# Noether current of the surface term of Einstein-Hilbert action, Virasoro
algebra and entropy

Bibhas Ranjan Majhi
IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India
E-mail<EMAIL_ADDRESS>
###### Abstract:

A derivation of Noether current from the surface term of Einstein-Hilbert
action is given. We show that the corresponding charge, calculated on the
horizon, is related to the Bekenstein-Hawking entropy. Also using the charge,
the same entropy is found based on the Virasoro algebra and Cardy formula
approach. In this approach, the relevant diffeomorphisms are found by imposing
a very simple physical argument: diffeomorphisms keep the horizon structure
invariant. This complements similar earlier results [27](arXiv:1204.1422)
obtained from York-Gibbons-Hawking surface term. Finally we discuss the
technical simplicities and improvements over the earlier attempts and also
various important physical implications.

Classical Theories of Gravity, Virasoro Algebra, Cardy formula, Entropy

##  1 Introduction

The thermodynamic properties of horizon arises from the combination of the
general theory of relativity and the quantum field theory. This was first
observed in the case of black holes [1, 2]. Now it is evident that it is much
more general and a local Rindler observer can attribute temperature and
entropy to the null surfaces in the context of the emergent paradigm of
gravity [3, 4]. Such a generality might provide us a deeper insight towards
the quantum nature of the spacetime. So far several attempts have been made to
know the microscopic origin of the entropy, but every method has its own
merits and demerits. Among others, Carlip made an attempt [5, 6] in the
context of Virasoro algebra to illuminate this aspect which is basically the
generalisation of the method by Brown and Hanneaux [7]. In brief, in this
method one first defines a bracket among the Noether charges and calculate it
for certain diffeomorphisms, chosen by some physical considerations. It turns
out that the algebra is identical to the Virasoro algebra. The central charge
and the zero mode eigenvalue of the Fourier modes of the charge are then
automatically identified which after substituting in the Cardy formula [8, 9]
one finds the Bekenstein-Hawking entropy 111For a complete list of works which
lead to further development of this method, see [10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25].. In all the previous attempts, the
Noether current was taken related to the Einstein-Hilbert (EH) action and the
analysis was on-shell, i.e. equation of motion has been used explicitly. Later
an off-shell analysis and a generalization to Lanczos-Lovelock gravity have
been presented in [26]. Earlier [27], based on the Virasoro algebra approach,
we showed that the entropy can also be obtained from the Noether current
corresponding to the York-Gibbons-Hawking surface term. But it is not clear if
the same can be achived from the surface term of the Einstein-Hilbert action,
since they are not exactly identical. So it is necessary to investigate this
issue in the light of Virasoro algebra context, particulary because both the
surface terms lead to the same entropy on the horizon. This will complement
our earlier work [27].

In this paper we will use the Noether current associated to the surface term
of EH action. Before going into the motivations for taking the surface term
only, let us first highlight some peculiar facts of EH action which are
essential for the present purpose. $\bullet$ It is an unavoidable fact that to
obtain the equation of motion in the Lagrangian formalism one has to impose
some extra prescription, like adding extra boundary term (in this case York-
Gibbons-Hawking term). This is because the action contains second order
derivative of metric tensor $g_{ab}$. But unfortunately the choice of the
surface term is not unique. This is quite different from other well known
field theories. $\bullet$ The EH action can be separated into two terms: one
contains the squires of the Christoffel connections ( i. e. it is in
$\Gamma\Gamma-\Gamma\Gamma$ structure) and the other one contains the total
derivation of $\Gamma$ ($\partial\Gamma-\partial\Gamma$ structure). We will
call them as $L_{quad}$ and $L_{sur}$, respectively. Interestingly, Einstein’s
equation of motion can be obtained solely from $L_{quad}$ by using the usual
variation principle where no additional prescription is not required [28].
$\bullet$ The most important one is that these two terms are related by an
algebraic relation, usually known as holographic relation [29, 30].
Interestingly, all the above features are happened to be common even for the
Lanczos-Lovelock theory [31]. For a recent review in this direction, see [32].

Although an extensive study on the Noether current of gravity has been done
starting from Wald [33], discussion on the current derived from $L_{sur}$ is
still lacking. To motivate why one should be interested, let us summarise
below the already observed facts. $\bullet$ It is expected that the entropy is
associated to the degrees of freedom around or on the relevant null surface
rather than the bulk geometry of spacetime. $\bullet$ This surface term
calculated on the Rindler horizon gives exactly the Bekenstein-Hawking entropy
[28]. $\bullet$ Extremization of the surface term with respect to the
diffeomorphism parameter whose norm is a constant, leads to the Einstein’s
equation [29]. $\bullet$ Another interesting fact is that in a small region
around an event, EH action reduces to a pure surface term when evaluated in
the Riemann normal coordinates. All these indicate that either the bulk and
the surface terms are duplicating all the information or the actual dynamics
is stored in surface term rather than in bulk term. To illuminate more on this
issue, one needs to study every aspect of the surface term.

In this paper, we shall discuss the Noether realization of the surface term of
the EH action, particularly we shall examine if the Noether current represents
the Virasoro algebra for a certain class of diffeomorphisms. This is necessary
to have a deeper understanding of the role of the surface term in the gravity.
Also it will give a further insight towards the earlier claim: the actual
information of the gravity is stored in the surface. To do this explicitly, we
shall consider the form of the metric close to the null surface in the local
Rindler frame around some event. This is given by the Rindler metric. The
reasons for choosing such metric are as follows. According to equivalence
principle, gravity can be mimicked by an accelerated observer and an uniformly
accelerated frame will have Rindler metric. Apart from that, it is a relevant
frame for an observer sitting very near to the black hole horizon. Hence any
thermodynamic feature of the null surface can be attributed by this metric and
it provides a general description which was originally obtained only for the
black hole horizon. Moreover, all the quantities will be observer dependent.

In this paper we shall proceed as follows. First a detailed derivation of the
Noether current for a diffeomorphism $x^{a}\rightarrow x^{a}+\xi^{a}$,
corresponding to $L_{sur}$, will be given. This is important because it has
not been done earlier and therefore the properties of the current have not
been explored. Here we will show that the corresponding charge $Q[\xi]$,
calculated on the null surface for $\xi^{a}$ to be Killing, yields exactly one
quarter of the horizon area after multiplying it by $2\pi/\kappa$ where
$\kappa$ is the acceleration of the observer or the surface gravity in the
case of a black hole. Next, a definition of the bracket among the charges will
be given. This will be done by taking variation of the charge $Q[\xi_{1}]$ for
another transformation $x^{a}\rightarrow x^{a}+\xi_{2}^{a}$. Finally, we need
to calculate all these quantities for a particular diffeomorphism. To identify
the relevant diffeomorphisms from which the algebra has to be constructed,
following our earlier work [27], we use the criterion that the diffeomorphism
should leave the near horizon form of the metric invariant in some non-
singular coordinate system. This will lead to a set of diffeomorphism vectors
for which the Fourier components of the bracket among the charges will be
exactly similar to Virasoro algebra. It is then very easy to identify the zero
mode eigenvalue and the central extension. Substitution of all these values in
the Cardy formula [8, 9] will yield exactly the Bekenstein-Hawking entropy [1,
2]. A similar analysis was done in [25] based on the Noether current
corresponding to $L_{bulk}$ [34, 35, 36]. In this calculation, to obtain the
correct value of the entropy, a particular boundary condition (Dirichlet or
Neumann) was used. But the physical significance of it is not well understood.

Before going into the main calculation, let us summarize the main features of
the present analysis. $\bullet$ First is the technical aspect. To obtain the
correct entropy, in most of the earlier works, one had to either shift the
zero mode eigenvalue [6] or choose a parameter contained in the Fourier modes
of $\xi^{a}$ as the surface gravity $\kappa$ [25] or both [26]. Here we shall
show that none of ad hoc prescriptions will be required. $\bullet$ The
important one is the simplicity of the criterion (near horizon structure of
the metric remains invariant in some non-singular coordinate system) to find
the relevant diffeoprphisms for which we obtain the Virasoro algebra. This was
first introduced by us [27] in this context. The significance of this choice
is that the full set of diffeomorphism symmetry of the theory is now reduced
to a subset which respects the existence of horizon in a given coordinate
system. Hence it may happen that some of the original gauge degrees of freedom
(which could have been eliminated by certain diffeomorphisms which are now
disallowed) now being effectively upgraded to physical degrees of freedom as
far as a particular class of observers are concerned. So all the thermodynamic
quantities, attributed to the horizon, become observer dependent. $\bullet$ In
our present analysis we will not need any use of boundary condition like
Dirichlet or Neumann to obtain the exact form of the entropy. $\bullet$ Since
our analysis will be completely based on $L_{sur}$ where no information about
$L_{bulk}$ is needed, it will definitely illuminate the emergent paradigm of
gravity, particularly the holographic aspects in the action. We will discuss
later more on different aspects and significance of our results.

The organization of the paper as follows. In section 2, the derivation of the
Noether current for the $L_{sur}$ will be presented explicitly. Next we shall
give the definition of the bracket among the charges and the relevant
diffeomorphims based on the invariance of horizon structure criterion. Section
4 will be devoted to show that the Fourier mode of the bracket is exactly like
the Virasoro algebra which by the Cardy formula will lead to Bekenstein-
Hawking entropy. Finally, we shall conclude.

