hep-ph/***

FTUAM-23-XXX IFT-UAM/CSIC-23-15

Physical charges versus conformal invariance in unimodular gravity

Enrique Alvarez, Jesus Anero and Irene Sanchez-Ruiz

Departamento de Física Teórica and Instituto de Física Teórica, IFT-UAM/CSIC,

Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain

E-mail: enrique.alvarez@uam.es,jesusanero<EMAIL_ADDRESS>
###### Abstract

Unimodularity can be implemented in different ways. In this paper we consider
only the formulation of Unimodular Gravity in which the unimodular metric is
obtained out of an unrestricted one as $\gamma_{\mu\nu}=|g|^{-{1\over
n}}g_{\mu\nu}$. This procedure induces an extra Weyl symmetry. Some physical
implications of this symmetry on the conserved currents are discussed. Finally
the results are illustrated for the Painlevé-Gullstrand extension of
Schwarzschild spacetime.

###### Contents

  1. 1 Introduction
  2. 2 Conserved currents in Unimodular Gravity.
    1. 2.1  Hilbert’s UG Lagrangian
    2. 2.2  Einstein’s UG Lagrangian
  3. 3  Schwarzschild’s UG spacetime.
  4. 4 Conclusions
  5. A  PG Hilbert’s current.
  6. B  PG Einstein’s current.

##  1 Introduction

Unimodular Gravity (UG) is a restriction of General Relativity to unimodular
metrics

$g\equiv\left|\det\,g_{\mu\nu}\right|=1$ (1)

A preliminary question is why Unimodular Gravity? The answer is that in this
theory a constant vacuum density does not weigh, so that it does not induce a
cosmological constant (cf. the first reference in [4]). Although that fact is
not a full solution of the cosmological constant problem, it does solve half
of it, namely why the vacuum energy usually believed to be generated in
several phase transitions in the history of the universe do not induce a huge
cosmological constant. It remains to explain the observed (small and positive)
value of the cosmological constant. UG has nothing new to say about this.

In this paper, we want to study the conserved currents for the Unimodular
Gravity in two equivalent formulation of General Relativity (GR), the Hilbert
Lagrangian

$S_{\text{\tiny{H}}}[\gamma]\equiv-\frac{1}{2\kappa^{2}}\int d^{n}x\hat{R}$
(2)

(our notation is that the metric $\gamma_{\mu\nu}$ is always unimodular and
geometrical quantities constructed out of it are represented by hatted
symbols) and the so-called Einstein Lagrangian111 We do not want here to take
issues on the priority of Hilbert over Einstein in coming up with this
Lagrangian. In fact this is usually (also by ourselves) called the Einstein-
Hilbert Lagrangian; we have chosen the names in order to be able to tell apart
two different candidates for the GR Lagrangian. where a surface term is added
that eliminates the second derivatives of the metric. Staring from the
identity[19]

$\gamma^{\mu\nu}\hat{R}_{\mu\nu}=\partial_{\rho}\left(\gamma^{\mu\nu}\hat{\Gamma}^{\rho}_{\mu\nu}\right)-\partial_{\nu}\left(\gamma^{\mu\nu}\hat{\Gamma}^{\rho}_{\mu\rho}\right)+\gamma^{\mu\nu}\left(\hat{\Gamma}^{\rho}_{\lambda\nu}\hat{\Gamma}^{\lambda}_{\mu\rho}-\hat{\Gamma}^{\rho}_{\lambda\rho}\hat{\Gamma}^{\lambda}_{\mu\nu}\right)$
(3)

and defining Einstein’s lagrangian from

$S_{\text{\tiny{E}}}\equiv\frac{1}{2\kappa^{2}}\int
d^{n}x\gamma^{\mu\nu}\left(\hat{\Gamma}^{\rho}_{\lambda\rho}\hat{\Gamma}^{\lambda}_{\mu\nu}-\hat{\Gamma}^{\rho}_{\lambda\nu}\hat{\Gamma}^{\lambda}_{\mu\rho}\right)$
(4)

$S_{\text{\tiny{H}}}=S_{\text{\tiny{E}}}+\int d^{n}x\,\partial_{\mu}V^{\mu}$
(5)

with

$V^{\mu}={1\over
2\kappa^{2}}\left(\hat{\Gamma}^{\mu}_{\rho\sigma}\gamma^{\rho\sigma}-\gamma^{\mu\nu}\hat{\Gamma}_{\nu\lambda}^{\lambda}\right)$
(6)

that is, Hilbert’s and Einstein’s Lagrangians differ by a total derivative
$\partial_{\mu}V^{\mu}$ , so that both lagrangians yield the same equations of
motion.

The reason for us to consider the Einstein Lagrangian stems from the fact that
it only involves first derivatives of the metric. In fact this is a well known
way to understand why Einstein equations are second order. A recent review
about Unimodular Gravity can be found in [2].

There seems not to be any free lunch in the universe, and the price to pay in
this case is that the restriction to unimodular metrics implies that the
theory is not invariant under the full diffeomorphism group (Diff), which is
the invariance group of General Relativity, but only under volume preserving
diffeomorphisms. This subgroup of the total diffeomorphism group, consists on
those diffeomorphisms (connected with the identity) that have unit Jacobian.

$x\rightarrow x^{\prime};\quad\text{det}\,{\partial x^{\prime}\over\partial
x}=1$ (7)

The tangent space of the identity component of volume preserving
diffeomorphisms is generated by 222 Diffeomorphism groups are rather tricky.
For example [16] there are diffeomorphisms arbitrarily close to the identity
which can not be obtained through the exponential map of the algebra.We shall
in fact only consider in this work those diffeomorphisms that can be generated
via the exponential map. We will always refer to this as the algebra of the
group and do not worry about topological considerations. transverse vectors
$\xi_{T}^{\mu}\in$ TDiff.

$\hat{\nabla}_{\mu}\xi_{T}^{\mu}=\partial_{\mu}\xi^{\mu}_{T}=0$ (8)

note, the covariant derivative is defined with the unimodular metric,
$\gamma_{\mu\nu}$, therefore

$\hat{\Gamma}^{\lambda}_{\lambda\mu}=\frac{1}{\sqrt{\gamma}}\,\partial_{\mu}\sqrt{\gamma}=0$
(9)

Once we have motivated our interest about Unimodular Gravity, a new question
appears, how we implement the unimodular constrait in the action. There are at
least three ways to do it

  * •

The first one is through a Lagrange multiplier [14] i.e., add to the
Lagrangian $\mathcal{L}$ the term

$S=\int d^{n}x\sqrt{|g|}\left[\mathcal{L}+\lambda(x)\left(g-1\right)\right]$
(10)

This option works fine at the classical level, but does not work for quantum
corrections [8].

  * •

Another option [10], it using an exponential representation of a unimodular
metric as the exponential of a traceless one

$g_{\mu\nu}(x)=e^{G_{\mu\nu}(x)}$ (11)

where $\text{tr}\,G_{\mu\nu}(x)=0$.

  * •

The one we will use, which is to work with an unrestricted metric,
$g_{\alpha\beta}$ and construct the unimodular one through

$\gamma_{\alpha\beta}=|g|^{-\frac{1}{n}}\,g_{\alpha\beta}$ (12)

This formalism has been used extensively in [6].

Recall that this third option induces a (tautological) Weyl invariance

$g_{\alpha\beta}\longrightarrow\Omega^{2}(x)g_{\alpha\beta}$ (13)

with $\Omega(x)=|g|^{-\frac{1}{2n}}$, therefore the total symmetry group of
the theory is now

$\text{WTDiff}=\text{Weyl}\rtimes\text{TDiff}$ (14)

Physically, what happens is that in the Weyl orbit of spacetimes that are
gauge equivalent to the physical one, see the figure 1, which by hypothesis is
the unimodular one, $\gamma_{\alpha\beta}$ most naively defined physical
characteristics are not constant (because they are not Weyl gauge invariant)
(cf. figure).

$\mathcal{M}$$g=1$ Figure 1: Weyl gauge orbits and gauge fixing for metrics
$|\gamma|=1$. TDiff orbits are not parallel to Weyl orbits.

The plan of the paper is as follows. In the second section, we briefly review
about Noether’s current, and we make explicit our results in Hilbert’s and
Einstein’s Lagrangian. We eventually find Weyl invariant currents, which
reduce to General Relativity ones in the Weyl gauge fixed $g=1$. Related work
on Noether currents in general Weyl invariant theories has recently appeared
[1] although the emphasis there is slightly different from ours. The section
three, we shall apply the previous results for the Painlevé-Gullstrand
extension of Schwarzschild spacetime. Finally we present our conclusions.

The main goal of this paper is the conformally invariant expression of the
physical currents in Unimodular Gravity.

##  2 Conserved currents in Unimodular Gravity.

Let us begin with a short review of the consequences of the full Diff
symmetry. After that, we study the special case of UG with only TDiff symmetry
and finally we derive our main results in the case of Weyl$\rtimes$TDiff
symmetry for Hilbert’s and Einstein’s Lagrangians.

Assume a Lagrangian, $\mathcal{L}$, scalar under the full Diff group. The
invariance of the action under an arbitrary diffeomorphism generated by
$\xi^{\mu}$ implies that the variation of the action reads

$\delta S=\int d^{n}x\,\partial_{\mu}(\mathcal{L}\xi^{\mu})$ (15)

On the other hand it is well-known that under a general variation of the
metric (without changing the domain of integration) the variation of the
action reads

$\delta S=\int d^{n}x\sqrt{|g|}\bigg{\\{}\frac{\delta S}{\delta
g_{\alpha\beta}}\delta g_{\alpha\beta}+\nabla_{\mu}j^{\mu}\bigg{\\}}$ (16)

on shell, that is, on the submanifold that obeys the equation of motion, this
variation reduces to

$\delta S=\int d^{n}x\sqrt{|g|}\nabla_{\mu}j^{\mu}=\int
d^{n}x\partial_{\mu}(\sqrt{|g|}j^{\mu})$ (17)

The fact that this two variations have to be consistent independently of the
integration domain means that the Noether current

$\partial_{\mu}j^{\mu}_{N}=\partial_{\mu}(\mathcal{L}\xi^{\mu}-\sqrt{g}j^{\mu})=0$
(18)

has got to be conserved.

