We observe that an analogue of the Positive Mass Theorem in the
time-symmetric case for three-space-time-dimensional general relativity follows
trivially from the Gauss-Bonnet theorem. In this case we also have that the
spatial slice is diffeomorphic to $\Real^2$.Comment: 3 pages, more or less trivial. Posting it because I cannot find it in
  the literature: comments and references to where this may have appeared
  before are welcome! Version 2: fixed some typos, added some remarks based on
  comments receive

Wong, Willie Wai-Yeung

English

arXiv

We observe that an analogue of the Positive Mass Theorem in the time-symmetric case for three-space-time-dimensional general relativity follows trivially from the Gauss-Bonnet theorem. In this case we also have that the spatial slice is diffeomorphic to $\Real^2$.PD

Infoscience - École polytechnique fédérale de Lausanne

A POSITIVE MASS THEOREM FOR TWO SPATIALDIMENSIONSWILLIE WAI-YEUNG WONGAbstract. We observe that an analogue of the Positive Mass Theorem in thetime-symmetric case for three-space-time-dimensional general relativity followstrivially from the Gauss-Bonnet theorem. In this case we also have that thespatial slice is diffeomorphic to R2.In this short note we consider Einstein’s equation without cosmological constant,that isRµν − 12Rgµν = Tµν ,on 1 + 2 dimensional space-times. This theory has long been considered as a toymodel with possible applications to cosmic strings and domain walls, or to quantumgravity. For a survey please refer to [Bro88, Car98] and references within. Thislow dimensional theory is generally considered as un-interesting [Col77] due to thefact that Weyl curvature vanishes identically in (1 + 2) dimensions, a fact ofteninterpreted in the physics literature as the theory lacking gravitational degrees offreedom. Furthermore, the theory does not reduce in a Newtonian limit [BBL86]:the exterior space-time to compact sources is necessarily locally flat and is typicallyasymptotically conical [Car98, DJtH84, Des85].For these space-times, by considering point-sources, it is revealed [DJtH84] thatthe mass should be identified with the angle defects near spatial infinity. For staticspace-times with spatial sections diffeomorphic to R3, it is also known that theasymptotic mass can be related to the integral of scalar curvature on the spatialslice, and hence under a dominant energy assumption must be positive.The purpose of this note is to remark that the topological assumption is unnec-essary.Throughout we shall assume that (Σ, g) is a complete two-dimensional Riemann-ian manifold which represents a time-symmetric spatial slice in a three-dimensionalLorentzian space-time (M, g¯) (that is, trace of the second fundamental form ofΣ ↪→ M vanishes identically; in other words, the slice is maximal). We assumethat the dominant energy condition holds for g¯, and in particular the ambient Ein-stein tensor satisfies G¯µνξµξν ≥ 0 for any time-like ξµ. The Gauss equation thenimmediately implies that g has non-negative scalar curvature.Definition 1. A complete two-dimensional Riemannian manifold (Σ, g) is said tobe asymptotically conical if there exists a compact subset K ( Σ where Σ \K hasfinitely many connected components, and such that if E is a connected componentof Σ\K, there exists a diffeomorphism φ : E → (R2\B¯(0, 1)) where in the Cartesian2010 Mathematics Subject Classification. 83C80.The author is supported in part by the Swiss National Science Foundation.Version rev84 of 2012-07-05 16:57:18 +0200 (Thu, 05 Jul 2012).12 W. W.-Y. WONGcoordinates on R2 the line element satisfiesds2 −[dx2 + dy2 − 1− P2x2 + y2(x dy − y dx)2]∈ O2((x2 + y2)−)for some  > 0 and P > 0. The notation f ∈ O2(r−2) is a shorthand for|f |+ r |∂f |+ r2 ∣∣∂2f ∣∣ ≤ Cr−2 .Remark 2. The decay condition is sufficient to imply that the scalar curvature Sof g is integrable on Σ. Note that in polar coordinates x = r cos θ and y = r sin θthe conical metric in the square brackets can be written as the conicaldr2 + P 2r2dθ2where we see that m = 2pi(1− P ) is the angle defect for parallel transport aroundthe tip of the cone.Theorem 3. If (Σ, g) is a complete asymptotically conical two-dimensional ori-entable Riemannian manifold with pointwise non-negative scalar curvature, then Σis diffeomorphic to R2 and m = 2pi(1 − P ) is non-negative. If furthermore m = 0then (Σ, g) is isometric to the Euclidean plane.Proof. Enumerate from 1 . . . N the asymptotic ends (Ei, gi) with diffeomorphismsφi and constant Pi. By the asymptotic structure, for sufficiently large Ri the curveγi ={φ−1i (x, y)∣∣ x2 + y2 = R2i } in the end Σi has positive geodesic curvature, ifwe choose the orientation so that the