Wang and Yau's Quasi-Local Energy for an Extreme Kerr Spacetime
Warner A. Miller Shannon Ray Mu-Tao Wang Shing-Tung Yau


Nuclear Magnetic Resonance Probe Head Design for Precision Strain Control
    N.J. Curro
    December 30, 2023
=========================================================================




There exist constant radial surfaces, 𝒮, that may not be globally embeddable in ℝ^3 for Kerr spacetimes with a>√(3)M/2.  To compute the Brown and York (B-Y) quasi-local energy (QLE), one must isometrically embed 𝒮 into ℝ^3.  On the other hand, the Wang and Yau (W-Y) QLE embeds 𝒮 into Minkowski space.  In this paper, we examine the W-Y QLE for surfaces that may or may not be globally embeddable in ℝ^3.  We show that their energy functional, E[τ], has a critical point at τ=0 for all constant radial surfaces in t=constant hypersurfaces using Boyer-Lindquist coordinates.  For τ=0, the W-Y QLE reduces to the B-Y QLE.  To examine the W-Y QLE in these cases,  we write the functional explicitly in terms of τ under the assumption that τ is only a function of θ. We then use a Fourier expansion of τ(θ) to explore the values of E[τ(θ)] in the space of coefficients.  From our analysis, we discovered an open region of complex values for E[τ(θ)].   We also study the physical properties of the smallest real value of E[τ(θ)], which lies on the boundary separating real and complex energies. 





§ INTRODUCTION

   
It is not possible to define a local measure of the gravitational energy associated with the curvature of spacetime due to the equivalence principle of general relativity.  However, it is possible to define a quasi-local energy (QLE) density with respect to a field of observers t⃗ and a 2-surface 𝒮 bounding some 3-volume in a spacetime manifold ℳ.  In 1993, Brown and York (B-Y) gave a natural method for devising such an energy using a Hamilton-Jacobi approach <cit.>.  To understand their expression for QLE, we first introduce Fig. <ref>, which includes notations for all submanifolds of ℳ and their respective metrics.  It also includes the notations for the normal and tangent vectors defined in ℳ.

Looking at equation 4.5 of <cit.>, the B-Y QLE is defined as 

    E=-1/8π∫_𝒮_t[N k - N^μ v^ν(K_μν - K g_μν)]√(σ_t) dx^2_physical  space  energy - 
    E^0_reference energy
 
where k is the mean curvature of 𝒮_t embedded in the spacelike hypersurface Σ_t, K is the extrinsic curvature tensor of Σ_t embedded in ℳ, K is the trace of K and E^0 is the reference energy that emerges from the freedom to choose the zero point energy in any Hamilton-Jacobi formulation. The lapse and shift are given by N and N⃗, respectively.  Hawking and Horowitz proposed a similar definition of QLE in 1996 <cit.>.  One choice for E^0 suggested by B-Y involves isometrically embedding 𝒮_t in some flat reference space and computing the corresponding reference energy.  This gives 

    E^0=-1/8π∫_𝒮_t[N k_0 - N^μ v_0^ν((K_0)_μν-K_0 η_μν)]√(σ_t) dx^2
 
where the N and N⃗ are the same as Eq. <ref> and η is the metric of the flat space.  Their reason for choosing the reference space to be flat is one would expect the QLE to be zero for a flat spacetime. 

Given 𝒮_t defined in a maximal hypersurface of a stationary spacetime, B-Y suggested that one uses the Eulerian observers defined by t⃗=u⃗ as their observers and ℝ^3 as their reference space.  Using these suggestions, the B-Y QLE reduces to

    E_BY = 1/8π∫_𝒮_t(k-k_0)√(σ_t) dx^2.
 
The surface isometric embedding theorem (proposed by Weyl and proved independently by Nirenberg <cit.> and Pogorelov <cit.>) states that a closed surface with
a Riemannian metric of positive Gaussian curvature can be uniquely isometrically embedded into ℝ^3.


In 1994, Martinez analyzed Eq. <ref> for Kerr spacetimes using a small angular momentum approximation <cit.>.  With this approximation, Martinez found that the B-Y QLE at the event horizon is given by

    E=2M_ir = √((M+√(M^2-a^2))^2+a^2)
 
where M_ir is the irreducible mass, a is the angular momentum per unit mass and M is the mass of the black hole.  In 1973, Larry Smarr showed that the event horizon of a Kerr black hole with a>√(3)M/2 has a region centered at the poles with negative Gaussian curvature <cit.>.  Since the Gaussian curvature is not positive everywhere, the theorem of Nirenberg and Pogorelov is not applicable. Thus an isometric embedding into ℝ^3 may not exist at all, and an existing isometric embedding may not be unique. This implies that the B-Y QLE energy is not well defined at the event horizon for spacetimes with large angular momentum. See Appendix  <ref> for a discussion on surface isometric embeddings. The existence of negative Gaussian curvature creates a demarcation between constant radial surfaces for which Eq. <ref> is well defined everywhere and those where it is only partially defined.  This demarcation is illustrated in Fig. <ref>.   

