Scalar Field Probes of Power-Law Space-Time Singularities

Abstract

We analyse the effective potential of the scalar wave equation near generic space-time singularities of power-law type (Szekeres-Iyer metrics) and show that the effective potential exhibits a universal and scale invariant leading x^{-2} inverse square behaviour in the ``tortoise coordinate'' x provided that the metrics satisfy the strict Dominant Energy Condition (DEC). This result parallels that obtained in hep-th/0403252 for probes consisting of families of massless particles (null geodesic deviation, a.k.a. the Penrose Limit). The detailed properties of the scalar wave operator depend sensitively on the numerical coefficient of the x^{-2}-term, and as one application we show that timelike singularities satisfying the DEC are quantum mechanically singular in the sense of the Horowitz-Marolf (essential self-adjointness) criterion. We also comment on some related issues like the near-singularity behaviour of the scalar fields permitted by the Friedrichs extension. Comment: v2: 21 pages, JHEP3.cls, one reference added

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arXiv:hep-th/0602207v2 1 Mar 2006
Preprint typeset in JHEP style - HYPER VERSION
Scalar Field Probes of Power-Law Space-Time
Singularities
Matthias Blau, Denis Frank, Sebastian Weiss
Institut de Physique, Universit´ e de Neuchˆ atel
Rue Breguet 1, CH-2000 Neuchˆ atel, Switzerland
Abstract: We analyse the effective potential of the scalar wave equation near generic
space-time singularities of power-law type (Szekeres-Iyer metrics) and show that the ef-
fective potential exhibits a universal and scale invariant leading inverse square behaviour
∼ x−2in the “tortoise coordinate” x provided that the metrics satisfy the strict Dominant
Energy Condition (DEC). This result parallels that obtained in [1] for probes consisting
of families of massless particles (null geodesic deviation, a.k.a. the Penrose Limit). The
detailed properties of the scalar wave operator depend sensitively on the numerical coeffi-
cient of the x−2-term, and as one application we show that timelike singularities satisfying
the DEC are quantum mechanically singular in the sense of the Horowitz-Marolf (essential
self-adjointness) criterion. We also comment on some related issues like the near-singularity
behaviour of the scalar fields permitted by the Friedrichs extension.

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Contents
1. Introduction1
2. Universality of the Effective Scalar Potential for Power-Law Singulari-
ties
2.1 Geometric Set-Up
2.2The Effective Scalar Potential for Power-Law Singularities
2.3The Significance of the (Strict) Dominant Energy Condition
2.4 Comparison with Massless Point Particle Probes (the Penrose Limit)
2.5Massive Scalar Fields and Geodesic Incompleteness
2.6Non-Minimally Coupled Scalar Fields
4
4
5
8
8
10
11
3.Self-Adjoint Physics of Power-Law Singularities
3.1 Functional Analysis Set-Up
3.2Essential Self-Adjointness and the Horowitz-Marolf Criterion
3.3The Horowitz-Marolf Criterion for Power-Law Singularities
3.4The Significance of the (Strict) Dominant Energy Condition
3.5The Friedrichs Extension and “Hospitable” Singularities
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11
14
15
16
17
References19
1. Introduction
The study of scalar field propagation in non-trivial curved (and possibly singular) back-
grounds is of fundamental importance in a variety of contexts including quantum field
theory in curved backgrounds, cosmology, the stability and quasi-normal mode analysis of
black hole metrics etc.
Typically, this is studied within the context of a particular metric or class of metrics.
For certain purposes, however, only the knowledge of the leading behaviour of the metric
near a horizon or the singularity is required. In that case, one can attempt to work with
a general parametrisation of the metric near that locus and, in this way, ascertain which
features of the results that have been obtained previously for particular metrics are special
features of those metrics or valid more generally.
