Cutoff $\rm AdS_3$ versus $\rm T\bar{T}$ $\rm CFT_2$ in the large central charge sector: correlators of energy-momentum tensor

In this article we probe the proposed holographic duality between $T\bar{T}$ deformed two dimensional conformal field theory and the gravity theory of $\rm AdS_3$ with a Dirichlet cutoff by computing correlators of energy-momentum tensor. We focus on the large central charge sector of the $T\bar{T}$ CFT in a Euclidean plane and in a sphere, and compute the correlators of energy-momentum tensor using an operator identity promoted from the classical trace relation. The result agrees with a computation of classical pure gravity in $\rm AdS_3$ with the corresponding cutoff surface, given a holographic dictionary which identifies gravity parameters with $T\bar{T}$ CFT parameters.


Introduction
The T T deformation of two dimensional quantum field theory has received intensive study in the past few years.As an irrelevant deformation, it leads to well-defined, albeit nonlocal, UV completion.In fact, it is a solvable deformation in many senses.It preserves integrability structures [1] [2], deforms the scattering matrix by multiplying CDD factors [3] [4], possesses solvable deformation of finite size spectrum [3] [5] and preserves modular invariance of conformal field theory torus partition function [6] [7].The non-locality and solvability of the T T deformation can be understood from a different perspective by reformulation to random geometry [8], which also neatly derives the flow equation of the partition function.In addition, the T T deformation can be re-interpreted as coupling to Jackiw-Teitelboim gravity of the quantum field theory, which leads to the same flow equation of the partition function and CDD factors of the scattering matrix [4] [9].Correlators of T T deformed QFT or CFT were studied in [10] [11] [12].While much of the work on the T T deformation has been done in the flat Euclidean plane or its quotient spaces such as cylinder and torus, the case for maximally symmetric spaces was considered in [13] [14].Further generalization to generic curved spaces was studied in [15] [16], which has remarkably reproduced lots of result of previous study.
For a holographic CFT 2 , it's natural to ask what the holographic dual of its T T deformation is.It was proposed by Mezei et al. [17] that for positive T T deformation parameter the holographic dual is a Dirichlet cutoff in the AdS 3 gravity, based on computation of signal propagation speed, quasi-local energy of BTZ blackhole and other physics quantities.It was followed by study on holographic entanglement entropy [18][19] [20][21] [22][23] [24][25], generalization to higher or lower dimensions [26][27] [28][29] [30] [31], and an interesting perspective from path integral optimization [32].In addition, the proposal was examined by holographic computation of correlators of energy-momentum tensor in [33].It was found that the large central charge perturbative correlators in T T CFT 2 agree with correlators of classical pure gravity in cutoff AdS 3 given a holographic dictionary that identifies gravity parameters with T T CFT parameters.But additional non-local double trace deformation must be supplemented to the T T deformation to reproduce correlators of scalar operators dual to matter fields added to gravity, in line with the general discussion of bulk cutoff in [34] [35].The possible limitation of the Dirichlet cutoff picture was echoed in [36], which showed that in the large central charge limit the holographic dual of T T CFT 2 in the Euclidean plane is in general AdS 3 gravity with mixed boundary condition, and only for positive deformation parameter and for pure gravity the mixed boundary condition can be reinterpreted as Dirichlet boundary condition at a finite cutoff, taking the original form proposed by Mezei et al..This article is to a large extent a follow-up of [33], and [37] which computed the correlators of energy-momentum tensor of T T CFT in a Euclidean plane beyond leading order in the large central charge limit.We start in Section 2 by a brief review of T T deformation which highlights a trace relation formula.In Section 3 we promote the trace relation to an operator identity and compute in the large central charge limit the correlators of energy-momentum tensor for T T CFT in a Euclidean plane, a sphere and a hyperbolic space.In Section 4 we compute correlators of energy-momentum tensor in classical pure gravity in Euclidean AdS 3 cut off by a Euclidean plane and a sphere.The gravity correlators are found to agree with T T CFT correlators given a dictionary between T T CFT parameters and gravity parameters.
In Section 5 we summarize our result and discuss related questions and possible directions of further research.