One can explicitly write r^* for a constant radial surface by finding the root of its Gaussian curvature at the poles.  We begin the derivation of r^* by first introducing the metric of the constant radial surface in Kerr.

The line element of Kerr in Boyer-Lindquist coordinates is given by

    dl^2_ℳ = g_tt dt^2 + 2 g_tϕ dt dϕ + g_rr dr^2 + g_θθ dθ^2 + g_ϕϕ dϕ^2

where

    g_tt    =     -( 1- 2M r/Ξ),
    
    g_tϕ    =     - 2 M r/Ξ  a sin^2θ,
    
    g_rr    =    Ξ/Δ,
    
    g_θθ    =    Ξ  and
    
    g_ϕϕ    =    ( r^2 + a^2 (1+ 2  M r sin^2θ/Ξ) ) sin^2θ

are the non-zero components of the Kerr metric.  The definitions of Ξ and Δ are

    Ξ    :=      r^2+ a^2 cos^2θ  and
    Δ    :=     r^2-2 M r+a^2.

The 2-surface 𝒮_t for which quasi-local energy is computed is defined in a t=constant hypersurface Σ with constant radius R.  The choice of t is inconsequential since the spacetime is stationary.  For this reason, we drop the subindex t from subsequent notation.  Inserting dt=dr=0 and r=R in Eq. <ref> gives the line element of 𝒮 as
 
    dl_𝒮^2 = (R^2+a^2 cos^2θ)_σ_θθ   dθ^2 +  
                 ( R^2+a^2+ 2 R a^2 M sin^2θ/Ξ) sin^2θ_σ_ϕϕ  dϕ^2

where σ_θθ and σ_ϕϕ are the non-zero components of the induced metric σ on 𝒮.  The Gaussian curvature of 𝒮 is given by

    𝒦 = σ_θθσ^2_ϕϕ,θ+σ_ϕϕ(σ_θθ,θσ_ϕϕ,θ-2σ_θθσ_ϕϕ,θθ)/4σ^2_θθσ^2_ϕϕ.

Solving for the root of Eq. <ref> at θ=0 gives

    r^*(a,M)=-3^1/3a^2+Γ^2/3/3^2/3Γ^1/3

where

    Γ=27 M a^2+a√(3)√(243 M^2 + a^4).

This is the only non-zero real root of the Gaussian curvature at the poles.  Recently, Yu and Liu studied QLE for r^*<R<√(3)a with unrestricted angular momentum <cit.>.  Their analysis remains in the regime of strictly positive Gaussian curvature.  

In this paper, we study the Wang and Yau (W-Y) QLE for constant radial surfaces in Kerr with r_+<R<r^*.   In the W-Y approach, one embeds 𝒮 into Minkowski space ℝ^3,1 instead of ℝ^3.  Because 𝒮 is a co-dimension 2 surface with respect to ℝ^3,1, the isometric embedding equations are underdetermined thus giving infinitely many embeddings. To solve this problem, W-Y introduced the scalar field τ on 𝒮, which determines a unique embedding into ℝ^3,1 given a choice of τ.  Choosing τ also chooses a unique field of observers on 𝒮.  Using the W-Y approach, Eq. <ref> is redefined as

    E[τ]=1/8π∫(-k̅√(1+|∇τ|^2) + ⟨∇τ | ∇v⃗̅⃗ | u⃗̅⃗⟩)√(σ) dx^2_physical space energy - 1/8π∫k̂√(σ̂) dx^2_reference energy

where k̅ is the mean curvature of 𝒮 embedded in ℳ with respect to the spacelike normal v⃗̅⃗ and k̂ is the mean curvature of the convex shadow 𝒮̂ embedded in ℝ^3.  The 2-metric of the convex shadow is written as σ̂.  All necessary information for this paper regarding the normal basis {u⃗̅⃗, v⃗̅⃗} and the convex shadow is contained in appendix <ref>; they are also defined in <cit.>.  The purpose of the appendix is to give the reader a self contained explanation for the physical motivations behind the W-Y formalism.  The W-Y QLE is defined as the minimum of Eq. <ref> with respect τ, which is equivalent to minimizing with respect to all possible observer fields.  

We are unaware of any research that explores QLE near the event horizon for extreme Kerr spacetimes using a Hamilton-Jacobi approach.  Given the generalization of the B-Y QLE by W-Y, we believe their definition is a good starting point to explore this area of research.  It can be shown for a Kerr spacetime that a critical point of the Eq. <ref> is found at τ=0 regardless of the value of R. Given τ=0, the W-Y QLE functional reduces to Eq. <ref>.  To gleam some insight on the behavior of E[τ] in this region, we explore the W-Y QLE using numerical techniques.  We restrict τ to only a function of θ to simplify the W-Y QLE functional and make the calculation more tractable.  Given this restriction on τ, the main results of this analysis are the following: (1) there exists a boundary separating admissible real energies from inadmissible complex energies and the minimum real value, E_min, of E[τ(θ)] lies on this boundary, (2) τ=0, which is a critical point of the W-Y QLE functional for constant radial surfaces with R<r^*, is not admissible within their formalism and (3) the physical behavior of E_min disagrees with the behavior one would expect from the analysis of Martinez.  