##  2 Derivation of Noether current from the surface term of Einstein-Hilbert
action

In this section, a detailed derivation of the Noether current and the
potential corresponding to the surface term of EH action will be presented.
Then we shall calculate the charge on the Rindler horizon.

The Lagrangian corresponding to the surface term is given by [28],

$\displaystyle L_{sur}=\partial_{a}(\sqrt{-g}S^{a})~{},$ (1)

where

$\displaystyle S^{a}=2Q^{ad}_{ck}g^{bk}\Gamma^{c}_{bd};\,\,\,\
Q^{ad}_{ck}=\frac{1}{2}(\delta^{a}_{c}\delta^{d}_{k}-\delta^{a}_{k}\delta^{d}_{c})~{}.$
(2)

Here the normalization $1/16\pi G$ is omitted and it will be inserted where
necessary. Now our task is to find the variations of both sides of (1) for a
diffeomorphism $x^{\prime a}=x^{a}+\xi^{a}$ and then equate them. The
variation we shall consider here as the Lie variation which is defined, in
general, as

$\displaystyle\delta A=A(x^{\prime})-A^{\prime}(x^{\prime})~{},$ (3)

where $A(x^{\prime})=A(x+\xi)=A(x)+\xi^{a}\partial_{a}A(x)$, $A(x)$ and
$A^{\prime}(x^{\prime})$ are the evaluated in two different coordinate systems
$x$ and $x^{\prime}$, respectively. In the following, for the notational
simplicity, we shall denote $A(x)$ as $A$.

The variation of the right hand side of (1) is given by,

$\displaystyle\delta L_{sur}$ $\displaystyle=$
$\displaystyle\partial_{a}[\delta(\sqrt{-g}S^{a})]=\partial_{a}\Big{[}S^{a}\delta(\sqrt{-g})+\sqrt{-g}\delta
S^{a}\Big{]}$ (4) $\displaystyle=$
$\displaystyle\partial_{a}\Big{[}\frac{S^{a}}{2}\sqrt{-g}g^{bc}\delta
g_{bc}+\sqrt{-g}\delta S^{a}\Big{]}~{}.$

Since $g_{ab}$ is a tensor, for the Lie variation, $\delta g_{ab}$ is
expressed by the Lie derivative and is given by

$\displaystyle\delta g_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}~{}.$ (5)

Therefore,

$\displaystyle\delta L_{sur}$ $\displaystyle=$
$\displaystyle\partial_{a}\Big{[}S^{a}\partial_{b}(\sqrt{-g}\xi^{b})+\sqrt{-g}\delta
S^{a}\Big{]}~{}.$ (6)

On the other hand, since $S^{a}$ is not a tensor, the variation of it can not
be expressed by simple Lie derivative. To find $\delta S^{a}$ we shall used
the general definition (3). Let us first calculate $S^{\prime a}(x^{\prime})$.
Under the change $x^{\prime a}=x^{a}+\xi^{a}$ we have,

$\displaystyle\frac{\partial x^{\prime a}}{\partial
x^{b}}=\delta^{a}_{b}+\partial_{b}\xi^{a}~{};$ $\displaystyle\frac{\partial
x^{b}}{\partial x^{\prime a}}=\delta^{b}_{a}-\partial_{a}\xi^{b}~{}.$ (7)

Here we considered infinitesimal change and so the terms from
$\partial\xi\partial\xi$ have been ignored. This will be followed in the later
analysis. Hence,

$\displaystyle\Gamma^{\prime
a}_{bc}(x^{\prime})=\Gamma^{a}_{bc}-\Gamma^{a}_{bd}\partial_{c}\xi^{d}-\Gamma^{a}_{cd}\partial_{b}\xi^{d}+\Gamma^{d}_{bc}\partial_{d}\xi^{a}-\partial_{b}\partial_{c}\xi^{a}~{},$
$\displaystyle g^{\prime
bk}(x^{\prime})=g^{bk}+g^{bf}\partial_{f}\xi^{k}+g^{kf}\partial_{f}\xi^{b}~{},$
$\displaystyle Q^{\prime ad}_{ck}(x^{\prime})=Q^{ad}_{ck}~{}.$ (8)

Substitution of these in $S^{\prime a}(x^{\prime})=2Q^{\prime
ad}_{ck}(x^{\prime})g^{\prime bk}(x^{\prime})\Gamma^{\prime
c}_{bd}(x^{\prime})$ lead to,

$\displaystyle S^{\prime
a}(x^{\prime})=S^{a}+S^{b}\partial_{b}\xi^{a}-g^{bd}\partial_{b}\partial_{d}\xi^{a}+g^{ab}\partial_{b}\partial_{c}\xi^{c}~{}.$
(9)

Other one is given by

$\displaystyle
S^{a}(x^{\prime})=S^{a}(x^{b}+\xi^{b})=S^{a}+\xi^{b}\partial_{b}S^{a}~{}.$
(10)

Therefore, according to (3), the Lie variation of $S^{a}$ due to the
diffeomorphism is

$\displaystyle\delta S^{a}=S^{a}(x^{\prime})-S^{\prime
a}(x^{\prime})=\xi^{b}\partial_{b}S^{a}-S^{b}\partial_{b}\xi^{a}+M^{a}~{},$
(11)

where

$\displaystyle
M^{a}=g^{bd}\partial_{b}\partial_{d}\xi^{a}-g^{ab}\partial_{b}\partial_{c}\xi^{c}~{}.$
(12)

Substituting this in (6) we obtain the variation of right hand side of (1) as,

$\displaystyle\delta
L_{sur}=\partial_{a}\Big{[}\partial_{b}(\sqrt{-g}S^{a}\xi^{b})-\sqrt{-g}S^{b}\partial_{b}\xi^{a}+\sqrt{-g}M^{a}\Big{]}~{}.$
(13)

Next we find the variation of left hand side of (1); i.e. $L_{sur}$. For this
we will start from the following relation:

$\displaystyle L_{sur}=\sqrt{-g}(L_{g}-L_{quad})~{},$ (14)

where

$\displaystyle L_{g}=R;\,\,\,\
L_{quad}=2Q^{bcd}_{a}\Gamma^{a}_{dk}\Gamma^{k}_{bc}~{},$ (15)

with
$Q^{bcd}_{a}=\frac{1}{2}\Big{(}\delta^{c}_{a}g^{bd}-\delta^{d}_{a}g^{bc}\Big{)}$.
Since $L_{g}$ is a scalar, by the definition of Lie derivative $\delta
L_{g}=\xi^{a}\partial_{a}L_{g}$. Therefore using (5) we find

$\displaystyle\delta L_{sur}$ $\displaystyle=$
$\displaystyle\delta(\sqrt{-g}L_{g})-\delta(\sqrt{-g}L_{quad})$ (16)
$\displaystyle=$
$\displaystyle\partial_{a}(\sqrt{-g}\xi^{a}L_{g})-\partial_{a}(\sqrt{-g}\xi^{a})L_{quad}-\sqrt{-g}\delta
L_{quad}$ $\displaystyle=$
$\displaystyle\partial_{a}\Big{[}\sqrt{-g}\xi^{a}(L_{g}-L_{quad})\Big{]}+\sqrt{-g}\xi^{a}\partial_{a}L_{quad}-\sqrt{-g}\delta
L_{quad}$ $\displaystyle=$
$\displaystyle\partial_{a}\Big{(}\xi^{a}L_{sur}\Big{)}+\sqrt{-g}\xi^{a}\partial_{a}L_{quad}-\sqrt{-g}\delta
L_{quad}~{}.$

To find $\delta L_{quad}$, we will proceed as earlier. Under the change
$x^{\prime a}=x^{a}+\xi^{a}$, $L^{\prime}_{quad}(x^{\prime})$ is calculated as

$\displaystyle L^{\prime}_{quad}(x^{\prime})$ $\displaystyle=$ $\displaystyle
2Q^{\prime bcd}_{a}(x^{\prime})\Gamma^{\prime
a}_{dk}(x^{\prime})\Gamma^{\prime k}_{bc}(x^{\prime})$ (17) $\displaystyle=$
$\displaystyle
L_{quad}+g^{bc}\Gamma^{k}_{bc}\partial_{d}\partial_{k}\xi^{d}+g^{bc}\Gamma^{d}_{dk}\partial_{b}\partial_{c}\xi^{k}-g^{bd}\Gamma^{c}_{dk}\partial_{b}\partial_{c}\xi^{k}~{},$

where (8) has been used. This can be expressed in terms of $M^{a}$ in the
following way. Second term on the right hand side can be expressed in the
following form

$\displaystyle\sqrt{-g}g^{bc}\Gamma^{k}_{bc}\partial_{d}\partial_{k}\xi^{d}$
$\displaystyle=$
$\displaystyle\Big{[}\sqrt{-g}g^{bc}g^{ak}\partial_{b}g_{ac}-g^{ak}\partial_{a}(\sqrt{-g})\Big{]}\partial_{d}\partial_{k}\xi^{d}$
(18) $\displaystyle=$
$\displaystyle-\partial_{a}(\sqrt{-g}g^{ak})\partial_{d}\partial_{k}\xi^{d}~{},$

where in the above we used
$g^{bc}g^{ak}\partial_{b}g_{ac}=-\partial_{a}g^{ak}$. Third term of (17)
reduces to