Let us derive Noether’s current for a general Lagrangian
$\mathcal{L}(g_{\alpha\beta},\partial_{\mu}g_{\alpha\beta},\partial_{\mu}\partial_{\nu}g_{\alpha\beta})$.
The variation of Lagrangian is

$\delta\mathcal{L}=\frac{\partial\mathcal{L}}{\partial g_{\alpha\beta}}\delta
g_{\alpha\beta}+\frac{\partial\mathcal{L}}{\partial\partial_{\mu}g_{\alpha\beta}}\delta\partial_{\mu}g_{\alpha\beta}+\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\nu}g_{\alpha\beta}}\delta\partial_{\mu}\partial_{\nu}g_{\alpha\beta}$
(19)

Euler-Lagrange’s equation reads

$\frac{\partial\mathcal{L}}{\partial
g_{\alpha\beta}}-\partial_{\mu}\frac{\partial\mathcal{L}}{\partial\partial_{\mu}g_{\alpha\beta}}+\partial_{\mu}\partial_{\nu}\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\nu}g_{\alpha\beta}}=0$
(20)

and Noether’s current is

$j_{N}^{\mu}=\Big{(}\frac{\partial\mathcal{L}}{\partial\partial_{\mu}g_{\alpha\beta}}\delta
g_{\alpha\beta}-\partial_{\nu}\Big{(}\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\nu}g_{\alpha\beta}}\Big{)}\delta
g_{\alpha\beta}+\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\nu}g_{\alpha\beta}}\partial_{\nu}\delta
g_{\alpha\beta}$ (21)

which for a Diff generated by the vector field $\xi^{\mu}$ can be written as

$j_{N}^{\mu}=\sqrt{|g|}\left(T_{\rho}^{\leavevmode\nobreak\
\mu}\xi^{\rho}+U_{\rho}^{\leavevmode\nobreak\
\mu\nu}\partial_{\nu}\xi^{\rho}+V_{\rho}^{\leavevmode\nobreak\
\mu(\lambda\tau)}\partial_{\lambda}\partial_{\tau}\xi^{\rho}\right)$ (22)

with

$\displaystyle T_{\rho}^{\leavevmode\nobreak\ \mu}$
$\displaystyle=\delta^{\mu}_{\rho}\mathcal{L}-\Big{(}\frac{\partial\mathcal{L}}{\partial\partial_{\mu}g_{\alpha\beta}}-\partial_{\nu}\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\nu}g_{\alpha\beta}}\Big{)}\partial_{\rho}g_{\alpha\beta}-\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\nu}g_{\alpha\beta}}\partial_{\nu}\partial_{\rho}g_{\alpha\beta}$
$\displaystyle U_{\rho}^{\leavevmode\nobreak\ \mu\nu}$
$\displaystyle=2\Big{(}\partial_{\tau}\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\tau}g_{\alpha\beta}}-\frac{\partial\mathcal{L}}{\partial\partial_{\mu}g_{\alpha\beta}}\Big{)}g_{\rho\sigma}I^{\nu\sigma}_{\alpha\beta}-\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\nu}g_{\alpha\beta}}\partial_{\rho}g_{\alpha\beta}-2\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\tau}g_{\alpha\beta}}I^{\nu\sigma}_{\alpha\beta}\partial_{\tau}g_{\rho\sigma}$
$\displaystyle V_{\rho}^{\leavevmode\nobreak\ \mu(\lambda\tau)}$
$\displaystyle=-\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\tau}g_{\alpha\beta}}g_{\rho\sigma}I^{\lambda\sigma}_{\alpha\beta}-\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\partial_{\lambda}g_{\alpha\beta}}g_{\rho\sigma}I^{\tau\sigma}_{\alpha\beta}$
(23)

with

$I^{\mu\nu}_{\alpha\beta}=\frac{1}{2}\Big{(}\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}+\delta^{\nu}_{\alpha}\delta^{\mu}_{\beta}\Big{)}$
(24)

What changes in UG, where we only have TDiff invariance? The fact that the
gauge parameters are not independent is the source of some small technical
complications. One way out is to write

$\xi_{T}^{\mu}=\nabla_{\lambda}\Omega^{\lambda\mu},$ (25)

where the tensor

$\Omega_{\mu\nu}=-\Omega_{\nu\mu}$ (26)

is antisymmetric.

Let us define in general the equations of motion as

$H_{\mu\nu}\equiv{\delta S\over\delta g^{\mu\nu}}$ (27)

TDiff invariance [3] implies that

$\delta S=\int
d^{n}x\sqrt{|g|}H_{\mu\nu}\left(\nabla^{\mu}\xi^{\nu}+\nabla^{\nu}\xi^{\mu}\right)=\int
d^{n}x\sqrt{|g|}H_{\mu\nu}\left(\nabla^{\mu}\nabla_{\alpha}\Omega^{\alpha\nu}+\nabla^{\nu}\nabla_{\alpha}\Omega^{\alpha\mu}\right)$
(28)

assuming that $\Omega_{\alpha\beta}$ vanishes at the boundary,
$\left.\Omega_{\alpha\beta}\right|_{\partial M}=0$, we can integrate by parts
[5]

$\delta S=\int
d^{n}x\sqrt{|g|}\Big{[}\nabla_{\alpha}\nabla^{\lambda}H_{\lambda\beta}-\nabla_{\beta}\nabla^{\lambda}H_{\lambda\alpha}\Big{]}\Omega^{\alpha\beta}$
(29)

therefore333 When there is ambiguity, we follow the rule of underlining the
indices which undergo antisymmetrization.

$\nabla_{[\underline{\alpha}}\nabla^{\lambda}H_{\lambda\underline{\beta}]}=0$
(30)

In conclusion, the Noether current (22), in function of the
$\Omega_{\alpha\beta}$ reads

$j^{\mu}_{N}=\sqrt{|g|}\left(T^{\mu}_{\rho}\nabla_{\alpha}\Omega^{\alpha\rho}+U_{\rho}^{\mu\nu}\nabla_{\nu}\nabla_{\alpha}\Omega^{\alpha\rho}+V_{\rho}^{\mu(\lambda\tau)}\nabla_{\lambda}\nabla_{\tau}\nabla_{\alpha}\Omega^{\alpha\rho}\right)$
(31)

and it satisfies

$\displaystyle 0=\int
d^{n}x\partial_{\mu}\Big{[}\sqrt{g}\left(T^{\mu}_{\rho}\nabla_{\alpha}\Omega^{\alpha\rho}+U_{\rho}^{\mu\nu}\nabla_{\nu}\nabla_{\alpha}\Omega^{\alpha\rho}+V_{\rho}^{\mu(\lambda\tau)}\nabla_{\lambda}\nabla_{\tau}\nabla_{\alpha}\Omega^{\alpha\rho}\right)\Big{]}$
(32)

Given our assumption on the behavior of $\Omega^{\mu\nu}$, we can integrate by
parts

$\displaystyle 0=\int
d^{n}x\sqrt{g}\Big{[}-(\nabla_{\alpha}\nabla_{\mu}T^{\mu}_{\rho})\Omega^{\alpha\rho}+(T^{\mu}_{\rho}+\nabla_{\lambda}U_{\rho}^{\lambda\mu})\nabla_{\mu}\nabla_{\alpha}\Omega^{\alpha\rho}+$
$\displaystyle+(U_{\rho}^{\mu\nu}+\nabla_{\lambda}V_{\rho}^{\lambda(\mu\nu)})\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\Omega^{\alpha\rho}+V_{\rho}^{\mu(\lambda\tau)}\nabla_{\mu}\nabla_{\lambda}\nabla_{\tau}\nabla_{\alpha}\Omega^{\alpha\rho}\Big{]}$
(33)

and finally

$\displaystyle\nabla_{[\underline{\alpha}}\nabla_{\mu}T^{\mu}_{\underline{\rho}]}+\nabla_{[\underline{\alpha}}\nabla_{\nu}\nabla_{\mu}U_{\underline{\rho}]}^{\mu\nu}+\nabla_{[\underline{\alpha}}\nabla_{\mu}\nabla_{\nu}\nabla_{\lambda}V_{\underline{\rho}]}^{\lambda(\mu\nu)}=0$
(34)

which is not a conservacion law as such, but rather a constraint on a
covariant derivative of a multi-index current. As has been already said,
Weyl’s invariance of Unimodular Gravity is, in some sense, tautological (this
is what Duff [9] calls pseudo-Weyl invariance), in the sense that there is a
field redefinition where the symmetry disappears.

We end this section with our main result, namely Noether’s current for the
Hilbert and Einstein Lagrangian, (The same Lagrangians are studied in the
paper [12] in the GR case).