We structure the paper in the following way. In Sec. <ref> we write the W-Y QLE functional in terms of τ.  In Sec. <ref> we show that τ=0 is a critical point of the W-Y QLE functional for Kerr regardless of the the value of R.  In Sec. <ref> we present our numerical analysis.  Finally, in Sec. <ref> we have further discussions and conclusions.



§ EXPRESSING THE W-Y QLE IN TERMS OF Τ


The purpose of this section is to write Eq. <ref> explicitly in terms of τ for constant radial surfaces.  This will be used in Sec. <ref> for our numerical analysis. To this end, we separate this section into two subsections.  The first derives the physical energy in terms of τ, while the second derives the reference energy in terms of τ.  Before we continue with our derivations,  we must define the mean curvature vector H⃗.  

Let X⃗(η^a) represent the spacetime coordinates of 𝒮 embedded in ℳ where η^a={θ,ϕ} are the Boyer-Lindquist coordinates of 𝒮.  At each point p∈𝒮 there also exists a spacelike tangent plane 𝒯_s(p) that is spanned by an orthogonal basis made of spacelike tangent vectors ζ⃗_a=∂X⃗/∂η^a.  Given an arbitrary normal basis {u⃗,v⃗} on 𝒮, the mean curvature vector can be written as

    H⃗= H_u⃗ u⃗ + H_v⃗ v⃗

where H_u⃗ is the fractional rate of expansion of 𝒮 along the timelike normal u⃗ and is given by

    H_u⃗ = σ^ab⟨ζ⃗_a | ∇u⃗ | ζ⃗_b⟩,

and H_v⃗ is the fractional rate of expansion of 𝒮 along the spacelike normal v⃗ and is given by

    H_v⃗ = σ^ab⟨ζ⃗_a | ∇v⃗ | ζ⃗_b⟩.

The covariant derivative is taken with respect to the Kerr metric g for both H_u⃗ and H_v⃗.  The mean curvature vector is the direction of maximal expansion of 𝒮 in ℳ and is independent of the normal basis in which it is computed. 



 §.§ The physical contribution to the W-Y QLE in terms of τ


In this subsection we follow the prescription given in <cit.> to compute the physical portion of QLE.  This is done in three steps:


  * Compute the normal basis {u⃗'⃗,v⃗'⃗} of 𝒮 that satisfies

    v⃗'⃗ = H⃗/|H⃗|.


  * Transform {u⃗'⃗,v⃗'⃗} to {u⃗̅⃗,v⃗̅⃗} using

    u⃗̅⃗   =   u⃗'⃗coshα+v⃗'⃗sinhα
    v⃗̅⃗   =   u⃗'⃗sinhα+v⃗'⃗coshα

where α is the hyperbolic angle that minimizes the physical energy in Eq. <ref> and is given by

    sinhα = -Δτ/|H⃗|√(1+|∇τ|^2).


  * Use {u⃗̅⃗,v⃗̅⃗} to express the physical energy in terms of τ.

We will refer to {u⃗'⃗,v⃗'⃗} and {u⃗̅⃗,v⃗̅⃗} as the non-preferred and preferred normals, respectively. 

For step 1, we begin with the non-preferred normal basis

    v⃗'⃗={0,1/√(g_rr),0,0}

and

    u⃗'⃗ = β{-g_ϕϕ/g_tϕ,0,0,1}

where

    β=1/√(g_ϕϕ(1-g_ϕϕg_tt/g^2_tϕ)).

Here, u⃗'⃗ is the timelike normal of Σ restricted to 𝒮.  Since Σ is a maximal hypersurface of Kerr and 𝒮⊂Σ, writing the mean curvature vector in terms of {u⃗'⃗,v⃗'⃗} gives

    H⃗ = 0H_u⃗'⃗   u⃗'⃗ + H_v⃗'⃗ v⃗'⃗,

which satisfies Eq. <ref> and completes step 1.  For step 2 we use cosh^2α-sinh^2α=1 to write

    coshα = √(1+(Δτ)^2/|H⃗|^2 (1+|∇τ|^2)).

Inserting sinhα and coshα into Eqs. <ref> and <ref> to transform from {u⃗'⃗,v⃗'⃗} to {u⃗̅⃗,v⃗̅⃗} completes step 2.

To complete step 3, we begin by inserting v⃗̅⃗ into  Eq. <ref> to compute k̅, which gives

    k̅=-√(g_rr)coshα̅(Γ^r_θθ/σ_θθ + Γ^r_ϕϕ/σ_ϕϕ).
 
