#### Non-relativistic geometry of holographic screens

Received: March
Non-relativistic geometry of holographic screens
Mudassir Moosa 0 1
Open Access 0 1
c The Authors. 0 1
Lawrence Berkeley National Laboratory,
0 University of California , Berkeley, CA 94720 , U.S.A
1 Center for Theoretical Physics and Department of Physics
We propose that the intrinsic geometry of holographic screens should be described by the Newton-Cartan geometry. As a test of this proposal, we show that the evolution equations of the screen can be written in a covariant form in terms of a stress tensor, an energy current, and a momentum one-form. We derive the expressions for the stress tensor, energy density, and momentum one-form using Brown-York action formalism.
Classical Theories of Gravity; Gauge-gravity correspondence
1 Introduction Holographic screens Newton-Cartan geometry 3.1
Global-time coordinates on M
Currents as response to the geometry
Newton-Cartan geometry on holographic screens
Covariant screen equations
Derivation of PA,
AB, and E0 from gravitational action
The holographic principle [1, 2], loosely speaking, asserts that the physical degrees of
freedom of quantum gravity in a region of area A can not be more that 4GA~ . This principle
was presented in a more formal setting in terms of covariant entropy bound, or Bousso
bound [3{7], which bounds the entropy on the lightsheets of any spatial codimension-2
surface by the area of this surface. The AdS-CFT correspondence [8] provides an example
of this principle. The correspondence states that the quantum state on a slice of AdS is
described by the state of the CFT on the boundary of that slice.
Despite the success of the AdS-CFT correspondence, not much progress has been
made in generalizing it to general spacetimes. The reason behind the lack of progress in
this direction is that it is not clear where the information of the state of the bulk theory
on a given slice of the spacetime lives. That is, it is not clear what codimenion-1 surface
plays the role of the conformal boundary of the AdS. Holographic screens [9] are believed
to be a candidate for this surface.
A holographic screens is constructed by starting with a null foliation, N (R), of the
spacetime. On each null hypersurface, we choose a spatial cross-section of largest area.
This cross-section, (R), called leaf, is a codimension-2 spatial surface with vanishing
expansion. That is,
where ka is the null generator of the null slice, N (R). Assuming the null energy condition,
the focussing theorem (see for e.g. [10]) guarantees that the portions of N (R) `outside'
and `inside' of (R) are two seperate lightsheets of (R). The covariant entropy bound
(k) = 0 ;
then implies that the entropy of the quantum state on the null slice is bounded by the
area of the corresponding leaf. The codimension-1 hypersuface foliated by these leaves,
by construction, bounds the entropy of the spacetime, slice by slice. This codimenion-1
hypersurface is called holographic screen.
Recently it has been shown that the area of the leaves obey an area law in the
spacetimes which satisfy few generic conditions [11, 12]. The area law states that the area of
the leaves is monotonic as we translate from one leaf to another. The fact that the area
of leaves is not constant seems problematic, as it indicates that the degrees of freedom
of the dual theory vary with time. This suggests that the dual theory might not be a
Some progress has been made in developing the holographic theory of the general
spacetimes. In [13], the holographic entropy formula of [14, 15] has been generalized beyond
AdS-CFT correspondence. They proposed that the entanglement entropy of the reduced
quantum state on some subregion of the leaf is related to the area of the stationary surface
anchored on the boundary of that subregion. This proposal has been used in [16, 17] to
study the Hilbert space structure of the holographic theory.
In [18], we have studied the dynamical equations of holographic screen. These are
actually the constraints equations of GR (see eqs. (2.9)), but we have decomposed them in
2 + 1 equations, using the additional structure of the screen in terms of foliation into leaves.
We will review these equations in section 2. These 2 + 1 equations are invariant under the
foliation preserving di eomorphism, but not under arbitrary coordinate transformation
on the screen. One of the main results of this paper is to write a covariant version of
these equations.
