# Synthetic methane for closing the carbon loop: Comparative study of three
carbon sources for remote carbon-neutral fuel synthetization

Michaël Fonder  Pierre Counotte  Victor Dachet  Jehan de Séjournet  Damien
Ernst

(September 2023)

###### Abstract

Achieving carbon neutrality is probably one of the most important challenges
of the 21st century for our societies. Part of the solution to this challenge
is to leverage renewable energies. However, these energy sources are often
located far away from places that need the energy, and their availability is
intermittent, which makes them challenging to work with. In this paper, we
build upon the concept of Remote Renewable Energy Hubs (RREHs), which are hubs
located at remote places with abundant renewable energy sources whose purpose
is to produce carbon-neutral synthetic fuels. More precisely, we model and
study the Energy Supply Chain (ESC) that would be required to provide a
constant source of carbon-neutral synthetic methane, also called e-NG, in
Belgium from an RREH located in Morocco. To be carbon neutral, a synthetic
fuel has to be produced from existing carbon dioxide (CO2) that needs to be
captured using either Direct Air Capture (DAC) or Post Combustion Carbon
Capture (PCCC). In this work, we detail the impact of three different carbon
sourcing configurations on the price of the delivered e-NG. Our results show
that PCCC is the most competitive from a cost perspective, and that e-NG could
be delivered to Belgium at a price of 136€/MWh when using existing CO2 sources
in Morocco. For comparison, this price is less than half of the one of regular
fossil natural gas delivered in Belgium at the peak of the 2022 energy crisis
in Europe.

[uliege]organization=University of Liège, city=Liège, country=Belgium
[tes]organization=Tree Energy Solutions, city=Zaventem, country=Belgium
[tpip]organization=Telecom Paris, Institut Polytechnique de Paris, city=Paris,
country=France

##  1 Introduction

Global warming is a climate change that is due to large anthropogenic
emissions of greenhouse gases, mainly carbon dioxide (CO2), in Earth’s
atmosphere. As its effects are detrimental for our societies and ecosystems in
general, it has appeared necessary to reduce, and even cancel, the emission of
these gases to minimise these effects. Since most of the CO2 is emitted by
burning fossil fuels to generate energy, transitioning our energy production
out of these fuels is mandatory.

The main alternatives to fossil fuels are renewable energies such as wind,
solar or hydropower, which can be harvested to produce electricity. However,
most renewable energy sources are intermittent, and the best sources are
located far away from places where this energy is the most needed [1]. As
electrical energy is challenging to transport over long distance and to store
in large quantities with current technology, innovations are needed to bridge
the gap between energy production and consumption locations.

One such innovation is the concept of Remote Renewable Energy Hub (RREH) for
carbon-neutral fuel synthesis that was first laid out by Hashimoto et al. [2].
The idea underlying RREH s is to install power-to-X facilities, that convert
electrical energy into chemical energy, at remote locations where renewable
energy sources are the most abundant [3, 4], and to transport the converted
energy to places where it is needed. A power-to-X facility converts the
electrical energy into chemical energy by synthesizing energy-dense molecules
[5], which are easier to store and transport back to the energy consumption
locations than electricity. The molecules typically considered for this kind
of application are dihydrogen, ammonia, or methane, each having its own
advantages and drawbacks [6].

Among these three molecules, synthetic methane, also called electric Natural
Gas (e-NG), holds a particular place. Indeed, it can be used as a simple drop-
in replacement within existing energy-demanding installations while decreasing
GHG emissions when produced from renewable energy sources, as shown in several
case studies [7, 8, 9]. More generally, existing life-cycle analysis of
methanation indicate that the Global Warming Potential (GWP) of e-NG produced
with renewable energy sources is smaller than fossil natural gas [10, 11].
Benefits of e-NG on GWP are even larger if the methanation is done through
catalysts through captured CO2 rather than from biomass [12, 13].

To be considered as carbon-neutral, catalyst-based e-NG has to be generated
from existing CO2, which must be actively captured prior to being used for
methanation. Capturing CO2 can be done either by filtering it out of ambient
air thanks to Direct Air Capture (DAC) technologies [14, 15] or by extracting
it directly at the source of emission, in industrial fumes, where it is the
most concentrated, by Post Combustion Carbon Capture (PCCC) [16, 17]. Fossil
fuel power plants, or cement and steel factories are examples of industries
that emit sufficiently large amounts of CO2 to be considered for PCCC. The
high concentration of CO2 in industrial fumes leads PCCC to be less expensive
than DAC for extracting the same amount of CO2 [18]. However, capturing CO2 by
PCCC needs to be done at the factory that uses the synthesized fuel. The CO2
then needs to be transported where needed, which induces additional costs.
This contrasts with DAC that can be performed anywhere on the planet directly
where needed, which eliminates the transportation costs. As changing the
source of CO2 impacts the price of the final commodity produced by an RREH, it
is necessary to know which CO2 source and capture method combination is the
most energy-efficient and/or the most cost-effective for a given RREH.

Multiple works have studied Power to Gas (PtG) RREH s in different
configurations. For example, Berger et al. [19] studied an RREH located in
Algeria designed to produce e-NG to be delivered in Belgium. Dachet et al.
[18] built on this work by adding a hub in Greenland and by studying the
impact of carbon pricing on the sizing of the system. Hampp et al. [1]
performed an extensive study on the opportunity to build RREH s at different
places on Earth to deliver energy to Germany. As opposed to the works of
Berger et al. [19] and Dachet et al. [18] that focus on e-NG, Hampp et al. [1]
consider and compare multiple carriers for energy. However, all these works
consider only a single configuration for sourcing the CO2 required for the
methanation process in their study. Therefore, it is impossible to know if the
CO2 sourcing chosen in each of these works is the most effective one.

This paper aims to compare the impact of changing the CO2 sourcing used to
close the carbon loop to the cost of the generated e-NG, and to the sizing of
the RREH, which has not been done before. More specifically, we study the
impact of three different configurations of CO2 sources and capture methods on
the cost and the sizing of an RREH located in Morocco and designed to produce
e-NG to be delivered in Belgium. We detail the exact scope and configurations
considered in our problem statement, in the next section. We detail our model
and methodology in Section 3, and analyse our results in Section 4. Section 5
concludes this paper.

The main contributions of this work are as follows:

  * 1.

We model the whole Energy Supply Chain (ESC) needed to deliver e-NG, produced
from salt water and renewable energy in Morocco, to Belgium;

  * 2.

We consider three different ways of sourcing the CO2 needed for the
methanation in the RREH;

  * 3.

We provide a detailed analysis of the advantages and drawbacks of each of
these sources.

##  2 Problem statement

In this section, we detail the scope of this work. We consider an RREH located
in Morocco whose purpose is to produce e-NG for the Belgian market. The RREH
has to produce e-NG from renewable energy sources (for electricity), from salt
water and from captured CO2. The e-NG produced by the hub has to be delivered
to Belgium by LNG carriers.

The choice of Morocco for the location of the hub is first motivated by the
quality of its renewable energy sources [20]. Second, an RREH requires a
significant land area to be deployed, especially for collecting renewable
energies. Morocco is also interesting in this regard, since the southern half
of the county is mostly a desert with an extremely low existing land use.

We consider three different configurations for capturing the CO2 required for
the methanation process in the RREH, namely: (i) DAC on site, in Morocco; (ii)
PCCC from Moroccan coal-fired power plants; and (iii) PCCC from e-NG use in
Belgium, with capture losses being compensated by DAC on site, in Morocco.

The purpose of this work is to model the ESC that corresponds to each of these
configurations, starting from renewable electricity and salt water in Morocco
up to the delivery of e-NG in Belgium, and to provide an insight on the impact
each configuration has on the price of the delivered e-NG and on the sizing of
the different elements making the ESC.

The ESC necessary to deliver e-NG to Belgium will be designed to be fully
autonomous and auto sufficient. To be comparable to previous works [19, 18],
the system will be designed to deliver 10TWh of e-NG uniformly over a year in
Belgium. The study will be performed with a Weighted Average Cost of Capita
(WACC) set to 7% , and with extra contingency costs added to all Capital
Expenditures (CAPEX) to factor in for feasibility uncertainties [21]. More
precisely, contingency costs of 10% and 30% are to be added to the CAPEX of
mature and unproven technologies respectively.

##  3 Material and method

In this section, we present the method used to carry the study described in
the previous section. We first detail the framework used to model and size the
desired ESC. We then fix the geographical parameters of our study. Finally, we
provide a detailed insight on the setup of the models used to analyse the
three CO2 sourcing configurations considered alongside the parameters used for
each part of the model.

###  3.1 Modelling framework

This study is built on the work of Berger et al. [19] which introduced the use
of a Graph-Based Optimization Modeling Language (GBOML) as an optimization
framework for multi-energy system models. The GBOML language [22] allows one
to model each part of a complex system as a set of nodes interconnected by
hyperedges that model the constraints existing between these nodes. In the
case of an RREH each node models a specific module of the hub, and each
hyperedge models the flow of a given commodity within the hub.

In the GBOML language, the nodes are, in part, modelled by a set of variables
that need to be tuned to minimize the objective function of the model while
verifying a set of linear constraints specific to each node. For this, each
node has to provide an objective to minimize that is a linear function of its
variables. The objective function of the whole model is then defined as the
sum of all the objectives of its nodes. In this study, we want to minimize the
cost of the e-NG delivered in Belgium, and therefore the cost of the hub. As a
result, our objective functions will be proportional to the CAPEX and
Operational Expenditures (OPEX) of the different modules of the hub.

