text:
Mitch McConnell Berates Obamacare
By: David Nelms
The Senate Republican leader from Kentucky, Mitch McConnell wrote an opinion piece for CNN.com where he lambasted President Obama's Obamacare law. The president pledged that Americans could keep their current coverage plans and that healthcare coverage would be stronger and more stable than it has ever been. McConnell believes that none of these pledges have been kept and he wrote that "As the horror stories continue to roll in, however, the emptiness of those promises and the human toll of this deeply misguided experiment have become increasingly difficult for Democrats to ignore."
The Senator claims that "millions of Americans" are speaking out "about their frustrations and heartbreak," but that no-one on the Democratic Party is listening or taking them seriously. He also says that promises of monthly premiums not increasing have been ignored by the current administration. He quotes this deception as being "the lie of the year" and that nothing is even "comparable to the scope of the deception involved in selling this law."
McConnell feels that the credibility of the Democratic Party is flailing badly because instead of having an open debate about the Obamacare policies, they are downplaying "statistics that cast it in a negative light." He is quoted as saying that "As for the President, he seems more interested in promoting his NCAA bracket and talking up Obamacare's elusive virtues on satirical talk shows like "Between the Ferns."
McConnell goes on to quote management consulting firm McKinsey and Co. who found that "only a fraction of the biggest local hospitals in a given coverage area will accept Obamacare patients." He also quotes an Associated Press poll where "only four of the 19 cancer centers it surveyed would give Obamacare patients access to its cancer care through the new Obamacare exchanges in their states." He claims that these statistics do not show any progress and that the law needs to be repealed for something more "patient-driven."
McConnell, however, does not realize that he is doing exactly what the Democrats are doing; he is just doing it in reverse. By talking only about the negative aspects of Obamacare, he is not making himself apart of the solution. The fact of the matter is that the United States ranks as number one in the world in healthcare costs, and twenty-fifth in the world in healthcare outcomes. That needs to change if the United States is going to remain the largest global economy in the world. And President Obama did introduce competitive bidding for durable medical equipment and has seen medical errors reduced since his healthcare law has been introduced. By not acknowledging some of the success of Obamacare, McConnell is just adding more heat to the fire that is the political landscape of the moment. Instead of actively working to fix negative aspects of the law, he would rather have it completely erased from memory. And that is not something that is likely to happen anytime soon
David is a political writer for issuehawk.com and latest.com who has been published in over ten different publications. He is also a business writer for ifreepress.com where he writes about business and financial news. A proponent for political justice and honesty, David hopes his writing is a positive contribution to the political landscape.


text:
Q: magento 2.3.1 inventory:reservation missing commands I have inventory_reservation inconsistencies that I need to clean up on M2 2.3.1 so following this wiki guide https://github.com/magento/inventory/wiki/CLI-Reference but when running it I get error
'there are no commands defined in the inventory:reservation namespace".

A: I was having trouble with that too. So, after some research, here are my findings:
There's no CLI module installed, so you need to install it or upgrade MSI to latest version. For Magento 2.3.1 in particular, use: composer require "magento/inventory-composer-metapackage": "1.1.3 as 1.1.1"
Hope this will help.

A: Make sure the Magento_InventoryReservationCli module is enabled:
bin/magento module:enable Magento_InventoryReservationCli

redpajama_set_name: RedPajamaStackExchange


text:
Youth Quake 2020
The Theme: BUILD MY LIFE
Jesus talks about building that foundation. He tells us to build our lives on the one thing that will give us the strength to survive the storms of life and thrive on the other side of them. He tells us that the key to building our lives is two-fold: Listening to His teaching and following it. At YQ20, join us and 2000 friends as we learn more about what it means to Build My Life on the only foundation that will last a lifetime.
24 "Everyone then who hears these words of mine and does them will be like a wise man who built his house on the rock. 25 And the rain fell, and the floods came, and the winds blew and beat on that house, but it did not fall, because it had been founded on the rock. 26 And everyone who hears these words of mine and does not do them will be like a foolish man who built his house on the sand. 27 And the rain fell, and the floods came, and the winds blew and beat against that house, and it fell, and great was the fall of it."
Emphasis on V24: "Anyone who listens to my teaching and follows it…" Our lives are built through the choices, experiences, and uncontrollable circumstances that we encounter. Each situation gives us a chance to make a decision, adding a piece to the foundation that we are building our life on. We don't get to choose when the storms in our life hit us and when they hit, but the foundation that we have built will dictate how we weather that storm.
Get Exclusive YQ20 News
Michael Chinchilla | Main Speaker
Michael is married to his beautiful wife Alicia and is on staff at Village Church, a fast-growing church on the west coast of Canada. He is currently pursuing a master's degree at Regent College. Michael is passionate about taking an active role in God's redemptive story because he has seen the grace of Jesus to transform lives.
Ryan Guerra
After spending almost a decade touring the world in a rock and roll band, Ryan hung up his Les Paul to serve as a Pastor at First Baptist Church in Lloydminster. He has a passion for engaging the next generation with biblical truth so that they will rise up to be the leaders that the church needs. When he is not speaking, playing practical jokes, or hanging out with his wife Talasi and daughter Avra, you can find him playing strategy board games with anyone and everyone who will indulge his obsession.
Jeremy Macdonald
Moncton, NB
Jeremy has served as a Youth Pastor in a large context reaching hundreds of students, a small context reaching dozens, and most recently at a multisite church running six separate youth groups. Now as part of the Canadian Youth Worker Community staff team, Jeremy oversees the operations of various projects that equip and encourage thousands of youth workers each year. He is married with three children and lives in Moncton, NB.
Cody Matchett
Cody Matchett serves the Church in the areas of teaching, spiritual formation, and discipleship. He is the former College Chaplain at Briercrest College, an apprenticing church planter with the Evangelical Covenant Church, and a partner in the Gospel with varying organizations across Canada. He loves books, coffee, music, philosophy, cultures, art, and hiking. He is married to Brianna, and they live in Calgary Alberta.
Chris has been the lead pastor at Calvary Baptist for ten years. He has recently resigned to pursue church planting in Vancouver. With a Masters in Christian Studies and a BA in Biblical studies, Chris' love for teaching the Bible, and bringing it to life in context of our everyday lives, can be found in his Sunday messages and extensive writing. He is an avid reader and author of "Radical Hope: Resurrection Hope in a Hurting World. He is also a contributor to the UK based Youth Alpha video teaching series and sometimes co-hosts a weekly podcast called Hidden City, which explores the hidden ideas that influence our lives today. Chris loves to spend time with his wife Diandra and two children, Kaeden and Mila.
1K Phew
1K Phew is among a new breed of Atlanta artists taking the trap sensibility and elevating it to great effect. His smooth delivery and evocative vocal tone pair well with the bouncy, energetic production that defines his sound. At just 23 years old, Phew's rise from local rapper with swagger to compelling artist equipped with a message, has been refreshing to watch. He's fun and engaging, no doubt, but he's so much more. As evidenced by his mixtapes Sunday Night (2015) and Life (2016), Phew's got plenty of swag and wisdom to impart, and people are beginning to connect with his candid approach. There's a sense of transparency that comes through in his music, a kind of refined rawness that can only be gained through experience and honing your craft. To let him tell it: "I'm not worried about being politically correct. I just want to give people the real. That's where the 1K comes from, always keeping it 1,000 no matter what."
Tehillah Worship
Tehillah Worship is a Canadian music collective based out of First Assembly Church in Calgary, Alberta. We aspire to equip and encourage the Canadian church to move God's heart with new songs and music. We believe the beauty and light of Jesus can change a nation and that music and the arts are a massive medium that God will use to spark renewal and revitalization in our land. Our third album "All Your Colours" was released in August of 2019.
The Young Escape
Born and raised in Southern California, THE YOUNG ESCAPE (Capitol) delivers a fresh shot of indie-pop with an energetic live show to match. Comprised of Ryan, McKenna, Kyrsten, and Luke Johns, the band's debut singles "good life" and "neverfade" are an alchemy of modern pop sheen and sophisticated edge, carried to otherworldly heights by McKenna's vocal. TYE speaks to the themes of insecurity, anxiety, and pressure facing every kid in the modern age with the hope and escape that's found in the kinetic spirit of youth.
Joshua Leventhal
Joshua is a bit of an anomaly, both as a worship leader and a songwriter. Meeting Jesus in power very, very young, he did not fall in love with music until his teens. Until that time he did not think of himself as an artist, let alone a musician. For that reason, Joshua's simultaneous satisfaction in and longings for his Redeemer are at the forefront of these songs—they forego what is expected of a modern worship chorus, and instead tread to deeper waters. In a diverse sonic landscape of everything from Black Gospel to Punk anthem to electronically influenced Indie Rock, he weaves images of bride and feast, sojourners and shores, hope in pain, all across the canvas of the Gospel. These are unapologetically Cross-centered, hope-awaiting works. And while their lyricism may not draw from the same wells as is common in contemporary worship, they are still fully intended for the Church to sing together—that we might better know Him, love Him, and stir in affection towards Him. To explore the marriage of meaningful art with Spirit-filled worship—to the delight of the King, and His satisfaction of us. This is art as catalyst for adoration. Leventhal is American-born but has spent most of his life in Canada. He is currently based in Vancouver BC and is the Senior Worship Leader at Village Church.
OJ The DJ
Hailing from Brampton, Ontario and joining us for our second year, OJ brings an electric and contagious vibe to our conference. Cue the dance party!
Refined Undignified
Refined/Undignified is a dance ministry that seeks to reveal true life through dance. Auditioning students from Briercrest College and Briercrest Christian Academy each year, RU creates an opportunity for students to be equipped to become godly leaders, to see these students excel in dance and lifestyle for the glory of God and and to allow these students to use the tools they have been given to engage a hurting world with gospel of Jesus Christ. Their shows are high energy and lots of fun, all while tackling relevant issue that need to be discussed. You won't want to miss them!
Briercrest Worship
Briercrest Worship is made up students involved in the Worship Arts program at Briercrest College. Each one of these students regular serves their community by leading worship at Briercrest chapels and worship nights. They also serve beyond the Briercrest community, touring and leading worship at churches and youth events. They are passionate about seeing people encounter the living God through authentic worship.
Youth Quake is open to anyone AGES 13-18!
All registration fees include meals, accommodation and all the events for the weekend. Payments must accompany registration forms in order to be processed.
Come as a group or as an individual.
Group Deal
Register 10 and get 1 youth worker for free!
Early-Bird: $175
Must register and pay before midnight Dec 1
Regular Rate: $189
Must register and pay before midnight Feb 8
Feb 9 - 14
All Inclusive!
Lodging on campus
YQ PHONE: 306-756-3251
GENERAL SCHOOL ENQUIRIES: 306.756.3200 (main office)
Briercrest College & Seminary
C/O Youth Quake
510 College Dr.Caronport, SK S0H 0S0
Briercrest College and Seminary is located in the town of Caronport, Saskatchewan, just 15 minutes west of the city of Moose Jaw (pop. 34,000) and an hour west of Regina (pop. 200,000). The campus is in a rural setting away from the noise and hectic pace of the city, and yet it offers an extensively developed 160-acre campus. Facilities include a 25,000 sq. ft. library, an auditorium that seats 2,000 people, student dormitories, a recently renovated dining hall, and a brand-new hockey arena.
For three days experience what life as a student at Briercrest College is like

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Gridview Dynamic linkButton event not fired I'm creating a GridView with dynamic controls(including itemtemplates).
Here is the html code.
<asp:GridView ID="gvItems" runat="server"
                        AutoGenerateColumns="False" CellPadding="4"
                        GridLines="None"
                        Width="95%" EmptyDataText="Records Not Found!!!!"
                        onselectedindexchanged="gvItems_SelectedIndexChanged"
                        onrowdatabound="gvItems_RowDataBound" onrowcommand="gvItems_RowCommand"
                        onselectedindexchanging="gvItems_SelectedIndexChanging">
                        <RowStyle CssClass="GVRowStyle" />
                        <Columns>

                        </Columns>
                        <PagerStyle CssClass="gridPager" />
                        <SelectedRowStyle BackColor="#DCCDDA" Font-Bold="true" ForeColor="#510030" />
                        <HeaderStyle CssClass="Gheader" />
                        <AlternatingRowStyle CssClass="AlternatingRowStyle" />
                        <EditRowStyle BackColor="#7C6F57" />
                        <EmptyDataRowStyle  CssClass="EmptyRowStyle" />
                    </asp:GridView>