Next we insert k̅ into the first term of the physical space energy giving

    k̅√(1+|∇τ|^2) = -√(g_rr(Γ^r_θθ/σ_θθ + Γ^r_ϕϕ/σ_ϕϕ)^2(1+|∇τ|^2)+Δτ^2)

where 

    |∇τ|^2=τ^2_,θ/σ_θθ

and

    Δτ   =   σ^ab∇_a ∇_b τ
       =   1/σ_θθ(τ_,θθ - Γ^θ_θθτ_,θ) - Γ^θ_ϕϕτ_,θ/σ_ϕϕ.

For the second term of the physical space energy, we map ∇τ from 𝒮 to ℳ using

    ∇τ = σ^abτ_,aζ⃗_b.

Inserting u⃗̅⃗ and v⃗̅⃗ into the second term of the physical space energy gives

    ⟨∇τ | ∇v⃗̅⃗ | u⃗̅⃗⟩ = τ_,θ/σ_θθ((v̅_t,θ-Γ^t_θ tv̅_t)u̅^t + (v̅_r,θ-Γ^r_θ rv̅_r)u̅^r-Γ^t_θϕv̅_tu̅^ϕ).

Combining Eq. <ref> and <ref>  and integrating over 𝒮 gives the physical contribution to the W-Y QLE in terms of derivatives of τ and completes step 3.




 §.§ The reference contribution to the W-Y QLE in terms of τ


In section 3 of <cit.>, it was shown that 

    E^0=-∫_𝒮[N k_0 - N^μ v_0^ν((K_0)_μν-K_0 η_μν)]√(σ) dx^2 = ∫k̂√(σ̂) dx^2

where k̂ is the mean curvature of the convex shadow embedded in ℝ^3.  Therefore, one only needs to isometrically imbed 𝒮̂ in ℝ^3 and integrate the mean curvature to find the reference energy.  Assuming τ is only a function of θ,  the metric components of 𝒮̂ are given by 

    σ̂_θθ   =   σ_θθ + τ^2_,θ
    σ̂_ϕϕ   =   σ_ϕϕ.

Let the Cartesian coordinates of 𝒮̂ be defined as

    x(θ, ϕ)   =   ρ(θ)cosϕ
    
    y(θ, ϕ)   =   ρ(θ)sinϕ
    
    z(θ)   =   f(θ)

where ρ( θ) and f( θ) are smooth real valued functions on the domain θ∈ [0,π].  Equating the line element on 𝒮̂ with that of Euclidean space, we get

    ρ(θ)   =   √(σ_ϕϕ)
    
    f_,θ(θ)   =   √(σ_θθ - σ^2_ϕϕ,θ/4 σ_ϕϕ+τ^2_,θ).

Now we can write the mean curvature in terms of the derivatives of ρ and f with respect to θ.  The principle curvatures in the θ and ϕ directions are

    k̂_θθ   =   f_,θρ_,θθ-ρ_,θf_,θθ/(f_,θ^2+ρ_,θ^2)^3/2 and
    k̂_ϕϕ   =    -f_,θ/ρ√(f_,θ^2+ρ_,θ^2),

respectively.  The mean curvature is the sum of the principle curvatures and is given by

    k̂=-(f_,θ^3+ ρ ρ_,θ   f_,θθ +  f_,θ(ρ_,θ^2-  ρ ρ_,θθ)/(f_,θ^2+ρ_,θ^2 )^3/2)  1/ρ.

We integrate Eq. <ref> over 𝒮̂ to get the contribution to QLE from the reference action.  With Eqs <ref>, <ref> and <ref>, the W-Y QLE functional is completely determined by τ_,θ, τ_,θθ and τ_,θθθ.




§ THE CRITICAL POINT OF THE W-Y QLE FUNCTIONAL FOR CONSTANT RADIAL SURFACES


In section 6 of <cit.>,  W-Y derived the Euler-Lagrange equation of E[τ], which is given by

    -(k̂σ̂^ab - σ̂^acσ̂^bdk̂_cd)∇_b ∇_a τ/√(1+|∇τ|^2) +_(1)
    σ^ab∇_a (∇_b τ/√(1+|∇τ|^2)coshα|H⃗|)_(2)-Δα_(3)-σ^ab∇_a ⟨ζ⃗_b | ∇v⃗̅⃗ | u⃗̅⃗⟩_(4)=0.

All covariant derivatives are taken with respect to the 2-metric on 𝒮 except for the covariant derivative on v⃗̅⃗, which is taken with respect to the spacetime metric g. To show that τ=0 is a solution to Eq. <ref>, we write each term explicitly in terms of τ.

The first term of Eq. <ref> written explicitly in terms of τ is given by

    (1)=-1/√(1+τ^2_,θ/σ_θθ)((τ_,θθ-Γ^θ_θθτ_,θ)(k̂-σ̂^θθk̂_θθ)/σ_θθ+τ^2_,θ-Γ^θ_ϕϕτ_,θ/σ_ϕϕ(k̂-σ̂^ϕϕk̂_ϕϕ)).