In particular, practically all explicitly known metrics with singularities are of “power-
law type” [2] in a neighbourhood of the singularity (instead of showing, say, some non-
analytic behaviour). In the spherically symmetric case, the leading behaviour of generic
metrics with such singularities of power-law type is captured by the 2-parameter family
ds2= ηxp(−dx2+ dy2) + xqdΩ2
d
(1.1)
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of Szekeres-Iyer metrics [2, 3, 4]. The singularity, located in these coordinates at x = 0, is
timelike for η = −1 and spacelike for η = +1. This class of metrics thus provides an ideal
laboratory for investigating the behaviour of particles, fields, strings, ...in the vicinity of
a generic singularity of this type.
A first investigation along these lines was performed in [5, 1] in the context of the
Penrose Limit, i.e. of probing a space-time via the geodesic deviation of families of massless
particles. There it was shown that the plane wave Penrose limits,
ds2= gµνdxµdxν→ 2dudv + Aab(u)xaxbdu2+ d? x2,(1.2)
of metrics with singularities of power-law type have a universal u−2-behaviour near the
singularity, Aab(u) ∼ u−2, provided that the near-singularity stress-energy (Einstein) tensor
satisfies the strict dominant energy condition (DEC). This behaviour, which is precisely
such that it renders the plane wave metric scale invariant [6], had previously been observed
in various particular examples and is thus now understood to be a general feature of this
class of singularities.
It is then natural to wonder whether a similar universality result can be established in
other circumstances or for other kinds of probes and if, analogously, some energy condition
plays a role in establishing this. If one considers e.g. the Klein-Gordon equation 2φ = 0 for
scalar fields, it is not difficult to see [7, 8] that under certain conditions the scalar effective
potential Vefffor general metrics with singularities of power-law type displays an inverse
square behaviour, Veff(x) ∼ x−2, near the singularity. This observation was then used in
[8] to study the quasi-normal modes for black holes with generic singularities of this type.
The purpose of this note is to study other aspects and consequences of this universality.
In particular, we will first show that the results obtained in [1], namely the scale invariant
inverse square behaviour of the wave profile Aab(u), as well as a crucial [9, 6] lower bound
on the coefficients, have a precise and rather striking analogue in the case of a scalar field.
Schematically, this analogy can be expressed as
strict DEC⇒
?
Aab(u) → caδabu−2scale invariant (ca≥ −1/4)
Veff(x) → cx−2
scale invariant (c ≥ −1/4)
(1.3)
Once again this shows that this inverse square behaviour, that had been observed before
in various specific examples in a variety of contexts, is a general feature of a large class
of space-time singularities. The precise statements are derived in sections 2.2 and 2.3 and
discussed in section 2.4, while sections 2.5 and 2.6 deal with minor variations of this theme.
We hasten to add that if such an inverse square behaviour were universally true without
any further qualifications then it would probably have to be true on rather trivial (dimen-
sional) grounds alone. What makes the results obtained here and in [1] more interesting
is that a priori in either case a more singular behaviour can and does occur and is only
excluded provided that some further (e.g. positive energy) condition is imposed.
The significance of the x−2-behaviour is that (as anticipated in (1.3)), the correspond-
ing Schr¨ odinger operator −∂2
operator, defines a scale invariant (c is dimensionless) “conformal quantum mechanics”
x+ cx−2, to which we will have reduced the Klein-Gordon
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[10] problem. Thus, here and in [1] we find a rather surprising emergence of scale invari-
ance in the near-singularity limit. One implication of this scale invariance in the plane wave
case, discussed in [11], is that it leads to a Hagedorn-like behaviour of string theory in this
class of backgrounds that is quite distinct from that in plane wave backgrounds with, say,
a constant profile and more akin to that in Minkowski space. It would be interesting to
explore other consequences of this near-singularity scale invariance.
This class of scale invariant models has recently also appeared and been discussed in
various other related settings, most notably in the analysis of the near-horizon (rather
than the near-singularity) properties of black holes, see e.g. [12, 13, 14, 15, 16], where the
emergence of scale invariance can largely be attributed to the near-horizon AdS geometry,
as well as in quantum cosmology [17].
