Born-Infeld Theory and Stringy Causality

Abstract

Fluctuations around a non-trivial solution of Born-Infeld theory have a limiting speed given not by the Einstein metric but the Boillat metric. The Boillat metric is S-duality invariant and conformal to the open string metric. It also governs the propagation of scalars and spinors in Born-Infeld theory. We discuss the potential clash between causality determined by the closed string and open string light cones and find that the latter never lie outside the former. Both cones touch along the principal null directions of the background Born-Infeld field. We consider black hole solutions in situations in which the distinction between bulk and brane is not sharp such as space filling branes and find that the location of the event horizon and the thermodynamic properties do not depend on whether one uses the closed or open string metric. Analogous statements hold in the more general context of non-linear electrodynamics or effective quantum-corrected metrics. We show how Born-Infeld action to second order might be obtained from higher-curvature gravity in Kaluza-Klein theory. Finally we point out some intriguing analogies with Einstein-Schr\"odinger theory.

Full text

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arXiv:hep-th/0008052v2 7 Oct 2000
Born-Infeld Theory and Stringy Causality
G W Gibbons∗& C A R Herdeiro†
(D.A.M.T.P.)
Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences,
University of Cambridge,
Wilberforce Road,
Cambridge CB3 0WA,
U.K.
August 2000
DAMTP-2000-71
Abstract
Fluctuations around a non-trivial solution of Born-Infeld theory have a limiting speed given
not by the Einstein metric but the Boillat metric. The Boillat metric is S-duality invariant
and conformal to the open string metric. It also governs the propagation of scalars and spinors
in Born-Infeld theory. We discuss the potential clash between causality determined by the
closed string and open string light cones and find that the latter never lie outside the former.
Both cones touch along the principal null directions of the background Born-Infeld field. We
consider black hole solutions in situations in which the distinction between bulk and brane is
not sharp such as space-filling branes and find that the location of the event horizon and the
thermodynamic properties do not depend on whether one uses the closed or open string metric.
Analogous statements hold in the more general context of non-linear electrodynamics or effective
quantum-corrected metrics. We show how Born-Infeld action to second order might be obtained
from higher-curvature gravity in Kaluza-Klein theory. Finally we point out some intriguing
analogies with Einstein-Schr¨ odinger theory.
04.50.+h; 04.70.-s; 11.25.-w; 11.10.Lm
1Introduction
A striking feature of much recent work on open string states in String/M-Theory is the considerable
insights afforded by the Born-Infeld [1] approximation. Reciprocally, String/M-Theory has provided
a rationale for some of the hitherto mysterious and only partially understood properties of this
remarkable theory.
∗Email address: gwg1@damtp.cam.ac.uk
†Email address: car26@damtp.cam.ac.uk
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The existence of a limiting electric field strength, which was originally the raison d’etre of
Born-Infeld theory, now finds a dynamical justification in the increasingly copious production of
electrically charged open string states as one approaches the critical value [8]. More subtly, the
electric-magnetic duality symmetry of Born-Infeld theory, a non-linear generalization of Hodge
duality first recognized by Schr¨ odinger [3], may be viewed as a special case of S-duality. In fact
electric-magnetic duality is a special case of Born-Reciprocity [10], a transformation which acts as
a rotation in phase space (p,q) → (pcosθ +q sinθ,qcosθ −psinθ). In non-linear electrodynamics,
the phase space variables might be considered to be B and D = ∂L/∂E, which are canonically
conjugate variables in the sense of the Poisson bracket1
?
Bi(x),Dj(y)
?
P.B= −ǫijk∂kδ(x − y).(1.1)
Born reciprocity applied to string theory gives rise to T-duality [34].
Townsend [7] T and S duality are included in the more general U-duality symmetry. This leads nat-
urally to the question of whether Born-Infeld theory is the only non-linear electrodynamic theory
admitting electric-magnetic duality. It is not [9].
Another striking feature of Born-Infeld theory, is that it admits BIon solutions. These are exact
solutions of the full non-linear theory with distributional sources with finite total energy. They can
now be understood in terms of strings ending on D-branes [11, 6].