 The second term is
 
    (2)=∂_θ(|H⃗|coshα/√(1+τ^2_,θ/σ_θθ))τ_,θ/σ_θθ + |H⃗|coshα/√(1+τ^2_,θ/σ_θθ)Δτ

 where coshα and Δτ are given by Eqs. <ref> and <ref>, respectively.
Term 3 is simply

    (3)=1/σ_θθ(α_,θθ - Γ^θ_θθα_,θ) - Γ^θ_ϕϕα_,θ/σ_ϕϕ.

Let 

    V_a=⟨ζ⃗_a | ∇v⃗̅⃗ | u⃗̅⃗⟩,

the last term in Eq. <ref> is given by

    (4)=σ^ab∇_a V_b=1/σ_θθ(V_θ,θ-Γ^θ_θθV_θ)+1/σ_ϕϕ(V_ϕ,ϕ-Γ^θ_ϕϕV_θ).

It is easy to see that the first three terms vanish for τ=0.  Next we show that the fourth term also vanishes for τ=0.  

Writing V_θ and V_ϕ, one gets

    V_θ   =   ∂_θv̅^νu̅_ν + Γ_θα^νv̅^αu̅_ν and
    
    V_ϕ   =   Γ_ϕ r^t v̅^r u̅_t.

From Eqs. <ref> and <ref>, it is clear that {u⃗̅⃗,v⃗̅⃗}={u⃗'⃗,v⃗'⃗} for τ=0. It is also clear that the first term of Eq. <ref> is equal to zero since v⃗'⃗ only has a radial component and the contravariant components of u⃗'⃗ are only non-zero for time.  Furthermore, the second term of Eq. <ref> reduces to Γ_θ r^t v'^r u'_t where Γ_θ r^t=0. This gives V_θ=0.  Inserting V_θ=0 into Eq. <ref> gives (4)=V_ϕ,ϕ/σ_ϕϕ.  Since V_ϕ is independent of ϕ, term 4 vanishes.  This shows explicitly that τ=0 is a critical point regardless of one's choice of R.   Indeed, it was shown in <cit.> that for any axi-symmetric surface, the fourth term of Eq. <ref> always vanishes and τ=0 is always a solution. However, τ=0 is not necessarily a local or global minimum, see <cit.> for a criterion for local minimum of a critical point in terms of a mean curvature inequality. 




§ NUMERICAL RESULTS


In this section, we apply the direct search algorithm developed by Torczon <cit.> to minimize E[τ(θ)], which is given by Eqs. <ref>, <ref> and <ref>, in the space of coefficients.  Without loss of generality, we will use a=M=1 for our numerical analysis unless stated otherwise.  The value of r^* is approximately 1.65 for this choice of a and M.  

To apply the direct search algorithm, we use a Fourier expansion to express τ_,θ as

    τ_,θ( θ) = F_0( θ) + ∑^κ_n=1a_nsinnθ

where θ is the polar angle in Boyer-Lindquist coordinates, F_0(θ) is an initial guess of the optimal τ_,θ and a_n are the Fourier coefficients.  Symmetry about the equator excludes all but the odd values of the Fourier coefficients of sin(nθ).  The expansion lacks cosine modes due to boundary conditions on the derivative of τ at the poles.  We choose our initial guess to be

    F_0(θ)=√(σ_ϕϕ/sin^2θ - σ_θθ) .

This function gives an integrand of E[τ(θ)] that is well behaved at the poles.  It also gives an initial guess reasonably close to a solution of the Euler-Lagrange equation for all radii.  The image of the convex shadow and its mean curvature at R=3/2  are shown in Fig. <ref>.

There is nothing notable about R=3/2;  we simply use this for as an illustrative example for surfaces with R < r^*.  We will continue to use this radius for further examples.  All statements made for R=3/2 apply equally to all radii below r^* unless specified otherwise.

The complexity of the space of coefficients increases with the dimension.  As one increases the number of coefficients used to minimize E[τ(θ)], the likelihood of getting caught in local minima increases.  To mitigate this difficulty, we begin with just one Fourier coefficient set to zero.  We then apply the direct search algorithm to find the smallest real value of E[τ(θ)] in the space of a_1.  Once a_1 is obtained, we add a_3=0 and search in the space of a_1 and a_3.  Here we allow both a_1 and a_3 to change until we find the minimum in two dimensions.  We iteratively increase the number of coefficients until the change in E_min is at least less than 10^-2 for each additional coefficient added.  The number of coefficients needed increases as one approaches r_+ due to increasing curvature gradients of 𝒮.  Our direct search algorithm was coded using Mathematica.  All integrals were done using the NIntegrate function.   




 §.§ The boundary separating admissible and non-admissible values of the W-Y QLE functional


There are three criteria within the W-Y QLE formalism that determine whether a choice of τ is admissible.  These criteria can be found in section 4 of <cit.> as well as section 5 of <cit.>.  The purpose of the second and third criteria is to ensure that the value of E[τ(θ)] is positive.  We will not focus on these since we do not obtain negative energies for any of our results.  Instead, we will focus on the first criteria, which is

    𝒦 - (1+|∇τ |^2)^-1 det( ∇_a∇_b τ) > 0
  
where 𝒦 is the Gaussian curvature of 𝒮 and all covariant derivatives are taken with respect to σ.  This requires that the Gaussian curvature of the convex shadow given a choice of τ is strictly positive everywhere. If this criteria is met, W-Y can guarantee the existence and uniqueness of E[τ(θ)].  Unfortunately, our analysis indicates that this criteria can not be met at τ=0. 