Having reduced the Klein-Gordon operator to the Schr¨ odinger operator −∂2
(after a separation of variables and a unitary transformation), one can then turn to a
more detailed spectroscopy of the Szekeres-Iyer metrics by analysing the properties of this
operator. Indeed, as is well known, the inverse square potential is a critical borderline case
in the sense that the functional analytic properties of this operator depend in a delicate way
on the numerical value of the coefficient c. This value, in turn, depends on the dimension
d (number of transverse dimensions) and the Szekeres-Iyer parameter q (it turns out to be
independent of p, while the corresponding coefficients cain the Penrose limit case typically
depend on (p,q) and d).
As one application, we will analyse the Horowitz-Marolf criterion [18] for general singu-
larities of power-law type. Horowitz and Marolf defined a static space-time to be quantum
mechanically non-singular (with respect to a certain class of test fields) if the evolution of
a probe wave packet is uniquely determined by the initial wave packet (as would be the
case in a globally hyperbolic space-time) without having to specify boundary conditions
at the classical singularity. This criterion can be rephrased as the condition that the (spa-
tial part of the) Klein-Gordon operator be essentially self-adjoint (and thus have a unique
self-adjoint extension).
While such a necessarily only semi-classical analysis is certainly not a substitute for a
full quantum gravitational analysis, it nevertheless has its virtues since one can learn what
kind of problems persist, can arise or can be resolved when passing from test particles to
test fields.
Intuitively one might expect a classical singularity with a sufficiently “positive” (in an
appropriate sense) matter content to remain singular even when probed by non-stringy test
objects other than classical point particles. This line of thought was one of the motivations
for analysing the Horowitz-Marolf criterion in this framework, and we will indeed be able
to show (section 3.4) that
x+ cx−2
metrics with timelike singularities of power-law type satisfying the strict Dom-
inant Energy Condition remain singular when probed with scalar waves.
A second issue we will briefly address is that of the allowed near-singularity behaviour of
the scalar fields for a given self-adjoint extension (section 3.5). A priori, one might perhaps
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expect a sufficiently repulsive singularity to be regular in the Horowitz-Marolf sense simply
because the corresponding unique self-adjoint extension forces the scalar field to be zero
at the singularity, thus in a sense again excluding the singularity from the space-time. It
is also possible, however, and potentially more interesting, to have a self-adjoint extension
with scalar fields that actually probe the singularity in the sense that they are allowed to
take on non-zero values there. We propose to call such singularities “hospitable”, establish
once again a relation, albeit not a strict correlation, with the DEC, and show among other
things that, in a suitable sense, half of the Horowitz-Marolf regular power-law singularities
are hospitable whereas the others are not.
2. Universality of the Effective Scalar Potential for Power-Law Singulari-
ties
2.1 Geometric Set-Up
Even though we will ultimately be interested in the properties of the scalar wave (Klein-
Gordon) equation (2 − m2)φ = 0 in the Szekeres-Iyer metrics (1.1), to set the stage it
will be convenient to begin the discussion in the more general setting of metrics with
a hypersurface orthogonal Killing vector. The general set-up here and in section 3.1 is
modelled on the approach of [19] (with minor adaptations to allow for both timelike and
spacelike singularities).
We begin with the n-dimensional metric
ds2= ηa2dy2+ hijdxidxj
(2.1)
where a and hijare independent of y, ξ = ∂yis a hypersurface orthogonal Killing vector
with norm ξµξµ= ηa2, and thus timelike (spacelike) for η = −1 (η = +1). Correspondingly
we assume that the metric hij induced on the hypersurfaces Σy∼= Σ of constant y is
Riemannian (Lorentzian) for η = −1 (η = +1).
Denoting the covariant derivatives with respect to the metric hij by Di, the wave
operator
1
√−detg∂µ
is easily seen to take the form
2 ≡
?
−detggµν∂ν
(2.2)
2 = a−2(η∂2
y+ aDiaDi) .(2.3)
Thus the massive wave equation (2 − m2)φ = 0 can be written as
∂2
yφ = −Aφ , (2.4)
where A is the operator
A = ηaDiaDi− ηa2m2. (2.5)
Assuming now spherical symmetry, the metric takes the warped product form
ds2= ηa(x)2dy2− ηb(x)2dx2+ c(x)2dΩ2
d
(2.6)
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