T T deformation and trace relation
The T T deformation with the continuous deformation parameter µ is defined by a flow of action in the direction of T T operator The T T operator is a covariant quadratic combination of energy-momentum tensor1 where the energy-momentum tensor is defined in the convention 3) It was shown in [5] that the composite T T operator has an unambiguous and UV finite definition modulo derivative of local operators by limit of point splitting for quantum field theory in the Euclidean plane with a conserved and symmetric energymomentum tensor.This point splitting definition can be generalized to maximally symmetric spaces by carrying over Zamolodchikov's argument, but it was found that the factorization property of the expectation value is lost in general [5][13].
We refer interested readers to Jiang's note [38] and other references for many interesting properties of T T CFT.Here we focus on the trace relation crucial for computation in the following sections When regarded as a classical field equation it was discovered in free scalar theory [3], and was later proved for T T CFT 2 in generic curved spaces in [15].Actually, we have a very basic argument for theories with Lagrangian density L as an algebraic function of the metric. 2For these theories, the energy-momentum tensor takes the form and we have the T T flow equation for the Lagrangian density And the trace relation takes the form Taking derivative of the left hand side of the equation above with respect to µ and using (2.8) 2 Free scalar falls into this category.
we get The trace relation holds at µ = 0 as a paraphrase that the energy-momentum tensor in CFT is traceless.By the first order differential equation above it must hold for all µ.For quantum theory we expect quantum corrections to the trace relation, it depends on how T T deformation is defined for quantum field theory in curve spaces. 3In our work we assume it holds as an operator identity within connected correlators, at least in the large central charge limit, and the T T operator is given by the point splitting definition since we work in maximally symmetric spaces.
3 Correlators of energy-momentum tensor of T T deformed CFT 2 in the large central charge limit In this section we use the trace relation (2.6) to compute the correlators of energy-momentum tensor in the large central charge limit, a limit of large degrees of freedom similar to the large N limit in gauge theory.More precisely it's a limit with a large central charge c of the undeformed CFT, but a finite µc where µ is the T T deformation parameter.A detailed discussion of the large c limit can be found in [37].Inspired by the work in [33] and [37], we first compute up to four point correlators of energy-momentum tensor for T T CFT in the two dimensional Euclidean plane E 2 .Then we consider T T CFT in the two dimensional sphere S 2 and the two dimensional hyperbolic space H 2 to compute up to three point correlators.

Large c correlators of T T CFT in E 2
In principle, our tools to compute correlators of energy-momentum tensor in this section are the trace relation, the conservation equation, dimensional analysis, Bose symmetry, CFT limit and other physical considerations.The conservation equation of energy-momentum tensor is It holds in a correlator except for contact terms, which plays no role if we only consider correlators at distinct points.In the Euclidean plane the metric takes the form in the complex coordinates z, z and the conservation equation is We have vanishing one point correlator and it's shown in [37] that two point correlators remain the same as in the undeformed CFT in the large c limit 4 5 6   T zz (w It's sometimes convenient to use the normalization of energy-momentum tensor in CFT 4 Here the superscript (0) on T indicates it's the energy-momentum tensor in the undeformed CFT, and the superscript (0) on the expectation value means it's evaluated in the undeformed CFT, for example, by path integral with the undeformed CFT action.By this convention we should add superscript like (µ) for the energy-momentum tensor and the expectation value in the T T deformed CFT with deformation parameter µ, but we choose to omit it for simplicity of the text. 5For simplicity we omit correlators that can be simply inferred by symmetry, e.g.
6 A bit abuse of notation, we use the equality sign even if it's only equal in the large c limit, because we exclusively work in this limit.
and the two point correlators now take the form To compute the three point correlators, we start with T ( w)Θ( v) T ( u) c where the superscript c means connected correlators. 7Using the trace relation 2.6 in the Euclidean plane we get Working in the large c limit in which connected correlators of energy-momentum tensor scale as c, the correlator on the right hand side only contribute in the large c limit by factorization into two correlators 4  (3.11) By the conservation equation ) modulo a holomorphic function in v.By Bose symmetry it must be holomorphic in w as well, then it cannot depend on u at all by translational symmetry, and it's further fixed to be zero by cluster decomposition principle.Other correlators can also be computed in this way except for T ( w)T ( v)T ( u) c and T ( w) T ( v) T ( u) c , we only know T ( w)T ( v)T ( u) c is holomorphic by the conservation equation and it has the CFT limit c However, it was proved in [37] that n point correlators are polynomial in µ of degree n − 2, so we can rule out possible additional terms dependent on µ like µ 3 c 4 1 (w−v) 4 (v−u) 4 (u−w) 4 .To summarize we list non-zero three point correlators Compared to previous work a clarification is needed.This result has been obtained in [33] as the leading order in µ result, by using the trace relation to the leading order in µ.Later in [37] it was derived for T T free scalars as large c result, that is, c times arbitrary function of µc.Here we derive it as large c result without assuming the specific model of the undeformed CFT, but we have to assume the operator identity promoted from the trace relation.In a similar way, we computed two four point correlators One can continue in this procedure to obtain all higher point correlators.

