Contents

# Contents

## Idea

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.

## Properties

### As Penrose limits

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 $AdS_4 \times S^7$ as well as the black M5-brane solution $AdS_7 \times S^4$ 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.

### Relation to BMN matrix model

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).

## Examples

### Maximally supersymmetric pp-wave in 11d

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 $\mathbb{R}^{11}$ with its canonical linear basis/coordinate functions $(x^a)_{a=0}^{10}$ and the corresponding light cone gauge basis, defined by

$\begin{array}{lcl} x^- &\coloneqq& (-x^0 + x^{10})/\sqrt{2} \\ x^+ &\coloneqq& \phantom{+}(x^0 + x^{10})/\sqrt{2} \\ x^i &\coloneqq& \;\phantom{+}x^a\; \;\text{for}\; a = i \in \{1,2,3\} \\ x^j &\coloneqq& \;\phantom{+}x^a\; \;\text{for}\; a = j \in \{4,5,6,7,8,9\} \end{array}$

in terms of which the Minkowski metric $\eta_{a b} \coloneqq \mathrm{diag}\big(-1,+1, +1, \cdots, +1\big)_{a b}$ has components

$\begin{array}{lcl} \eta_{+ +} \,=\, \eta_{- -} &=& 0 \\ \eta_{- +} \,=\, \eta_{+ -} &\coloneqq& 1 \\ \eta_{i_1 i_2} &\coloneqq& diag(1,1,1)_{i_1 i_2} \\ \eta_{j_1 j_2} &\coloneqq& diag(1,1,1,1,1,1)_{j_1 j_2} \,. \end{array}$

Now on $\mathbb{R}^{1,10}$ regarded as a coordinate chart, the pp-wave under consideration has metric tensor of the form

(1)$\begin{array}{lcl} \mathrm{d}s^2 &\equiv& 2 \, \mathrm{d}x^- \otimes \mathrm{d}x^+ \,-\, \big( (\mu/3)^2 x^i x_i + (\mu/6)^2 x^j x_j \big) \, \mathrm{d}x^- \otimes \mathrm{d}x^- \\ && \,+\, \mathrm{d}x^i \otimes \mathrm{d}x_i \,+\, \mathrm{d}x^j \otimes \mathrm{d}x_j \,, \end{array} \;\; \text{where} \;\; \begin{array}{l} i \,\in\, \{1,2,3\} \mathrlap{\,,} \\ j \,\in\, \{4,5,6,7,8,9\} \mathrlap{\,,} \end{array}$

supported by a C-field flux density of the form

(2)$G_4 \;\equiv\; \mu \; \mathrm{d}x^- \, \mathrm{d}x^1 \, \mathrm{d}x^2 \, \mathrm{d}x^3 \,,$

for given parameter $\mu \in \mathbb{R}$.

The Cartan connection. A compatible choice of orthonormal coframe field for (1) is [cf. Bandos 2012 (3.4)]

$\begin{array}{lcl} e^+ &\coloneqq& \mathrm{d}x^+ \,-\, \tfrac{1}{2} \big( (\mu/3)^2 \, x^i x_i \,+\, (\mu/6)^2 \, x^j x_j \big) \, \mathrm{d}x^- \\ e^- &\coloneqq& \mathrm{d}x^- \\ e^i &\coloneqq& \mathrm{d}x^i \\ e^j &\coloneqq& \mathrm{d}x^j \end{array}$

in that

$\mathrm{d}s^2 \;=\; 2 \, e^- \otimes e^+ \,+\, e^1 \otimes e^1 \,+\, e^2 \otimes e^2 \,+\, \cdots \,+\, e^{9} \otimes e^9 \,.$

From this one finds that the torsion-free “spin connection$\omega$ for this co-frame field, characterized by

$\mathrm{d} e^a - \omega^a{}_b \, e^b \;=\; 0 \,,$

has as only non-vanishing components [cf. Bandos 2012 (3.5)]:

