arXiv:1307.5618v3 [gr-qc] 18 Nov 2013 parameter etro h edsccnetn event connecting geodesic the of vector steSnewrdfnto isaa 5 and [5] bi-scalar function world Synge the is ∗ neato ssitdaa rmtelgtcns noa onto cones, light size the of from hyperboloid away spacelike shifted gravitational is momentum effective the large interaction of of singularity the limit rel- that the of transfer) (in resummation implies that diagrams found evant amplitudes lad- and the Bethe-Salpeter in approximation, particles the der result scalar between studied earlier exchange graviton he quan- his for which for by in motivated action was effective [4] which Bryce an gravity, particular, proposed In tum 1981 theory. field in effective idea DeWitt of this quantify level Various to the past [3]. the at in GR made of been have solutions and attempts classical QFT in in divergences singularities for the regulator universal a as serve ueo h rpgtraqie oicto fteform the of modification 2 a acquires propagator the of ture e eivdt eo h re fPac egh i.e., length, Planck of- of is order scale the of length be This and L to (QFT) believed [1]. theory ten (GR) most field funda- Relativity the quantum the of General of combine one to principles is attempts mental scale of length implications minimal generic a of Existence 1 lcrncaddress: Electronic G 0 nvre 2.ItakPo.Nria o onigti u t out this the pointing of for “response Narlikar from Prof. comes thank I cut-off QFT [2]. universe” relevant the action-at-a-distance which Hoyle-Narlikar in the is exception One Ω( ∞ = ,P p, ( O ,P p, (1) = ) ) ℓ ≡ λ ⋆ aygnrcagmnsspotteeitneo iiu sp minimum a of existence the support arguments generic Many on”lnt a entrlyitoue nalclyLoren locally a in introduced Ω( naturally bi-scalar function be can length point” P oaiyi eoee hnΩ( when recovered is Locality ASnmes 04.60.-m w numbers: as PACS propagators QFT for spacetime of structure small-scale opigo erct h isaa Ω( bi-scalar the to metric of coupling nidpnetln fwr,bsdon based work, of line independent An . 2 1 where [Ω( hwta hr xssa exists there that show I . ( λ .INTRODUCTION I. ,P p, ( P [email protected] ℓ ) eateto hsc,Ida nttt fTcnlg Madr Technology of Institute Indian Physics, of Department iia eghadSalSaeSrcueo Spacetime of Structure Scale Small and Length Minimal ⋆ )] − ( = − λ 1 ( → G p )) ~ /c λ Ω( λ L ( Z p ( 3 0 P ) ,P p, ) ,P p, pcfial,testruc- the Specifically, . 1 ) / g 2 hc esrssurdgoei nevlbtenspaceti between interval geodesic squared measures which ) ab ) 10 = t − a P t ,P p, 2 b L − to non-local ( 0 2 33 x ) p t − ( Ωae:Nvme 9 2013) 19, November (ΩDated: ≫ ihaffine with , cm. a λ 1 ,P p, )d )) stangent is where , L aodKothawala Dawood 0 2 1 / λ ,wihyed edscitra of interval geodesic a yields which ), theory, .Idsussvrlcneta mlctoso h resulta the of implications conceptual several discuss I 2. eomto fsaeiegoer ie ya by given geometry spacetime of deformation and , me. o (1) rsne n[] Hneot,frcaiy ilwr with work will distance I clarity, geodesic for the (Henceforth, [7]. in presented h anipiaino hs le eut a ecap- be and can geometric, variant results following older the these in tured of implication main The e nte nlsslaigt h aersl,bsdon based field result, [6]). gravitational same e.g., over the for path-integral to (see, leading result analysis similar another to Yet lead to metric, shown spacetime also the was of fluctuations conformal quantum itnebtentesaeiepoints spacetime the between distance ntemnsrp r w.r.t. are manuscript the in of neighbourhood convex normal lycoordinates ploy pctm vns ilas suethat assume also will I events. spacetime akrudmetric background utosla to lead tuations n ( and ersn utbept nerlaeaeoe h fluc- the over average integral tuations path suitable a represent g eaiitcpitpril cin d action: (infinitesimal) particle and of point (2) transformation relativistic Eq. par- “duality” a between so-called that exists the mentioning worth connection is over elegant it averaging ticularly context, for this of prescription In form precise the