The goal of this project is to understand the intrinsic geometry of holographic screen,
H. The motivation behind this is that if we hope to put a holographic
eld theory on
H, then we need to understand how to couple this theory with the geometry of H. For
instance, we have to de ne the notion of parallel transport and covariant derivative on H.
If H were a hypersurface which is nowhere null, with the normal vector normalized such
that n^an^a =
1, then the induced metric on the screen would have been
ab = gab
In this paper, we will assume the following index notation: the rst half of the Latin
letters (a; b; c) denote the directions in the full 4-dimensional spacetime. The second half
of the Latin letters (i; j; k) denote the directions on the codimension-1 hypersurface (such
as H). The upper case Latin letters (A; B; C) denote the directions on the codimenison-2
surface (such as (R)). Let eia be the pull-back operator that pull backs an one-form from
the spacetime to the codimension-1 hypersurface. Similarly, we de ne eaA and eiA. The
pull-back of the spacetime metric on H is
This metric would have completely described the intrinsic geometry of H. For instance,
if we were only given H with this metric, and had no information of the geometry of
background spacetime, we could have used this metric to de ne various geometric quantities
on H, like connection and intrinsic Riemann curvature for example.
However, there are known examples where H does not have a xed signature. As a
result, the induced metric becomes degenerate at points where the hypersurface changes
signature. Therefore, we can not invert this metric, which means we can not use the
standard formula of various geometric quantities, such as Christo el symbols.1 This concludes
that the metric in eq. (1.2) is not suitable to describe the intrinsic geometry of H.
Furthermore, holographic screens are not arbitrary hypersurfaces of inde nite
signature. Rather they have an additional structure of foliation into leaves. It was shown in [12]
that this foliation is unique. This hints to the non-relativistic structure of holographic
screen. Another reason to expect the non-relativistic structure of holographic screen comes
from the fact that the quantization on the light front is non-relativistic (see for e.g. [19]).2
Motivated by these observations, we propose that the geometry of holographic screen
should be described by a non-relativistic geometry. In section 3, we will review the
NewtonCartan (NC) geometry, an example of a non-relativistic geometry. In section 4, we will
argue that the Riemannian metric on the leaves, the foliation structure of the screen, and
the existence of the vector eld normal to the leaves but tangent to the screen, allow us to
de ne NC data on H, which we can use to de ne connection and curvature on H.
In section 4.1, we will use the di eomorphism invariance of the screen to derive a
covariant version of the dynamical equations of the holographic screens. This will be a
test of our proposal that the geometry of the screen is described by the NC formalism.
These covariant equations will be in terms of a screen `stress tensor', `energy current', and
`momentum one-form'. We will make an ansatz of these tensors in section 4.1. Finally, we
will follow [20] to perform the treatment similar to that of Brown and York [21] to verify
Holographic screens
A future/past holographic screen [11, 12] is constructed by starting with the world line of
an observer and shooting future/past light cones from it [9]. These light cones provide us
with an one-parameter null foliation, N (r), of some region of the spacetime. On each null
hypersurface, we choose the spatial cross-section of maximum area. These spatial surfaces
are called leaves, (r). The Bousso bound [3, 5] guarantees that entropy on N (r) is bounded
by the area of (r). The union of these leaves is called the future/past holographic screen.
By construction, holographic screen is a codimension-1 hypersurface, which is foliated
by the codimension-2 spacelike surfaces,
(r), where r is the foliation parameter. Since
the background manifold is Lorentzian, there are two future directed null vectors normal
1. The
de ning property of holographic screen is that the expansion of the null congruence in ka
1Of course, one can still de ne various quantities as the pull back of their analogues from the full
spacetime. However, this would require the knowledge of the background spacetime, where as we want to
study the geometry of H assuming we know nothing about the metric of the spacetime.
2We thank R. Bousso for emphasizing these points.
direction vanishes, that is
is the induced (inverse) metric on the leaves. For future/past holographic screens, the
leaves are marginally trapped/anti-trapped [11, 12]. This means that the expansion in the
la direction is negative for future holographic screens,
where as the opposite inequality holds for past holographic screen.
