement from
colleagues, special work by technical staff or financial support from
organizations should do so in an unnumbered `Acknowledgments' section
immediately following the last numbered section of the paper. In \verb"iopart.cls" the
command \verb"\ack" sets the acknowledgments heading as an unnumbered
section.

Please ensure that you include all of the sources of funding and the funding contract reference numbers that you are contractually obliged to acknowledge. We often receive requests to add such information very late in the production process, or even after the article is published, and we cannot always do this. Please collect all of the necessary information from your co-authors and sponsors as early as possible.

\subsection{Appendices}
Technical detail that it is necessary to include, but that interrupts
the flow of the article, may be consigned to an appendix.
Any appendices should be included at the end of the main text of the paper, after the acknowledgments section (if any) but before the reference list.
If there are
two or more appendices they should be called Appendix A, Appendix B, etc.
Numbered equations will be in the form (A.1), (A.2), etc,
figures will appear as figure A1, figure B1, etc and tables as table A1,
table B1, etc.

The command \verb"
\section{Introduction}
\label{sec:Intro}

The classical-relativistic Alcubierre drive solution allows timelike observers to travel superluminally, although at the expense of using material with negative rest-mass energy \cite{Alcubierre1994}; for recent reviews see also \cite{Lobo2007,Alcubierre2017}. This solution is given by the following asymptotically-flat metric:
\begin{equation}
    {\rm d}s^2=-c^2 {\rm d} t^2 + ({\rm d}x-f(r_s)v_s {\rm d}t)^2+{\rm d}y^2 + {\rm d}z^2,
    \label{eq:Alc}
\end{equation}
where $r_s=\sqrt{(x-v_st)^2 + y^2 + z^2}$. The metric describes a spherical warp bubble (a region deviating from the flat metric) moving along the $x$-axis with an arbitrary velocity $v_s$, which may be larger or smaller than the speed of light $c$.

The shape function $f(r_s)$, present in the metric, defines the size and profile of the warp bubble. For large distances $r_s$ from the bubble, $f(r_s)=0$ and the spacetime is flat. For small distances, $r_s\approx0$, the shape function $f(r_s) = 1$, and the metric describes the flat internal region of the bubble. In a coordinate system with $x\rightarrow x^\prime = x-v_st$, this internal region is described explicitly by the Minkowski flat metric. The intermediate region, for which $f(r_s)\gtrsim 0$, corresponds to the spherical boundary of the warp. In the original study, \cite{Alcubierre1994} chose function $f(r_s)$, somewhat arbitrarily, as:
\begin{equation}
\label{eq:ShFun0}
f_{\rm Alc} (r_s) = \frac{\tanh (\sigma (
r_s + R )) - \tanh (\sigma ( r_s - R ))}{2 \tanh( \sigma R)},
\end{equation}
where parameters $R$ and $\sigma^{-1}$ define the radius and the thickness of the transition from the internal to the external region, correspondingly.

In the case of superluminal motion, the metric possesses a black hole-like event horizon behind the bubble and a white hole-like event horizon in front of it \cite{Finazzi2009}. These event horizons arise because timelike observers cannot exit the superluminal ship in the direction ahead of it, and cannot enter it from behind. In both cases, the timelike observers would have to move superluminally when outside of the ship.

The energy density for the Alcubierre drive, as measured by Eulerian observers ($u_\mu = (1,0,0,0)$), is given by:
\begin{equation}
\label{eq:AlcT}
    T^{00}=-\frac{1}{8\pi}\frac{\rho^2 v_s^2}{4r_s^2}\left(\frac{{\rm d} f}{{\rm d} r_s}\right)^2,
\end{equation}
where $\rho^2 \equiv y^2 + z^2$ is the cylindrical coordinate.

Despite its interesting properties which allow timelike observers to travel at arbitrary velocities, the Alcubierre drive solution possesses several drawbacks. As noted earlier, it requires negative energy densities, Equation~(\ref{eq:AlcT}), and thus violates the weak energy condition. Although negative energy densities are a general property of any superluminal drive \cite{Olum1998,Visser2000},  the energy density is also negative at subluminal speeds for the Alcubierre drive, even in the weak-field approximation \cite{Lobo2004}. Additionally, superluminal motion allows for closed timelike loops, e.g. leading to grandfather paradox, and violates the null energy condition and causality, e.g. \cite{Everett1996}, although the latter may be recovered at the expense of Lorentz invariance \cite{Liberati2002}. When moving superluminally, the drive has an additional problem. It leads to quantum instabilities related to pair production near the horizon behind the warp as well as accumulation of particles at the horizon in the front part of the warp \cite{Finazzi2009}.

The Alcubierre drive is also problematic at sub-relativistic speeds. Firstly, it requires unphysically large amounts of (negative) energy. For instance, it would require an amount of negative energy comparable to the mass of the Sun to produce relativistic bubbles of $\approx \rm{meter}$ sizes \cite{Alcubierre1994}. Furthermore, such high {\it negative} energy densities do not appear even theoretically feasible. There are no known materials which would allow for gathering large amounts of negative energy in a controlled way. While zero-point vacuum fluctuations may produce negative energies in curved spacetimes, for Alcubierre drives this situation is only possible if the walls of the bubble had thicknesses comparable to Planck scales. Such thin walls, however, require extreme amounts of energy -- comparable to the rest-mass energy of the Universe -- as may be seen from Equation~(\ref{eq:AlcT}). Therefore, there is no physical way to create an Alcubierre drive \cite{Pfenning1997CQG,Ford1997}.

Finally, there is no proposed way of creating an Alcubierre drive, even if negative energy were available. In the original study, \cite{Alcubierre1994} suggested that the velocity may be time-dependent, $v_s = v_s(t)$. Indeed, Equation~(\ref{eq:AlcT}) retains its form even for time-dependent velocities. And, since $v_s=0$ corresponds to flat spacetime, it was assumed that the Alcubierre drive might be generated through acceleration. However, the metric in Equation~(\ref{eq:Alc}) with time-variable $v_s = v_s(t)$ corresponds to a time-variable stress-energy tensor which does not satisfy continuity equations. Alternatively, such solutions may be said to require an implicit dynamical field to effectively provide propulsion for the object, e.g. \cite{Bassett2000}. Generally, there are no self-consistent warp drive solutions proposed in the literature which can self-accelerate at all from zero velocities, not to mention gain superluminal speeds.

Despite the rather extensive work on the properties of the Alcubierre drive solution, it remains unclear which of the above issues are features of the Alcubierre solution specifically or more fundamental properties of warp drives as such. New warp drive solutions have been introduced only in very few studies. \cite{vandenBroeck1999} reduced the energy requirements of the Alcubierre drive to about the mass of the Sun while satisfying the vacuum energy inequalities. The reduction was realized by decreasing the externally measured size of the warp bubble down to $10^{-15}\,{\rm m}$ while keeping the internal volume constant. This solution satisfies the weak energy conditions, although it requires that classical gravity remains applicable down to such small scales, at which it was never tested. However, as we show in \ref{sec:AppA} through a coordinate transformation, this solution is equivalent to the Alcubierre solution.

\cite{Natario2002} constructed a warp drive solution without space contraction or expansion, contrary to the earlier assumption that it facilitated the movement of warp drives. \cite{Natario2006} constructed a new subluminal warp drive solution in the weak-field regime, which required negative energies. \cite{Loup2001} had previously introduced a modified version of the Alcubierre drive intended to alter the rate of time for the observers inside the bubble. However, their modification reduces to the original Alcubierre metric, as we also show in \ref{sec:AppA}. Finally, \cite{Lentz2020} has recently proposed a warp drive metric claiming to have purely positive energy everywhere in both subluminal and superluminal regimes, although without providing means to reproduce the study.

The works above, to our knowledge, summarize all the modifications of the Alcubierre drive available in the literature. Superluminal travel had also been studied by \cite{Krasnikov1998} and \cite{Everett1997}. In these studies, the authors introduced Krasnikov tubes. Krasnikov tubes are `spacetime tunnels' which allow for superluminal travel without violating causality, but only for round trips and with much larger energy requirements than the Alcubierre drive. Superluminal travel has also been discussed in the context of wormholes, e.g. \cite{Garattini2007}, and time-machine metrics, e.g. \cite{Fermi2018}, in all cases requiring negative energies. Finally, modified gravity theories may provide some desirable properties for the Alcubierre drive. For instance, conformal gravity allows for construction of Alcubierre solutions with positive energy only \cite{Varieschi2013}, while extra-dimensional theories of gravity may reduce the energy requirements of the drive \cite{White2013}.