While minimizing in the space of coefficients, we discovered a boundary separating τ's with real values of E[τ(θ)] from those with complex values.  This can be seen in Fig. <ref> where we use two Fourier coefficients, a_1 and a_3, to visualize the QLE landscape.

The white gap in the middle of the plot represents complex values of E[τ(θ)] whose existence can be understood by examining Eq. <ref>.  Here, complex energies arise for choices of τ_,θ that satisfy

    τ^2_,θ < σ^2_ϕϕ,θ/4σ_ϕϕ - σ_θθ.
    
We will show that these choices of τ are inadmissible using our numerical results.

To demonstrate that choices of τ with complex energies are not admissible, we compare the Gaussian curvature of 𝒮 and 𝒮̂ for the initial guess and τ_min in Fig. <ref>.

From Fig. <ref>a, we see that the Gaussian curvature of the convex shadow for the initial guess is strictly positive and significantly different than the curvature of 𝒮.  On the other hand, Fig. <ref>b shows that the Gaussian curvature of the convex shadow at τ_min is similar to the curvature of 𝒮 within the interval of positive Gaussian curvature.  Outside of this interval, the Gaussian curvature of 𝒮 becomes negative while the shadow's curvature is flat.  This indicates that the optimization algorithm tends toward a τ that embeds 𝒮 into ℝ^3 as much as possible.  In fact, if we allow the algorithm to cross the boundary of admissible solutions by taking the real part of the QLE functional, it converges to τ=0.  This implies that choices of τ within the boundary do not have shadows with strictly positive Gaussian curvature.  We believe this is due to the unnecessary restriction that τ is a function of θ only. In general, we should allow τ to be dependent of both θ and ϕ and solve both the isometric embedding equation and the Euler-Lagrange equation. 





 §.§ The physical relevance of E_min


In this section we analyze the physical behavior of E_min and compare it to what one would expect based on the results of Martinez.  We begin by plotting E_min as a function of R in Fig. <ref>.  

The vertical dividing line is placed at the critical radius r^*≈1.65.  Above r^*, E_min is equivalent to the B-Y QLE.  Below the plot in Fig. <ref> are the convex shadows at τ_min associated with radii R={1.05, 1.1, 1.2, 1.4, 1.6}.  The mean curvature of these shadows can be see in Fig. <ref>.

These plots show a Gibbs' phenomena that occurs when the Gaussian curvature of 𝒮 becomes negative, which, as we showed in Fig. <ref>, is also when the Gaussian curvature of 𝒮̂ at τ_min becomes zero.  To physically interpret the results in Fig. <ref>,  we analyze E_min at the outer event horizon for black holes with increasing angular momentum.  This will give us a reference on how E_min should behave at the event horizon once the angular momentum exceeds a=√(3)M/2.  We will also interpret the results by looking at the field of observers associated with E_min and compare them to the Eulerian observers chosen by B-Y.

In Fig. <ref>a, we plot E_min, which is equivalent to the B-Y QLE for a<√(3)M/2, at various values of a between (r_+,0) and (r_+,√(3)M/2). 

We see that these two plots agree for a ≤ 0.4.  As the angular momentum grows, the low angular momentum approximation starts to deviate from the B-Y QLE.  The most important feature of this plot is the fact that the B-Y QLE decreases as angular momentum increases.  Looking at Fig. <ref>b, we plot E_min at r_+ + ϵ, where ϵ=10^-5, for values of a between points (r_+,√(3)M/2) and (r_+,1).  Here we see that the E_min predicts a growing energy with increased angular momentum.  So why is E_min significantly greater than the predicted 2M_ir and why does it disagree with the trend of decreasing QLE with increased angular momentum?  The reason is due to the field of observers that are chosen at τ_min.

Assuming the isometric embedding of 𝒮 is a surface of revolution, the interval for which 𝒮 is embeddable in ℝ^3 is determined by Eq. <ref> with τ=0.  For surfaces in the regime of strictly positive Gaussian curvature, σ_θθ is strictly greater than or equal to σ^2_ϕϕ,θ/4 σ_ϕϕ.  Surfaces with regions of negative Gaussian curvature can only be partially embedded in ℝ^3 between θ^** <θ <π-θ^**. Here, θ^** is the smaller root of Eq. <ref> with τ=0.  We will refer to this interval as the “interval of embeddability".  In Fig. <ref>, we plot the inner product between the field of observers given by W-Y at τ_min with the Eulerian observers that would be chosen by B-Y as a function of θ. 