Here is the class for Adding ItemTemplate...................................
public class GridViewTemplate : ITemplate
{
ListItemType _templateType;
string _columnName;

public GridViewTemplate(ListItemType type, string colname)
 {
    _templateType = type;
    _columnName = colname;
 }

void ITemplate.InstantiateIn(System.Web.UI.Control container)
{
    switch (_templateType)
    {
        case ListItemType.Header:
            Label lbl = new Label();
            lbl.Text = _columnName;
            container.Controls.Add(lbl);
            break;

        case ListItemType.Item:
            LinkButton Lb1 = new LinkButton();
            Lb1.CommandName = "Select";
            Lb1.DataBinding += new EventHandler(tb1_DataBinding);
            container.Controls.Add(Lb1);
            break;

        case ListItemType.EditItem:

            break;

        case ListItemType.Footer:
            CheckBox chkColumn = new CheckBox();
            chkColumn.ID = "Chk" + _columnName;
            container.Controls.Add(chkColumn);
            break;
    }
 }

 void tb1_DataBinding(object sender, EventArgs e)
 {
    LinkButton LinkData = (LinkButton)sender;
    GridViewRow container = (GridViewRow)LinkData.NamingContainer;
    object dataValue = DataBinder.Eval(container.DataItem, _columnName);
    if (dataValue != DBNull.Value)
    {
        LinkData.Text = dataValue.ToString();
        LinkData.ForeColor = System.Drawing.Color.Red;
        LinkData.CommandName = "Select";
    }
}

}
Here is the page code.........................................
for (int i = 0; i < dtGrid.Columns.Count; i++)
        {
            string columnName = dtGrid.Columns[i].ColumnName;
            BoundField bField = new BoundField();
            TemplateField tField = new TemplateField();
            if (i == 0)
            {
                tField.HeaderTemplate = new GridViewTemplate(ListItemType.Header,    columnName);
                tField.ItemTemplate = new GridViewTemplate(ListItemType.Item, columnName);
                tField.HeaderStyle.HorizontalAlign = HorizontalAlign.Left;
                tField.ItemStyle.ForeColor = System.Drawing.Color.Red;
                gvItems.Columns.Add(tField);

            }
            else
            {

                bField.DataField = columnName;
                bField.HeaderText = columnName;
                bField.HeaderStyle.HorizontalAlign = HorizontalAlign.Left;
                gvItems.Columns.Add(bField);
            }

Now Problem is when i click on linkbutton it disappear and no event other than rowdatabound is raised.

A: Since you're creating these columns dynamically, you'll need to recreate the controls at every postback. As for the event handler not firing, make sure you create the columns early enough in the page lifecycle. I would suggest creating the dynamic columns OnInit, and see if that fixes the problem.

A: Are you re-binding the data on postbacks?
Please make sure that your data-binding logic is surrounded with if(!IsPostback).
The re-binding of the grid means re-creation of your button, and therefore no events are fired.


If you want a truly unique look for your hotel's reception, bedrooms and dining rooms you need to choose bespoke furniture pieces handcrafted with love. At Jacobs Furniture we have decades of furniture making experience that we can draw on. This enables us to design, illustrate and create exceptional bespoke furniture.
When it comes to upholstered furniture the fabric you pick matters. We give you free reign, you can pick from our huge selection or we will search further afield to find exactly what you're looking for. Nothing is too much trouble as we want you to be 100% happy with the beautiful furniture we craft for you.
Having stylish seating in your reception area is a must as your guests will want somewhere to sit, even if it's just for five minutes while they wait for a taxi or to check in during busy periods. If you like traditional furniture, this side chair with matching foot stool could be ideal. We love it in this warm and welcoming tartan fabric but you could choose a fabric that matches your décor instead. (It would also look great next to an open fire or in a bedroom, the solid beech wood tapered legs are stunning in their own right).
If your boutique is all about making a statement and being different, these bespoke pieces could be just what you're looking for. They would look great in a contemporary styled bedroom or in your reception area and are definitely Instagrammable.
When it comes to your dining room, a well-made, one-off solid oak table can be a great statement piece. We can polish it to any required colour and it can be complemented by Oak Carver chairs, upholstered in a fabric to match your feature wall. We love both the table and chairs polished natural with a white antiqued finish.
Take a look at our full range to get a taste of what we can offer you as a hotelier. You'll be left speechless by the options available to you. We have some wonderful one-off high-end pieces that you can reserve or we can create something bespoke for you. Get in touch with us today.


23/04/2011 · Steps. Part 1. Training for Bigger Calves . 1. Do seated calf raises. This movement targets the soleus calf muscles. Sit on the machine with your toes on the lower portion of the platform and your heels extending off. Your lower thighs go under the lever pad, and your hands go on top of the lever pad to keep it in place. Lift the lever by pushing your heels up, then slowly lower your heels by... Lift both your left leg and your left arm up to the ceiling at about 90 degrees, keeping your hips stacked without rocking backwards and with your elbow pressed into your side. Hold at the top for 2 seconds before returning to start.
The calves are the fleshy areas in the back of the lower legs below the knees. The upper part of this muscle group is known as the gastrocnemius and the lower part is the soleus. Both sections are responsible for plantar flexion, which causes your toes to point downward. The gastroc is also involved with knee flexion.


text:
\part{title}---- Definición de las opciones ------------------------------------- %

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\title{Covariant phase space for gravity with boundaries: metric vs tetrad formulations}

\author[c,d]{J. Fernando Barbero G.}
\author[a,c]{Juan Margalef-Bentabol}
\author[c,d]{Valle Varo}
\author[b,c]{Eduardo J.S.~Villaseñor}

\emailAdd{fbarbero@iem.cfmac.csic.es}
\emailAdd{juanmargalef@psu.edu}
\emailAdd{valle@cvb.es}
\emailAdd{ejsanche@math.uc3m.es}

\affiliation[a]{Institute for Gravitation and the Cosmos and Physics Department. Penn State
	University. PA 16802, USA.
	\vspace*{1ex} \mbox{}}
\affiliation[b]{Departamento de Matemáticas, Universidad Carlos III de Madrid. Avda. de la
	Universidad 30, 28911 Leganés, Spain.
	\vspace*{1ex} \mbox{}}
\affiliation[c]{Grupo de Teorías de Campos y Física Estadística. Instituto Gregorio Millán (UC3M).
	Unidad Asociada al Instituto de Estructura de la Materia, CSIC, Madrid, Spain.}
\affiliation[d]{Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006, Madrid, Spain}

\abstract{We use covariant phase space methods to study the metric and tetrad formulations of General Relativity in a manifold with boundary and compare the results obtained in both approaches. Proving their equivalence has been a long-lasting problem that we solve here by using the cohomological approach provided by the relative bicomplex framework. This allows us to prove the equivalence without having to make any \emph{ad hoc} choices. We also compute several relevant charges in both schemes and show that, as expected, they are equivalent.}

\arxivnumber{}

\begin{document}

\maketitle

\flushbottom

\section{Introduction}

Space-time boundaries play a prominent role in classical and quantum General Relativity (GR). Their applications range from  black hole thermodynamics \cite{Wald,WaldLR} to the study of radiative modes at \textit{scri} \cite{P1,P2,AS,ACL}. They are also fundamental to describe isolated and dynamical horizons \cite{AK,AshtekarFairhurstKrishnan2000,AshtekarEngleSloan2008,EKPP10,EKPP11} or in the definition of physical charges \cite{IyerWald,CorichiReyes2015,MargalefVillasenor2020}. Boundary terms also control the dynamical properties of some midisuperspace models with ``conical singularities'' at spatial infinity such as Einstein-Rosen waves coupled to massless scalar fields \cite{AM,ALarge,BGV,BV}.\vspace*{2ex}

Classical GR defined in a space-time without boundary may be alternatively described by the first order Hilbert-Palatini (or Holst) action in terms of tetrads or the Einstein-Hilbert Lagrangian in metric variables. In particular, the corresponding Hamiltonian descriptions are equivalent. Moreover, by employing standard covariant phase space (CPS) methods, it is possible to see that their respective symplectic potentials differ by an exact form. Thus, when this form is integrated over a Cauchy slice $\Sigma$, the result vanishes as a consequence of Stokes' theorem. This shows that the presymplectic forms corresponding to the metric and the tetrad formalisms are equivalent in the space of solutions.

The situation changes substantially when boundaries (possible at conformal infinity) are present. Some of the concerns regarding the equivalence of the tetrad and metric formulations may be attributed to the choice of bulk and boundary Lagrangians. In fact, many discrepancies found and discussed in the literature \cite{JacobsonMohd2015,OliveriSpeziale2020a,PaoliSpeziale2018,FGP1,FGP2,FGP3} are to be expected since the boundary dynamics is not always carefully studied and, consequently, the solution spaces are not fully specified. To avoid these problems, it is crucial to choose the variational principles such that the equivalence of the dynamics they describe, both in the bulk and at the boundaries, is guaranteed. The choice of the bulk and boundary Langrangians will determine the presymplectic structure in the solution space as well as the Noether charges.\vspace*{2ex}

The purpose of this paper is to prove that the metric and tetrad formulations are equivalent in the CPS with Dirichlet and Neumann boundary conditions (the same methods can be applied to other boundary conditions). We do this by relying on the relative bicomplex framework, a formalism developed in \cite{MargalefVillasenor2020} which is cohomological in nature so no \emph{ad hoc} choices are required. This formalism will allow us to establish the equality of the metric and tetrad symplectic potentials in the \emph{relative cohomology} (equality up to a \emph{relative} exact form as explained in the appendix). Consequently, the respective presymplectic forms in the solution space are equivalent and, as expected, there is a precise correspondence between the Noether charges in both formulations. As a direct result of our discussion, we will consider the asymptotically flat case and rederive the well-known formula for the ADM-energy.\vspace*{2ex}

The paper is structured as follows. In the next section, we present some of the geometric concepts relevant for the implementation of the CPS-algorithm developed in \cite{MargalefVillasenor2020}.  In \ref{section: metric}, we apply this algorithm to the metric formulation of GR. Section \ref{section: tetrads} presents an analogous study for the tetrad formalism. In Section \ref{section: metricvstetrad} both formulations are compared and shown to be equivalent. We present our conclusions in the last section of the paper. We have also included in appendix \ref{section: appendix} a short summary of the relative bicomplex framework together with some computational details.

\section{The geometric arena}

\subsection[\texorpdfstring{$M$}{M} as a space-time]{\texorpdfstring{$\boldsymbol{M}$}{M} as a space-time}\label{subsection: M space-time}
Let $M$ be a connected and oriented  $n$-manifold admitting a foliation by Cauchy hypersurfaces. Without loss of generality, $M=I\times \Sigma$ for some interval $I=[t_i,t_f]$, ($t_i<t_f$) and some $(n-1)$-manifold $\Sigma$ with boundary $\partial\Sigma$ (possibly empty). Denoting $\Sigma_i=\{t_i\}\times\Sigma$ and $\Sigma_{\!f}=\{t_f\}\times\Sigma$, we split $\partial M$ into three distinguished parts
\[\partial M=\Sigma_i\cup \peqsub{\partial}{L} M\cup \Sigma_{\!f}\]
where $\Sigma_i,\Sigma_{\!f}$ are the lids and $\peqsub{\partial}{L} M:=I\times \partial \Sigma$ the ``lateral boundary''. $M=I \times \Sigma$ is a manifold with corners $\partial\Sigma_i\cup\partial\Sigma_{\!f}$, which, as a set are $\partial(\peqsub{\partial}{L} M)$. The following diagram summarizes relevant notational information about embeddings and the induced geometric objects.