In this study, we show that the properties of the Alcubierre metric -- in particular, its negative energy density and the accompanying immense energy requirements -- are not a necessary feature of warp drive spacetimes. In Section~2, we discuss that any general warp drive, including the Alcubierre metric, may be thought of as a shell of positive- or negative-energy density material which modifies the state of spacetime in the flat vacuum region inside it. In Section~3, we introduce, for the first time, the most general spherically symmetric warp drives. We show that the reason for the negative energy requirements of the Alcubierre metric and all the warp drives introduced in the literature is, likely, the truncation of the gravitational field outside of the metric, and the accelerated rate of time compared to the comoving free observer. In turn, the most general positive-energy (spherical) warp drive solutions must slow down the time compared to the comoving observer and have a gradual (Schwarzschild) fall off of the gravitational field. In Section~4, we introduce a simple method to construct warp drive metrics by choosing the state of spacetime inside them. Additionally, we show that the Alcubierre metric is only a particular member of a much more general class of warp drive solutions admitting negative energy densities. Subsequently, we construct several spacetimes with a range of new properties. We introduce a macroscopic negative-energy drive which satisfies quantum inequalities, as well as several metrics with negative-energies which allow one to control the rate of time and spatial contraction inside the drive. In Section~5, we review the interpretation of warp drives in light of this study.

\section{General class of warp drives}
\label{sec:GenDrives}
\subsection{Definition of a general warp drive spacetime}

\begin{figure}
\begin{center}
    \includegraphics[width=\linewidth]{./WarpSketchFinal2.png}
\end{center}
\begin{flushleft}
    \caption{A schematic illustration of a warp-drive spacetime. The spacetime consists of three regions: Asymptotically-flat vacuum background $\mathcal{D}_{\rm out}$ (background), general stationary curved region $\mathcal{D}_{\rm warp}$ with a spherical topology (the warping region) and a flat inner region $\mathcal{D}_{\rm in}$ (`passenger' space). Any such spacetime, including the Alcubierre drive metric, is realised through a shell of ordinary or exotic negative energy density material filling the warping region $\mathcal{D}_{\rm warp}$. Axis $x$ shows the direction of motion, while $\rho$ is the cylindrical radius. As we discuss in Section~\ref{sec:AxiClasses}, flattened disk-shaped metrics minimise energy requirements of the particular Alcubierre, but not necessarily other, warp drive spacetimes. As we also discuss in Section~\ref{sec:Disc}, warp drive spacetimes require some form of propulsion in order to accelerate. For this reason, in physical realisations of such spacetimes, the front and rear parts are likely asymmetric. \label{fig:DriveScheme}}
\end{flushleft}
\end{figure}

We define a general warp drive spacetime, as shown in Figure~\ref{fig:DriveScheme}, as an asymptotically-flat vacuum region $\mathcal{D}_{\rm out}$ (background) which encloses a compact arbitrarily curved region $\mathcal{D}_{\rm warp}$ with a spherical topology (the warping region); $\mathcal{D}_{\rm warp}$, in turn, encloses a flat extended\footnote{To avoid effectively including general asymptotically-flat spacetimes in the defintion, we require that the extent of the region $\mathcal{D}_{\rm in}$ should be non-vanishing and physically interesting. In other words, the inner region $\mathcal{D}_{\rm in}$ should be large enough to allow observers to conduct physical experiments of desired scales.} compact region $\mathcal{D}_{\rm in}$ with a trivial topology (the `passenger' area). This definition covers both all the existing warp drive spacetimes, such as the one by \cite{Alcubierre1994}, as well as trivial solutions, such as weak-field perturbations of the Minkowski spacetime. To formalize the difference between extreme and mild cases like these, we shall consider the warped region $\mathcal{D}_{\rm warp}$ by using comoving reference frames located inside and outside it, which will allow us to quantify how strong its gravitational influence is.

We shall focus on warp drive spacetimes which, intuitively, are stationary or non-changing from the `passenger' point of view. More formally, we consider warp drive spacetimes which{, by definition,} admit a global Killing vector field, ${\bf \xi}$, which is aligned with the four-velocity of the boundary of the region $\partial\mathcal{D}_{\rm in}$ (the inner boundary of the shaded region in Figure~\ref{fig:DriveScheme}). Such a field establishes the global frame of rest with respect to the warped region. Subsequently, whenever we discuss a physical motion of an observer relative to the warp drive, we consider, locally, the motion relative to the global reference frame defined by the field ${\bf \xi}$. Furthermore, using this vector field will allow us to apply the well-known techniques suitable for spacetimes possessing a timelike Killing vector to analyse and classify warp drive spacetimes.

Since the internal region $\mathcal{D}_{\rm in}$ is flat, vector field ${\bf \xi}$ is constant in that region. Therefore, in the internal region, this vector field ${\bf \xi}$ defines a reference frame (tetrad) for the internal observer\footnote{While, in relativity, physical observers are timelike by definition, we shall occasionally be mentioning timelike and spacelike observers for convenience. By these, we shall understand physical observers and spacelike frames, correspondingly.}  $\mathcal{O}_{\rm in}$. Similarly, in the asymptotic infinity, vector field ${\bf \xi}$ defines the frame of the remote comoving observer $\mathcal{O}_{\rm out, co}$. This comoving reference frame $\mathcal{O}_{\rm out, co}$, as we further argue, is the most natural frame against which one can compare the frame $\mathcal{O}_{\rm in}$ inside the warped region. Finally, this remote comoving observer $\mathcal{O}_{\rm out, co}$ may be moving with constant three-velocity ${\bf v}_s$ with respect to a timelike observer $\mathcal{O}_{\rm out}$, the latter of which we shall consider to be at rest. Everywhere in this study, apart from the general discussion in this section, we shall consider warp drives spacetimes which are axisymmetric along the direction of motion.

The three-velocity of the comoving observer ${\bf v}_s$, { which represents the velocity of the warped region relative to the remote observer $\mathcal{O}_{\rm out}$}, may be slower or faster than the speed of light with respect to the observer. This corresponds to four-velocity of the comoving observer $\mathcal{O}_{\rm out, co}$ being timelike, null or spacelike. Both observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out, co}$, therefore, may in general, at least formally, be timelike, null, or spacelike. If region $\mathcal{D}_{\rm warp}$ contained vanishing stress-energy tensor, the whole spacetime would be close to Minkowski spacetime and the two observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out}$ would both be timelike, null, or spacelike at the same time. However, it is also possible for region $\mathcal{D}_{\rm warp}$ to be sufficiently curved such that, as we show further, the norms of the four-velocities of observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out,co}$ may be different from each other. Therefore, all stationary warp drive spacetimes may be split into four distinct classes, based on whether the remote comoving observer  $\mathcal{O}_{\rm out,co}$  moves subluminally (subluminal drives) or superluminally (superluminal drives), and whether the internal observer $\mathcal{O}_{\rm in}$ has the same norm as the comoving observer  $\mathcal{O}_{\rm out,co}$ (mild warp drives) or different norm from the comoving observer (extreme warp drives):
\begin{description}
    \item[Class I: Mild subluminal warp drives:] These spacetimes are defined by the vector field ${\bf \xi}$ being timelike everywhere. Consequently, three-velocities of such drives are subluminal, i.e. $v_s<c$. Spacetimes of this class approach the flat Minkowski spacetime in the trivial limit, and observers $\mathcal{O}_{\rm in}$, $\mathcal{O}_{\rm out, co}$ reduce to a pair of co-moving timelike Lorentz observers. Non-trivial members of this class contain spacetimes with region $\mathcal{D}_{\rm warp}$ sufficiently curved, so that tetrads of observers $\mathcal{O}_{\rm in}$ and  $\mathcal{O}_{\rm out, co}$ differ significantly from each other, i.e. the observers read off different rates of clocks and lengths of rulers. At the same time, such spacetimes also contain weak-field solutions corresponding to classical shell-like objects moving with subluminal velocities and weakly modifying the state of the spacetime inside them. Such solutions are possible because the $\mathcal{D}_{\rm warp}$ region may be set arbitrarily close to being flat, rendering the whole spacetime arbitrarily close to Minkowski spacetime.

