A heuristic review of Lanczos potential

César Mora and Rubén Sánchez Research Center on Applied Science and Advanced Technology of National Polytechnic Institute, Av. Legaría No. 694. Col. Irrigación, CP 11,500, México City.

E-mail: [email protected]

(Received 20 July 2007; accepted 26 August 2007)

Abstract We present a review of 3-tensor potential Labc proposed by Lanczos for the Weyl conformal curvature tensor. We show the role that plays Lanczos tensor in General Relativity for theoretical physics postgraduate students. In the same way that the electromagnetic vector potential can be used to compute the Maxwell field, the Lanczos potential can also be employed to compute the Weyl curvature tensor of a gravitational field.

Key words: Lanczos potential theory, Weyl-Lanczos equati ons, Lanczos coefficients.

Resumen Realizamos una revisión del potencial 3-tensorial propuesto por Lanczos Labc para el tensor de Weyl de curvatura conforme. Mostramos el papel que juega el tensor de Lanczos en Relatividad General para estudiantes de posgrado de física teórica. En la misma forma que el vector potencial puede ser usado para calcular el campo de Maxwell, el potencial de Lanczos también puede ser empleado para calcular el tensor de curvatura de Weyl de un campo gravitacional.

Palabras clave: Teoría de potenciales de Lanczos, ecuaciones de Weyl-Lanczos, coeficientes de Lanczos.

PACS: 04.20.-q, 04.90.+e, 95.30.Sf

I. INTRODUCTION we show the method for the vacuum space-times; in Sect. V, we show a brief example for Schwarzschild space-time. In the recent years there has been a renewed interest in the Finally, in Sect. VI, we present our conclusions. Appendix 3-tensor potential Labc proposed by Lanczos for the Weyl A.I is devoted to spinor formalism. curvature tensor [1]. However, in the General Relativity and gravitation most popular textbooks, like Misner, Thorne and Wheeler [2], Wald [3], Hawkings et al. [4], II. ALGEBRAIC PROPERTIES OF LANCZOS Weinberg [5], Penrose and Rindler [6], Stephani et al. [7], POTENTIAL etc. there is not any treatment about Lanczos potential theory. Nevertheless, it is important in theoretical and In General Relativity, we often need to work in a given physical aspects because we acknowledge that Einstein space-time, or to derive one in the form of an exact equations can be written in terms of the covariant solution to Einstein’s gravity equations. In 1962 Lanczos derivative of Lanczos Potential in its Jordan form; also, suggested an auxiliary potential [9]; through it and the there is a possibility of the existence of an Aranov- covariant derivative, we can obtain the conformal Bhom’s gravitational quantum equivalent effect to the curvature Cabcd of decomposition of Riemann curvature traditional one [8]. In the educational field, the tensor importance of Lanczos potential is clear because of its RCEG=++, (1) analogy with the electromagnetic 4-vector potential. abcd abcd abcd abcd Our aim in this work is to present an heuristic point where the following abbreviations are followed of view of Lanczos potential that reaffirms the last mentioned analogy between gravity and electromagnetism. This is focused to postgraduate General Relativity students. The paper is organized as follows: in Sect. II, we present some algebraic properties of Lanczos potential; in Sect. III, we mentioned the physical interpretation of Lanczos potential; en Sect. IV,

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An heuristic review of Lanczos potential

1 way that we derive the electromagnetic tensor F from A , EgSgSgSgS≡+−−(), ab a abcd 2 ac bd ad ac ad bc bc ad we can derive the electromagnetic field Fab as:

RR Fab==−WA ab () Aa;; b A b a , (12) Gggggg≡−≡(), (2) abcd 12 ac bd ad bc 12 abcd we can also derive the Weyl curvature tensor from a potential; nevertheless, this could not be a vector potential SR≡−1 RgRR,. ≡a ab ab 4 ab a as in the electromagnetic case, instead it must be a 3-rank The Lanczos potential is studied in relation to a Weyl tensor called the Lanczos potential Labc , with similar candidate, i.e., a 4-rank tensor with the following properties, see equation (3). expression Also, assuming that some space-time M admits an WL=−+LL−L electromagnetic field Fab = - Fba , then this field obeys abcd abc;;;; d abd c cdb a cda b certain rules. For example