Given an arbitrary codimension-1 hypersurface of inde nite signature, one might ask
what data is required to describe the intrinsic geometry of the hypersurface. One
candidate is to use the induced metric on the hypersurface, de ned as the pull-back of the
spacetime metric
q~abrakb = 0 ;
q~ab = gab + kalb + lakb ;
= eaAebBq~AB ;
Though this works for the hypersurfaces that are nowhere null, it is not suitable for the
hypersurfaces which do not have a
xed signature. For these hypersurfaces, the metric in
eq. (2.5) becomes degenerate, and hence non-invertible, wherever the signature changes.
As a result, we can not de ne connection, curvature, or any other intrinsic quantity that
requires the inverse metric, on these hypersurfaces.
As holographic screens do not have a de nite signature, it is interesting to understand
how to describe their intrinsic geometry. In this section, we will review some properties
of the holographic screens that were proven in [12]. We will see in section 4 that these
properties allow us to propose that the intrinsic geometry of the holographic screens can
be described by the Newton-Cartan geometry. We will also review the evolution equations
of the holographic screens, that were studied in [18], in the current section.
Let hi be the vector tangent to the screen, but normal to the leaves. This means that
we can write it as a linear combination [11, 12]
eiahi = ha = la + ka :
Since the signature of the screen is not xed, we can not normalize the vector ha by xing
This implies that the normal vector can be written as
Assuming some mild generic conditions, an area law for the holographic screens was
proven in [11, 12]. An important ingredient in the proof of this area law in [12] was the
na =
proof of the fact that
in eq. (2.6) does not vanish. In particular, it was shown that
< 0 for future holographic screens, while
> 0 for past screens. This important result,
together with eq. (2.1) and eq. (2.6), immediately yields the area law: \the area of the
leaves always increases as we move along the screen". That is,
(h) =
this result plays a signi cant role in the consistency of our proposal to describe the intrinsic
geometry of the screen using Newton-Cartan geometry.
Another important theorem proven in [12] guarantees the uniqueness of the foliation
of a holographic screen into leaves. In particular, it was shown that if we take a screen and
choose a di erent foliation into codimension-2 closed surfaces, these surfaces will not have
vanishing null expansion. This implies that the foliation into leaves is important for the
de ning property of the holographic screens, as was also emphasized in [20]. Without this
foliation, (k) = 0 condition is not well-de ned.
Imposing Einstein equations on the spacetime puts constraints on the extrinsic
geometry of a codimesion-1 hypersurface,
For hypersurfaces with
xed signature, Gauss-Codazzi equations (see for e.g. [22]) can be
used to write these constraint equations as
Gabnaeib = 8 Tabnaeib :
ij Kb ) = Tabn^aeib ;
where Dbj is the covariant derivative compatible with the metric ij in eq. (2.5), and Kbij
is the extrinsic curvature of the hypersurface.
However, eq. (2.10) is not well-de ned
for the holographic screens as both the covariant derivative compatible with ij and the
extrinsic curvature tensor are de ned using the inverse induced metric, ij . To resolve this
problem, it was suggested in [18] to use the foliation structure of the screen, and to use
the geometric quantities de ned using the (Riemannian) metric on the leaves, such as the
extrinsic curvature of the leaves
BA(hB) = eaAeaBBa(hb) ;
BA(nB) = eaAeaBBa(nb) ;
Ba(hb) =
Ba(nb) =
The extrinsic geometry of the screen is not completely
xed by that of the leaves,
as BA(nB) has only three components while there are six components of Kbij . The rest of
the information of the extrinsic geometry of the screen is given in terms of the normal
which can be decomposed into a scalar and a one-form on the leaf,
~ = hi!i =
A = eiA!i =
In terms of these geometric quantities, and using the foliation structure of the screen,
eq. (2.9) can be decomposed in 2 + 1 `evolution' equations of the holographic screens [18]
(also see [23{25] for similar equations for dynamical or trapping horizons [26{29])
A = 8 Tabnbha ;
= 8 TabnbeaA ;
where DeA is the covariant derivative compatible with q~AB, and bA
These evolution equations are invariant under di eomorphism on the screen that
preserve the foliation structure of the screen. These di eomorphism include the di
eomorphism on the leaves, plus the reparameterization of the foliation parameter [18]. In
section 4.1, we will use the Newton-Cartan structure on the screen to write these equations
in a covariant form, that is in a form which is invariant under the full di eomorphism on
Newton-Cartan geometry
The Newton-Cartan geometry is an example of a non-relativistic geometry, originally
developed by Cartan as the geometric formulation of Newtonian gravity [30, 31]. This has
gained popularity in the recent years owing to its many applications. For instance, it has
been used to write the e ective eld theory and to study the covariant Ward identities of
the quantum Hall e ect in [32, 33]. It has also been used to couple non-relativistic eld
theories with the background spacetime in [34{38], and to present the covariant
formaulation of non-relativistic hydrodynamics in [37, 39, 40]. It was shown in [41] that making
the Newton-Cartan geometry dynamical gives rise to Horava-Lifshitz gravity [42], and it
has been used to study Lifshitz holography in [36, 43{45].