$\displaystyle\sqrt{-g}g^{bc}\Gamma^{d}_{dk}\partial_{b}\partial_{c}\xi^{k}=\partial_{k}(\sqrt{-g})g^{bc}\partial_{b}\partial_{c}\xi^{k}~{}.$
(19)

Similarly, the last term can be expressed as

$\displaystyle 2\sqrt{-g}g^{bd}\Gamma^{c}_{dk}\partial_{b}\partial_{c}\xi^{k}$
$\displaystyle=$
$\displaystyle\sqrt{-g}g^{bd}g^{ca}\partial_{k}g_{ad}\partial_{b}\partial_{c}\xi^{k}$
(20) $\displaystyle=$
$\displaystyle-\sqrt{-g}\partial_{k}(g^{bc})\partial_{b}\partial_{c}\xi^{k}~{},$

where in the last line $g^{bd}g^{ca}\partial_{k}g_{ad}=-\partial_{k}g^{bc}$
has been used. Substituting all these in (17) we obtain

$\displaystyle L^{\prime}_{quad}(x^{\prime})$ $\displaystyle=$ $\displaystyle
L_{quad}-\frac{1}{\sqrt{-g}}\Big{[}\partial_{a}(\sqrt{-g}g^{ak})\partial_{d}\partial_{k}\xi^{d}+\partial_{k}(\sqrt{-g}g^{bc})\partial_{b}\partial_{c}\xi^{k}\Big{]}$
(21) $\displaystyle=$ $\displaystyle
L_{quad}+\frac{1}{\sqrt{-g}}\partial_{a}(\sqrt{-g}M^{a})$

On the other hand,

$\displaystyle
L_{quad}(x^{\prime})=L_{quad}(x^{a}+\xi^{a})=L_{quad}+\xi^{a}\partial_{a}L_{quad}~{}.$
(22)

Hence

$\displaystyle\sqrt{-g}\delta
L_{quad}=\sqrt{-g}L_{quad}(x^{\prime})-\sqrt{-g}L^{\prime}_{quad}(x^{\prime})=\sqrt{-g}\xi^{a}\partial_{a}L_{quad}-\partial_{a}(\sqrt{-g}M^{a})~{}.$
(23)

Substituting this in (16) we obtain

$\displaystyle\delta
L_{sur}=\partial_{a}\Big{(}\xi^{a}L_{sur}+\sqrt{-g}M^{a}\Big{)}~{}.$ (24)

Now equating (13) and (24) we obtain $\partial_{a}J^{a}[\xi]=0$, where the
conserved Noether current $J^{a}[\xi]$ is given by

$\displaystyle
J^{a}[\xi]=-\partial_{b}(\sqrt{-g}S^{a}\xi^{b})+\sqrt{-g}S^{b}\partial_{b}\xi^{a}+\xi^{a}L_{sur}~{}.$
(25)

Finally, using $L_{sur}=\partial_{a}(\sqrt{-g}S^{a})$ in the above, we can
express the current as the divergence of a anti-symmetric two index quantity:

$\displaystyle
J^{a}[\xi]=\partial_{b}\Big{[}\sqrt{-g}(\xi^{a}S^{b}-\xi^{b}S^{a})\Big{]}=\partial_{b}\Big{[}\sqrt{-g}J^{ab}[\xi]\Big{]}~{}.$
(26)

It is evident that the anti-symmetric object $J^{ab}[\xi]$ is not a tensor and
it is usually called the Noether potential. Therefore, inserting the proper
normalization, the charge is given by

$\displaystyle Q[\xi]=\frac{1}{32\pi
G}\int_{\cal{H}}d\Sigma_{ab}\sqrt{h}J^{ab}[\xi]~{},$ (27)

where $d\Sigma_{ab}=-d^{2}x(N_{a}M_{b}-N_{b}M_{a})$ is the surface element of
the $2$-dimensional surface $\cal{H}$ and $h$ is the determinant of the
corresponding metric. Since our present discussion will be near the horizon,
we choose the unit normals $N_{a}$ and $M_{a}$ as spacelike and timelike
respectively.

Now we shall calculate the charge (27) explicitly on the horizon. This will be
done by considering the form of the metric near the horizon,

$\displaystyle ds^{2}=-2\kappa xdt^{2}+\frac{1}{2\kappa
x}dx^{2}+dx^{2}_{\perp}~{},$ (28)

where $x_{\perp}$ represents the transverse coordinates. The metric has a
timelike Killing vector $\chi^{a}=(1,0,0,0)$ and the Killing horizon is given
by $\chi^{2}=0$; i.e. $x=0$. The non-zero Christoffer connections are

$\displaystyle\Gamma^{t}_{tx}=\frac{1}{2x};\,\,\,\
\Gamma^{x}_{tt}=2\kappa^{2}x;\,\,\,\ \Gamma^{x}_{xx}=-\frac{1}{2x}~{}.$ (29)

For the metric (28) we find

$\displaystyle N^{a}=(0,\sqrt{2\kappa x},0,0);\,\,\
M^{a}=(\frac{1}{\sqrt{2\kappa x}},0,0,0)~{},$ (30)

and hence $d\Sigma_{tx}=-d^{2}x$. Also, (2) yields

$\displaystyle S^{t}=0;\,\,\ S^{x}=-2\kappa~{}.$ (31)

Therefore,

$\displaystyle J^{tx}=(\xi^{t}S^{x}-\xi^{x}S^{t})=-2\kappa\xi^{t}~{}.$ (32)

Now if $\xi^{a}$ is a Killing vector, then $\xi^{t}=\chi^{t}=1$ and so
calculating the charge (27) explicitly we find

$\displaystyle Q[\xi=\chi]=\frac{\kappa A_{\perp}}{8\pi G}~{},$ (33)

where $A_{\perp}=\int_{\cal{H}}d^{2}x$ is the horizon cross-section area.
Multiplying it by the periodicity of time coordinate $2\pi/\kappa$ we obtain
exactly the entropy: one quarter of horizon area. Moreover, the above can be
expressed as $Q[\xi=\chi]=TS$, where $T=\kappa/2\pi$ is the temperature of the
horizon and $S=A_{\perp/4G}$ is the entropy. Therefore one can call it as the
Noether energy. Such interpretaion was done earlier in [37, 38].

So far we found that the Noether charge corresponding to the surface term of
EH action alone led to the entropy of the Rindler horizon. This was shown
earlier for the charge coming from the total EH action [33]. Therefore, the
present analysis reveled that it may be possible that the information is
actually encoded in the surface term rather than the bulk term. Then the
natural question arises: What are the degrees of freedom responsible for this
entropy? So far it is not known. In the next couple of sections we shall give
an idea on the nature of the possible degrees of freedom in the context of
Virasoro algebra and Cardy formula.

##  3 Bracket among the charges and the diffeomorphism generators

In the previous section, we have given the expression for the charge (see Eq.
(27)) for an arbitrary diffeomorphism. Here we shall define the bracket among
the charges. The relevant diffeomorphisms will be chosen by imposing a minimum
condition on the spacetime metric. The charge and the bracket will be then
expressed in terms of these generators.

We shall find the bracket following our earlier works [26, 27]. For this let
us first calculate the following:

$\displaystyle\delta_{\xi_{1}}(\sqrt{-g}J^{ab}[\xi_{2}])$ $\displaystyle=$
$\displaystyle\delta_{\xi_{1}}(\sqrt{-g})J^{ab}[\xi_{2}]+\sqrt{-g}\delta_{\xi_{1}}(J^{ab}[\xi_{2}])$
(34) $\displaystyle=$
$\displaystyle-\frac{1}{2}\sqrt{-g}g_{mn}\delta_{\xi_{1}}g^{mn}J^{ab}[\xi_{2}]$
$\displaystyle+$
$\displaystyle\sqrt{-g}\Big{[}(\delta_{\xi_{1}}\xi_{2}^{a})S^{b}+\xi_{2}^{a}(\delta_{\xi_{1}}S^{b})-(a\leftrightarrow
b)\Big{]}~{}.$

Using

$\displaystyle\delta_{\xi}g^{ab}$ $\displaystyle=$
$\displaystyle\pounds_{\xi}g^{ab}=-\nabla^{a}\xi^{b}-\nabla^{b}\xi^{a};$
$\displaystyle\delta_{\xi}\Gamma^{a}_{bc}$ $\displaystyle=$
$\displaystyle\nabla_{b}\nabla_{c}\xi^{a}+R^{a}_{cmb}\xi^{m}~{},$ (35)

and in addition the expression for $S^{a}$, given by (2), we obtain,

$\displaystyle\delta_{\xi_{1}}(\sqrt{-g}J^{ab}[\xi_{2}])$ $\displaystyle=$
$\displaystyle\sqrt{-g}\Big{[}\nabla_{m}\xi_{1}^{m}J^{ab}[\xi_{2}]+\\{(\xi_{1}^{m}\nabla_{m}\xi_{2}^{a}-\xi_{2}^{m}\nabla_{m}\xi_{1}^{a})S^{b}$
(36) $\displaystyle+$
$\displaystyle\xi_{2}^{a}\Big{(}-2\Gamma^{b}_{mn}\nabla^{m}\xi_{1}^{n}+\nabla_{m}\nabla^{m}\xi_{1}^{b}+2R^{b}_{m}\xi_{1}^{m}$
$\displaystyle-$
$\displaystyle\Gamma^{n}_{nm}(\nabla^{b}\xi_{1}^{m}+\nabla^{m}\xi_{1}^{b})-\nabla_{m}\nabla^{b}\xi_{1}^{m}\Big{)}-(a\leftrightarrow
b)\\}\Big{]}~{}.$