them. the and both fluctuations specify to the expected course, of is, ltd nain ne h neso d am- inversion path the the under makes invariant essentially plitude which arc-length), the is l unu utain ftebcgon metric background the of fluctuations quantum ble a one u yPdaahni 8,adi’ con- was theory it’s string and in T-duality [8], of in notion the Padmanabhan with by nection out pointed was hr “ where 2 ab h factor The esaddt h pctm nevl.Atog eeatfo this relevant absorb Although will I propagator, intervals. into the spacetime of the structure singularity to added gets rdcdb hs utain.Tebakt “ brackets The fluctuations. these by produced g L
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( ,P p, ), ,P p, all ntelimit the in = L X x 0 2Ω( = ) σ eevents me saa,tno possi- ) . . . tensor (scalar, | uha“zero- a Such . .) g 2 = ( ab ,P p, X eoe h geodesic the denotes ) P ( disformal s l differentiations All . P | ,P p, h g p → t.t represent to etc. ) ab ab oal oet in- Lorentz locally p → s + ) and ,adotnem- often and ), = p d → P nt p s L and g . ǫL 0 2 + N ∈ L ab → 0 2 0 2 L − / P ǫL 0 2 d ( η / s P ab 0 2 d This . nthe in . ,the ), s was , factor h ... g (d (2) ab L i ” s 0 2 r , 2 explored in [9]; the latter was again shown to yield result modification is being sought; this metric may then be consistent with Eq. (2). interpreted as an “effective metric” near X.
Given it’s relevance, it is therefore of significance to ask (x) 2 To generalize to curved spacetime, note that: a ση = whether, given a spacetime metric gab, there exists some b b ∇ 2ηab(x X ) in flat spacetime. This allows us to rewrite deformation of it which directly leads to the result in − (4) as [replacing η g , σ σg]: ⋆ ab → ab η → Eq. (2). That is, whether one can find a metric gab such that 2 2 2 ⋆ 2 2 2 L0 2+ L0/σg g (x; σ ) = 1+ L /σ gab(x) ⋆ ab g 0 g 2 2 2 2 2 − 4 1+ L0/σg σ (p, q gab) = σ (p, q gab)+ L0 (3) ! | | 2 2 ∂a ln σg ∂b ln σg (5) The idea that existence of a minimal length might re- × | | | | quire modification of geometry is not new [10], although This is the desired covariant form of Eq. (4), and I will a precise, mathematical characterization of the modifi- ⋆ now show that gab can be recast into a remarkably simple cation has not been attempted before. In this Letter, I form, which is essentially a disformal transform [11] of the show that not only does there exist an “effective space- ⋆ original metric involving the bi-scalar Ω(x, X). time metric” gab which yields the desired result, but it turns out to have an unexpectedly simple form in terms To do this, recall that the affinely parametrized tangent of a disformal coupling of the spacetime metric to the to the geodesic from p to P at p (and pointing away from 2 2 bi-scalar [σ; L0]=1+ L /σ (p, q). Several conceptual 2 √ 2 2 A 0 P ) is given by ta = ∂aσ /2 ǫσ , where t = ǫ = 1. implications of this result are discussed. Using this, we finally obtain ± ⋆ The main results of this Letter are contained in Eqs. (6), g (p; P ) = g (p) ǫ −1 t t (7), (11) & (12). ab A ab − A − A a b ⋆ − − gab(p; P ) = 1 gab(p)+ ǫ 1 qaqb A A − A where [σ; L ] = 1+ L2/σ2 (6) A 0 0 g II. CONSTRUCTING THE MODIFIED METRIC ⋆ ⋆ where g gcb = δb , gab g−1 and qa = gabt . Note ac a ≡ ab b 2 2 ⋆ab ab that, for σg L0, 1 and g (p; P ) g (p). It is easiest to begin with flat spacetime, since this is ≫ A → → the context in which the results quoted above were first I must emphasize that the above modification is non- ⋆ derived. In flat spacetime, the propagator of a massive trivial in that g / Diff[g] but rather describes a genuinely ∈ ⋆ scalar field gets modified to different spacetime – i.e., in general, [g] = [g], where represents some curvature invariant.K It6 isK only when 2 1/2 4m 1/2 K ⋆ G(x, X)= H(2) m σ2 L2 Riem[g]=0 that we have g Diff[g]. σ2 L2 1 − η − 0 ∈ − η − 0 h i Although a direct proof of this is difficult, involving the 2 a a b b where ση = ηab(x X )(x X ). I take this as my formidable task of computing curvature invariants for starting point, and− ask if it− is