    \item[Class II: Mild superluminal warp drives:] These spacetimes are characterized by the vector field ${\bf \xi}$ being spacelike or null everywhere. Consequently, such warp drives have luminal or superluminal velocities, i.e., $v_s\geq c$. These spacetimes also admit a trivial limit, wherein they reduce to flat Minkowski spacetime, with the tetrads $\mathcal{O}_{\rm in}$, $\mathcal{O}_{\rm out, co}$ corresponding to a pair of comoving null or spacelike (superluminal) `observers'. Weak-field members of the class correspond to small amounts of `superluminal matter' in the region $\mathcal{D}_{\rm warp}$ introducing small differences in the measurements of frames $\mathcal{O}_{\rm in}$, $\mathcal{O}_{\rm out, co}$. By `superluminal matter' we understand the matter at rest with respect to a space-like reference frame. In the case of the stress-energy tensor for a perfect fluid, such matter violates the dominant energy condition. A general spacetime of this class introduces non-trivial differences between frames $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out, co}$. Since superluminal matter cannot be produced from physical matter, and since null or spacelike tetrads cannot be associated with physical observers, the spacetimes of this class have limited interest.

    \item[Class III: Extreme superluminal warp drives:] These spacetimes are defined by the vector field ${\bf \xi}$ being timelike in the inner region $\mathcal{D}_{\rm in}$, but null or spacelike in the asymptotic infinity of the outer region $\mathcal{D}_{\rm out}$. The remote comoving observers in such spacetimes move luminally or superluminally relative to the resting timelike observer $\mathcal{O}_{\rm out}$, i.e., $v_s\geq c$. This class of spacetimes does not contain trivial solutions and the warped region $\mathcal{D}_{\rm warp}$ is sufficiently curved to allow timelike observers $\mathcal{O}_{\rm in}$ to be moving superluminally relative to the timelike observer $\mathcal{O}_{\rm out}$ (and as a consequence, the timelike observer $\mathcal{O}_{\rm in}$ will be travelling back in time from the point of view of yet another remote timelike observer $\mathcal{O}_{\rm out}^\prime$). At the same time, the comoving observer $\mathcal{O}_{\rm out, co}$ is formally superluminal; in other words, remote timelike observers cannot be comoving with a warp drive of this class. For such spacetimes, we can define a Killing horizon $\partial\mathcal{D}_{\rm K}$ as the minimal boundary surface where the Killing vector field ${\bf \xi}$  becomes null. A necessary, but not sufficient, condition for warp drives of this class to be physical is that the Killing horizon $\partial\mathcal{D}_{\rm K}$ does not intersect with region $\mathcal{D}_{\rm warp}$. If the Killing horizon did intersect with region $\mathcal{D}_{\rm warp}$, some part of the matter in that region would be at rest with respect to spacelike tetrads aligned with vector field ${\bf \xi}$. In other words, a fraction of matter in the region $\mathcal{D}_{\rm warp}$ would have to be superluminal (in the same sense, as for Class II above) and violate the dominant energy condition.

    \item[Class IV: `Extreme' subluminal warp drives:] Spacetimes of this class are defined by the vector field  ${\bf \xi}$ being null or spacelike in the inner region $\mathcal{D}_{\rm in}$, but timelike at the asymptotic infinity of the outer region $\mathcal{D}_{\rm out}$. Since the comoving observer $\mathcal{O}_{\rm out, co}$ is timelike, such spacetimes are subluminal, i.e. $v_s < c$. Since the Killing vector is spacelike in the inner region $\mathcal{D}_{\rm in}$, no timelike internal observers can be at rest relative to the inner boundary of the drive $\partial\mathcal{D}_{\rm in}$. This property bears similarity to black hole spacetimes, except for that the inner region $\mathcal{D}_{\rm in}$ for this class of spacetimes is flat everywhere. A necessary condition for spacetimes of this class to be physical is that the Killing horizon coincides with the boundary of the inner region, $\partial\mathcal{D}_{\rm in}$. Otherwise, some fraction of matter in the region $\mathcal{D}_{\rm warp}$ will be superluminal, as in the previous class.
\end{description}

In practical terms, as follows from our definition, any warp drive spacetime may be seen as a shell of ordinary or exotic material which fills region $\mathcal{D}_{\rm warp}$ and moves with some constant velocity relative to an external timelike observer $\mathcal{O}_{\rm out}$. The presence of the shell inevitably modifies the space and time in the inner region $\mathcal{D}_{\rm in}$. This modification leads (in addition to other effects) to a difference in the measurements of times and lengths between an observer in the inner region $\mathcal{D}_{\rm in}$ and a comoving remote observer $\mathcal{O}_{\rm out, co}$. General warp drive spacetimes, therefore, form a continuous family which includes both trivial (flat or nearly-flat) and non-trivial (strongly curved) spacetimes.

\subsection{How do existing warp drive spacetimes fit in the classification?}
\label{sec:ClassReview}
The Alcubierre drive is a truncated warp drive spacetime defined by meeting the three requirements. First, that the external spacetime $\mathcal{D}_{\rm out}$ is Minkowskian. In other words, the matter in the warped region $\mathcal{D}_{\rm warp}$ does not exhibit gravitational influence outside of it (Requirement 1). Second, observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out}$ are always timelike, and the tetrads of these observers are equal to each other. In other words, the clock rates and the lengths of the rulers of observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out}$ are synchronized and put equal to each other (Requirement 2). Finally, the warped region $\mathcal{D}_{\rm warp}$ is functionally limited to Equations~(\ref{eq:Alc}) and (\ref{eq:ShFun0}) in Section~\ref{sec:Intro} (Requirement 3). For subluminal velocities, the Alcubierre drive belongs to Class I, mild subluminal warp drives. For superluminal velocities it belongs to Class III, extreme superluminal warp drives.

Compared to general warp drives, this set of constraints may seem artificial. For example, requiring that the material of the Alcubierre drive does not gravitate in region $\mathcal{D}_{\rm out}$, which is non-typical for massive objects, as we discuss further in Section~\ref{sec:SphGen}, unnecessarily puts strong constraints on the stress-energy tensor in region $\mathcal{D}_{\rm warp}$. Similarly, requiring that the time of observers $\mathcal{O}_{\rm in}$ is synchronized with the time of observers $\mathcal{O}_{\rm out}$ means that, at least in the subluminal regime, the time of internal observers $\mathcal{O}_{\rm in}$ is accelerated compared to the time of remote comoving observers $\mathcal{O}_{\rm out, co}$. In comparison, general warp drive spacetimes do not need to be truncated and may allow an arbitrary relation between the measurements of observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out, co}$.

A more general class of warp drive metric was proposed by \cite{Natario2002}. However, their most general warp drive, similarly to the Alcubierre solution, still imposes Requirements 1 and 2 from the Alcubierre metric. That is, is also makes region $\mathcal{D}_{\rm out}$ strictly flat outside of the drive and synchronizes the time of the internal and the resting (not comoving) observers. Similarly to the Alcubierre solution, the \cite{Natario2002} class of spacetimes also belongs to Class I in the subluminal regime and Class III in the superluminal regime.

Further, in Sections~\ref{sec:Axi} and \ref{sec:SphGen}, we show that general warp drive spacetimes are diverse and can have a variety of properties that are often more appealing compared to either the Alcubierre spacetime or to the \cite{Natario2002} class of spacetimes.

\subsection{Properties of warp drive spacetimes}
\label{sec:DriveProps}

In this section, we summarise the standard tools applicable for analyzing general warp drives, in order to apply them further in Sections~\ref{sec:Axi} and \ref{sec:SphGen}.

The internal physical volume of a warp drive measured by observer $\mathcal{O}_{\rm in}$, inside the warped region, is given by $V_{\rm in}\equiv\int_{\mathcal{D}_{\rm in}}{\rm d}^3x^i_{\rm in}$. For truncated warp drive spacetimes, i.e. spacetimes where the outside vacuum region $\mathcal{D}_{\rm out}$ is flat, the external volume may be defined as $V_{\rm out}\equiv\int_{\mathcal{D}_{\rm in \cup warp}}{\rm d}^3x^i_{\rm out}$. The integral is calculated over the volume enclosing the warped region and assuming, during the integration, that the volume's interior is flat. Such a construction provides a measure of the size of the warp bubble as observed from outside. It may be formally done with the help of the Cartesian coordinate grid of the remote observer $O_{\rm out}$, analytically extended to cover the warped region. The comoving external volume $V_{\rm out, co}$ may be similarly defined for the comoving observer. For general asymptotically-flat metrics, we do not consider external volumes as they are coordinate-dependent.

