Lorentzian angles and trigonometry including lightlike vectors
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Lorentzian angles and trigonometry including lightlike vectors (cid:63)
Rafael D. SorkinRaman Research Institute, Bengaluru, Karnataka, IndiaandPerimeter Institute, 31 Caroline Street North, Waterloo ON, N2L 2Y5 CanadaandDepartment of Physics, Syracuse University, Syracuse, NY 13244-1130, U.S.A.address for email: [email protected]
Abstract
We define a concept of Lorentzian angle that works even when one or bothof the directions involved is null (lightlike). Such angles play a role inRegge-Calculus, in the boundary- and corner- terms for the gravitationalaction, and in the Lorentzian Gauss-Bonnet theorem (for which we providea proof).
Keywords and phrases : Lorentzian trigonometry, lightlike angles, nullboundary, corner-terms in the gravitational action, Lorentzian Gauss-Bonnet theoremImagine that you have sliced a pizza into several “wedges”, and now you want to reassemblethem. Imagine also that you have numbered and marked the individual pieces. When youput two consecutive wedges together, their edges will align perfectly without any specialeffort on your part. Moreover the opening angles of the wedges will add up to 2 π or 360degrees, no matter where you made the cuts. In fact, the reassembly would have succeededequally well if the pizza’s radius had been infinite: only the opening angle of each wedgematters. These familiar facts could be summarized by saying that addition of angles iswell-defined in the Euclidean plane. (cid:63) also available at http://arxiv.org/abs/1908.100221ow imagine that the plane is not Euclidean but Lorentzian. At first it might seemthat nothing much has changed. Some of the cuts will now be timelike instead of spacelike,but consecutive wedges will fit together unambiguously just as before, because distancesneed to match up along the common boundary between adjacent wedges. But there’s anexception! The unambiguous matching of each wedge to its neighbor will fail (assuminga pizza of infinite radius) if the line along which the cut was made was null (lightlike).For the distance-matching in that case reduces tautologically to 0 = 0. In fact, therecorresponds to any radial null-line in the Minkowskian plane M a Lorentz transformationthat rescales it by an arbitrary factor, and the identification across the adjacent boundariesis therefore ambiguous by such a factor. (cid:63) Thanks to this ambiguity the reassembled pizzacan differ from the original. When you put the last wedge in place you might find thata lacuna remains, or vice versa that the last wedge overlaps the first one. In this senseopening angles are not necessarily additive in M .The difference from the Euclidean case is of course that a Lorentzian “angle” param-eterizes a boost-transformation, not a rotation, and no boost can take one edge of a wedgeto the other edge if the latter is null. Angles involving a null edge fail to be additive becausethey fail to be defined at all! Nevertheless, we can still bring about a unique matchingby restoring the rescaling information that was lost when we chose to cut our “Lorentzianpizza” along a null ray. To preserve this information, it suffices to mark both edges at thesame point along the cut, and then to require that the marks be opposite each other whenthe wedges are brought together.This suggests that, although one cannot define the angle between, for example, aspacelike ray (half-line) and a null ray, one might be able to define the angle between aspacelike ray and a truncated null ray (the truncation being equivalent to a marking asillustrated in figure 1). In other words, the angle between two vectors in M might bedefinable even when the angle between the corresponding rays is ambiguous because one (cid:63) What it more precisely means to “join two wedges together” can be explicated as follows.The wedges are to be embedded isometrically into M in such a way that the edges to bejoined coincide. The union of the images of the wedges then gives the geometry of thecombined wedge. 2f them is lightlike. We will see that a definition of this sort is indeed possible, and thatwhen one adopts it, full additivity is achieved.Such additivity is basic to simplicial gravity (Regge calculus), where spacetime istreated as a simplicial complex Σ built from flat simplices. The action S , ignoring bound-aries, is then a sum over the interior subsimplices σ of dimension two (more generally ofcodimension two), the summand being the product, Aθ , where A is the area of σ and θ is a defect-angle that measures the deviation of Σ from flatness at σ (“conical singular-ity”). More specifically, θ represents the difference between the sum of the dihedral anglesformed by the 4-simplices that meet at σ and the value this sum would have were Σ flatat σ . In making this definition, one is almost literally following the steps described aboveto reassemble a pizza, with the opening angles in the pizza corresponding to the dihedralangles here. Thus the Regge-action S rests on a general definition of Lorentzian angle(although strictly speaking, the net defect can be defined by parallel transport withoutactually needing to define separately the individual opening angles.)The above pertains to 2-simplices σ that are interior to Σ. When σ lies on ∂ Σ, theboundary of Σ, one must replace θ by the analogous angle that measures the deviationfrom extrinsic flatness of ∂ Σ, viz. the difference between the sum of the dihedral anglesformed by the 4-simplices that meet at σ and the value this sum would have were ∂ Σextrinsically flat at σ .One can think of this defect in terms of the dihedral angle between the two boundary3-simplices that meet at σ , and when one does so, the so-called corner terms needed(in the continuum) to supplement the Einstein-Hilbert action emerge automatically, thisbeing apparently the simplest way to derive them. It is plausible, moreover, that the entireboundary action in the continuum, including even the contribution of the null boundaries,could be interpreted as being the sum of an infinite number of infinitesimal corner terms.As a simple consequence of these considerations of “Lorentzian trigonometry”, we willbe able in Section 9 to deduce a Lorentzian Gauss-Bonnet theorem, in probably its mostgeneral form.In what follows, we explore to what extent one can define, in the Lorentzian plane,angles which are finite and add up consistently, even when one or both of the directionsinvolved is lightlike. Taking additivity as our guide, we will begin with a pair of spacelike3ectors and progress finally to the various null cases, whose analysis will be the main goalof our investigation.