Schr¨ odinger recognized yet another remarkable property of Born-Infeld theory: viewed as a
non-linear optical theory, Born-Infeld theory exhibits uncommon properties with respect to the
scattering of light by light. He constructed exact wave like solutions of the full non-linear equa-
tions representing light pulses with solitonic properties. They pass through one another without
scattering [4, 5]. This can be also understood from a string theory point of view [11].
Some time after Shr¨ odinger’s work it was realized that the propagation of Born-Infeld fluctua-
tions around a background solution has exceptional causal properties [15, 22] and that the theory
also admits exact solutions exhibiting these exceptional properties [18]. From the String Theory
perspective, this relates to the recent interest in open string theory in a constant background Kalb-
Ramond potential Bµνand thus with gauge theory in a flat non-commutative spacetime [31]. The
Kalb-Ramond potential Bµνappears in the Born-Infeld action in the combination Fµν+Bµνand so
from the Born-Infeld point of view, a constant Bµνfield may be regarded as a background solution
of Born-Infeld theory. Some of the open string states propagating around the constant background
Bµνfield may be identified (at least in the abelian case) with fluctuations of the Born-Infeld theory.
Thus we need to understand the causal structure of their propagation. It turns out that this is
governed by a metric, Gµν = gµν− BµαgαβBβν2, which differs from the usual spacetime metric
gµν [31]. In fact we have two light-cones: the usual light-cone given by gµν which governs the
propagation of closed string states such as the graviton and that given by Gµνwhich governs the
propagation of open string states such as the Born-Infeld photon. In general, the former lies outside
the latter except in two privileged directions corresponding to the two principal null directions or
eigenvectors of the background two-form field. To put things provocatively, gravitons almost always
travel faster than light.
According to Hull and
1We use the convention {xi,pj}P.B. = δi
2Bµν may be taken to stand for the background field or for the constant Kalb-Ramond 2-form. In string language
we are using units in which 2πα′= 1
j.
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One immediate consequence is that if the closed string metric admits no closed timelike curves
then neither will the open string metric. Another obvious consequence is that if the closed string
metric contains an event horizon then the open string metric will also contain an event horizon
which lies inside or on the closed string event horizon.
The comments in the last two paragraphs encode the global theme of this paper. Within many
physical theories, two non-conformal metric structures arise,
gµν
andgµν+ Sµν,(1.2)
where Sµν is some symmetric two-tensor. Geometrically they represent two sets of light-cones.
Questions regarding causality or the existence of event horizons might then become particularly
subtle. It is our purpose to discuss such issues in several contexts. For the aforementioned open ver-
sus closed string theory causal structures, there is some advantage in placing these properties in the
general context of non-linear electrodynamic theories, just as in the case of electric-magnetic duality.
Particularly so because a quite separate strand of recent research has been concerned with analogues
of black holes, closed timelike curves and circular null-geodesics in non-linear electrodynamics [37].
There is also considerable interest in black holes in theories of non-linear electrodynamics coupled
to Einstein gravity (some recent references are [38]).
In section 2 we look at general non-linear electrodynamics in a four-dimensional flat background.
Hence gµνis the Minkowski metric. The study of the propagation of fluctuations of the electromag-
netic field in a given electromagnetic background introduces naturally a second metric structure:
the Boillat metric ABoillat
µν
. We emphasize the special properties of Born-Infeld theory in at least
four senses: the existence of both electric-magnetic and Legendre duality and the absence of both
bi-refringence and shocks.
In section 3, we specialize to the case when the electrodynamics theory is Born-Infeld. One
can deal with a general curved background. So the natural background geometry is described by
gµν, the closed string metric. But open strings propagating in a non-trivial electromagnetic field or
Kalb-Ramond potential see a different metric: the open string metric Gµν, conformal to ABoillat
We shall then see that if the closed string metric is static and the Born-Infeld field is pure electric
or pure magnetic then the open string metric cannot have a non-singular event horizon distinct
from the one given by the closed string metric, because on it the electric field E must either equal
its limiting value or the magnetic field B must diverge. Note that the metric Gµνis not invariant,
even up to a conformal factor, under Hodge duality δBµν= ⋆Bµνbut, as we shall see, it is invariant
up to a conformal factor under electric-magnetic duality rotations.