We calculate the energy density, the momentum flux, and their volume integrals for warp drive spacetimes, and the mechanical stress distributions, in the coordinate system adapted to the remote comoving observer ${\mathcal{O}}_{\rm out, co}$. The coordinate system adapted to the comoving observer is aligned with the Killing vector field ${\bf \xi}$, which allows for a $3+1$ decomposition of the spacetime and thus provides coordinate-invariant measures for the total energy, mass and momentum of the drive. We also consider a system of Eulerian observers defined through four-velocities $u_{\mu}=\frac{1}{\sqrt{-g^{00}}}(1,0,0,0)$, similar to how it was done in \cite{Alcubierre1994}. These observers are imperatively timelike and are normal to the surfaces of constant time, but are coordinate-dependent.

The energy density $w$ in each of the coordinate systems is given by $w=(-g^{00})^{-1}T^{00}$. The momentum flux $j^i$ and the pressure $P$ in the same reference system are given by $j^i=u_{\mu} T^{\mu i}=\frac{1}{\sqrt{-g^{00}}} T^{0i}$ and $P=T^{\mu\nu}(\delta_{\mu\nu}-u_{\mu}u_{\nu})= T + (g^{00})^{-1}T^{00}$, where $T\equiv T^{\mu\nu}g_{\mu\nu}$. Notably, the total energy and momentum contain the contribution only from the warped region $\mathcal{D}_{\rm warp}$: $E=\int_{\mathcal{D}_{\rm warp}}(-g^{00})^{-1}T^{00}\sqrt{-g}{\rm d}^3 x^i_{\rm out}$, $J^i=\int_{\mathcal{D}_{\rm warp}}\frac{1}{\sqrt{-g^{00}}} T^{0i}\sqrt{-g}{\rm d}^3 x^i_{\rm out}$.

Warp drive solutions may be compared based on how they satisfy different energy conditions. Among these are the strong and weak energy conditions. Furthermore, we also consider quantum inequalities, e.g. \cite{Pfenning1997CQG}. These inequalities, at least in the weak-field regime, potentially allow the spacetime curvature to modify the quantum vacuum and, this way, allow for some negative-energy density:
\begin{equation}
\label{eq:EnCond}
    \frac{\tau_0}{\pi}\int_{-\infty}^{\infty}\frac{\langle T_{\mu\nu}u^\mu u^\nu\rangle}{\tau^2+\tau_0^2}{\rm d}\tau\ge -\frac{3}{32\pi^2\tau_0^4},
\end{equation}
where $\tau$ is the proper time of the observer and $\tau_0$ is an arbitrary constant smaller than the local radius of curvature.

\section{General spherically symmetric warp drives}
\label{sec:SphGen}

\subsection{General subluminal spherically symmetric drives}

One may gain considerable intuition into the nature of warp drive spacetimes by considering spacetimes which are spherically symmetric in the comoving reference frame. Since spherically symmetric spacetimes are fully solvable in relativity, we may achieve full characterization of such warp drive spacetimes. More formally, in this section, we limit ourselves to spacetimes which contain an ${\rm SO}(3)$ group whose orbits are orthogonal to the Killing vector field ${\bf \xi}$ introduced in Section~\ref{sec:GenDrives}. The Killing vector field ${\bf \xi}$ defines the local frame of rest in the comoving reference frame. As we comment further, the presence of such a group is possible only for class I and IV spacetimes, i.e., subluminal warp drives.

Class I of subluminal warp drives -- mild subluminal warp drives -- contains trivial solutions and may be associated with ordinary objects, such as a thin shell of non-exotic material. However, the well-known Alcubierre and Natário drives, in the subluminal regime, also belong to the same class. By considering general spherically symmetric subluminal warp drives, we shall gain considerable intuition about these spacetimes and provide a possible explanation for their negative energy requirement.

The most general metric for a spherically symmetric stationary spacetime can be written in Schwarzschild coordinates as:
\begin{equation}
    {\rm d}s^2 = - N(r) c^2{\rm d}t^2+\Lambda(r){\rm d}r^2+r^2({\rm d}\theta^2+\sin^2\theta{\rm d}\varphi^2),
\label{eq:SphMetric}
\end{equation}
where functions $N(r)$ and $\Lambda(r)$ are general. Therefore, energy density of a spherically symmetric warp drive spacetime, in the comoving reference frame, is given by:
\begin{equation}
    w(r)=\frac{1}{8\pi r^2}\left(1-(\frac{r}{\Lambda(r)})^\prime\right)
    \label{eq:Sph}
\end{equation}
A necessary condition for the spherically symmetric metric in Equation~(\ref{eq:SphMetric}) (above) to describe a warp drive spacetime is that the energy density $w$ in both the inner region $\mathcal{D}_{\rm in}$ and outer region $\mathcal{D}_{\rm out}$ be equal to zero. In particular, as follows from Equation~(\ref{eq:Sph}), for any spherically symmetric warp drive, $\Lambda(r)=1$ in region $\mathcal{D}_{\rm in}$. Furthermore, integrating Equation~(\ref{eq:Sph}), we can express the metric function $\Lambda(r)$ directly through the energy density distribution:
\begin{equation}
    \Lambda(r)=\frac{1}{1-\frac{2}{r}\int_0^r 4\pi w(r^\prime) r^{\prime 2}{\rm d}r^{\prime}}
\label{eq:Sph2}
\end{equation}

Equations~(\ref{eq:Sph}) and (\ref{eq:Sph2}) allow us to analyze the energy density distribution of spherically symmetric solutions and to construct such spacetimes from any desired energy distribution. Thus, we may see that for purely positive energy spacetimes, $\Lambda(r)>1$ in the outside region $\mathcal{D}_{\rm out}$. This result is in agreement with the Birkhoff's theorem, and states that the metric in the region $\mathcal{D}_{\rm out}$ is described by the Schwarzschild solution. In comparison, for truncated warp drive spacetimes, such as the Alcubierre or Natário solutions, the metric in the external region is assumed to be strictly flat. This assumption necessarily requires that the integral $\int_0^{r_{\rm out}} 4\pi w(r^\prime) r^{\prime 2}{\rm d}r^{\prime}=0$. As a result, any non-trivial truncated spherically symmetric warp drive must contain regions where the energy density is negative. Therefore, the fact that Alcubierre and Natário solutions are truncated may partly explain why they require negative energy even in subluminal regimes. Finally, for any spherically symmetric warp drive spacetime, since $\Lambda(0)=\Lambda(r\rightarrow\infty)=1$, coordinate $r$ describes the same physical length scale both inside the drive and at the asymptotic infinity. In other words, unlike the drives discussed in \cite{vandenBroeck1999}, spherically symmetric warp drives cannot contain objects which significantly exceed the sizes of warp drives, as measured by external observers.

We may further calculate the spatial component of the stress energy tensor. To do so, we assume that the material in region $\mathcal{D}_{\rm warp}$ is an isotropic fluid, i.e. $T_{rr}=P(r) \Lambda(r)$. This way, we obtain the pressure of the material which constitutes the warp drive:
\begin{equation}
    P(r)\Lambda(r) = \frac{1-\Lambda(r)+r N^\prime(r)/N(r)}{r^2}
    \label{eq:Sph3}
\end{equation}
By requiring that $P=0$ in the inner vacuum region $\mathcal{D}_{\rm in}$, and by remembering that $\Lambda(r)=1$ in that region, we can see that $N(r)$ is constant in $\mathcal{D}_{\rm in}$. In the external region $\mathcal{D}_{\rm out}$, from Equation~(\ref{eq:Sph3}), $N(r) = (\Lambda(r))^{-1} = 1-\frac{2}{r}\int_0^r 4\pi w(r^\prime) r^{\prime 2}{\rm d}r^{\prime}$.  This result also follows from Birkhoff's theorem, according to which spherically symmetric spacetimes must be static and asymptotically flat. In the vacuum region, such setup, by definition, corresponds to the Schwarzschild solution (assuming the central object is electrically neutral), which matches by the form the equation for $N(r)$. In this case, $\int_0^r 4\pi w(r^\prime) r^{\prime 2}{\rm d}r^{\prime}$ will have the effective meaning of enclosed mass.