M = R
In this section, we present a review of the Newton-Cartan (NC) geometry. There
are di erent conventions used in the literature when describing NC geometry. We will be
using the conventions used in [32]. Consider a 2 + 1-dimensional manifold M such that
are 2-dimensional Riemannian manifolds (generalization to higher
dimensions is trivial). The NC structure on M
means that there is a one-form
^ d = 0 ;
a degenerate inverse metric hij such that
and a vector eld vi such that
The condition in eq. (3.1) implies that
is hypersurface orthogonal. The volume element
on M is given by
where e is the volume element on .
Note that we are not provided with a metric on M. This means that the connection,
and hence covariant derivative, on M can not be de ned using the standard formulae from
any textbook of GR, such as [10]. To remedy this, we rst de ne a unique tensor hij
Demanding `metric compatibility' conditions
hij vj = 0 ;
hikhkj = ij
Di j = 0 ;
Dihjk = 0 ;
and `curl-freeness' of vi
yields the unique connection [33]
Dj vi = 0 ;
bijk = vi@j k +
Note that this connection has torsion unless i is closed:
The Lie derivative of hij along vi
is transverse, that is viBi(jv) = 0 :
Global-time coordinates on M
= vi(d )jk :
2 Lvhij ;
All the tensors and equations in the last section holds in any coordinate system on M.
coordinates' (GTC) in [32, 33]. In these coordinate systems, i is of the form
hij j = 0 ;
vi i = 1 :
i = N (dt)i ;
is hypersurface
orthoginverse Riemannian metric on . That is
The condition vi i = 1 means that
Eq. (3.5) implies that hij is of the form
hij =
vi =
hij =
= dt ^ d2x N pq~:
Currents as response to the geometry
For a relativistic eld theory, the energy momentum tensor is de ned as the variation of
the action with respect to the metric to which the theory is coupled. Similarly, the currents
in a non-relativistic theory can be de ned as the variation of the action with respect to the
NC geometry [32, 33, 35{37]. Assume that we have an action as a functional of the NC
data, INC[ ; h; v], and we want to compute its variation under a di eomorphism generated
by the vector eld i. The variations of i, vi, and hij are
means that we can keep
i unconstrained, but demand that [33]
momentum one-form Pi, are then de ned in terms of the variation of the action, INC[ ; h; v],
= j Di j
vi = L vi =
hij = L hij = Di j + D
vi =
hij =
dtd2x N pq~
Note that the stress tensor and momentum one-form are transverse:
Inserting the variations from eqs. (3.17){(3.22) in eq. (3.23), and performing the integration
by parts lead to [33, 35, 37]
ij i = 0 ;
Pivi = 0 :
INC =
This is an important result that we will use in section 4.1 to write the covariant form of
the screen equations, eqs. (2.18){(2.19). We will also use the general form of the variation
in eq. (3.23) to identify the currents on holographic screens.