++−Lg()ad bc Lg () bc ad Lg () ac bd (3) FFFab;;; c+ bc a+= ca b 0, 2 (13) −+Lg Lrs ( gg − gg), FJcd= c . ()bd ac 3 r; s ac bd ad bc ;d where we define Therefore, the gravitational Lanczos potential Labc rr physically corresponds with the electromagnetic vector LLad≡− a d;; r L a r d . (4) potential A . In a certain point of view, A could be The above equation is equivalent to the dual form a a derived in a covariant way to get the electromagnetic WLabcd =+ab[; c d ] L cd [ a;]b tensor F (as is suggested by (12)). Also, the Lanczos (5) ab ** ** potential could be derived in a covariant way to get the −−LL. ab[;] c d cd[;] a b Weyl gravitational field as it is correspondingly suggested Initially, an arbitrary tensor of third order has 43 free by (12). The source of the electromagnetic equation components, but further we have to impose the following (second equation of (13)) corresponds to (11) Lanczos 40 algebraic symmetries equation with sources.

LL=− , (6) abc bac the four (gauge algebraic conditions of Lanczos) r Lar= 0, (7) IV. METHOD FOR THE VACUUM SPACE- and the dual four conditions TIMES

*L r = 0, (8) ar If we have a space-time with a global Killing vector field or equivalently ξa [11], it is sometimes possible to generate a Lanczos potential with the following method: LLLabc++= bca cab 0, (9) If ξa is a non-null Killing vector that satisfies Killing then, his initially sixty four degrees of freedom are equation reduced to sixteen. Further six differential Lanczos gauge conditions ξξab;;+= ba 0, (14) r and also is hyper surface orthogonal vector (i.e. satisfies Lab; r = 0 . (10) the following mathematical relation) In fact, the Lanczos potential also admits a wave equation [2], the tensorial form of this wave equation has proved to ξξ[;ab c ]= 0. (15) be very useful to find Lanczos potential by some Then, it is possible to take an unit vector ua from the group interesting methods due to Velloso and Novello [3]. The of motions, such that complete form of the tensorial equation for Lanczos ξ 2 a ⎧ 1 potential is a ussaa==>=,0ξξξ ,⎨ . (16) rrr ξ ⎩−1 LRLRLRL+−−2 , abc c abr a bcr b car u (11) rs rs 1 Thus, the Killing equation (14) guarantees that ua is −+−=gRL gRL RL J . ac rbs bc ras2 abc abc expansion-less and shear-free, i.e. uuu()(δaa− δ b−uub)0,= (17) (;)ab m m n n and by means of the hyper surface orthogonallity condition of ξa we have III. PHYSICAL INTERPRETATION OF uu= 0, (18) LANCZOS POTENTIAL [;ab c ] therefore, The role of Lanczos potential L with respect to the abc usauab; = a b, (19) Weyl tensor W is the same that plays the vector abcd where we have defined the first curvature vector of the potential A for the Maxwell tensor F . Then, in the same a ab group of congruence (also called and known as

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César Mora and Rubén Sánchez

b acceleration) to be aa=ua;bu which for all group of we continue the present example employing the null tetrad motions even those not satisfying (18), this is a gradient provided by Kinnersley, which are aa=(ln(1/ξ)),a. Then, a candidate of Lanczos potential is aa1122 lr=Δ( , ,0,0), n =( r , −Δ ,0,0), given by Δ 2r2 Lauauu=−() abc a b b a c Δ≡rMr2 −2, (20) 1 (28) −−sag(), ag a 1 3 abc bac mi= (0,0,1, csc(θ )), 2r which satisfies (6), (7) and (9). Then, verifying the a 1 Lanczos gauge (10) and the (3) condition of the potential, mi=−(0, 0,1, csc(θ )). we can fix completely our candidate as a full-fledged 2r Lanczos potential. We can trace the spinorial analog of Here, the non-vanishing spin coefficients and Weyl scalar (3) to fix our candidate. In the following section, we show components are an example for the Schwarzschild space-time. 1cot(θ )2rM− ρβαμ=−,,, =− = =− r 22r 2r2 (29) MM γ =Ψ=,, V. AN EXAMPLE FOR THE SCHARZSCHILD 22 3 SPACE-TIME. 2rr then, the intrinsic derivatives are