For the present metric (28), $g=-1$, $R^{a}_{b}=0$ and hence
$\sqrt{-g}\Gamma^{n}_{nm}=\partial_{m}(\sqrt{-}g)=0$. Therefore

$\displaystyle\delta_{\xi_{1}}(\sqrt{-g}J^{ab}[\xi_{2}])$ $\displaystyle=$
$\displaystyle\Big{[}\nabla_{m}\xi_{1}^{m}J^{ab}[\xi_{2}]+\frac{1}{16\pi
G}\\{(\xi_{1}^{m}\nabla_{m}\xi_{2}^{a}-\xi_{2}^{m}\nabla_{m}\xi_{1}^{a})S^{b}$
$\displaystyle+$
$\displaystyle\xi_{2}^{a}\Big{(}-2\Gamma^{b}_{mn}\nabla^{m}\xi_{1}^{n}+\nabla_{m}\nabla^{m}\xi_{1}^{b}-\nabla_{m}\nabla^{b}\xi_{1}^{m}\Big{)}-(a\leftrightarrow
b)\\}\Big{]}$ $\displaystyle\equiv K^{ab}_{12}~{}.$ (37)

Finally we define a bracket as:

$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]:=\frac{1}{2}\int_{\cal{H}}d\Sigma_{ab}\sqrt{h}\Big{[}K^{ab}_{12}-(1\leftrightarrow
2)\Big{]}~{},$ (38)

which for the present metric (28) reduces to

$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]:=-\int_{\cal{H}}d^{2}x\Big{[}K^{tx}_{12}-(1\leftrightarrow
2)\Big{]}~{}.$ (39)

To calculate the above bracket we need to know about the generators $\xi^{a}$.
We shall determine them by using the condition that the horizon structure
remains invariant in some nonsingular coordinate system. For that let us first
express the metric (28) in Gaussian null (or Bondi like) coordinates,

$\displaystyle du=dt-\frac{dx}{2\kappa x};\,\,\ dX$ $\displaystyle=$
$\displaystyle dx~{}.$ (40)

In these coordinates the metric reduces to the following form:

$\displaystyle ds^{2}=-2\kappa Xdu^{2}-2dudX+dx^{2}_{\perp}~{}.$ (41)

Now impose the condition that the metric coefficients $g_{XX}$ and $g_{uX}$ do
not change under the diffeomorphism, i.e.

$\displaystyle\pounds_{\tilde{\xi}}g_{XX}=0;\,\,\,\
\pounds_{\tilde{\xi}}g_{uX}=0~{},$ (42)

where $\pounds_{\tilde{\xi}}$ is the Lie derivative along the vector
$\tilde{\xi}$. These lead to,

$\displaystyle\pounds_{\tilde{\xi}}g_{XX}=-2\partial_{X}\tilde{\xi}^{u}=0;$
$\displaystyle\pounds_{\tilde{\xi}}g_{uX}=-\partial_{u}\tilde{\xi}^{u}-2\kappa
X\partial_{X}\tilde{\xi}^{u}-\partial_{X}\tilde{\xi}^{X}=0~{}.$ (43)

The solutions are:

$\displaystyle\tilde{\xi}^{u}=F(u,x_{\perp});$
$\displaystyle\tilde{\xi}^{X}=-X\partial_{u}F(u,x_{\perp})~{}.$ (44)

The condition $\pounds_{\tilde{\xi}}g_{uu}=0$ automatically satisfied near the
horizon, because use of the above solutions lead to
$\pounds_{\tilde{\xi}}g_{uu}={\cal{O}}(X)$. These conditions were appeared
earlier in [39] in the context of late time symmetry near the black hole
horizon. Finally expressing (44) in the old coordinates ($t,x$) we find

$\displaystyle\xi^{t}=T-\frac{1}{2\kappa}\partial_{t}T;\,\,\,\
\xi^{x}=-x\partial_{t}T~{},$ (45)

where $T(t,x,x_{\perp})=F(u,x_{\perp})$.

Next we calculate $K^{tx}_{12}$ from (37) for our present case. Since
$S^{t}=0$, we find

$\displaystyle K^{tx}_{12}$ $\displaystyle=$
$\displaystyle\nabla_{m}\xi_{1}^{m}\xi_{2}^{t}S^{x}+(\xi_{1}^{m}\nabla_{m}\xi_{2}^{t}-\xi_{2}^{m}\nabla_{m}\xi_{1}^{t})S^{x}$
(46) $\displaystyle+$
$\displaystyle\xi_{2}^{t}\Big{(}-2\Gamma^{x}_{mn}\nabla^{m}\xi_{1}^{n}+\nabla_{m}\nabla^{m}\xi_{1}^{x}-\nabla_{m}\nabla^{x}\xi_{1}^{m}\Big{)}$
$\displaystyle-$
$\displaystyle\xi_{2}^{x}\Big{(}-2\Gamma^{t}_{mn}\nabla^{m}\xi_{1}^{n}+\nabla_{m}\nabla^{m}\xi_{1}^{t}-\nabla_{m}\nabla^{t}\xi_{1}^{m}\Big{)}~{}.$

Now since the integration (39) will ultimately be evaluated on the horizon, we
shall find the value of each term of the above very near the horizon.
Therefore, using (29), (31) and the form of the generators (45) we obtain the
values of each term of the above expression near the horizon $x=0$ as,

$\displaystyle\nabla_{m}\xi_{1}^{m}\xi_{2}^{t}S^{x}=T_{2}\partial_{t}^{2}T_{1}-\frac{1}{2\kappa}\partial^{2}_{t}T_{1}\partial_{t}T_{2}$
$\displaystyle\xi_{1}^{m}\nabla_{m}\xi_{2}^{t}S^{x}=-\kappa
T_{1}\partial_{t}T_{2}+\kappa
T_{2}\partial_{t}T_{1}+T_{1}\partial^{2}_{t}T_{2}-\frac{1}{2\kappa}\partial_{t}T_{1}\partial^{2}_{t}T_{2}$
$\displaystyle-\xi_{2}^{m}\nabla_{m}\xi_{1}^{t}S^{x}=\kappa
T_{2}\partial_{t}T_{1}-\kappa
T_{1}\partial_{t}T_{2}-T_{2}\partial^{2}_{t}T_{1}+\frac{1}{2\kappa}\partial_{t}T_{2}\partial^{2}_{t}T_{1}$
$\displaystyle-2\xi_{2}^{t}\Gamma^{x}_{mn}\nabla^{m}\xi_{1}^{n}=-T_{2}\partial^{2}_{t}T_{1}+\frac{1}{2\kappa}\partial^{2}T_{1}\partial_{t}T_{2}$
$\displaystyle\xi_{2}^{t}\nabla_{m}\nabla^{m}\xi_{1}^{x}=\frac{1}{2\kappa}T_{2}\partial^{3}_{t}T_{1}-\frac{1}{4\kappa^{2}}\partial^{3}_{t}T_{1}\partial_{t}T_{2}-2\kappa
T_{2}\partial_{t}T_{1}+\partial_{t}T_{1}\partial_{t}T_{2}+T_{2}\partial^{2}_{t}T_{1}-\frac{1}{2\kappa}\partial^{2}_{t}T_{1}\partial_{t}T_{2}$
$\displaystyle-\xi_{2}^{t}\nabla_{m}\nabla^{x}\xi_{1}^{m}=0$ $\displaystyle
2\xi_{2}^{x}\Gamma^{t}_{mn}\nabla^{m}\xi_{1}^{n}=-\frac{1}{2\kappa}\partial^{2}_{t}T_{1}\partial_{t}T_{2}-2\kappa
x\partial_{x}T_{1}\partial_{t}T_{2}$
$\displaystyle-\xi_{2}^{x}\nabla_{m}\nabla^{m}\xi_{1}^{t}=\frac{1}{4\kappa^{2}}\partial^{3}_{t}T_{1}\partial_{t}T_{2}$
$\displaystyle\xi_{2}^{x}\nabla_{m}\nabla^{t}\xi_{1}^{m}=-\frac{1}{4\kappa^{2}}\partial^{3}_{t}T_{1}\partial_{t}T_{2}$
(47)

So near the horizon (46) reduces to

$\displaystyle K^{tx}_{12}$ $\displaystyle=$ $\displaystyle-2\kappa
T_{1}\partial_{t}T_{2}+T_{1}\partial_{t}^{2}T_{2}-\frac{1}{2\kappa}\Big{(}\partial_{t}T_{1}\partial^{2}_{t}T_{2}+\partial^{2}_{t}T_{1}\partial_{t}T_{2}\Big{)}+\frac{1}{2\kappa}T_{2}\partial^{3}_{t}T_{1}+\partial_{t}T_{1}\partial_{t}T_{2}$
(48) $\displaystyle-\frac{1}{4\kappa^{2}}\partial^{3}T_{1}\partial_{t}T_{2}$

