Fermi Coordinates and Penrose Limits

Abstract

We propose a formulation of the Penrose plane wave limit in terms of null Fermi coordinates. This provides a physically intuitive (Fermi coordinates are direct measures of geodesic distance in space-time) and manifestly covariant description of the expansion around the plane wave metric in terms of components of the curvature tensor of the original metric, and generalises the covariant description of the lowest order Penrose limit metric itself, obtained in hep-th/0312029. We describe in some detail the construction of null Fermi coordinates and the corresponding expansion of the metric, and then study various aspects of the higher order corrections to the Penrose limit. In particular, we observe that in general the first-order corrected metric is such that it admits a light-cone gauge description in string theory. We also establish a formal analogue of the Weyl tensor peeling theorem for the Penrose limit expansion in any dimension, and we give a simple derivation of the leading (quadratic) corrections to the Penrose limit of AdS_5 x S^5. Comment: 25 pages

Full text

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arXiv:hep-th/0603109v1 14 Mar 2006
Fermi Coordinates and Penrose Limits
Matthias Blau, Denis Frank, Sebastian Weiss
Institut de Physique, Universit´ e de Neuchˆ atel
Rue Breguet 1, CH-2000 Neuchˆ atel, Switzerland
Abstract
We propose a formulation of the Penrose plane wave limit in terms of null Fermi coor-
dinates. This provides a physically intuitive (Fermi coordinates are direct measures of
geodesic distance in space-time) and manifestly covariant description of the expansion
around the plane wave metric in terms of components of the curvature tensor of the
original metric, and generalises the covariant description of the lowest order Penrose
limit metric itself, obtained in [1]. We describe in some detail the construction of null
Fermi coordinates and the corresponding expansion of the metric, and then study vari-
ous aspects of the higher order corrections to the Penrose limit. In particular, we observe
that in general the first-order corrected metric is such that it admits a light-cone gauge
description in string theory. We also establish a formal analogue of the Weyl tensor
peeling theorem for the Penrose limit expansion in any dimension, and we give a simple
derivation of the leading (quadratic) corrections to the Penrose limit of AdS5× S5.