In section 4 we show that scalar and spinor fluctuations around a Born-Infeld background are
governed by the open string metric.
In section 5 the bi-metric theme takes another perspective. Recently [36] examples have been
given of how dimensional reduction can alter the causal structure of stringy black holes. Consid-
ering a trivial dilaton field, the relation between the lower dimensional metric gµνand the higher
dimensional one ˆ gµνis of the form (1.2), with ˆ gµν= gµν+AµAν. With this motivation, we look to
higher-order Kaluza-Klein theory. We notice that it is possible to obtain Born-Infeld theory to sec-
ond order and still avoid ghosts, as long as the higher dimensional graviton is only excited along the
compact dimensions. We similarly show that the effective theory for QED, the Euler-Heisenberg
theory, may be obtained in this fashion, and discuss some properties of the theory obtained by
µν
.
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starting with an Einstein-Hilbert plus Gauss-Bonnet action in higher dimensions [27]. This gives
an application of the general concepts discussed in section 2.
In section 6 we start by reviewing the results of section 2 considering the gravitational effects
of the electromagnetic background field. One is then led to consider besides the usual Einstein
metric gµνan effective co-metric of the form gµν+ARµν. Another possible origin for such effective
metric is quantum renormalization of the propagator of test fields in a fixed background. We then
discuss the universality of black holes event horizons and thermodynamic properties, by applying
a result derived in the context of quantum renormalized metrics to the case of the Boillat metric
for non-linear electrodynamics coupled to gravity.
In section 7, we review an old attempt of Einstein and Schr¨ odinger to construct a unified theory
of gravity and electromagnetism (see [43] for original references). One then introduces a metric
which has an antisymmetric part. The symmetric part gµνand the inverse of the symmetric part
of the inverse of the full metric, AEins−Schro
µν
have remarkable similarities with the closed and open
string metric, as first noticed by Boillat [14]. We discuss some exact solutions found by Papapetrou
[30].
We close with a discussion.
2 Causality in Non-Linear Electrodynamics
2.1Characteristics and effective geometry
We consider a general Lagrangian L = L(x,y) depending on the Lorentz invariants x =1
and y =
These are the only independent Lorentz invariants in four spacetime
dimensions. The energy momentum tensor is given by
4FµνFµν
1
4Fµν⋆ Fµν.3
Tµν= −LxTMaxwell
µν
+1
4Tgµν, (2.1)
where the trace and the Maxwell energy-momentum tensor are given by
T ≡ Tµ
µ= −4(L − xLx− yLy),TMaxwell
µν
= −FµαFνβgαβ+ xgµν.(2.2)
Since both TMaxwell
tion, and the set of energy momentum tensors satisfying the dominant energy condition is a convex
cone, a sufficient requirement for Tµνto satisfy the dominant energy condition is that Lx< 0 and
T ≥ 0. An argument of Hawking and Ellis [2] then shows that propagation in the full non-linear
theory is causal in the sense that if at time zero all fields vanish outside some compact set, then
they will vanish outside the future of that set. In general one expects the fields to advance into
empty space with no background field at the speed of light and this expectation is supported by the
observation (originally due to Schr¨ odinger [4]) that any solution of Maxwell’s linear electrodynamics
with vanishing invariants, x = y = 0, will also be an exact solution of non-linear electrodynamics.
Among such so-called self-conjugate solutions are the usual plane wave solutions which have unit
speed.
µν
and gµν(with mainly minus signature) satisfy the dominant energy condi-
3We use a mainly minus metric signature and, contrary to Boillat and some other references who use the opposite
sign we choose L to have the standard sign such that for Maxwell theory L = −x. Subscripts indicate partial
differentiation.