Large c correlators of T T CFT in S 2 and H 2
Now we study correlators of T T CFT in a two dimensional sphere of radius r or a hyperbolic space of radius r.In a maximally symmetric space, one point correlator of energy-momentum tensor is proportional to the metric The coefficient can be determined by the trace relation in vacuum expectation value supplemented by a trace anomaly term [17][18], and by using large c factorization, we get For sphere with radius r the scalar curvature is R = 2 r 2 , we find For hyperbolic space with radius r the scalar curvature is R = − 2 r 2 , we find We note a square root singularity occurs at µ = 24πr 2 c .Higher point correlators are a bit more complicated in a curved space.They are multipoint tensors based on the (co)tangent spaces at those points.Because the sphere and the hyperbolic space are maximally symmetric, two point correlators must be maximally symmetric bi-tensors, that is, bi-tensors invariant under the stabilizer of the two points in the isometry group, and covariant when the isometry moves the two points.Maximally symmetric bi-tensor has been studied in [39] exactly in the context of tensorial two point functions, and it has already been used in [40] to study correlators of energy-momentum tensor in maximally symmetric spaces.Recently it was reviewed in [13] to study expectation value of T T operator in maximally symmetric spaces in general dimensions.Following their analysis and assuming the energy-momentum tensor is traceless in connected correlators in the undeformed CFT, we get two point correlators of undeformed CFT in S 2 and H 2 .Details of computation are left to the Appendix A. Two point correlators of energy-momentum tensor of CFT in S 2 take the form in the complex stereographic projection coordinates of the sphere8 ,in which the metric is It's related to the spherical coordinates by z = 2 cot θ 2 e iφ z = 2 cot θ 2 e −iφ .9And two point correlators of energy-momentum tensor of CFT in H 2 take the form 4  (3.20) in the complex Poincare disk coordinates of the hyperbolic space, in which the metric is In an alternative coordinate system z = 2 tanh σ 2 e iφ z = 2 tanh σ 2 e −iφ , the metric takes the form For T T CFT in S 2 and H 2 , we can use trace relation to show the energy-momentum tensor is traceless in connected two point correlators in the large c limit, so the analysis in Appendix A can be carried over to show two point correlators are determined up to a factor as a function for S 2 and for H 2 .For S 2 , the factor can be determined by using the one point correlator of energymomentum tensor in the replica sphere obtained in [18] to compute Renyi entropy of antipodal points Taking a variation in n, the replica number, which can be viewed as a variation of the metric, we have Setting n = 1 we return to the regular sphere, and by plugging in With the known correlator , and by repeated use of Ward identity of conservation of energy momentum tensor we obtain where δ( w − v) is the delta function with respect to the measure i 2 dv ∧ dv. 10 Plugging in these correlators and completing the integration, we finally get By the same token, we need to work out one point correlator of energy-momentum tensor in the replica hyperbolic space to find the factor g(µ) for H 2 .We play the same trick as in [18], that is, we solve (3.15) together with the conservation equation in the replica hyperbolic space with the conical singularity smoothed. 11We find we get we have Similarly for H 2 we get 4 Correlators of energy-momentum tensor of Einstein gravity in cutoff AdS 3 In this section we compute correlators of energy-momentum tensor of Einstein gravity in cutoff The action for the Euclidean Einstein gravity is The first term is the Einstein-Hilbert action, the second term is the Gibbons-Hawking term where K = h ij K ij is the trace of the extrinsic curvature K ij on the boundary surface, and the third term is the counter term with other possible addition of local functions of the boundary metric omitted.Taking a functional derivative of (4.1) with respect to the boundary metric, we get one point correlator of energy-momentum tensor in T T CFT on the left hand side, and the Brown-York tensor on the right hand side which depends on the extrinsic curvature and the boundary metric.Multi-point connected correlators of energy-momentum tensor can be computed by taking functional derivative of the one point correlator with respect to the metric Therefore in order to compute gravity correlators of energy-momentum tensor, we have to compute functional derivatives of the extrinsic curvature with respect to the boundary metric.
To this end, we solve the variation of the bulk metric in response to variation of the boundary metric, then compute the extrinsic curvature from the bulk metric.
To begin with, we gauge-fix the metric to be in Gaussian normal coordinates by diffeomorphism, that is, the radial coordinate is the arclength parameter along the geodesic normal to the cutoff surface.For a variation of the boundary metric δh ij = f ij where is the infinitesimal parameter, the bulk metric takes the form where Here ρ is the radial coordinate and x i 's are transverse coordinates.In this gauge there are only three independent components of the metric.At the cutoff surface ρ = ρ 0 , the extrinsic curvature is given by The Einstein's equation for the AdS 3 gravity is12 It's shown in the Appendix B that the Einstein's equation for AdS 3 can be decomposed into three equations, the Gauss equation and the radial equation Solving these three equations order by order, we obtain the Brown York tensor order by order, thus the correlators of energy-momentum tensor.In fact, the Einstein's equation for AdS 3 can be further simplified to partial differential equations in the transverse two dimensional space, because the form of the radial dependence of the metric can be solved independently from the boundary metric, following the spirit of [41].Here we list the results of the gravity correlators and compare them to the correlators in T T CFT, leaving details of the computation to Appendix C.