possible to deform/re- ⋆ gab in terms of those of gab, one can see why it must parametrize the standard flat Minkowski metric ηab such be true in at least two different ways. First, the exam- that the above propagator naturally arises as the kernel of ple given below (see Sec. II B) of maximally symmetric the modified d’Alembartian operator. Then, if this mod- space(time)s clearly illustrate the point, since these are ification is expressible in a covariant form, generalization the simplest space(time)s which exhibit all the symme- to curved spacetime will be immediate. In fact, for flat tries of a flat space(time) but have constant, non-zero spacetime, one can achieve the desired modification by curvature. Further, Appendix A gives the modified Ricci introducing a “hole” of size L0 around the event X. This scalar, evaluated in the manner described in Sec. II B, requires a diffeomorphism which is singular at X, and for a deformed 2-sphere (a case which Maple can handle). yields the following modified metric (see Appendix A for Since a 2-sphere represents the simplest possible curved details) space, it very well illustrates the fact that the metrics ⋆ ⋆ gab and gab will generically have different curvature in- η (x; X) = 1+ L2/σ2 η Ξ (x X )(x X ) ab 0 η ab − a − a b − b variants, unless the original metric is flat. For the second −1 argument, which is somewhat indirect, see the comment Ξ[σ ; L ] = 1/σ2 L2/σ 2 1+ 1+ L2/σ2 (4) η 0 η 0 η 0 η below Eq. (11). ⋆ The argument X in ηab(x; X) is important; it is a re- It is now straightforward to prove the following identity: minder that the modified metric depends on two-points, ⋆ab 2 2 2 2 2 2 the field point x and the base point X around which the g ∂a σg + L0 ∂b σg + L0 =4 σg + L0 (7) 3
2 and recall that the defining equation for σg is the B. Spacetime singularities ab 2 2 2 Hamilton-Jacobi equation g ∂aσg ∂bσg = 4σg. Eq. 2 2 (7) therefore gives us our key result. It shows that σg +L0 ⋆ We now turn to the second relevant implication of min- is indeed the geodesic distance for metric g . That such ab imal length: the modification to invariant properties of a simple result should follow from a non-trivial modifica- spacetime such as it’s curvature invariants. Once the tion of the metric (which, in a genuine curved spacetime, world function is known in a given spacetime, a symbolic will not even have the same curvature as the original package such as Maple can be used to obtain expressions metric) is in itself remarkable. ⋆ for, say, any modified curvature invariant (p; P,L0). Relationship with conformal fluctuations: We can iden- It’s behaviour at any spacetime event P canK then be de- √ ⋆ ⋆ tify Ta = ta/ as the properly normalized tangent vec- duced from (P,L0) = lim (p; P,L0). Unfortunately, ⋆ A K p→P K tor w.r.t. gab. This allows us to write exact expression for the world function is rarely obtain- ⋆ able, and it’s approximate expansion in coordinate inter- g ǫTaTb = [σ; L0] (gab ǫtatb) (8) ab − A − vals (which is what is often resorted to in the literature) is 2 which shows that it is the induced geometry on σg =const often inadequate near singularities. However, the follow- surface which undergoes a conformal deformation. Ex- ing results can be obtained in a straightforward manner: trinsic curvature of this surface can be shown to be ⋆ 3/2 1/2 k 1. Flat spacetime: By construction, for a flat space- Kµν = Kµν + (1/2) q ∂k hµν (9) ⋆ ⋆ A A A time, Rabcd(p; P ) = 0, and hence Rabcd(P,L0) = and can be used to study L0 corrections to focussing of ⋆ lim Rabcd(p; P,L0) = 0. geodesics [14]. p→P