\begin{equation}\label{eq: diagrama}
\begin{tikzcd}[row sep = scriptsize]
\Big(\Sigma,\gamma,D,\{a,b,\ldots\}\Big)\arrow[rr,"{(\imath,n^\alpha)}",hook]  & &
\Big(M,g,\nabla,\{\alpha,\beta,\ldots\}\Big)\\
\Big(\partial\Sigma,\overline{\gamma},\overline{D},\{\overline{a},\overline{b},\ldots\}\Big)
\arrow[u, "{(\overline{\jmath},\mu^a)}",hook]
\arrow[rr,"{(\overline{\imath},\overline{m}^{\overline{\alpha}})}"', hook] & &
\Big(\partial M,\overline{g},\overline{\nabla},\{\overline{\alpha},\overline{\beta},\ldots\}\Big)
\arrow[u,"{(\jmath,\nu^\alpha)}"', hook]
\end{tikzcd}
\end{equation}
The entries in each $4$-tuple in (\ref{eq: diagrama}) are: the manifold, the (non-degenerate) metric or pulled-back metric, its associated Levi-Civita connection, and the abstract indices used to describe tensors in the manifold. The arrow labels specify the notation used for embeddings and normal unitary vector fields. Horizontal arrows are associated with future pointing normal unit vector fields and vertical arrows are used for outward pointing normal unit vector fields at the boundary. The notation just introduced is not consistent for the bottom lid $\Sigma_i$, as it may be thought of as a spacelike hypersurface embedded by $\imath_i$ or considered as part of $\partial M$\!. $\Sigma_i$ often appears as the boundary of $M$\!, so we choose the outwards (past pointing) convention in this case. An overline is often used to denote objects that live exclusively at the boundary, such as $\overline{g}$, which in index notation reads $\overline{g}_{\overline{\alpha}\overline{\beta}}$. Notice that the embeddings $\jmath$ and $\overline{\jmath}$ are fixed since the boundaries are also fixed.
However, $\imath$ and $\overline{\imath}$, which embed $(\Sigma,\partial\Sigma)$ into $(M,\partial M)$, can be chosen among the Cauchy embeddings satisfying $\imath(\partial\Sigma)\subset\peqsub{\partial}{L} M$\!. \vspace*{2ex}

As $(M,g)$ is oriented, we have the metric volume form $\mathrm{vol}_g$ that assigns the value $1$ to every positive orthonormal basis. We orient $\Sigma$ and $\peqsub{\partial}{L} M$ with $\mathrm{vol}_{\gamma}$ and $\mathrm{vol}_{\overline{g}}$, respectively given by
\begin{align}\label{eq: orientation sigma}
\imath^*(\iota_{\vec{U}}\mathrm{vol}_g)=-n_\alpha U^\alpha\mathrm{vol}_\gamma\qquad\qquad\qquad \jmath^*(\iota_{\vec{U}}\mathrm{vol}_g)=\nu_\alpha U^\alpha\mathrm{vol}_{\overline{g}}
\end{align}
for every $\vec{U}\in\mathfrak{X}(M)$. These orientations are the ones for which Stokes' theorem holds in its usual form. Finally, $\partial\Sigma$ can be oriented as the boundary of $\Sigma$. Thus $\mathrm{vol}_{\overline{\gamma}}$ is given by
\begin{align}\label{eq: reverse orientation}
\overline{\jmath}^*(\iota_{\vec{V}}\mathrm{vol}_\gamma)=\mu_a V^a\mathrm{vol}_{\overline{\gamma}}\qquad\quad\longrightarrow\quad\qquad\overline{\imath}^*(\iota_{\vec{W}}\mathrm{vol}_{\overline{g}})=+\overline{m}_{\overline{\alpha}} W^{\overline{\alpha}}\mathrm{vol}_{\overline{\gamma}}
\end{align}
Notice that if we use the Stokes' theorem from $\peqsub{\partial}{L} M$ to $\partial(\peqsub{\partial}{L} M)=\partial\Sigma_i\cup\partial\Sigma_{\!f}$, a minus sign appears in the integral over $\partial\Sigma_i$.

\subsection[From \texorpdfstring{$M$}{M}-vector fields to \texorpdfstring{$\mathcal{F}$}{F}-vector fields]{From \texorpdfstring{$\boldsymbol{M}$}{M}-vector fields to \texorpdfstring{$\boldsymbol{\mathcal{F}}$}{F}-vector fields}
Let $\mathcal{F}$ be  a space of tensor fields on $M$ (sections of a certain bundle $E\stackrel{\pi}{\to}M$) and think of it as an infinite dimensional differential manifold where we assume that the usual differential objects---like tensor fields, the exterior derivative $\dd$, the interior derivative $\raisebox{-.04ex}{\includegraphics[height=1.1\alturaI]{./ii.pdf}}$, the wedge $\mathrel{\!{\setbox0\hbox{\ensuremath{\mathrel{\wedge}}}\rlap{\hbox to \wd0{\hss\,\ensuremath\wedge\hss}}\box0}\!}$\,, or the Lie derivative $\includegraphics[height=1.1\alturaL]{./LL.pdf}$---are well defined (see the appendix of \cite{MargalefVillasenor2020} for more details about this and other details). As a reminder for the reader, most objects defined over the space of fields are denoted with a double font. Given a vector field $\xi\in\mathfrak{X}(M)$ on $M$,  let us associate to it a canonical vector field $\mathbb{X}_\xi\in\raisebox{-.075ex}{\includegraphics[height=1.05\alturaX]{./XX.pdf}}(\mathcal{F})$ on $\mathcal{F}$. To that purpose, we assume that $\xi$ is tangent to $\peqsub{\partial}{L} M$ but not necessarily to $\Sigma_i$ and $\Sigma_{\!f}$ (notice that we can always extend the interval $[t_i,t_f]$ to avoid problems at the lids) and take advantage of the fact that a field $\phi^r$ ($r$ ranging over the number of fields of the theory) can be interpreted in two ways:
\begin{itemize}
	\item As a tensor field on $M$\!. That is, a section $\phi^r:M\to E^r$ of some bundle $E^r\overset{\pi}{\to}M$ such that $\phi^r_p:=\phi^r(p)\in E^r_p:=\pi^{-1}(p)$. In particular, we can take its Lie derivative $(\L_\xi\phi^r)_p=\partial_\tau|_{0}(\varphi^\xi_\tau)^*\phi^r_p$, where $\{\varphi^\xi_\tau\}_\tau\subset\mathrm{Diff}(M)$ is the flow of $\xi$. If $\mathcal{F}$ is reasonable enough, as it is the case in this paper, we will have $\L_\xi\phi\in\mathcal{F}$.
	\item As a point of $\mathcal{F}$. A vector field $\mathbb{X}_\xi\in\raisebox{-.075ex}{\includegraphics[height=1.05\alturaX]{./XX.pdf}}(\mathcal{F})$ is a section of the tangent bundle $T\mathcal{F}$ i.e.\ $(\mathbb{X}_\xi)_\phi:=\mathbb{X}_\xi(\phi)\in T_\phi\mathcal{F}\cong\mathcal{F}$. The last isomorphism comes from the fact that the field space $\mathcal{F}$ is linear in this case. The non-linear case is not as straightforward but, in concrete examples, it is possible to carry out similar constructions.
\end{itemize}
Taking into account these remarks, we define
\begin{equation}\label{eq: definicion XX_xi}
(\mathbb{X}_\xi^r)_{\phi}=\L_\xi\phi^r\qquad\qquad\equiv\qquad\qquad\includegraphics[height=1.1\alturaL]{./LL.pdf}_{\mathbb{X}_\xi}\phi^r=\L_\xi\phi^r
\end{equation}
Notice that $\dd\phi^r$ (which appears in the definition of the Lie derivative $\includegraphics[height=1.1\alturaL]{./LL.pdf}$ through Cartan's magic formula) has to be interpreted as the exterior derivative of the evaluation function.

\subsection{CPS-Algorithm}\label{section: summary of algorithm}
This paper relies heavily on the results of \cite{MargalefVillasenor2020}.  For convenience, a summary has been included in the appendix but we list here the main steps of the CPS-algorithm  since it will be used throughout the paper. This algorithm provides an ambiguity-free method to construct a presymplectic structure  in the space of solutions canonically associated with the action of the theory.

\begin{enumerate}\setcounter{enumi}{-1}
	\item Given the action $\SS:\mathcal{F}\to\R$ describing the dynamics of a particular field theory, choose any Lagrangian pair $(L,\overline{\ell})$ such that
	\[\SS=\int_M L-\int_{\partial M}\overline{\ell}\]
	\item Compute $\dd L=E_r\wedge\dd\phi^r+\d\Theta$ and choose any $\Theta$ compatible with the previous expression (uniquely defined up to a $\d$-exact form).
	\item Compute $\dd\overline{\ell}-\jmath^*\Theta=\overline{b}_r\wedge\dd\phi^r-\d\overline{\theta}$  over $\peqsub{\partial}{L} M$\! and choose any admissible $\overline{\theta}$.
	\item Define $\mathrm{Sol}(\SS)=\{\phi\in\mathcal{F}\ /\ E_r(\phi)=0, \ \ \overline{b}_r(\phi)=0\}$ and the inclusion $\peqsub{\raisebox{-.37ex}{\includegraphics[height=.9\alturaJ]{./jj.pdf}}}{\SS}:\mathrm{Sol}(\SS)\hookrightarrow\mathcal{F}$.
	\item Compute the presymplectic structure canonically associated with $\S$
	\[\includegraphics[height=\alturaO]{./OOmega.pdf}_\SS^\imath:=\dd\!\left(\int_\Sigma\imath^*\Theta-\int_{\partial\Sigma}\overline\imath^*\overline{\theta}\right)\qquad\longrightarrow\qquad\includegraphics[height=\alturaO]{./OOmega.pdf}_\SS=\peqsub{\raisebox{-.37ex}{\includegraphics[height=.9\alturaJ]{./jj.pdf}}}{\SS}^*\includegraphics[height=\alturaO]{./OOmega.pdf}^\imath_\SS\]
	where $\imath:\Sigma\hookrightarrow M$ is any Cauchy embedding and $\overline{\imath}:=\imath|_{\partial\Sigma}:\partial\Sigma\hookrightarrow\peqsub{\partial}{L} M$ its restriction. $\includegraphics[height=\alturaO]{./OOmega.pdf}_\SS$ does not depend on $\imath$.
\end{enumerate}
It is important to notice that it is not always possible to perform the second step. If $\overline{b}_r$ and $\overline{\theta}$ do not exist (their existence does not depend on the chosen $\Theta$), that means that the theory is not well defined and hence we will have to change $\mathcal{F}$ and/or the action (hence $(L,\overline{\ell})$) to ensure that the equality holds. Once we have the presymplectic structure, the following two steps provide important additional information about the theory at hand.

\begin{enumerate}\setcounter{enumi}{4}
	\item Study symmetries; i.e. find out if $\mathbb{X}_\xi$ is a $\underline{\d}$-symmetry and obtain the $\xi$-currents and $\xi$-charges.
	\[\begin{array}{l}
	J_\xi:=\iota_\xi L-\raisebox{-.04ex}{\includegraphics[height=1.1\alturaI]{./ii.pdf}}_{\mathbb{X}_\xi}\Theta\\
	\overline{\jmath}_{\xi}:=-\iota_{\overline{\xi}}\overline{\ell}-\raisebox{-.04ex}{\includegraphics[height=1.1\alturaI]{./ii.pdf}}_{\mathbb{X}_\xi}\overline{\theta}
	\end{array}\qquad\qquad\qquad \mathbb{Q}^{\imath}_\xi:=\int_\Sigma\imath^*J_\xi-\int_{\partial\Sigma}\overline{\imath}^*\overline{\jmath}_{\xi}\]
	\item Compare $\includegraphics[height=\alturaO]{./OOmega.pdf}_\SS$ with the presymplectic structure obtained in the  standard Hamiltonian formulation.
\end{enumerate}

\section{General Relativity in terms of metrics}\label{section: metric}

\subsection*{Step 0: Action}
Let us consider the following spaces of metrics on $M$
\begin{align*}
&\mathcal{F}^{(m)}_N=\mathrm{Met}(M)&&\mathcal{F}^{(m)}_D=\left\{g\in\mathrm{Met}(M)\ /\ \overline{g}:=\jmath^*\!g\text{ fixed}\right\}
\end{align*}
that we will refer to as Neumann and Dirichlet metrics respectively. Consider also the actions $\SS^{(m)}_N$ and $\SS^{(m)}_D$ defined by
\[\SS^{(m)}(g)\!=\!\!\int_M\! L_{\mathrm{EH}}(g)-\int_{\partial_L\!M}\!\overline{\ell}_{\mathrm{GHY}}(g)\qquad\qquad\begin{array}{l}
L_{\mathrm{HE}}(g)=(R_g-2\Lambda)\mathrm{vol}_g\\[1.2ex]\overline{\ell}_{\mathrm{GHY}}(g)=-2 \mathrm{Tr}_{\overline{g}}(\xoverline{K})\mathrm{vol}_{\overline{g}}
\end{array}\]
suitably restricted to these spaces of metrics. Our Lagrangian pair $(L_{\mathrm{EH}},\overline{\ell}_{\mathrm{GHY}})$ consists of  the Einstein-Hilbert term with cosmological constant in the bulk and the Gibbons-Hawking-York term on the lateral boundary. As usual, $R_g$ denotes the $g$-scalar curvature, $\mathrm{vol}_g$ and $\mathrm{vol}_{\overline{g}}$ the metric volume forms, and $\xoverline{K}$ the extrinsic curvature of  $\partial_L\!M\subset M$ given by
\[\xoverline{K}:=\frac{1}{2}\jmath^*\L_{\vec{\nu}}g\qquad\equiv\qquad \xoverline{K}_{\overline{\alpha}\overline{\beta}}=\jmath^\alpha_{\!\overline{\alpha}}\jmath^\beta_{\overline{\beta}}\nabla_{\alpha}\nu_\beta\]
where $\vec{\nu}$ is the $g$-normal vector field to $\partial_L\!M$ with $\nu_\alpha\nu^\alpha=+1$.