1. Two identities useful in defining and adding angles
In the Euclidean plane the existence of opening-angles which add consistently under juxta-position of wedges can be traced to the properties of the rotation group SO (2), in particularthe fact that no matter how wide or narrow a wedge is, one can always find a rotationtaking one of its edges to the other. In M this is in general not possible, as emphasizedabove. Only in certain cases can one define an opening angle by reference to the group SO (1 , θ = a · b/ | a || b | determines the Euclidean angle between (nonzero) vectors a and b , and so the fact that θ ( a, b ) + θ ( b, c ) = θ ( a, c ) when b lies between a and c must correspond to an equation thatrelates a · c to a · b and b · c . Modulo a choice of signs and “analytic branches” thisequation is clearly cos − ( a · c ) = cos − ( a · b ) + cos − ( b · c ), which can also be written interms of logarithms and square roots. Underlying this equation is a simple identity whoseLorentzian counterpart will lead us to definition of angle that works for all pairs of vectors,be they timelike, spacelike, or lightlike.In fact we will need two identities in the Lorentzian case which we now state andprove. To that end let us define, for any two vectors, a and b , in M , Z ( a, b ) = a · b + || a ∧ b || (1) Z ( a, b ) = a · b − || a ∧ b || where || a ∧ b || is the norm of a ∧ b , namely the positive square root of ( a · b ) − ( a · a )( b · b ).Notice here that ( a · b ) − ( a · a )( b · b ) is always a positive real number because the square4f the timelike two-form a ∧ b is on one hand always negative, and on the other hand equalto ( a · a )( b · b ) − ( a · b ) . Our first, trivially proven identity is then Z ( a, b ) Z ( a, b ) = | a | | b | (2)where | v | is by definition v · v , the inner product of v with itself. (We will adopt theLorentzian signature for which v · v is positive for spacelike v and negative for timelike v .Neither of our identities will depend on this choice, however.)The second, less trivial identity involves three vectors a , b , c , of which the second isbetween the other two, in the sense that it is a linear combination of the other two withpositive coefficients (see figure2): b = αa + γc α, γ ≥ Z ( a, b ) Z ( b, c ) = | b | Z ( a, c ) (3)To prove it, let us observe first that since (3) is homogeneous in the vectors involved, wecan without loss of generality assume that α = γ = 1, whence b = a + c . From this followsalso a ∧ b = a ∧ ( a + c ) = a ∧ c , and similarly b ∧ c = a ∧ c , so all three norms || a ∧ b || , || a ∧ c || , and || b ∧ c || are equal to || a ∧ c || . Substituting then a + c for b in (3), using thedefinition (1), and expanding terms, we arrive at 0 = 0.
2. Opening-angle of a spacelike wedge in the first quadrant
The two null lines through the origin in M divide the plane into four “quadrants” asshown in figure 3. Consider, to begin with, the case where a and b lie in the first quadrant.There is then a unique Lorentzian boost that takes the ray through a to that through b ,and the corresponding boost-parameter θ with its standard normalization has, as is wellknown, the magnitude θ = cosh − (cid:98) a · (cid:98) b , where (cid:98) a = a/ | a | and similarly for (cid:98) b . (One caneasily verify this explicitly in null coordinates, u = x − t and v = t + x .) Without anyloss of generality, we could of course constrain a and b to be normalized in this case, but5t will be useful below to have left them general. We therefore write the formula for theLorentzian angle in the present case as θ ( a, b ) = cosh − a · b | a || b | (4)Notice here that we are taking the opening-angle θ of a spacelike wedge to be real andpositive by convention. This is the first of several conventions we will be led to make indefining Lorentzian angles. Notice also that (4) is symmetric in a and b , consistent withthe fact that we are not tying the definition of angle to any choice of orientation in M .To put (4) into a form better suited to our identities, let us recall thatcosh − z = log( z + (cid:112) z − , (5)in virtue of which we can rewrite (4) as θ ( a, b ) = log Z ( a, b ) | a || b | (6)In conjunction with the key requirement of additivity, this equation will determine all otherangles θ ( a, b ) almost uniquely.
3. Opening angle of a wedge with one edge in quadrant I and the other inquadrant II
The next case to consider is that of the angle between a spacelike vector a and a timelikevector b . For example a could be in the first quadrant and b in the second, as in figure 4.Recalling that we have adopted additivity of opening-angles as our guiding light, letus observe that with the definition (6), additivity within quadrant I is guaranteed bythe identity (3). Since such identities are preserved under analytic continuation, let uscontinue to employ the analytic form (6) when b moves from the first quadrant to thesecond. This however will only determine θ fully when we decide which sheet of thecorresponding Riemann surface θ should lie on, or equivalently which “branches” of ‘log’and of | b | = √ b · b should be chosen. 6magine now that a remains fixed, while b moves continuously from b = a to a point inthe second quadrant. Then Z ( a, b ) will remain strictly positive, but | b | will pass throughzero when b crosses the lightcone. The ratio z = Z ( a, b ) / | a || b | will thus trace a paththrough the point at infinity in the Riemann sphere, where the logarithm has a branch-point. To resolve the resulting ambiguity in log z we can adopt the “ iε prescription” ofreference [1], where the metric was given a positive-definite imaginary part. In applicationto the present problem, this simply means that a · a will acquire a positive imaginary part,or equivalently, a · a → a · a + iε , meaning that a · a will circle the origin in a positive oranticlockwise direction. In consequence, z will circle the origin in a negative direction, andlog z will pick up an imaginary piece, − iπ/
2. When b completes its journey, we will thushave | b | = √ b · b = i (cid:112) | b · b | , together with θ ( a, b ) = log Z ( a, b ) | a | | b | = log Z ( a, b ) || a || || b || − iπ/ , (7)where || v || denotes the absolute value of | v | , and where in the middle expression one shouldinterpret log(1 /i ) as − iπ/ || a || || b || and defining (cid:98) b = b/ || b || and similarly for (cid:98) a . The result is θ = log (cid:18)(cid:98) a · (cid:98) b + (cid:113) ( (cid:98) a · (cid:98) b ) + 1 (cid:19) − iπ/ , an equation which can also be written as θ ( a, b ) = sinh − ( (cid:98) a · (cid:98) b ) − iπ/ − ( x ) = log( x + √ x + 1). Notice here that the plus sign underthe square root came about because b was timelike. Remark
The steps leading to (8) had the effect of substituting y = (cid:98) a · (cid:98) b in the identity,cosh − ( − iy ) = sinh − ( y ) − iπ/
2, which holds for a suitable identification of the Riemannsurfaces of cosh − and sinh − with each other. To prove this identity, recall that cos y =cosh( iy ), sin y = − i sinh( iy ), and make these substitutions in the further identity sin x =cos( x − π/ y = sinh( ix ).We have in this section made a second choice of convention in taking | b | to be positive-imaginary rather than negative, and therefore taking the imaginary parts of (7) and (8) to7e negative rather than positive. The physical meaning of this sign shows up in connectionwith topology-changing spacetimes, where the defect angle enters into the gravitationalaction-functional (see [1]). Additivity alone would not have forced the angle to be complex,but the requirement that the usual formulas of trigonometry continue to hold for Lorentzianmetrics does demand it (see [2]), and the Lorentzian Gauss-Bonnet theorem also requiresit. Whether it has further significant consequences remains to be seen.