Above comments connect and contrast the present anal- 2. Symmetric spacetimes: For maximally symmetric ysis with an older one based on quantum conformal fluc- spaces, Ω(p, P ) is known in a closed form, and a Maple tuations [15], in which quantum fluctuations of the con- ⋆ computation gives: (P,L ) = (P ) 1+ cK (P )L2 , formal factor simply lead to K 0 K K 0 where cK is a numerical coefficient. (q.c.f.) g = [σ; L0] gab (10) ab A 3. Schwarzschild singularity: World function for this case is not known in closed form, but if one restricts to radial timelike geodesics near the singularity (r = 0), one finds A. Green’s function that the modified Kretschmann scalar behaves as 1/(1 + 4 2 6 ǫ)L0 instead of M /r . Although still divergent since ǫ = 1, the divergence is of a very different character and, − Using identities given in Appendix B we can show that being M independent, can presumably be regularized. ⋆ 2 2 2 2 2 σg + L0 2D = 1+ L0/σg σg 2D (11) For non-singular spacetime events, it should be possible − − to employ Riemann Normal Coordinates (RNC) and es- 2 Now, in a general curved spacetime, σ satisfies the equa- tablish (1) and (2) generically. Investigation is currently 2 tion σ = 2D + (terms involving curvature). Eq. (11) in progress as to whether something generic can be said therefore provides a quick confirmation of the fact that about curvature singularities. the modified spacetime is Riemann flat if (and only if) the original spacetime is Riemann flat. Using again Ap- pendix B, we get another remarkable result
D−2 D−2 III. DISCUSSION ⋆ ⋆ − − g σ2 + L2 2 = √ g σ2 2 − g 0 − g q ⋆ ⋆ (D=2) g ln σ2+ L2 = √ g ln σ2 (12) − g 0 − g The modified metric derived here has several conceptual q D−2 connotations. I begin with the most relevant one - the 2 2 in an arbitrary curved spacetime. Since 1/σg is the analytic structure of the metric in the coincidence limit leading term in the scalar propagator G(p, P gab) in any (this has also been emphasized by Brown [12, 13]): the metric g , and since the propagator satisfies | −1 2 2 ab term (1 + Ψ) , with Ψ = L0/σ in the modified metric ∞ n D can be represented as a series = n=0( Ψ) when √ g gG(p, P gab)= δ (p, P ), (13) 2 2 S − − | Ψ < 1, i.e., when σ > L0. It is quite possible that Eq. (12) immediately implies that the leading modifica- |one| arrives at some such series by doingP a perturbative 2 tion to the propagator is obtained by replacing σg analysis for small Ψ. Any such series would not be valid 2 2 → σg + L0, which generalizes the flat spacetime result for beyond it’s region of convergence ( Ψ = 1), and there which, of course, this replacement gives exact propagator. might be many different ways to extract| | from it a term 4 non-analytic at σ2 = 0. In the present analysis, the term and so on. One can therefore “re-construct” spacetime (1 + Ψ)−1 which arises naturally is the analytic contin- from Ω(p, P ), and study how modifying Ω(p, P ) affects uation of the series beyond it’s region of convergence. the re-constructed spacetime, which is essentially what (See also [12].) S we have done. (This might have overlap with some in- teresting recent ideas, mainly due to Achim Kempf [17].) As already stated, when spacetime is flat, the modified In this sense, the “local” nature of familiar gravitational metric can be mapped back, via a singular coordinate actions also seems to be an illusion which would break 2 transformation, to the original flat spacetime, but with a down when 2Ω(p, P ) . L0. region (a “hyperboloid”) of size L0 around the base point (X above) removed. (Incidentally, this also naturally Finally, I must mention that the entire analysis presented −1 suggests existence of a maximal acceleration, amax L0 here is for timelike/spacelike separated events. The case in flat spacetime. See Appendix C.) Usually,≡ for a of null separated events bring in several subtleties, both smooth curved manifold with metric gab, one employs technical, and more importantly conceptual, since all ar- the standard metric expansion in RNC [yi = xi Xi] guments in the literature for minimal length deal with − bounds on timelike/spacelike intervals only, and there are c d 3 gab(x; X) ηab (1/3)Racbd(X) y y + O(y ) (14) no such arguments for null separated events. We hope to ≈ − study this case in subsequent work. However, there are (at least) two important cases when such an approximation is unjustified: (a) for extremely Acknowledgements – I thank T. Padmanabhan, L. Sri- small