inward normal is toward the compact set K.Let Σ0 ) K denote the compact manifold with boundary in Σ that is bounded bythe γi. Applying Gauss-Bonnet theorem, using the fact that the geodesic curva-tures are all signed and the scalar curvature is non-negative, we have that Σ0 haspositive Euler characteristic. As Σ0 is orientable and connected, and has nonemptyboundary, it must be diffeomorphic to a disc. Hence Σ has only one asymptotic endand is diffeomorphic to R2. For sufficiently large R we let ΣR denote the compactregion bounded by x2 + y2 = R2. Using the Gauss-Bonnet theorem again, alongwith the decay properties of the metric, we see that m, the angle defect, is in factgiven bym = limR→∞12∫ΣRS dvolg =12∫ΣS dvolg .Hence m is necessarily nonnegative, with equality to 0 only in the case S ≡ 0. Remark 4. One can analogously define the “quasilocal mass” mγ associated to γ ( Σa simple closed curve by lettingmγ be the angle defect for parallel transport aroundγ. Then it is easy to see the this quantity has a monotonicity property: if γ1 is tothe “outside” of γ2, let Σ1,2 be the annular region bounded by the two curves, wemust havemγ1 −mγ2 =12∫Σ1,2S dvolg .Remark 5. It was pointed out to the author by Julien Cortier that some similarconsiderations in the asymptotically hyperbolic case was mentioned by Chruścieland Herzlich; see Remark 3.1 in [CH03].Indeed, Theorem 3 follows also from some more powerful classical theoremsin differential geometry. The topological classification can be deduced from, e.g.Proposition 1.1 in [LT91]. One can also deduce the theorem (with some work) fromShiohama’s Theorem A [Shi85]. As shown above, however, in the very restrictedPOSITIVE MASS THEOREM IN 2D 3case considered in this note the desired result can be obtained with much lessmachinery.Remark 6. The author would also like to thank Gary Gibbons for pointing outthat a similar argument to the proof of Theorem 3 was already used by Comtetand Gibbons (see end of section 2 of [CG88]) to establish a positive mass conditionon cylindrical space-times about a cosmic string; the main difference is that in theabove theorem we contemplate, and rule out, the possibility of multiple asymptoticends, as well as non-trivial topologies inside a compact region.Remark 7. One can also ask about asymptotically cylindrical spaces, which canbe formally viewed as a limit of cones. Indeed, if in Definition 1 we replace theasymptotic conditionds2 → dr2 + P 2r2dθ2withds2 → dr2 + (P 2r2 + p2)dθ2 ,then the limit P = 0 is no longer degenerate, and in fact corresponds to a spatialslice that is cylindrical at the end. In this case, however, the topological statementin Theorem 3 is no longer true: the standard cylinder S1 × R1 is flat, is asymp-totically cylindrical, and has two asymptotic ends. However, it is easily checkedusing the same method of proof as Theorem 3 that this is the only multiple-endedasymptotically cylindrical surface to support a non-negative scalar curvature.References[BBL86] John D. Barrow, A. B. Burd, and David Lancaster, Three-dimensional classical space-times, Classical Quantum Gravity 3 (1986), no. 4, 551–567. MR 851288 (87h:83023)[Bro88] J. D. Brown, Lower dimensional gravity, World Sci. Publishing, 1988.[Car98] Steven Carlip, Quantum gravity in 2 + 1 dimensions, Cambridge Monographs onMathematical Physics, Cambridge University Press, Cambridge, 1998. MR 1637718(99e:83026)[CG88] A. Comtet and G. W. Gibbons, Bogomol’ny˘ı bounds for cosmic strings, Nuclear Phys.B 299 (1988), no. 4, 719–733. MR 936758 (89h:83042)[CH03] Piotr T. Chruściel and Marc Herzlich, The mass of asymptotically hyperbolic Riemann-ian manifolds, Pacific J. Math. 212 (2003), no. 2, 231–264. MR 2038048 (2005d:53052)[Col77] Peter Collas, General relativity in two- and three-dimensional space-times, AmericanJournal of Physics 45 (1977), no. 9, 833.[Des85] S. Deser, On the breakdown of asymptotic Poincaré invariance in D = 3 Einsteingravity, Classical Quantum Gravity 2 (1985), no. 4, 489–495. MR 795095 (86k:83020)[DJtH84] S. Deser, R. Jackiw, and G. ’t Hooft, Three-dimensional Einstein gravity: dynamics offlat space, Ann. Physics 152 (1984), no. 1, 220–235. MR 734213 (85k:83004)[LT91] Peter Li and Luen-Fai Tam, Complete surfaces with finite total curvature, J. DifferentialGeom. 33 (1991), no. 1, 139–168. MR 1085138 (92e:53051)[Shi85] Katsuhiro Shiohama, Total curvatures and minimal areas of complete surfaces, Proc.Amer. Math. Soc. 94 (1985), no. 2, 310–316. MR 784184 (86h:53047)École Polytechnique Fédérale de Lausanne, SwitzerlandE-mail address: willie.wong@epfl.ch

A positive mass theorem for two spatial dimensions

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