4. Opening angle of a timelike wedge in quadrant II
We could obtain θ ( a, b ) in this case by starting with (8) and analytically continuing a fromquadrant I to quadrant II, but since we can deduce it directly from the spacelike case byinvoking additivity, it seems simpler and more instructive to proceed that way.In figure 5, the pairs of vectors, a cum b and a (cid:48) cum b (cid:48) respectively delineate two wedgesrelated by a Lorentz-boost, illustrating the familiar fact that spacelike and timelike vectorsrotate in opposite directions under a boost. But since angles are defined by the intrinsicgeometry, they are Lorentz-invariant, and θ ( a, b ) must equal θ ( a (cid:48) , b (cid:48) ). On the other hand,additivity of opening-angles requires that θ ( a, b ) = θ ( a, a (cid:48) ) + θ ( a (cid:48) , b (cid:48) ) + θ ( b (cid:48) , b ) from whichit follows that θ ( a, a (cid:48) ) + θ ( b, b (cid:48) ) = 0.In other words, the angle separating two timelike vectors in quadrant II has a mag-nitude equal to that of the boost relating them, but its sign is negative since it has to beopposite to that of angles within quadrant I: timelike wedges have negative opening-angles.In simplicial gravity, this opposite sign is what guarantees that the defect-angle that entersinto the Regge action is defined consistently (see [2] and [3]).Now the boost-angle between two timelike vectors in the same quadrant is given,similarly to (4), by cosh θ = | (cid:98) a · (cid:98) b | , so we have for this case (and for our choice of signaturethat makes (cid:98) a · (cid:98) b negative) θ ( a, b ) = − cosh − ( − (cid:98) a · (cid:98) b ) , (9)which can also be written with the aid of (5) and the definition of Z as θ ( a, b ) = − log Z ( a, b ) | a | | b | (10)8otice here that because a and b are timelike, | a | and | b | are pure imaginary, and so both Z ( a, b ) and | a | | b | are negative.
5. Opening angle of a wedge with one null edge
Our formulas derived so far allow one to deduce the opening angle of any wedge W whoseedges are either spacelike or timelike but not null. This includes wedges whose edges lie inopposite quadrants, as well as non-convex wedges like that shown in figure 6, and wedgeswhich overlap themselves. In all such cases, it suffices to subdivide W into sub-wedgeseach of which is convex and fits into one of the cases analyzed above. Additivity will thenguarantee that summing the angles of the sub-wedges will produce a value independent ofhow the subdivision was done.The only outstanding situation, therefore, is that where either or both edges of W arenull (lightlike). One might think that no useful angle could be defined at all then, becausethe opening angle of, for example, a spacelike wedge diverges as one of its edges approachesthe light cone: a boost that could take a non-null direction to a null direction would haveto be infinite. It is therefore surprising that — as explained earlier — one can actuallydefine finite and additive opening-angles in this situation if one works with wedges whosenull edges are marked . That is, one can successfully define an angle between a non-nullray and a null vector, or between two null vectors. And these definitions arise naturally inthe context of quantum gravity.To see how this comes about, consider a triplet of vectors as seen in figure 7. Let a and b lie in quadrant I and c in quadrant II, with b near to, but not actually on, thelightcone that divides the quadrants from each other. We know that for these non-nullvectors, θ ( a, b ) + θ ( b, c ) = θ ( a, c ). Let us try to rearrange the latter equation in such a waythat it will remain well defined when b approaches the null vector n .To that end, let us call on the identity (3), which expresses additivity of angles interms of inner products of vectors: Z ( a, b ) Z ( b, c ) = | b | Z ( a, c ) (3)As b → n , | b | →
0, and (3) must trivialize to 0 = 0. Hence either Z ( a, n ) or Z ( n, c ) mustvanish, and it is easy to see that (with our signature of ( − + ++)) Z ( n, c ) does so, because9 · c < c being timelike. The equation 0 = 0 is not very useful, of course, but if wemultiply through by Z ( b, c ) before taking the limit, the zeros will cancel and what remainswill suggest appropriate definitions of θ ( a, n ) and θ ( n, c ).Proceeding this way and taking note of (2), we obtain successively Z ( a, b ) Z ( b, c ) Z ( b, c ) = | b | Z ( a, c ) Z ( b, c ) Z ( a, b ) | b | | c | = | b | Z ( a, c ) Z ( b, c ) Z ( a, b ) | c | = Z ( a, c ) Z ( b, c ) (11)Now let b → n , obtaining in the limit (which is now smooth) Z ( a, n ) | c | = Z ( a, c ) Z ( n, c )But because n · n = 0 and a · n >
0, we have firstly Z ( a, n ) = a · n + (cid:112) ( a · n ) − ( a · a )( n · n ) = 2 a · n and secondly (because also c · n < Z ( n, c ) = n · c − (cid:112) ( n · c ) − ( n · n )( c · c ) = 2 n · c Equation (11) thus becomes Z ( a, c ) = 2 a · n c · n | c | (12) Remark
It is obvious geometrically that a · c must be determined by a · n and n · c , andindeed, one can show that with n null, this relationship takes the pretty form,2 a · c = c · na · n − a · nc · n Equation (12) is nothing but a convenient form of this last equality.In (12) we have the result we need, but it is not quite in the form we need. In orderto make contact with θ ( a, c ), we can divide through by | a | | c | , obtaining thereby Z ( a, c ) | a | | c | = 2 a · n/ | a | c · n/ | c |
10r with reference to equation (7), θ ( a, c ) = log Z ( a, c ) | a | | c | = log 2 a · n/ | a | c · n/ | c | , (13)which since a · n and | a | are both positive real numbers, can be written without any lossof phase information as θ ( a, c ) = log(2 a · n/ | a | ) − log(2 c · n/ | c | ) (14)In (14), θ ( a, c ) is expressed as a sum of two terms, the first involving only a and n , and thesecond only b and n — a form suited perfectly to angle additivity! The suggestion springingfrom (14) then, is to define θ ( a, n ) to be log(2 a · n/ | a | ) and θ ( c, n ) to be − log(2 c · n/ | c | ).(We could of course have dropped the factor of 2 from these formulas, but it turns outthat one obtains a more uniform set of angle-definitions by retaining it.)This is basically the course we will follow, but with one or two amendments. The firstproblem is that the arguments of the logarithms are not dimensionless. † For dimensionalconsistency it seems necessary to introduce a reference length (cid:96) , leading to the amendeddefinitions, θ ( a, n ) = log 2 a · n | a | (cid:96) (provisional) (15 a ) θ ( c, n ) = − log 2 c · n | c | (cid:96) (provisional) (15 b )Here, in the second equation, one is meant to interpret | c | as a positive imaginary num-ber and to interpret log i as + iπ/
