Gravitational duality in General Relativity and Supergravity theories

0.07 MB·English

by  François Dehouck

Gravitational duality in General Relativity and Supergravity theories.
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0 F. Dehouck a
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aService de physique math´ematique et interactions fondamentales.

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Universit´e Libre de Bruxelles, Campus Plaine CP-231, 1050 Bruxelles, Belgium
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We quickly review the current status of gravitational duality in General Relativity. We summarize and com-
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supergravity theories.

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1. Introduction tity. The Riemann tensors are related by:
e
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[ equItaitsioanws,elwlkhneonwnrefdaucctetdhattoGtehnreerealdRiemlaetnisviiotny’ss, Rµ(2ν)ρσ = 2ǫµναβR(1)αβρσ, (2)
1 possess an hidden SL(2,R) symmetry known as and generic SO(2) transformations of those ten-
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Ehler’ssymmetry. Gravitationalduality refersto sors, together with rotations of the stress-energy
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2 anSO(2)subgroupofit. Inthissense,itwasfirst tensors, keep the equations of motion invariant.
0 understood as a symmetry of the space of solu- This symmetry wasmade manifestatthe levelof
4 tions with a Killing vector. The famous example theactionin[1]wheretheexistenceofagenerator
1. is the Schwarzschildsolution, with mass M, that of duality and its associated conserved Noether
0 is mapped under duality to the Taub-NUT solu- charge was unveiled.
1 tion,withNUTchargeN. Surprisingly,thestory This work was generalized in the presence of
1 is not over as it was recently shown in [1] that a cosmological constant in [4]. In this case, the
:
v the Pauli-FierzHamiltonian, describing the four- duality relates the on-shell electric and magnetic
Xi dimensional linearized theory, also possesses this partsoftheWeyltensor. Onerecoverstheresults
symmetry, without any reference to the presence oftheasymptoticallyflatcaseinthelimitΛ→0.
r
a of Killing vectors. However, the duality is still
3. Existence of dual charges
very hypothetical at the non-linear level in the
absence of Killing vectors (See also [2]).
Asmagneticchargesarebydefinitiontopologi-
cal,onecannotproceedthroughtheNoetherpro-
cedure to define them. In [3], we defined ten ad-
2. Gravitational duality for linearized gen-
ditionnal dual charges. These were postulated in
eral relativity
analogy with the analysis of electromagnetic du-
In the linearized theory, gravitational duality ality. DualmomentawererecoveredbyaWitten-
interchanges the equation of motion with the Nester construction in [5] 1. The twenty charges
cyclic identity. The Bianchi identity stays invari- are (8πG=1):
ant. Thisisobviouswhenthe equationsarewrit-
tenas(seeforexample[3]andreferencestherein): Pµ = Z G(01µ)d3xµ, Lµν =2Z G(01[µ)xν]d3xµ,
Gµ(aν) =Tµ(aν), (1) Kµ = Z G(02µ)d3xµ, L˜µν =2Z G(02[µ)xν]d3xµ(,3)
and we also found a way to express them as sur-
where a = 1,2 describes the electric or magnetic
faceintegrals. LookingattheKerr-NUTsolution,
linearized Einstein tensor and we introduced a
magnetic stress-energy tensor in the cyclic iden- 1 Seealso[6]and[7]
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we find P = M,K = N,L = Ma,L˜ = Na Thisanalysisshowsthatsupersymmetryseemsto
0 0 xy xy
when appropriate delta singularities are taken be preserved under duality.
into account. It is clear, however, that a proper In[8],wealsostudiedthisphenomenainN =1
treatment of the Lorentz charges in GR should supergravity where shock pp-waves are half-BPS
include information about non-linearities. In solutions. Under gravitational duality, pp-waves
the common approach, usual (electric) Poincar´e whosemetricsarecharacterizedbytheknowledge
charges are recovered by setting the magnetic of an harmonic function F are sent to supersym-
part of the Weyl tensor to zero. Allowing it to metric (dual) pp-waves characterizedby the con-
be non-zero permits to define the dual momenta jugate harmonic function of F. The dual of the
at first order. However, it is not clear how relax- Aichelburg-Sexl pp-wave was denoted the NUT-
ing this still makes sense at non-linear order. We wave.
hope to report on this in the near future [10]. ForthegaugedN =2supergravitywithΛ<0,
IntheAdSlinearizedcase,theNUTchargecan supersymmetry of the Plebanski-Demianskisolu-
be computed using the Cotton tensor [11]. tionwasstudiedin[9]. Supersymmetricsolutions
withNUTchargewerealsoconsidered. Although
4. Supergravity electromagneticdualityisbrokenandalsoappar-
ently the gravitationalduality, the BPS bound is
InpureN =2supergravity,thechargedTaub-
invariantunderamixedU(1)symmetry(seealso
NUT solution has a BPS bound M2 + N2 =
[11]).
Q2+H2. This solution is supersymmetric as we
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