(3)$\begin{array}{l} \omega^{+ i} \;=\; - \omega^{i +} \;=\; (\mu/3)^2 \, x^i \, e^- \\ \omega^{+ j} \;=\; - \omega^{j +} \;=\; (\mu/6)^2 \, x^j \, e^- \,, \end{array}$

with which indeed

$\begin{array}{ccl} \mathrm{d}e^+ &\equiv& \,-\, (\mu/3)^2 x_i \, \mathrm{d}x^i \otimes e^- \,-\, (\mu/6)^2 x_j \, \mathrm{d}x^j \otimes e^- \\ &=& \;\;\;\;\;\;\;\;\;\;\;\; \omega^+{}_{i} \, \mathrm{e}^i \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \,+\, \;\;\;\; \omega^+{}_{j} \, \mathrm{e}^j \,. \end{array}$

The curvature and Einstein tensor. From this, the only non-vanishing contribution in

• the curvature 2-form $R^{a b} \,\equiv\, \mathrm{d}\omega^{a b} - \omega^a{}_c \omega^{c b}$ is [cf. Bandos 2012 (3.6)]:

$\begin{array}{l} R^{+ i} \,=\, - R^{i +} \;=\; \mathrm{d} \omega^{+ i} \;=\; (\mu/3)^2 \, e^i \, e^- \\ R^{+ j} \,=\, - R^{j +} \;=\; \mathrm{d} \omega^{+ j} \;=\; (\mu/6)^2 \, e^j \, e^- \,, \end{array}$
• the Riemann tensor ($R^{a b} = \tfrac{1}{2}R^{a b}{}_{c_1 c_2} e^{c_1} e^{c_2}$) is [cf. Bandos 2012 (3.7)]:

$\begin{array}{l} R^{+ i_1}\,{}_{i_2 -} \;=\; \delta^{i_1}_{i_2} (\mu/3)^2 \\ R^{+ j_1}\,{}_{j_2 -} \;=\; \delta^{j_1}_{j_2} (\mu/6)^2 \,, \end{array}$
• the Ricci tensor $R_{a b} \coloneqq \eta_{a a'} R^{a' c}{}_{b c}$ is

(4)$Ric_{- -} \;=\; -\mu^2/3 -\mu^2/6 \,=\, -\tfrac{1}{2}\mu^2 \,,$
• the scalar curvature $\mathrm{R} \equiv \eta^{a b}Ric_{a b}$ is

$\mathrm{R} \;=\; 0 \,,$
• the Einstein tensor $G_{a b} = Ric_{a b} - \tfrac{1}{2} \mathrm{R} \eta_{a b}$ is again

(5)$G_{- -} \;=\; -\mu^2/3 -\mu^2/6 \,=\, -\tfrac{1}{2}\mu^2 \,.$

The energy-momentum and Einstein equation. On the other hand, the non-vanishing contribution in the energy momentum tensor

$T_{\mu \nu} \;=\; - \tfrac{1}{12} \big( (G_4)_{\mu \, \mu_1 \cdots \mu_{s-1}} (G_4)_{\nu}{}^{ \mu_1 \cdots \mu_{s-1} } - \tfrac{1}{8} (G_4)_{\mu_1 \cdots \mu_s} (G_4)^{\mu_1 \cdots \mu_s} \, g_{\mu \nu} \big)$

of the C-field (2) is

(6)$\begin{array}{lcl} T_{- -} &=& - \tfrac{1}{12} \mu^2 \epsilon_{i_1 i_2 i_3} \, \epsilon^{i_1 i_2 i_3} \\ &=& - \tfrac{1}{2} \mu^2 \,. \end{array}$

Comparison of (5) with (6) shows that the Einstein equation of D=11 supergravity [cf. here]

$G_{a b} \;=\; T_{a b}$

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 $\mathbb{R}^{11}$ to the supermanifold $\mathbb{R}^{1,10\vert \mathbf{32}}$ with odd coordinate functions $\big(\theta^\alpha\big)_{\alpha = 1}^{32}$.