We can now obtain $N(r)$ in region $\mathcal{D}_{\rm warp}$, and this way determine $N(r)$ in the inner region $\mathcal{D}_{\rm in}$. We do this by writing down the equation for continuity of the stress-energy tensor, $T^{\mu}_{~r;\mu}=0$, similarly to how it is done for Tolman–Oppenheimer–Volkoff (TOV) equation. As a result, we obtain:
\begin{equation}
    \frac{N^\prime(r)}{N(r)}=-\left(\frac{P+\rho}{2}\right)^{-1}P^\prime(r)
    \label{eq:Sph4}
\end{equation}
{ Firstly, we make a simplifying assumption that density $\rho$ is a function of only pressure $P$, as, for example, is the case for the polytropic equation of state. Then, by using the condition that $P=0$ at the inner and outer boundary of the warp region $\mathcal{D}_{\rm warp}$, we find that the integral of the right-hand side over $\mathcal{D}_{\rm warp}$ vanishes. As a result, we conclude that $N(r_{\rm in})=N(r_{\rm out})$, where $r_{\rm in}$ and $r_{\rm out}$ are the locations of the inner and outer boundary of the region $\mathcal{D}_{\rm warp}$.}

From the solution for $N(r)$ in the outside region $\mathcal{D}_{\rm out}$, we see that the time-time coefficient of the metric $N(r)$ at the asymptotic infinity can only be larger than inside the warped region. In other words, for a subluminal warp drive based on non-exotic matter, the time inside the drive can only pass more slowly than it does for a remote Minkowski observer comoving with the drive. For scale, an Earth-mass shell of $10$ meter radius will slow down the rate of time by a small fraction of $4\cdot 10^{-4}$. If we require that the time in the inner region goes faster than in the reference frame of the comoving observer, the material in the warp drive would have to contain negative energy. This is the case for the Alcubierre and Natário drives. In these drives, the time is passing more quickly in the inside region than it is in the reference frame of the remote comoving observer (due to the clocks in the inner region being synchronized to the remote observer at rest). This is another related possible reason why the Alcubierre drive requires negative energy even in the subluminal mode.

\subsection{Impossibility of superluminal spherically symmetric spacetimes}

The method from the previous section could potentially have been useful for analysing superluminal warp drives. However,  superluminal classes of warp drives (Classes II and III) cannot be spherically symmetric. This fact is closely related to the impossibility of superluminal spherical objects in special relativity, e.g. \cite{Fayngold2002}. In Section~\ref{sec:GenDrives}, we have introduced a Killing vector field ${\bf \xi}$, which defines a local frame of rest with respect to the drive. In superluminal warp drives, vector field ${\bf \xi}$ becomes spacelike whether inside or outside of the drive. In order both to maintain spherical symmetry and for the field ${\bf \xi}$ to remain orthogonal to the orbits of the ${\rm SO}(3)$ group, the spatial component of the field ${\bf \xi}$ can only be radial with respect to the drive. Therefore, at asymptotic infinity, comoving observers for such a spacetime would have to be moving radially towards or away from a spherically symmetric drive. In other words, warp drives cannot be both spherically symmetric and move superluminally.

Similarly, Class IV drives cannot be spherically symmetric. As follows from Equation~(\ref{eq:Sph2}), in order for the radial basis vector inside the drive to be timelike, one would need to have non-zero energy density present in the vacuum inner region $\mathcal{D}_{\rm in}$. Therefore, all spherically symmetric positive-energy warp drive solutions belong to Class I. These solutions are always subluminal, satisfy the energy conditions, and are devoid of causality paradoxes present in superluminal metrics. Overall, the subluminal spherically symmetric solution presented in this section is the first example of manifestly positive energy warp drive spacetimes.

\section{Axisymmetric warp drives with a general internal region}
\label{sec:Axi}

In this section, we explore the diversity of axisymmetric warp drive solutions. There are some limitations on possible axisymmetric solutions. Explicitly expressing a metric through a desired stress-energy tensor in a closed form is not possible for a general axisymmetric spacetime. Therefore, we leave generalizing the positive energy spherically symmetric warp drive solutions to the axisymmetric case to future studies. However, by constructing sufficiently broad classes of metrics, we show that interesting solutions are possible even within these less general classes.

\subsection{Method for constructing axisymmetric warp drive solutions}
\label{sec:AxiClasses}

Below we introduce a method, by which one can construct new metrics for  a warp drive spacetime. For constructing new solutions, following the discussion in Section \ref{sec:SphGen}, we focus on choosing how the spacetime properties of the observers inside the inner region $\mathcal{O}_{\rm in}$ relate to those of remote observers $\mathcal{O}_{\rm out}$. This choice may put significant constraints on the energy content of the spacetime, and may potentially be exploited to find spacetimes with lower energy requirements. Additionally, having an explicit way to define the properties of spacetime inside the inner region $\mathcal{D}_{\rm in}$ allows us to explore and demonstrate the diversity of possible warp drive solutions.

As discussed in Section~\ref{sec:GenDrives}, any stationary warp drive spacetime may be associated with a coordinate system $x^\mu_{\rm co}$, adapted to the Killing vector ${\bf \xi}$ which defines the rest frame of the craft. We also adopt a global coordinate system $x^\mu$ which asymptotically, at infinity, approaches the coordinate system of the resting observer $\mathcal{O}_{\rm out}$ in region $\mathcal{D}_{\rm out}$. Since the two charts $x^\mu_{\rm co}$ and $x^\mu$ cover the whole spacetime and overlap, it is, in principle, possible to introduce a mapping from one to another in the regions $\mathcal{D}_{\rm in}$ and $\mathcal{D}_{\rm warp}$. The procedure for constructing an axisymmetric warp drive spacetime from this subclass is as follows:
\begin{enumerate}
    \item[1.] Choose a one-to-one mapping $x^{\mu}_{\rm co}(x^{\nu})$ between the coordinate system $x^{\mu}_{\rm co}$ adopted to the comoving observer inside the inner region $\mathcal{O}_{\rm in}$ and the coordinate system $x^{\nu}$ adopted to the external observer at rest $\mathcal{O}_{\rm out}$.
    \item[2.] Choose functions  $f_x (x^i_{\rm co})$, $f_y (x^i_{\rm co})$, $f_z (x^i_{\rm co})$, $f_t (x^i_{\rm co})$ which are equal to $1$ in region $\mathcal{D}_{\rm in}$ and are equal to $0$ in region $\mathcal{D}_{\rm out}$. These functions define the shape and size of the warp region $\mathcal{D}_{\rm warp}$ from the point of view of observer $\mathcal{O}_{\rm in}$.
    \item[3.] Formulate the metric of the spacetime as:
    \begin{equation}
    \label{eq:GenTrans}
    {\rm d}s^2=-c^2  ({\rm d}t(1-f_t)+f_t{\rm d} t_{\rm co})^2 + ({\rm d}x(1-f_x)+f_x {\rm d} x_{\rm co})^2+\qquad\
    \end{equation}
\[
 \qquad + ({\rm d}y(1-f_y)+f_y{\rm d} y_{\rm co})^2 + ({\rm d}z(1-f_z)+f_z{\rm d} z_{\rm co})^2,
\]
Express the metric in terms of { a common coordinate system $x^{\mu \prime\prime}$, by expressing $x^\mu_{\rm co}$ and $x^\nu$ explicitly through $x^{\mu \prime\prime}$}.
    \item[4.] Analyze the metric with the methods introduced in Section~\ref{sec:DriveProps}
\end{enumerate}

The procedure above is based on the idea of explicitly comparing the measurements of observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out}$. Indeed, in region $\mathcal{D}_{\rm in}$ all the functions $f_{\eta}=1$, and the metric corresponds to the coordinate system of observer $\mathcal{O}_{\rm in}$ (${\rm d}s^2=-c^2 {\rm d} t_{\rm co}^2 + {\rm d} x_{\rm co}^2 + {\rm d} y_{\rm co}^2 + {\rm d} z_{\rm co}^2$). Similarly, asymptotically, in region $\mathcal{D}_{\rm out}$ all the functions $f_{\eta}=0$ and the metric corresponds to the coordinate system of the remote observer at rest $\mathcal{O}_{\rm out}$ (${\rm d}s^2=-c^2 {\rm d} t^2 + {\rm d} x^2 + {\rm d} y^2 + {\rm d} z^2$). Several studies in the literature, e.g. \cite{Loup2001}, have proposed modifications of the Alcubierre metric without explicitly considering the measurements of inner and outer observers. Such modifications may be erroneous and may reduce to coordinate transformations, as happens in \cite{Loup2001} case. We discuss these studies further in \ref{sec:AppA}.

The procedure developed here assumes that the observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out}$ are timelike, and therefore leads to warp drive spacetimes of Class I in the subluminal regime, and of Class III in the superluminal regime. While these classes of spacetimes are arguably most interesting, the procedure may be easily generalized to cover the spacetimes of Class II and Class IV. We also comment that while the procedure ensures a relationship for the measurements of observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out}$, one may also use it to obtain different coordinate representations and, potentially, different spacetimes satisfying the same relationships between the observers. Subsequently, one may choose, for example, the most interesting spacetimes of the class.

