Bending of Light and Inhomogeneous Picard-Fuchs Equation
BBending of Light and Inhomogeneous Picard-Fuchs Equation
Tadashi Sasaki ∗ and Hisao Suzuki † Institute for the Advancement of Higher Education,Hokkaido University, Sappro 060-0817, Japan Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
Abstract
Bending of light rays by gravitational sources is one of the first evidences of the general relativity.When the gravitational souce is a stationary massive object such as a black hole, the bending anglehas an integral representation, from which various series expansions in terms of the parametersof orbit and the background spacetime has been derived. However, it is not clear that it has anyanalytic expansion. In this paper, we show that such an analytic expansion can be obtained forthe case of a Schwarzschild black hole by solving an inhomogeneous Picard-Fuchs equation, whichhas been applied to compute effective superpotentials on D-branes in the Calabi-Yau manifolds.From the analytic expression of the bending angle, both weak and strong deflection expansions areexplicitly obtained. We show that the result can be obtained by the direct integration approach.We also discuss how the charge of the gravitational source affects the bending angle and show thata similar analytic expression can be obtained for the extremal Reissner-Nordstr¨om spacetime.
The theory of general relativity was proposed by Einstein in 1915. One of the important predictionsis bending of light ray in the presence of gravitational fields. In particular, deflection by astrophys-ical gravitational sources such as stars, black holes, or galaxies has been studied both theoreticallyand observationally. α = 4 Mb , (1)where M is the mass of the lensing object and b is the impact parameter of the light trajectory. Inthe weak deflection limit, b (cid:29) M is assumed, which is satisfied for the lensing by a star such as thesun. Note that although deflection of light can be derived even in Newtonian gravity, the bendingangle in general relativity is almost twice as much as that in Newton’s theory. This prediction wasconfirmed observationally in 1919 during the total solar eclipse[1].When the lensing objects are very massive, it can give rise to a deviation from the aboveexpression. The extension to include the higher order corrections in terms of the power seriesexpansion with M/b has been studied[5, 6]. Also, generalizations to the case with nonvanishingspin and electric charge are found for example in ref.[7].On the other hand, in the strong deflection limit, i.e. the impact parameter and the Schwarzschildradius are comparable, the light ray can wind around the object arbitrary times producing an in-finite number of images, called relativistic images. This behavior is related to the circular orbitof photons, whose coordinate radius is r = 3 M for the Schwarzschild spacetime. Analytically, theexistence of the relativistic images can be understood by observing the logarithmic divergence of α when the impact parameter b approaches the critical value[2]. The strong deflection expansion of α beyond the leading divergence for the Schwarzschild case was performed in ref.[6]. For the caseswith spin and/or electric charge, only numerical calculations using expressions with various ellipticintegrals can be found in the strong field limit[8, 10]. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] S e p he strong deflection limit of the bending angle has been of interest also because it is relevantto the optical appearance, or the shadow of a black hole[3], which has been recently observed bythe Event Horizon Telescope[4].In deriving the results mentioned above, the starting point is usually expressions of the deflectionangle α (more precisely, Θ := α + π ) in terms of the standard elliptic integrals, and the parameters ofthese integrals such as the modulus are complicated functions of the parameters of the backgroundspacetime and the trajectory. Therefore we cannot read off from these expressions arbitrarilyhiger order terms in both weak and strong deflection limits although the first few terms have beenobtained.One of the aims of this paper is to obtain full order expansion in terms of M/b of the deflectionangle for the Schwarzschild case. Our strategy is to consider a differential equation satisfied by Θas a function of
M/b , which can be seen as an inhomogeneous Picard-Fuchs equation. Picard-Fuchsequations are differential equations with respect to the moduli of algebraic manifolds satisfied bythe periods integrals. The notion of Picard-Fuchs equations has been used in physics. For example,it was applied to study the dependence of some Feynman integrals on the external variables[9].Also, it has been applied to analyze the dependence of D-brane superpotentials on the complexmoduli in various Calabi-Yau manifolds in the context of mirror symmetry[19, 20, 21, 22, 23], andin these cases the differential equations arise with inhomogeneous terms.The differential equation for Θ, which is derived in Sec.3 after introducing our notation in Sec.2,turns out to be a hypergeometric one with an inhomogeneous term. The explicit solution is givenin terms of hypergeometric functions, from which the coefficients of arbitrarily higher order termsin the weak deflection limit can be easily read off. By using analytic continuation formulas forthe hypergeometric functions and the inhomogeneous Picard-Fuchs equation, the strong deflectionexpansion is also derived. The result is completely consistent with the previous results in refs.[5, 6].In Sec.4, the same result is rederived directly performing the defining integral of Θ with the helpof analytic continuation to confirm our result. Furthermore, we consider a generalization of themethod used in Sec.3 to the Reissner-Nordstr¨om black hole, and show that for the extremally-charged case Θ is given in a similar expression to the uncharged case in Sec.5. Finally, we concludethis paper with some discussions.