2, following the conventions we have been adhering tothroughout. If we do so then θ ( a, n ) will be real, while θ ( n, c ) will, in common with θ ( a, c ),have − iπ/ θ ( a, c ) = θ ( a, n ) + θ ( n, c ) (16) † This statement presupposes that an inner product like a · n has the dimensions of length-squared, as one would normally expect it to do. However if n and/or a had other dimensionsthan the usual ones, as a normalized vector like v/ || v || does for example, then the argumentsof the logarithms would not necessarily be dimensionful.11he second problem, or rather ambiguity, is that the provisional pair of definitions(15) is only one among many equally consistent possibilities. Just as we could have omittedthe factor of 2 in both (15a) and (15b), we could, without in any way spoiling (16), haveadded any complex constant c to θ ( a, n ) and subtracted it from θ ( c, n ). To do justice tothis freedom, we should perhaps have included such a constant c explicitly in equations(15). Complicating our equations that way can be avoided, however, if we notice that anysuch c can be absorbed into (cid:96) , provided that we are willing to let (cid:96) become complex. If weagree to keep in mind that (cid:96) might in principle be complex, then we can treat the choiceof c as purely a question of notational convenience. (Or still better, could we identify aconvincing reason why one value of c should be selected as the “right one”? Perhaps, giventhat a change of c would modify the “corner terms” in the gravitational action, quantumgravity could provide such a reason, but for the moment, I know of none.)What then would be the most convenient choice? On one hand, (15) looks simple,on the other hand it introduces a hard-to-remember asymmetry into our angles. Whereas θ ( a, n ) in (15a) is purely real, as if n had been displaced infinitesimally into quadrant-I, θ ( c, n ) in (15b) has an imaginary piece of − iπ/
2, as if n had been displaced infinitesi-mally into quadrant-II. More symmetrically we could imagine n as falling precisely on thelightcone, exactly midway between quadrant I and quadrant II, in which case we wouldattribute equal contributions of − iπ/ θ ( a, n ) and θ ( c, n ). With this conventionevery wedge with a single lightlike edge will acquire an imaginary contribution of − iπ/ real parts of θ ( a, n ) and θ ( c, n ) will begiven by the same expressions: Re θ ( a, n ) = log 2 | a · n ||| a || (cid:96) (17 a )Re θ ( c, n ) = − log 2 | c · n ||| c || (cid:96) (17 b )With the conventions we have adopted the full formulas in the present case will be asfollows. 12hen a is spacelike and n is null (both in same closed quadrant) θ ( a, n ) = log 2 | a · n ||| a || (cid:96) − iπ/ b is timelike and n is null (both in same closed quadrant) θ ( b, n ) = − log 2 | b · n ||| b || (cid:96) − iπ/ (cid:96) , the other being how to apportion the imaginary contribution − iπ/ n and a and the “almost timelike”wedge that lies between n and c .That we have had to introduce the length (cid:96) means that, insofar as null vectorsare involved, angles have lost their familiar conformal invariance. Under rescaling of themetric they will now change by a logarithmic additive constant. Perhaps it should not be asurprise that when one manages to convert an angle that a priori would have been infiniteinto something finite, an additive ambiguity arises. On the other hand, it is also true thatthe ambiguity cancels in certain sums or differences of angles. (Indeed its cancellation inthe sum θ ( a, n ) + θ ( n, b ) was precisely the requirement that led us to our definitions in thefirst place.) It cancels in particular if one compares the angle θ ( a, b ) between two vectorswith the angle between the same two vectors with respect to a different metric, providedthat neither a nor b goes from being null to non-null or the reverse. It is this fact whichguarantees that the “double path integral” is unambiguously defined in gravity, even forregions with null boundary-portions (cf. [4] [5]). In light of these indications, we mightbe tempted to declare that only quantities from which the reference length l has droppedout are meaningful. Heuristic derivation of equation (17a)
Given the somewhat circuitous route we took to reach the definitions (18) and (19), itmight be reassuring to see how one could have arrived at the same result by a differentpath which, though it takes some liberties with logic, is more direct and intuitive. In thefollowing we will be using the familiar fact that, when it acts on a null vector n , a boostΛ of rapidity-parameter θ takes n to Λ n = e θ n .13e seek to deduce the angle between the same two vectors as before, a (spacelike)and n (null), assuming for simplicity that a is normalized ( a · a = 1). Now let b = Λ a bethe result of applying to a a boost transformation Λ of angle η which will carry it towardthe lightcone, as in figure 8.Since Λ is an isometry, b is also normalized. We know that b can never truly reachthe lightcone, but by taking η arbitrarily great it can come as “close” as desired. Imaginethen, that b has come so close to being lightlike that it is “for all practical purposes” amultiple of n , a very great multiple since the boost was so large. In other words b ≈ n (cid:48) forsome λ (cid:29) n (cid:48) = λn . (For definiteness, we can determine n (cid:48) by the conditionthat b − n (cid:48) be lightlike.) Because n and n (cid:48) are both on the lightcone, a unique boosttakes n to n (cid:48) ; let its parameter be γ , so that n (cid:48) = e γ n . Because Lorentzian angles aretaken by definition to be boost-parameters whenever this makes sense, we can say that θ ( n, n (cid:48) ) = γ . Combining this with the approximate equality, θ ( a, b ) ≈ θ ( a, n (cid:48) ), we learnthat θ ( a, n ) = θ ( a, n (cid:48) ) − θ ( n, n (cid:48) ) ≈ θ ( a, b ) − θ ( n, n (cid:48) ) = η − γ .Now let us put this plan into action. Let u and v be null vectors such that u · v = 1 / v pointing toward the future and u toward the past. Without loss of generality, wecan take a = u + v and n = αv . Then (for η (cid:29) b = Λ a = e η v + e − η u ≈ e η v , n (cid:48) = e η v = ( e η /α ) n , γ = log( e η /α ) = η − log α , θ ( a, n ) ≈ η − γ = log α .Now compare this with, a · n = ( u + v ) · ( αv ) = α/ θ = log α = log(2( a · n )) , (20)in perfect agreement with (15a). (cid:91)(cid:91) Even the factor of 2 is the same! That no dimensionful constant like l appears in (20)illustrates the point made in a previous footnote. Because b · b = a · a = 1 is a pure number,and because we have assumed n = b/α with α a pure number, the combination 2( a · n )is also a pure number, and is already dimensionless without the need for any conversionfactor. 14 . Opening angle of a wedge with two null edges Consider a wedge W with two lightlike edges marked by vectors a and b . If we confineourselves to convex wedges, there will be three sub-cases to consider according as W fillsout a spacelike quadrant, a timelike quadrant, or an entire half-space (figure 9). In allthree situations, the required opening angle θ ( a, b ) follows uniquely via additivity from theformulas we have already derived. (In addition — if we want to include it — there is afourth sub-case of an “infinitely thin null wedge”, but we will postpone its considerationto the next section.)Referring to the first situation depicted in the figure, let a and b be null vectors on theboundaries of quadrant I, and let w be a unit vector between them. It is easy to verify (e.g.by introducing orthonormal vectors (cid:98) x and (cid:98) t , and taking w = (cid:98) x , a = λ ( (cid:98) x − (cid:98) t ), b = µ ( (cid:98) x + (cid:98) t ))that (2 a · w )(2 w · b ) = (2 a · b ) , (21)the logarithm of which says (all the factors being positive)log(2 a · w ) + log(2 w · b ) = log(2 a · b )Comparing with (18) and remembering that | w | = 1, we conclude that for a and b bothnull and in the closure of quadrant I, θ ( a, b ) = log 2 | a · b | (cid:96) − iπ/ a and b both null and in the closedquadrant II, θ ( a, b ) = − log 2 | a · b | (cid:96) − iπ/ , (23)exactly the same formula as (22) except for the sign of the real part, which as we knowwill always flip when we go from a spacelike to a timelike wedge.Turn now to the third situation illustrated in the figure, where a and b are anti-parallelnull vectors and the wedge W is a half-space, and take m to be a null vector which liesbetween a and b within W . The calculation is even simpler in this case, since no non-null15dges are involved. Starting from θ ( a, b ) = θ ( a, m ) + θ ( m, b ) and substituting the values(22) and (23), we obtain θ ( a, b ) = log | a · m || b · m | − iπ = log( − a : b ) − iπ (24)In this last equality, a : b is the ratio of a to b , defined as that number λ (necessarilynegative when a and b are antiparallel) for which a = λ b .How does it happen that a : b and not b : a occurs in (24)? At first sight, it mightseem that a and b play symmetrical roles, but an examination of the different signs in(22) and (23) reveals the relevant difference: a forms a spacelike wedge together with m whereas the wedge bounded by b and m is timelike. Or to put the distinction another way, a would become spacelike if it were to move into W , whereas b would become timelike wereit to do so.The calculations here illustrate the mnemonic that each null edge contributes − iπ/
