Formalism
Definition
Spacetime configurations
Properties
Spacetimes
black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
Quantum theory
A pp-wave spacetime (for parallel plane waves) is a spacetime, i.e. an exact solution to Einstein's equations for gravity, containing nothing but radiation (gravitational waves and/or electromagnetic waves) with one fixed wave vector.
The Ricci flat pp-wave spacetimes are examples of universal spacetimes.
Every spacetime looks like a pp-wave near a lightlike geodesic. This is due to Penrose 1976 and hence came to be called the Penrose limit (review includes Blau 11).
Notice that Penrose limits of Einstein manifolds are Ricci flat (Philip 2005, Prop. 3.4.1, Philip 2006, Prop. 3, review in Blau 2011, p. 41).
For example, the Penrose limit of both the black M2-brane solution as well as the black M5-brane solution of D=11 supergravity are [BFHP 2002, BMN 2002, §2] the (same) pp-maximally supersymmetric pp-wave spacetime originally found by Kowalski-Glikman 1984 (9), Figueroa-O’Farrill & Papadopoulos 2001 (12,14), discussed below.
The BMN matrix model (BMN 02, Section 5 and Appendix B) is a deformation of the BFSS matrix model by mass-terms which correspond to a deformation of the 11d Minkowski spacetime background to the maximally supersymmetric pp-wave (discussed below).
The maximally supersymmetric pp-wave solution of D=11 supergravity is given as follows [Kowalski-Glikman 1984 (9), Figueroa-O’Farrill & Papadopoulos 2001 (12,14)]:
Consider on the vector space/smooth manifold with its canonical linear basis/coordinate functions and the corresponding light cone gauge basis, defined by
in terms of which the Minkowski metric has components
Now on regarded as a coordinate chart, the pp-wave under consideration has metric tensor of the form
supported by a C-field flux density of the form
for given parameter .
The Cartan connection. A compatible choice of orthonormal coframe field for (1) is [cf. Bandos 2012 (3.4)]
in that
From this one finds that the torsion-free “spin connection” for this co-frame field, characterized by
has as only non-vanishing components [cf. Bandos 2012 (3.5)]:
with which indeed
The curvature and Einstein tensor. From this, the only non-vanishing contribution in
the curvature 2-form is [cf. Bandos 2012 (3.6)]:
the Riemann tensor () is [cf. Bandos 2012 (3.7)]:
the Ricci tensor is
the scalar curvature is
the Einstein tensor is again
The energy-momentum and Einstein equation. On the other hand, the non-vanishing contribution in the energy momentum tensor
Comparison of (5) with (6) shows that the Einstein equation of D=11 supergravity [cf. here]
is indeed satisfied by the pp-wave spacetime (1) with the C-field flux (2).
Extension to super-spacetime. We discuss lifting the maximally supersymmetric 11d pp-wave spacetime to a super spacetime.
For that purpose, enhance the underlying coordinate chart to the supermanifold with odd coordinate functions .
The gravitino super-torsion. The odd co-frame component of the gravitino field strength
in a D=11 supergravity solution is fixed as a function of the flux density as (see this Prop.):
We need to complete the coframe field to a super coframe field and find a superfield extension the spin connection (3) such that this equation (7) holds.
To this end, define [DSV 2002 (2.14), Bandos 2012 (3.17)]
where in the second line we inserted (3) and (2) via (7), and set [DSV 2002 (2.10), Bandos 2012 (3.14)]
(…)
A half BPS pp-wave solution of D=11 supergravity. See at M-wave.