###  2.1  Hilbert’s UG Lagrangian

The Unimodular Hilbert action is

$S_{\text{\tiny{H}}}[\gamma]\equiv-\frac{1}{2\kappa^{2}}\int d^{n}x\hat{R}$
(35)

where the metric is unimodular $\gamma=|\text{det}\gamma_{\mu\nu}|=1$. As has
been already remarked, it is possible to formulate the theory in such a way
that it has an added Weyl invariance by writing

$\gamma_{\mu\nu}=|g|^{-\frac{1}{n}}g_{\mu\nu}$ (36)

if we apply this transformation, the action for this new unrestricted metric
reads

$S_{\text{\tiny{H}}}[g]=-\frac{1}{2\kappa^{2}}\int
d^{n}xg^{\frac{1}{n}}\Big{[}R-\frac{(n-1)(5n-2)}{4n^{2}}g^{\mu\nu}\frac{\nabla_{\mu}g\nabla_{\nu}g}{g^{2}}+\frac{(n-1)}{n}\frac{\Box
g}{g}\Big{]}$ (37)

Under a general variation $\delta g_{\mu\nu}\equiv h_{\mu\nu}$ (forgetting the
prefactor $-{1\over 2\kappa^{2}}$ but keeping total derivatives)

$\delta S_{\text{\tiny{H}}}=\int d^{n}x\sqrt{|g|}\bigg{\\{}\frac{\delta
S}{\delta g_{\alpha\beta}}\delta
g_{\alpha\beta}+\nabla_{\mu}j^{\mu}\bigg{\\}}$ (38)

explicitly

$\displaystyle\delta
S_{\text{\tiny{H}}}=g^{1/n}\Bigg{\\{}\frac{1}{n}Rg_{\mu\nu}-R_{\mu\nu}+\frac{(n-2)(2n-1)}{4n^{2}}\left(\frac{\nabla_{\mu}g\nabla_{\nu}g}{g^{2}}-\frac{1}{n}\frac{\nabla_{\alpha}g\nabla^{\alpha}g}{g^{2}}g_{\mu\nu}\right)-\frac{n-2}{2n}\left(\frac{\nabla_{\mu}\nabla_{\nu}g}{g}-\frac{1}{n}\frac{\Box
g}{g}g_{\mu\nu}\right)\Bigg{\\}}\,h^{\mu\nu}+$
$\displaystyle+\sqrt{g}\Bigg{\\{}\nabla_{\mu}\Big{[}g^{\frac{2-n}{2n}}\nabla_{\nu}h^{\mu\nu}\Big{]}-\frac{1}{2}\nabla_{\nu}\Big{[}h^{\mu\nu}g^{\frac{2-n}{2n}}\frac{\nabla_{\mu}g}{g}\Big{]}-\frac{1}{n}\left(\nabla_{\lambda}\Big{[}g^{\frac{2-n}{2n}}\nabla^{\lambda}h\Big{]}-\frac{1}{2}\nabla_{\lambda}\Big{[}hg^{\frac{2-n}{2n}}\frac{\nabla^{\lambda}g}{g}\Big{]}\right)\Bigg{\\}}$
(39)

The corresponding equation of motion is Weyl invariant and, up to total
derivatives, traceless. This means that

$\displaystyle j^{\mu}$
$\displaystyle=g^{\frac{2-n}{2n}}\Big{[}\nabla_{\nu}h^{\mu\nu}-\frac{1}{2}h^{\mu\nu}\frac{\nabla_{\nu}g}{g}-\frac{1}{n}\left(\nabla^{\mu}h-\frac{1}{2}h\frac{\nabla^{\mu}g}{g}\right)\Big{]}$
(40)

and using the previous results, the total TDiff current reads

$j^{\mu}_{N}=\sqrt{g}\left(T^{\mu}_{\rho}\nabla_{\alpha}\Omega^{\alpha\rho}+U_{\rho}^{\mu\nu}\nabla_{\nu}\nabla_{\alpha}\Omega^{\alpha\rho}+V_{\rho}^{\mu(\lambda\tau)}\nabla_{\lambda}\nabla_{\tau}\nabla_{\alpha}\Omega^{\alpha\rho}\right)$
(41)

where

$\displaystyle T^{\mu}_{\rho}$
$\displaystyle=\delta^{\mu}_{\rho}g^{\frac{2-n}{2n}}\Big{[}R-\frac{(n-1)(5n-2)}{4n^{2}}g^{\alpha\beta}\frac{\nabla_{\alpha}g\nabla_{\beta}g}{g^{2}}+\frac{(n-1)}{n}\frac{\Box
g}{g}\Big{]}$ $\displaystyle U_{\rho}^{\mu\nu}$
$\displaystyle=\frac{1}{2}g^{\frac{2-3n}{2n}}\Big{[}g^{\mu\nu}\nabla_{\rho}g+\delta^{\mu}_{\rho}\nabla^{\nu}g-\frac{1}{n}\delta^{\nu}_{\rho}\nabla^{\mu}g\Big{]}$
$\displaystyle V_{\rho}^{\mu(\lambda\tau)}$
$\displaystyle=g^{\frac{2-n}{2n}}\Big{[}-g^{\lambda\tau}\delta^{\mu}_{\rho}-\frac{n-2}{2n}\left(\delta^{\lambda}_{\rho}g^{\mu\tau}+\delta^{\tau}_{\rho}g^{\mu\lambda}\right)\Big{]}$
(42)

In the unimodular Weyl gauge

$g_{\mu\nu}\rightarrow\gamma_{\mu\nu}\hskip 14.22636pt\text{with}\hskip
14.22636pt\gamma=|\text{det}\gamma_{\mu\nu}|=1$ (43)

these expressions reduce to

$\displaystyle\hat{T}^{\mu}_{\rho}$ $\displaystyle=\hat{R}\delta^{\mu}_{\rho}$
$\displaystyle\hat{U}_{\rho}^{\mu\nu}$ $\displaystyle=0$
$\displaystyle\hat{V}_{\rho}^{\mu(\lambda\tau)}$
$\displaystyle=-\gamma^{\lambda\tau}\delta^{\mu}_{\rho}-\frac{n-2}{2n}\left(\delta^{\lambda}_{\rho}\gamma^{\mu\tau}+\delta^{\tau}_{\rho}\gamma^{\mu\lambda}\right)$
(44)

we continue to denote the unimodular objects with hatted symbols.

As a trivial check, under a Weyl transformation

$\delta g_{\alpha\beta}=h_{\alpha\beta}=2\omega g_{\alpha\beta}$ (45)

the Weyl current (40) yields

$\displaystyle j_{W}^{\mu}$
$\displaystyle=g^{\frac{2-n}{2n}}\Big{[}2\nabla_{\nu}(g^{\mu\nu}\omega)-\omega
g^{\mu\nu}\frac{\nabla_{\mu}g}{g}-2\nabla^{\mu}\omega+\omega\frac{\nabla^{\mu}g}{g}\Big{]}=0$
(46)

which vanishes identically. This is of course due to the tautological
character of Weyl symmetry in our problem.

###  2.2  Einstein’s UG Lagrangian

The Unimodular Einstein action [18] reads

$S_{\text{\tiny{E}}}\equiv\frac{1}{2\kappa^{2}}\int
d^{n}x\gamma^{\mu\nu}\left(\hat{\Gamma}^{\rho}_{\lambda\rho}\hat{\Gamma}^{\lambda}_{\mu\nu}-\hat{\Gamma}^{\rho}_{\lambda\nu}\hat{\Gamma}^{\lambda}_{\mu\rho}\right)$
(47)

In terms of the unrestricted metric

$\displaystyle S_{\text{\tiny{E}}}$ $\displaystyle=\frac{1}{2\kappa^{2}}\int
d^{n}xg^{1/n}\Big{[}g^{\mu\nu}\left(\Gamma^{\rho}_{\lambda\rho}\Gamma^{\lambda}_{\mu\nu}-\Gamma^{\rho}_{\lambda\nu}\Gamma^{\lambda}_{\mu\rho}\right)-\frac{(n-2)}{2n}\left(g^{\mu\nu}\Gamma^{\lambda}_{\mu\nu}\frac{\nabla_{\lambda}g}{g}-\Gamma^{\lambda}_{\lambda\tau}\frac{\nabla^{\tau}g}{g}\right)-$
(48)
$\displaystyle-\frac{(n-1)(n-2)}{4n^{2}}\frac{\nabla_{\lambda}g\nabla^{\lambda}g}{g^{2}}\Big{]}$

Under a general variation $\delta g_{\mu\nu}\equiv h_{\mu\nu}$ (keeping total
derivatives)

$\delta S_{\text{\tiny{H}}}=\int d^{n}x\sqrt{|g|}\bigg{\\{}\frac{\delta
S}{\delta g_{\alpha\beta}}\delta
g_{\alpha\beta}+\nabla_{\mu}j^{\mu}\bigg{\\}}$ (49)

to be specific

$\displaystyle\delta
S_{E}=g^{1/n}\Bigg{\\{}\frac{1}{n}g_{\mu\nu}g^{\alpha\beta}\left(\Gamma^{\rho}_{\lambda\rho}\Gamma^{\lambda}_{\alpha\beta}-\Gamma^{\rho}_{\lambda\alpha}\Gamma^{\lambda}_{\beta\rho}\right)-\left(\Gamma^{\rho}_{\lambda\rho}\Gamma^{\lambda}_{\mu\nu}-\Gamma^{\rho}_{\lambda\nu}\Gamma^{\lambda}_{\mu\rho}\right)+\nabla_{\lambda}\Gamma^{\lambda}_{\mu\nu}-\frac{1}{n}g_{\mu\nu}g^{\alpha\beta}\nabla_{\lambda}\Gamma^{\lambda}_{\alpha\beta}-$
$\displaystyle-\nabla_{\nu}\Gamma^{\lambda}_{\lambda\mu}+\frac{1}{n}g_{\mu\nu}\nabla^{\tau}\Gamma^{\lambda}_{\lambda\tau}+\frac{(n-2)}{2n}\Big{(}\frac{\nabla_{\mu}\nabla_{\nu}g}{g}-\frac{1}{n}g_{\mu\nu}\frac{\Box
g}{g}\Big{)}-$
$\displaystyle-\frac{(n-2)(2n-1)}{4n^{2}}\Big{(}\frac{\nabla_{\mu}g\nabla_{\nu}g}{g^{2}}-\frac{1}{n}g_{\mu\nu}\frac{\nabla_{\lambda}g\nabla^{\lambda}g}{g^{2}}\Big{)}\Bigg{\\}}h^{\mu\nu}+$
$\displaystyle+\sqrt{|g|}\Bigg{\\{}-\nabla_{\lambda}\Big{[}g^{1/n-1/2}\Big{(}\Gamma^{\lambda}_{\mu\nu}h^{\mu\nu}-\frac{1}{n}g^{\alpha\beta}\Gamma^{\lambda}_{\alpha\beta}h\Big{)}\Big{]}+\nabla_{\nu}\Big{[}g^{1/n-1/2}\Big{(}\Gamma^{\lambda}_{\lambda\mu}h^{\mu\nu}-\frac{1}{n}g^{\mu\nu}\Gamma^{\lambda}_{\lambda\mu}h\Big{)}\Big{]}-$
$\displaystyle-\frac{(n-2)}{2n}\nabla_{\nu}\Big{[}g^{1/n-3/2}\Big{(}h^{\mu\nu}\nabla_{\mu}g-\frac{1}{n}hg^{\mu\nu}\nabla_{\mu}g\Big{)}\Big{]}\Bigg{\\}}$
(50)

the corresponding equation is Weyl invariant and, up to total derivatives,
traceless. The total derivative term is generated by