7. “Opening angle” of a “sliver” with parallel null edges
In Section 5 we observed that the boost-angle between two parallel null vectors n and n (cid:48) is given by the log of their ratio, i.e. log λ if n and λn are the two vectors in question.But the sign of such an expression is in general ambiguous, unless we can decide whetherto form the ratio as n : n (cid:48) or as n (cid:48) : n (whether the answer should be log λ or log λ − ). Itmight seem that this question was moot because the type of “infinitely thin null wedge”to which it refers would never arise in practise anyway. (After all, why would you wantto slice a pizza twice in the same place?) In fact, however, such wedges or “slivers” canbecome relevant whenever something akin to the extrinsic curvature of a null boundaryplays a role, as it does in surface terms for the gravitational action and in connection withthe Lorentzian Gauss-Bonnet theorem. Let us therefore examine the question more closely.As earlier, let W be a wedge in quadrant I with edges a (spacelike) and n (null), n being adjacent to quadrant II. Let N be a null sliver, conceived of as a very thin wedgewith null edges “marked” by n and n . There are then two ways in which we could sew16 to N , depending on whether we attach its n -edge to the n edge of N or to the n edge.(See figure 10.) And unlike all the cases we have considered up until now where there wassuch a choice, this choice makes a difference.Suppose we glue the n edge of N to W , matching n with n . The combined wedge willthen have edges n and a , with n lying “between” n and a . Additivity in this situationwould require θ ( a, n ) = θ ( a, n ) + θ ( n , n ). Substituting the known values for θ ( a, n )and θ ( a, n ) into this equality and cancelling equal terms from the right- and left-handsides of the equation yields log( a · n ) = log( a · n ) + θ ( n , n ). Hence θ ( n , n ) = log a · n a · n = log( n : n ) , (25)where n : n = λ if n = λn . But had we sewn the sliver in the other way, matching its n edge with W , we would have obtained instead an angle of θ = log( n : n ) = − log( n : n ).The upshot is, as we anticipated, that the sign of the angle between n and n (i.e. thatof the “opening angle” of N ) remains indefinite until one specifies how N lies on the planenext to W . If you flip it over, the sign of the angle also flips.This looks rather confusing, but it can seemingly be encapsulated in a relatively simplerule: θ = log( n : n ) if the n edge is the one that faces “downward” toward quadrant I.Another way to say this rule is that θ is positive when the edge with the shorter markingfaces downward and negative when it faces upward. (Stated like this, the rule assumesthat the null vectors point upward. More generally, we would replace “facing downward”with “facing toward the spacelike quadrant of the plane adjacent to N ”.) (cid:63) Along with the case of a single “sliver”, corresponding to a single pair of parallel nullvectors, our rule generalizes naturally to the case of multiple slivers, corresponding to asuccession of null vectors. If, as in figure 10, the null vectors point toward the future, andif we number them so that, proceeding from past to future, the slivers are delineated bythe pairs ( n , n ), ( n , n ), ( n , n ), etc, and if we take θ ( n j , n j +1 ) = log( n j +1 : n j ), thenit’s evident from (25) that the angles will add up correctly. (cid:63) Compare the remarks following equation (24)17 . Opening angle of an arbitrary wedge in M We have now analysed enough special cases that the opening angle of any wedge whatsoevercan be deduced straightforwardly from angles we already know. It suffices to subdividethe given wedge W into sub-wedges W k which are narrow enough that a formula fromone of the previous sections will furnish θ ( W k ). The opening angle of W is then simply θ ( W ) = (cid:80) k θ ( W k ). We have already seen this procedure at work in Section 6, and bythe same method we could derive an explicit formula for any other case of interest. Infact there are really only five primitive cases, from which all others can be deduced bysummation, namely those treated in Sections 2, 4, 5, and 7.This seems a good place to point out how the concept of “opening angle of a wedge”differs from that of “angle between two vectors”. The two are closely related of course,but the key difference which explains why we have worked mainly with the former conceptis that it contains information that the latter lacks. In the Euclidean context for example,consider two orthogonal vectors, a and b , and ask what angle θ they subtend. Is it 90degrees or 270 or any one of an infinite number of other possibilities? All we really know isthat its cosine vanishes. The question cannot be answered starting solely from the vectorsthemselves, but it can be answered if we associate each vector with an edge of a specifiedwedge W , i.e. if we specify how to fill in the space between a and b (or equivalently if wegive a path that connects a to b while remaining within W ).This is also a good place to call attention to the fact that the opening angles we havedefined pay no attention to any orientation that a wedge might or might not carry. As onesees clearly from the sign-rules exposed above in Sections 2–8, the positive or negative signthat a Lorentzian angle like θ ( W ) carries stems from the distinction between spacelike andtimelike directions; it has nothing to do with any orientation of W or of the vector-space inwhich it resides. Thanks to this independence of orientation, the Lorentzian Gauss-Bonnettheorem we will prove in the next section will encounter no difficulty in unorientable spaceslike R P . 18 . Applications and Implications Regge Action (dihedral angle, boundary term, additivity, continuum limit)