Substituting this in (39) and inserting the normalization factor, we obtain
the expression for the bracket

$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]:$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi
G}\int_{\cal{H}}d^{2}x\Big{[}2\kappa\Big{(}T_{1}\partial_{t}T_{2}-T_{2}\partial_{t}T_{1}\Big{)}-\Big{(}T_{1}\partial^{2}_{t}T_{2}-T_{2}\partial^{2}_{t}T_{1}\Big{)}$
(49) $\displaystyle+$
$\displaystyle\frac{1}{2\kappa}\Big{(}T_{1}\partial^{3}_{t}T_{2}-T_{2}\partial^{3}_{t}T_{1}\Big{)}+\frac{1}{4\kappa^{2}}\Big{(}\partial^{3}_{t}T_{1}\partial_{t}T_{2}-\partial^{3}_{t}T_{2}\partial_{t}T_{1}\Big{)}\Big{]}~{}.$

Similarly, (27) yields,

$\displaystyle Q[\xi]=\frac{1}{8\pi G}\int d^{2}x\Big{(}\kappa
T-\frac{1}{2}\partial_{t}T\Big{)}~{}.$ (50)

A couple of comments are in order. It must be noted that in finding the
expression for the bracket (49), no use of boundary conditions (Dirichlet or
Neumann) has been used. Earlier this was used for the case of $L_{bulk}$ to
throw away the non-covariant terms in the bracket without giving any physical
meaning [25]. Also, we did not use the condition
$\delta_{\xi_{1}}\xi^{a}_{2}=0$ (see Eq. (34)) which was adopted in earlier
works. For instance, see [6, 25]. This is logically correct since
$\delta_{\xi_{1}}\xi^{a}_{2}=0$ contradicts the algebra among the Fourier
modes of the diffeomorphisms (see Eq. (52), in next section).

##  4 Virasoro algebra and entropy

In this section, the Fourier modes of the bracket and the charge will be found
out. We shall show that for a particular ansatz for the the Fourier modes of
the generators will lead to the Virasoro algebra. Finally using the Cardy
formula, the entropy will be calculated.

Consider the Fourier decompositions of $T_{1}$ and $T_{2}$:

$\displaystyle T_{1}=\displaystyle\sum_{m}A_{m}T_{m};\,\,\
T_{2}=\displaystyle\sum_{n}B_{n}T_{n}~{},$ (51)

where $A^{*}_{m}=A_{-m}$; $B^{*}_{n}=B_{-n}$. The Fourier modes $T_{m}$ will
be chosen such that the Fourier modes of the diffeomorphisms (44) obey one
sub-algebra isomorphic to Diff. $S^{1}$:

$\displaystyle i\\{\xi_{m},\xi_{n}\\}^{a}=(m-n)\xi_{m+n}^{a}~{},$ (52)

where $\\{,\\}$ is the Lie bracket. Now with the use of (51), let us first
find the Fourier modes of the bracket (49) and the charge (50). Substitution
of (51) in (49) yields,

$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]:$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{m,n}\frac{C_{m,n}}{16\pi
G}\int_{\cal{H}}d^{2}x\Big{[}2\kappa\Big{(}T_{m}\partial_{t}T_{n}-T_{n}\partial_{t}T_{m}\Big{)}-\Big{(}T_{m}\partial^{2}_{t}T_{n}-T_{n}\partial^{2}_{t}T_{m}\Big{)}$
(53) $\displaystyle+$
$\displaystyle\frac{1}{2\kappa}\Big{(}T_{m}\partial^{3}_{t}T_{n}-T_{n}\partial^{3}_{t}T_{m}\Big{)}+\frac{1}{4\kappa^{2}}\Big{(}\partial^{3}_{t}T_{m}\partial_{t}T_{n}-\partial^{3}_{t}T_{n}\partial_{t}T_{m}\Big{)}\Big{]}~{},$

where $C_{m,n}=A_{m}B_{n}$ and so $C^{*}_{m,n}=C_{-m,-n}$. Next defining the
Fourier modes of $[Q[\xi_{1}],Q[\xi_{2}]]$ as

$\displaystyle[Q[\xi_{1}],Q[\xi_{2}]]=\displaystyle\sum_{m,n}C_{m,n}[Q_{m},Q_{n}]~{}.$
(54)

we find

$\displaystyle[Q_{m},Q_{n}]:$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G}\int_{\cal{H}}d^{2}x\Big{[}2\kappa\Big{(}T_{m}\partial_{t}T_{n}-T_{n}\partial_{t}T_{m}\Big{)}-\Big{(}T_{m}\partial^{2}_{t}T_{n}-T_{n}\partial^{2}_{t}T_{m}\Big{)}$
(55) $\displaystyle+$
$\displaystyle\frac{1}{2\kappa}\Big{(}T_{m}\partial^{3}_{t}T_{n}-T_{n}\partial^{3}_{t}T_{m}\Big{)}+\frac{1}{4\kappa^{2}}\Big{(}\partial^{3}_{t}T_{m}\partial_{t}T_{n}-\partial^{3}_{t}T_{n}\partial_{t}T_{m}\Big{)}\Big{]}~{}.$

Similarly from (50), the Fourier modes of the charge are given by

$\displaystyle Q_{m}=\frac{1}{8\pi G}\int d^{2}x\Big{(}\kappa
T_{m}-\frac{1}{2}\partial_{t}T_{m}\Big{)}~{},$ (56)

where $Q[\xi]=\displaystyle\sum_{m}A_{m}Q_{m}$. It must be noted that the
present expression (56) is exactly identical to that obtained in [27] for the
York-Gobbons-Hawking surface term whereas the other expression (55) is
different by some terms. This may be because these two surface terms are not
exactly same. But we shall show that final result for the bracket is identical
to the earlier analysis.

To calculate the above expressions (55) and (56) explicitly we need to have
$T_{m}$’s. Following the earlier arguments, we choose

$\displaystyle T_{m}=\frac{1}{\alpha}e^{im\Big{(}\alpha
t+g(x)+p.x_{\perp}\Big{)}}$ (57)

such that they satisfy the algebra (52). Here $\alpha$ is a constant, $p$ is
an integer and $g(x)=G(X=x)=-\alpha\int\frac{dx}{2\kappa x}$. This is a
standard choice in these computations and has been used several times in the
literature [5, 6, 25]. It must be noted that the transverse directions are
non-compact due to our Rindler approximations and so we will assume that
$T_{m}$ is periodic in the transverse coordinates with the periodicities
$L_{y}$ and $L_{z}$ on $y$ and $z$, respectively. Now substituting (57) in
(55) and (56) and then integrating over the cross-sectional area
$A_{\perp}=L_{y}L_{z}$ we obtain

$\displaystyle Q_{m}=\frac{A_{\perp}}{8\pi
G}\frac{\kappa}{\alpha}\delta_{m,0};$ (58) $\displaystyle
i[Q_{m},Q_{n}]:=\frac{A_{\perp}}{8\pi
G}\frac{\kappa}{\alpha}(m-n)\delta_{m+n,0}+n^{3}\frac{A_{\perp}}{16\pi
G}\frac{\alpha}{\kappa}\delta_{m+n,0}~{}.$ (59)

Using (58), (59) can be re-expressed as

$\displaystyle i[Q_{m},Q_{n}]:=(m-n)Q_{m+n}+n^{3}\frac{A_{\perp}}{16\pi
G}\frac{\alpha}{\kappa}\delta_{m+n,0}~{}.$ (60)

This is exactly identical to Virasoro algebra with the central charge $C$ is
identified as

$\displaystyle\frac{C}{12}=\frac{A_{\perp}}{16\pi G}\frac{\alpha}{\kappa}~{}.$
(61)

The zero mode eigenvalue is evaluated from (58) for $m=0$:

$\displaystyle Q_{0}=\frac{A_{\perp}}{8\pi G}\frac{\kappa}{\alpha}~{}.$ (62)

Finally using the Cardy formula [8, 9], we obtain the entropy as

$\displaystyle S=2\pi\sqrt{\frac{CQ_{0}}{6}}=\frac{A_{\perp}}{4G}~{},$ (63)

which is exactly the Bekenstein-Hawking entropy.

##  5 Conclusions

It has already been observed that several interesting features and
informations can be obtained from the surface term without incorporating the
bulk term of the gravity action. In this paper we studied the surface term of
the Einstein-Hilbert (EH) action in the context of Noether current. So far we
know, this has not been attempted before. First the current was derived for an
arbitrary diffeomorphism by using Noether prescription. Then we showed that
the charge evaluated on the horizon for a Killing vector led to the
Bekenstein-Hawking entropy after multiplied it by $2\pi/\kappa$. But till now,
it is not known about the degrees of freedom responsible for the entropy. Here
we addressed the issue and try to shed some light. This has been discussed in
the context of Virasoro algebra and Cardy formula.