spacetime intervals, for which continuous struc- ramkumar, and J. Narlikar for useful comments and sug- tures such as Rabcd are unlikely to make much sense, gestions, and also A. Kempf for pointing out the second and (b) near spacetime singularities, where the curvature reference in [17]. Part of this research was supported tensor and/or it’s derivatives might blow up. Motivated by post-doctoral fellowship from NSERC and AARMS, by several earlier results on minimal length, we have ar- Canada. I also thank IUCAA, Pune and CTP Jamia, rived at a metric which can serve as an effective metric at New Delhi for hospitality during completion of this work. small scales. The non-locality of this metric is best un- All symbolic computations relevant for this work have derstood by comparing it with the above RNC expansion: been done in Maple using GRTensorII [18]. in the modified case, the metric at x (X) depends not only on curvature at X, but also∈ on N the spacetime interval between x and X. The resultant non-locality is therefore natural, an outcome of our starting assumption Appendix A: Derivation of Eq. (4) that spacetime intervals have a lower bound. This also suggests analysing the modified metric in the context of “spacetime foam”. (See also [16].) We look for a re-parametrization of flat space(time) such that geodesic distances have a natural lower bound, and It would be insightful to set up an effective action which we do so by introducing a “hole” in flat space(time). The is extremized by the modified metric. Such an action will analysis is most easily done in flat Euclidean space, and necessarily be non-local, but should be relatively straight- then analytically continuing back to Lorentzian signa- forward to set up given the disformal form of the metric. ture. Consider therefore an event in a D dimensional One earlier work with somewhat similar motivation is by flat Euclidean space (for simplicity,P I set it’s coordinates DeWitt (see the second reference in [4]). Xk = 0 without loss of generality; they can be easily restored in final expressions). The result in Eq. (4) is As already emphasized, our result captures the essence derived via following steps: start with standard Carte- of several earlier works on the topic. More importantly, sian coordinates for flat space, transform to spherical- the closed form expression for the modified metric should polar like coordinates with the “radial” coordinate be- make it possible to perform explicit computations in the ing the geodesic distance σ , introduce a new coordi- modified geometry. All one needs is an expression for E nate via σ σ2 + L2, and then revert to Cartesian the world function Ω(p, P ) in the background geometry, E E 0 like coordinates→ using standard transformations. (For the either under some (valid) approximation, or exact if avail- Lorentzian case,p the only difference is the appearance of able. In fact, our analysis also suggests a conceptually hyperbolic, instead of standard trig, function in one of appealing possibility in which the world function might the transformations.) These steps are straightforward, play a more fundamental role than the spacetime metric and lead to the following re-parametrization itself. This is very much possible, since almost all the information about spacetime geometry can be shown to 2 be encoded in the coincidence limit (denoted below by k L0 k xE 1 a b xE (A1) “[...]”) of covariant derivatives of Ω(p, P ). For e.g. [5], → s − δabxExE ′ ′ ga′b′ = gab = [ a bΩ(x, x )] = [ a′ b′ Ω(x, x )] which must be treated as an active diffeomorphism (no- ∇ ∇ ∇ ∇′ R ′ ′ ′ ′ = (3/2) [ Ω(x, x )] tice that it becomes singular in the coincidence limit). a (c d )b ∇a∇b∇c∇d 5