\subsection*{Step 1 and 2: Variations}\label{subsection: variations metric}
Following the computations given in the appendix, we have
\[\dd L_{\mathrm{HE}}=E_{(m)}^{\alpha\beta}\wedge \dd g_{\alpha\beta}+\d\Theta_{(m)}\qquad\qquad \dd\overline{\ell}_{\mathrm{GHY}}-\jmath^*\Theta_{(m)}=\overline{b}_{(m)}^{\overline{\alpha}\overline{\beta}}\wedge\dd\overline{g}_{\overline{\alpha}\overline{\beta}}-\d\overline{\theta}_{(m)}\]
where
\begin{align*}
    &E_{(m)}^{\alpha\beta}(g)=-\left(\mathrm{Ric}^{\alpha\beta}-\left(\frac{R_g}{2}-\Lambda\right)g^{\alpha\beta}\right)\mathrm{vol}_g&&\overline{b}_{(m)}^{\overline{\alpha}\overline{\beta}}(g)=-\left(\xoverline{K}^{\overline{\alpha}\overline{\beta}}-\xoverline{K}\overline{g}^{\overline{\alpha}\overline{\beta}}\right)\mathrm{vol}_{\overline{g}}\\
    &\Theta_{(m)}(g)=\iota_{\vec{W}}\mathrm{vol}_g&&\overline{\theta}_{(m)}(g)=\iota_{\vec{V}}\mathrm{vol}_{\overline{g}}\\
    &W_\alpha=\nabla^\beta\dd g_{\alpha\beta}-\nabla_{\!\alpha}\dd g&&\overline{V}_{\overline{\alpha}}=-\jmath^\alpha_{\overline{\alpha}}(\iota_{\vec{\nu}}\dd g)_{\!\alpha}
\end{align*}
where $\dd g:=g^{\alpha\beta}\dd g_{\alpha\beta}$ stands for the $g$-trace of $\dd g_{\alpha\beta}$ (not to be confused with the variation of the determinant of $g$, which we do not use in this paper). Besides, we use the notation $(\iota_{\vec{\nu}}\dd g)_{\!\beta}:=\nu^\alpha\dd g_{\alpha\beta}$ even though $\dd g_{\alpha\beta}$ is not a $2$-form.

\subsection*{Step 3: Space of solutions}
Once we know the basic variations, we can compute the variation of the action
\begin{equation}\label{eq: dd SS = 0}\dd_g\S\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{=}}}\rlap{\hbox to \wd0{\hss\raisebox{1.2ex}{\ensuremath{\scriptscriptstyle\mathcal{L}}}\hss}}\box0}}\int_M E_{(m)}^{\alpha\beta}(g)\mathrel{\!{\setbox0\hbox{\ensuremath{\mathrel{\wedge}}}\rlap{\hbox to \wd0{\hss\,\ensuremath\wedge\hss}}\box0}\!} \dd g_{\alpha\beta}-\int_{\partial_L\!M}\overline{b}_{(m)}^{\overline{\alpha}\overline{\beta}}(g)\mathrel{\!{\setbox0\hbox{\ensuremath{\mathrel{\wedge}}}\rlap{\hbox to \wd0{\hss\,\ensuremath\wedge\hss}}\box0}\!}\dd\overline{g}_{\overline{\alpha}\overline{\beta}}
\end{equation}
where $\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{=}}}\rlap{\hbox to \wd0{\hss\raisebox{1.2ex}{\ensuremath{\scriptscriptstyle\mathcal{L}}}\hss}}\box0}}$ denotes equality up to integrals over the lids. Notice that those lid integrals are unimportant because the values of the dynamical fields on the lids are kept fixed when obtaining the equations of motion of the theory. The solutions obtained from $\SS^{(m)}_N$ are given by metrics $g\in\mathcal{F}^{(m)}_N$ satisfying Einstein's equations $E_{(m)}(g)=0$ and the Neumann boundary condition $\overline{b}_{(m)}(g)=0$, whereas those from $\SS^{(m)}_D$ correspond to metrics $g\in\mathcal{F}^{(m)}_D$ satisfying Einstein's equation and no additional boundary conditions. Indeed, in the later case the boundary integral of \eqref{eq: dd SS = 0} vanishes as a consequence of the Dirichlet boundary conditions introduced \emph{a priori} on the definition of $\mathcal{F}^{(m)}_D$.

\subsection*{Step 4: Symplectic form}
Given a Cauchy embedding $\imath:\Sigma\hookrightarrow M$, we compute the presymplectic structure canonically associated with the action
\[\includegraphics[height=\alturaO]{./OOmega.pdf}_{(m)}^\imath=\dd\left(\int_\Sigma\imath^*\Theta_{(m)}-\int_{\partial\Sigma}\overline{\imath}^*\overline{\theta}_{(m)}\right)\]
Using the results of the previous sections and the computations of the appendix, we obtain:
\begin{equation}
\begin{split}
    \includegraphics[height=\alturaO]{./OOmega.pdf}_{(m)}^\imath&=\frac{1}{2}\!\int_\Sigma n^\alpha\delta^{\lambda\eta\sigma}_{\alpha\beta\zeta}g^{\beta\rho}g^{\zeta\phi}\dd g_{\eta\rho}\mathrel{\!{\setbox0\hbox{\ensuremath{\mathrel{\wedge}}}\rlap{\hbox to \wd0{\hss\,\ensuremath\wedge\hss}}\box0}\!}\nabla_{\!\lambda}\dd g_{\sigma\phi}\mathrm{vol}_\gamma\\
&\phantom{=}-\frac{1}{2}\!\int_{\partial\Sigma}\!\Big(n^\alpha g^{\beta\eta}\mu^\lambda +m^\alpha\nu^\beta \overline{g}^{\overline{\eta}\overline{\lambda}}\jmath^\eta_{\overline{\eta}}\jmath^\lambda_{\overline{\lambda}}\Big)\dd g_{\alpha\beta}\mathrel{\!{\setbox0\hbox{\ensuremath{\mathrel{\wedge}}}\rlap{\hbox to \wd0{\hss\,\ensuremath\wedge\hss}}\box0}\!}\dd g_{\eta\lambda}\mathrm{vol}_{\overline{\gamma}}
\end{split}
\end{equation}
As indicated in \eqref{eq: diagrama}, $n^\alpha$ is the $g$-normal to $\imath(\Sigma)\subset M$, $\nu^\beta$ is the $g$-normal to $\partial_LM\subset M$, $m^\alpha$ is the pushforward of the $\overline{g}$-normal to $\overline{\imath}(\partial\Sigma)\subset\partial_L\!M$, and $\mu^\lambda$ is the pushforward of the $\gamma$-normal  to $\partial\Sigma\subset\Sigma$. We have also used the generalized Kronecker delta \[\tensor*{\delta}{*_{\alpha_1}^{\beta_1}_\cdots^\cdots_{\alpha_s}^{\beta_s}}=\begin{vmatrix}
\delta_{\alpha_1}^{\beta_1} & \cdots & \delta_{\alpha_1}^{\beta_s}\\ \vdots & &\vdots\\ \delta_{\alpha_s}^{\beta_1}&\cdots&\delta_{\alpha_s}^{\beta_s}\end{vmatrix}\]

\subsection*{Step 5: Charges}
The Noether currents associated with an arbitrary vector field $\xi\in\mathfrak{X}(M)$ tangent to the lateral boundary are computed in the appendix. The result is
\begin{align*}
    &J_\xi^{(m)}=\d Q_\xi^{(m)}+2\star_g(\iota_\xi\tilde{E}_{(m)})&&\overline{\jmath}_\xi^{(m)}=\jmath^* Q_\xi^{(m)}-2\star_{\bar{g}}(\iota_{\overline{\xi}}\tilde{b}_{(m)})
\end{align*}
where $Q_\xi^{(m)}=\star_g\d\xi$ is the $\xi$-Noether potential and $(\tilde{E}_{(m)},\tilde{b}_{(m)})$ are the $2$-tensors that multiply the top-forms in $(E_{(m)},\overline{b}_{(m)})$. As usual, $\star_g$ is the $g$-Hodge dual operator and, in order to ease the notation, we represent also by $\xi$ the $1$-form metrically equivalent to the vector field $\xi$.  The $\xi$-charge is then given by
\begin{equation}\label{eq: xi-charge}
\Q^\imath_{\xi,(m)}=2\int_\Sigma\imath^*\star_g\big(\iota_\xi\tilde{E}_{(m)}\big)+2\int_{\partial\Sigma}\overline{\imath}^*\star_{\bar{g}}\big(\iota_{\bar{\xi}}\tilde{b}_{(m)}\big)
\end{equation}
The last expression, when pulled back to $\mathrm{Sol}(\SS^{(m)}_N)\subset\mathcal{F}^{(m)}_N$, vanishes as expected because the theory is Diff-invariant \cite{MargalefVillasenor2020}. However, over $\mathrm{Sol}(\SS^{(m)}_D)\subset\mathcal{F}^{(m)}_D$, the boundary integral is not necessarily zero as $\overline{b}_{(m)}$ does not vanish in general. Notice that fixing the metric on the boundary breaks the Diff-invariance.

\subsection*{Step 6: comparing with the Hamiltonian formulation}
If we consider $\partial\Sigma=\varnothing$, the equivalence between the CPS presymplectic form obtained in step 4 and the canonical ADM presymplectic form has been established in \cite{AshtekarMagnon1982,FrauendienerSparling1992}. The case with boundaries is not as straightforward and will be studied elsewhere.

\subsection*{The asymptotically flat case}
As a further application of the CPS-algorithm, let us obtain the ADM energy of an asymptotically flat space-time (see \cite{AshtekarBombelliReula1991,HarlowWu2020} for similar approaches). For that purpose, consider $M=\Sigma\times\R$ endowed with the Minkowski metric $g_0=-\d t^2+\sigma$. Then $\Sigma$ is a $3$-manifold with Euclidean metric $\sigma$ and boundary $\partial\Sigma=\S^2_r$, a $2$-sphere of $\sigma$-radius $r_0$. Take a small perturbation $h\in T_{g_0}\mathrm{Met}(M)$, such that $\overline{h}:=\jmath^*h=0$ (compatible with the Dirichlet condition, otherwise the charge is zero as the theory is Diff-invariant). The $\xi$-charge given by \eqref{eq: xi-charge} restricted to the space of solutions is given by
\[\Q_{\xi}=2\int_{\partial\Sigma}\overline{\xi}_{\overline{\alpha}}\overline{m}_{\overline{\beta}}(\xoverline{K}^{\overline{\alpha}\overline{\beta}}-\xoverline{K}\overline{g}^{\overline{\alpha}\overline{\beta}})\mathrm{vol}_{\overline{\gamma}}\]
Expanding all these terms in $h$, and labeling with a circle the quantities associated with $g_0$ (the ``background'' objects), we obtain
\[\Q_{\xi}=2\int_{\partial\Sigma}\overline{\xi}_{\overline{\alpha}}(\accentset{\circ}{\overline{m}}_{\overline{\beta}}+\includegraphics[height=1.1\alturaL]{./LL.pdf}_h \overline{m}_{\overline{\beta}})\Big(\accentset{\circ}{\xoverline{K}}^{\overline{\alpha}\overline{\beta}}+\includegraphics[height=1.1\alturaL]{./LL.pdf}_h\xoverline{K}^{\overline{\alpha}\overline{\beta}}-(\accentset{\circ}{\xoverline{K}}+\includegraphics[height=1.1\alturaL]{./LL.pdf}_h \xoverline{K})(\accentset{\circ}{\overline{g}}^{\overline{\alpha}\overline{\beta}}-\includegraphics[height=1.1\alturaL]{./LL.pdf}_h\overline{g}^{\overline{\alpha}\overline{\beta}})\Big)(\mathrm{vol}_{\overline{\sigma}}+\includegraphics[height=1.1\alturaL]{./LL.pdf}_h\mathrm{vol}_\gamma)+\cdots
\]
where we omit higher order terms in $h$ as they vanish when $r_0\to\infty$ if we consider $\xi:=n$ and the inertial foliation $\{\{t\}\times\Sigma\}_t$ (in particular, we take $\vec{n}=\partial_t$).\vspace*{2ex}