$\displaystyle j^{\mu}$
$\displaystyle=-g^{1/n-1/2}\Big{(}\Gamma^{\mu}_{\alpha\beta}h^{\alpha\beta}-\frac{1}{n}g^{\alpha\beta}\Gamma^{\mu}_{\alpha\beta}h\Big{)}+g^{1/n-1/2}\Big{(}\Gamma^{\lambda}_{\lambda\nu}h^{\mu\nu}-\frac{1}{n}g^{\mu\nu}\Gamma^{\lambda}_{\lambda\nu}h\Big{)}-$
(51)
$\displaystyle-\frac{n-2}{2n}g^{1/n-3/2}\Big{(}h^{\mu\nu}\nabla_{\nu}g-\frac{1}{n}hg^{\mu\nu}\nabla_{\nu}g\Big{)}$

and the Noether current is given by

$j^{\mu}_{N}=\sqrt{g}\left(T^{\mu}_{\rho}\nabla_{\alpha}\Omega^{\alpha\rho}+U_{\rho}^{\mu\nu}\nabla_{\nu}\nabla_{\alpha}\Omega^{\alpha\rho}+V_{\rho}^{\mu(\lambda\tau)}\nabla_{\lambda}\nabla_{\tau}\nabla_{\alpha}\Omega^{\alpha\rho}\right)$
(52)

with

$\displaystyle T^{\mu}_{\rho}$
$\displaystyle=\delta^{\mu}_{\rho}g^{\frac{2-n}{2n}}\Big{[}g^{\alpha\beta}\left(\Gamma^{\tau}_{\lambda\tau}\Gamma^{\lambda}_{\alpha\beta}-\Gamma^{\tau}_{\lambda\beta}\Gamma^{\lambda}_{\alpha\tau}\right)-\frac{(n-2)}{2n}\left(g^{\alpha\beta}\Gamma^{\lambda}_{\alpha\beta}\frac{\nabla_{\lambda}g}{g}-\Gamma^{\lambda}_{\lambda\tau}\frac{\nabla^{\tau}g}{g}\right)-$
$\displaystyle-\frac{(n-1)(n-2)}{4n^{2}}\frac{\nabla_{\lambda}g\nabla^{\lambda}g}{g^{2}}\Big{]}$
$\displaystyle U_{\rho}^{\mu\nu}$
$\displaystyle=g^{\frac{2-n}{2n}}\Big{[}2\,\Gamma^{\mu}_{\alpha\rho}g^{\alpha\nu}-\frac{2}{n}g^{\alpha\beta}\Gamma^{\mu}_{\alpha\beta}\delta^{\nu}_{\rho}-\Gamma^{\lambda}_{\lambda\rho}g^{\mu\nu}-\Gamma^{\lambda}_{\lambda\alpha}g^{\alpha\nu}\delta^{\mu}_{\rho}+$
$\displaystyle+\frac{2}{n}g^{\mu\tau}\Gamma^{\lambda}_{\lambda\tau}\delta^{\nu}_{\rho}-\frac{n-2}{2n}\left(g^{\mu\nu}\frac{\nabla_{\rho}g}{g}+\delta^{\mu}_{\rho}\frac{\nabla^{\nu}g}{g}-\frac{2}{n}\delta^{\nu}_{\rho}\frac{\nabla^{\mu}g}{g}\right)\Big{]}$
$\displaystyle V_{\rho}^{\mu(\lambda\tau)}$ $\displaystyle=0$ (53)

In the Weyl unimodular gauge these expressions reduce to

$\displaystyle\hat{T}^{\mu}_{\rho}$
$\displaystyle=-\delta^{\mu}_{\rho}\gamma^{\alpha\beta}\hat{\Gamma}^{\tau}_{\lambda\beta}\hat{\Gamma}^{\lambda}_{\alpha\tau}$
$\displaystyle\hat{U}_{\rho}^{\mu\nu}$
$\displaystyle=2\hat{\Gamma}^{\mu}_{\alpha\rho}\gamma^{\alpha\nu}-\frac{2}{n}\gamma^{\alpha\beta}\hat{\Gamma}^{\mu}_{\alpha\beta}\delta^{\nu}_{\rho}$
$\displaystyle\hat{V}_{\rho}^{\mu(\lambda\tau)}$ $\displaystyle=0$ (54)

Under a Weyl rescaling also in this case Weyl’s current

$j_{W}^{\mu}=0$ (55)

vanishes identically. It is interesting that this result holds true in spite
of the fact that Einstein’s and Hilbert’s lagrangians differ by a total
derivative. Again, this is true due to the tautological character of Weyl’s
symmetry in this problem.

Incidentally, the canonical energy-momentum tensor associated to
$\mathcal{L}_{E}$ is precisely Einstein’s energy-momentum pseudotensor.

$t^{\text{\tiny{E}}}_{\mu\nu}\equiv\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}g_{\alpha\beta})}\partial^{\nu}g_{\alpha\beta}-g^{\mu\nu}\mathcal{L}$
(56)

and it is conserved in the sense that

$\partial_{\mu}t^{\mu\nu}_{\text{\tiny{E}}}=0$ (57)

in detail

$\displaystyle t_{E}^{\mu\nu}$ $\displaystyle\equiv
g^{1/n}\Big{[}\Gamma^{\lambda}_{\alpha\beta}\Gamma^{\tau}_{\tau\lambda}g^{\alpha\mu}g^{\beta\nu}-\Gamma^{\lambda}_{\lambda\alpha}\Gamma^{\tau}_{\tau\beta}g^{\alpha\mu}g^{\beta\nu}+\frac{1}{2}\Gamma^{\lambda}_{\lambda\beta}\Gamma^{\mu}_{\rho\sigma}g^{\rho\sigma}g^{\beta\nu}+\frac{1}{2}\Gamma^{\lambda}_{\lambda\beta}\Gamma^{\nu}_{\rho\sigma}g^{\rho\sigma}g^{\beta\mu}-$
(58)
$\displaystyle-\Gamma^{\rho}_{\beta\lambda}\Gamma^{\mu}_{\rho\tau}g^{\lambda\tau}g^{\beta\nu}-\Gamma^{\rho}_{\beta\lambda}\Gamma^{\nu}_{\rho\tau}g^{\lambda\tau}g^{\beta\mu}+\frac{1}{2}\Gamma^{\rho}_{\rho\lambda}\Gamma^{\mu}_{\beta\tau}g^{\lambda\tau}g^{\beta\nu}+\frac{1}{2}\Gamma^{\rho}_{\rho\lambda}\Gamma^{\nu}_{\beta\tau}g^{\lambda\tau}g^{\beta\mu}-$
$\displaystyle-\frac{(n-2)}{2n}\Big{(}\Gamma^{\lambda}_{\alpha\beta}g^{\alpha\mu}g^{\beta\nu}\frac{\nabla_{\lambda}g}{g}+\frac{1}{2}\Gamma^{\mu}_{\beta\lambda}g^{\beta\nu}\frac{\nabla^{\lambda}g}{g}+\frac{1}{2}\Gamma^{\nu}_{\beta\lambda}g^{\beta\mu}\frac{\nabla^{\lambda}g}{g}-$
$\displaystyle-\Gamma^{\lambda}_{\beta\lambda}g^{\beta\nu}\frac{\nabla^{\mu}g}{g}-\Gamma^{\lambda}_{\beta\lambda}g^{\beta\mu}\frac{\nabla^{\nu}g}{g}\Big{)}\Big{]}-g^{\mu\nu}\mathcal{L}$

Finally, in the unimodular Weyl gauge, these expressions reduce to

$\hat{t}_{E}^{\mu\nu}=-\hat{\Gamma}^{\rho}_{\beta\lambda}\hat{\Gamma}^{\mu}_{\rho\tau}\gamma^{\lambda\tau}\gamma^{\beta\nu}-\hat{\Gamma}^{\rho}_{\beta\lambda}\hat{\Gamma}^{\nu}_{\rho\tau}\gamma^{\lambda\tau}\gamma^{\beta\mu}+\gamma^{\mu\nu}\gamma^{\alpha\beta}\hat{\Gamma}^{\rho}_{\lambda\alpha}\hat{\Gamma}^{\lambda}_{\beta\rho}$
(59)

this result coincide with the classical one in the Schrödinger book
[Schrodinger:2011gqa].

We have studied the conserved current to Hilbert and Einstein Lagrangian,
next, we want to apply our results in a particular case, the Schwarzschild
metric.

##  3  Schwarzschild’s UG spacetime.

As a simple application of the above, let us work out an explicit example,
namely Schwarzschild spacetime from the Unimodular Gravity viewpoint.
Schwarschild spacetime is still a solution of the equation of motion for
Unimodular Gravity, although more general spherically symmetric vacuum
solutions are possible. The reason is that for the usual Unimodular Hilbert
Lagrangian

$S_{\text{\tiny{H}}}[\gamma]\equiv-\frac{1}{2\kappa^{2}}\int d^{n}x\hat{R}$
(60)

the equation of motion (2.1), in the Weyl unitary gauge $\gamma=1$, reduces to
the simple form of

$\hat{R}_{\mu\nu}-{1\over n}\hat{R}\gamma_{\mu\nu}=0$ (61)

allowing Schwarzschild-(anti)-de Sitter spacetimes, in such a way that the
naive version of Birkhoff’s theorem is not valid.