The Action for a massless particle is given by S = N (cid:90) dτ g µν ( x ( τ )) dx µ dτ dx ν dτ , (2)where N is a Lagrangian multiplier field (we will set N = 1 later). We consider the Schwarzschildgeometry ds = − (cid:18) − Mr (cid:19) dt + (cid:18) − Mr (cid:19) − dr + r dθ + r sin θ dφ (3)and (2)is written as S = N (cid:90) dτ (cid:34) − (cid:18) − Mr (cid:19) (cid:18) dtdτ (cid:19) + (cid:18) − Mr (cid:19) − (cid:18) drdτ (cid:19) + r (cid:18) dθdτ (cid:19) + r sin θ (cid:18) dφdτ (cid:19) (cid:35) . (4)The variation with respect to N and setting N = 1 leads to the null condition, − (cid:18) − Mr (cid:19) (cid:18) dtdτ (cid:19) + (cid:18) − Mr (cid:19) − (cid:18) drdτ (cid:19) + r (cid:18) dθdτ (cid:19) + r sin θ (cid:18) dφdτ (cid:19) = 0 . (5)Variation with respect to t, θ, φ leads to(1 − Mr ) dtdτ = ε, r dθdτ = L, r sin θ dφdτ = m, (6)where ε, L, m are constants. By setting the polar coordinates, we set φ = 0 ( m = 0) and inserting(6) into (5), we find 1 L (cid:18) drdτ (cid:19) + (cid:18) − Mr (cid:19) r = 1 b , (7) here b = L/ε . We will write the trajectory as a function of θ , we find1 r (cid:18) drdθ (cid:19) + (cid:18) − Mr (cid:19) r = 1 b . (8)By setting the variable x as x = 1 r (9)we find (cid:18) dxdθ (cid:19) = − (1 − M x ) x + 1 b . (10)Note that at far distant region ( ⇔ x (cid:28)
1) where the gravitational field of the source can benegligible, the solution of the equation can be approximated by r sin θ = b . Therefore b is theimpact parameter of the photon trajectory as can be seen in Fig.1. Equation (10) is the first orderequation and we can easily obtain the integral form of the deflection angle Θ asΘ = 2 (cid:90) x dx (cid:112) /b − x + 2 M x , (11)where x is the turning point of the orbit and given by one of the roots of the equation 1 /b − x +2 M x = 0, which reduces to x = 1 /b when M = 0.We can perturbatively obtain the series of expansionΘ = π + 4 Mb + 15 π (cid:18) Mb (cid:19) + 1283 (cid:18) Mb (cid:19) + 3465 π (cid:18) Mb (cid:19) + O (( M/b ) ) , (12)which was obtained in ref.[5] . Our aim of this paper is to obtain the series of the expansion for allorder. We change the variable of integration to t = bx and set the constants α, β , and γ so that Figure 1: Trajectory of light ray. − t + 2 Mb t = (1 − αt )(1 − βt )(1 − γt ) . (13)The photon can reach the observer, which is assumed to be at the infinity, when the impactparameter is sufficiently large, i.e. 2 M/b ≤ (4 / . In this case, the roots of the above equationare all real and we can take the constants so that γ < < β < α . Then we getΘ = 2 (cid:90) /α dt (1 − αt ) (1 − βt ) (1 − γt ) (14)Changing the variable to s = αt , we haveΘ = 2 α (cid:90) ds − s ) (1 − βα s ) (1 − γα s ) . (15) In ref.[5], the explicit coefficients are given up to the order of (
M/b ) . aking the power series and making the integration, we getΘ = 2 α ∞ (cid:88) m,n =0 (1 / m (1 / n m ! n ! Γ( m + n + 1)Γ(1 / m + n + ) (cid:18) βα (cid:19) m (cid:16) γα (cid:17) n = 4 α F (cid:18) , ,
12 ; 32 ; βα , γα (cid:19) , (16)where F is the Appell function[11] and the pochhammer symbol is defined as ( x ) n := Γ( x + n ) / Γ( x ).However, α, β , and γ are complicated functions of M/b so this expression is not adequate to obtainthe series expansion in terms of
M/b . In the next section, we try a different approach.