Using the decomposition of the normal $\nu_\alpha=-\nu_\perp n_\alpha+\nu^\top_\alpha$ and some of the results of section \ref{subsection: some metric computations}, we have
\begin{align*}
\includegraphics[height=1.1\alturaL]{./LL.pdf}_h\overline{m}_{\overline{\alpha}}&=\jmath^\alpha_{\overline{\alpha}}\includegraphics[height=1.1\alturaL]{./LL.pdf}_h\!\left(\frac{n_\alpha+\nu_\perp \nu_\alpha}{|\nu^\top|}\right)=\raisebox{-.04ex}{\includegraphics[height=1.1\alturaI]{./ii.pdf}}_h\left(\frac{1}{|\nu^\top|}\jmath^\alpha_{\overline{\alpha}}\dd n_\alpha-\overline{m}_{\overline{\alpha}}\frac{\nu_\perp}{|\nu^\top|}\dd\nu_\perp\right)\\
&=\frac{1}{|\accentset{\circ}{\nu}^\top|}\jmath^\alpha_{\overline{\alpha}}\raisebox{-.04ex}{\includegraphics[height=1.1\alturaI]{./ii.pdf}}_h\left(\frac{-1}{2} n_\alpha\iota^2_{\vec{n}}\dd g\right)-\overline{m}_{\overline{\alpha}}\frac{\accentset{\circ}{\nu}_\perp}{|\accentset{\circ}{\nu}^\top|}\includegraphics[height=1.1\alturaL]{./LL.pdf}_h\nu_\perp\\
&=-\frac{1}{2} \overline{\xi}_{\overline{\alpha}}\iota^2_{\xi}\,h+0=-\frac{1}{2} \overline{\xi}_{\overline{\alpha}}\iota^2_{\overline{\xi}}\,\overline{h}=0
\end{align*}
where we have used that $\accentset{\circ}{\nu}_\perp=0$, $\xi=n$ and the fact that $\xi$ is tangent to the lateral boundary (hence we can replace $h$ with $\overline{h}$, which is zero). We have also used the (not very standard) notation $\iota^2_{V}T=V^\alpha V^\beta T_{\alpha\beta}$. Notice that the previous computation is performed at the point $g_0$ in the space of fields and at the lateral space-time boundary.\vspace*{2ex}

This computation together with  $\accentset{\circ}{\overline{m}}_{\overline{\beta}}=\overline{\xi}_{\overline{\beta}}$ takes care of the first parenthesis of $\mathbb{Q}_\xi$. The last one can also be easily computed by using the variation of the volume given in section \ref{subsection: some metric computations}. Let us now deal with the term in the second parenthesis. First notice that $\accentset{\circ}{\xoverline{K}}^{\overline{\alpha}\overline{\beta}}$ vanishes when we contract it with $\overline{\xi}_{\overline{\alpha}}\overline{\xi}_{\overline{\beta}}$ (in coordinates this would be $\accentset{\circ}{\xoverline{K}}^{tt}$) while $\includegraphics[height=1.1\alturaL]{./LL.pdf}_h\overline{g}^{\overline{\alpha}\overline{\beta}}=\overline{h}=0$. A long but straightforward computation gives the following variations
\begin{align*}
\overline{\xi}_{\overline{\alpha}}\overline{\xi}_{\overline{\beta}}\includegraphics[height=1.1\alturaL]{./LL.pdf}_h\xoverline{K}^{\overline{\alpha}\overline{\beta}}&=\frac{1}{2}\xi^\alpha\xi^\beta\Big(\accentset{\circ}{\nabla}_{\!\vec{\nu}}h_{\alpha\beta}-2\nu^\gamma\accentset{\circ}{\nabla}_{\!\alpha}h_{\beta\gamma}\Big)\\
\includegraphics[height=1.1\alturaL]{./LL.pdf}_h\xoverline{K}&=\frac{1}{2}\imath^\alpha_a\imath^\beta_b\gamma^{ab}\Big(\accentset{\circ}{\nabla}_{\!\vec{\nu}}h_{\alpha\beta}-\nu^\gamma\accentset{\circ}{\nabla}_{\!\alpha}h_{\beta\gamma}\Big)-\frac{1}{2}\xi^\alpha\xi^\beta\Big(\accentset{\circ}{\nabla}_{\!\vec{\nu}}h_{\alpha\beta}-2\nu^\gamma\accentset{\circ}{\nabla}_{\!\alpha}h_{\beta\gamma}\Big)-\\
&\phantom{=}-\frac{1}{2}\accentset{\circ}{\overline{D}}_{\overline{a}}\Big(\overline{\gamma}^{\overline{a}\overline{b}}\overline{\jmath}^{\overline{\alpha}}_{\overline{b}}\jmath^\alpha_{\overline{\alpha}}(\iota_{\vec{\nu}}h)_\alpha\Big)+\frac{1}{2}\xi^\alpha\nu^\beta\L_{\vec{n}}g_{\alpha\beta}\xi^\gamma\nu^\mu h_{\gamma\mu}
\end{align*}
As the result is independent of the Cauchy slice, we introduce coordinates $\{t,x_1,x_2,x_3\}$ and take, as we mentioned before, a Cauchy slice given by $\Sigma=\{t=t_0\}$, then
\[\vec{\nu}=\frac{1}{r_0}(0,x_1,x_2,x_3)\]
Notice that everything is constant along the $t$-direction (which is the $\xi$ direction because $\vec{n}=\partial_t$). Thus we have
\[ \accentset{\circ}{\xoverline{K}}_{\overline{\alpha}\overline{\beta}}=\jmath^\alpha_{\overline{\alpha}}\jmath^\beta_{\overline{\beta}}\accentset{\circ}{\nabla}_{\!\alpha}\nu_\beta\qquad\qquad\accentset{\circ}{\xoverline{K}}_{AB}=\frac{1}{r_0^3}(\delta_{AB}r_0^2-x_{A}x_{B})\qquad\qquad\accentset{\circ}{\xoverline{K}}=\frac{D-2}{r_0}\]
where here $A,B=1,2,3$ label the coordinates. Finally, notice that the integral of $h\accentset{\circ}{\xoverline{K}}\mathrm{vol}_{\overline{\sigma}}$ goes to zero in the limit and $\iota_\xi \xoverline{K}=0$ because the normal is constant along the foliation. Putting everything together, we obtain
\[\Q_{\xi}=\int_{\partial\Sigma}\Big(\accentset{\circ}{\xoverline{K}}+\gamma^{BC}\nu^{A}\Big(\accentset{\circ}{\nabla}_{\!A}h_{BC}-\accentset{\circ}{\nabla}_{\!B}h_{CA}\Big)\Big)\mathrm{vol}_{\overline{\sigma}}+\cdots
\]
A last comment is in order now. As we can see we have a divergent constant term in the limit $r_0\to\infty$. To remove it, we introduce the following $r_0$-dependent boundary Lagrangian
\[\overline{\ell}_{(r_0)}=2\Big(\mathrm{Tr}_{\overline{g}}(\accentset{\circ}{\xoverline{K}})-\mathrm{Tr}_{\overline{g}}(\xoverline{K})\Big)\mathrm{vol}_{\overline{g}}\]
By doing this, the constant term does not appear, the limit is well defined, and we recover the well-known expression for the ADM-energy.

\section{General relativity in terms of tetrads}\label{section: tetrads}
\subsection*{Notation}
In order to introduce the tetrad formulation for GR, we will use internal abstract indices as discussed, for instance, in \cite{Penrose,AHM,Romano}. For the convenience of the unfamiliar reader, we have included this short subsection with the basic ingredients. The more familiar reader can go directly to equation \eqref{eq: ledge def}, where we introduce some new notation.\vspace*{2ex}

Let $\mathcal{V}\to M$ be a  Minkowski vector bundle i.e\ it has typical fiber $V$\!, an $n$-dimensional vector space with internal indices $\{I,J,\ldots\}$, endowed with a Minkowskian metric $\eta$, which in abstract index notation reads $\eta_{IJ}$. This metric allows us to define the $\eta$-metric volume form $\varepsilon$, the $\eta$-Hodge dual $\star_\eta$, and the $\eta$-trace $\peqsub{\mathrm{Tr}}{\eta}$ (see \cite{Tecchiolli} for a recent review).\vspace*{2ex}

Tetrads are bundle isomorphisms $TM\rightarrow \mathcal{V}$. In abstract index notation they look like $e_{\alpha}^I$, so it is clear that a tetrad, together with the non-degenerate metrics $g_{\alpha\beta}$ and $\eta_{IJ}$, provides a way to swap space-time and internal indices. It is also clear from the index notation that tetrads can be identified with sections of the product bundle $\mathcal{V}\otimes T^*\!M$ i.e. elements of $\Gamma(\mathcal{V}\otimes T^*\!M)$, the set of $\mathcal{V}$-valued $1$-forms over $M$, although in practice it would be more useful to identify them with  $V$-valued $1$-forms over $M$. This is the case, for instance, if $M$ is parallelizable (an example would be a non-compact manifold admitting a spinor structure \cite{Geroch}). In that case it is enough to work with coframes which, loosely speaking, are the local version of a tetrad. Indeed, a coframe $e^I_\alpha(p)$ at $p\in M$ is a linear ismorphism $T_pM\rightarrow V$ satisfying
\[
g_{\alpha\beta}(p)=\eta_{IJ}e^I_\alpha(p)e^J_\beta(p)
\]
Equivalently, a coframe can be thought of as an orthonormal basis in $T^*_pM$. When $M$ is  parallelizable, coframes can be identified with global covector fields but, otherwise, they can only be considered as local covector fields, \cite{Romano}.\vspace*{2ex}

It is also useful to introduce objects with more indices, the so-called \textbf{generalized tensor fields} $t\in \Gamma(\mathcal{V}^{\otimes p} \otimes \mathcal{V}^{*\otimes q}\otimes TM^{\otimes r}\otimes T^*\!M^{\otimes s})$, where $\mathcal{V}^*\to M$ is the dual bundle of the  Minkowski bundle. In the index notation, generalized tensor fields look like $t_{\alpha_1\cdots \alpha_s\, I_1\cdots I_q  }^{\beta_1\cdots \beta_p\,J_1\cdots J_r}$. We denote as $\Omega^k_r(M)$ the space of $k$-forms of $M$ with $r$ totally upper antisymmetric internal indices. Using Greek abstract indices $\{\alpha,\beta,\ldots\}$ for $M$ and capital Latin indices $\{I,J,\ldots\}$ for the internal indices, an element of $\Omega^k_r(M)$ will be written as \[\omega_{\alpha_1\cdots\alpha_k}^{I_1\cdots I_r}=\omega_{[\alpha_1\cdots\alpha_k]}^{[I_1\cdots I_r]}\]
To ease the notation, space-time indices will often be  omitted and we will simply write $\omega^{I_1\cdots I_r}$. Using the generalized Kronecker delta with internal indices, $\tensor*{\delta}{*_{I_1}^{J_1}_\cdots^\cdots_{I_m}^{J_m}}$, we define the graded wedge $(\cdot{}\wedge\cdot{}):\Omega^k_r(M)\times\Omega^m_s(M)\to\Omega^{k+m}_{r+s}(M)$
\begin{equation}
    (A\wedge B)_{I_1\cdots I_rJ_1\cdots J_s}:=\frac{1}{r!s!}\tensor*{\delta}{*_{I_1}^{K_1}_\cdots^\cdots_{I_r}^{K_r}_{J_1}^{L_1}_\cdots^\cdots _{J_s}^{L_s}}A_{K_1\cdots K_r}\wedge B_{L_1\cdots L_S}
\end{equation}
and the graded bilinear product $[\cdot{}\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}}\cdot{}]:\Omega^k_r(M)\times\Omega^m_s(M)\to\Omega^{k+m}_{r+s-2}(M)$
\begin{equation}\label{eq: ledge def}[A\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}} B]_{I_1\dots I_{r-1}J_1\dots J_{s-1}}:=\tensor*{\delta}{*_{I_1}^{K_1}_{\dots}^{\dots}_{I_{r-1}}^{K_{r-1}}_{J_1}^{L_1}_{\dots}^{\dots}^{L_{s-1}}_{J_{s-1}}}    A_{K_1\dots K_{r-1}M}\wedge B^M_{\phantom{M}L_1\dots L_{s-1}}
\end{equation}
where the symbol $\,\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}}\,$ will be called the \emph{ledge} (Lie wedge). The previous operation is performed by contracting the last index of $A$ with the first one of $B$ and anti-symmetrizing the remaining ones. Although not necessary for our purposes, it is worth mentioning that $\Omega^k_2(M)$ with the ledge product
\[[A\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}} B]_{IJ}=A_{IK}\wedge \tensor{B}{^K_J}-A_{JK}\wedge \tensor{B}{^K_I}\]
 is a Lie algebra. The following easy to derive formula will be used in the rest of the paper
\begin{align}
\label{eq: tr(a (b c))=-tr([a b] c)}
\peqsub{\mathrm{Tr}}{\eta}\big(\alpha\wedge(\beta\wedge\gamma)\big)=-\peqsub{\mathrm{Tr}}{\eta}([\alpha\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}}\beta]\wedge\gamma)\qquad\qquad\alpha\in\Omega^*_2(M),\ \ \beta,\gamma\in\Omega^*_1(M)
\end{align}