The most direct application of our formulas, one to which we’ve referred repeatedly, con-cerns the calculation of the gravitational action S in a spacetime presented as a piecewiseflat simplicial complex Σ ( S being called in this setting the Regge-action). As describedearly on in this paper, each interior 2-simplex σ within Σ contributes to S the productof its area A by a defect-angle θ which is computed by adding up the dihedral angles θ j formed at σ by the 4-simplices which meet at σ (the 4-simplices of its so-called “star”, σ (cid:63) )and subtracting the result from the corresponding result for flat spacetime: θ = flat-value − (cid:88) θ j (26)(See [6][2][3].)Of course a dihedral angle formed by a pair of 3-simplexes (tetrahedra), is not imme-diately the same thing as a wedge in M . That the discussion in this paper neverthelessallows us to define and evaluate the θ j becomes clear when one realizes that the dihedralangle to which θ j corresponds lives in effect in a quotient space of dimension two. Thus,let ρ be the j th σ (cid:63) and let the two faces of ρ that meet at σ be ϕ (cid:48) and ϕ (cid:48)(cid:48) (eachbeing a 3-simplex). If we project σ to zero, ρ projects down to a 2-simplex (a triangle), ϕ (cid:48) and ϕ (cid:48)(cid:48) project down to a pair of edges of the triangle, and σ itself projects down tothe vertex at which these two edges meet. † (See figure 11.) These edges in turn can beidentified with the vectors, a , b , n , etc, which feature in formulas like (18), (19), and (23)above, and the dihedral angle θ j is then nothing but the opening angle of the triangle at(the projection of) σ . For clarity, I have described the projection in 3 + 1 dimensions, butthe conclusion is valid in general: σ will always be of codimension 2, and ρ will projectdown to a triangular wedge in M whose opening angle will furnish θ j .So far, we have not said whether the so-called “hinge simplex” σ is spacelike, timelike,or null. When it is spacelike, the two dimensions that get lost when σ is projected away will † We can regard ρ as a subset of the vector-space M , and then the projection in questioncollapses M down to M /V , where V is the subspace of M spanned by σ .19lso be spacelike, and our triangular wedge will live in a quotient space which is Lorentzianand isomorphic to M . (Think of ignoring Cartesian coordinates x and x in Minkowskispace.) We are in this case brought back to the situation studied in earlier sections ofthis paper, and all of our formulas derived there are applicable. As we have seen the totalangle surrounding a point in M equals − πi with the conventions we have adopted. Forthe defect angle of a spacelike hinge σ , we thus obtain from (26), θ = − πi − (cid:88) θ j (27)(Notice incidentally that in this case of a spacelike σ , an edge of our wedge is spacelike,timelike, or null precisely when the 3-simplex, ϕ (cid:48) or ϕ (cid:48)(cid:48) , of which it is the projection isspacelike, timelike, or null.)The hinge-simplex σ can also be timelike, in which case the quotient space is isomor-phic to the Euclidean plane. Defining the opening angles θ j and the resulting defect-angleat σ is then routine and presents no difficulty. The contribution of σ to the Regge-action isonce again in this case θ A , the area A of σ being taken by definition to be a non-negativereal number, and the defect-angle being given by θ = 2 π − (cid:88) θ j (28)There is also a third possibility, which in some ways is the most interesting mathe-matically: σ could be null. In this case the geometry of the quotient space falls somewherebetween Lorentzian and Euclidean, and seems to be characterized by a degenerate con-travariant “metric” h ab . It seems that opening angles are not well-defined in such a space,but that for certain pairs of wedges, ratios of opening angles remain meaningful. It alsoseems, however, that to the extent a defect angle can be defined at all, it must vanish (seealso [5]). This third case could bear further analysis, but as far as the Regge-action isconcerned, the question is moot. A null hinge does not contribute to S , since its area isby definition zero. [3]Along with a “bulk” contribution, the gravitational action-functional contains alsoa boundary term which in a simplicial manifold takes almost the same form as the bulk20erm. As mentioned earlier, it is a sum of terms Aθ , one for each codimension-two simplex σ that lies on ∂ Σ. Instead of (26) however, one now defines θ by θ = flat-half-value − (cid:88) θ j , (29)where flat-half-value denotes − iπ when σ is spacelike and π when σ is timelike.The significance of the change from “flat-value” in (26) to “flat-half-value” in (29), i.e.from − πi or 2 π to − iπ or π , emerges when one imagines Σ as part of a larger spacetimewhich contains Σ in its interior. Let this larger spacetime be Σ ∪ Σ (cid:48) , where Σ ∩ Σ (cid:48) = ∂ Σ.From the definitions, (26) and (29), it is obvious that in this situation S (Σ ∪ Σ (cid:48) ) = S (Σ) + S (Σ (cid:48) ) (30)because the two boundary terms (29) coming from Σ and Σ (cid:48) combine to give the singlebulk term (26) for Σ ∪ Σ (cid:48) . Conversely, this additivity is ultimately the raison d’etre for theboundary term (29) and the explanation of its particular form. If you start with (26) forthe “interior” hinges, and you ask yourself, What action could I attribute to the boundaryhinges so that the full action will be additive?, you will be led inevitably to (29) as theobvious answer. Remark
Equation (30) will fail if Σ and Σ (cid:48) share a boundary simplex σ that remainson the boundary of their union. In order for it to hold true, one needs that ∂ Σ ∩ ∂ Σ (cid:48) bedisjoint from ∂ (Σ ∪ Σ (cid:48) ). In a similar way, it will fail in general when the larger manifold isthe union of three or more pieces. [7], [8]The action-additivity expressed implicitly by (29) and explictly by (30) has two im-portant implications for the continuum theory. First of all, (29) implies the existence ofso-called corner terms in the action S , whose form it also furnishes. Second of all, (30)implies that when all boundary terms are included, the total action S (Σ) will be stationaryunder small variations about a solution that holds fixed the induced metric of ∂ Σ, an im-plication we might summarize by saying, angle-additivity implies action-stationarity . Letus take these implications in turn.The “boundary defect angle” θ of (29) is already a corner-term for the simplicialspacetime Σ, being supported on the codimension-two simplices where one boundary 3-simplex meets another. In a limit where the simplices of Σ become infinitely fine in such21 way as to converge to a smooth manifold M with corners, the term (29) will remainas a corner term where M has corners, while it will converge to some smooth boundaryterm on the rest of ∂M . Accordingly, the surviving corner terms (per unit area) will bedetermined directly in terms of the opening angles we have derived in this paper. Thischain of reasoning, if followed through for all the various types of corners, should providea complete explanation of, and a recipe for, the corner terms found in [9] and [5] includingespecially the novel type of corner where a null boundary-segment meets another null ornon-null portion of the boundary. (It would make a good project to confirm this claim indetail!)It is also very plausible that the smoother boundary terms, i.e. those that do notpertain to corners, could be understood in the same way. In fact it is not hard to verify (cf.[7] [10]) that at a hinge-simplex σ where two simplices belonging to a spacelike portion of ∂ Σ meet, the trace of the extrinsic curvature takes the form of a δ -function supported on σ (the 3-simplexes themselves being internally flat), which when integrated reproduces (29).It would be interesting to try to derive the well-known Tr K boundary-term rigorouslyfrom these observations, and even more interesting to understand similarly the “surfacegravity” contributions from the null portions of ∂M , starting from equations (24) and (25).In those cases where a null boundary is involved, the ambiguity in parameterizing itsnull generators should correspond to the “marking ambiguity” that arises when one wishesto compute the opening angle of a wedge, either of whose edges is lightlike. additivity and extremality The second thing that needs fleshing out is my claim above that the additivity of S impliesits extremality. In fact this is easily demonstrated by embedding M into a larger spacetime M ∪ M (cid:48) as we did above simplicially with Σ. Let this be done and let the metric of M ∪ M (cid:48) solve the Einstein equations. Then by definition, S ( M ∪ M (cid:48) ) will be stationary undersmall variations of the metric. But by hypothesis (i.e. with the appropriate boundaryterms included) we also know that S ( M ∪ M (cid:48) ) = S ( M ) + S ( M (cid:48) ), whether or not we areat a solution. Now restrict the variation so that the metric on M (cid:48) remains unchanged,whence δS ( M (cid:48) ) = 0. The metric on M can still vary freely, except that for the sake of22ontinuity with M (cid:48) , the induced metric on ∂M will also have to remain unchanged. (cid:91) Underthese conditions, 0 = δS ( M ∪ M (cid:48) ) = δS ( M ), and we have proven that δS ( M ) = 0 underarbitrary variations which fix the induced metric on ∂M . Remark