Since the lifetime of each module is specific, raw CAPEX are not
representative of real costs when studying the cost of the hub over a limited
time horizon. One method to address this concern, which we employ in this
study, involves the use of annualized CAPEX. The annualized CAPEX $\zeta_{m}$
of a module $m$ can be computed from the raw CAPEX, from the life-time $L_{m}$
of the module and from the WACC $w$ as follows:

$\zeta_{m}=\text{\lx@glossaries@gls@link{acronym}{CAPEX}{\leavevmode{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}CAPEX}}}_{m}\times\frac{w}{1-(1+w)^{-L_{m}}}$
(1)

Given that the optimization problem of interest has already been
comprehensively formalized by Berger et al. [19], we shall only remind ourself
of the set of assumptions made by prior studies to model and size an RREH
within this framework:

  * 1.

All technologies and their interactions are modelled using linear equations;

  * 2.

The infrastructures and networks needed to transport and process commodities
within a single node are not modelled;

  * 3.

Curves modelling the boundaries of the model, such as the renewable energy
load factors and the energy demand, are assumed to be known in advance for the
whole optimization time horizon;

  * 4.

The sizing of the modules is assumed to be constant over the whole time
horizon considered;

  * 5.

Sizing the whole hub with an optimization framework assumes that all planning,
investment and operation decisions are made by a single entity.

###  3.2 Geographical parameters

Figure 1: Geographical location of the different parts of the ESC studied in
this work.

One of the aspects that heavily impact any study of an RREH is the location of
its different components. As mentioned in the problem statement, this study
focuses on an RREH located in Morocco with a e-NG terminal in Belgium. In this
part of the document we detail the choice of location of all the parts of the
hub and the constraints that result from this choice.

The precise location of the different modules of the hub is mapped in Fig. 1.
The delivery terminal is set to be in the Belgian harbour of Zeebrugge as it
already features LNG terminals. We build our study on a renewable energy
capture hub located on the Atlantic coat in the Western Sahara desert. We
consider this location because it has excellent wind and solar energy sources,
and because it is mostly empty, which leaves a lot of room to deploy large
fields of windmills and solar PV panels. Finally, we set our e-NG production
hub to be near the Moroccan city of Safi. By doing so, the e-NG production hub
is located near a pool of available workforce, which is required to run the
hub, near the coast, which is needed to export the produced e-NG by carriers,
and near a large coal-fired power plant [23], which can provide a valuable
source of CO2 for PCCC. It is worth noting that the Safi power plant is sized
to deliver 10TWh of electricity over a year [23], and is therefore a source of
CO2 large enough to feed the methanation process of the hub studied in this
work. In addition, the area of Safi has a lot of available space for large
industrial projects, which is a requirement due to the sheer scale of the hub
considered.

With such a setup, the power and e-NG production hubs are separated by 650 km,
and need to be connected by a High Voltage DC (HVDC) line. Since the power
production hub is also on the coast, the HVDC line can be installed offshore
to get the shortest connection distance. The e-NG production hub and the LNG
terminal in Belgium are separated by 2600 km, and require connection by LNG
carriers. We estimate that 100 hours are required to connect Safi to Zeebrugge
by carrier.

Figure 2: Illustration of three major modules that are part of our ESC and
that are common for all CO2 sourcing configurations. The LNG hub features the
infrastructures necessary to receive liquefied e-NG from carriers for the end
delivery. The module will be installed in Belgium. The power production hub
groups all the elements required to provide renewable power coming from
energies to other elements of the ESC. Finally, the e-NG production hub takes
power and CO2 as inputs and produces liquid e-NG, water, hydrogen, and heat as
outputs. These outputs can be consumed by other elements of the ESC, such as
DAC units.

###  3.3 Model

Despite being close to the works of Berger et al. [19] and Dachet et al. [18],
the RREH model used in this study features some differences. In this section,
we first explain the fundamental principles of the hub, and then highlight the
key differences with existing works. We would like to emphasize that we open-
sourced the code used for the hub to provide all the details required to
reproduce this
work111https://gitlab.uliege.be/smart_grids/public/gboml/-/tree/master/examples/synthetic_methane_morocco.

####  3.3.1 Hub principle

As detailed in the previous section, the whole ESC necessary to deliver e-NG,
synthesized from renewable energy sources in Morocco, in Belgium is made up of
the three main hubs illustrated in Fig. 2. First, the power production hub is
made of three nodes; the solar PV panel farm, the windmill farm, and a pack of
batteries used to smooth out variations inherent to renewable energy sources.
The power production hub is used to deliver the electricity needed to capture
CO2 in Morocco, and to synthesize e-NG. The power profiles used to model the
renewable energy production over the optimization time horizon were obtained
from the renewable.ninja website [24, 25, 26].

Second, the e-NG production hub, which takes electricity and a source of CO2
as inputs, is responsible for synthesizing e-NG from salt water and CO2. This
hub gets the clean water required for electrolysis from a reverse osmosis
water desalination module. This fresh water then passes through an
electrolysis module to produce the hydrogen needed by the hub. The core of
this hub, the methanation unit, consumes both hydrogen and CO2 to synthesize
e-NG with the Sabatier reaction. This node has two byproducts that can be used
at other places in the ESC: fresh water and low-grade steam heated to 300°C.
Lastly, the synthesized e-NG is liquefied for transportation. In addition,
water, hydrogen, and liquefied e-NG storage nodes are added to buffer flow
variations of each commodity.

Third, the LNG hub in Belgium is simply made of two nodes: one liquefied e-NG
storage and one e-NG regasification node to deliver the e-NG in a gaseous
form.

The model of the complete ESC necessary to produce e-NG from CO2 capture by
DAC units in Morocco is illustrated in Fig. 3. The one for CO2 sourced from
coal-fired power plants PCCC is given in Fig. 4, while Fig. 5 illustrates the
model of the CO2 sourcing configuration implying PCCC in Belgium and DAC in
Morocco. In all of these models, CO2 transits as a gas, excepted for storage
and carrier transportation where it needs to be liquefied.

Figure 3: Illustration of the hub design used for our first CO2 sourcing
configuration. The CO2 is captured by DAC units at the place where the
methanation is performed. Figure 4: Illustration of the hub design used for
our second CO2 sourcing configuration. The CO2 is captured by a PCCC unit
placed at a local coal-fired power plant, in Morocco. The CO2 is transferred
to the methanation plant by pipeline. Figure 5: Illustration of the hub design
used for our third CO2 sourcing configuration. The CO2 is captured by PCCC at
places where e-NG is used in Belgium. This carbon is transferred to the
methanation hub by liquefied CO2 carriers. The electricity required for the
PCCC units is provided by a small renewable energy power hub. Since PCCC
cannot capture all the emitted CO2, the missing CO2 required for methanation
is captured by DAC units on site, in Morocco.

####  3.3.2 Our adaptations

Although we use the same nodes with the same parameters as Berger et al. [19]
and Dachet et al. [18] for most nodes of the model, we need to adapt some
parameters of the reference model for our study. Since Berger et al. [19]
already detail all the parameters of their model, we only focus on our
adaptations to these parameters hereafter.

As required by our problem statement, we add a contingency cost to the CAPEX
of all nodes. This contingency cost, which was not considered in previous
works, is equal to 10% of the CAPEX for well-established technologies, and 30%
of the CAPEX for less mature technologies. The technologies that we consider
to be less mature regarding the scale of the RREH are the following: large-
scale battery packs, HVDC lines, electrolysis units, methanation units, and
CO2 capture technologies.

Modifications are also carried out to the methanation node when compared to
the original node proposed by Berger et al. [19]. First, we align our CAPEX
with the $300\text{k\texteuro/}(\text{MWH CH${}_{4}$/h})$ proposed by Gorre et
al. [27]. Second, we add an output for the residual heat produced by the
Sabatier reaction as it can also be valued. Indeed, Coppitters et al. [28]
showed that this heat can be recycled in the hub to lower the heat energy
required by solid sorbent DAC units. According to their work, the Sabatier
reaction produces 2.1MWh of usable heat energy per ton of synthesized CH4.

Using a heat source of 300°C to regenerate a sorbent is only possible for
solid sorbents. Indeed, solid sorbents only require to be heated to 100°C to
regenerate [28], whereas liquid sorbents need to be heated to 900°C [19].
Therefore, as opposed to preceding works [19, 18], we use solid sorbent DAC
units in our model, and rely on the data of the Danish Energy Agency [29] for
their specifications. In addition to using recirculated heat, the system has
the ability to burn hydrogen to provide heat for the sorbent regeneration.

A last minor change of in our model when compared to previous works is the
cost of liquefaction, storage and regasification of CO2. The source used in
previous works [30] dates from 2004, which is significantly older than most of
our sources. To uniformize cost estimates, we adjusted the estimates of
Mitsubishi Heavy Industries, LTD [30] by taking inflation between 2004 and
2021 into account.

ECS Elements |  CH4 Production Free CO2 (€/MWh CH4) |  CO2 Sourcing DAC MA (€/MWh CH4) |  CO2 Sourcing PCCC MA (€/MWh CH4) |  CO2 Sourcing PCCC BE + DAC MA (€/MWh CH4)
---|---|---|---|---
Solar PV Field | 10.43 | 1.23 | 0.37 | 0.11
Windmill farm | 43.00 | 4.02 | 1.9 | 0.20
Batteries | 4.06 | 1.49 | -0.05 | 0.33
HVDC | 11.86 | 1.12 | 0.50 | 0.06
CO2 capt. + transport. | 0.00 | 21.61 | 8.39 | 20.09
Desalination plant | 0.11 | 0.66 | 0.00 | 0.16
Water storage | 0.05 | 0.00 | 0.00 | 0.00
Electrolysis plant | 35.42 | 2.89 | 0.29 | 0.05
H2 storage | 4.55 | 0.44 | 0.17 | 0.02
Methanation plant | 7.75 | 0.00 | 0.00 | 0.01
CH4 liquefaction | 5.09 | 0.00 | 0.00 | 0.01
CH4 storage (Morocco) | 0.23 | 0.00 | 0.00 | 0.00
CH4 carriers | 0.67 | 0.00 | 0.00 | 0.00
CH4 storage (Belgium) | 0.23 | 0.00 | 0.00 | 0.00
CH4 regasification | 1.02 | 0.00 | 0.00 | 0.00
Total cost | 124.47 | 33.46 | 11.57 | 21.04
CH4 cost | 124.47 | 157.93 | 136.04 | 145.51
Table 1: Breakdown of the cost for delivering 1MWh of CH4 in Belgium depending
on the source of CO2 used for the methanation process. The cost given for the
”CO2 capt. + transport.” line encompasses all the infrastructures required to
capture and transport the CO2 back to the e-NG production hub. This line is
detailed in Table 2 for each CO2 sourcing configuration individually. The
other costs induced by a CO2 sourcing, which listed in the table, detail the
costs of feeding the appropriate commodities (electricity, water,…) to a given
CO2 sourcing configuration with the hub.