The above diffeomorphism then yields the Euclidean ana- where ⋆ logue of the metric ηab given in Eq. (4) of the paper (in 2 k x + 3 (x cot x 1) which we have re-introduced the coordinates X of ). f(x) = 4 − After some further manipulations, we can also obtainP the x 1 Euclidean analogue of the form given in Eq. (6): lim f(x) = (A9) x→0 −15 ⋆ (A10) −1 E E δab = Aδab A A ta tb (A2) − − Using this, we finally obtain E b m n 2 where ta = δabxE / δmnxE xE with tE = +1 is the nor- ⋆ ⋆ malized geodesic tangent vector connecting Xk to xk . R(P,L0) = lim R(p; P,L0) p E p→P 2 1 L2 Analytic continuation to Lorentzian signature: The fi- = 1 0 (A11) nal step is the analytic continuation from Euclidean to L2 − 15 L2 Lorentzian signature, in which the only non-triviality E E concerns the ta tb term in the metric. For the Lorentzian signature, the normalized geodesic tangent vector is given by Appendix B: Some Identities
b ηabx ta = The following identities, which are relevant for the anal- √ǫ η xmxn mn ysis of Green’s function in a D dimensional curved t2 = ǫ = 1 (A3) ± spacetime, can be proved in a straightforward (although lengthy) manner using the modified metric. To be- The correct analytic continuation, tEtE ǫt t , is most a b → a b gin with, one may use the matrix determinant lemma: easily deduced by starting from det M + uvT = (det M) 1+ vTM−1u , where M is an invertible square matrix,× and u, v are column vec- ⋆ −1 η (x; X)= Aη Kǫ A A t t (A4) ⋆ ab ab − − a b tors (of same dimension as M), to obtain: g = − ⋆ (D−2)/2 q and showing that the desired result requires K = 1. More g 1+ L2/σ2 . Using this, one can show that − 0 g specifically, evaluate the inverse metric q(with Ψ L2/σ2) ≡ 0 ⋆ab −1 ab −1 a b η (x; X) = A η + Fǫ A A q q (A5) ⋆ − m m − m 2 2 2 − 2 2 2 − − σ + L = (1+Ψ) σ −1 2 1 a ab g 0 g F = 1 + (K 1) A ; q = η tb " − − − m Ψ 1 2 To fix F , we appeal to the Hamilton-Jacobi equation, m(m D + 2) σ2 − − 1+Ψ g which yields # ⋆ Ψ 4 2D 2 2 2 ⋆ab 2 2 2 2 2 1 F ln σg + L0 = ln σg + − 2 η ∂ (σ + L )∂ (σ + L )=4σ F A + − (A6) 1+Ψ σg a 0 b 0 A 2 2 2 To get the desired result 4(σ + L0)=4Aσ on the RHS, we must therefore have F = 1, which fixes K = 1. Appendix C: Minimal length and Maximal acceleration It is instructive to explicitly analyse what our modifica- tion gives when applied to a 2-sphere, the simplest pos- sible example of a curved space. The geodesic distance That our Lorentz invariant modification of geodesic between points p and P on a 2 sphere of radius L is given 2 2 2 distances σ σ + L0 has implications for max- by imal acceleration→ in flat spacetime is essentially a consequence of the fact that uniformly accelerated −1 σ = L cos Θ observers with acceleration g in flat spacetime are − Θ = cos θ cos θ′ + sin θ sin θ′ cos(φ φ′) (A7) given by contours of σ2(p, P ) = g 2 where P is the − origin (the points on the contours are therefore con- An explicit Maple computation gives the Ricci “bi”-scalar nected to P by spacelike geodesics). A lower bound for the modified metric as on σ2 should therefore also imply an upper bound on acceleration. One can explicitly demonstrate this in ⋆ 2 L2 f(σ/L) flat spacetime as follows. For simplicity, we will work R(p; P,L ) = 1+ 0 (A8) 0 L2 L2 in (1 + 1)D which suffices for our purpose. Set up 6 the deformed metric in a spacelike neighbourhood of set up a Maple routine to evaluate the acceleration of ∗ P , in the so called right Rindler wedge, in Rindler 2 2 2 a x =const. curve; this yields a = g/ N(x) + g L0. coordinates. The standard Rindler metric is given by ∗ Since N(x)=0at x = 1/g, we get a =1/L . ds2 = N(x)2dt2 + dx2, where N(x) = 1+ gx and − maxp 0 − x [ 1/g, ). The acceleration of any x =const. curve Note that this result hinges on the fact that uniformly is∈ given− by ∞a = g/(1 + gx), which diverges at x = 1/g, − accelerated observers in flat spacetime move along orbits the Rindler horizon. To construct the deformed metric, of the boost Killing field, which are hyperbola’s given by we need the geodesic distance between two events 2 2 contours of the geodesic interval σ . It’s generalization in Rindler coordinates; this is given by σ (p, P ) = to curved spacetime is currently under investigation. g−2 N(x)2 + N(X)2 2N(x)N(X)cosh g∆t . Al- though the final form− of the metric is unwieldy, one may
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