As a demonstration, we show how the above procedure may be used to construct the Alcubierre metric given by Equation~(\ref{sec:Intro}).

\begin{itemize}
    \item[1.] We choose the one-to-one mapping as follows:
    \begin{equation}
\left\{
\begin{array}{lll}
    {\rm d}t_{\rm co}&={\rm d}t\\
    {\rm d}x_{\rm co}&={\rm d}x - v_s{\rm d}t\\
    {\rm d}y_{\rm co}&={\rm d}y\\
    {\rm d}z_{\rm co}&={\rm d}z\\
    \end{array}
    \right.
\end{equation}
    \item[2.] We select $f_x=f$, where $f$ is given by Equation~(\ref{eq:ShFun0}), whereas functions $f_y$, $f_z$ and $f_t$ cancel out in Equation~(\ref{eq:GenTrans}) and will not contribute to the metric.
    \item[3.] We arrive at the metric ${\rm d}s^2=-c^2 {\rm d} t^2 + ({\rm d}x-f(r_s)v_s {\rm d}t)^2+{\rm d}y^2 + {\rm d}z^2$, as given by Equation~(\ref{eq:Alc}).
\end{itemize}

This example once again highlights the fact that the Alcubierre metric is based on a rather artificial identification of spacetime properties between the two observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out}$, in particular increasing the rate of the time passing for the observer $\mathcal{O}_{\rm in}$. In the following sections, we explore other possible axisymmetric spacetimes and show that they have more appealing properties than the Alcubierre metric.

\subsection{Reducing $E_{\rm tot}$ by flattening the Alcubierre metrics}
\label{sec:AlcDeformed}

We start by considering the flattened Alcubierre metrics. The longitudinal extent is a simple property of the Alcubierre metric. However, it has not been studied in the literature, despite having several interesting properties.

The longitudinal extent has several peculiarities for the Alcubierre metric. Indeed, in Equation~(\ref{eq:Alc}), for sufficiently large distances $r_{s0}$ from the center of the warped region, function $f(r_s)$ vanishes in the Alcubierre solution. Therefore, the external observer $\mathcal{O}$ perceives the boundaries of the warped region $\mathcal{D}_{\rm warp}$ as a sphere of radius $r_{s0}$ moving with velocity $v_s$. If the velocity $v_s$ is subluminal due to Lorentz contraction, the boundary of the warped region $\mathcal{D}_{\rm warp}$ must be elongated for the comoving observers. Moreover, when the Alcubierre warp drive approaches the speed of light while at the same time preserving its energy content, given by Equation~(\ref{eq:AlcT}), the warped region must appear infinitely elongated. The divergent elongation of the solution in the comoving frame puts into question the possibility of accelerating Alcubierre metrics beyond the speed of light.

To analyze the solutions deformed along the axis of motion, we switch to more convenient cylindrical coordinates $(y,z)\rightarrow (\rho,\theta)$. For the transition regions described by Equation~(\ref{eq:Alc}), a more general non-spherically symmetric metric may be obtained by replacing $f(r_s)\equiv f(\sqrt{(x-x_s)^2+\rho^2}) \longrightarrow f(x-x_s, \rho)$. Notably\footnote{We believe, this form has not been found in the literature so far}, for the Alcubierre drive, recalculating the energy density with $f = f(x-x_s, \rho)$ simplifies the expression, compared to Equation~(\ref{eq:ShFun0}):
\begin{equation}
   w=-\frac{1}{8\pi}\frac{v_s^2}{4}\left(\frac{{\partial} f}{{\partial} \rho}\right)^2
\end{equation}

This form leads to three useful implications. The first implication is that, for a given velocity of the warp drive $v_s$, the most optimal way of reducing the total energy, as measured by Eulerian observers, is by flattening the shape of the warp drive. Indeed, as $\sqrt{-g}=\rho$ for the Alcubierre metric, the total energy
$E=-\frac{v_s^2}{16}\int_{-\infty}^{\infty}{\rm d}x \int_0^{\infty}{\rho{\rm d}\rho}\left(\frac{{\partial} f}{{\partial} \rho}\right)^2$. And indeed, flattening the warp drive by a factor of $\alpha_X$ ($\alpha_X>1$) -- i.e., by replacing $f(x-x_s, \rho) \rightarrow f(\alpha_X(x-x_s), \rho)$, as may be shown through variable change $\alpha_X(x-x_s) \rightarrow x^\prime$ -- leads to the energy reduction by $E\rightarrow E/\alpha_X$. Similarly, elongating the drive (choosing $\alpha_X<1$) increases its energy requirements. Therefore, putting aside the fundamental issues related to negative energy of Alcubierre drives, an optimal implementation of such spacetimes would likely be flattened in shape.

The second implication of Equation~(\ref{eq:ShFun0}) is that the flattening of the warped region may be adjusted with velocity so that the drive preserves the same total energy. Indeed, setting $\alpha_X=1+v^2$ leads to $E\rightarrow E/(1+v^2)$, asymptotically removing the dependency of the total energy on velocity. As discussed earlier in Section~\ref{sec:Intro}, the fact that the total energy of the Alcubierre drive depends on velocity is problematic. This is because the energy and momentum conservation, applicable to asymptotically-flat spacetimes, implies that the Alcubierre drive must be changing its already very large energy (and mass) as it accelerates. Removing or softening the dependence of the total energy on velocity may, in principle, lead to more efficient ways of accelerating the drive to large velocities.

Finally, the third implication of Equation~(\ref{eq:ShFun0}) is that it allows one to construct superluminal solutions which satisfy the quantum inequalities given by Equation~(\ref{eq:EnCond}). Indeed, selecting a sufficiently large $\alpha_X$ allows one to reduce the thickness of the warp in $x$-direction down to nearly-Planck scale size (correspondingly, allowing only for extremely thin physical observers inside the warp bubble). As a result, the local curvature radius in Equation~(\ref{eq:EnCond}) may be arbitrarily small, thus satisfying the quantum inequalities given by Equation~(\ref{eq:EnCond}). At the same time, such superluminal drives still maintain a macroscopic size in $\rho$-direction perpendicular to the direction of motion and, more importantly, do not increase their densities due to contraction. Such solutions, may offer an exciting possibility of superluminal physical solutions. Perhaps, they may help probe the physics of superluminal motion and the problems associated with it, e.g. the violation of causality \cite{Krasnikov1998}, \cite{Everett1997}. However, they more likely probe the limits of applicability of the quantum inequalities~(\ref{eq:EnCond}), which are derived { in semiclassical gravity approximation. Indeed, superluminal motion violates the averaged null energy condition \cite{Visser2000}, and the latter does not have the same dependency on the dimensions of the bubble as the quantum inequalities, e.g. \cite{Graham2007}. As a result, even the extremely flattened version of the Alcubierre drive, as discussed here, does not satisfy the averaged null energy conditions.}

\subsection{Lorentz drive}

The Alcubierre metric is artificial in the sense that it forces the clocks and rulers of observers $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm out}$ to be synchronized. An arguably more natural choice would be to require that the observer $\mathcal{O}_{\rm in}$ should experience the same time dilation and space contraction as a Lorentz observer would experience when moving with velocity $v_s<c$. Since Lorentz transformations are defined for subluminal speeds, in this section, we consider subluminal warp drives of Class I. To construct the spacetime, we choose:
\begin{equation}
\label{eq:LorDrive}
    {\rm d}t_{\rm co}=\gamma({\rm d}t-\frac{v_s {\rm d}x}{c^2})
\end{equation}
\[
{\rm d}x_{\rm co}=\gamma ({\rm d}x-v_s {\rm d}t),
\]
where $\gamma$ is the Lorentz gamma-factor. Using these definitions to construct the warp drive metric with Equation~(\ref{eq:GenTrans}) leads to a diagonal metric:
\begin{equation}
{\rm d}s^2=-c^2 F^2{\rm d} t^2 + F^2{\rm d}x^2+{\rm d}\rho^2 + \rho^2{\rm d}\theta^2,
\end{equation}
where $F^2\equiv 1+2f(1-f)(\gamma -1)$. Moreover, any diagonalizable warp drive spacetimes are described by Equation~(\ref{eq:LorDrive}) (the diagonal form is retained if $\gamma$ is replaced by any constant).