##  4 Results

With the study and the model being defined, we can analyse the results of the
optimization process. In this section, we first analyse the three CO2 sourcing
configurations from a cost perspective. We then analyze them from an energy
perspective. Finally, we discuss the results obtained in this work.

###  4.1 Costs analysis

The cost for delivering CH4 to Belgium for our three CO2 sourcing
configurations are detailed in Tables 1 and 2. The price for the commodities
produced by the hub is given in Table 3. Producing and delivering 1MWh of CH4
costs 124.47€ without factoring in costs related to CO2 sourcing. The largest
part of this cost is due to the renewable energy capture (53.43€/MWh CH4),
electricity transportation (11.86€/MWh CH4), and water electrolysis
(35.42€/MWh CH4).

Capturing and transporting CO2 to the e-NG production hub creates additional
costs that range from 11.57€/MWh CH4 for the PCCC in Morocco to 33.46€/MWh CH4
for DAC on site, which is the most expensive way to source CO2 to the hub.
Therefore, delivering 1MWh of e-NG in Belgium costs a total of 136.04€ when
sourcingCO2 from PCCC in Morocco, 145.51€ when sourcing CO2 from PCCC in
Belgium, and 157.93€ when sourcing CO2 from DAC on site. For comparison,
regular fossil natural gas has been exchanged at prices well above these costs
for several months during the 2022 energy crisis in Europe (up to 342€/MWh in
August 2022).

CO2 coming from DAC on site seems to also be more expensive than PCCC CO2
imported from Belgium. As a result, the methanation uses as much CO2 from
Belgium as possible in the configuration where CO2 can be sourced either from
DAC on site or from PCCC in Belgium. This leads the fraction of CO2 coming
from Belgium used for methanation to be equal to the efficiency of the PCCC
process (90% in our model).

In order to be exhaustive, we tested an alternate model where CH4 and CO2 are
transported through offshore pipelines for the configuration where part of the
CO2 comes from PCCC in Belgium. In this case, the price of e-NG delivered in
Belgium rises to 151.24€/MWh, which is more expensive than transportation by
carriers.

DAC MA CO2 sourcing elements |  Cost (€/Mwh)
---|---
DAC (Morocco) | 21.61
CO2 storage (Morocco) | 0.0
Total cost | 21.61
(a) DAC in Morocco

PCCC MA CO2 sourcing elements |  Cost (€/Mwh)
---|---
PCCC (Morocco) | 8.08
CO2 storage (Morocco) | 0.3
CO2 pipe | 0.01
Total cost | 8.39
(b) PCCC from coal-fired power plants in Morocco

PCCC BE + DAC MA CO2 sourcing elements |  Cost (€/Mwh)
---|---
Solar PV field (Belgium) | 1.12
Windmill farm (Belgium) | 2.36
Batteries (Belgium) | 1.16
HVDC (Belgium) | 0.24
PCCC (Belgium) | 7.07
CO2 liquefaction (Belgium) | 0.14
CO2 storage (Belgium) | 0.44
CO2 carriers | 1.77
DAC (Morocco) | 5.28
CO2 storage (Morocco) | 0.45
CO2 regasification (Morocco) | 0.06
Total cost | 20.09
(c) PCCC in Belgium completed by DAC in Morocco

Table 2: Breakdown of the cost for sourcing CO2 to the methanation process for our three different sourcing configurations. The electricity required for the PCCC in Belgium is provided by a small renewable energy hub installed in Belgium. Commodity | Cost [€]
---|---
Hydrogen [t] | 3210.69
Water [t] | 1.17
CO2\- DAC MA [t] | 256.28
CO2\- PCCC MA [t] | 72.28
CO2\- PCCC BE + DAC MA [t] | 136.62
Wind power - MA [MWh] | 30.19
Solar power - MA [MWh] | 22.89
Wind power - BE [MWh] | 65.47
Solar power - BE [MWh] | 58.51
Table 3: Cost of the commodities produced by our ESC model. Due to
limitations, the price given for CO2 sources using DAC correspond to the upper
bound price that happens when all the heating energy comes from hydrogen. The
production costs for power, water, and hydrogen are independent of the CO2
sourcing configuration.

###  4.2 Energy analysis

CO2 Sourcing Configuration |  ESC\- Global Efficiency [%] |  Wind - Fraction of Curtailment [%] |  PV - Fraction of Curtailment [%] |  DAC - Part of Recycled Heat[%]
---|---|---|---|---
DAC MA | 48.44 | 25.17 | 0.0 | 65.41
PCCC MA | 51.10 | 25.17 | 0.0 | N/A
DAC MA + PCCC BE | 51.26 | 25.14 | 0.0 | 99.99
Table 4: Insight of some key figures related to energy within our model. The
first column gives the ratio between the energy delivered in Belgium in the
form of CH4 and the captured renewable energy. The second column gives the
fraction of available wind energy that was not used by the model. The third
column gives the fraction of available solar energy that was not used by the
model. The last column gives the fraction of heat energy required by DAC that
comes from recirculated heat rather than from hydrogen.

Table 4 gives some key statistics related to energy within our model. This
table shows that the efficiency of the hub, that is the ratio between the
energy contained in the e-NG delivered in Belgium and the energy captured by
windmills and solar panels, lies around 50% with small variations depending on
the CO2 sourcing configuration. The highest efficiency, 51.26%, is obtained by
the CO2 sourcing that combines PCCC in Belgium and DAC in Morocco and beats
the efficiency of simple PCCC in Morocco. This observation, which may be
surprising at first glance, can be explained by the use of heat recirculation.
As shown in the last column of the table, a large part of the heat required
for DAC can be provided by the Sabatier reaction. In the case of joint PCCC
and DAC, the Sabatier reaction can even provide all the heat required for DAC,
which lowers the energy input required for carbon capture when compared to
PCCC alone. However, it is worth noting that the CO2 sourcing involving PCCC
in Morocco has residual heat that is not used in our model, but that could be
valued for other applications, which would virtually increase the efficiency
of the hub.

The Table 4 also shows the fraction of available renewable energy that is not
used by the hub. For windmills, the energy curtailment rises to 25%, while all
the available solar energy is used. This observation is identical for all CO2
sourcing configurations. We believe that the discrepancy between the
curtailments of solar and wind power can be explained by a difference in
regularity of the energy sources. Indeed, the available solar power is pretty
regular from a day to the other whereas wind power fluctuates more.

In addition to these insights, we computed the load factor for different parts
of the system. Windmills and PV panels have a load factor of 41.4% and 25.5%
respectively. Electrolysis units have a load factor of about 81%, while
desalination and methanation units are used at full capacity all the time by
design. These load factors are similar for all CO2 sourcing configurations.

Finally, Table 5 shows the capacity that needs to be installed for delivering
an average power of 1MW of e-NG to Belgium for various modules of the hub.
Since the model is fully linear, these capacities scale linearly with the
desired power of e-NG to be delivered. It is interesting to note that the
biggest consumers of fresh water are by farDAC units according to numbers
given in this table.

ECS Elements |  CH4 Production Free CO2 [/MW CH4 ] |  CO2 Sourcing DAC MA [/MW CH4 ] |  CO2 Sourcing PCCC MA [/MW CH4 ] |  CO2 Sourcing PCCC BE + DAC MA [/MW CH4 ]
---|---|---|---|---
PV Capacity [MW] | 2.1389 | 0.2514 | 0.0769 | 0.0226
Windmill capacity [MW] | 3.4660 | 0.3238 | 0.1528 | 0.0162
Battery capacity [MWh] | 1.1266 | 0.4182 | -0.0188 | 0.0909
Battery throughput [MW] | 0.1469 | 0.0469 | 0.0007 | 0.0110
HVDC line capacity [MW] | 2.3105 | 0.2178 | 0.0983 | 0.0110
Desalination units [MW] | 0.1508 | 0.9468 | 0.0000 | 0.2257
Electrolysis units [MW] | 2.1717 | 0.1774 | 0.0179 | 0.0029
Table 5: Capacity of several modules of the hub required to deliver an average
power of 1MW of e-NG in Belgium.

###  4.3 Results discussion

All the results presented in this section are directly linked to our
hypothesis and to our model. Our observations and analysis may be slightly
different when considering some additional factors. In this section, we
discuss how some factors that could impact the results presented in this work.

First, our model does not factor in for the economies of scale that could come
with the deployment of a large RREH. Indeed, most of the current cost
estimated are projections based on existing prototypes or units that are of
small scale or still at experimental stage. Therefore, it seems reasonable to
assume that the overall cost of modules would fall if manufactured in large
quantities.

Second, the efficiency of PCCC can vary depending on the industry on which CO2
capture is performed [8]. As little data are available on the type and
capacity of industries that may use the delivered e-NG in Belgium, we had to
make an educated assumption on the efficiency of PCCC. Real-world deployments
may lead to efficiencies different to the assumptions made in this work, and
could change our observations. Indeed, if the efficiency of PCCC drops to
steeply, costs associated to PCCC may rise to a point where the balance
between PCCC in Belgium and DAC in Morocco has a different optimum.