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If a background field is present however these arguments require re-examination. One approach
might be to look at the energy momentum tensor of the fluctuations. We shall not do this here
but begin by considering the characteristics, which by definition are hypersurfaces along which
weak discontinuities propagate. Assuming Fµνto be discontinuous across the S(xµ) = constant the
characteristics are given by [13, 15, 22, 24]
(Tµν
Maxwell+ µgµν)∂µS∂νS = 0. (2.3)
This has the form of a relativistic Hamilton-Jacobi equation for massless particles with effective co-
metric Tµν
the propagation of weak, but not necessarily discontinuous fluctuations around a background. Later
we will turn to the propagation of shocks and the behavior of fully non-linear fluctuations. The
function µ = µ(x,y) satisfies
̟µ2+ µ + ω − ̟(x2+ y2) = 0,
where,
LxxLyy− L2
Lx(Lxx+ Lyy),
Maxwell+µgµν, and where S would be the action function. This effective metric also governs
(2.4)
̟ =
xy
(2.5)
and
ω =Lx+ x(Lxx− Lyy) + 2yLxy
Lxx+ Lyy
.(2.6)
In the general case the characteristics exhibits bi-refrigence: µ(x,y), which for convenience
we parameterize as µ(x,y) = x + ζ±(x,y), can take two values, depending upon the polarization
state and the background field. Thus there are, in general, two metrics. The interpretation of
the quantities ζ±is that they correspond to critical electric field strengths above which the theory
breaks down. For exceptional theories the two values of ζ±coincide and there is a single light-cone
and no bi-refringence. Exceptional theories fall into two classes. The first has ̟ = 0. This happens
for instance if the Lagrangian L is independent of y, which includes Maxwell’s theory as a special
case. But not all theories with ̟ = 0 are exceptional in this sense. In fact, although (2.4) still
encodes relevant information for this case, it does not contain all the information anymore. One
example will be given in section 5.
For ̟ ?= 0, the only exceptional theory is Born-Infeld [15]. The latter is also very special in
that ζ±is a constant independent of x and y. It is the only theory for which this is true. We shall
use units in which this constant is taken to be one.
The condition that the theory admit electric-magnetic duality rotations is rather weaker. It
suffices that B · E = D · H [9], which implies that the Lagrangian satisfy the first order Hamilton-
Jacobi type equation
y(L2
y) − 2xLxLy= y.
The characteristics or wave surfaces may be thought of as null hypersurfaces of a metric whose
null geodesics correspond to the rays. Note that the characteristics and the rays depend only a con-
formal equivalence class of metrics, defined by (2.3). A particular choice of conformal representative
used by Boillat, which we shall refer to as the Boillat metric and co-metric, is given by
x− L2
(2.7)
ABoillat
µν
=
1
?µ2− x2− y2(µgµν− TMaxwell
5
µν
)(2.8)

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Cµν
Boillat=
1
?µ2− x2− y2(µgµν+ Tµν
µ. As we shall see in detail later, in the case of Born-Infeld theory, the
open-string metric Gµνand the Boillat metric ABoillat
µν
Because
ABoillat
µν
=
?µ2− x2− y2(gµν−
the Boillat metric has a remarkable expression as a sort of square root:
Maxwell),(2.9)
so that4ABoillat
µα
Cαν
Boillat= δν
are conformal.
µ − x
1
µ − xFµαgαβFβν), (2.10)
ABoillat=
µ − x
?µ2− x2− y2(g +
?
1
√µ − xF)g−1(g −
1
√µ − xF).(2.11)
It follows easily that
−detABoillat
µν
=?−detgµν,(2.12)
in other words, the Boillat metric and the spacetime metric induce the same volume element.
The two principal null vectors common to both cones are annihilated by g + F/√µ − x or by
g − F/√µ − x .
We record for later use that if the background Einstein metric g is flat, then up to the conformal
factor 1/?µ2− x2− y2the Boillat metric is
(µ − x)(dt2− dx2) − E2dt2+ (E · dx)2± 2E × B · dxdt − B2dx2+ (B · dx)2. (2.13)
In the generic case, one may diagonalise the Boillat metric with respect to the usual spacetime
metric gµν. This gives the speeds of propagation of the fluctuations in the associated inertial frame.
In this frame the Poynting vector E × H = −LxE × B vanishes. The velocities, i.e. the ratio of
spacelike to timelike eigenvalues, turn out to be
(1,µ −
µ +
?x2+ y2
?x2+ y2,µ −
?x2+ y2
µ +
?x2+ y2). (2.14)
Thus in general there are two directions in which the Boillat-cone touches the usual Einstein light-
cone, corresponding to the first component of (2.14). These are the principal null directions of Fµν.