The gravitino super-torsion. The odd co-frame component of the gravitino field strength

$\rho \;\coloneqq\; \mathrm{d} \psi \,-\, \tfrac{1}{4} \omega^{a b} \Gamma_{a b} \psi$

in a D=11 supergravity solution is fixed as a function of the flux density $(G_4)$ as (see this Prop.):

(7)$\begin{array}{l} \rho \;=\; \tfrac{1}{2} \rho_{a b} \, e^a \, e^b + H_a \psi \, e^b \\ H_a \;\coloneqq\; \tfrac{1}{6} \tfrac{1}{3!} (G_4)_{a\, b_1 b_2 b_3} \Gamma^{b_1 b_2 b_3} - \tfrac{1}{12} \tfrac{1}{4!} (G_4)^{b_1 \cdots b_4} \Gamma_{a\, b_1 \cdots b_4} \,. \end{array}$

We need to complete the coframe field $e$ to a super coframe field $(e,\psi)$ and find a superfield extension the spin connection $\omega$ (3) such that this equation (7) holds.

To this end, define [DSV 2002 (2.14), Bandos 2012 (3.17)]

$\begin{array}{l} \mathrm{D} \theta^\alpha \;\coloneqq\; \mathrm{d}\theta^\alpha - \tfrac{1}{4} \omega^{a b} (\Gamma_{a b} \theta)^\alpha \pm H_a \theta \, e^a \\ \;=\; \mathrm{d}\theta^\alpha \,-\, \tfrac{1}{2} (\mu/3)^2 x^i (\Gamma_{+ i} \theta)^\alpha \, e^- \,-\, \tfrac{1}{2} (\mu/6)^2 x^j (\Gamma_{+ j} \theta)^\alpha \, e^- \,-\, \big( \tfrac{1}{6} \tfrac{1}{3!} (G_4)_{a\, b_1 b_2 b_3} \Gamma^{b_1 b_2 b_3} - \tfrac{1}{12} \tfrac{1}{4!} (G_4)^{b_1 \cdots b_4} \Gamma_{a\, b_1 \cdots b_4} \big) \theta\, e^a \,, \end{array}$

where in the second line we inserted (3) and (2) via (7), and set [DSV 2002 (2.10), Bandos 2012 (3.14)]

$\begin{array}{l} \psi \;\coloneqq\; D \theta + \cdots \,. \end{array}$

(…)

### The M-wave in 11d

A half BPS pp-wave solution of D=11 supergravity. See at M-wave.

## References

### General

Introducing what came to be known as the Penrose limit for producing pp-wave spacetimes:

Lecture notes

Detailed mathematical discussion:

### Examples

The maximally supersymmetric pp-wave solution of D=11 supergravity is due to:

Its realization as a Penrose limit of both the black M2-brane solution $AdS_4 \times S^7$ as well as the black M5-brane solution $AdS_7 \times S^4$ is due to

### pp-Wave Super spacetimes

On the enhancement of pp-wave spacetimes to super spacetimes:

### pp-Waves as Penrose limits of $AdS_p \times S^q$ spacetimes

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:

Review:

• Michael Gutperle, Nicholas Klein, A Penrose limit for type IIB $AdS_6$ solutions (arXiv:2105.10824)

### BMN limit: AdS/CFT in the pp-wave limit

The AdS/CFT dual to the pp-wave Penrose limit of $AdS_5 \times S^5$-spacetimes is the BMN limit of super Yang-Mills theory (governed by spin chain-dynamics), due to:

Review:

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 $AdS_7 \times S^4/\Gamma_{ADE}$ and spin chains in the dual D=6 N=(2,0) SCFT:

Last revised on July 3, 2024 at 21:29:13. See the history of this page for a list of all contributions to it.