2 Now, we show a brief example of the method [11] using a r ∂∂ Dl=∇= + , the Schwarzschild line element in the coordinates, (t, r, θ, a Δ∂tr ∂ ø) as follows a 1 ∂Δ∂ 2 Dn',=∇=a − 2 ∂∂tr2 22⎛⎞2Mdr 2r ds =−⎜⎟1 dt − (30) ⎝⎠r 2M a 1 ⎛⎞∂∂i 1− (21) δ =∇=m a ⎜⎟ + , r 2r ⎝⎠∂∂θθφsin( ) −−rd22θ r 2sin(θ )22dφ a 1 ⎛⎞∂∂i . δ ',=∇=m − a ⎜⎟ aa 2r ⎝⎠∂∂θ sin(θφ ) The time-like Killing vector ξ = δ has squared norm 0 from these set of relations we have that the only non-

2 ⎛⎞2M vanishing components of Lanczos tensor are ξ ==−g00 ⎜⎟1 , (22) −1 ⎝⎠r 12M ⎛⎞MM LL16=−⎜⎟1, − =− , (31) clearly this vector fields are hypersurface-orthogonal. If 3rr22⎝⎠ 6r we introduce the corresponding unit vector field, where we used the following relations between the null

ξa a tetrad and the Weyl scalars uu==,1,u (24) aaξ abcd abcd Ψ=01Clmlmabcd ,,Ψ= Clmlnabcd

then a b cd ab cd b Ψ=2Cabcd lmmn,,Ψ=3 Cabcd lnmn (32) uauab;;== a b,,with auua ab (25) Ψ=Cmnabmncd, now, with the velocity vector ua and the acceleration aa, 4 abcd it is possible to write a Lanczos potential in the following and Lanczos scalars abc abc form LLlmlLLlmm01==abc ,,abc

1 abc ab c LLmnlLLmnm==,, Lauauuagagabc=−()(), a b b a c − a bc − b ac (26) 23abc abc 3 (33) LLlmmLLlmn==abc,,abc this is the 3-rank tensor that Novello and Velloso [10] 45abc abc prove to be a Lanczos potential. We start to manage the LLmnmLLmnn==ab c,.abc example by using a preferred null tetrad in the Newmann- 67abc abc Penrose convention, say This is an example of a Lanczos potential that has been computed from the Novello and Velloso’s Lanczos {}emmlk= ( , ,, ), aaaaa gravitational potential 3-rank tensor that illustrates the use ⎛⎞01 0 0 of the null tetrad and spinorial coefficients from the ⎜⎟ Newmann- Penrose formalism [10]. 10 0 0 gmmkl=−=22⎜⎟, (27) ab ()a b ()a b ⎜⎟00 0− 1 ⎜⎟ VI. CONCLUSIONS ⎝00−1 0⎠ m mklaa==−1, 1, It has been found that the method of Novello and Velloso aa [10] described in section IV, could be used to compute a Lanczos gravitational potential, and it has been pointed out

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An heuristic review of Lanczos potential

that we can use it in the derivation of the Weyl curvature tensor for the simple case of a Schwarzschild spherically A.I NEWMANN-PENROSE CONVENTION symmetric vacuum space-time. The Lanczos tensor is FOR THE SPIN-COEFFICIENTS important in the derivation of the Einstein field equations in his Jordan form. We have used the spin-coefficients and In this section we show the symbology employed by the Newmann-Penrose formalism in this method to Newmann and Penrose to denote several spin-coefficients achieve our objective of showing an easy and convenient that we have already computed for the vacuum space-time. way to get this important quantity. This notation has been proved to be very useful. The Christoffel symbols of second kind can be written in terms of spinorial coefficients as AKNOWLEDGEMENTS cCC C Γ=γ εγε + , (34) This work was supported by EDI, COFAA-IPN SNI- ab AABB AABC  B CONACyT grants and Research Project SIP-20071482. where we followed the convention of decomposition of the null tetrad in terms of the canonical spinorial frame REFERENCES

aAAaAA [1] Bergqvist, G., A Lanczos potential in Kerr geometry, mo==ιι, m o,  (35) J. Math. Phys. 38, 3142-3154 (1997); Edgar, S. B. and lkaAA==ιι ,, aoo AA Hoglund, a., The Lanczos potential for the Weyl curvature A A tensor: existence, wave equation and algorithms, Proc. the basis spinors o and ι from a dyad in the sense that Roy. Soc. Lond. A 453, 835-851 (1997); López-Bonilla, satisfies the normalized relation of inner product: J. L., Ovando, G. and Peña, J. J., Lanczos potential for ooιιAAB= ε = 1, plane gravitational waves, Found. Phys. Lett. 12, 401-405 AAB