Newton-Cartan geometry on holographic screens
This section is the most important part of this paper. In this section, we will combine the
ideas from the last two sections to propose that the geometry of the holographic screen,
H, can be described by the NC geometry. In particular, we suggest that the Riemannian
metric on the leaves, (r), the vector eld, hi, normal to the leaves, and a non-vanishing
in eq. (2.6), play the role of NC data on H.
The foliation structure of H into leaves allows us to pick a coordinate system on H:
fr; xAg, where r is the foliation parameter, and xA are angular coordinates on (r). These
are the GTC of section 3. The foliation parameter, r, acts as the `screen time'. One can
always de ne a one-form, (dr)i, which is normal to the leaves. We use this to write a
This means that
the vector eld hi to de ne vi
plays the role of the lapse function, N . This proposal would have been
were allowed to vanish anywhere. Recall that
is indeed non-vanishing,
as was proven in [12] assuming certain generic conditions.
1hi. The coordinate representation of this vector eld
Finally, we use the metric on the leaves, q~AB, and its inverse q~AB, to construct a (2; 0)
symmteric tensor on H, whose coordinate representation is
We combine i in eq. (4.1), vi in eq. (4.2), and hij in eq. (4.3) to put a NC structure on
the screen. This allows us to de ne geometric quantities like connection and covariant
i = (dr)i :
vi =
hij =
derivatives on H. The non-zero components of the connection in eq. (3.8) are
b0i0 = @i log ;
A
where e BC are the Christo el symbols of the metric q~AB.
Note that we have only presented the NC data in the GTC. The representation of i,
vi, and hij in any arbitrary coordinate system can be determined using the transformation
law of tensors. This is an important result, as this allows us to describe the geometry of
H without using the foliation structure of H.
Also note that the scalar
in eq. (2.6), unlike , does not appear in eqs. (4.1){(4.3).
This means that the intrinsic geometry of H is not dependent on . This is consistent with
the interpretation of
as the \slope" of H [18]. That is, it tells us how much the screen
bends in the spacetime it is embedded in. Therefore,
is an extrinsic quantity.
nish this section, we want to compare the NC geometry with the geometry
described by the induced metric for screens which are nowhere null. The advantage of NC
geometry is that the foliation structure of screen is implicit in it through one-form
Another advantage of using NC geometry will be apparent in section 5, where we will nd
that the boundary conditions on NC data are more suitable for holographic screens than
the boundary conditions on the induced metric.
Covariant screen equations
The evolution equations of holographic screen, eqs. (2.18){(2.19), are the decomposition of
constraint equations, eq. (2.9), into directions normal and transverse to the leaves. These
equations are invariant under the foilation-preserving di eomorphisms. In this section, we
will show that we can use the NC geometry to write a covariant version of these equations.
Before we derive the covariant screen equations, recall how the constraint equations,
eq. (2.10), for the hypersurface of a
xed signature arise from di eomorphism
invariance. Imagine taking the spacetime and truncating it at the hypersurface. As a result,
the Einstein-Hilbert (EH) action has to be compensated with the Gibbons-Hawking-York
(GHY) boundary term living on the hypersurface,
I = IEH[g] + IGHY [g; ] :
The variation of this action is [21]
Now we take a vector eld tangent to the hypersurface, i, and generate di eomorphism
by it. The variation of the total action under this transformation is
I =
I =
1 Z
1 Z
I =
1 Z
I =
I = IEH[g]
IH [g; ; h; v] :
has a well-de ned variation, and is given by
where we have used eq. (3.23). If we take the variation to be a di eomorphism generated
by a vector eld i, on H, then we get
integration by parts yields
I =
1 Z
Demanding that the action is invariant under di eomorphisms generated by i gives us
Bianchi identity and Gauss-Codazzi constraint. The latter when combined with the
Einstein equations becomes eq. (2.10).