\subsection*{Constructing the geometric objects}
Consider $\{e_\alpha^I\}$ a co-frame, such that $\varepsilon_{I_1\cdots I_n}e^{I_1}\wedge\cdots\wedge e^{I_n}$ defines a volume form on $M$. We define the Lorentzian $e$-metric
\[g=\peqsub{\mathrm{Tr}}{\eta}(e\otimes e)\qquad\equiv\qquad g_{\alpha\beta}=\eta_{IJ}e^I_\alpha e^J_\beta\]
together with its $g$-Levi-Civita connection $\nabla$. We can now define the dual frame $\{E_I^\alpha\}$, given by the relations
\[E_I^\alpha e^I_\beta=g^\alpha_\beta\qquad\qquad E_I^\alpha e^J_\alpha=\eta^J_I\]
as well as the $e$-connection $\omega$ defined by
\[\nabla_{\!X} E=\omega(X)E\qquad\equiv\qquad\nabla_{\!\mu} E^\alpha_I=\omega_{\mu\,\,\,I}^{\,\,K} E^\alpha_K\qquad\equiv\qquad
\omega_{\mu\,\,\,I}^{\,\,K} =e_\alpha^K\nabla_{\!\mu}E^\alpha_I\]
which can be proven to characterize $\omega$ uniquely. Notice that the internal indices of $\omega$ are antisymmetric once the second one is raised $\omega_\mu^{IJ}=-\omega_\mu^{JI}$. Notice also that it is now possible to rewrite the covariant derivative of forms as
\[\mathcal{D}\alpha=\d\alpha+[\omega\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}} \alpha]\qquad\qquad \alpha\in\Omega^r_s(M)\]
where here and in the following, we will sometimes omit the internal indices when they are not needed. Let us also denote the associated $e$-Christoffel as $\Gamma$, the $e$-torsion as $T$, the $e$-curvature as $R$, and the $\omega$-curvature as $F$, which are given by
\begin{align*}
&\nabla_{\!E_I}E_J=\tensor{\Gamma}{^K_I_J}E_K&&\equiv&&e^I\tensor{\Gamma}{^K_I_J}=\tensor{\omega}{^K_J
\\
&\mathrm{Tor}(E_I,E_J)=\tensor{T}{^L_I_J}E_L&&\equiv&&\tensor{T}{^L_I_J}=e^L\Big(\mathrm{Tor}(E_I,E_J)\Big)\\
&\textrm{Riem}(E_I,E_J)E_K=\tensor{R}{^L_I_J_K}E_L&&\equiv&&\tensor{R}{^L_I_J_K}=e^L\Big(\textrm{Riem}(E_I,E_J)E_K\Big)\\
&F=\d\omega+\frac{1}{2}[\omega\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}}\omega]&&\equiv&&F_{IJ}=\d\omega_{IJ}+\omega_{IK}\wedge\tensor{\omega}{^K_J}
\end{align*}
where $\mathrm{Tor}$ and $\textrm{Riem}$ are the torsion and the Riemannian curvature of the $g$-Levi-Civita connection. In particular $\mathrm{Tor}=0$. It is interesting to notice that, as mentioned in the introduction of the section, we are using $E^\alpha_I$ and $e^I_\alpha$ to transform the space-time indices in $\mathrm{Tor}$ and $\textrm{Riem}$ into internal indices to define $T$ and $R$. From the previous equations, the following relations are easy to derive
\begin{equation}\label{eq: F=R}
\tensor{F}{^I_J}=\frac{1}{2}\tensor{R}{^I_J_K_L}e^K\wedge e^L\qquad\qquad\qquad \tensor{T}{^L_I_J}=\mathcal{D}e^L(E_I,E_J)
\end{equation}
It is also straightforward to show that the following properties hold
\[(\mathcal{D}^2 \alpha)_{I_1\cdots I_r}=[F\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}} \alpha]_{I_1\cdots I_r}\qquad\qquad\qquad\qquad\mathcal{D}F_{IJ}=0\qquad\qquad\qquad\qquad\mathcal{D}e^I=0\]

\subsection*{Step 0: Action}
Let us consider the following spaces of non-degenerate tetrads on $M$
\begin{align*}
&\mathcal{F}^{(t)}_N=\{e_I\in\Omega^1(M)\text{ non-degenerate}\}\\&\mathcal{F}^{(t)}_D=\left\{e_I\in\Omega^1(M)\text{ non-degenerate}\ /\ \overline{e}:=\jmath^*\!e\text{\; fixed}\right\}
\end{align*}
that we will refer to as Neumann and Dirichlet tetrads, respectively. The actions $\SS^{(t)}_N$ and $\SS^{(t)}_D$ are then defined by
\[\SS^{(t)}(e)=\int_M L(e)-\int_{\partial_LM}\overline{\ell}(e)\qquad\qquad\begin{array}{l}
L(e):=L_{\mathrm{EH}}\Big(\peqsub{\mathrm{Tr}}{\eta}(e\otimes e)\Big)
\\[1.2ex]\overline{\ell}(e):=\overline{\ell}_{\mathrm{GHY}}\Big(\peqsub{\mathrm{Tr}}{\eta}(e\otimes e)\Big)
\end{array}\]
suitably restricted to these spaces of tetrads. Notice that, essentially, we have only performed a ``change of variables'' from $g$ to $e$ through the (surjective but not injective) map
\[\Phi(e)_{\alpha\beta}=\eta_{IJ}e^I_\alpha e^J_\beta\]
Indeed, we have $\S^{(t)}=\S^{(m)}\circ\Phi$. Since $\Phi$ is Lorentz invariant, the tetrad action is also Lorentz invariant i.e. if $\Psi\in SO(1,3)$, then $\SS(\Psi.e)=\SS(e)$. By using \eqref{eq: F=R}, it is possible to show that the Lagrangian admits the following explicit expression
\[L(e)=\peqsub{\mathrm{Tr}}{\eta}\left( \left(F-\frac{\Lambda}{6}\right)\wedge\star_\eta(e\wedge e)\right)=\frac{1}{2}\varepsilon_{IJKL}F^{IJ}\wedge e^K\wedge e^L-\frac{1}{12}\varepsilon_{IJKL}e^I \wedge e^J\wedge e^K\wedge e^L
\]
Finding a useful explicit expression for the boundary Lagrangian requires more work. Let us define the ``internal'' normal $N^I:=\iota_{\vec{\nu}}e^I\in\Omega^0_1(M)$, which satisfies $N_I\overline{e}^I_{\overline{\alpha}}=0$ and $N_IN^I=1$, and the ``internal'' projector $\gamma^I_J:=\eta^I_J-N^IN_J$.
With these elements at hand, together with $\varepsilon_{IJKL}=\mathrm{vol}_{\alpha\beta\gamma\delta}E^\alpha_I E^\beta_J E^\gamma_K E^\delta_L$ and \eqref{eq: orientation sigma}, it is easy to rewrite $\overline{\ell}$ as
\[\overline{\ell}(e)=-2N^I\overline{\mathcal{D}}N^J\wedge\star_\eta(\overline{e}\wedge \overline{e})_{IJ}\]
where
\begin{equation}\label{eq: overline D}
\overline{\mathcal{D}}A=\d A+[\overline{\omega}\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}} A]\
\end{equation}
is the induced covariant derivative over the boundary given by the pullback connection $\overline{\omega}=\jmath^*\omega$. Recall also that $\overline{e}:=\jmath^*e$ which, in turn, allows us to define  its dual $\overline{E}_I^{\overline{\alpha}}:=\overline{g}^{\overline{\alpha}\overline{\beta}}\eta_{IJ}\overline{e}^J_{\overline{\beta}}$ which satisfies
\[\overline{E}^{\overline{\alpha}}_I\overline{e}^I_{\overline{\beta}}=\overline{g}^{\overline{\alpha}}_{\overline{\beta}}\qquad\qquad\overline{E}^{\overline{\alpha}}_I\overline{e}^J_{\overline{\alpha}}=\gamma^K_I\]
With these ingredients at hand, it is possible to obtain a better expression for $\overline{\ell}$ by considering
\begin{align*}
\Big\{(2N^I\d N^J&{}-{}\overline{\omega}^{IJ})-2N^I\overline{\mathcal{D}}N^J\Big\}\wedge\star_\eta(\overline{e}\wedge\overline{e})_{IJ}=\\
&=\overline{\omega}^{JK}(\eta^I_K-2N^IN_K)\wedge\star_\eta(\overline{e}\wedge\overline{e})_{IJ}=\\
&=\frac{1}{2}\gamma^R_K\gamma^S_J\gamma^M_A\gamma^N_B\varepsilon_{RSMN}\overline{\omega}^{JK}\wedge\overline{e}^A\wedge\overline{e}^B
\end{align*}
which is zero as all four internal indices of $\varepsilon$ are projected. This identity allows us to obtain, as it has been done already for example in \cite{BodendorferNeiman2013,CorichiRubalcavaVukasinac2016}, the alternative expression for $\overline{\ell}$ given by
\begin{equation}\label{eq: l(e)}
\overline{\ell}(e)=-\peqsub{\mathrm{Tr}}{\eta}\left(\left\{\rule{0ex}{2.5ex}2N\d N-\overline{\omega}\right\}\wedge\star_\eta(\overline{e}\wedge\overline{e})\right)
\end{equation}

\subsection*{Step 1 and 2: Variations}
As before, we rely on the results and computations of the appendix to obtain
\[\dd L=E^{(t)}_I\wedge \dd e^I+\d\Theta^{(t)}\qquad\qquad \dd\overline{\ell}-\jmath^*\Theta^{(t)}=\overline{b}^{(t)}_I\wedge\dd\overline{e}^I-\d\overline{\theta}^{(t)}\]
where
\begin{align*}
&E^{(t)}_I(e)=-\big[\star_\eta\widetilde{F}\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}} e\big]_I&&\overline{b}^{(t)}_I(e)=2\big[\star_\eta(2 N\d N-\overline{\omega})\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}}\overline{e}\big]_I+4\peqsub{\mathrm{Tr}}{\eta}\big(\big[(\star_\eta\overline{\omega})\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}}\overline{e}\big]\wedge N\big)N_I\\
&\Theta^{(t)}=\peqsub{\mathrm{Tr}}{\eta}\Big(\!\star_\eta(e\wedge e)\wedge\dd\omega\Big)&&\overline{\theta}^{(t)}=2\peqsub{\mathrm{Tr}}{\eta}\Big(\!\star_\eta(\overline{e}\wedge\overline{e})\wedge(N\dd N)\Big)\\
&\widetilde{F}:=F-\frac{\Lambda}{3}e\wedge e
\end{align*}

\subsection*{Step 3: Space of solutions}
The solutions derived from  $\SS^{(t)}_N$ are non-degenerate tetrads $e\in\mathcal{F}^{(t)}_N$ satisfying $E_I^{(t)}(e)=0$ and the ``Neumann'' boundary condition  $\overline{b}_I^{(t)}(e)=0$. The solutions obtained from $\SS^{(t)}_D$ are tetrads $e\in\mathcal{F}^{(t)}_D$ satisfying $E_I^{(t)}(e)=0$ and no additional condition at the boundary (the Dirichlet boundary conditions are part of the definition of $\mathcal{F}^{(t)}_D$).