Third, we modelled the hub to be self-sufficient. As shown in the previous
section, this implies to synthesize hydrogen to provide heat to DAC units when
the heat resulting from the Sabatier reaction is not sufficient, which
increases the cost of the hub. However, the low-grade heat required to
regenerate solid sorbents in DAC units is a common industrial byproduct that
is often considered as a waste due to a lack usecase, but that can find a use
within the hub. As a result, importing heat from a neighbouring industry could
make the price of the configuration relying on DAC units more competitive
price-wise when compared to other configurations of CO2 sourcing.

Last, the ability of DAC units to provide a CO2 source that does not need a
dedicated supply chain is a practical advantage over PCCC that does not appear
when simply looking at numbers. A simpler supply chain for the production of
e-NG is also probably a more robust one as fewer elements come into play. This
could lead to a higher reliability of the whole supply chain, which has an
economic value that is not taken into account in our model.

##  5 Conclusion

In this paper, we consider the sizing and the cost of an ESC designed to
synthesize e-NG from renewable energy sources in an RREH in Morocco and to
deliver it in Belgium. Synthesizing e-NG requires a source of CO2. In this
work, we considered three different configurations for sourcing the required
CO2 to the RREH, and studied their impact on the sizing and the cost of the
RREH, and by extension on the cost of the e-NG delivered in Belgium. The three
different configurations considered for capturing the CO2 are (i) DAC on site,
in Morocco; (ii) PCCC from Moroccan coal-fired power plants; and (iii) PCCC
from e-NG use in Belgium, with capture losses being compensated by DAC on
site, in Morocco.

We modelled and optimized the ESC corresponding to the three configurations
using the GBOML framework. Results show that DAC is more expensive than PCCC
generally speaking. This is in part due to the heat required to regenerate the
sorbent of DAC units. Indeed, the CO2 sourcing relying only on DAC units leads
to a price of 157.93€/MWh of e-NG delivered in Belgium, while the
configuration relying on PCCC from Moroccan coal-fired power plants lead to of
price of 136.04€/MWh. The hybrid configuration, which implies PCCC in Belgium
and DAC in Morocco, leads to an intermediate price of 145.51€/MWh of e-NG
delivered in Belgium. The price for CO2 induced by this configuration is still
lower than the one of CO2 captured by DAC on site despite the additional costs
induced by the transportation of CO2 between Belgium and Morocco. Overall, the
cost for delivering e-NG in Belgium with the configurations presented in this
work seems reasonable since regular fossil natural gas has already been sold
for higher prices during the 2022 energy crisis in Europe.

## Glossary

## Acronyms

CAPEX

    Capital Expenditures
CO2

    carbon dioxide
DAC

    Direct Air Capture
e-NG

    electric Natural Gas
ESC

    Energy Supply Chain
GWP

    Global Warming Potential
HVDC

    High Voltage DC
OPEX

    Operational Expenditures
PCCC

    Post Combustion Carbon Capture
PtG

    Power to Gas
RREH

    Remote Renewable Energy Hub
WACC

    Weighted Average Cost of Capita

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# Spherically Symmetric Configurations in Unimodular Gravity

Júlio C. Fabris<EMAIL_ADDRESS>Núcleo Cosmo-ufes & Departamento
de Física, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514,
Goiabeiras, 29060-900, Vitória, ES, Brazil  National Research Nuclear
University MEPhI (Moscow Engineering Physics Institute), 115409, Kashirskoe
shosse 31, Moscow, Russia.  Mahamadou Hamani Daouda<EMAIL_ADDRESS>Département de Physique - Université de Niamey, Niamey, Niger.  Hermano Velten
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de
Ouro Preto (UFOP), Campus Universitário Morro do Cruzeiro, 35.400-000, Ouro
Preto, Brazil

###### Abstract

Unimodular gravity (UG) is considered, under many aspects, equivalent to
General Relativity (GR), even if the theory is invariant under a more
restricted diffeomorphic class of transformations. We discuss the conditions
for the equivalence between the two formulations by applying the UG to the
static and spherically symmetric configurations being the energy-momentum
tensor sourced by a scalar field or by the electromagnetic field. We argue
that the equivalence between UG and GR may be broken when analyzing the
stability of the solutions at perturbative level.

Unimodular gravity, black holes, stability

##  I Introduction

General Relativity (GR) is the modern theory of gravitational interaction. The
gravitational phenomena is considered as the structure of the space-time
itself induced dynamically by matter. GR is considered a very successful
theory: all local tests confirms the predictions of GR within high precision.
At cosmological scales, it leads to the Standard Cosmological Model (SCM)
which also addresses consistently all available observations, from scales of
galaxies up to the larger structures of the universe. It accounts also
successfully to the different phases of the evolution of the universe,
including the primordial phases at least to the primordial nucleosynthesis
scales. See from this point of view, the SCM, based on GR, is an almost
perfect model to describe the entire evolution of the universe.

However, seen from a different perspective, the SCM is at least problematic.
To account for the observations at the different scales, it demands the
introduction of two until now undetected components in the matter/energy
content of the universe. The dynamics of galaxies and cluster of galaxies, and
even the formation of such structures, asks for an additional pressureless
component, dubbed dark matter, which manifests only indirectly. Moreover, to
explain the present accelerated phase of the universe, the CMB spectrum and to
obtain an age of the universe consistent with the age of globular clusters,
the SCM asks for another component, with negative pressure, which does not
agglomerate, dubbed dark energy.

Dark energy is now frequently associated to the vacuum energy as predicted by
quantum field theory. However, its observed value seems not consistent with
the theoretical predictions by dozens of order of magnitude [1, 2, 3]. There
are many proposals to cope with this problem. One of them is to replace it by
a self-interacting scalar field, called quintessence [4]. However, it must be
explained, in the quintessence program, why the vacuum energy must be exactly
or, at least, nearly zero. Therefore, the vacuum energy must, somehow,
degravitate [5, 6, 7]. There are many mechanisms to implement such
degravitating mechanism, but until now such proposals are, in some sense, in
construction. For a general overview of the dark energy problem, see Ref. [8].

One interesting approach to the cosmological constant problem described above
is through the unimodular gravity (UG) class of theories [9, 10, 11, 12] where
the determinant of the metric, $g$, is fixed. Originally, $g=1$, but other
possibilities can be explored, see next section. UG leads to traceless
gravitational equations. The energy-momentum tensor does not conserve
necessarily anymore, since UG is not invariant by the full diffeomorphism
group, but by a more restricted structure called transverse diffeomorphism
[13]. If the conservation of the energy-momentum tensor is imposed, GR is
recovered with a cosmological term that appears as an integration constant. If
the conservation of the energy-momentum tensor is not imposed, as we will
prove below, a class of dynamical vacuum theories is obtained, implying an
interaction of the matter sector with the decaying cosmological term.

In previous works we have explored the distinction between GR and UG mainly at
perturbative level in the cosmological context, see [14] and references
therein (see also [15]). Here we will extend such studies to the static,
spherically symmetric configurations. In UG, with the imposition of the
conservation of the energy-momentum tensor, the static, spherically symmetric
solutions are identical to those of GR, but now containing a cosmological
constant. The non-conservation of the energy-momentum tensor, on the other
hand, can be mapped in the GR structure with a dynamical cosmological term.
Indeed, the non-conservation of the energy-momentum tensor is allowed in this
context, leading to a new formulation of the UG theory. It is also worth
mentioning that such a non-conservation mechanism appears in many other
situations. For a review, see Ref. [16]. All these aspects are discussed in
the next section. In section III the general equations for a static,
spherically symmetric configuration are settled out. Some examples of
interacting models, resulting from the non-conservation of the energy-momentum
tensor, will be shown in section IV, both in presence of an electromagnetic
field as well as of a self-interacting scalar field in section V. For the
latter case, we perform, in section VI, a perturbative analysis aiming to show
how the usual results of GR change in the unimodular context. In particular,
the unimodular condition on the determinant of the metric implies in vanishing
perturbations at linear level. The results obtained are discussed in section
VII.

##  II Field equations

The Einstein-Hilbert action, in presence of the cosmological term and the
matter Lagrangian,

$\displaystyle{\cal S}=\int d^{4}x\sqrt{-g}\biggr{\\{}\frac{R}{16\pi
G}+2\Lambda+{\cal L}_{m}\biggl{\\}},$ (1)

implies in the following field equations:

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi
GT_{\mu\nu}+g_{\mu\nu}\Lambda.$ (2)

The application of the Bianchi identities leads to the energy-momentum tensor
$T^{\mu\nu}$ conservation:

$\displaystyle{T^{\mu\nu}}_{;\mu}=0.$ (3)

The conservation laws related to the energy-momentum tensor can be
alternatively deduced from the invariance of the Einstein-Hilbert Lagrangian
by diffeomorphic transformations [17].

In order to obtain the UG equations, we introduce a constraint in the action
via a Lagrange Multiplier $\chi$ and an external field $\xi$ [15]:

$\displaystyle{\cal S}=\int
d^{4}x\biggr{\\{}\sqrt{-g}R-\chi(\sqrt{-g}-\xi)\biggl{\\}}+\int
d^{4}x\sqrt{-g}{\cal L}_{m}.$ (4)

The presence of the external field allows one to use a suitable coordinate
system according to the problem under analysis, for example, the usual
spherical coordinates or the quasi-global coordinates employed in spherical
symmetric space-time.

The final field equations for this case are

$\displaystyle R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle 8\pi G\biggr{(}T_{\mu\nu}-\frac{1}{4}g_{\mu\nu}T\biggl{)},$ (5)
$\displaystyle\frac{R^{;\nu}}{4}$ $\displaystyle=$ $\displaystyle 8\pi
G\biggr{(}{T^{\mu\nu}}_{;\mu}-\frac{1}{4}T^{;\nu}\biggl{)}.$ (6)

The above equation (6) is obtained by using the Bianchi identities in (5).