In this paper, we defined the bracket among the charges. It was done, in the
sprite of our earlier works [26, 27], by taking the variation of the Noether
potential $J^{ab}[\xi_{1}]$ for a different diffeomorphism $x^{a}\rightarrow
x^{a}+\xi^{a}_{2}$, then an anti-symmetric combination between the indices $1$
and $2$ and integrating over the horizon surface. To achieve the final form,
we did not use Einstein equation of motion or any ambiguous prescription, like
vanishing of the variation of diffeomorphism parameter $\xi^{a}$, certain
boundary conditions (e.g. Dirichlet or Neumann), etc. For explicit evaluation
of our bracket, the spacetime was considered as the Rindler metric. The
relevant diffeomorphisms were identified by using a very simple, physically
motivated condition: the diffeomorphisms keep the horizon structure of the
metric invariant in some non-singular coordinate system. It turned out that
the Fourier modes of the bracket is similar to the Virasoro algebra.
Identifying the central charge and the zero mode eigenvalue and then using
these in Cardy formula we obtained exactly the Bekenstein-Hawking entropy.

Let us now discuss in details what we have achieved in this paper. We first
tabulate couple of technical points. $\bullet$ To obtain the exact expression
for entropy we did not need any hand waving prescriptions, like shifting of
the value of the zero mode eigenvalue or the specific choice of the value of
the parameter $\alpha$ appeared in Fourier modes of $T$ or both. $\bullet$ The
relevant diffeomorphisms for invariance of the horizon structure can be
obtained various ways. Here our idea was to impose minimum constraints so that
the bracket led to Virasoro algebra. It is also possible to have other choices
of constraints to find the vectors $\xi^{a}$. For instance, the whole metric
is invariant and the diffeomorphisms come out to be the Killing vectors which
in general do not exist for a general spacetime. Finally, we discuss several
conceptual aspects. The analysis presents a nice connection between the
horizon entropy and the degrees of freedom which are responsible for it. In
the usual cases, one always find that the concepts of degrees of freedom and
entropy are absolute. These does not have any observer dependent description.
But in the case of gravity, as we know, the notion of temperature and entropy
is observer dependent and hence one can expect that the degrees of freedom may
not be absolute. Here we showed that a certain class of observers which can
see the horizon and keep the horizon structure invariant, always attribute
entropy. This signifies the fact that among all the diffeomorphisms, some of
them upgraded to real degrees of freedom which were originally gauge degrees
of freedom and they have observer dependent notion. Also, everything what we
achieved here, was done from surface term. This again illustrates the
holographic nature of the gravity actions - either the bulk and the surface
terms may duplicate the same information or the surface term alone contains
all the information about the theory of gravity. Moreover, the methodology is
general enough to discuss other theories of gravity [40].

Acknowledgement
I thank T. Padmanabhan for several useful discussions.

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mortality from COVID-19 disease: the AstraZeneca case for France and Italy

Davide Faranda<EMAIL_ADDRESS>Laboratoire des Sciences du Climat et de
l’Environnement, CEA Saclay l’Orme des Merisiers, UMR 8212 CEA-CNRS-UVSQ,
Université Paris-Saclay & IPSL, 91191, Gif-sur-Yvette, France.  London
Mathematical Laboratory, 8 Margravine Gardens, London, W6 8RH, UK.  LMD/IPSL,
Ecole Normale Superieure, PSL research University, 75005, Paris, France.
Tommaso Alberti  National Institute for Astrophysics-Institute for Space
Astrophysics and Planetology (INAF-IAPS), via del Fosso del Cavaliere 100,
00133, Rome, Italy.  Maxence Arutkin  UMR CNRS 7083 Gulliver, ESPCI Paris, 10
rue Vauquelin, 75005 Paris, France.  Valerio Lembo  National Research Council
of Italy, Institute of Atmospheric Sciences and Climate (CNR-ISAC), Bologna,
Italy.  Valerio Lucarini  Department of Mathematics and Statistics, University
of Reading, Reading, UK.  Centre for the Mathematics of Planet Earth,
University of Reading, Reading, UK.

###### Abstract

Several European countries have suspended the inoculation of the AstraZeneca
vaccine out of suspicion of causing deep vein thrombosis. In this letter we
report some Fermi estimates performed using a stochastic model aimed at making
a risk-benefit analysis of the interruption of the delivery of the AstraZeneca
vaccine in France and Italy. Our results clearly show that excess deaths due
to the interruption of the vaccination campaign injections largely overrun
those due to thrombosis even in worst case scenarios of frequency and gravity
of the vaccine side effects.

††preprint: CHAOS/FARANDA

##

We analyze, in the framework of epidemiological modelling, the stop in the
deployment of the AstraZeneca vaccine due to some suspected side effects.
Indeed, few dozen suspicious cases of Deep Vein Thrombosis (DVT) over 5
millions vaccinations have arisen in Europe and pushed several European
countries to suspend AstraZeneca injection. Using both an epidemiological
Susceptible-Exposed-Infected-Recovered (SEIR) model and statistical analysis
of publicly available data, we estimate the excess deaths resulting from
missing inoculations of the vaccine and those potentially linked to DVT side
effects in France and Italy. We find that, despite the many simplifications
and limitations in our analysis, the excess deaths differ by at least an order
of magnitude in the two strategies, that the relative benefits are wider in
situations where the reproduction number is larger, and they increase with the
temporal duration of the vaccine ban.

##  I Introduction

As of March 2021, the spread of the SARS-CoV-2 virus (Wu  _et al._ , 2020) has
caused more than 120 millions infections worldwide with a total death toll of
more than 2 millions. Up to the end of 2020, the only effective measures to
contain the spread of the virus were based on social distancing, wearing face
masks and more/less stringent lockdown (Anderson  _et al._ , 2020; Chinazzi
_et al._ , 2020; Yuan  _et al._ , 2020). Later on, a massive vaccination
campaign kicked off in several countries thanks to the availability of a
variety of vaccines (e.g., AstraZeneca, Johnson&Johnson, Moderna,
Pfizer/BionTech, Sputnik V). Such vaccines differ substantially in terms of
efficacy, legal status, availability, and logistics needed for their delivery
to patients. According to various estimates (Sridhar and Gurdasani, 2021),
vaccinations would produce a reduction in infections, and eventually yield to
”herd immunity” when $\approx 70\%$ of the population gets fully vaccinated.
When such a large fraction of the population becomes immune to the disease,
its spread from person to person becomes very unlikely, and the whole
community becomes protected. By allowing for an earlier easing of non-medical
measures against the SARS-CoV-2 virus, vaccination is also expected to
significantly reduce the economical, social and psychological impacts of
lockdown measures (Fernandes, 2020). Those estimates assume that there is no
break in the supply of vaccines or any other suspension in the procedure due
to side effects from vaccination. Unfortunately, on March 15th 2021 several
European countries suspended the use of AstraZeneca COVID-19 vaccine as a
precaution in order to investigate the death of a few dozens of patients
developing blood clots - associated with Deep Vein Thrombosis (DVT) Cushman
(2007) \- after such vaccine, despite no proof has been found yet of causal
link between vaccination and DVT 111Bloomberg, March 2021:
https://tinyurl.com/2zk29abr. Health personnel who inoculated the vaccine to
those who died as a result of DVT are being investigated in Italy for
manslaughter 222ANSA, March 2021: https://tinyurl.com/4ywxt5kp. The contingent
situation with the widespread COVID-19 pandemic naturally raises the question
of whether a prolonged stop in vaccinations coming from adopting the
precautionary principle (Steele, 2006) could cause an excess mortality beyond
that caused by hypothetical side effects of the vaccines. The European
Medicines Agency (EMA) is currently assessing whether the vaccine can continue
to be used despite possibly causing this very rare side effect. In this
Letter, we aim at exploring this issue by computing future COVID-19 epidemic
scenarios by comparing i) the excess mortality caused by reducing the
vaccinations using the stochastic Susceptible-Exposed-Infected-Recovered
(SEIR) model (Faranda and Alberti, 2020), and ii) the estimates of the
possible casualties caused by side effects of a vaccine, namely those
associated with DVT. We remark that the additional, longer-term effect of the
presence of higher infection rates, e.g. the increased risk of virus mutations
leading to possibly more malignant and/or more infectious variants, is not
included in our treatment. Our analysis focuses on France and Italy, which
have been among the countries that have been most severely impacted by the
COVID-19 pandemic (Alberti and Faranda, 2020). An important remark follows.
Our goal is not to provide an exact estimate of both i) and ii) but rather to
perform an order-of-magnitude comparison between excess deaths resulting from
different scenarios of vaccination policy. We proceed in the spirit of
complexity science, where simple models are useful for elucidating the main
mechanisms behind complex behaviour and provide useful inputs for the
deployment of more advanced modelling suites and data collection strategies
Held (2005); Pascual  _et al._ (2011); Gähde  _et al._ (2013); Almaraz (2014);
Ghil and Lucarini (2020). In other words, we will approach the problem by
performing Fermi estimates (Mahajan, 2014) where the classical back-of-the-
envelope calculations are performed via the SEIR model, allowing to take into
account the uncertainties in both model parameters and data. In nuce, we
perform a counterfactual analysis based on a story-line approach, which has
become a powerful investigation method for assessing risks coming from extreme
events Shepherd (2016). While the quantitative consolidation of our results
clearly requires extensive data analysis and modelling, our findings show with
a large confidence that excess deaths due to the interruption of the
vaccination campaign largely override those due to DVT even in the worst case
scenarios of frequency and gravity of the vaccine side effects. Fermi
estimates can provide valuable inputs for an efficient and pragmatic
application of the precautionary principle able to reduce the negative impacts
of hazards of various nature, as done in economics Anderson and Sherman
(2010).