\subsection*{Step 4: Symplectic form}
Given a Cauchy embedding $\imath:\Sigma\hookrightarrow M$, we have
\[\includegraphics[height=\alturaO]{./OOmega.pdf}_{(t)}^\imath=\dd\!\left(\int_\Sigma\imath^*\Theta^{(t)}-\int_{\partial\Sigma}\overline{\imath}^*\overline{\theta}^{(t)}\right)\]
Using the results of the previous section, we immediately obtain
\begin{align*}
\includegraphics[height=\alturaO]{./OOmega.pdf}_{(t)}^\imath=&\int_\Sigma \peqsub{\mathrm{Tr}}{\eta}\Big(\!\star_\eta \dd(e\wedge e)\mathrel{\!{\setbox0\hbox{\ensuremath{\mathrel{\wedge}}}\rlap{\hbox to \wd0{\hss\,\ensuremath\wedge\hss}}\box0}\!}\dd\omega\Big)-\\
&-2\int_{\partial\Sigma}\peqsub{\mathrm{Tr}}{\eta}\Big(\!\star_\eta\dd(\overline{e}\wedge\overline{e})\mathrel{\!{\setbox0\hbox{\ensuremath{\mathrel{\wedge}}}\rlap{\hbox to \wd0{\hss\,\ensuremath\wedge\hss}}\box0}\!}(N\wedge\dd N)+\star_\eta(\overline{e}\wedge\overline{e})\wedge(\dd N\mathrel{\!{\setbox0\hbox{\ensuremath{\mathrel{\wedge}}}\rlap{\hbox to \wd0{\hss\,\ensuremath\wedge\hss}}\box0}\!}\dd N)\Big)
\end{align*}

\subsection*{Step 5: Charges}
A direct computation using the definition of the $\xi$-current $J_\xi^{(t)}$ leads to
\[
J_\xi^{(t)}=\peqsub{\mathrm{Tr}}{\eta}\Big(2[\star_\eta\widetilde{F}\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}} e]\wedge\iota_\xi e-\star_\eta\mathcal{D}(e\wedge e)\wedge\iota_\xi\omega+\star_\eta(e\wedge e)\wedge(\L_\xi-\includegraphics[height=1.1\alturaL]{./LL.pdf}_{\mathbb{X}_\xi})\omega\Big)-\d\peqsub{\mathrm{Tr}}{\eta}\Big(\!\star_\eta(e\wedge e)\wedge\iota_\xi\omega\Big)
\]
where we have used $\iota_\xi F=(\L_\xi-\mathcal{D}\iota_\xi)\omega$, which follows from Cartan's magic formula. The second term of $J_\xi^{(t)}$ vanishes as the torsion $\mathcal{D}e$ is zero, while the third one is also zero using \eqref{eq: definicion XX_xi} and the fact that the only background object is $\eta$, which is ``constant''. Thus we obtain
\begin{align*}
J_\xi^{(t)}=\peqsub{\mathrm{Tr}}{\eta}\Big(\iota_\xi e\wedge E^{(t)}\Big)-\d\peqsub{\mathrm{Tr}}{\eta}\Big(\!\star_\eta(e\wedge e)\wedge\iota_\xi\omega\Big)\qquad\longrightarrow\qquad Q_\xi^{(t)}:=-\star_\eta(e\wedge e)\wedge\iota_\xi\omega
\end{align*}
$J_\xi^{(t)}$ is then exact over the space of solutions with $\xi$-potential $Q_\xi^{(t)}$. Similarly, we have for the boundary
\begin{align*}
  \overline{\jmath}_\xi^{(t)}-\jmath^*Q_\xi^{(t)}&=\mathrm{Tr}\Big(2\star_\eta(\overline{e}\wedge\overline{e})\wedge\Big\{n(\L_\xi-\includegraphics[height=1.1\alturaL]{./LL.pdf}_{\mathbb{X}_\xi})N\Big\}-2(\iota_{\overline{\xi}}\overline{e}\wedge\overline{e})\wedge\star_\eta(2N\d N-\overline{\omega})\Big)\\
  &=\mathrm{Tr}\Big(0-2\iota_{\overline{\xi}}\overline{e}\wedge[\overline{e}\mathrel{{\setbox0\hbox{\ensuremath{\mathrel{\cdot}}}\rlap{\hbox to \wd0{\hss\ensuremath\wedge\hss}}\box0}}\star_\eta(2N\d N-\overline{\omega})]\Big)\\
  &=\peqsub{\mathrm{Tr}}{\eta}(\iota_{\overline{\xi}}\overline{e}\wedge\overline{b}^{(t)})
\end{align*}
In the last equality, a term may seem to be missing according to the definition of $\overline{b}^{(t)}$ but, in fact, such term vanishes as a consequence of $\overline{e}^IN_I=0$. We then conclude that the $\xi$-charges are given by
\begin{equation}\label{eq: xi charge tetrad}\mathbb{Q}^\imath_{\xi,(t)}=\int_\Sigma\imath^*\peqsub{\mathrm{Tr}}{\eta}\Big(\iota_\xi e\wedge E^{(t)}\Big)-\int_{\partial\Sigma}\overline{\imath}^*\peqsub{\mathrm{Tr}}{\eta}\Big(\iota_{\overline{\xi}}\overline{e}\wedge\overline{b}^{(t)}\Big)
\end{equation}
This expression, when pulled-back to $\mathrm{Sol}(\SS^{(t)}_N)\subset\mathcal{F}^{(t)}_N$, vanishes as expected because the theory is Diff-invariant \cite{MargalefVillasenor2020}. However, over $\mathrm{Sol}(\SS^{(t)}_D)\subset\mathcal{F}^{(t)}_D$ the boundary integral will not be zero in general, as $\overline{b}^{(t)}$ might not vanish (fixing the tetrad on the boundary breaks the invariance under diffeomorphisms).

\subsection*{Step 6: comparing with the Hamiltonian formulation}
From Step 6 of the CPS-algorithm in the metric formalism and the following section, the equivalence is assured when no boundaries are present. The case with boundaries is again not as straightforward and it will be studied elsewhere.

\section{Metric vs Tetrad formulation}\label{section: metricvstetrad}
We have obtained in the previous sections the presymplectic structure over the space of solutions for both the metric and tetrad formulations together with their $\xi$-charges. In this section we prove that the spaces of solutions are in correspondence, that their symplectic structures are equivalent (without considering the internal gauge freedom), and that the $\xi$-charges are equal.

\subsection{Space of solutions}
Let us first show the correspondence between the solution spaces of metric and tetrad gravity. For this, define the maps
\[\Phi_N:\mathcal{F}^{(t)}_N\to\mathcal{F}^{(m)}_N\qquad\qquad\qquad\Phi_D:\mathcal{F}^{(t)}_D\to\mathcal{F}^{(m)}_D\]
both obtained by assigning the following metric to a given tetrad
\[\Phi(e)=\peqsub{\mathrm{Tr}}{\eta}(e\otimes e)\qquad\qquad\equiv\qquad\qquad\Phi(e)_{\alpha\beta}=\eta_{IJ}e^I_\alpha e^J_\beta\]
Of course, in the Dirichlet case the compatibility condition $\overline{g}_{\overline{\alpha}\overline{\beta}}=\eta_{IJ}\overline{e}^I_{\overline{\alpha}}\overline{e}^J_{\overline{\beta}}$ must hold for $\Phi_D$ to be well defined. On one hand, it is well known that $\Phi$ is surjective but not injective. For instance, $\Phi(-e)=\Phi(e)$. In fact, it can be proved that $\Phi(e)=\Phi(e')$ if and only if $e'_I=\tensor{\Psi}{_I^J}e_J$ for some $\Psi\in SO(3,1)$. \vspace*{2ex}

On the other hand, we have that $\S^{(t)}=\S^{(m)}\circ\Phi$ so
\[\dd_e\SS^{(t)}=\dd_e(\SS^{(m)}\circ\Phi)=\dd_{\Phi(e)}\SS^{(m)}\circ\dd_e\Phi\]

It is easy to check that $\dd_e\Phi$ is surjective. Hence, the relation between the spaces of solutions is clear: if $e\in\mathrm{Sol}(\SS^{(t)})$, then $\Phi(e)\in\mathrm{Sol}(\SS^{(m)})$ and if $g\in\mathrm{Sol}(\SS^{(t)})$, then every $e\in\Phi^{-1}(\{g\})$ belongs to $\mathrm{Sol}(\SS^{(t)})$.

\subsection{Presymplectic structures}
Let us now compare $\includegraphics[height=\alturaO]{./OOmega.pdf}^\imath_{(m)}$ and $\includegraphics[height=\alturaO]{./OOmega.pdf}^\imath_{(t)}$ by looking at their symplectic potentials. For that purpose, we perform the ``change of variable'' $g=\Phi(e):=e^Ie_I$ in $(\Theta_{(m)},\overline{\theta}_{(m)})$. First, we notice that instead of working with $\Theta_{(m)}$, it is more convenient to work with its $g$-dual
\begin{equation}\label{star Theta (m)}
(\star_g\Theta_{(m)})^\mu=(\star_g\iota_{\vec{W}}\mathrm{vol}_g)^\mu=W^\mu
\end{equation}
where $W_\mu:=\nabla^\beta\dd g_{\mu\beta}-\nabla_{\!\mu}\dd g$ was obtained in step 1  of section \ref{section: metric}. Using the variation
\[\dd\omega^{KL}_\mu=(\dd e^K_\alpha)\nabla_{\!\mu} E^{\alpha L}+E^{\alpha K}\nabla_{\!\mu}\dd e_\alpha^L-(e_{\alpha I}\dd e_\beta^I+e_{\beta I}\dd e_\alpha^I)E^{\alpha K}\nabla_{\!\mu}E^{\beta L}-E^{\beta K} e^L_\gamma(\dd\nabla)^\gamma_{\ \mu\beta}\]
which follows from the definition of  $\omega^{IJ}_\mu$, and the definition of $W^\mu$, we obtain on one hand
\begin{equation}\label{eq: W= cosas}
W^\mu=2E^\mu_I\iota_{E_J}\dd\omega^{IJ}+(\delta\mathcal{U})^\mu\qquad\qquad \mathcal{U}^{\alpha\mu}:=\big(g^{\beta\mu}E^\alpha_L-g^{\beta\alpha}E^\mu_L\big)\dd e^L_\beta
\end{equation}
where $(\delta\mathcal{U})^\mu:=-\nabla_{\!\alpha}\mathcal{U}^{\alpha\mu}$ is the codiferential. On the other hand, a standard computation using the definition of the $g$-Hodge star operator leads to
\begin{equation}\label{eq: star Theta (t)}
(\star_g\Theta^{(t)})^\mu=2E^\mu_I\iota_{E_J}\dd\omega^{IJ}
\end{equation}
Taking the Hodge dual of \eqref{eq: W= cosas}, using \eqref{star Theta (m)} and \eqref{eq: star Theta (t)}, and the fact that for $k$-forms we have $\star_g\star_g=(-1)^{k(n-k)+1}\mathrm{Id}$ and $\star_g\delta=(-1)^k\d\star_g$, we finally obtain
\[\Theta_{(m)}=\star_gW=\star_g\star_g\Theta^{(t)}+\star_g\delta\mathcal{U}=\Theta^{(t)}+\d\star_g\mathcal{U}\]

