Quasi-local energy and compactification

Based on the quasi-local energy definition of Brown and York, we compute the integral of the trace of the extrinsic curvature over a codimension-2 hypersurface. In particular, we study the difference between the uncompactified Minkowski spacetime and the toroidal Kaluza-Klein compactification. For the latter, we find that this quantity interpolates between zero and the value for the uncompactified spacetime, as the size of the compact dimension increases. Thus, the quasi-local energy is able to discriminate between the two spacetimes.


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The aim of the present paper is to begin the exploration of the QLE in toroidal spacetimes and compare it to the corresponding uncompactified spacetime. In this preliminary investigation we are going to discuss very simple examples, for which we believe the discussion of the zero-point energy to be less relevant.
To set up our notation, consider a codimension-p hypersurface Σ embedded in an ambient spacetime M of dimension n and Lorentzian signature, Φ : Σ → M. Let y α and x a be two coordinate systems on M and Σ, respectively, with α = 1, . . . , n and a = 1, . . . , m, where m = n − p is the dimension of Σ. The embedding is defined by the equations Φ : y α = y α (x a ). (1.2) Denoting by g µν (y) the metric in the ambient manifold, the induced metric on the hypersurface is given by The m vectors on the tangent space to the ambient manifold, T (M), tangent to the hypersurface are given by In the useful reference [15] it is proved the fact that if h ≡ det(h ab ) = 0, then there are p (as many as the codimension of the hypersurface) real mutually orthogonal vectors normal to Σ, none of which are null. Let us denote them by n A ∈ T (M), A = 1, . . . , p.
where (A) = ±1. 2 The generalization of the second fundamental form is the set of p symmetric tensors given by where the orthogonality n A · t a = 0 has been used. We are interested in the integral where dS is the surface element of the hypersurface Σ and The paper is organized as follows. In the next section we compute (1.8) in the case of 5-dimensional flat spacetime. In section 3 we repeat the calculation for the compactified spacetime M 4 × S 1 . Then, in section 4 we study the stationary points of the QLE before we end up with some conclusions.

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2 Codimension-2 spheres in M 5 Consider a codimension-2 spacelike hypersurface in 5-dimensional flat space (2.1) Let the hypersurface be a 3-sphere defined by the embedding where latin indices i, j, . . . denote spatial coordinates. The normal vectors are given by In this setup, it is plain that the only normal vector with non-vanishing derivative is the last one It yields We have to project this on the tangent space using the tangent vectors t α a . In spherical coordinates, the hypersurface can be parametrized as follows y 2 = L sin θ 1 sin θ 2 sin θ 3 y 3 = L sin θ 1 sin θ 2 cos θ 3 y 4 = L sin θ 1 cos θ 2 y 5 = L cos θ 1 (2.6) so that the induced metric reads It follows that the tangent vectors t α a take the form and satisfy y α t α a = 0. (2.9) Their norm t 2 = η αβ t α t β is given by (2.10)

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Hence, 3 The integrand of (1.8) is then given by The integration measure in this coordinates takes the form so that the integral finally yields The metric of the ambient space is now where the last coordinate is compact and has periodicity and l is the radius of the compact dimension. We assume the same algebraic surface as in the previous case, that is In this case, the spacelike normal vector depends on the compact coordinate, but its expression is the same as in the previous case Also, the covariant derivative of the normal vector can still be written as The Gauss-Codazzi equations, which relate the ambient Riemann tensor R αβγδ projected on the hypersurface with the Riemann tensor corresponding to the induced metric, provide a useful check of our computations.

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For this computation, we parametrize the hypersurface in cartesian coordinates The induced metric then reads The tangent vectors still obey Explicitly, they read (3.9) The second fundamental form K A ab then takes the form The integrand of (1.8) is again given by One has to be careful with the integration range over the compactified coordinate. For small 3-spheres that completly lie within the compact dimension, that is with L < lπ, the integration is done over the full hypersurface, so that −L ≤ y 5 ≤ L. On the other hand, when L > lπ, there are self intersections of the hypersurface, due to the periodicity of the compact dimension. Thus, the integration range is restricted to −lπ ≤ y 5 ≤ lπ, as can be seen in figure 1. We obtain for L ≤ lπ, Q M 4 ×S 1 = 12π 2 l L 2 − π 2 l 2 + 12πL 2 tan −1 πl √ L 2 − π 2 l 2 for L > lπ.  As expected, in the decompactification limit where l → ∞ for any finite L, the QLE for M 4 × S 1 is that of M 5 . In fact, this happens whenever L ≤ lπ, since the hypersurface does not see the periodicity of the compact dimension; thus, the QLE cannot distinguish between M 4 × S 1 and M 5 . When L > lπ, as can be seen in figure 2, the QLE monotonically decreases to zero as l → 0.

Stationary points of the QLE
Let us study the stationary points of the QLE integral under variations of the spacetime ambient metric, keeping fixed the equation for the embedding

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From the normalization of the normal vectors we have where from now onwards we will omit the label A in the normal vectors. Orthogonality between normal and tangent vectors implies δ n α g αβ t β a = 0, (4.3) and note that δn α = δ g αβ n β = 0. (4.4) Let us define the auxiliary tensor (remember that the tangent vectors we are using are not normalized), in such a way that The determinant of the induced metric also varies because Thus, the variation of the QLE reads It is not possible in general to integrate by parts, because (4.10) It would be interesting to study classes of solutions to those integral-differential equations.
In the particular case where the variation of the metric is assumed to be covariantly constant the equations reduce to the much simpler condition where K = K α n α . For umbilic surfaces where the extrinsic curvature is proportional to the induced metric K ab = λh ab , (4.13)

Conclusions
We have begun to apply some preliminary ideas on quasi-local energy to the simplest instances of toroidal compactifications, and we have found somewhat surprisingly, that this observable can be sensitive to it. We take this fact as an encouragement to pursue this set of ideas, with the final objective in mind of being able to apply energetic arguments to the study of the ground state of fundamental physics including gravity.
In particular, when one dimension is allowed to compactify, we find that there is a runaway behavior of sorts, and the configuration that minimizes the QLE corresponds to this dimension disappearing completely. It is even possible that this behavior is not unrelated to the old problem of stabilization of extra dimensions in a Kaluza-Klein setting (confer [16,17], and references therein).
We have also determined the equations that make stationary the QLE under arbitrary variations of the spacetime metric. They are quite complicated, but seem worthy of further consideration.
We are aware that we are exploring uncharted waters here. Ours are only preliminary ideas. The main difference between Brown and York's definition of quasi-local energy and the more evolved definitions (there are several of them) of Yau and collaborators [13,14], lies in the term that is substracted to the extrinsic curvature. This is crucial in order to get positivity and to define the zero of energy in flat spacetimes. We have evaded the consideration of those thorny issues by working in flat spacetimes, but we will tackle them in future work. Previous definitions, for example the one of Hawking [8], have not been considered since they yield a non-vanishing energy for certain hypersurfaces in flat space [18].
There are several lines of further work that can be pursued. It would be interesting to analyze the cases where not all the compact dimensions are contained in the hypersurface, as well as to study more general compact geometries. This includes, in particular, the energetics of fluxes in non-trivial cycles [19] as compared with the same geometry without the fluxes. Although we considered a simple example, it should be straightforward to generalize it to higher dimensional compact spaces. The holy grail we are looking for is to estimate the definition of gravitational energy by just considering a small portion of the interior of the spacetime, without necessarily worrying about its asymptotic behavior. This will hopefully lead to a quasi-local Wheeler-de Witt equation. Work on those issues is also in progress.