The energy density for this metric is given by:
\begin{equation}
   w = -\frac{1}{8\pi}\frac{(\rho F^\prime_\rho)^\prime_\rho}{\rho F}
\end{equation}

For this metric, independent of the choice of function $f(x-x_s,\rho)$, the region $\mathcal{D}_{\rm warp}$ contains areas of both positive and negative energy density. This result is likely related to the fact that the spacetime is effectively flat beyond a certain radius, rather than asymptotically approaching the flat spacetime, similarly to the Schwarzschild solution. In the case of spherically symmetric solutions, as discussed in Section~\ref{sec:SphGen}, such truncation of the gravitational field also requires the warp region $\mathcal{D}_{\rm warp}$ to contain regions of positive and negative energy.

The Lorentz drive metric may be generalized to porduce a continuous limit to flat spacetime. Indeed, if region $\mathcal{D}_{\rm warp}$ were replaced by Minkowski spacetime, the whole spacetime would be Minkowski space. Therefore, one may introduce a continuum of solutions parametrized by a dimensionless parameter $\lambda\in[0,1]$, by using $F^2\equiv 1+2\lambda f(1-f)(\gamma -1)$. For $\lambda = 0$ the whole spacetime reduces to a flat spacetime, and for $\lambda = 1$ the Lorentz warp drive solution is recovered. The intermediate values of $\lambda$ allow for solutions with smaller energy requirements. To our knowledge, this is the first example of a family of warp drive solutions containing near-Minkowski metrics.

\subsection{An improved Van Den Broeck's drive}

The warp drive solution by \cite{vandenBroeck1999} is intended to reduce the energy requirements of warp drives by significantly expanding the volume of the Alcubierre drive inside radius $r_s$ while significantly decreasing the externally measured size of the craft.
In this section, we realize the idea of \cite{vandenBroeck1999} in a simpler and, importantly, coordinate-independent form. We select the relation between clocks of the internal and the external observers to be the same as in the Alcubierre metric, ${\rm d}t_{\rm co}={\rm d}t$, but choose the internal spacetime to be expanded in $x$-direction: ${\rm d}x_{\rm co}=A({\rm d}x
-v_s{\rm d}t)$, ${\rm d}\rho_{\rm co}={\rm d}\rho$, ${\rm d}\theta_{\rm co}={\rm d}\theta$, where $A>1$. Substituting these relations into Equation~(\ref{eq:GenTrans}), we obtain the resulting spacetime:
\begin{equation}
    {\rm d}s^2=-c^2 {\rm d}t^2 + ({\rm d}x + f(A-1){\rm d}x-fAv_s{\rm d}t)^2 + {\rm d}\rho^2 + \rho^2{\rm d}\theta^2
\end{equation}
This spacetime leads to the energy density distribution of:
\begin{equation}
    w=-\frac{1}{8\pi}\left[\frac{v_s^2}{4}\frac{A^2f^{\prime 2}_{\rho}}{(1+(A-1)f)^2}+(A-1)\frac{(\rho f^\prime_\rho)^\prime_\rho}{(1+(A-1)f)}\right]
\end{equation}
The expression contains a velocity-independent component, which also leads to regions of positive and negative energy density. We see here that while stretching the space inside the warp drive does lead to higher energy density, the metric may be optimized to achieve the highest internal volume per unit energy needed to construct the spacetime.

\subsection{Warp drive metric with modified time}
In this section, we consider spacetimes, in which the clocks in the region $\mathcal{D}_{\rm in}$ run at a different rate compared to the clocks in the region $\mathcal{D}_{\rm out}$. This is done by letting ${\rm d}t_{\rm co}=A^{-1}{\rm d}t$, where $A>1$ corresponds to the clocks going slower in the region $\mathcal{D}_{\rm in}$. Substituting this relation into Equation~(\ref{eq:GenTrans}) and otherwise using the same definitions as in the Alcubierre solution, we obtain:
\begin{equation}
    {\rm d}s^2=-c^2 ((1-f){\rm d} t + A^{-1}f{\rm d} t)^2 + ({\rm d}x-f v_s {\rm d}t)^2 + {\rm d}\rho^2 + \rho^2{\rm d}\theta^2
    \label{eq:AlcV}
\end{equation}
The resulting energy density for the Eulerian observers is:
\begin{equation}
    w=-\frac{1}{8\pi}\frac{ (1-(A^{-1}-1)f)^2}{(1+(A^{-1}-1)f)^4}\cdot\frac{v_s^2}{4}\left(\frac{{\partial} f}{{\partial} \rho}\right)^2
    \label{eq:TChangeDrive}
\end{equation}
This demonstrates that the rate of clocks used in the original Alcubierre solution ($A=1$) in fact leads to the simplest possible expression for the total energy of the warp drive, other parameters being constant. From Equation~(\ref{eq:TChangeDrive}) it follows that slowing down the clocks inside the Alcubierre drive leads to higher amounts of energy density from the point of view of Eulerian observers.

\subsection{Spinning warp drive metric}

Finally, our method can be used to construct relatively complex relations between the inner and outer regions. For example, the following metric corresponds to the two Minkowski regions $\mathcal{D}_{\rm in}$ and $\mathcal{D}_{\rm out}$ spinning with respect to each other. In other words, the observers at rest in region $\mathcal{D}_{\rm in}$ will be rotating with respect to observers in region $\mathcal{D}_{\rm out}$ without experiencing any centrifugal or Coriolis forces typical to rotating systems. Such settings are impossible to realize in the absence of the transition region $\mathcal{D}_{\rm warp}$. The settings may be achieved by relating ${\rm d}\theta_{\rm co}={\rm d}\theta - \omega_s {\rm d}t$, in addition to the definition used for the Alcubierre drive, where $\omega_s$ is a constant parameter defining the angular velocity. Substituting these relations into Equation~(\ref{eq:GenTrans}) leads to the following metric:
\begin{equation}
    {\rm d}s^2=-c^2 {\rm d} t^2 + ({\rm d}x-v_s f {\rm d}t)^2+{\rm d}\rho^2 + (\rho{\rm d}\theta-f\rho \omega_s {\rm d}t)^2,
    \label{eq:AlcAlt}
\end{equation}
The energy density for Eulerian observers then is:
\begin{equation}
    w=-\frac{1}{8\pi}\left[\frac{v_s^2}{4}\left(\frac{{\partial} f}{{\partial} \rho}\right)^2 + \frac{\rho^2\omega_s^2}{4}\left(\left(\frac{{\partial} f}{{\partial} \rho}\right)^2 + \left(\frac{{\partial} f}{{\partial} x}\right)^2\right)\right]
\end{equation}
We see that the energy density is negatively defined independent of the spin or velocity of the inner region. Additionally, the contributions from the velocity in $x$-direction and the velocity of angular rotation have a similar form. The main interest for such spacetimes may be that they offer a stationary, dissipationless way of storing energy or angular momentum.

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

\subsection{What is a warp drive?}

One of the main conclusions of our study is that warp drives are simpler and much less mysterious objects than the broader literature has suggested when citing \cite{Alcubierre1994}. Warp drives are inertially moving shells of positive or negative energy material which enclose a `passenger' region with a flat metric. The main feature distinguishing warp drives from trivial inertially moving low-mass shells is that the large amount of energy contained in the warp shell allows one to modify the state of spacetime inside it. In particular, as shown in Sections~\ref{sec:SphGen} and~\ref{sec:Axi}, the time in the inner region may go faster or slower than it would go without the shell. Similarly, the spatial volume may be stretched, compressed, or even be rotating compared to its normal state. Further, more complex, modifications are likewise possible.

Warp drives can move superluminally only in the same sense as any ordinary inertial mass, test mass, or any other object. Namely, there is no known way of accelerating regular material beyond the speed of light. However, one may postulate a test particle which moves faster than light in relativity, in which case it may continue moving inertially. In the same way, as warp drives are shells of material, there is no known way of accelerating a warp drive beyond the speed of light. However, one may also postulate the warp drive shell to be in superluminal motion, just like the hypothetical test particles, and the shell-like object will continue moving in the same fashion. In this sense, superluminal warp drives are at least as hypothetically possible as any other superluminal objects.

An interesting feature of warp drives, from the theoretical point of view, is that the modifications of the spacetime in the internal region may be sufficiently strong so as to allow superluminal objects to move subluminally or vice versa. The different possibilities are embodied in four different classes of warp drives, presented in Section~\ref{sec:GenDrives}. In particular, there are solutions possible (Class IV) wherein a shell of subluminal material may contain regions where no subluminal material can remain at rest no matter what velocity it has. In this aspect, the shells of these hypothetical Class IV warp drives would share properties with black holes. Further, such a spacetime could effectively stop and contain inside a hypothetical superluminal test-mass object, or make it move slower than the speed of light.