Once we have taken care of the bulk terms, we focus on the boundary ones. If we use  $\varepsilon_{IJKL}=\mathrm{vol}_{\alpha\beta\gamma\delta}E^\alpha_I E^\beta_J E^\gamma_K E^\delta_L$, $N_I=\nu_\alpha E^\alpha_I$, and $\dd N^I=-N_J\iota_{\overline{E}^I}\dd\overline{e}^J$, we can obtain on one hand
\begin{align*}
\big(\overline{\theta}^{(t)}\big)_{\overline{\alpha}\overline{\beta}}&=\tensor{\varepsilon}{^I^J_K_L}(\overline{e}^K\wedge\overline{e}^L)_{\overline{\alpha}\overline{\beta}}N_I\dd N_J\\
&=-\tensor{(\mathrm{vol}_g)}{^\alpha^\beta_\gamma_\delta}e^I_\alpha e^J_\beta E^\gamma_K E^\delta_L(\overline{e}^K_{\overline{\alpha}}\overline{e}^L_{\overline{\beta}}-\overline{e}^K_{\overline{\beta}}\overline{e}^L_{\overline{\alpha}})\nu_\sigma E^\sigma_IN_R\iota_{\overline{E}_J}\dd\overline{e}^R\\
&=-2N_R(\jmath^*\iota_{\vec{\nu}}\mathrm{vol}_g)_{\overline{\sigma}\,\overline{\alpha}\overline{\beta}}\overline{g}^{\overline{\sigma}\,\overline{\mu}}\dd\overline{e}^R_{\overline{\mu}}=(\iota_{\vec{U}}\mathrm{vol}_{\overline{g}})_{\overline{\alpha}\overline{\beta}}
\end{align*}
where $\overline{U}^{\overline{\mu}}:=-2N_R\overline{g}^{\overline{\mu}\,\overline{\alpha}}\dd\overline{e}^R_{\overline{\alpha}}$. On the other hand, using \eqref{eq: star 3+1}, we compute the following expression
\[\overline{\theta}_{(m)}-\overline{\theta}^{(t)}-\jmath^*(\star_g\mathcal{U})=\iota_{\vec{V}}\mathrm{vol}_{\overline{g}}-\iota_{\vec{U}}\mathrm{vol}_{\overline{g}}-\star_{\overline{g}}\jmath^*\iota_{\vec{\nu}}\mathcal{U}=\iota\big(\vec{V}-\vec{U}-\jmath^*\iota_{\vec{\nu}}\mathcal{U}\big)\mathrm{vol}_g\]
where $\overline{V}_{\overline{\alpha}}:=-\jmath^\alpha_{\overline{\alpha}}(\iota_{\vec{\nu}}\dd g)_{\!\alpha}$ was obtained in the step 2 of section \ref{section: metric}. Notice that we have made a small abuse of notation because $\jmath^*\iota_{\vec{\nu}}\mathcal{U}$ is a $1$-form so a raise of index with $\overline{g}$ is to be understood. Finally, we prove that this last expression is in fact zero
\begin{align*}\overline{V}^{\overline{\alpha}}&-\overline{U}^{\overline{\alpha}}-\overline{g}^{\overline{\alpha}\overline{\beta}}\jmath^\beta_{\overline{\beta}}g_{\mu\beta}\nu_\alpha\mathcal{U}^{\alpha\mu}\\
&=-\overline{g}^{\overline{\alpha}\overline{\beta}}\jmath^\beta_{\overline{\beta}}\nu^\alpha\eta_{IJ}(e_\alpha^I\dd e_\beta^J+e_\beta^I\dd e_\alpha^J)+2N_R\overline{g}^{\overline{\alpha}\overline{\beta}}\dd\overline{e}^R_{\overline{\beta}}-\overline{g}^{\overline{\alpha}\overline{\beta}}\jmath^\beta_{\overline{\beta}}g_{\mu\beta}\nu_\alpha(g^{\sigma\mu}E^\alpha_L-g^{\sigma\alpha}E^\mu_L)\dd e^L_\sigma\\
&=-N_J\overline{g}^{\overline{\alpha}\overline{\beta}}\dd \overline{e}^J_{\overline{\beta}}-\overline{g}^{\overline{\alpha}\overline{\beta}}\eta_{IJ}\overline{e}^I_{\overline{\beta}}\nu^\alpha\dd e_\alpha^J+2N_R\overline{g}^{\overline{\alpha}\overline{\beta}}\dd\overline{e}^R_{\overline{\beta}}-\overline{g}^{\overline{\alpha}\overline{\beta}}N_L\dd \overline{e}^L_{\overline{\beta}}+\overline{g}^{\overline{\alpha}\overline{\beta}}\nu^\sigma \eta_{LK}\overline{e}_{\overline{\beta}}^K \dd e^L_\sigma\\
&=0
\end{align*}
which proves that
\begin{equation}\overline{\theta}_{(m)}=\overline{\theta}^{(t)}+\jmath^*(\star_g\mathcal{U})
\end{equation}
Putting everything together, we finally obtain the main result of the paper
\begin{equation}\label{eq: (Theta,theta)=(Theta,theta)+d(A,a)}\Big(\Theta_{(m)},\overline{\theta}_{(m)}\Big)=\Big(\Theta^{(t)},\overline{\theta}^{(t)}\Big)+\underline{\d}\big(\!\star_g\mathcal{U},0\big)
\end{equation}
Hence, the symplectic potentials are equal up to a relative exact form. Notice that there is no problem in the fact that the LHS comes from the metric formalism while the RHS comes from the tetrad formalism because the LHS is implicitly evaluated at the $e$-dependent metric $\Phi(e):=\eta_{IJ}e^Ie^J$.\vspace*{2ex}

Taking now the $\dd$-exterior derivative of \eqref{eq: (Theta,theta)=(Theta,theta)+d(A,a)}, integrating over a Cauchy slice $(\Sigma,\partial\Sigma)$, and using the relative Stokes' theorem \eqref{eq: relative stokes}, we obtain the desired equality of the two presymplectic forms $\includegraphics[height=\alturaO]{./OOmega.pdf}_{(t)}$ and $\includegraphics[height=\alturaO]{./OOmega.pdf}_{(m)}$ (see appendix \ref{subsection: appendix mathematical background} for a brief account of the relative framework). A final comment is in order now: we say that the presymplectic structures are equivalent but of course we mean up to the gauge freedom in the tetrad formalism, coming from the $SO(1,3)$-invariance of $\Phi$, which has no counterpart in the metric formulation. This means in particular that $\includegraphics[height=\alturaO]{./OOmega.pdf}_{(t)}$ has more degenerate directions than $\includegraphics[height=\alturaO]{./OOmega.pdf}_{(m)}$. Finally, the equivalence of the charges \eqref{eq: xi-charge} and \eqref{eq: xi charge tetrad} is straightforward as both theories are equivalent, implying that $\overline{b}_{(m)}$ and $\overline{b}^{(t)}$ are equivalent as well.

\section{Conclusions}

In this paper, we have studied the metric and tetrad formulations of gravity on a manifold with boundary. By considering the appropriate bulk and boundary Lagrangians, we have shown that both theories are equivalent and hence, as one would expect, they are symplectically equivalent in the covariant phase space. Here we have focused on Dirichlet and Neumann boundary conditions but any other boundary conditions will give equivalent results as long as the metric and tetrad actions are in correspondence.

It has been known for some time that, in the absence of boundaries, the metric symplectic current $\Omega_{(m)}:=\dd\Theta_{(m)}$ is equal to the tetrad symplectic current $\Omega^{(t)}:=\dd\Theta^{(t)}$ up to an exact form $\d A$ (and thus cohomologically equal). Therefore, their presymplectic forms over the space of solutions are equivalent since the integral of $\d A$ over a Cauchy slice $\Sigma$ is zero according to Stokes' theorem. On the other hand, if one considers a space-time with boundaries, the boundary-free covariant phase space procedure fails. This is due to some ambiguities that arise in the construction of the presymplectic form that hinder the direct comparison between metric and tetrad formulations. This has caused some confusion in previous works where some \emph{ad hoc} solutions have been proposed in order to solve this discrepancy \cite{PaoliSpeziale2018,FGP1}. Other authors have also argued about the possible inequivalence of these formulations \cite{JacobsonMohd2015}. As we have shown, these difficulties arise because the traditional covariant phase space methods are only suited for the boundary-free case.\vspace*{2ex}

When boundaries are present, it is necessary to use more sophisticated techniques, like the relative bicomplex framework \cite{MargalefVillasenor2020}. Indeed, following the ideas of that formalism, we obtain the main result of the paper: the metric symplectic currents $(\Omega_{(m)},\overline{\omega}_{(m)}):=(\dd\Theta_{(m)},\dd\overline{\theta}_{(m)})$ and the tetrad symplectic currents  $(\Omega^{(t)},\overline{\omega}^{(t)}):=(\dd\Theta^{(t)},\dd\overline{\theta}^{(t)})$ are equal up to a \emph{relative} exact form $\underline{\d}(A,\overline{a})$. This implies, in particular, that they are equal in the relative cohomology (see appendix \ref{subsection: appendix mathematical background} for the relevant definitions) and that their presymplectic forms over the space of solutions are equivalent. Indeed, the relative Stokes' theorem tells us that the integral of $\underline{\d}(A,\overline{a})$ over a relative Cauchy slice $(\Sigma,\partial\Sigma)$ is zero. We have also checked that the Noether charges are equivalent, as one would expect for equivalent theories. Finally, we have applied the covariant phase space methods to the asymptotically flat case recovering the well-known formula for the ADM-energy.

\section*{Acknowledgments}
This work has been supported by the Spanish Ministerio de Ciencia Innovaci\'on y Uni\-ver\-si\-da\-des-Agencia Estatal de Investigaci\'on FIS2017-84440-C2-2-P grant. Juan Margalef-Bentabol is supported by the Eberly Research Funds of Penn State, by the NSF grant PHY-1806356 and by the Urania Stott fund of Pittsburgh foundation UN2017-92945. E.J.S. Villase\~nor is supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).

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Definition - What does Breakout Wrench mean?
A breakout wrench is a machine that clamps onto drilling pipe and breaks it loose at the joint of two pipes so that they can be separated. It can require tremendous force to perform this action. A breakout wrench is used during trenchless drilling operations to decouple pipes in the drill string.
Even without bolts and fasteners, coupled pipe can be difficult to separate. Anyone who has tried to separate two small joined pipes by hand knows that sometimes the bonding between the surfaces can become quite strong. In that case, a pipe wrench may be needed to grip the pipe and apply the force required to twist and remove it.
Breakout wrenches are used on drill rigs and horizontal directional drilling projects to perform a similar function. These machines may be automated or powered hydraulically. The specifications of a breakout wrench include size, torque, weight, and hydraulic capacity.


text:
January, 4th, 2023 posts
Cornwall Air Ambulance and Carbis Bay Estate announce two-year partnership
4th January 2023 By Sarah Yeoman
Cornwall Air Ambulance Trust has announced a new two-year partnership with Carbis Bay Estate.


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I personalize my teaching style for the student. I want people to actually learn Spanish and be able to use it/ understand it. If you are a visual learner or a by-the-ear learner, I can teach you what you want to know. I have spent the last 15 years of my life studying the Spanish language and culture. I spent my senior year of high school in Chile, and all of my classes were in Spanish. Since then I have worked as an interpreter in several lines of business.
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ESPAS Report 'Global Trends 2030 - Citizens in an Interconnected and Polycentric World'
Álvaro de Vasconcelos
EU policy formulation depends on the timely identification of long-term global trends. Strategic thinking is essential both to drive EU domestic policies and to give coherence and consistency to the EU's external action, bringing together common institutions and individual states. The EUISS, in its role as the European agency for strategic studies, was identified as the core institution of the pilot trend-monitoring European Strategy and Policy Analysis System (ESPAS). In leading the ESPAS project, the EUISS sought the close involvement of policy experts and regional representatives from around the world. The product of this pilot project was the ESPAS research report 'Global Trends 2030 - Citizens in an Interconnected and Polycentric World' which assesses the long-term political and economic environment facing the European Union over the next 20 years. By 2030, the demands and concerns of citizens in many different countries of the world are likely to converge, with a major impact on national politics and international relations. This will be the result mainly of greater awareness among the world's citizenry that their aspirations and grievances are shared. This awareness will create a global citizens' agenda that emphasises fundamental freedoms, economic and social rights and, increasingly, environmental issues. The future of this agenda will depend greatly on national and global governmental decisions. However, in a world characterised by the diffusion of power, meeting the challenges of human development will depend increasingly on non-state actors, be they private companies, civil society non-governmental organisations (NGOs), or philanthropic organisations. Some extremist non-state actors are likely to present a threat to the well-being of human communities.
By 2030, the information revolution will likely cover the entire planet. This revolution will empower citizens, including a large and growing global middle class. It can therefore be assumed that the citizens of 2030 will want a greater say in their future than previous generations. They are likely to be more aware of the threats they face together, and that they are part of a single human community in a highly interconnected world. But the emergence of a global public is also bound to generate increased expectations. This "gap" between what citizens want from their national governments and what these governments are able to deliver may become a source of friction, conflict and even revolt.
Global Governance 2025: at a critical juncture
The New Global Puzzle. What World for the EU in 2025?


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