We started with a particular set of coordinates for the Schwarzschild metric,
the Painlevé-Gullstrand (PG) coordinates [17]

$ds^{2}=\left(1-\frac{r_{s}}{r}\right)\,dt^{2}\mp 2\sqrt{\frac{r_{s}}{r}}dtdr-
dr^{2}-r^{2}\Big{[}\frac{d\psi^{2}}{1-\psi^{2}}+(1-\psi^{2})d\phi^{2}\Big{]}$
(62)

where $r_{s}=2GM$. In fact in [13] it was shown that with the upper sign this
coordinate system covers regions I and II in Kruskal’s diagram; whereas with
the lower sign it covers regions $I^{\prime}$ and $II^{\prime}$. This is
worked out in detail in [15].

This metric can be easily written in UG by means of a new coordinate $x$ such
that the function $r(x)$, such that

$r^{4}[x](r^{\prime}[x])^{2}=1$ (63)

which general solution is

$r=(3x+b)^{1/3}$ (64)

then the Painlevé-Gullstrand metric in unimodular gauge reads

$ds^{2}=\left(1-\frac{r_{s}}{(3x+b)^{1/3}}\right)\,dt^{2}-\frac{2\sqrt{r_{s}}}{(3x+b)^{5/6}}dtdx-\frac{1}{(3x+b)^{4/3}}dx^{2}-(3x+b)^{2/3}\Big{[}\frac{d\psi^{2}}{1-\psi^{2}}+(1-\psi^{2})d\phi^{2}\Big{]}$
(65)

The role of the constant $b$ is rather curious. As explained in [11]
Schwarzschild chose $b=r_{s}^{3}$, in order that the metric be regular
everywhere except for a singularity at $x=0$. This contributed to the initial
confusion on the Schwarzschild singularity.

In spite of the fact that they are one of our main results we have relegated
the explicit formulae to the appendix, owing to their unwieldly character.

##  4 Conclusions

In the Weyl invariant formulation of Unimodular Gravity the unimodular metric
is represented as

$\gamma_{\alpha\beta}\equiv g^{-\frac{1}{4}}\,g_{\alpha\beta}$ (66)

where the metric $g_{\alpha\beta}$ is unrestricted. This introduces a Weyl
abelian symmetry under

$g_{\alpha\beta}\rightarrow\Omega^{2}(x)\,g_{\alpha\beta}$ (67)

The manifold of solutions is then a set of Weyl gauge orbits, with one
representative in each orbit with a unimodular metric $\gamma_{\alpha\beta}$
and the other metrics in the orbit related to that one by Weyl rescalings

$g_{\alpha\beta}=g(x)^{1\over n}\gamma_{\alpha\beta}$ (68)

(with $g(x)$ an arbitrary function).

This Weyl invariance is a bit peculiar; we call it tautological, in the sense
that it disappears under a field redefinition. This translates into the fact
that the associated Weyl current vanishes identically (in Hilbert’s version)

$j_{W}^{\mu}=0$ (69)

Let us make a comment on Weyl invariance. The horizon $r=r_{S}$ in
Schwarzschild’s spacetime has an associated Bekenstein-Hawking entropy [7]
given by

$S={A\over 4G}=\pi{r_{s}^{2}\over G}$ (70)

which according to Wald [20] can be interpreted as the value of the Noether
charge associated to diffeomorphism invariance. In the Weyl orbit there are
static spacetimes without a Killing horizon (of course there are also non-
static spacetimes without any Killing). Given the fact that in the vicinity of
the horizon

$g_{00}\sim\frac{x-x_{s}}{r_{s}^{3}}+\ldots$ (71)

(with $x_{s}\equiv\frac{r_{s}^{3}-b}{3}$) the rescaling

$g(x)\sim\frac{1}{(x-x_{s})^{4}}$ (72)

is singular at $x=x_{S}$, and there are corresponding singularities in the
derivatives of $g(x)$. This is the the price to pay to get rid of the horizon.
If only nonsingular rescalings are accepted, then $g_{00}=0$ all along Weyl’s
orbit.

The issue treated in this paper is that physical characteristics of different
spacetimes (like total mass in the asymptotically flat case) are not Weyl
invariant. This means that in some sense naive TDiff conserved currents are
not constant on Weyl orbits. It seems desirable to have a definition of TDiff
conserved currents that is Weyl invariant.

What we provide in this paper is precisely such a Weyl gauge invariant
formulation of the TDiff currents, although we claim that the physical value
is the one corresponding to the unimodular gauge fixing. This is more or less
obvious owing to the way that Weyl invariance has been introduced in the
theory.

## Acknowledgments

This work has received funding from the Spanish Research Agency (Agencia
Estatal de Investigacion) through the grant IFT Centro de Excelencia Severo
Ochoa MCIN CEX2020-001007-S. This project has also received funding from the
European Union s Horizon 2020 research and innovation programme under the
Marie Sklodowska -Curie grant agreement No 860881-HIDDeN.

##  Appendix A  PG Hilbert’s current.

In this case the total current under TDiff reads

$j^{\mu}_{N}=\sqrt{g}\left(T^{\mu}_{\rho}\nabla_{\alpha}\Omega^{\alpha\rho}+U_{\rho}^{\mu\nu}\nabla_{\nu}\nabla_{\alpha}\Omega^{\alpha\rho}+V_{\rho}^{\mu(\lambda\tau)}\nabla_{\lambda}\nabla_{\tau}\nabla_{\alpha}\Omega^{\alpha\rho}\right)$
(73)

in the unimodular Weyl gauge reduces to

$\displaystyle\hat{T}^{\mu}_{\rho}$ $\displaystyle=0$
$\displaystyle\hat{U}_{\rho}^{\mu\nu}$ $\displaystyle=0$
$\displaystyle\hat{V}_{\rho}^{\mu(\lambda\tau)}$
$\displaystyle=-\gamma^{\lambda\tau}\delta^{\mu}_{\rho}-\frac{n-2}{2n}\left(\delta^{\lambda}_{\rho}\gamma^{\mu\tau}+\delta^{\tau}_{\rho}\gamma^{\mu\lambda}\right)$
(74)

owing to the fact that the scalar of curvature vanishes. In full detail, by
components

$\displaystyle V^{t(tt)}_{t}$ $\displaystyle=-2+\frac{2}{n}$ $\displaystyle
V^{t(tx)}_{t}$ $\displaystyle=V^{t(xt)}_{t}=\frac{(n-2)\sqrt{r(b+3x)}}{n}$
$\displaystyle V^{t(xx)}_{t}$
$\displaystyle=\frac{-2(b+3x)\left[(b+3x)^{1/3}-r\right]+n\Big{[}-1-3rx+3x(b+3x)^{1/3}+b\left[(b+3x)^{1/3}-r\right]\Big{]}}{n}$
$\displaystyle V^{t(\psi\psi)}_{t}$
$\displaystyle=\frac{-bn+2(b+3x)^{1/3}(-1+\psi^{2})+n\Big{[}-3x-(b+3x)^{1/3}(-1+\psi^{2})\Big{]}}{n(b+3x)}$
$\displaystyle V^{t(\phi\phi)}_{t}$
$\displaystyle=\frac{2(b+3x)^{1/3}+bn(1-\psi^{2})-n\Big{[}(b+3x)^{1/3}+3x(-1+\psi^{2})\Big{]}}{n(b+3x)(-1+\psi^{2})}$
(75) $\displaystyle V^{x(tt)}_{x}$
$\displaystyle=-1+\frac{2}{n}-3xr+3x(b+3x)^{1/3}+b\Big{[}(b+3x)^{1/3}-r\Big{]}$
$\displaystyle V^{x(tx)}_{x}$
$\displaystyle=V^{x(xt)}_{x}=\frac{(n-2)\sqrt{r_{s}(b+3x)}}{n}$ $\displaystyle
V^{x(xx)}_{x}$
$\displaystyle=\frac{2(n-1)(b+3x)\Big{[}(b+3x)^{1/3}-r\Big{]}}{n}$
$\displaystyle V^{x(\psi\psi)}_{x}$
$\displaystyle=(b+3x)\Big{[}(b+3x)^{1/3}-r\Big{]}-\frac{(n-2)(-1+\psi^{2})}{n(b+3x)^{2/3}}$
$\displaystyle V^{x(\phi\phi)}_{x}$
$\displaystyle=-3rx+b\Big{[}(b+3x)^{1/3}-r\Big{]}+\frac{3x(b+3x)+\frac{n-2}{n(-1+\psi^{2})}}{(b+3x)^{2/3}}$
(76) $\displaystyle V^{\psi(tt)}_{\psi}$
$\displaystyle=-\frac{b(n-2)+3(n-2)x+n(b+3x)^{1/3}(-1+\psi^{2})}{n(b+3x)}$
$\displaystyle V^{\psi(tx)}_{\psi}$
$\displaystyle=V^{\psi(xt)}_{\psi}=\frac{(n-2)\sqrt{r_{s}(b+3x)}}{n}$
$\displaystyle V^{\psi(xx)}_{\psi}$
$\displaystyle=\frac{r+\frac{1}{n}(n-2)(b+3x)^{2}\Big{[}(b+3x)^{1/3}-r\Big{]}-r\psi^{2}-\Big{[}(b+3x)^{1/3}-r\Big{]}(-1+\psi^{2})}{b+3x}$
$\displaystyle V^{\psi(\psi\psi)}_{\psi}$
$\displaystyle=-\frac{2(n-1)(-1+\psi^{2})}{n(b+3x)^{2/3}}$ $\displaystyle
V^{\psi(\phi\phi)}_{\psi}$
$\displaystyle=\frac{2-n(2-2\psi^{2}+\psi^{4})}{n(b+3x)^{2/3}(-1+\psi^{2})}$
(77) $\displaystyle V^{\phi(tt)}_{\phi}$
$\displaystyle=\frac{-n(b+3x)^{1/3}+b(n-2)(-1+\psi^{2})+3(n-2)x(-1+\psi^{2})}{n(b+3x)(-1+\psi^{2})}$
$\displaystyle V^{\phi(tx)}_{\phi}$
$\displaystyle=V^{\phi(xt)}_{\phi}=\frac{(n-2)\sqrt{r_{s}(b+3x)}}{n}$
$\displaystyle V^{\phi(xx)}_{\phi}$
$\displaystyle=\frac{\frac{1}{n}(n-2)(b+3x)^{2}\Big{[}(b+3x)^{1/3}-r\Big{]}-\frac{r}{-1+\psi^{2}}+\frac{1}{-1+\psi^{2}}\Big{[}(b+3x)^{1/3}-r\Big{]}}{b+3x}$
$\displaystyle V^{\phi(\psi\psi)}_{\phi}$
$\displaystyle=\frac{-1-\frac{1}{n}(n-2)(-1+\psi^{2})^{2}}{(b+3x)^{2/3}(-1+\psi^{2})}$
$\displaystyle V^{\phi(\phi\phi)}_{\phi}$
$\displaystyle=\frac{-2(n-1)}{n(b+3x)^{2/3}(-1+\psi^{2})}$ (78)

$V^{t(\lambda\tau)}_{x}=V^{x(\lambda\tau)}_{t}=\sqrt{r(b+3x)}$ (79)

##  Appendix B  PG Einstein’s current.

We are sorry for the formulas of this paragraph, but we did not find a nice
geometrical way to write them.