Note that Fµν and any duality rotation of it have the same principal null directions. In Figure
1, we represent the light cones for the effective plus Einstein geometry. The left cones illustrate
the case with bi-refringence; we then have the Einstein plus two effective geometry cones. For the
causal case, the Einstein cone will be C1. All the cones touch in two points, along the principal
null directions of Fµν. The cones in the centre of Figure 1 illustrate the exceptional case (like the
Born-Infeld or open string theory case) where the effective geometry only possesses one light cone.
It may happen that the two principal null directions coincide. This occurs if and only if x =
0 = y. In this case the metric takes the form,
ABoillat
µν
= gµν− lµlν,(2.15)
4Throughout this paper all indices will be raised or lowered using the usual Einstein or closed string metric gµν
with the exception of the open string metric Gµν whose inverse is denoted by Gµνin accordance with string theory
conventions.
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C3
C2
C1
BI-REFRINGENCE
C1
C2
PRINCIPAL NULL
DIRECTIONS
PRINCIPAL NULL
DIRECTIONS
EXCEPTIONAL CASE
DEGENERATE PRINCIPAL
DIRECTIONNULL
DEGENERATE CASE
C1
C2
EINSTEIN PLUS EFFECTIVE GEOMETRIES
Figure 1: Cones for the Einstein geometry and effective geometry describing the propagation of
fluctuations in a non-trivial background Fµνfield. If condition (2.16) is obeyed, C1 is the Einstein
cone, C2 (and C3 for the non-exceptional case) the effective geometry cones. The cone on the right
represents the exceptional degenerate case.
where lµis parallel to the principal null direction. The characteristic cone touches the Einstein cone
along a single generator. This degenerate case is illustrated for exceptional theories in Figure 1
right. For a generic electromagnetic field the principal null directions will coincide on a submanifold
of dimension (and also co-dimension) two, N. The complement M \ N in the spacetime manifold
M, may not be simply connected. This gives rise to ambiguities in defining the ‘complexion’
1
2arctany/x of the electromagnetic field. In many ways, particularly if it is timelike, N behaves
rather like a cosmic string [33].
In string theory, if a dilaton Φ is present, one distinguishes between the Einstein metric gµν
and the (closed)-string metric e−2Φgµν. However both have the same (Einstein) light-cone, i.e.,
they are conformal. This is because the dilaton is a state of the closed string. It seems therefore,
at least at the level of approximation we are considering, that there are just two causal structures
and two sets of cones: the open and the closed. Of course from the strict string theory point of
view one refers brane and the other to bulk propagation but we have in mind situations where the
distinction is not sharp, such as for example in the case of space-filling branes, or when considering
gravitons confined to, or at least moving parallel to, the surface of a brane.
A sufficient condition that the Boillat-cone does not lie outside the usual Einstein light-cone,
i.e. that the speeds never exceed unity is that both µ’s must satisfy
µ >
?
x2+ y2≡ r.(2.16)
In terms of the coefficients in (2.4) this requirement reads ω < −r, −1/(2r) < ̟ < 0. Specialized
to the Born-Infeld case (2.16) yields positive the quantity under the square root in the Born-Infeld
action.
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2.2Wave surfaces and Ether drift
Boillat [15] has calculated the wave-front produced by waves moving outwards from a point source
with respect to an inertial frame in which the Poynting vector E×H = −LxE×B does not vanish.
If
w =1
2(E2+ B2),
µ + w
a =
?
µ +
?x2+ y2
,b =
?
µ −
?x2+ y2
µ + w
,(2.17)
and c = ab, so that 1 > a ≥ b > c, he finds that it is given by the family of ellipsoids (in cartesian
coordinates (x,y,z))
x2
a2+y2
where the drift velocity is given by
vdrift=E × B
b2+(z − tvdrift)2
c2
= t2,(2.18)
µ + w.