The integral (11) can be written as an incomplete elliptic integral. It is related to the algebraiccurve y = 1 − t + 2 Mb t , (17)which is the defining equation of an algebraic torus in the complex t − y plane. Holomorphicone-form on the torus can be defined as ω = (cid:73) dy ∧ dt πi y − (1 − t + Mb t )= dt (cid:113) − t + Mb t (18)It is known that the holomorphic one-form ω satisfies the Picard-Fuchs equation with respect tothe moduli 2 M/b . Let the Picard-Fuchs operator be D ( ∂ z ) (the definition of z is given soon). Thenthe holomorphic one-form ω satisfies D ( ∂ z ) ω = − dβ, (19)where β is a zero form. In fact, the Picard-Fuchs operator and the zero-form β can be obtainedby this requirement. Taking the cyclic integral of this identity, the Picard-Fuchs equation can beobtained, D ( ∂ z ) (cid:73) dω = 0 . (20)The integral (11) can be considered as the integral with the boundary t = 0. Therefore acting D will leads to the inhomogeneoust Picard-Fuchs equation, D ( ∂ z ) (cid:90) dω = β ( t = 0) . (21)Now let us define the moduli parameter z Mb = 13 z (22)and the diffential operator θ = z ddz .We find that the deflection angle satisfies the following Picard-Fuchs equation: P Θ = 13 z , (23)where the operator P is given by P = θ − z (cid:18) θ + 16 (cid:19) (cid:18) θ + 56 (cid:19) . (24)It is easy to get the following special solution of this equation:43 z F (cid:20) / , , / / , / z (cid:21) . (25) wo independent solutions of the homogeneous equation are hypergeometric functions F [1 / , /
6; 1; z ]and F ∗ [1 / , /
6; 1; z ] := F [1 / , /
6; 1; 1 − z ], the latter of which contains the logarimic singularityaround z = 0. The general solution is then given byΘ = c · F (cid:20) ,
56 ; 1; z (cid:21) + c · F ∗ (cid:20) ,
56 ; 1; z (cid:21) + 43 z F (cid:20) / , , / / , / z (cid:21) , (26)where c and c are integration constants. Since the orbit becomes a straight line in the limit z →
0, i.e. Θ( z = 0) = π , these constants are fixed uniquely,Θ = π F (cid:20) ,
56 ; 1; z (cid:21) + 43 z F (cid:20) / , , / / , / z (cid:21) , (27)which can be expressed by the original variables asΘ = π F (cid:20) ,
56 ; 1; 27 M b (cid:21) + 4 Mb F (cid:20) / , , / / , / M b (cid:21) , (28)= π ∞ (cid:88) n =0 (1 / n (5 / n ( n !) (cid:18) M b (cid:19) n + 4 Mb ∞ (cid:88) n =0 (2 / n (4 / n ((3 / n ) (cid:18) M b (cid:19) n . (29)It is easy to check that the first few series expansion agrees with (12).To understand the strong deflection limit of this solution, we need analytic continuation formulaefor hypergeometric functions. For F , such an identity is a classical result[11], F [ a, b ; a + b ; z ] = Γ( a + b )Γ( a )Γ( b ) ∞ (cid:88) n =0 ( a ) n ( b ) n ( n !) [ k n − log(1 − z )] (1 − z ) n , (30)where k n = 2 ψ ( n + 1) − ψ ( n + a ) − ψ ( n + b ) with ψ ( z ) = Γ (cid:48) ( z ) / Γ( z ) being the digamma function.For F , we can use the following identity given in ref.[12] Γ( a )Γ( a )Γ( a )Γ( b )Γ( b ) F (cid:20) a , a , a b , b ; z (cid:21) = ∞ (cid:88) n =0 ( a ) n ( a ) n ( n !) (cid:40) n (cid:88) k =0 ( − n ) k ( a ) k ( a ) k A (2) k ( ψ ( n − k + 1) + ψ ( n + 1) − ψ ( n + a ) − ψ ( n + a ) − log(1 − z ))+( − n n ! ∞ (cid:88) k = n +1 ( k − n − a ) k ( a ) k A (2) k (cid:41) (1 − z ) n , (31)where A (2) k = ( b − a ) k ( b − a ) k /k !. Note that this formula is valid only when F is the so-calledzero-balanced series, meaning that the parameters satisfy a + a + a = b + b .Although these formulas are enough to understand the strong deflection limit, the inhomoge-neous Picard-Fuchs equation (23) helps us simplify the expansion about w := 1 − z as we shownow. Observe that Θ takes the following form:Θ( w ) = p ( w ) + q ( w ) log w, (32)where p ( w ) and q ( w ) have power series expansions around w = 0. Since the inhomogeneous termin Eq.(23) is free of logarithmic singularity log w , q ( w ) must be a homogeneous solution which isregular at w = 0. As a result, q ( w ) is proportional to F [1 / , /