As highlighted in Ref. [17], it is important to note that in UG, the
conservation of the energy-momentum tensor cannot be derived through the
conventional diffeomorphism invariance because the theory only exhibits
invariance with respect to a limited set of diffeomorphisms, referred to as
transverse diffeomorphisms. The latter implies that the energy-momentum
divergence tensor is equal to the gradient of a (undetermined) scalar
function:

$\displaystyle{T^{\mu}_{\nu}}_{;\mu}=\Theta_{;\nu},$ (7)

On one hand, it is entirely permissible to set the gradient of $\Theta$ to
zero. If this is done, we recover (2), with $\Lambda$ appearing as an
integration constant. On the other hand, one can also choose,

$\displaystyle\Theta=\frac{R}{4}+2\pi GT.$ (8)

From now on, we will identify $\Theta\equiv-\Lambda$. With this
identification, $\Lambda$ becomes a dynamical term. If $\Lambda$ is constant,
as already stressed, we return to the GR equations in presence of a
cosmological constant. But, if $\Lambda$ is a function of the space-time
coordinates, we end up with the following set of equations,

$\displaystyle R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle 8\pi G\biggr{\\{}T_{\mu\nu}-\frac{1}{4}g_{\mu\nu}T\biggl{\\}},$
(9) $\displaystyle{T^{\mu}_{\nu}}_{;\mu}=-\Lambda_{;\nu}.$ (10)

This is equivalent (up to the restriction in the diffeomorphic class of
transformation) to the RG equations in presence of a dynamical cosmological
term:

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle 8\pi GT_{\mu\nu}+g_{\mu\nu}\Lambda,$ (11)
$\displaystyle{T^{\mu}_{\nu}}_{;\mu}=-\Lambda_{;\nu},$ (12)

provided that $\Lambda$ is identified with $\Theta$ as given by (8). Hence,
the non-conservation of the energy-momentum tensor allows to map the UG theory
into GR equipped with a dynamical cosmological term, implying in an
interacting like model in the GR context.

It is also convenient, for reasons that will become clear later on in the
work, to write down the UG equations in a more compact form such as

$\displaystyle E_{\mu\nu}=8\pi G\,\tau_{\mu\nu},$ (13)

with the definitions,

$\displaystyle E_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R=G_{\mu\nu}+\frac{1}{4}g_{\mu\nu}R,$ (14)
$\displaystyle\tau_{\mu\nu}$ $\displaystyle=$ $\displaystyle
T_{\mu\nu}-\frac{1}{4}g_{\mu\nu}T.$ (15)

We will call $E_{\mu\nu}$ the unimodular gravitational tensor and
$\tau_{\mu\nu}$ the unimodular energy-momentum tensor.

##  III Equations for a symmetric and static configuration

In this section, we will write down the general expressions for a symmetric
and static configuration. In the appendix A the corresponding expressions with
a time dependence will be derived, which are necessary to perform the
perturbative analysis to be described later.

Let us consider the metric,

$\displaystyle
ds^{2}=e^{2\gamma}dt^{2}-e^{2\alpha}du^{2}-e^{2\beta}d\Omega^{2}.$ (16)

The non-vanishing Christoffel symbols are the following.

$\displaystyle\Gamma^{0}_{10}=\gamma^{\prime}\quad,\quad\Gamma^{1}_{00}=e^{2(\gamma-\alpha)}\gamma^{\prime},$
(17)
$\displaystyle\Gamma^{1}_{11}=\alpha^{\prime}\quad,\quad\Gamma^{1}_{22}=-e^{2(\beta-\alpha)}\beta^{\prime}\quad,\quad\Gamma^{1}_{33}=-e^{2(\beta-\alpha)}\beta^{\prime}\sin^{2}\theta,$
(18)
$\displaystyle\Gamma^{2}_{12}=\Gamma^{3}_{12}=\beta^{\prime},\quad\Gamma^{2}_{33}=-\sin\theta\cos\theta\quad,\quad\Gamma^{3}_{23}=\cot\theta.$
(19)

Also, the non-vanishing components of the Ricci tensor and the Ricci scalar
are the following.

$\displaystyle R_{00}$ $\displaystyle=$ $\displaystyle
e^{2(\gamma-\alpha)}[\gamma^{\prime\prime}+\gamma^{\prime}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})],$
(20) $\displaystyle R_{11}$ $\displaystyle=$
$\displaystyle-\gamma^{\prime\prime}-2\beta^{\prime\prime}+\gamma^{\prime}(\alpha^{\prime}-\gamma^{\prime})+2\beta^{\prime}(\alpha^{\prime}-\beta^{\prime}),$
(21) $\displaystyle R_{22}$ $\displaystyle=$ $\displaystyle
1-e^{2(\beta-\alpha)}[\beta^{\prime\prime}+\beta^{\prime}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})],$
(22) $\displaystyle R_{33}$ $\displaystyle=$ $\displaystyle
R_{22}\sin^{2}\theta,$ (23) $\displaystyle R$ $\displaystyle=$
$\displaystyle-2e^{-2\beta}+2e^{-2\alpha}[\gamma^{\prime\prime}+2\beta^{\prime\prime}+3\beta^{\prime
2}+\gamma^{\prime}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})-2\alpha^{\prime}\beta^{\prime}].$
(24)

Consequently, the non-vanishing components of the unimodular gravitational
tensor defined in (14) are the following:

$\displaystyle E_{00}$ $\displaystyle=$ $\displaystyle
e^{2(\gamma-\alpha)}\biggr{[}\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{3}{2}\beta^{\prime
2}+\frac{\gamma^{\prime}}{2}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})+\beta^{\prime}\alpha^{\prime}\biggl{]}+\frac{e^{2(\gamma-\beta)}}{2},$
(25) $\displaystyle E_{11}$ $\displaystyle=$
$\displaystyle-\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{\beta^{\prime
2}}{2}-\frac{\gamma^{\prime}}{2}(\gamma^{\prime}-\alpha^{\prime})+\beta^{\prime}(\alpha^{\prime}+\gamma^{\prime})-\frac{e^{2(\alpha-\beta)}}{2},$
(26) $\displaystyle E_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{e^{2(\beta-\alpha)}}{2}[\gamma^{\prime\prime}-\beta^{\prime
2}+\gamma^{\prime}(\gamma^{\prime}-\alpha^{\prime})],$ (27) $\displaystyle
E_{33}$ $\displaystyle=$ $\displaystyle E_{22}\sin^{2}\theta.$ (28)

The left-hand side of the UG field equations for the symmetric and static
configuration has been set up. The next step is to characterize the source
field. In the next couple of sections, the electromagnetic field and a scalar
field will be considered as sources of the gravitational field.

##  IV The electromagnetic field

For the case of a electromagnetic field as the source of the energy-momentum
tensor one has,

$\displaystyle 8\pi
GT^{EM}_{\mu\nu}=-2\biggr{\\{}F_{\mu\rho}F_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\biggl{\\}}.$
(29)

It is worth mentioning that it has zero trace:

$\displaystyle T^{EM}=0.$ (30)

Equations (5) and (6) become,

$\displaystyle R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle-2\biggr{\\{}F_{\mu\rho}{F_{\nu}}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\biggl{\\}},$
(31) $\displaystyle{{F^{\mu\rho}}_{;\mu}}F_{\nu\rho}$ $\displaystyle=$
$\displaystyle-\frac{R_{;\nu}}{8}.$ (32)

Remark that, contrarily to GR, the traceless character of the energy-momentum
tensor does not imply $R=0$, unless the Maxwell equations are obeyed.

Imposing the spherical symmetry, the only non-vanishing component is
$F^{01}=E\equiv E(r)$. Then, the equations are:

$\displaystyle\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{3}{2}\beta^{\prime
2}+\frac{\gamma^{\prime}}{2}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})+\beta^{\prime}\alpha^{\prime}+\frac{e^{2(\alpha-\beta)}}{2}$
$\displaystyle=$ $\displaystyle e^{2\gamma+4\alpha}E^{2},$ (33)
$\displaystyle-\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{\beta^{\prime
2}}{2}+\frac{\gamma^{\prime}}{2}(\alpha^{\prime}+2\beta^{\prime}-\gamma^{\prime})+\beta^{\prime}\alpha^{\prime}-\frac{e^{2(\alpha-\beta)}}{2}$
$\displaystyle=$ $\displaystyle-e^{2\gamma+4\alpha}E^{2},$ (34)
$\displaystyle\frac{1}{2}\biggr{[}\gamma^{\prime\prime}-\beta^{\prime
2}+\gamma^{\prime}(\gamma^{\prime}-\alpha^{\prime})\biggl{]}+\frac{e^{2(\alpha-\beta)}}{2}$
$\displaystyle=$ $\displaystyle e^{2\gamma+4\alpha}E^{2},$ (35)
$\displaystyle(E^{2})^{\prime}+2(\alpha^{\prime}+\gamma^{\prime}+2\beta^{\prime})E^{2}$
$\displaystyle=$ $\displaystyle\frac{e^{-2(\alpha+\gamma)}}{4}R^{\prime},$
(36)

with $R$ given by (24).

Until now, no coordinate condition has been imposed. Adding (33) and (34), we
obtain,

$\displaystyle\beta^{\prime}(\alpha^{\prime}+\gamma^{\prime})-\beta^{\prime\prime}-\beta^{\prime
2}=0.$ (37)

The use of the quasi-global coordinates, with $\alpha=-\gamma$, leads to,

$\displaystyle\beta=\log r.$ (38)

As in the usual Reissner-Nordström (RN) solution in GR, there is a center at
$r=0$. Equation (8), with $T=0$ and with the identification of $\Theta$ with
$-\Lambda$ implies in $R=-4\Lambda$. The equations of motion reduce to,

$\displaystyle\gamma^{\prime\prime}+2\gamma^{\prime
2}-\frac{1}{r^{2}}+\frac{e^{-2\gamma}}{r^{2}}$ $\displaystyle=$ $\displaystyle
2e^{-2\gamma}E^{2},$ (39) $\displaystyle(E^{2})^{\prime}+4\frac{E^{2}}{r}$
$\displaystyle=$ $\displaystyle-\Lambda^{\prime},$ (40)

In order to proceed further, we must impose a condition. This is a crucial
step in working with UG as already stressed in [14]. One possibility it to fix
$R=-4\Lambda\equiv$ constant. This leads to the Reisnner-Nordström-de Sitter
(RNdS) solution. In fact, this implies to recover the conservation law
${F^{\mu\nu}}_{;\mu}=0$. If $\Lambda=0$, we re-obtain the RN solution. If
$\Lambda>0$, the RNdS solution is obtained, and if $\Lambda<0$, the Reisnner-
Nordström-(Anti) de Sitter (RNAdS) solution is recovered, as it will be seen
below. On the other hand, there also also other possibilities that to be
explored since $\Lambda$ can be non constant, covering the possibility of a
dynamical cosmological term.