##  II Methods

The model (Faranda  _et al._ , 2020) with time-dependent control parameters
can mimic the dependence on additional/external factors such as variability in
the detected cases, different physiological response to the virus, release or
reinforcement of distancing measures (Faranda and Alberti, 2020). Our
compartmental model (Brauer, 2008) divides the population into four groups,
namely Susceptible (S), Exposed (E), Infected (I), and Recovered (R)
individuals, according to the following evolution equations:

$\displaystyle S_{t+1}$
$\displaystyle=S_{t}-\lambda\left(1-\alpha\right)\dfrac{I_{t}S_{t}}{N_{t}}-\lambda\alpha\left(1-\sigma\right)\dfrac{I_{t}S_{t}}{N_{t}}$
$\displaystyle+\left(1-\sigma\alpha\right)S_{t}$ (1) $\displaystyle E_{t+1}$
$\displaystyle=E_{t}+\lambda\left(1-\alpha\right)\dfrac{I_{t}S_{t}}{N_{t}}+\lambda\alpha\left(1-\sigma\right)\dfrac{I_{t}S_{t}}{N_{t}}$
$\displaystyle+\left(1-\epsilon\right)E_{t}$ (2) $\displaystyle I_{t+1}$
$\displaystyle=I_{t}+\epsilon E_{t}+(1-\alpha-\beta)I_{t}$ (3) $\displaystyle
R_{t+1}$ $\displaystyle=R_{t}+\sigma\alpha S_{t}+\beta I_{t}$ (4)

In the SEIR model above, the classical parameters are the recovery rate
($\beta$), the inverse of the incubation period ($\epsilon$), and the
infection rate ($\lambda$). Here we have generalized the model presented in
Faranda and Alberti (2020) by introducing two additional parameters able to
succinctly mimic the strategies of a vaccination campaign, namely the
vaccination rate per capita $\alpha$ and the vaccine efficacy $\sigma$, see
Sun and Hsieh (2010). In order to consider uncertainties in long-term
extrapolations and time-dependent control parameters, a stochastic approach is
used through which the control parameters
$\kappa\in\\{\alpha,\beta,\epsilon,\lambda,\sigma\\}$ are described by an
Ornstein-Uhlenbeck process Uhlenbeck and Ornstein (1930) with drift as
follows:

${\mathrm{d}\kappa}=-\kappa(t)\mathrm{d}t+\kappa_{0}\mathrm{d}t+\varsigma_{\kappa}dW_{t},$
(5)

where
$\kappa_{0}\in\\{\alpha_{0},\beta_{0},\epsilon_{0},\lambda_{0},\sigma_{0}\\}$,
$dW_{t}$ is the increment of a Wiener process. We remind that the basic
reproduction number (Paul L. Delamater et al., 2019) is written as
$R_{0}=\beta_{0}/\lambda_{0}$. In Eqs. (1)-(5) we set $dt=1$, which is the
highest time resolution available for official COVID-19–related counts and is
relatively small compared to the characteristic times associated with COVID-19
infection, incubation, and recovery/death.

$\alpha_{0}$ | $\beta_{0}$ | $\epsilon_{0}$ | $\sigma_{0}$ | $m_{0}$
---|---|---|---|---
0.0015 (see Ref. Salvioli, 2021) | 0.37 (see Ref. Faranda and Alberti, 2020) | 0.27 (see Ref. Lauer  _et al._ , 2020) | 0.59 (see Ref. Merryn Voysey et al., 2021) | 0.015 (see Ref. Fanelli and Piazza, 2020)
$\varsigma_{\alpha}$ | $\varsigma_{\beta}$ | $\varsigma_{\epsilon}$ | $\varsigma_{\sigma}$ | $\varsigma_{m}$
0.25 (see Ref. Salvioli, 2021) | 0.2 (see Ref. Faranda  _et al._ , 2020) | 0.2 (see Ref. Salvioli, 2021) | 0.1 (see Ref. Merryn Voysey et al., 2021) | 0.0
Table 1: Model parameters used for our simulations with corresponding
references.

Initialising parameters with their associated reference are shown in Table 1.
The mortality rate $m_{0}$ is also shown, set to $0.015$ (Fanelli and Piazza,
2020). While $\beta_{0}$ and $\epsilon_{0}$ and the associated $\varsigma$ are
the same as in (Faranda and Alberti, 2020), the values of $\sigma_{0}$ and
respective $\varsigma$ are derived from the range given for the AstraZeneca
vaccine phase 3 tests for the first dose (Merryn Voysey et al., 2021), and
$\alpha_{0}$ and $\varsigma_{\alpha}$ are given supposing that both Italy and
France keep vaccinating $10^{5}$ individuals per day with a 20% daily
fluctuation (Salvioli, 2021). As in Faranda and Alberti (2020), we also set
$\varsigma_{\lambda}=0.2$, allowing for 20% daily fluctuations in the
infection rate. Note that here we restrict to Gaussian fluctuations: as shown
in Faranda and Alberti (2020), allowing for log-normal fluctuations of the
parameters does not change the average results but slightly enhance their
dispersion. See Supplementary Material for the numerical code.

##  III Estimate of the Excess Deaths due to Stopping AstraZeneca Vaccine
Inoculation

Figure 1 reports the daily number of deaths $m_{0}\times I_{t}$ as a function
of time for Italy (a) and France (b). Initial conditions are set for both
countries to the values reported on March 15th as follows: for Italy, we set
$N=60\cdot 10^{6}$ population, $E_{t=1}=I_{t=1}=20\cdot 10^{4}$ as the
infected and exposed populations, $R_{t=1}=11\cdot 10^{6}$ as the sum of
$9\cdot 10^{6}$ recovered estimated from serologic tests and $2\cdot 10^{6}$
immunized from 2 doses of either Pfizer/BioNTech, Moderna or AstraZeneca
vaccines and $R_{0}=1.16$. For France, we set $N=67\cdot 10^{6}$,
$E_{t=1}=I_{t=1}=25\cdot 10^{4}$, $R_{t=1}=13.2\cdot 10^{6}$ as the sum of
$11\cdot 10^{6}$ recovered estimated from serologic tests and $2.2\cdot
10^{6}$ immunized from vaccines and $R_{0}=1.02$. For both France and Italy,
we assume that the virus, after the second wave, has infected the 15% of the
population. This estimates are based on Pullano  _et al._ (2021) who reported
a 7%$\pm$ 3% total infections for France after the first wave, assuming that
the second wave had a similar magnitude for both countries. We remark however,
that our results are basically insensitive to oscillation of $S(1)$ of order
of 5 millions individuals (cfr. Supplementary Material Figure S1). Rather than
integrating the Fokker-Planck equation (Risken, 1996) corresponding to the
system of equations given above, we follow a Monte Carlo approach and we
perform two sets of $N_{r}=1000$ realizations (see supplementary material
Figure S2 for a justification of this value): stopping (red) and continuing
(blue) the vaccination campaign at the same rate. The model is integrated for
500 days, that is about the time it would take to vaccine the rest of the
susceptible population with AstraZeneca at the rate of 105 individuals per
day.

Figure 1: The number of daily deaths $m\times I(t)$ as a function of time (300
out of 500 days shown) for Italy (a) and France (b) using the values of
$R_{0}=1.16$ (Italy) and $R_{0}=1.02$ reported respectively for the 15th of
March countries. Solid lines show the ensemble average, dotted lines extend to
one standard deviation of the mean. Red and blue curves refer respectively to
no vaccination and a vaccination campaign whose efficacy is 59%.

First, we observe a monotonic decrease in the daily deaths for all scenarios
considered from the initial date $t=t_{0}$ corresponding to March 15, 2021.
This is in agreement with actual estimates that for Italy and France the so-
called third wave should reach its peak in the second half of March, 2021
333The Guardian, March 2021: https://tinyurl.com/vvwhcydzMoreover, we observe
that the cumulative number of deaths significantly (we take the width of the
error bars as level of significance) reduces if vaccinations are continued at
100000 doses per day with respect to the scenario where vaccination is
stopped. For Italy (France) completely halting the vaccination, at the actual
epidemic rate, the number of excess deaths from COVID19 would amount to $9\pm
3\cdot 10^{3}$ ($1.2\pm 0.4\cdot 10^{3}$) excess deaths from COVID19. The
difference between the two countries is largely due to the value of $R_{0}$,
which is larger for Italy. This suggests that halting vaccination in a growing
epidemics phase (Italy) has more dramatic consequences than in a more
controlled scenario of $R_{0}\approx 1$ (France).

Our previous analysis is based on a total stop of AstraZeneca vaccination.
However, a more realistic scenario is to assume that AstraZeneca vaccination
will resume after a limited number of days used for verification. We
investigate this effect in Fig. 2. There, we consider the average excess
deaths as a function of the interruption length in number of days (x-axis) and
$R_{0}$ (y-axis) for Italy (a) and France (b). The excess deaths are computed
with respect to a base scenario where vaccine injections are never interrupted
and they are averaged over 1000 realizations of the SEIR model. Figure 2 shows
that the longer is the vaccine injections disruption, the higher is the number
of excess deaths. The impact is stronger for higher values of $R_{0}$. While
waiting the advice of EMA about AstraZeneca safety, many national health
agencies also announced that, when allowed, they would resume the vaccination
at a higher rate than before to override the effects of the stop. In the
supplementary Figure S3 we therefore present a set of simulation where, for a
number of days equal to those of the vaccination interruption, injections are
performed at a double rate than originally planned, i.e., 2$\cdot 10^{5}$
individuals/day, in order to compensate for the lost vaccinations. Although
reduced, the number of excess deaths is still high and of the same order of
magnitude as the one estimated in Fig. 2, as a result of the nonlinear cascade
effect of the extra infections occurred in the period when vaccinations were
interrupted. A focus on the actual values of $R_{0}$ for Italy and France is
reported in Fig. 3. Here we compare the two countries and we also show the
effect of doubling vaccination rates. This shows that excess deaths scale down
by a factor two but they remain of the same order of magnitude as for the case
of a business-as-usual vaccination rate, namely $10^{5}$ vaccinations/day.