(1999); Cartin, D., The Lanczos potential as a spin-2 and ε AB is the “local metric” of the spin space SL(2,C), field, hep-th/0311185v1 (2003); O’Donnell, P., Lanczos which has the components tensor potential for conformally flat space-times 119, ⎛⎞01 341-345 (2004); Mena, F. C and Tod, P., Lanczos AB . εεAB ==⎜⎟ potentials and a definition of gravitational entropy for ⎝⎠−10 perturbed FLRW space-times, gr-qc/0702057v1 (2007). The spinor indices are raised and lowered according to the [2] Misner, Ch. W., Thorne, K. S. and Wheeler, J. A. rules (Gravitation, Freeman, New York, 1995). α A ==εαAB , α εαB . (36) [3] Wald, Robert M. (General Relativity, The University BAAB of Chicago Press, Chicago and London, 1984). The spin inner product is antisymmetric in the sense that [4] Hawkings, S. W., Ellis, G. F. R. (The Large Scale the spinorial indices A, B,… do not commute, i.e. Structure of Space-Time, Cambridge Universty Press, αεAB≠ ε BA α, Cambridge, 1973). AA (37) AB A A [5] Weinberg, S. (Gravitation and Cosmology: Principles ε ABα βαβαβ==−A A . and Aplications of the General Theory of Relativity, John The diagrams of Newmann-Penrose are the following Wiley, New York, 1972). [6] Penrose, R. and Rindler, W. (Spinors and space-time, TABLE I. Symbology of Newmann-Penrose for the spinor vol. 1 “Two-Spinor Calculus and Relativistic Fields”, C γ . Cambridge University Press, Cambridge, 1984). AAB 0 1 1 1 [7] Stephani, H., Kramer, D., Maccallum, M., C 0 0 0 1 Hoenselaers, C., Herlt, E. (Exact Solutions of Einstein’s AAB \ Field Equations, Cambridge University Press, Cambridge, 00 ε −κ −τ ' γ ' 2006). 10 α −ρ −σ ' β ' [8] Holgersson, D. (Lanczos potentials for perfect fluid  β −ρ ' cosmologies, Linköping University Thesis, 2004); on-line 01 −σ α ' at: http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-2582 11 γ −τ −κ ' ε [9] Lanczos, C., The splitting of the Riemann tensor, Rev. Mod. Phys. 34, 379-389 (1962). TABLE II. Symbology of Newmann-Penrose for the spinor [10] Novello, M. and Velloso, A. L., The connection . γ AABC between general observers and Lanczos potential, Gen. AABC \ 00 1010= 11 Rel. Grav. 19, 1251-1265 (1987).  [11] Dolan, P. and Kim, C. W., Some solutions of the 00 κ ε =−γ ' π =−τ ' Lanczos vacuum wave equation, Proc. R. Soc. Lond. 447, 10 ρ α =−β ' λ=−σ ' 577-585 (1994). 01 σ β =−α ' μ =−ρ ' 11 τ γ =−ε ' ν =−κ '

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César Mora and Rubén Sánchez

There are also two more tables for the complex conjugate Notice that in these tables the second and third rows, of spin-coefficients symbols. C the complex conjugate of spin-coefficient γ AAB and

TABLE III. Symbology of Newmann-Penrose for the spinor γ AABC are interchanged. Because of the obvious relations C . γ AAB  ______C 0 1 1 1  CCC (38) AA \ B 0 0 0 1 γγγAAB== AAB AAB ,  00 ε −κ −τ ' γ ' and correspondingly 01 α −ρ −σ ' β '

 β −σ −ρ ' α ' 10 ______ 11 γ −τ −κ ' ε ' γγγAABC== AABC AABC. (39) (We also can think that the capital latin spinorial indices TABLE IV. Symbology of Newmann-Penrose for the spinor with dot AB, ,... can not see the spinorial indices without γ  . AABC a dot AB, ,... and then, can permute with this last set AABC\  00 0110= 11 without interference. But the capital indices without the 00 ε =−γ ' π =−τ ' κ dot A, B,... can not commute between them, because in 01 ρ α=−β' λ =−σ ' doing so, they could interfere with the value of the  10 σ β =−α ' μ =−ρ ' spinorial quantity.) 11 τ γ =−ε ' ν =−κ '

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