E 2 as the cutoff surface
Pure gravity in AdS 3 with a cutoff y = y 0 in the Poincare patch was proposed to be the holographic dual to T T CFT in the cutoff Euclidean plane.In the Appendix C, we computed one point correlators two point correlators three point correlators 4  (4.15) and four point correlators After a rescaling of the coordinates z → y 0 l z z → y 0 l z to bring the metric in the plane ds 2 = l 2 dzdz y 2 0 back to form ds 2 = dzdz, we find the gravity correlators agree with the T T CFT correlators given the holographic dictionary c = 3l 2G µ = 16πGl (4.17)

S 2 as the cutoff surface
Pure gravity in AdS 3 with a cutoff ρ = ρ 0 in the patch is proposed to be the holographic dual to T T CFT in the cutoff sphere.We computed one point correlator, which is just the Brown-York tensor two point correlators and three point correlators We find the gravity correlators agree with the T T CFT correlators given the dictionary13 c = 3l 2G µ = 16πGl (4.22) which takes the same form as T T in a Euclidean plane.The sphere has its intrinsic scale r, so the second line can also be replaced by which relates T T deformation parameter to the location of the bulk cutoff.

Summary and discussion
In this article we have computed large c correlators of energy-momentum tensor for T T CFT in a Euclidean plane, a sphere and a hyperbolic space using an operator identity version of the trace relation.To examine the cutoff AdS holographic proposal by Mezei et al. [17], we have computed correlators in pure Einstein gravity in Euclidean AdS 3 cut off by the Euclidean plane and the sphere, and found agreement with the T T CFT correlators given the same dictionary for both cases relating gravity parameters G, l to T T CFT parameters c, µ.The cutoff AdS picture was derived from first principle by Guica et al [36] as a pure gravity special case of more general holographic description of AdS 3 with mixed boundary condition, for T T CFT in a Euclidean plane in the large c limit.Our computation suggests a generalization of Guica's derivation to the case of a sphere.It's also natural to consider correlators of other operators or add matter fields to the bulk for the case of a sphere, and we will not be particularly surprised to see the Dirichlet cutoff to yield to a mixed boundary condition for the AdS gravity.
Apart from holography, T T CFT in a sphere and hyperbolic space deserves further study in its own right.T T deformation in a Euclidean plane was shown to be an integrable deformation, but the holographic proposal by Mezei et al. [17], the work on partition function and entanglement entropy in [18] and our computation of two point correlators of energymomentum tensor seems to indicate that large c T T flows to trivial in a sphere.On the other hand, correlators of energy-momentum tensor in T T CFT in the hyperbolic space blow up and run into a square root singularity when µ = 24πr 2 c , that may be an indication of failure of the notion of a local energy-momentum tensor.In general, we expect T T in curved spaces to be qualitatively different from T T in a Euclidean plane in many ways, even though for maximally symmetric spaces the definition of T T is somewhat similar.Further study on correlators and entanglement entropy will shed more light on this issue.
We have restricted our work to maximally symmetric spaces.The symmetry does not only greatly reduces the complexity of the computation, but also provides an unambiguous definition of the T T operator, assuming the existence of a conserved symmetric energy-momentum tensor.Perhaps the most important open question is to generalize T T to generic curved spaces, which has been studied in [15][16] and some good results have been given like a derivation of Guica's mixed boundary condition and the large c sphere partition function.It would be interesting to see how the new formalism works at the level of correlators, of energy-momentum tensor and other operators, in and beyond large c limit.