Figure 2: (a,c) Average and (b,d) standard deviation over $N_{r}=1000$
realizations of the stochastic SEIR model showing the excess deaths $m\times
I(t)$ as a function of the number of the days of interruption of AstraZeneca
vaccinations (x-axis) and $R_{0}$ (y-axis) for Italy (a,b) and France (c,d).
The excess deaths are computed with respect to a base scenario where vaccine
injections are never interrupted. Note that $x$-axis starts at $N=1$. Each
realization of the SEIR model is integrated for 500 days. Figure 3: Average
over $N_{r}=1000$ realizations of the stochastic SEIR model showing the excess
deaths $m\times I(t)$ as a function of the number of the days of interruption
of AstraZeneca vaccinations for Italy $R_{0}=1.16$ (black and France
$R_{0}=1.02$ (magenta). Simulations are smoothed with a moving average filter
with window size 10 days. Stars indicate simulations where vaccination are
resumed at the same rate, dots simulations where the vaccination rate is
doubled for a number of days equivalent to those of interruption. Error bars
are computed as the mean relative error.

##  IV Worst case scenarios for AstraZeneca Side Effects

The final step in our investigation is to compare the previous estimates of
excess deaths with an order of magnitude estimate of deaths due to DVT
resulting from side effects of the AstraZeneca vaccine. In order to make a
meaningful comparison, in a case where uncertainties are very large and hard
to quantify, we will consider a worst case scenario for the impacts of the
side effects. This scenario relies on the unrealistic hypothesis that the
totality of susceptible population to DVT suffers from DVT shortly after being
vaccinated, and the lethality rate is similar to the one observed in the
overall population.

As of March 15th 2021, few dozens suspect cases of DVT have been reported over
a number of 5 millions vaccinated people with AstraZeneca in Europe444here, we
use European data accessible via the website of the European Medicines Agency
at https://tinyurl.com/ht8y98kr to average out the large spread of national
data.. By suspect cases we mean people who have developed DVT in the few days
following the vaccination. This leads us to an estimate of a frequency of 6
cases per million of vaccines. Let us call this rate $r_{AZ}^{DVT}$. Let us
also consider that, in the case of France, the incidence of DVT has been
estimated to 1800 people per 1 million inhabitants per year (Bouée  _et al._
(2016)), with a lethality rate after three months of 5% Heit (2015), raising
to 30% when a period of 5 years is considered Cushman (2007). This leads to
estimating a total of the order of 10000 deaths per year as a result of DVT.
Even assuming that all DVT cases following the inoculation of the AstraZeneca
vaccine would have not manifested themselves in absence of the injection, we
have that $N$ vaccinations would lead to an extra $N\times r_{AZ}^{DVT}$ DVT
cases. Let us assume that all of these cases result into death555Current data
suggest that this is manifestly a gross worst case approximation.. We then
have that $10^{5}$ daily vaccinations would result into a maximum of 0.6 daily
deaths. In 500 days, which is the time needed to cover the entirety of the
French population, this leads to an upper bound of 300 deaths. Considering a
death rate of $30\%$, the number scales down to approximately 100, while
considering a death rate of $5\%$ the number scales down to approximately 15.
Similar figures apply for Italy.

##  V Conclusion

Decision-making in presence of strong uncertainties associated with health and
environmental risks is an extremely complex process, resulting from the
interplay between science, politics, stakeholders, activists, lobbies, media,
and society at large Anonymous (2010); Benessia and De Marchi (2017); Reis and
Spencer (2019). In this letter, we have aimed at contributing to the debate on
different strategies for combating, in conditions of great uncertainties in
terms of health and social response, pandemic like the current one caused by
the SARS-CoV-2 virus. We have focused on the case of the AstraZeneca COVID-19
vaccine and on the locales of Italy and France, for the period starting on
March 15th 2021. The goal is providing a semi-quantitative comparison, based
on Fermi estimates informed by a simple yet robust stochastic model, between
the excess deaths due to temporal restriction in the deployment of a still
experimental vaccine and the excess deaths due to its possible side effects.
Given the many uncertainties on the (possible) side effects of the vaccine, we
have resorted to making worst case scenario calculations in order to provide a
robust upper bound to the related excess deaths. Our results are preliminary
and should be supplemented by more detailed modelling and data collection
exercises. Indeed: i) we assume a single vaccine with the nominal AstraZeneca
efficacy, neglecting the other available vaccines, ii) we consider a fixed
vaccination rate, iii) for AstraZeneca DVT side effects we consider French
data and rescale them for the Italian populations, iv) we focused our analysis
on DVT side effects, but other pathologies could be considered with the same
approach. Yet, these results clearly suggest - see a useful summary in Table 2
\- that the benefits of deploying the vaccine greatly outweigh the associated
risks, and that the relative benefits are wider in situations where the
reproduction number is larger, and they increase with the temporal duration of
the vaccine ban. We have also analysed the case of resuming the vaccinations
at a double rate ($2\cdot 10^{5}$ vaccinations/day) for an amount of days
equal to vaccine interruption period (Fig. 3 and Fig. S3). This analysis has
pointed out that excess deaths are still of the same order of magnitude as
those observed by resuming vaccinations with $10^{5}$ vaccinations/day
injection rate but scale down by a factor 2. This is an evident outcome of the
nonlinear effects of epidemiological dynamics: those who have not been
vaccinated can contaminate other individuals before vaccination resume, as a
result of a cascade mechanism also observed in turbulent flows: there, energy
injected in large scales vortex is transferred to small scales via nonlinear
interactions between scales (Kolmogorov, 1941). Here, in analogy, a few non-
vaccinated individuals can produce a large number of infected individuals. The
process can only stop if a huge number of daily vaccinations (much larger than
a factor 2) is performed. Nevertheless, this still requires a characteristic
recovery timescale $T$ that is larger than the typical immunization scale
$\eta$ (e.g., a few months for AstraZeneca (Merryn Voysey et al., 2021)).
Finally, even if several countries have resumed, or are going to resume,
AstraZeneca vaccinations, the effect of the interruption is hard to
counterbalance and require vaccination efforts difficult to set-up in due
times. Furthermore, at least for large countries where AstraZeneca vaccination
could resume, the confidence of the population in the vaccines is reduced by a
non negligible percentage 666The Economist, March 2021:
https://tinyurl.com/83cbr4d3. In this sense, our estimates are likely to be
conservative and might possibly underestimate the excess deaths deriving from
the disbelief in the vaccination policies observed in the largest European
countries. The analysis presented here has been performed with a parsimonious
but well-posed and tested model and we hope that the results we obtain might
be the starting point for more detailed, more advanced, and more mature
investigations with sophisticated models and data collection exercises.

Excess Deaths | Italy | France
---|---|---
Stop AZ for $t=500$ days | $9000\pm 3000$ | $1200\pm 400$
Stop AZ for $t=14$ days | $1700\pm 500$ | $430\pm 70$
Stop AZ for $t=7$ days | $790\pm 90$ | $160\pm 30$
Stop AZ for $t=3$ days | $260\pm 50$ | $130\pm 20$
Worst case DVT deaths due to AZ | $\approx 280$ | $\approx 300$
High fatality DVT deaths due to AZ | $\approx 90$ | $\approx 100$
Standard fatality DVT deaths due to AZ | $\approx 13$ | $\approx 15$
Table 2: The first 4 lines of the table indicate the excess deaths due to the
interruption of AstraZeneca compared to a reference scenario where the vaccine
injections are never interrupted. The SEIR model is integrated for 500 days
with $R_{0}$ = 1.16 for Italy and $R_{0}$ = 1.02 for France. The last 3 rows
of the table show the deaths from deep vein thrombosis (DVT) that could be due
to the vaccine in three different scenarios: the worst case (100% mortality
rate), a high mortality scenario (death rate of 30%) and a standard mortality
scenario (5% mortality rate) assuming a period of 500 days.

## Acknowledgements

This work has been greatly supported by the London Mathematical Laboratory and
we acknowledge the logistic support of SCuP. VLu acknowledges the support
received from the EPSRC project EP/T018178/1 and from the EU Horizon 2020
project TiPES (Grant no. 820970). This is TiPES’ contribution #97. We thank A
Veber, A Mazaud, FM Breon, the Modcov19 CNRS community and two anonymous
reviewers for useful comments and suggestions.

## Data Availability

Raw data that support the findings of this study are openly available in Johns
Hopkins University Center for Systems Science at
https://systems.jhu.edu/research/public-health/ncov/. Derived data supporting
the findings of this study are available from the corresponding author upon
reasonable request.

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