In seeking a rationale for choosing boundary terms in the gravitational (or anyother) action, one often invokes the requirement that the action be additive as in (30) whenone builds a spacetime from two pieces whose induced boundary-metrics agree (which inturn is tantamount to asking that the action be the integral of a Lagrangian density thatis of first differential order). One also often asks that the action be stationary undervariations that preserve the boundary-geometry. We have seen here how closely relatedthese two conditions are. an unfamiliar familiar fact about triangles
An amusingly familiar, but also very useful fact that follows directly from our definitions isthe Lorentzian counterpart of the Euclidean theorem that the interior angles of a triangleadd up to π . In order to get some practice in Lorentzian Trigonometry in the spirit of [11],let us prove this counterpart in M , paying special attention to the possibility that someof the edges can be lightlike. To help bring out the parallelism between the Euclideanand Lorentzian cases, I will in the following let h be the value of a “straight angle” in therespective cases: h = π if Euclidean and h = − iπ if Lorentzian. Our goal is then to provethat the angles of a triangle sum to h .To set up the proof, start with a triangle with vertices A , B , C , and extend the edgesas indicated in figure 12, so that A = ( C + C (cid:48) ) / C and C (cid:48) . In terms ofvectors, we have then [ AC ] = − [ AC (cid:48) ], or [ CA ] = [ AC (cid:48) ], where [ XY ] stands for the vectorfrom X to Y . Similarly [ AB ] = [ BA (cid:48) ] and [ BC ] = [ CB (cid:48) ], Now consider the situation at (cid:91) One might think that continuity would also force the transverse derivatives of the met-ric to be unvaried. To analyze this issue fully, one needs to be careful about definingthe differentiable structures of M , M (cid:48) and M ∪ M (cid:48) and tracing out the consequences ofdiffeomorphism-invariance, but suffice it to say that in the end there is no such restriction.In particular the extrinsic curvature can vary freely, as one sees with particular clarity inthe simplicial context. 23ertex A . The angle of interest, indicated by α in the diagram, is the opening angle of thewedge BAC . By additivity, this angle is related to α (cid:48) by α + α (cid:48) = h, (31)from which we conclude, by adding this to the analogous equations for the other twovertices, that ( α + β + γ ) + ( α (cid:48) + β (cid:48) + γ (cid:48) ) = 3 h (32)But we also know that α (cid:48) = θ ([ AC (cid:48) ] , [ AB ]) = θ ([ CA ] , [ AB ]), which when added to itscounterparts at B and C becomes α (cid:48) + β (cid:48) + γ (cid:48) = θ ([ AB ] , [ BC ]) + θ ([ BC ] , [ CA ]) + θ ([ CA ] , [ AB ]) (33)This last expression, being the angle accrued in going full circle from [ AB ] back to [ AB ](or equivalently the net opening angle of three wedges that fit together with neither gapnor overlap), is plainly 2 h (namely 2 π or − πi , respectively). Combining (32) with (33),produces finally α + β + γ = h (34)This completes the proof, but where were the subtleties involving null vectors hiding?First of all in (31), which when AC is lightlike, relies on (24), which in turn holds onlybecause − [ AC ] : [ AC (cid:48) ] = 1 (and of course because opening angles are additive even whennull edges are involved). And secondly in the conclusion that the right hand side of (33)is 2 h , which holds only because, the vector [ AB ] (for example) is not only the same ray inboth its occurrences in (33) but literally the same vector. In effect we have construed theangle BAC as a marked wedge by using the two other vertices, B and C , as marks. Ina simplicial complex, consequently, every wedge that occurs is automatically marked. Sothere were subtleties, but if we avert our gaze from them, we will see no difference betweenthe Euclidean and Lorentzian proofs. Remark
The “complementary angle” α (cid:48) at vertex A (which in another context is equal tothe exterior defect-angle) can also be interpreted as the angle between the normal vectorsto sides AB and AC of the triangle, but one would have to be careful about signs. Ifwe chose the normals to be outward when interpreted as covectors, and if the resultingsign-rules could be sorted out, we might obtain an alternative, and even slightly simpler,way to deduce that α (cid:48) + β (cid:48) + γ (cid:48) = 2 h . 24 orentzian Gauss-Bonnet theorem for any topology By taking advantage of the triangle-theorem just demonstrated, we can, simply by mim-icking the analogous deductions for Euclidean signature, prove with almost no extra effort,a simplicial Lorentzian Gauss-Bonnet theorem! To that end, let Σ be a two-dimensional(Lorentzian) simplicial complex as above, which we can take for definiteness to be thetriangulation of a two-dimensional manifold (not necessarily orientable and possibly withboundary). Let χ be the Euler number of Σ, and let S be its Regge-action as definedearlier. (cid:63) We want to show that the action is nothing but the Euler number, or moreprecisely that S = − πiχ (35)which we can write as S = 2 hχ if, as before, we define h to be − iπ . Also write V for thenumber of vertices (0-simplices) of Σ, E for the number of edges (1-simplices), and F forthe number of triangles (2-simplices). By definition, χ = V − E + F .We will need to distinguish “interior” simplices from “boundary” simplices, the latterbeing simplices which are subsets of ∂ Σ. Let V o denote the number of interior vertices,and V ∂ the number of boundary vertices, and similarly for E o and E ∂ . We will establishthe following three equations, which together will yield the desired equality. S/h = 2 V o + V ∂ − F (36) − E o − E ∂ + 3 F = 0 (37) V ∂ − E ∂ = 0 (38)The first equation (36) comes from summing (27) and (29) over all the 0-simplices of Σ.In two dimensions, a hinge-simplex, being of codimension two, is simply a 0-simplex orvertex, and its area is A = 1 by convention. Hence, the total action S is nothing but thesum over all vertices of their defect-angles, as given by either (27) for interior vertices or(29) for boundary vertices. In this sum, each interior angle of each triangle appears exactlyonce, and since the sum of the angles for any given triangle is h , the contribution of the θ j terms in (27) and (29) to S/h is simply − F h/h = − F where F is the total number of (cid:63) The normalization is that of (1 / (cid:82) RdV h in(27) and h in (29) contribute 2 hV o + hV ∂ to S . This explains the first two terms in (36).The second equation (37) reflects the facts that each interior edge is incident on exactlytwo triangles, while each boundary edge is incident on exactly one, while each triangleis incident on exactly three edges; therefore 2 E o + 1 E ∂ = 3 F . The third equation (38)records that, because the boundary is simply a cycle (or a disjoint union of cycles) of theform, vertex-edge-vertex-edge-vertex-etc, there are exactly as many boundary vertices asboundary edges. Finally, equation (35) results if one adds the left-hand sides of (37) and(38) to the right-hand side of (36). QED.The Gauss-Bonnet theorem we have just proven presupposes remarkably little aboutthe topology of Σ. Let us apply it for example to the “trousers cobordism” M that mediatesthe splitting of one circle into two. We can assume that the metric on M vanishes at asingle “Morse point”, but is invertible everywhere else. If our theorem (35) persists inthe continuum limit sketched above, it will imply immediately (given that the trousersis homeomorphic to a sphere with three disks removed, and therefore has Euler number, χ = 2 − −