A Maximally symmetric bi-tensor and CFT correlators of energymomentum tensor in S 2 and H 2 In this appendix we briefly discuss maximally symmetric bi-tensor and derive two point correlators of energy-momentum tensor of CFT in S 2 and H 2 , loosely following the notation in [13].
Roughly speaking, the direction along the geodesic connecting the two points is the only special direction in the (co)tangent spaces of the two points.As a result, it was shown in [39] that the natural basis for maximally symmetric bi-tensors based on two points w and v are the operators of parallel transport along the geodesic I ij ( w, v), the metric at each point g ij ( w), g k l ( v) and the unit tangent vectors to the geodesic at each point where L( w, v) denotes the geodesic length and the differentiations are with respect to the point w and v, respectively. 14As is shown in [40], two point correlator of energy-momentum tensor in a d-dimensional maximally symmetric space is a linear combination of five independent bi-tensor structures with coefficients being functions of the geodesic length This bi-tensor structure is further constrained by conservation of energy-momentum tensor which by identities reduces to three equations and the solution for H 2 is where a 1 and b 5 are two constants.Because the energy-momentum tensor is symmetric and traceless within connected correlators, it's natural to use the complex stereographic projection coordinates for the sphere, in which the metric takes the form and complex Poincare disk coordinates for the hyperbolic space, in which the metric takes the form for H 2 .Plugging these quantities in A.1, we find the two point correlators of energy-momentum tensor of CFT in S 2 take the form To have the correct flat limit, we must have a 1 = c 8π 2 r 4 and b 5 = 0, that is Similarly for H 2 we find

B Geometry of hypersurfaces and Einstein's equation in cutoff
The first fundamental form is given by the induced metric for X, Y ∈ T p Σ, where P µν = g µρ P ρ ν .The Weingarten map is defined as and the second fundamental form, also known as the extrinsic curvature, is given by for X, Y ∈ T Σ, with the assumption that the connection is Levi-Civita, that is metric compatible and torsion free An alternative definition of the extrinsic curvature is given by for X, Y ∈ T Σ.In fact we have To work out the extrinsic curvature in coordinate basis, we have to do projection onto T Σ first since the coordinate basis doesn't all lie in T Σ.We find Now we study the relation between the intrinsic and extrinsic geometry of hypersurfaces.
A covariant derivative of a vector can be decomposed into a sum of the part in T p Σ and the part in For X, Y ∈ T Σ, we define the covariant derivative in the hypersurface as Because the projection operator P commutes with linear combination over C ∞ (M ) and tensor product, ∇ is also a connection.Furthermore, for so ∇ is also Levi-Civita.Needless to say, it coincides with the unique Levi-Civita connection we would have derived from the intrinsic geometry, namely the induced metric.It's natural to define the Riemann curvature tensor in the hypersurface The decomposition of the equation above into T p Σ and N p gives us Gauss and Codazzi equation, respectively.For Or in coordinate basis The Einstein's equation for the AdS d+1 gravity takes the form where l is the AdS radius.We choose a Gaussian normal coordinate patch in which the metric takes the form By definition Using n = ∂ ρ and setting X = ∂ i , Y = ∂ j , we find a simple formula for the extrinsic curvature in this coordinate system By a double contraction the Gauss equation is reduced to15 By a single contraction the Codazzi equation is reduced to To derive the radial equation, we proceed as where