Similarly, one may hypothesize a spherical shell moving faster than the speed of light, but which contains an inner region where no (hypothetical) superluminal objects can be at rest relative to the shell (Class III). Or, in another interpretation, subluminal normal objects inside such shell could be at rest with respect to them, despite their motion being superluminal. This class of warp drives, which also includes the Alcubierre drive, however, remains entirely hypothetical, just in the same way as no other object can be set to move superluminally. { Using the methods in Section~\ref{sec:GenDrives}, one can examine whether any other particular spacetime belongs to this class of warp drives.}

\subsection{Constructing warp drives}

Warp drives, being inertially moving shells of normal or exotic material, do not have any natural way of changing their velocities. They are just like any other types of inertially moving objects. Similarly, just like for any other massive objects, achieving a certain velocity for a warp drive requires an externally applied force or, more practically, some form of propulsion. Propulsion may be realized, for example, by an interaction with a bosonic field, or regular gaseous or plasma material.

Whatever is the acceleration mechanism, it must obey the conservation of 4-momentum. This is because all warp drive spacetimes are asymptotically-flat. An unfortunate error, introduced in \cite{Alcubierre1994}, was to postulate the velocity in Equation~(\ref{eq:Alc}) to be time-variable. An Alcubierre spacetime with time-variable velocity also changes its energy and momentum with time, and, this way, such a construction violates energy conservation. More technically, the metric given by Equation~(\ref{eq:Alc}) does not satisfy the continuity equations, unless additional dynamical fields are implicitly introduced to compensate for that. In view of this, no metric which describes an accelerating warp drive solution has so far been presented in the literature.

A more subtle point is that the Alcubierre and Natário drives, as well as the spacetimes constructed in Section~\ref{sec:Axi}, represent classes of different objects parametrized by velocity $v$ rather than the same object changing its velocity $v$. In particular, different warp drive solutions with different values of $v$ have different mass, different energy content, and often different shapes in their reference frame of rest. Any realistic object should at least conserve its ADM-mass in the subluminal regime and its analogue in the superluminal regime. A natural way of constructing such spacetimes is by defining them explicitly in their frame of rest, as discussed in Section~\ref{sec:GenDrives} and implemented for spherically symmetric drives in Section~\ref{sec:SphGen}. Implementing metrics for such accelerating objects and more general axisymmetric objects which preserve their shape and mass in the comoving frame as they change their velocity remains a subject for future studies. As a simple compromise, one may also adjust the shape of the warp drive with velocity so as to conserve the mass of the drive, as suggested in Section~\ref{sec:AxiClasses}. Finally, among all classes of subluminal warp drive solutions, the particularly interesting ones in the practical sense are those classes which contain a continuous set of solutions ranging from trivial to highly curved.

At least in the subluminal case, warp drive spacetimes may be constructed by using purely positive energy density, as presented in Section~\ref{sec:SphGen} for the spherically symmetric case. They can likewise be constructed using purely negative energy density, as is the case for the Alcubierre solution, or constructed using both positive and negative energy density. In Section~\ref{sec:SphGen} we showed, for the first time, that the only type of modification to the internal spacetime that is achievable with purely positive energy for spherically symmetric warp drives is slowing down the rate of time inside the craft.

In Section~\ref{sec:Axi} we demonstrated that, by using both positive and negative energy density, one may achieve a variety of modifications for the spacetime inside more general axisymmetric subluminal warp drives. The range of all the possible modifications achievable with purely positive energy in the general axisymmetric case, and whether the class of axisymmetric spacetimes in Section~\ref{sec:Axi} may lead to purely positive energy metrics, remain important open questions in the field.  Similarly, in Section~\ref{sec:GenDrives} we have provided a new argument why superluminal warp drive solutions may always violate weak energy conditions resulting in their negative energy density requirements. While this is an established fact \cite{Olum1998,Visser2000}, a strict independent proof of this based on our argument is another important avenue for future studies. Our conclusions do not support the recent claim in \cite{Lentz2020} of superluminal purely positive energy warp drive solutions, which merits further investigation.

\subsection{Optimal members in the class}

An interesting question remaining is: assuming that a practical realization of a warp drive spacetime is possible, what would the optimized versions look like? As we showed in Section~\ref{sec:AlcDeformed}, a more optimal implementation of the Alcubierre warp drives would be flattened in shape, since such shapes are more efficient in terms of energy requirements. In particular, flattening the shape by a factor of $10$ would lead to proportionally smaller energy requirements. This conclusion likely holds for the, more physical, purely positive energy subluminal warp drives as well. Curiously, as we discussed in Section~\ref{sec:AxiClasses}, extreme flattening of Alcubierre drives may allow for superluminal solutions which satisfy quantum inequalities, without reaching extreme energy densities.

For the Alcubierre solution, one may similarly optimize the energy requirement by finding the most suitable shape function $\bar{f}(r_s)$. The originally proposed function given by Equation~(\ref{eq:ShFun0}), is not optimized and was originally chosen solely for demonstration. By applying the variational method to the expression for the total energy of the Alcubierre drive, we find that the shape function optimizing the energy is given by $\bar{f}(r_s)=\min(\frac{r_0}{r_s},1)$, where $r_0$ is a free parameter determining the inner size of the region $\mathcal{D}_{\rm warp}$. Using this slower-decreasing shape reduces the energy requirement for a similarly sized Alcubierre drive by about a factor of three. The physical reason for it, as we discuss in Section~\ref{sec:SphGen}, is perhaps related to the fact that truncating the gravitational fields of a warp drive, as done in the Alcubierre solution, may increase the (absolute) amount of necessary negative energy compared to the more slowly falling off solutions. We provide the details of this derivation in \ref{sec:AppA3}, and also remark that optimizing the shape or matter distribution in region $\mathcal{D}_{\rm warp}$ can be equally well performed for all the other warp drive solutions.

Given the wide range of possible states of spacetime achievable inside a warp drive, it is also possible to imagine more complex and instrumental optimizations. For example, one may speculate that it is possible, at least in principle, to form a region inside a subluminal warp drive which is similar to ergospheres of spinning black holes. In this case, such a region would be used as an efficient energy storage. The energy could then potentially be extracted through a Penrose process applied to the propellant of the craft, when passing through the ergoregion. Similarly to the Penrose process for spinning black holes, the extracted energy would likely be coming from the rotation of some regions of the spacetime.

Finally, since all warp drive objects require propulsion in order to accelerate, any practical implementation of such objects would have to be asymmetric in shape, since the back part would have to accommodate a propellant exhaust system. One may further hypothesise on setups, wherein black hole-like regions of the spacetime may be used to produce accretion power. Accretion of material onto black holes is known to be a few tens of times more efficient at extracting rest-mass energy in the form of electromagnetic radiation from the material than nuclear burning \cite{Frank2002}. Such a process could potentially provide both a source of energy and a source of propulsion.

\subsection{Towards physical warp drives}

In Section~\ref{sec:SphGen}, we demonstrated that it is possible to construct non-trivial warp drive solutions with purely positive energy. In other words, at least in principle, one can construct objects of progressively larger masses and with progressively more salient modifications to the internal spacetime. While the mass requirements needed for such modifications are still enormous at present, our work suggests a method of constructing such objects based on fully understood laws of physics.

The most promising way of practically probing such spacetimes is through laboratory experiments -- most importantly -- through analogue gravity experiments, e.g. \cite{Barcelo2005}. Another important avenue of exploring such spacetimes, especially the accelerating solutions, is through numerical relativity. Such experiments may bring a better understanding of purely positive energy drives, and negative energy solutions, as well as the possibility of accelerating objects superluminally.

Since the introduction of \cite{Alcubierre1994}, much theoretical effort has been put into unveiling the unphysical nature of the Alcubierre solution. Our work shows that there is a variety of warp solutions, each with properties often much more physical and interesting than the originally proposed spacetime. Through this, we suggest there is a need for broader theoretical and experimental investigation to uncover the full diversity and properties of physical warp drives.

\section*{Acknowledgements}
We would like to thank Parsa Ghorbani, Lorenzo Pieri, Adam Lewis, Philip Chang, Hrant Gharibyan and Stefano Liberati for their insightful comments and helpful discussions at different stages of this work.

\newpage

redpajama_set_name: RedPajamaArXiv


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