6; 1; w ]. By inserting Eqs.(30)and (31) into the expression (28) and picking up ( n = 0)-term, we can identify the proportionalityconstant to conclude q ( w ) = − F (cid:20) ,
56 ; 1; w (cid:21) . (33) p ( w ) is determined by solving the inhomogeneous hypergeometric equation assuming a powerseries expansion around w = 0, but it is more convenient to define ˜ p ( w ) as follows:Θ( w ) = ˜ p ( w ) + 2 π F (cid:20) ,
56 ; 1; 1 − w (cid:21) , (34) − ψ ( n + a ) is missing in Eq.(5.1) in this refference article. hich is equivalent to p ( w ) = ˜ p ( w ) + r ( w ), where r ( w ) = ∞ (cid:88) n =0 (cid:0) (cid:1) n (cid:0) (cid:1) n ( n !) (2 ψ ( n + 1) − ψ ( n + 1 / − ψ ( n + 5 / w n . (35)Then, ˜ p ( w ) is the solution of P ( p ) = √ − w/ √ p ( w ) = (cid:80) ˜ p n w n ,of which coefficients obey the following reccurrence relation: n ˜ p n = (cid:18) n − (cid:19) (cid:18) n − (cid:19) ˜ p n − + 13 √ (cid:0) (cid:1) n − ( n − , n ≥ . (36)The general solution for n ≥ p n = (cid:0) (cid:1) n (cid:0) (cid:1) n ( n !) ˜ p + (cid:0) (cid:1) n − (cid:0) (cid:1) n − ( n !) n − (cid:88) j =0 j ! (cid:0) (cid:1) j (cid:0) (cid:1) j (cid:0) (cid:1) j , (37)where ˜ p is an arbitrary constant. The first term gives a contribution ˜ p F [1 / , /
6; 1; w ]. Sim-ilarly, by using the identity ψ ( x + 1) = ψ ( x ) + 1 /x we can subtract the hypergeometric functionfrom r ( w ) as r ( w ) = (2 ψ (1) − ψ (1 / − ψ (5 / F (cid:20) ,
56 ; 1; w (cid:21) + ∞ (cid:88) n =1 (cid:0) (cid:1) n (cid:0) (cid:1) n ( n !) n (cid:88) j =1 (cid:18) j − j − / − j − / (cid:19) w n . (38)The digamma function at rational numbers can be expressed in terms of elementary functions[11]so that the coefficient of the hypergeometric function is given by2 ψ (1) − ψ (1 / − ψ (5 /
6) = log 432 . (39)Combining above results, we obtainΘ = (cid:18) ˜ p + log 432 w (cid:19) F (cid:20) ,
56 ; 1 , w (cid:21) + ∞ (cid:88) n =1 (cid:0) (cid:1) n (cid:0) (cid:1) n ( n !) n (cid:88) j =1 (cid:32) j − j − / − j − / − √ j − (cid:0) − (cid:1) j (cid:0) (cid:1) j (cid:0) (cid:1) j (cid:33) w n . (40)The remaining task is to determine ˜ p . From the analytic continuation formula for F and F given above, one of the expressions for ˜ p is found to be˜ p = − log 4 −
32 + 964 F (cid:20) , , / , / / , , / (cid:21) . (41)Another expression can be derived by using the method of variation of constant to obtain an integralexpression for Θ in terms of the homogeneous solutions. After some calculations we found˜ p = − π √ (cid:90) dz √ z F (cid:20) ,
56 ; 1; z (cid:21) = − π √ F (cid:20) / , / , / , / (cid:21) . (42)In fact, one can directly expand the original expression of Θ to obtain˜ p = log(7 − √ . (43)The equality of Eqs.(42) and (43) can be proved by using the Watson’s formula[13] with the helpof contiguous relations[14] for F . In [6], a strong deflection limit of ˆ α := Θ − π was consideredand a first few coefficient in b (cid:48) -expansion was obtained, where b (cid:48) = 1 − √ − w . From our resultEq.(40), the expansion given in ref.[6] is completely recovered. Direct evaluation of the integral via analytic continuation