Three cases will be considered, namely a constant and two dynamical
cosmological terms, corresponding to either the usual or the modified
conservation laws.

###  IV.1 Constant cosmological term

If $\Lambda=$ constant,

$\displaystyle E=\frac{Q}{r^{2}},$ (41)

after identifying an integration constant with the total charge $Q$. The
Coulomb law is recovered, as in the RN solution.

Using the quasi-global coordinate condition and the solution for the electric
field $E$, equation (33) becomes:

$\displaystyle e^{2\gamma}(\gamma^{\prime\prime}+2\gamma^{\prime
2})-\frac{e^{2\gamma}}{r^{2}}=\frac{1}{r^{2}}+2\frac{Q^{2}}{r^{4}}.$ (42)

Defining $A=e^{2\gamma}$, the equation takes the form,

$\displaystyle
A^{\prime\prime}-2\frac{A}{r^{2}}=\frac{2}{r^{2}}+4\frac{Q^{2}}{r^{4}}$ (43)

This is a second order, linear, non-homogeneous differential equation whose
solution is

$\displaystyle A=1+\frac{C_{1}}{r}+\frac{Q^{2}}{r^{2}}+\frac{C_{2}}{3}r^{2},$
(44)

$C_{1,2}$ being integration constants. Inserting this solution into the
condition $R=-\Lambda$, we obtain that it is satisfied provided
$C_{2}=-\Lambda$, while $C_{1}$ remains arbitrary, being fixed by using the
newtonian limit.

The final solution is given by,

$\displaystyle A=1-2\frac{GM}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}.$
(45)

This is the RNdS solution. It coincides with the static and spherically
symmetric solution in GR with an electromagnetic field and a cosmological
constant. This could be expected from the beginning since UG (satisfying the
usual conservation laws) leads to the same field equations as RG with a
cosmological term, with the only (but important, as we will see later)
difference that UG is restricted to transverse diffeormophism instead of the
full diffeomorphism.

###  IV.2 Varying cosmological term

For a varying cosmological term, it is necessary to impose an ansatz on the
behaviour of the function $\Lambda$. This is also true in GR when the
cosmological term is dynamical. Since a static and spherically symmetric
configuration is considered, the cosmological term must be a function on the
coordinate $r$ only: $\Lambda\equiv\Lambda(r)$.

Let us restrict ourselves again to the condition $R=-4\Lambda$. Using the
previous results and also identifying $\beta=\ln r$, $\alpha=-\gamma$ and
$A=e^{2\gamma}$, then:

$\displaystyle
A^{\prime\prime}+4\frac{A^{\prime}}{r}+2\frac{A}{r^{2}}=\frac{2}{r^{2}}-4\Lambda(r).$
(46)

The solution for the homogenous equation is,

$\displaystyle A_{h}=\frac{C_{1}}{r}+\frac{C_{2}}{r^{2}}.$ (47)

To obtain the inhomogeneous solution, we write,

$\displaystyle A=\frac{f}{r^{2}},$ (48)

obtaining,

$\displaystyle f^{\prime\prime}=2-4r^{2}\Lambda(r).$ (49)

with a solution which depends on $r$:

$\displaystyle
f=r^{2}-4\int\biggr{[}\int^{r}{r^{\prime}}^{2}\Lambda(r^{\prime})dr^{\prime}\biggl{]}dr.$
(50)

We will consider two different configurations for the function $\Lambda(r)$,
corresponding to two distinct behavior both asymptotically as well as at the
center ($r=0$).

####  IV.2.1 Case A

First, it is imposed a power law behavior for $\Lambda(r)$,

$\displaystyle\Lambda(r)=\Lambda_{0}+\Lambda_{1}r^{p},$ (51)

with $\Lambda_{0,1}$ constants.

The final solution is given by the following expressions.

  * •

$p\neq-4$:

$\displaystyle A$ $\displaystyle=$ $\displaystyle
1-\frac{2GM}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda_{0}}{3}r^{2}-\frac{4\Lambda_{1}}{(p+3)(p+4)}r^{p+2},$
(52) $\displaystyle E^{2}$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{r^{4}}-\frac{p}{p+4}\Lambda_{1}r^{p};$ (53)

  * •

$p=-4$:

$\displaystyle A$ $\displaystyle=$ $\displaystyle
1-\frac{2GM}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda_{0}}{3}r^{2}-\frac{16\Lambda_{1}}{9r}\biggr{\\{}3(\ln
r)^{2}+\ln r\biggl{\\}},$ (54) $\displaystyle E^{2}$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{r^{4}}+4\frac{\Lambda_{1}}{r^{4}}\ln r.$ (55)

The case $p=-4$ is clearly pathological since the electric field becomes
imaginary near $r=0$ when $\Lambda_{1}>0$ or for large $r$ if $\Lambda_{1}<0$.
For $p\neq-4$ a change of sign of $E^{2}$ can be avoided by choosing
$\Lambda_{1}>0$ for $-4<p<0$, or $\Lambda_{1}<0$ for $p<-4$ or $p>0$. The
values $p=0,-3$ correspond to the cases already included in the constants
$C_{1}$ and $C_{2}$ of the homogenous solution.

The solution $p\neq 0$, with the required conditions to avoid an imaginary
electric field, contains either multiple horizon black holes, with a
singularity at $r=0$, or naked singularities similarly to the dSRN solution in
RG but the metric functions, in the UG case, may present a different shape
mainly near the singularity. These solutions are asymptotically non-flat
except if $\Lambda_{0}=0$ and $p>-2$. The corresponding equations in GR
equipped with a cosmological term with the same functional dependence, using
the same symmetries, lead to the same solution as it can be explicitly
verified.

####  IV.2.2 Case B

We will exploit now the functional form,

$\displaystyle\Lambda(r)=\Lambda_{0}+\frac{\Lambda_{1}}{(r^{2}+a^{2})^{2}}.$
(56)

If $\Lambda_{0}=0$, this functional form represents an asymptotically constant
cosmological term near the origin, which becomes zero at infinity. Following
the same steps of the previous case, the final form of the metric function is:

$\displaystyle A$ $\displaystyle=$ $\displaystyle
1-\frac{2GM}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda_{0}}{3}r^{2}$ (57)
$\displaystyle-$ $\displaystyle
2\frac{\Lambda_{1}}{r^{2}}\biggr{[}\frac{r}{a}\arctan\frac{r}{a}-\ln\biggr{(}1+\frac{r^{2}}{a^{2}}\biggl{)}\biggl{]},$
$\displaystyle E^{2}$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{r^{4}}+\Lambda_{1}\biggr{\\{}-\frac{1}{(r^{2}+a^{2})^{2}}+2\biggr{[}\frac{1}{r^{2}a^{2}}-\frac{1}{r^{4}}\ln\biggr{(}1+\frac{r^{2}}{a^{2}}\biggl{)}\biggl{]}\biggl{\\}}.$
(58)

Again, the same solution is obtained in the GR with a varying cosmological
term giving by (56). There are multiple horizons and naked singularities, as
in the previous case. No change of sign in the $E^{2}$ term can be assured by
imposing $\Lambda_{1}>0$.

In all the cases discussed above, the presence of the a cosmological term,
constant or not, introduces new features in the solutions with respect to the
usual RN solution but does not remove the singularity at $r=0$.

##  V Scalar field

The energy-momentum tensor for a self-interacting scalar field is,

$\displaystyle
T_{\mu\nu}=\epsilon\biggr{(}\phi_{;\mu}\phi_{;\nu}-\frac{1}{2}g_{\mu\nu}\phi_{;\rho}\phi^{;\rho}\biggl{)}+g_{\mu\nu}V(\phi).$
(59)

The ordinary scalar field is denoted by $\epsilon=+1$ and the phantom scalar
field by $\epsilon=-1$. In GR, in four dimensions, black holes exist only for
the phantom case [19, 20].

Inserting the expression for the energy-momentum tensor (59) in the UG
equations one obtains,

$\displaystyle R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle\epsilon\biggr{(}\phi_{;\mu}\phi_{;\nu}-\frac{1}{4}g_{\mu\nu}\phi_{;\rho}\phi^{;\rho}\biggl{)},$
(60) $\displaystyle\frac{R_{;\nu}}{4}$ $\displaystyle=$
$\displaystyle\epsilon\biggr{(}\phi_{;\nu}\Box\phi+\frac{\phi^{;\rho}\phi_{;\nu;\rho}}{2}\biggl{)}.$
(61)

One distinguishing feature of the above equations is the absence of the
potential (or, as before, the cosmological term, which is the particular case
of a constant potential): it naturally disappears due to the traceless
structure of the UG equations. Equation (61) can be written as,

$\displaystyle\biggr{(}\frac{R}{4}-\frac{\epsilon}{4}\phi_{\rho}\phi^{;\rho}\biggl{)}_{;\nu}$
$\displaystyle=$ $\displaystyle\epsilon\phi_{;\nu}\Box\phi.$ (62)

Identifying,

$\displaystyle\frac{R}{4}-\frac{\epsilon}{4}\phi_{\rho}\phi^{;\rho}$
$\displaystyle=$ $\displaystyle-V(\phi),$ (63)

equations (60) and (61) take the following form,

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle\epsilon\biggr{(}\phi_{;\mu}\phi_{;\nu}-\frac{1}{2}g_{\mu\nu}\phi_{;\rho}\phi^{;\rho}\biggl{)}+g_{\mu\nu}V(\phi),$
(64) $\displaystyle\Box\phi=-\epsilon V_{\phi}(\phi).$ (65)

In this way, we recover the GR equations equipped with a self-interacting
scalar field.