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Contents
1
Introduction
2
2
Lightning Review of the Penrose Limit
6
3
Brinkmann Coordinates are Null Fermi Coordinates
8
4
Null Fermi Coordinates: General Construction
10
5
Expansion of the Metric in Null Fermi Coordinates
14
6
Covariant Penrose Limit Expansion via Fermi Coordinates
16
7
Example: AdS5× S5
19
8
A Peeling Theorem for Penrose Limits
21
A Higher Order Terms
23
A.1 Expansion of the Metric in Fermi Coordinates to Quartic Order . . . . . . . . . . . . . .23
A.2 Expansion around the Penrose Limit to O(λ2) . . . . . . . . . . . . . . . . . . . . . . . . . .24
References
24
1Introduction
Following the observations in [2, 3, 4, 5] regarding the maximally supersymmetric type
IIB plane wave background, its relation to the Penrose limit of AdS5×S5, and the corre-
sponding BMN limit on the dual CFT side1, the Penrose plane wave limit construction
[7] has attracted a lot of attention. This construction associates to a Lorentzian space-
time metric gµνand a null-geodesic γ in that space-time a plane wave metric,
(ds2= gµνdxµdxν,γ)
→
d¯ s2
γ= 2dx+dx−+Aab(x+)xaxbdx+2+δabdxadxb, (1.1)
the right hand side being the metric of a plane wave in Brinkmann coordinates, char-
acterised by the wave profile Aab(x+).
The usual definition of the Penrose limit [7, 8, 9] is somewhat round-about and in general
requires a sequence of coordinate transformations (to adapted or Penrose coordinates,
from Rosen to Brinkmann coordinates), scalings (of the metric and the adapted coor-
dinates) and limits.2And even though general arguments about the covariance of the
1see e.g. [6] for a review and further references.
2For sufficiently simple metrics and null geodesics it is of course possible to devise more direct ad
hoc prescriptions for finding a Penrose limit.
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Penrose limit [9] show that there is of course something covariant lurking behind that
prescription, after having gone through this sequence of operations one has probably
pretty much lost track of what sort of information about the original space-time the
Penrose limit plane wave metric actually encodes.
This somewhat unsatisfactory state of affairs was improved upon in [1, 10]. There it
was shown that the wave profile Aab(x+) of the Penrose limit metric can be determined
from the original metric without taking any limits, and has a manifestly covariant
characterisation as the matrix
Aab(x+) = −Ra+b+|γ(x+)
(1.2)
of curvature components (with respect to a suitable frame) of the original metric, re-
stricted to the null geodesic γ. This will be briefly reviewed in section 2.
The aim of the present paper is to extend this to a covariant prescription for the expan-
sion of the original metric around the Penrose limit metric, i.e. to find a formulation of
the Penrose limit which is such that
• to lowest order one directly finds the plane wave metric in Brinkmann coordinates,
with the manifest identification (1.2);
• higher order corrections are also covariantly expressed in terms of the curvature
tensor of the original metric.
We are thus seeking analogues of Brinkmann coordinates, the covariant counterpart of
Rosen coordinates for plane waves, for an arbitrary metric. We will show that this is
provided by Fermi coordinates based on the null geodesic γ. Fermi normal coordinates
for timelike geodesics are well known and are discussed in detail e.g. in [11, 12]. They
are natural coordinates for freely falling observers since, in particular, the corresponding
Christoffel symbols vanish along the entire worldline of the observer (geodesic), thus
embodying the equivalence principle.
In retrospect, the appearance of Fermi coordinates in this context is perhaps not par-
ticularly surprising. Indeed, it has always been clear that, in some suitable sense, the
Penrose limit should be thought of as a truncation of a Taylor expansion of the metric
in directions transverse to the null geodesic, and that the full expansion of the metric
should just be the complete transverse expansion. The natural setting for a covariant
transverse Taylor expansion are Fermi coordinates, and thus what we are claiming is
that the precise way of saying “in some suitable sense” is “in Fermi coordinates”.
In order to motivate this and to understand how to generalise Brinkmann coordinates,
in section 3 we will begin with some elementary considerations, showing that Brinkmann
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coordinates are null Fermi coordinates for plane waves. Discussing plane waves from this
point of view, we will also recover some well known facts about Brinkmann coordinates
from a slightly different perspective.
In section 4 we introduce null Fermi coordinates in general, adapting the construction
of timelike Fermi coordinates in [12] to the null case. These coordinates (xA) = (x+,x¯ a)
consist of the affine parameter x+along the null geodesic γ and geodesic coordinates x¯ a
in the transverse directions. We also introduce the covariant transverse Taylor expansion
of a function, which takes the form
f(x) =
∞
?
n=0
1
n!
?Eµ1
¯ a1...Eµn
¯ an∇µ1...∇µnf?(x+) x¯ a1...x¯ an, (1.3)
where Eµ
transformation from arbitrary adapted coordinates (i.e. coordinates for which the null
geodesic γ agrees with one of the coordinate lines) to Fermi coordinates is nothing
other than the transverse Taylor expansion of the coordinate functions in terms of
Fermi coordinates.
Ais a parallel frame along γ. As an application we show that the coordinate
In section 5, we discuss the covariant expansion of the metric in Fermi coordinates in
terms of components of the Riemann tensor and its covariant derivatives evaluated on
the null geodesic. We explicitly derive the expansion of the metric up to quadratic order
in the transverse coordinates and show that the result is the exact null analogue of the
classical Manasse-Misner result [13] in the timelike case, namely
ds2
=2dx+dx−+ δabdxadxb
?
O(x¯ ax¯bx¯ c)
−
R+¯ a+¯bx¯ ax¯b(dx+)2+4
3R+¯b¯ a¯ cx¯bx¯ c(dx+dx¯ a) +1
3R¯ a¯ c¯b¯dx¯ cx¯d(dx¯ adx¯b)
?
+(1.4)
where (x¯ a) = (x−,xa) and all the curvature components are evaluated on γ.
expansion up to quartic order in the transverse coordinates is given in appendix A.1.
The
In section 6, we show how to implement the Penrose limit in Fermi coordinates. To that
end we first discuss the behaviour of Fermi coordinates under scalings gµν → λ−2gµν
of the metric. Since Fermi coordinates are geodesic coordinates, measuring invariant
geodesic distances, Fermi coordinates will scale non-trivially under scalings of the metric,
and we will see that the characteristic asymmetric scaling of the coordinates that one
performs in whichever way one does the Penrose limit arises completely naturally from
the very definition of Fermi coordinates. Combining this with the expansion of the
metric of section 5, we then obtain the desired covariant expansion of the metric around
its Penrose limit.
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The expansion to O(λ), for which knowledge of the expansion of the metric in Fermi
coordinates to cubic order is required, reads
ds2
=2dx+dx−+ δabdxadxb− Ra+b+xaxb(dx+)2
?
O(λ2) .
+λ
−2R+a+−xax−(dx+)2−4
3R+bacxbxc(dx+dxa) −1
3R+a+b;cxaxbxc(dx+)2?
+(1.5)
where the first line is the Penrose limit metric (1.1). In particular, if the characteristic
covariantly constant null vector ∂/∂x−of (1.1) is such that it remains Killing at first
order it is actually covariantly constant and the first-order corrected metric is that of a
pp-wave which is amenable to a standard light-cone gauge description in string theory
[14]. Moreover, in general the above metric is precisely such that it admits a modified
light-cone gauge in the sense of [15]. The expansion to O(λ2) is given in appendix A.2.
We illustrate the formalism in section 7 by giving a quick derivation of the second order
corrections to the Penrose limit of AdS5× S5. These corrections have been calculated
before in other ways [16, 17], and the point of this example is not so much to advocate
the Fermi coordinate prescription as the method of choice to do such calculations (even
though it is geometrically appealing and transparent in general, and the calculation
happens to be extremely simple and purely algebraic in this particular case). Rather,
the interest is more conceptual and lies in the precise identification of the corrections
that have already been calculated (and subsequently been used in the context of the
BMN correspondence) with particular components of the curvature tensor of AdS5×S5.
In section 8 we return to the general structure of the λ-expansion of the metric. The
leading non-trivial contribution to the metric is the λ0-term Ra+b+(1.2) of the Penrose
limit, and higher order corrections involve other frame components of the Riemann
tensor, each arising with a particular scaling weight λw. In the four-dimensional case
it was shown in [18], using the Newman-Penrose formalism, that the complex Weyl
scalars Ψi, i = 0,...,4 scale as λ4−i. This is formally analogous to the scaling of the
Ψias (1/r)5−iwith the radial distance, the peeling theorem [19] of radiation theory in
general relativity. We will show that the present covariant formulation of the Penrose
limit significantly simplifies the analysis of the peeling property in this context (already
in dimension four) and, using the analysis in [20] of algebraically special tensors and
the (partial) generalised Petrov classification of the Weyl tensor in higher dimensions,
allows us to establish an analogous result in any dimension.
We hope that the covariant null Fermi normal coordinate expansion of the metric devel-
oped here will provide a useful alternative to the standard Riemann normal coordinate
expansion, in particular, but not only, in the context of string theory in plane wave
backgrounds and perturbations around such backgrounds.
5