Now we repeat the similar analysis for hologrpahic screens, H. That is, we take a
spacetime in which the holographic screen is embedded in, and truncate it at H. The
variational principle requires that we add a boundary action on H, whose variation cancels
the boundary term in the variation of the EH action. In the next section, we will follow [20]
and study this variational problem. For now, simply assume that there exists a screen action
as a functional of NC data such that the combination
Pi =
ij =
E i =
where we have used eq. (3.25) in the second term. Performing the integration by parts,
combined with Einstein equations becomes
Di(Pj vi) + PiDj vi + (Di
(d )ij E i =
1 Tabnaejb :
In the following, we will show that this tensor equation is a covariant version of the screen
equations, eqs. (2.18){(2.19). In particular, we will show that this equation reduces to
eqs. (2.18){(2.19) when we expand it in the GTC. Note that eq. (3.24) implies that the
momentum one-form Pi, stress tensor
sentation in the GTC
ij , and energy current E i have the following
repreThe contraction of eq. (4.14) with vi is
1 Tabnavb =
Tabnahb =
Similarly, the contraction of eq. (4.14) with hij is
1 Tabnaebhij = hij
Writing this equation in the GTC leads to
With the identi cations
TabnaebA = q~ABLhpB + pA
PA =
0 =
A =
AB =
(l) =
Eq. (4.20) and eq. (4.22) reduces to eq. (2.18) and eq. (2.19) respectively. This con rms
our claim that eq. (4.14) is a covariant generalization of the screen evolution equations. We
view this as an application of the Newton-Cartan formalism to describe the geometry of the
screen. However at this point, the identi cations that we have made in eqs. (4.23){(4.26)
seem arbitrary. In the next section, we will follow [20] to write the variation of the total
action in a form similar to eq. (4.12). This will allow us to verify the identi cations that
we have made for PA, E 0, and
Derivation of PA,
0 from gravitational action
In the spacetime with boundaries, the Einstein-Hilbert action has to be compensated with
a boundary term which depends on the boundary conditions. For instance, Dirichlet
conditions on a boundary of xed signatures leads to Gibbons-Hawking-York term. Brown
and York studied this problem in [21], where they assumed that the spacetime is bounded
by four boundaries: an outer timelike, an inner timelike, a future spacelike, and a past
spacelike boundaries. This analysis was generalized in [46], where the corner contributions
from the interaction of two di erent boundaries were also considered. This was further
generalized in [20], where the inner timelike boundary was replaced with the codimension-1
hypersurface which
is foliated by the codimenion-2 marginally trapped or anti-trapped closed surfaces,
is of inde nite signature.
Note that these are exactly the properties of holographic screens. In this section, we will
follow the calculations of [20]. However, our analysis will be di erent from that of [20] in
two ways. As we only care about the boundary contributions to the Einstein-Hilbert action
coming from the holographic screen, we will assume that there are no other boundaries of
the spacetime. The other, and more important, di erence in the two analysis is that we
will be assuming slightly di erent boundary conditions.
We now discuss the boundary conditions that we are imposing on the holographic
screens. The Dirichlet condition on the induced metric is, of course, not suitable as the
signature of screens is allowed to change. There is another reason for why Dirichlet conditions
are inapplicable. Note that we are only allowed to impose six boundary conditions on the
screen. If we x the induced metric on the screen, then we can not x any other geometric
data. However, holographic screens are special as the leaves have vanishing null-expansion,
Rather, we assume a Dirichlet condition on (k). This boundary condition also xes the
foliation structure of the holographic screen.
We still need to choose
ve more boundary conditions. We
x three of those by
imposing Dirichlet condition on the metric on the leaves, q~AB, which also
xes its inverse,
q~AB. So far, our four boundary conditions are same as those of [20]. Next, we choose to
impose boundary conditions on the variation of the evolution vector eld, hi. Note that
hi = eiA sA :
We now impose the Dirichlet conditions on sA. These are our last two boundary conditions.
Note that these conditions are di erent from those of [20] where they had imposed Dirichlet
A. As we will see below, our boundary conditions not only result in the
well-de ned variation of the action, it will also help us to compare the variation of the total
action with eq. (4.12).