In a static and spherically symmetric configuration, the UG field equations
(60) read,

$\displaystyle\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{3}{2}\beta^{\prime
2}+\frac{\gamma^{\prime}}{2}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})+\beta^{\prime}\alpha^{\prime}+\frac{e^{2(\alpha-\beta)}}{2}$
$\displaystyle=$ $\displaystyle\epsilon\frac{\phi^{\prime 2}}{4},$ (66)
$\displaystyle-\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{\beta^{\prime
2}}{2}-\frac{\gamma^{\prime}}{2}(\gamma^{\prime}-\alpha^{\prime})+\beta^{\prime}(\alpha^{\prime}+\gamma^{\prime})-\frac{e^{2(\alpha-\beta)}}{2}$
$\displaystyle=$ $\displaystyle\epsilon\frac{3}{4}\phi^{\prime 2},$ (67)
$\displaystyle\frac{1}{2}[\gamma^{\prime\prime}-\beta^{\prime
2}+\gamma^{\prime}(\gamma^{\prime}-\alpha^{\prime})]+\frac{e^{2(\alpha-\beta)}}{2}$
$\displaystyle=$ $\displaystyle-\epsilon\frac{\phi^{\prime 2}}{4}.$ (68)

Combining these equations, we have the following relations:

$\displaystyle\gamma^{\prime\prime}-\beta^{\prime\prime}-2\beta^{\prime
2}+\gamma^{\prime}(\gamma^{\prime}+\beta^{\prime}-\alpha^{\prime})+\alpha^{\prime}\beta^{\prime}+e^{2(\alpha-\beta)}$
$\displaystyle=$ $\displaystyle 0,$ (69)
$\displaystyle-\beta^{\prime\prime}-\beta^{\prime}+\beta^{\prime}(\gamma^{\prime}+\alpha^{\prime})$
$\displaystyle=$ $\displaystyle\frac{\phi^{\prime 2}}{2}.$ (70)

Remark that in the UG equations, there is no potential, even if it appears in
the energy-momentum tensor. Moreover, there are three metric functions (which
can be reduced to two functions by gauging the radial coordinate) and the
scalar field to be determined, and just two independent equations, (69) and
(70). Hence an ansatz must be introduced. From the conservation law, we have
the relation,

$\displaystyle R+e^{-\alpha}\phi^{\prime 2}=-4V(\phi),$ (71)

where $V(\phi)$ is a function to be determined. We have slightly changed the
notation ($V$ instead of $\Lambda$) to identify the unknown function with the
potential. With this identification, the UG equations become identical to the
GR equations with a potential. In GR, the potential must be chosen. In UG a
functional form for the scalar field (or for one of the metric function) must
be chosen. There is a correspondence between the choice of the functional form
of the scalar field and the choice of the potential in GR.

Two possible examples are the following.

  1. 1.

If the scalar field is chosen such that,

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle-\epsilon\frac{C}{2k}P,$
(72) $\displaystyle P$ $\displaystyle=$ $\displaystyle 1-2\frac{k}{\rho},$
(73)

we find,

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
P^{a}dt^{2}-P^{-a}d\rho^{4}-P^{1-a}\rho^{2}d\Omega,$ (74) $\displaystyle
a^{2}$ $\displaystyle=$ $\displaystyle 1-\epsilon\frac{C^{2}}{k^{2}}.$ (75)

Using (71) we find $V=0$. This solution represents a black hole only if
$\epsilon=-1$. This solution has been determined in the GR context in Ref.
[18].

  2. 2.

The regular black hole determined in Ref. [20], is also solution in the UG
case, without a potential. Imposing that the scalar field is given by,

$\displaystyle\psi=\frac{\phi}{\sqrt{2}}=\arctan\frac{\rho}{b},$ (76)

the metric is then given, in the quasi-global coordinates, by,

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
Adt^{2}-\frac{d\rho^{2}}{A}-r^{2}(\rho)d\Omega^{2},$ (77) $\displaystyle A$
$\displaystyle=$ $\displaystyle
1+\frac{c}{b^{2}}r^{2}+\frac{\rho_{0}}{b^{3}}\biggr{(}b\rho+r^{2}\arctan\frac{\rho}{b}\biggl{)}.$
(78)

In these expressions $b$, $c$ and $\rho_{0}$ are constants. Using relation
(71) the potential in GR context is given by,

$\displaystyle
V-\frac{c}{b^{3}}\biggr{(}3-2\cos^{2}\psi\biggl{)}-\frac{\rho_{0}}{b^{3}}\biggr{[}3\sin\psi\cos\psi+\psi\biggr{(}3-2\cos^{2}\psi\biggl{)}\biggl{]},$
(79)

the same used in the GR context.

##  VI Remarks on the Birkhoff theorem and the stability of the solutions

Initially, we present a straightforward argument to demonstrate that, in the
cases of both electromagnetism and scalar fields, the Birkhoff theorem holds
the same significance as it does in GR. For electric charged static solutions
in GR, the Birkhoff theorem is valid. The same occur for the corresponding
solution in the UG. This can be seen by supposing radial time dependent
configurations. The argumentation follows the same reasoning used in GR, see
for example [21]. From the expression presented in the appendix, and
considering only a radial electric field, the $0-1$ component of the field
equations, using the Schwarzschild coordinate system, with $\beta=\ln r$,
implies that $\alpha$ must be time independent, since the right hand side of
the equation is zero for a pure radial electric field. Combining equations
$0-0$ and $1-1$, it comes out that $\alpha=-\gamma$. Hence, all metric
functions are time independent.

For the scalar field case, the Birkhoff theorem is not valid because the right
hand side contains a term of the time $\dot{\phi}\phi^{\prime}$ which forbids
to consider the metric function $\alpha$ as time independent, as it happens in
the GR case. The Birkhoff theorem is verified only if the scalar field is
static [22].

In the two examples discussed in the previous section, having the scalar field
as the source of the geometry, and considering the GR context, the solutions
are unstable, except for the regular solution in the very special case the
minimum of the areal function coincides with the horizon [23, 24]. However,
this result can change in the UG context since the unimodular condition
implies in new relations for the perturbed functions that are absent in GR.

We will illustrate the special features of the perturbative analysis
considering the case of black holes with a scalar field. Only radial
perturbations will be considered. In the GR context, this is enough to
conclude about the instability of the solution [23]. We will show that in the
UG, if if we try to follow the same procedure as in GR, the perturbations at
first order are strictly zero due to the unimodular condition.

The unimodular condition implies,

$\displaystyle g=\mbox{det}{g_{\mu\nu}}=e^{\alpha+\gamma+2\beta}=\xi.$ (80)

Since the function $\xi$ is fixed, the unimodular condition leads, at linear
perturbative order,

$\displaystyle\delta\alpha+\delta\gamma+2\delta\beta=0.$ (81)

There is still the freedom to impose a coordinate condition due to the
diffeomorphic (even if transverse) invariance. The choice $\delta\beta=0$ is
related to the gauge invariant variables [23]. Hence, we end up with the
conditions,

$\displaystyle\delta\alpha=-\delta\gamma,\quad\delta\beta=0.$ (82)

We write down the perturbations in a generic way as

$\displaystyle\delta f(x,t)=f(x)e^{-i\omega t}.$ (83)

The perturbed equations, under the conditions above, are the following:

$\displaystyle\delta\gamma^{\prime\prime}+4\gamma^{\prime}\delta\gamma^{\prime}-\biggr{\\{}\omega^{2}.e^{-4\gamma}+2e^{-2(\gamma+\beta)}\biggl{\\}}\delta\gamma$
$\displaystyle=$ $\displaystyle\phi^{\prime}\delta\phi^{\prime},$ (84)
$\displaystyle\delta\gamma^{\prime\prime}+4\gamma^{\prime}\delta\gamma^{\prime}-\biggr{\\{}\omega^{2}.e^{-4\gamma}+2e^{-2(\gamma+\beta)}\biggl{\\}}\delta\gamma$
$\displaystyle=$ $\displaystyle-3\phi^{\prime}\delta\phi^{\prime},$ (85)
$\displaystyle\delta\gamma^{\prime\prime}+4\gamma^{\prime}\delta\gamma^{\prime}-\biggr{\\{}\omega^{2}.e^{-4\gamma}+2e^{-2(\gamma+\beta)}\biggl{\\}}\delta\gamma$
$\displaystyle=$ $\displaystyle-\phi^{\prime}\delta\phi^{\prime},$ (86)
$\displaystyle-2\beta^{\prime}\delta\gamma$ $\displaystyle=$
$\displaystyle\phi^{\prime}\delta\phi.$ (87)

It is clear that the equations are consistent only in the trivial case:
$\delta\phi=\delta\gamma=0$. Hence, it is not possible to obtain informations
on the stability of the solution, at least at linear level and following a
procedure close to that used in GR. This is a distinguishing feature of
unimodular gravity in comparison with GR.

##  VII Conclusion

Unimodular gravity (UG) is one of the first alternatives to General Relativity
(GR). It is a geometric theory which is invariant with respect to a restricted
diffeomorphic class of transformations, the transverse diffeomorphism, due to
the imposition of a constraint on the determinant of the metric. In UG the
usual conservation of the energy-momentum tensor is not assured: The
conservation of the energy-momentum tensor is a choice. If it is imposed, UG
becomes in principle equivalent to GR with a cosmological term. However, the
restriction on the determinant of the metric may lead to some important new
features at perturbative level. We have shown here that if the conservation of
the energy-moment tensor is relaxed, UG becomes equivalent to GR with a
dynamical cosmological term, with still the same important difference due to
the UG constraint which can manifested at perturbative level.