$\displaystyle\hat{T}^{\mu}_{\rho}$
$\displaystyle=-\delta^{\mu}_{\rho}\gamma^{\alpha\beta}\hat{\Gamma}^{\tau}_{\lambda\beta}\hat{\Gamma}^{\lambda}_{\alpha\tau}$
$\displaystyle\hat{U}_{\rho}^{\mu\nu}$
$\displaystyle=2\hat{\Gamma}^{\mu}_{\alpha\rho}\gamma^{\alpha\nu}-\frac{2}{n}\gamma^{\alpha\beta}\hat{\Gamma}^{\mu}_{\alpha\beta}\delta^{\nu}_{\rho}$
$\displaystyle\hat{V}_{\rho}^{\mu(\lambda\tau)}$ $\displaystyle=0$ (80)

in the Weyl unimodular gauge

$\hat{T}^{\mu}_{\rho}=\begin{pmatrix}\frac{2r}{b+3x}&\frac{2\sqrt{r}\left[(b+3x)^{1/3}-r\right]}{\sqrt{b+3x}}&0&0\\\
\frac{2\sqrt{r}}{(b+3x)^{3/2}}&\frac{2\left[(b+3x)^{1/3}-r\right]}{(b+3x)}&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ (81) $\displaystyle
U_{\rho}^{\mu\nu}=\text{by components}$ $\displaystyle U^{tt}_{t}$
$\displaystyle=\frac{r^{3/2}}{n(b+3x)^{5/6}}$ $\displaystyle U^{xx}_{t}$
$\displaystyle=\frac{-\sqrt{r}\left[r+(n-1)(b+3x)^{1/3}\right]}{n(b+3x)^{5/6}}$
$\displaystyle U^{\psi\psi}_{t}$
$\displaystyle=U^{\phi\phi}_{t}=\frac{2(n-1)\sqrt{r}}{n\sqrt{b+3x}}$
$\displaystyle U^{tx}_{t}$
$\displaystyle=-\frac{r}{n}\Big{[}-1+n+\frac{r}{(b+3x)^{1/3}}\Big{]}$
$\displaystyle U^{xt}_{t}$ $\displaystyle=\frac{r}{n(b+3x)^{4/3}}$

$\displaystyle U^{tt}_{x}$
$\displaystyle=\frac{r\left[(n-5)(b+3x)+4r(b+3x)^{2/3}\right]}{n(b+3x)}$
$\displaystyle U^{xx}_{x}$
$\displaystyle=\frac{1}{n}\Big{[}(7-3n)r-\frac{4r^{2}}{(b+3x)^{1/3}}+4(n-1)(b+3x)^{1/3}\Big{]}$
$\displaystyle U^{\psi\psi}_{x}$
$\displaystyle=U^{\phi\phi}_{x}=\frac{2(n-1)\left[(b+3x)^{1/3}-r\right]}{n}$
$\displaystyle U^{tx}_{x}$
$\displaystyle=\frac{-4\sqrt{r}(b+3x)^{1/6}\left[(b+3x)^{1/3}-r\right]^{2}}{n}$
$\displaystyle U^{xt}_{x}$
$\displaystyle=\frac{4\sqrt{r}\left[r+(n-1)(b+3x)^{1/3}\right]}{n(b+3x)^{5/6}}$

$\displaystyle U^{x\psi}_{\psi}$
$\displaystyle=\frac{2(-1+\psi^{2})}{(b+3x)^{5/3}}$ $\displaystyle U^{\psi
t}_{\psi}$ $\displaystyle=-\frac{2\sqrt{r}}{\sqrt{b+3x}}$ $\displaystyle
U^{\psi x}_{\psi}$ $\displaystyle=2\left[(b+3x)^{1/3}-r\right]$ $\displaystyle
U^{\psi\psi}_{\psi}$
$\displaystyle=U^{\phi\phi}_{\psi}=\frac{-2(n-1)\psi}{n(b+3x)^{2/3}}$

$\displaystyle U^{x\phi}_{\phi}$
$\displaystyle=\frac{2}{(b+3x)^{5/3}(-1+\psi^{2})}$ $\displaystyle
U^{\psi\phi}_{\psi}$
$\displaystyle=\frac{2\psi}{(b+3x)^{2/3}(-1+\psi^{2})^{2}}$ $\displaystyle
U^{\phi t}_{\phi}$ $\displaystyle=-\frac{2\sqrt{r}}{\sqrt{b+3x}}$
$\displaystyle U^{\phi x}_{\phi}$ $\displaystyle=2\left[(b+3x)^{1/3}-r\right]$
$\displaystyle U^{\phi\psi}_{\phi}$ $\displaystyle=\frac{2\psi}{(b+3x)^{2/3}}$

Finally, in the unimodular Weyl gauge, the energy-momentum pseudotensor

$\hat{t}_{\text{\tiny{E}}}^{\mu\nu}=-\hat{\Gamma}^{\rho}_{\beta\lambda}\hat{\Gamma}^{\mu}_{\rho\tau}\gamma^{\lambda\tau}\gamma^{\beta\nu}-\hat{\Gamma}^{\rho}_{\beta\lambda}\hat{\Gamma}^{\nu}_{\rho\tau}\gamma^{\lambda\tau}\gamma^{\beta\mu}+\gamma^{\mu\nu}\gamma^{\alpha\beta}\hat{\Gamma}^{\rho}_{\lambda\alpha}\hat{\Gamma}^{\lambda}_{\beta\rho}$
(82)

for the the Painlevé-Gullstrand metric conducts to

$\hat{t}_{\text{\tiny{E}}}^{\mu\nu}=\begin{pmatrix}\hat{t}^{tt}_{\text{\tiny{E}}}&0&0&0\\\
0&\hat{t}^{rr}_{\text{\tiny{E}}}&\hat{t}^{r\psi}_{\text{\tiny{E}}}&0\\\
0&\hat{t}^{\psi r}_{\text{\tiny{E}}}&\hat{t}^{\psi\psi}_{\text{\tiny{E}}}&0\\\
0&0&0&\hat{t}^{\phi\phi}_{\text{\tiny{E}}}\end{pmatrix}$ (83)

in full detail

$\displaystyle\hat{t}^{tt}_{\text{\tiny{E}}}$
$\displaystyle=\frac{1}{2}\Bigg{\\{}-\frac{r_{s}^{2}}{(1+3x)^{4/3}}-\frac{r_{s}^{2}}{(2+3x)^{2}\bigl{(}(2+3x)^{1/3}-r_{s}\bigr{)}^{2}}+\frac{(3x+b)^{1/3}}{(3x+b)^{1/3}-r_{s}}f[x,\psi,r_{s}]\Bigg{\\}}$
$\displaystyle\hat{t}^{rr}_{\text{\tiny{E}}}$
$\displaystyle=-\frac{r_{s}^{2}}{2(1+3x)^{4/3}}-\frac{\bigl{(}8+12x-3(2+3x)^{2/3}r_{s}\bigr{)}^{2}}{2(2+3x)^{10/3}\bigl{(}(2+3x)^{1/3}-r_{s}\bigr{)}^{2}}-\frac{2\bigl{(}3+3x-(3+3x)^{2/3}r_{s}\bigr{)}}{3(1+x)(-1+\psi^{2})}-$
$\displaystyle-\frac{2\bigl{(}(4+3x)^{1/3}-r_{s}\bigr{)}(-1+\psi^{2})}{(4+3x)^{1/3}}-\frac{1}{2}(3x+b)\left[(3x+b)^{1/3}-r_{s}\right]f[x,\psi,r_{s}]$
$\displaystyle\hat{t}^{\psi\psi}_{\text{\tiny{E}}}$
$\displaystyle=-\frac{2}{(2+3x)^{2}}+2\psi^{2}-\frac{2\psi^{2}}{(-1+\psi^{2})^{2}}-\frac{2\bigl{(}3+3x-(3+3x)^{2/3}r_{s}\bigr{)}}{3(1+x)(-1+\psi^{2})}+\frac{(-1+\psi^{2})}{2(3x+b)^{2/3}}f[x,\psi,r_{s}]$
$\displaystyle\hat{t}^{\phi\phi}_{\text{\tiny{E}}}$
$\displaystyle=2-\frac{2}{(2+3x)^{2}}-\frac{2\psi^{2}}{(-1+\psi^{2})^{2}}+\frac{2r_{s}(-1+\psi^{2})}{(4+3x)^{1/3}}+\frac{1}{2(3x+b)^{2/3}(-1+\psi^{2})}f[x,\psi,r_{s}]$
$\displaystyle\hat{t}^{r\psi}_{\text{\tiny{E}}}$ $\displaystyle=\hat{t}^{\psi
r}_{\text{\tiny{E}}}=\psi\Bigg{\\{}-4-3x+(4+3x)^{2/3}r_{s}+\frac{3+3x-3^{2/3}(1+x)^{2/3}r_{s}}{(-1+\psi^{2})^{2}}+$
$\displaystyle+\frac{1}{(3+3x)(-1+\psi^{2})}-\frac{1-\psi^{2}}{4+3x}\Bigg{\\}}$