We see that the observers at τ_min and the Eulerian observers agree within some ϵ around 0.7<θ<π-0.7.  For this choice of angular momentum and radius, θ^** is approximately equal to 0.64.  This shows that E_min chooses the Eulerian observers within the interval of embeddability and smoothly transitions to observers that are boosted with respect to the Eulerian observers outside of this interval.  

As one approaches (r_+,M), the interval of embeddability decreases.  This implies that more observers chosen by the W-Y QLE procedure at τ_min are boosted with respect to the Eulerian observers. We also found that the magnitude of the boosts increases as one approaches (r_+,M).  This is why E_min has a growing energy with increased angular momentum.  It also explains why the E_min at the event horizon is significantly greater than twice the irreducible mass when a>√(3)M/2. 



§ CONCLUSION AND FURTHER DISCUSSION


In this paper, we analyzed the W-Y QLE functional with the restriction that τ is only a function of θ for constant radial surfaces with R<r^*.  These surfaces may not be embeddable in ℝ^3, but they are embeddable in ℝ^3,1.   We discovered an open region of complex values for E[τ(θ)] while minimizing the functional in the space of coefficients.  Our results suggest that the smallest real value of E[τ(θ)] lies on the boundary separating real and imaginary energies. Our results also suggest that there does not exist a convex shadow whose Gaussian curvature is strictly non-negative for choices of τ within the region of complex energies.   

We also analyzed the behavior of E_min to gleam some insight on its possible physical relevance.  In Fig. <ref>, we saw a sudden increase in E_min for surfaces with R<r^*≈1.65.  It is uncertain if these energies are physically meaningful since no results exists for such surfaces.  To gain some clarity, we examined E_min at the event horizon as a function of angular momentum.  For a<√(3)M/2, the results of Martinez suggest that the QLE is comparable to the irreducible mass of the black hole, which decreases with increasing angular momentum.  Above √(3)M/2, E_min increases with increasing angular momentum.  We attributed this change in behavior to the difference between the Eulerian observers chosen by B-Y  and the field of observers chosen by W-Y at τ_min.  In Fig. <ref>, we showed that the W-Y observers at τ_min agree with the Eulerian observers within the interval of embeddability and transition to boosted observers outside of this interval.  Our results are contingent on τ being a function of θ alone.  For a true understanding of the W-Y QLE applied to extreme Kerr spacetimes near the event horizon, one must allow τ to be a function of both θ and ϕ.  This is an interesting avenue for future research.
 



§ ACKNOWLEDGEMENTS

We wish to thank Po-Ning Chen, Rory Conboye, Matthew Corne, and Ye-Kai Wang for stimulating discussions. We also thank the the Information Directorate of the Air Force Research
  Laboratory and the Griffiss Institute for providing us with an
  excellent environment for research. This work was supported in part
  through the VFRP and SFFP program, as well as AFRL grant
  #FA8750-15-2-0047. 
 



 §.§ Physical motivations behind the W-Y QLE formalism


Let ℳ be an arbitrary spacetime manifold.  The B-Y QLE energy given by Eq. <ref> does not give a general description on how to choose the field of observers in ℳ, nor does it give the reference space.  For stationary spacetimes, B-Y suggested the Eulerian observers associated with maximal hypersurfaces as their observers and ℝ^3 as their reference space.  This choice is reasonable when the extrinsic curvature of 𝒮 along u⃗ vanishes and the embedding of 𝒮 exists.  However, it was shown that these choices do not work for surfaces in general and can give non-zero values of QLE for flat spacetimes <cit.>.  This is due to the second term in Eq. <ref>.  If the extrinsic curvature of 𝒮_t along u⃗ does not vanish, there is no way to account for this curvature in ℝ^3 when computing the reference energy.  To address this problem, W-Y used the flat spacetime as their reference space.  This extends the application of the B-Y QLE to dynamical spacetimes.

The extension of the reference space from ℝ^3 to Minkowski space ℝ^3,1 creates a new challenge. Since 𝒮_t is a co-dimension two surface with respect to ℝ^3,1, the isometric embedding equations are underdetermined. To address this problem, W-Y introduced the scalar field τ on 𝒮_t which allows them to define a unique embedding into Minkowski space up to a choice of τ.  They then construct a procedure to associate each choice of τ with two observer fields which are used to compute QLE.  One field exists in the physical space and is denoted as t⃗=Nu⃗̅⃗+N⃗ while the other exists in Minkowski space and is denoted as t⃗_0=Nu⃗_0+N⃗. These observers are chosen such that the extrinsic curvature of 𝒮_t along u⃗̅⃗ embedded in ℳ is equal to the extrinsic curvature of 𝒮_t along u⃗_0 embedded in ℝ^3,1.  The notation u⃗̅⃗ is used to distinguish the unique timelike normal on 𝒮_t whose extrinsic curvature agrees with u⃗_0 as opposed to an arbitrary timelike normal u⃗.  This matching of extrinsic curvature along timelike normals is given by the constraint

    ⟨u⃗̅⃗,H⃗⟩ = ⟨u⃗_0,H⃗_0 ⟩

where H⃗_0 is the mean curvature vector of 𝒮_t embedded in ℝ^3,1.
This addresses the problem of the second term in Eq. <ref>.  Next we discuss the isometric embedding into Minkowski space and how the lapse and shift are chosen.  