We have discussed, in this context, the static and spherically symmetric
solutions in UG. For the vacuum configuration, Schwarzschild solution is also
verified in UG. The same occurs with the Reissner-Nordström solution, but only
if the energy-momentum tensor is conserved. If not, the dynamical cosmological
term induces new features, but it does not prevent the appearance of the
singularity at $r=0$. Similar features appear in the case when a scalar field
appears as the main source. In this case, the potential term, representing the
self-interaction of the scalar field, disappears in the UG context and an
ansatz must be imposed in order to close the set of equations. This mounts, in
the GR context, to choose a given potential for the scalar field. For a
discussion of the UG in static, spherical configurations but focusing compact
objects, see Ref. [25].

We have shown that the Birkhoff theorem follows the same features as in GR,
being satisfied for a charged solution, being possibly violated for a
dynamical scalar field. The linear radial perturbations have been analyzed
when a scalar field is present. Once more, GR black hole solutions are
generically unstable in the latter case. In UG, using the gauge invariant
approach employed in GR and restricting to radial perturbations, the condition
on the determinant of the metric leads to vanishing perturbations at linear
order, and possibly also for higher order. As already discussed in the
cosmological context, this result seems to point out for a breaking of the
equivalence of UG and GR at perturbative level. There are other viewpoints on
the implementation of the UG constraints in performing a perturbative
analysis, see for example Ref. [26]. However, the results reported here
indicates that a direct application of the procedures used in GR combined with
the unimodular constraint may lead to conclusions different from those
obtained in GR.

Acknowledgements: We thank CNPq, FAPES and FAPEMIG for partial financial
support. We thank K.A. Bronnikov for enlighting discussions on some aspects of
the problem treated in this work and L.F. de Oliveira Guimarães for his
remarks on the text.

## Appendix A: The spherically symmetric non static metric

A dynamical spherically symmetric metric, admitting radial oscillations, is
given by,

$\displaystyle
ds^{2}=e^{2\gamma(t,u)(}dt^{2}-e^{2\alpha(t,u)}du^{2}-e^{2\beta(t,u)}d\Omega^{2}.$
(88)

The non-vanishing Christoffel symbols are the following.

$\displaystyle\Gamma^{0}_{00}=\dot{\gamma},\quad\Gamma^{0}_{10}=\gamma^{\prime},\quad\Gamma^{0}_{11}=e^{2(\alpha-\gamma)}\dot{\alpha},$
(89)
$\displaystyle\Gamma^{0}_{22}=e^{2(\beta-\gamma)}\dot{\beta},\quad\Gamma^{0}_{33}=\Gamma^{0}_{22}\sin^{2}\theta,$
(90)
$\displaystyle\Gamma^{1}_{00}=e^{2(\gamma-\alpha)}\gamma^{\prime},\quad\Gamma^{1}_{01}=\dot{\alpha},\quad\Gamma^{1}_{11}=\alpha^{\prime},$
(91)
$\displaystyle\Gamma^{1}_{22}=-e^{2(\beta-\alpha)}\beta^{\prime},\quad\Gamma^{1}_{33}=\Gamma^{1}_{22}\sin^{2}\theta,$
(92)
$\displaystyle\Gamma^{2}_{02}=\Gamma^{3}_{03}=\dot{\beta},\quad\Gamma^{2}_{12}=\Gamma^{3}_{12}=\beta^{\prime},$
(93)
$\displaystyle\Gamma^{2}_{33}=-\sin\theta\cos\theta\quad,\quad\Gamma^{3}_{23}=\cot\theta.$
(94)

The non-vanishing components of the Ricci tensor and the Ricci scalar are the
following.

$\displaystyle R_{00}$ $\displaystyle=$
$\displaystyle-\ddot{\alpha}-2\ddot{\beta}+\dot{\gamma}(\dot{\alpha}+2\dot{\beta})-\dot{\alpha}^{2}-2\dot{\beta}^{2}$
(95) $\displaystyle+$ $\displaystyle
e^{2(\gamma-\alpha)}[\gamma^{\prime\prime}+\gamma^{\prime}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})],$
$\displaystyle R_{11}$ $\displaystyle=$ $\displaystyle
e^{2(\alpha-\gamma)}\biggr{\\{}\ddot{\alpha}+\dot{\alpha}(\dot{\alpha}-\dot{\gamma}+2\dot{\beta})\biggl{\\}}$
(96) $\displaystyle-$
$\displaystyle\gamma^{\prime\prime}-2\beta^{\prime\prime}+\gamma^{\prime}(\alpha^{\prime}-\gamma^{\prime})+2\beta^{\prime}(\alpha^{\prime}-\beta^{\prime}),$
$\displaystyle R_{22}$ $\displaystyle=$ $\displaystyle
1+e^{2(\beta-\gamma)}[\ddot{\beta}+\dot{\beta}(\dot{\alpha}+2\dot{\beta}-\dot{\gamma})]$
(97) $\displaystyle-$ $\displaystyle
e^{2(\beta-\alpha)}[\beta^{\prime\prime}+\beta^{\prime}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})],$
$\displaystyle R_{33}$ $\displaystyle=$ $\displaystyle R_{22}\sin^{2}\theta,$
(98) $\displaystyle R_{01}$ $\displaystyle=$ $\displaystyle
2\biggr{\\{}\dot{\beta}^{\prime}+\dot{\beta}(\gamma^{\prime}-\beta^{\prime})+\dot{\alpha}\beta^{\prime}\biggl{\\}},$
(99) $\displaystyle R$ $\displaystyle=$
$\displaystyle-2e^{-2\beta}+2e^{-2\alpha}[\gamma^{\prime\prime}+2\beta^{\prime\prime}+3\beta^{\prime
2}+\gamma^{\prime}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})-2\alpha^{\prime}\beta^{\prime}]$
(100) $\displaystyle-$ $\displaystyle
2e^{-2\gamma}[\ddot{\alpha}+2\ddot{\beta}+3\dot{\beta}^{2}+\dot{\alpha}(\dot{\alpha}+2\dot{\beta}-\dot{\gamma})-2\dot{\gamma}\dot{\beta}]$

The non-vanishing components of the unimodular gravitational tensor,

$\displaystyle E_{\mu\nu}=R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R,$ (101)

are the following:

$\displaystyle E_{00}$ $\displaystyle=$
$\displaystyle-\frac{\ddot{\alpha}}{2}-\ddot{\beta}+\frac{\dot{\gamma}}{2}(\dot{\alpha}+2\dot{\beta})-\frac{\dot{\alpha}^{2}}{2}-\frac{\dot{\beta}^{2}}{2}+\dot{\alpha}\dot{\beta}$
(102) $\displaystyle+$ $\displaystyle
e^{2(\gamma-\alpha)}\biggr{[}\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{3}{2}\beta^{\prime
2}+\frac{\gamma^{\prime}}{2}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})+\beta^{\prime}\alpha^{\prime}\biggl{]}+\frac{e^{2(\gamma-\beta)}}{2},$
$\displaystyle E_{11}$ $\displaystyle=$ $\displaystyle
e^{2(\alpha-\gamma)}\biggr{[}\frac{\ddot{\alpha}}{2}-\ddot{\beta}-\frac{3}{2}\dot{\beta}^{2}+\frac{\dot{\alpha}}{2}(\dot{\alpha}+2\dot{\beta}-\dot{\gamma})+\dot{\gamma}\dot{\beta}\biggl{]}$
(103) $\displaystyle-$
$\displaystyle\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{\beta^{\prime
2}}{2}-\frac{\gamma^{\prime}}{2}(\gamma^{\prime}-\alpha^{\prime})+\beta^{\prime}(\alpha^{\prime}+\gamma^{\prime})-\frac{e^{2(\alpha-\beta)}}{2},$
$\displaystyle E_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{e^{2(\beta-\gamma}}{2}[\ddot{\alpha}-\dot{\beta}^{2}+\dot{\alpha}(\dot{\alpha}-\dot{\gamma})]$
(104) $\displaystyle+$
$\displaystyle\frac{e^{2(\beta-\alpha)}}{2}[\gamma^{\prime\prime}-\beta^{\prime
2}+\gamma^{\prime}(\gamma^{\prime}-\alpha^{\prime})],$ $\displaystyle E_{01}$
$\displaystyle=$ $\displaystyle
2[\dot{\beta}^{\prime}+\dot{\beta}(\gamma^{\prime}-\beta^{\prime})+\dot{\alpha}\beta^{\prime}],$
(105) $\displaystyle E_{33}$ $\displaystyle=$ $\displaystyle
E_{22}\sin^{2}\theta.$ (106)

For the static case, the above expressions reduce to,

$\displaystyle E_{00}$ $\displaystyle=$ $\displaystyle
e^{2(\gamma-\alpha)}\biggr{[}\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{3}{2}\beta^{\prime
2}+\frac{\gamma^{\prime}}{2}(\gamma^{\prime}+2\beta^{\prime}-\alpha^{\prime})+\beta^{\prime}\alpha^{\prime}\biggl{]}+\frac{e^{2(\gamma-\beta)}}{2},$
(107) $\displaystyle E_{11}$ $\displaystyle=$
$\displaystyle-\frac{\gamma^{\prime\prime}}{2}-\beta^{\prime\prime}-\frac{\beta^{\prime
2}}{2}-\frac{\gamma^{\prime}}{2}(\gamma^{\prime}-\alpha^{\prime})+\beta^{\prime}(\alpha^{\prime}+\gamma^{\prime})-\frac{e^{2(\alpha-\beta)}}{2},$
(108) $\displaystyle E_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{e^{2(\beta-\alpha)}}{2}[\gamma^{\prime\prime}-\beta^{\prime
2}+\gamma^{\prime}(\gamma^{\prime}-\alpha^{\prime})],$ (109) $\displaystyle
E_{33}$ $\displaystyle=$ $\displaystyle E_{22}\sin^{2}\theta.$ (110)

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