where

$\displaystyle f[x,\psi,r_{s}]=$
$\displaystyle\frac{2}{(2+3x)^{2}}+\frac{1}{(3+3x)^{2}}+\frac{1}{(4+3x)^{2}}+\frac{3r_{s}^{2}}{4(1+3x)^{4/3}}+\frac{r_{s}^{2}}{4(2+3x)^{4/3}}+$
(85)
$\displaystyle+\frac{r_{s}^{2}}{4(1+3x)^{2}\bigl{(}(1+3x)^{1/3}-r_{s}\bigr{)}^{2}}+\frac{r_{s}^{2}}{4(2+3x)^{2}\bigl{(}(2+3x)^{1/3}-r_{s}\bigr{)}^{2}}+$
$\displaystyle+\frac{\bigl{(}8+12x-3(2+3x)^{2/3}r_{s}\bigr{)}^{2}}{2(2+3x)^{10/3}\bigl{(}(2+3x)^{1/3}-r_{s}\bigr{)}^{2}}-4\psi^{2}+\frac{4\psi^{2}}{(-1+\psi^{2})^{2}}+$
$\displaystyle+\frac{3+3x-3^{2/3}(1+x)^{2/3}r_{s}}{(1+x)(-1+\psi^{2})}+\frac{2+3x-(2+3x)^{2/3}r_{s}}{(2+3x)(-1+\psi^{2})}+$
$\displaystyle+\frac{\bigl{(}(2+3x)^{1/3}-r_{s}\bigr{)}(-1+\psi^{2})}{(2+3x)^{1/3}}+\frac{3\bigl{(}(4+3x)^{1/3}-r_{s}\bigr{)}(-1+\psi^{2})}{(4+3x)^{1/3}}$

It this formalism, just by construction, the values of the physical energy
/mass

$m=\int t^{0}_{0}dx\wedge d\psi\wedge d\phi$ (86)

are the same in the whole Weyl orbit generated by $\Omega(x)$ (that is, they
are independent of the Weyl rescaling $g$).

## References

  * [1] A. Alonso-Serrano, L. J. Garay and M. Liška, “Noether charge formalism for Weyl invariant theories of gravity”, Phys. Rev. D 106, no.6, 064024 (2022) doi:10.1103/PhysRevD.106.064024 [arXiv:2206.08746 [gr-qc]].
”Noether charge formalism for Weyl transverse gravity”, Class.Quant.Grav. 40
(2023) 2, 025012.

  * [2] E. Álvarez and E. Velasco-Aja, “A Primer on Unimodular Gravity,” [arXiv:2301.07641 [gr-qc]].
R. Carballo-Rubio, L. J. Garay and G. García-Moreno, “Unimodular gravity vs
general relativity: a status report,” Class. Quant. Grav. 39 (2022) no.24,
243001 doi:10.1088/1361-6382/aca386 [arXiv:2207.08499 [gr-qc]].

  * [3] E. Álvarez, “Windows on Quantum Gravity”, Fortsch. Phys. 69, no.1, 2000080 (2021) doi:10.1002/prop.202000080 [arXiv:2005.09466 [hep-th]].
E. Álvarez and J. Anero, “Covariant Techniques in Quantum Field Theory”,
Springer, 2022, ISBN 978-3-031-13725-9, 978-3-031-13726-6
doi:10.1007/978-3-031-13726-6

  * [4] E. Álvarez and J. Anero, “Unimodular Cosmological models,” [arXiv:2109.08077 [gr-qc]].
“Some remarks on the Hamiltonian for unimodular gravity”, Phys. Rev. D 104
(2021) no.8, 084096 doi:10.1103/PhysRevD.104.084096 [arXiv:2107.08893 [gr-
qc]].

  * [5] E. Álvarez and M. Herrero-Valea, “Unimodular gravity with external sources”, JCAP 01, 014 (2013) doi:10.1088/1475-7516/2013/01/014 [arXiv:1209.6223 [hep-th]].
  * [6] E. Álvarez, S. González-Martín, M. Herrero-Valea and C. P. Martín, “Quantum Corrections to Unimodular Gravity,” JHEP 08, 078 (2015) doi:10.1007/JHEP08(2015)078 [arXiv:1505.01995 [hep-th]].
  * [7] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7, 2333-2346 (1973) doi:10.1103/PhysRevD.7.2333
S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43,
199-220 (1975) [erratum: Commun. Math. Phys. 46, 206 (1976)]
doi:10.1007/BF02345020

  * [8] E. Álvarez, J. Anero, C. P. Martin and E. Velasco-Aja, “The quantum dynamics of Lagrange multipliers,” [arXiv:2304.05188 [hep-th]].
  * [9] M. J. Duff, “Twenty years of the Weyl anomaly,” Class. Quant. Grav. 11, 1387-1404 (1994) doi:10.1088/0264-9381/11/6/004 [arXiv:hep-th/9308075 [hep-th]].
  * [10] A. Eichhorn, “On unimodular quantum gravity,” Class. Quant. Grav. 30, 115016 (2013) doi:10.1088/0264-9381/30/11/115016 [arXiv:1301.0879 [gr-qc]].
  * [11] P. Fromholz, E. Poisson and C. M. Will, “The Schwarzschild metric: It’s the coordinates, stupid!,” Am. J. Phys. 82 (2014), 295 doi:10.1119/1.4850396 [arXiv:1308.0394 [gr-qc]].
  * [12] B. Julia and S. Silva, Class. Quant. Grav. 15 (1998), 2173-2215 doi:10.1088/0264-9381/15/8/006 [arXiv:gr-qc/9804029 [gr-qc]].
  * [13] P. Kraus and F. Wilczek, “Some applications of a simple stationary line element for the Schwarzschild geometry”, Mod. Phys. Lett. A 9, 3713-3719 (1994) doi:10.1142/S0217732394003567 [arXiv:gr-qc/9406042 [gr-qc]].
  * [14] M. Henneaux and C. Teitelboim, “The Cosmological Constant and General Covariance,” Phys. Lett. B 222 (1989), 195-199 doi:10.1016/0370-2693(89)91251-3
T. Kugo, R. Nakayama and N. Ohta, “Covariant BRST quantization of unimodular
gravity: Formulation with antisymmetric tensor ghosts,” Phys. Rev. D 105,
no.8, 086006 (2022) doi:10.1103/PhysRevD.105.086006 [arXiv:2202.03626 [hep-
th]].

  * [15] J. P. S. Lemos and D. L. F. G. Silva, “Maximal extension of the Schwarzschild metric: From Painlevé-Gullstrand to Kruskal-Szekeres”, Annals Phys. 430, 168497 (2021) doi:10.1016/j.aop.2021.168497 [arXiv:2005.14211 [gr-qc]].
  * [16] Kathryn Mann, “Minicourse: The algebraic structure of diffeomorphism groups”, https://math.berkeley.edu/ kpmann/algdiff.pdf.
  * [17] Paul Painlevé, ”La m canique classique et la th orie de la relativit ”, C. R. Acad. Sci. (Paris) 173, 677-680(1921).
Gullstrand, Allvar (1922). ”Allgemeine L sung des statischen Eink rperproblems
in der Einsteinschen Gravitationstheorie”. Arkiv f r Matematik, Astronomi och
Fysik. 16 (8): 1-15.

  * [18] W. Pauli, ”Theory of relativity” (Dover, 2013)
  * [19] Schrödinger, Erwin, “Space-Time Structure”, 978-0-521-31520-3, Cambridge University Press, (2011)
  * [20] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D 48, no.8, R3427-R3431 (1993) doi:10.1103/PhysRevD.48.R3427 [arXiv:gr-qc/9307038 [gr-qc]].
V. Iyer and R. M. Wald, “A Comparison of Noether charge and Euclidean methods
for computing the entropy of stationary black holes,” Phys. Rev. D 52,
4430-4439 (1995) doi:10.1103/PhysRevD.52.4430 [arXiv:gr-qc/9503052 [gr-qc]].
“Some properties of Noether charge and a proposal for dynamical black hole
entropy,” Phys. Rev. D 50, 846-864 (1994) doi:10.1103/PhysRevD.50.846
[arXiv:gr-qc/9403028 [gr-qc]].