As emphasized in [20], we also have to x the `length' of the null vector ka. Note that
this is not an additional boundary condition. We need to impose this condition because
the extrinsic quantities like
A depend on the `length' of the null vectors [18]. This
is equivalent to the case of the boundary with a
xed signature, where we x the norm of
the normal vector to be
1. Without xing the norm of the normal vector, the extrinsic
curvature and hence the GHY boundary term have ambiguities, and are not well de ned.
We can x the `length' of the null vectors by imposing the Dirichlet condition on .
After discussing the boundary conditions in detail, we now consider the variation of EH
action on the spacetime with the holographic screen as the boundary. We copy from [20]
IEH[g] =
1 Z
which we re-write as [20]
IEH[g] =
1 Z
Note that if we impose our boundary conditions, then the second line in the above equation
vanishes. The second term in the rst line is a total variation, and should be added to EH
action as the boundary term. Therefore, the total gravitational action in the spacetime
with holographic screen as the boundary is
I =
1 Z
The variation of this total action follows from eq. (5.3)
I =
By comparing the above expression for the total variation with eq. (4.12) (or eq. (3.23)),
we can deduce the expressions for momentum one-form Pi, stress tensor
current E i. However, all the calculation that we have done in this section are in the GTC
ij , and energy
gauge, eq. (3.12){(3.15). In this gauge, the variation
i, ui in eq. (3.21), Hij in eq. (3.22),
are of the form
With these variations, eq. (4.12) becomes
I =
Comparison of this expression with eq. (5.5) leads us to conclude that
0 =
A = 0 ;
ui =
Hij = eiAeiB q~AB :
(l) =
PA =
0 =
AB =
which agrees with the identi cations that we made in eqs. (4.23){(4.24) and in eq. (4.26).
However, note that we have failed to derive the expression for E
The reason behind this failure is that we have been working in the GTC gauge, in which
action of the theory with respect to A. This requires going away from the GTC at least to
the rst order. It is not clear how to perform the above analysis in a fully covariant way.
A using the above formalism.
We leave understanding this task to future work.
We have seen that the induced metric on a holographic screen is degenerate, and is not
applicable to de ne geometric structure such as connection and covariant derivative. We have
proposed that the intrinsic geometry should, instead, be described by the Newton-Cartan
geometry. We have used some results from [12] to argue that this proposal is consistent.
Using the NC connection on H, we have presented the screen evolution equations of [18]
into a covariant form. There are many possible directions which we intend to pursue in
the future. We brie y discuss these in the following.
The natural next step would be to couple a (non-relativistic) eld theory to holographic
screen. The coupling of eld theories to Newton-Cartan geometry has been studied in [35{
37]. It would be interesting to know if there are any constraints on the eld theory that we
can de ne on the screen. One possible way to
nd these constraints might be to compare
the entanglement structure of the eld theory with the screen entanglement conjecture
In AdS spacetime, metric near the conformal boundary can be written as a power series
in the bulk coordinate [47]. This type of expansion has also been studied in [43] for Lifshitz
gravity, where the geometry of the boundary is Newton-Cartan. It would be fascinating if
a metric of a spacetime can be expanded in a similar fashion near the screen. One-to-one
correspondence between the screen and the null foliation of the spacetime suggests that
the a ne parameter on null slices can be used as a bulk coordinate.
As discussed in section 5, we also plan to understand how to generalize the calculations
of [20] without using the GTC gauge, eq. (3.12){(3.15). This would be important as this
will allow us to derive the spatial components of the energy currents, E A, which we failed
to do in section 5.
Acknowledgments
We would like to thank R. Bousso, Z. Fisher, S. Leichenauer, N. Obers, M. Rangamani,
A. Shahbazi, and Z. Yan for helpful discussions, and to Z. Fisher and S. Shaukat for useful
feedbacks on a draft of this manuscript. This work was supported in part by the
Berkeley Center for Theoretical Physics, by the National Science Foundation (award numbers
1521446 and 1316783), by FQXi, and by the US Department of Energy under Contract
DE-AC02-05CH11231.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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