In the previous section, we have obtained the deflection angle by using the inhomogeneous Picard-Fuchs equation. The next natural question is whether we can obtain the result by direct integration.In ref., the integral of the holomorphic forms for planes with boundaries has been performed viaanalytic continuation. We will apply the method. Note that the integral can be written asΘ = (cid:90) C dt (1 − t + Mb t ) , (44)where C is the line starting from 0 and encircling around the root and coming back to 0 (Fig.2).We will use the following representation1(1 − t + Mb t ) = (cid:90) ds Γ( − s )( 2 Mb ) s t s Γ( s + )Γ( ) (1 − t ) − s + − , (45)where s takes the pole of non-negative integers. The original integral have a cut structure at thepole but now it has a cut structure at t = 1. Therefore the integral can be evaluated by two timesthe line integral from 0 to 1. Namely, Figure 2: The contour of t which can be evaluated by 2 times the line integral from t = 0 to t = 1. Θ = 2 (cid:90) ds Γ( − s )( 2 Mb ) s Γ( s + )Γ( ) (cid:90) dtt s (1 − t ) − s + − , (46)We take line integral by using beta integral and findΘ = 2 (cid:90) ds Γ( − s )( 2 Mb ) s Γ( s + )Γ( )Γ( s + 1) cos πs . (47)Taking the pose of s , we get Θ = π ∞ (cid:88) n =0 ( ) n (1) n (1) n ( 2 Mb ) n (48)This reslult coincide with (28). We have discussed the bending angle for Shwartzshild geometry. As we easily expected, the bendingangle glows as the mass M increases with the impact parameter b kept fixed. The next questionwe consider is the effect of the charge for the bending angles. If the spherical object have electriccharge, the spacetime metric is described by the Reissner-Nordstr¨om solution. We can apply ourdirect integration method even for this geometry but it will lead an expansion with two variables.However, such an expression will not be so illuminating. Instead, we apply the method used inSec.3 to the charged case. he Reissner-Nordstr¨om metric is given by ds = − (cid:18) − Mr + Qr (cid:19) dt + (cid:18) − Mr + Qr (cid:19) − dr + r θ + r sin θdφ , (49)where m is the mass of the star and Q is the charge.In this case, photon trajectories are described by1 r (cid:18) drdθ (cid:19) + (cid:18) − Mr + Q r (cid:19) r = 1 b . (50)Introducing dimensionless parameters u = 2 M/r and q = Q/ (2 M ), this equation can be reducedto the following elliptic form: (cid:18) dudθ (cid:19) = − q u + u − u + 4 M b , (51)Before integrating this equation, we have to specify the parameter region of our interest. In orderthat the photon can reach the observer at infinity without crossing the event horizon, the parametersmust obey 4 M b < f ( u − ) , f ( u ) := q u − u + u , (52)where u ± = (3 ± (cid:112) − q ) / (8 q ) are the positions of extrema of f ( u ). In this case, all the roots u i ( i = 1 , , ,
4) of 4 M /b = f ( u ) are real so that we can assume u < < u < u < u . Withthis convention, integration of the differential equation gives the integral expression of the deflectionangle Θ,Θ = 2 (cid:90) u du (cid:112) − q u + u − u + 4 M /b = 2 (cid:90) u du (cid:112) q ( u − u )( u − u )( u − u )( u − u ) . (53)In addition to the parameter M /b , Θ depends on the squared background charge q . Asa result, the Picard-Fuchs equation now becomes a partial differential equation which contains aterm proportional to ∂ q Θ. Interestingly, the coefficient of the additional term is proportional to q ( q − / q = 1 /