* (55) D. Anninos and B. Mühlmann, “The semiclassical gravitational path integral and random matrices (toward a microscopic picture of a dS2 universe),” JHEP 12 (2021) 206, arXiv:2111.05344 [hep-th].
  * (56) B. Mühlmann, “The two-sphere partition function from timelike Liouville theory at three-loop order,” JHEP 05 (2022) 057, arXiv:2202.04549 [hep-th].
  * (57) B. Mühlmann, “The two-sphere partition function in two-dimensional quantum gravity at fixed area,” JHEP 09 no. 189, (2021) 189, arXiv:2106.04532 [hep-th].
  * (58) D. Anninos, F. Denef, Y. T. A. Law, and Z. Sun, “Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions,” JHEP 01 (2022) 088, arXiv:2009.12464 [hep-th].
  * (59) D. Anninos, “De Sitter Musings,” Int. J. Mod. Phys. A 27 (2012) 1230013, arXiv:1205.3855 [hep-th].
  * (60) M. Spradlin, A. Strominger, and A. Volovich, “Les Houches lectures on de Sitter space,” in Les Houches Summer School: Session 76: Euro Summer School on Unity of Fundamental Physics: Gravity, Gauge Theory and Strings, pp. 423–453.  10, 2001.  arXiv:hep-th/0110007.
  * (61) E. Schrodinger, Expanding universes.  Cambridge University Press, 1956.
  * (62) P. A. M. Dirac, “The Electron Wave Equation in De-Sitter Space,” Annals Math. 36 (1935) 657–669.
  * (63) P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory.  Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997\.
  * (64) S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2023.
  * (65) U. Ottoson, “A classification of the unitary irreducible representations of ${\rm SO}_{0}\,(N,\,1)$,” Communications in Mathematical Physics 8 no. 3, (1968) 228 – 244. https://doi.org/.
  * (66) F. Schwarz, “Unitary irreducible representations of the groups so0(n, 1),” Journal of Mathematical Physics 12 no. 1, (1971) 131–139, https://doi.org/10.1063/1.1665471. https://doi.org/10.1063/1.1665471.
  * (67) M. Boers, “Group theory and de Sitter QFT: The concept of mass,” Master’s thesis, Groningen University, 2013.
  * (68) K. Hinterbichler and A. Joyce, “Manifest Duality for Partially Massless Higher Spins,” JHEP 09 (2016) 141, arXiv:1608.04385 [hep-th].
  * (69) S. Deser and R. I. Nepomechie, “Anomalous Propagation of Gauge Fields in Conformally Flat Spaces,” Phys. Lett. B 132 (1983) 321–324.
  * (70) A. Higuchi, “Symmetric Tensor Spherical Harmonics on the $N$ Sphere and Their Application to the De Sitter Group SO($N$,1),” J. Math. Phys. 28 (1987) 1553. [Erratum: J.Math.Phys. 43, 6385 (2002)].
  * (71) D. Anninos, T. Bautista, and B. Mühlmann, “The two-sphere partition function in two-dimensional quantum gravity,” JHEP 09 (2021) 116, arXiv:2106.01665 [hep-th].
  * (72) A. Higuchi, “Symmetric tensor fields in de Sitter spacetime,” YTP-85-22, Yale preprint (November, 1985) .
  * (73) N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1982.
  * (74) O. Babelon and C.-M. Viallet, “The geometrical interpretation of the faddeev-popov determinant,” Physics Letters B 85 no. 2, (1979) 246–248. https://www.sciencedirect.com/science/article/pii/0370269379905896.
  * (75) P. O. Mazur and E. Mottola, “The path integral measure, conformal factor problem and stability of the ground state of quantum gravity,” Nuclear Physics B 341 no. 1, (1990) 187–212. https://www.sciencedirect.com/science/article/pii/055032139090268I.
  * (76) Z. Bern, S. K. Blau, and E. Mottola, “General covariance of the path integral for quantum gravity,” Phys. Rev. D 43 (Feb, 1991) 1212–1222. https://link.aps.org/doi/10.1103/PhysRevD.43.1212.
  * (77) D. Vassilevich, “Heat kernel expansion: user’s manual,” Physics Reports 388 no. 5, (2003) 279–360. https://www.sciencedirect.com/science/article/pii/S0370157303003545.
  * (78) Harish-Chandra, “On the characters of a semisimple Lie group,” Bulletin of the American Mathematical Society 61 no. 5, (1955) 389 – 396.
  * (79) Harish-Chandra, “Invariant eigendistributions on semisimple Lie groups,” Bulletin of the American Mathematical Society 69 no. 1, (1963) 117 – 123.
  * (80) M. Atiyah, M. F. Atiyah, R. Bott, S. Helgason, D. Kazhdan, B. Kostant, and G. Lustztig, “The Harish-Chandra character”, in Representation Theory of Lie Groups.  London Mathematical Society Lecture Note Series. Cambridge University Press, 1980.
  * (81) T. Hirai, “The characters of irreducible representations of the Lorentz group of $n$-th order,” Proceedings of the Japan Academy 41 no. 7, (1965) 526 – 531. https://doi.org/10.3792/pja/1195522333.
  * (82) T. Hirai, “On irreducible representations of the Lorentz group of $n$-th order,” Proceedings of the Japan Academy 38 no. 6, (1962) 258 – 262. https://doi.org/10.3792/pja/1195523378.
  * (83) T. Hirai, “On infinitesimal operators of irreducible representations of the Lorentz group of $n$-th order,” Proceedings of the Japan Academy 38 no. 3, (1962) 83 – 87. https://doi.org/10.3792/pja/1195523460.
  * (84) W. Donnelly and A. C. Wall, “Unitarity of Maxwell theory on curved spacetimes in the covariant formalism,” Phys. Rev. D 87 no. 12, (2013) 125033, arXiv:1303.1885 [hep-th].
  * (85) S. Giombi, I. R. Klebanov, and G. Tarnopolsky, “Conformal QEDd, $F$-Theorem and the $\epsilon$ Expansion,” J. Phys. A 49 no. 13, (2016) 135403, arXiv:1508.06354 [hep-th].
  * (86) H. Casini, M. Huerta, J. M. Magán, and D. Pontello, “Logarithmic coefficient of the entanglement entropy of a Maxwell field,” Phys. Rev. D 101 no. 6, (2020) 065020, arXiv:1911.00529 [hep-th].
  * (87) J. S. Dowker, “Entanglement entropy for even spheres,” arXiv:1009.3854 [hep-th].
  * (88) L. Lindblom, N. W. Taylor, and F. Zhang, “Scalar, Vector and Tensor Harmonics on the Three-Sphere,” Gen. Rel. Grav. 49 no. 11, (2017) 139, arXiv:1709.08020 [gr-qc].
  * (89) G. W. Gibbons, “Spectral Asymmetry and Quantum Field Theory in Curved Space-time,” Annals Phys. 125 (1980) 98.
  * (90) A. Higuchi, D. Marolf, and I. A. Morrison, “On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT,” Phys. Rev. D 83 (2011) 084029, arXiv:1012.3415 [gr-qc].
  * (91) D. Schlingemann, “Euclidean field theory on a sphere,” arXiv:hep-th/9912235.
  * (92) G. Sengör and C. Skordis, “Unitarity at the Late time Boundary of de Sitter,” JHEP 06 (2020) 041, arXiv:1912.09885 [hep-th].
  * (93) G. Sengor and C. Skordis, “Principal and Complementary Series Representations at the Late-Time Boundary of de Sitter,” Springer Proc. Math. Stat. 396 (2022) 269–276, arXiv:2205.11550 [hep-th].
  * (94) G. Şengör, “Particles of a de Sitter Universe,” Universe 9 no. 2, (2023) 59, arXiv:2212.10626 [hep-th].
  * (95) E. Witten, “On S duality in Abelian gauge theory,” Selecta Math. 1 (1995) 383, arXiv:hep-th/9505186.
  * (96) E. Witten, “Quantum Field Theory and the Jones Polynomial,” Commun. Math. Phys. 121 (1989) 351–399.
  * (97) D. Anninos and E. Harris, “Three-dimensional de Sitter horizon thermodynamics,” JHEP 10 (2021) 091, arXiv:2106.13832 [hep-th].
  * (98) A. Kitaev and J. Preskill, “Topological entanglement entropy,” Phys. Rev. Lett. 96 (2006) 110404, arXiv:hep-th/0510092.
  * (99) G. Tian and S.-T. Yau, “Kähler-einstein metrics on complex surfaces withc1¿0,” Communications in Mathematical Physics 112 no. 1, (1987) 175–203. https://doi.org/10.1007/BF01217685.
  * (100) G. Tian, “On calabi’s conjecture for complex surfaces with positive first chern class,” Inventiones mathematicae 101 no. 1, (1990) 101–172. https://doi.org/10.1007/BF01231499.
  * (101) G. Kallen, “On the definition of the Renormalization Constants in Quantum Electrodynamics,” Helv. Phys. Acta 25 no. 4, (1952) 417.
  * (102) H. Lehmann, “On the Properties of propagation functions and renormalization contants of quantized fields,” Nuovo Cim. 11 (1954) 342–357.
  * (103) M. Loparco, J. Penedones, K. Salehi Vaziri, and Z. Sun, “The Källén-Lehmann representation in de Sitter spacetime,” arXiv:2306.00090 [hep-th].
  * (104) S. Hollands, “Massless interacting quantum fields in deSitter spacetime,” Annales Henri Poincare 13 (2012) 1039–1081, arXiv:1105.1996 [gr-qc].
  * (105) L. Di Pietro, V. Gorbenko, and S. Komatsu, “Analyticity and unitarity for cosmological correlators,” JHEP 03 (2022) 023, arXiv:2108.01695 [hep-th].
  * (106) J. Bros, “Complexified de Sitter space: Analytic causal kernels and Kallen-Lehmann type representation,” Nucl. Phys. B Proc. Suppl. 18 (1991) 22–28.
  * (107) J. Bros and U. Moschella, “Two point functions and quantum fields in de Sitter universe,” Rev. Math. Phys. 8 (1996) 327–392, arXiv:gr-qc/9511019.
  * (108) J. Bros, H. Epstein, M. Gaudin, U. Moschella, and V. Pasquier, “Triangular invariants, three-point functions and particle stability on the de Sitter universe,” Commun. Math. Phys. 295 (2010) 261–288, arXiv:0901.4223 [hep-th].
  * (109) P. V. Buividovich and M. I. Polikarpov, “Numerical study of entanglement entropy in SU(2) lattice gauge theory,” Nucl. Phys. B 802 (2008) 458–474, arXiv:0802.4247 [hep-lat].
  * (110) D. Anninos, V. Letsios, A. Rios Fukelman, M. Sempe, and G. Silva, “In preparation,,”.
  * (111) J. R. David and J. Mukherjee, “Partition functions of p-forms from Harish-Chandra characters,” JHEP 09 (2021) 094, arXiv:2105.03662 [hep-th].
  * (112) D. Anninos, F. Denef, R. Monten, and Z. Sun, “Higher Spin de Sitter Hilbert Space,” JHEP 10 (2019) 071, arXiv:1711.10037 [hep-th].
  * (113) P. Goldbart and M. Stone, “Mathematics for physics: A guided tour for graduate students,” Cambridge University Press (2009) .
  * (114) G. W. Gibbons, “Part III: Applications of Differential Geometry to Physics,” DAMTP Lecture notes, Unpublished .
  * (115) K. Kumar and O. Lechtenfeld, “On rational electromagnetic fields,” Phys. Lett. A 384 (2020) 126445, arXiv:2002.01005 [hep-th].
  * (116) U. H. Gerlach and U. K. Sengupta, “Homogeneous Collapsing Star: Tensor and Vector Harmonics for Matter and Field Asymmetries,” Phys. Rev. D 18 (1978) 1773–1784.
  * (117) I. R. Klebanov, S. S. Pufu, S. Sachdev, and B. R. Safdi, “Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors,” JHEP 05 (2012) 036, arXiv:1112.5342 [hep-th].