By a single contraction over roman indices of the Gauss equation we have
Using the fact that Rij = R 2 g ij in two dimensional space, we eliminate Rij to get a radial equation more practical for computation We use these three equations (B.25)(B.26)(B.29), the same set of equations used in [33], to compute gravity correlators.However further simplifications are possible.Following the spirit of [41], we can fix the radial dependence of the bulk metric and reduce the Einstein's equation to partial differential equations in the two dimensional transverse space.For three dimensional space, the Einstein's equation (B.21) fixes the metric to be locally AdS We set l = 1 for simplicity here and from now on in the appendix.Using (B.24), the radial equation now reads where " " denotes derivative with respect to ρ.It's straightforward to verify, by changing to Fefferman-Graham coordinates ρ = e −2ρ that the radial equation and the uncontracted Gauss and Codazzi equation become Equation ( 7),( 8) and ( 9) in [41].The radial equation can be integrated to give so these three equations are further reduced to Equation ( 15) in [41] as partial differential equations in the two dimensional transverse space.In the standard context of AdS/CFT, g (0) as the metric on the conformal boundary is given, we solve for g (2) to compute various holographic physics quantities as we study holographic Weyl anomaly, holographic renormalization etc. [42][43].In our context of cutoff AdS/T T CFT, we fix the metric at a finite cutoff surface as a combination of g (0) and g (2) , still three equations for three independent variables.

C Perturbative solutions to Einstein gravity in cutoff AdS 3 and correlators of energy-momentum tensor
When the cutoff surface is the two dimensional Euclidean plane E 2 , it's natural to use Poincare patch for AdS 3 with the cutoff surface at y = y 0 .Consider a variation of the boundary metric where η is the flat metric, which takes the form η ij = δ ij in the Cartesian coordinates and η z z = η zz = 1 2 η zz = η z z = 0 in the complex coordinates.In response to the boundary perturbation, the bulk metric now takes the form where subject to the boundary condition ij (y 0 , x) = 0 . . .
Now we work out g ij (y, x) order by order by solving the Einstein's equation.We will give explicit formula for computation to the second order, while computation to the third order and higher is too complicated to give explicit and complete expression. 16The inverse of the metric is computed to be kl + 2 y 6 η ik η mn η lj g km g nl − 2 y 4 η ik η jl g kl + . . .
To compute three point correlators we need to obtain the bulk metric to the second order.
Plugging the expression of g (1) into the radial equation (C.10) we obtain So g (2) takes the form Plugging this expression into the Codazzi equation (C.12) we get where Therefore we have with the integrability condition Therefore the solution for g (2) can be written as We now use the solution of g (2) and the formula to compute three point correlators, where the extrinsic curvature is given by Computing with assistance of Mathematica, we find the radial equation for g (3)   (y∂ y + 1 2 y∂ y y∂ y )(g kl ) + y 2 L ij ( x) = 0 (C.54) where L ij , as a function of g (1) and g (2) , is traceless.Explicit expressions of L ij and other quantities in the third order perturbation are too long to be written down here.From the radial equation we have where η ij J ij = 0, η ij K ij = 0. Furthermore we find the Codazzi equation to take the form where Π z , Π z , Θ z , Θ z are functions of g (1) and g (2) .So the trace part of g (3) takes the form where X, Y, Z, Ω are functions of g (1) and g (2) .By counting the powers in y, this equation reduces to four equations, three being consistency equations satisfied by the solution of g (1)   and g (2) , and one being the propagation equation

AdS 3 .
In the holographic setup, the large c partition function of the T T CFT living on the cutoff surface as the boundary of the bulk gravity, as a functional of the boundary metric h, is related to the on-shell action of the gravity by log Z[h] = −I on−shell [h] (4.1)

AdS 3 For
self-containedness we offer a basic introduction to the geometry of hypersurface to derive the equations used to compute correlators of energy-momentum tensor in Einstein gravity in cutoff AdS 3 .A hypersurface Σ in a (Pseudo)Riemannian manifold M can be defined as the zero set of a smooth function Σ = {p ∈ M, f (p) = 0}.The canonical normal vector is defined by ζ = (g µν ∂ ν f )∂ µ (B.1)If ζ is a null vector, then it's also a tangent vector of the hypersurface.If ζ is either spacelike or timelike, the tangent space can be decomposed as the direct sum of the tangent space of the hypersurface and the one-dimensional space N spanned by ζ, T p M = T p Σ N p .In this case we can also define the unit normal n = ζ √ |g(ζ,ζ)| which is normalized to g(n, n) = with = 1 for spacelike normal and = −1 for timelike normal.Now we consider the extrinsic geometry of the hypersurface.The operator of projection to T p Σ, denoted simply by P , takes the form in the coordinate basis