1) that S = +2 πi . We can even do better than this, because one canfurnish M with a metric which is flat everywhere except at the Morse-point (see e.g. [12]),and for such a metric the simplicial approximation is already exact, so that our theoremapplies as is. Now in the quantum-gravity amplitude e iS , an action of 2 πi yields a dampingfactor of e − π , in agreement with what was found in [1] by another method. Of course theagreement in sign (damping vs. enhancement) is not entirely accidental. It stems from thechoice we made in analytically continuing (6) past the branch point at b · b = 0 to obtain(8), and our choice involved a closely related kind of complexified metric to that employedin [1].In any case, we now have a convenient mnemonic to help remember the choice of signmade herein. It is the one for which the trousers-cobordism is dynamically suppressed,while the “yarmulke-cobordism” is enhanced. For the yarmulke there’s no null directionat all at the Morse-point, whence we get − πi from (27); for the trousers there are eightof them, whence we get − πi − × − iπ/ πi . Remark
An interesting question is where (apart from any Morse-points that might bepresent) an imaginary action like that of (35) could come from in a continuum calculation.In part it could come from the corner terms, but what if the boundary were entirely26mooth? In that situation it would seem to have to arise where the tangent vector passedfrom spacelike or timelike to null, but how to define the boundary integrand there? Or issuch a boundary inadmissible in a Lorentzian spacetime? final comments
One implication of our work in this paper is that the seemingly natural proposal to defineangles involving lightlike directions by first “regulating” the metric and then taking somesort of renormalized limit seems to be untenable. The problem is that even if, for example,we add a positive-definite imaginary part to the metric of M so that the denominator of(6) no longer vanishes when a or b becomes lightlike, we will still end up with a conceptof angle that refers to unmarked rays. Only if the renormalization somehow smuggled in“marking” could such an approach succeed. Nevertheless, deforming the metric into thecomplex did prove helpful in connection with our analytic continuation in Section 3.Finally, a few comments of a general nature on what was done above. A surprisingfeature of the expressions we have derived is that angles involving null vectors are easierto write down than angles involving only spacelike or timelike vectors (likewise for thecorresponding corner terms in the gravitational action). On the other hand, in order tobe dimensionally correct we had to introduce a reference length l into our definitions ofcertain angles involving null vectors, and the physical significance of this length, if any, isnot evident. For angles not confined to a single (open) quadrant, we also had to make alargely arbitrary choice of where to put the imaginary contributions of − iπ/
4. But perhapsthe biggest (and certainly most welcome) surprise is that, once the appropriate definitionsand conventions are in place, Lorentzian trigonometry appears almost indistinguishablefrom its Euclidean cousin.It’s a pleasure to thank Sumati Surya and Fay Dowker for comments on these topics, andspecial thanks go to Safiya Sivjee for converting my scribbled diagrams into the beautifulfigures that the paper now enjoys. This research was supported in part by NSERC throughgrant RGPIN-418709-2012. This research was supported in part by Perimeter Institutefor Theoretical Physics. Research at Perimeter Institute is supported by the Governmentof Canada through Industry Canada and by the Province of Ontario through the Ministryof Economic Development and Innovation. 27 eferences [1] Jorma Louko and Rafael D Sorkin, “Complex Actions in two-dimensional topologychange”,
Class. Quant. Grav.
14 : [2] Rafael D. Sorkin,
Development of Simplicial Methods for the Metrical and Electromag-netic Fields , Ph.D. Thesis, California Institute of Technology (1974), https://thesis.library.caltech.edu/2978/1/Sorkin rd 1974.pdf [3] Rafael Sorkin, “The Time-evolution Problem in Regge Calculus”,
Phys. Rev. D
12 :
23 :
565 (1981)[4] Rafael D. Sorkin, “The null-surface boundary term in the variation of the gravitationalaction-functional”, lecture delivered at:1. Perimeter Institute, Meeting of Quantum-Gravity group (15 april 2015)2. “Peyresq Physics 20: Micro and macro structure of spacetime”, conference heldPeyresq, Haute Provence, France (June 2015)3. “Mann Fest”, conference in honour of Robert Mann, held University of Waterloo,Waterloo, Canada (December 2015)[5] Ian Jubb, Joseph Samuel, Rafael D. Sorkin, Sumati Surya, “Boundary and Cornerterms in the Action for General Relativity”, Class. Quantum Grav. 34, 065006 (2017)arxiv:1612.00149[6] Tullio Regge, “General relativity without coordinates”,
Nuovo Cimento
19 :
Gen. Rel. Grav.
13 :
Phys. Rev.D
50 : arXiv:gr-qc/9403018 [9] Luis Lehner, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin, “Gravitationalaction with null boundaries”
Phys. Rev. D
94 : http://arxiv.org/abs/1609.00207 [10] J.W. Barrett and T.J. Foxon, “Semi-classical Limits of Simplicial Quantum Gravity”,
Class. Quant. Grav.
11 :
The American Mathematical Monthly
Class. Quantum Grav.
34 : arXiv:1609.03573 IAGRAMS
Figures 1 and 2.
1. A wedge marked by the vectors [ OA ] and [ OB ].2. Three vectors with b between a and c igure 3. The four quadrants and a wedge in quadrant I30 igure 4.
A wedge W spanning two quadrants. The edge marked by a isin quadrant I, the b -edge is in quadrant II. We analytically continue theupper edge of W from a to b igure 5. Illustrating why timelike angles have to be negative if spacelikeones are positive 32 igure 6.
A wedge need not be convex.33 igure 7.
The vectors that enter the derivation of (18) and (19).34 igure 8.
The boosted vector, b , approaches the light-cone as “closely” asdesired 35 igure 9. The opening angle of a wedge with two lightlike edges: threesub-cases 36 igure 10.
Two ways to glue an “infinitesimal wedge” or “sliver” to thewedge W . (a) the sliver. (b) the wedge W . (c) the combined wedge if n rather than n is matched with n . 37 igure 11. The projection that converts a dihedral angle to a wedge in R is illustrated in 2 + 1 dimensions. The hinge simplex σ collapses to asingle point, preserving the opening angle θ j .38 igure 12.igure 12.