4. In this case, the differential equation again turns outto be an inhomogeneous hypergeometric equation, (cid:20) x (1 − x ) d dx + (1 − x ) ddx − (cid:21) Θ = 14 √ x , (54)where the independent variable is differently normalized, x = 16 M /b . The solution satisfying theboundary condition Θ( x = 0) = π is uniquely identified asΘ = π F (cid:20) ,
34 ; 1; x (cid:21) + √ x F (cid:20) / , , / / , / x (cid:21) , (55)or in terms of the original variables,Θ = π F (cid:20) ,
34 ; 1; 16 M b (cid:21) + 4 Mb F (cid:20) / , , / / , / M b (cid:21) . (56)Explicit coefficients up to 4th order terms are given byΘ = π + 4 Mb + 3 πM b + 80 M b + 105 πM b + O (( M/b ) ) , (57)which is consistent with the previous result in ref.[7].As in the case of Schwarzschild spacetime, strong deflection limit of Θ can be derived. We hereshow only the result,Θ = √ F (cid:20) ,
34 ; 1; y (cid:21) log (cid:32) − √ y (cid:33) + ∞ (cid:88) n =1 (cid:0) (cid:1) n (cid:0) (cid:1) n ( n !) n (cid:88) j =1 (cid:40) √ (cid:18) j − j − / − j − / (cid:19) + 43 ( j − (cid:0) (cid:1) j − (cid:0) (cid:1) j − (cid:0) (cid:1) j − (cid:41) y n , (58) here y = 1 − x . In ref.[8], the leading logarithmic divergence of α = Θ − π for generic value of elec-tric charge was numerically studied. There, they assumed the asymptotic form α = − A log( B(cid:15) ) − π ,where A and B are constants depending on q and (cid:15) represents deviation of the closest approachdistance normalized by the Schwarzschild radius x = r / M from the photon sphere x ps , i.e. x = x ps + (cid:15) . In the strong deflection limit, (cid:15) and the variable y are related as y = 2 (cid:15) , which, withthe help of Eq.(58), gives Θ = − √ √ / (cid:15) + o (1) , (cid:15) → . (59)This is completely consistent with the numerical result in ref.[8] (see TABLE I). So far we have derived exact and explicit expressions of the bending angles of photon trajectoriesin Schwarzschild and extremal Reissner-Nordstr¨om spacetimes in terms of the impact parameterby means of solving inhomogeneous Picard-Fuchs equations. Our results are generalizations ofpreviously obtained expansions of the bending angles[6, 7, 8], and are confirmed to be consistentwith them. Both weak and strong deflection expansions for the bending angles now becomesavailable up to an arbitrary order. The method used here can give another analytical tool to studysimilar problems in other geometries such as Reissner-Nordstr¨om spacetime for an arbitrary valueof charge or Kerr-Newman spacetime.By comparing the coefficients of the power series for the deflection angles Eqs.(28) and (56),it can be seen that the bending angle for the Schwarzschild spacetime is larger than that for theextremal Reissner-Nordstr¨om spacetime, for every fixed value of
M/b . This is due to the repulsiveeffect caused by electric charge as shown numerically in ref.[10].It is known that there are several transformation formulas for Appell functions[15, 16, 17, 18]but we could not get an adequate transformation to relate the two expressions (16) and (28). Itis very interesting if we can find an appropriate transformation to obtain the deflection angle notonly for Schwarzschild geometry but also for Kerr-Newman geometry.
Acknowledgement
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