Introducing what came to be known as the Penrose limit for producing pp-wave spacetimes:
Lecture notes
See also:
Detailed mathematical discussion:
José Figueroa-O'Farrill, Patrick Meessen, Simon Philip, Homogeneity and plane-wave limits, Journal of High Energy Physics 2005 JHEP05 (2005) [arXiv:hep-th/0504069, doi:10.1088/1126-6708/2005/05/050]
Simon Philip, Plane-wave limits and homogeneous M-theory backgrounds, PhD thesis, Edinburgh (2005) [pdf, pdf]
Simon Philip, Penrose limits of homogeneous spaces, Journal of Geometry and Physics 56 9 (2006) 1516-1533 [doi:10.1016/j.geomphys.2005.08.002, arXiv:math/0405506]
The maximally supersymmetric pp-wave solution of D=11 supergravity is due to:
Jerzy Kowalski-Glikman, Vacuum states in supersymmetric Kaluza-Klein theory, Physics Letters B 134 3–4 (1984) 194-196 [doi:10.1016/0370-2693(84)90669-5, InSpire:202143]
José Figueroa-O’Farrill, George Papadopoulos, Homogeneous fluxes, branes and a maximally supersymmetric solution of M-theory, JHEP 0108:036 (2001) [arXiv:hep-th/0105308, doi:10.1088/1126-6708/2001/08/036]
Its realization as a Penrose limit of both the black M2-brane solution as well as the black M5-brane solution is due to
Matthias Blau, José Figueroa-O’Farrill, Christopher Hull, George Papadopoulos, Penrose limits and maximal supersymmetry, Class. Quant. Grav. 19 (2002) L87-L95 [arXiv:hep-th/0201081, doi:10.1088/0264-9381/19/10/101]
On the enhancement of pp-wave spacetimes to super spacetimes:
Keshav Dasgupta, Mohammad M. Sheikh-Jabbari, Mark Van Raamsdonk, §2.2 in: Matrix Perturbation Theory For M-theory On a PP-Wave, J. High Energy Physics 2002 JHEP05 (2002) [arXiv:hep-th/0205185, doi:10.1088/1126-6708/2002/05/056]
Igor A. Bandos, p. 7 of: Multiple M0-brane equations in eleven dimensional pp-wave superspace and BMN matrix model, Phys. Rev. D 85 126005 (2012) [arXiv:1202.5501, doi:10.1103/PhysRevD.85.126005]
Dedicated discussion of pp-wave spacetimes as Penrose limits (Inönü-Wigner contractions) of AdSp x S^q spacetimes and of the corresponding limit of AdS-CFT duality:
David Berenstein, Juan Maldacena, Horatiu Nastase, Section 2 of: Strings in flat space and pp waves from Super Yang Mills, JHEP 0204 (2002) 013 (arXiv:hep-th/0202021)
N. Itzhaki, Igor Klebanov, Sunil Mukhi, PP Wave Limit and Enhanced Supersymmetry in Gauge Theories, JHEP 0203 (2002) 048 (arXiv:hep-th/0202153)
Nakwoo Kim, Ari Pankiewicz, Soo-Jong Rey, Stefan Theisen, Superstring on PP-Wave Orbifold from Large-N Quiver Gauge Theory, Eur. Phys. J. C25:327-332, 2002 (arXiv:hep-th/0203080)
E. Floratos, Alex Kehagias, Penrose Limits of Orbifolds and Orientifolds, JHEP 0207 (2002) 031 (arXiv:hep-th/0203134)
E. M. Sahraoui, E. H. Saidi, Metric Building of pp Wave Orbifold Geometries, Phys.Lett. B558 (2003) 221-228 (arXiv:hep-th/0210168)
Review:
Darius Sadri, Mohammad Sheikh-Jabbari, The Plane-Wave/Super Yang-Mills Duality, Rev. Mod. Phys. 76:853, 2004 (arXiv:hep-th/0310119)
Badis Ydri, Section 3.1.10 of: Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory (arXiv:1708.00734), published as: Matrix Models of String Theory, IOP 2018 (ISBN:978-0-7503-1726-9)
See also:
The AdS/CFT dual to the pp-wave Penrose limit of -spacetimes is the BMN limit of super Yang-Mills theory (governed by spin chain-dynamics), due to:
Review:
Rodolfo Russo, Alessandro Tanzini, The Duality between IIB String Theory on PP-wave and SYM: a Status Report, Class. Quant. Grav. 21 (2004) S1265-2196 [arXiv:hep-th/0401155, doi:10.1088/0264-9381/21/10/001]
Jan Plefka, Lectures on the Plane-Wave String/Gauge Theory Duality, Fortsch. Phys. 52 (2004) 264-301 [arXiv:hep-th/0307101, doi:10.1002/prop.200310121]
Further references as of 2010 are listed at:
Review in the context of the AdS/CMT correspondence:
Analogous phenomena claimed for pp-wave limits of and spin chains in the dual D=6 N=(2,0) SCFT:
Florent Baume, Jonathan J. Heckman, Craig Lawrie, 6D SCFTs, 4D SCFTs, Conformal Matter, and Spin Chains, Nuclear Physics B 967 (2021) 115401 [doi:10.1016/j.nuclphysb.2021.115401, arXiv:2007.07262]
Jonathan J. Heckman, Qubit construction in 6D SCFTs, Physics Letters B 811 (2020) 135891 [doi:10.1016/j.physletb.2020.135891, arXiv:2007.08545]
Last revised on July 3, 2024 at 21:29:13. See the history of this page for a list of all contributions to it.