Let i:𝒮_t ↪ℝ^3,1 represent an isometric embedding of 𝒮_t into Minkowski space.  In principle, one would compute the reference energy using a field of observers who are at rest with respect to i(𝒮_t).  If we work in the rest frame of these observers, at each point p ∈ i(𝒮_t) we have t⃗_0 = {1,0,0,0}.  Let τ be the time component of i(𝒮_t),  then the embedding takes the form x⃗_0={τ,x^1,x^2,x^3}.  One can alternatively write the embedding as

    x⃗_0 = x⃗̂⃗ + τt⃗_0
 
where x⃗̂⃗={0,x^1,x^2,x^3} are the spatial coordinates of i (𝒮_t) that lie in a three dimensional Euclidean plane orthogonal to t⃗_0.  This projection 𝒮̂ onto ℝ^3 is defined as the shadow of i (𝒮_t) with respect to t⃗_0.  Vectors with hats exist on the shadow, while vectors with the zero subscript exist on i (𝒮_t).  

Starting from Eq. <ref>, the metric of the shadow σ̂ is given by

    ⟨x⃗_,a | x⃗_,b⟩   =   ⟨x⃗̂⃗_,a+τ_,at⃗_0 | x⃗̂⃗_,b+τ_,bt⃗_0 ⟩
       =   ⟨x⃗̂⃗_,a | x⃗̂⃗_,b⟩ -τ_,aτ_,b,

which implies

    σ̂_ab = (σ_t)_ab + τ_,aτ_,b.

The isometric embedding of 𝒮_t into ℝ^3,1 using 𝒮̂ and τ is shown in Fig. <ref>.  

A necessary condition for choosing τ requires the shadow 𝒮̂ to be a smooth convex surface in ℝ^3.  This condition is used to prove the existence and uniqueness of i(𝒮_t) given the observer field t⃗_0.  It can be seen from the embedding theorem of Nirenberg and Pogorelov and Eq. <ref> that any isometric embeddings of 𝒮_t in Minkowski space with the same convex shadow and scalar field τ must be congruent.  This completes the discussion on embedding 𝒮_t into ℝ^3,1. 

Since the field of observers in Minkowski is defined as being at rest with respect to i(𝒮_t), the lapse and shift are chosen such that

    t⃗_0=Nu⃗_0 + N⃗.

Using the embedding as described in Fig. <ref>, it can be shown that

    t⃗_0={1,0,0,0}=√(1+|∇τ|^2) u⃗_0 - ∇τ

where

    ∇τ = σ_t^abτ_,ax⃗_0,b.

Here we see that N=√(1+|∇τ|^2) and N⃗= -∇τ.  The corresponding field of observers in ℳ is

    t⃗=√(1+|∇τ|^2) u⃗̅⃗ - ∇τ

where the coordinates x⃗_0 in Eq. <ref> are replaced with the coordinates of 𝒮_t in ℳ.  With the observer fields and the isometric embedding into ℝ^3,1 written in terms of τ, the discussion on the physical motivations behind the W-Y formalism is complete. 



 §.§ A discussion on the isometric embedding theorem



There are several common misconceptions about isometric embedding of a closed surface into ℝ^3.  We take this opportunity to address these issues. 




1. Isometric embeddings do not preserve symmetry: 
One reason why the current formalism does not work is because of the assumption that τ is a function of θ only, or τ is axi-symmetric. The Killing field of a Riemannian metric does not extend to the embedding, or does not extend to be a Killing field of the ambient space. In particular, it is possible that an axi-symmetric metric admits an isometric embedding into ℝ^3 that is not a surface of revolution. 




2. Non-embeddability: 
The surface isometric embedding theorem guarantees the existence and uniqueness of a global isometric embedding if the Gaussian curvature is positive everywhere. However, there does not seem to be any non-trivial non-embeddability theorem. In particular, for a surface with a metric that has negative Gauss curvature at some point, isometric embedding into ℝ^3 is still possible. There are many closed surfaces in ℝ^3 with negative Gauss curvature somewhere, but these isometric embeddings are not expected to be unique. 




3. Global isometric embedding vs. local isometric embedding: 
The theorem of Frolov on the non-embeddability near a point of negative Gauss curvature <cit.> seems to contradict a well-known local isometric embedding theorem <cit.> that states if a surface has negative Gauss curvature at a point, then there exists a neighborhood near the point that can be isometrically embedded into ℝ^3. This is the local isometric embedding theorem which holds as long as the Gauss curvature is positive, negative, or changes sign cleanly. This violation implies that Frolov's theorem does not necessarily eliminate the existence of embeddings into ℝ^3 for these surfaces.  In particular, one can not rule out embeddings that are not surfaces of revolution.   











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