(2.19)
Since v2
the background electromagnetic field causes the drift of the origin of disturbances and establishes
preferred directions in spacetime; in a sense plays the role of ‘ether’.
drift= (1 − a2)(1 − b2), the drift velocity is always less than one. Therefore, the presence of
2.3 Convexity of the Hamiltonian Function
The energy density or Hamiltonian density T00should be considered as a function H(D,B) of the
canonically conjugate variables (D,B) (in the sense of (1.1)). Their time evolution is obtained by
taking the curl of (H,−E) where the constitutive relation (E,H) = (∂H/∂D,∂H/∂B) holds. In
other words (E,H) and (D,B) are related by a Legendre transformation and in this sense one may
regard the variables (E,H) as canonically conjugate to the variables (D,B). Of course this is a
different sense of canonically conjugate than that in which B and D are canonically conjugate. It
is a covariant sense in which one thinks of the space of Faraday tensors Fµν(possibly subject to the
closure constraint ∂[µFντ]= 0) as the covariant configuration space rather than the non-covariant
configuration space of magnetic induction fields B subject to the constraint divB = 0.
The Legendre transformation will be well defined and invertible if and only if the Hamiltonian
density H(B,D) is a convex function of it’s arguments. In other words the matrix of second
derivatives or Hessian is positive definite. Note that in general H may be defined only in a portion
of the six-dimensional space of possible D and B ’s and the Legendre transform may only map into
part of six-dimensional space of possible E and H ’s. Thus for example, in the case of Born-Infeld
theory
H =
?
which is, in fact, defined for all B and D. However the inverse Legendre transformation is effected
by means of the function
1 −
which is defined only over the domain of (E,H) given by
(1 + B2)(1 + D2) − (B · D)2− 1, (2.20)
?
(1 − H2)(1 − E2) − (E · H)2.(2.21)
E2+ H2< 1 + (E × H)2.(2.22)
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B
E
L
GIBBS
SURFACE
SLOPE = D
CONTACT POINT DEFINING E
Figure 2: The Gibbs surface is the Lagrangian function. Inverting the constitutive relations, to
find E = E(B,D), corresponds to finding the E coordinate of the contact point of the Gibbs
surface with a plane with slope D along a line of constant B. Convexity is necessary for the inverse
constitutive relations to be well defined.
Born-Infeld theory is one with the same constant upper bound for both the electric (at zero magnetic
field) and magnetic field strengths (at zero electric field).
Of course it should be borne in mind that singling out a particular pair of variables is rather
artificial. The underlying invariant geometric structure is the 12-dimensional symplectic vector
space with symplectic form dB · ∧dH + dD · ∧dE and a Lagrangian submanifold which defines
the constitutive relations. If one wishes one may pass to a 13 dimensional contact manifold with
contact form dL−D·dE+H·dE. Then the constitutive relation provides a Legendre submanifold,
which of course on projection onto the L coordinate gives back the Lagrangian submanifold. One
may instead perform a projection onto any pair of the 12 vector coordinates to obtain a “Gibbs
surface” in a seven-dimensional space. Picking for example the pair (E,B) the Gibbs surface is
given by
L = 1 −
This is defined only in the domain D ⊂ R6connected to the origin for which det(g+F) < 0, that is
E2−B2+(B·E)2< 1. Geometrically for example, to find E as a function of D and B one brings
up a 6-plane parallel to the B axis whose slope is given by D until it touches the Gibbs surface.
The point of contact defines E. If the Gibbs surface is convex there will be only one such contact
point. This is illustrated in figure 2.
Convexity will guarantee that all these projections are well defined over the relevant domain
and that, the surface has no folds for example as it would if the system exhibited some sort of
hysteresis phenomenon. For a general non-linear electrodynamic theory the Hessian will only be
positive definite over some domain in (B,D) space. Outside that domain the constitutive relation
is just that: a relation rather than a function.
The components of the Hessian are just the electric permitivities and magnetic permeabilities.
They govern the behaviour of small disturbances around a background. Thus the background will
be stable as long as the Hessian is positive definite. The equations for small fluctuations will also
?
1 − E2+ B2− (B · E)2.(2.23)
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be hyperbolic as long as the Hessian is positive definite [19].
2.4Shock Waves and Exceptionality
In Maxwell theory, in flat space E3,1, there exist traveling wave solutions of the form
Fµν= f(S)F0
µν,(2.24)
where f is an arbitrary function of it’s argument, S = n·x−vt, and n is a constant unit 3-vector.
For fixed n these represent a train of parallel waves moving with unit speed in a fixed direction.
The arbitrary function f allows us to pick the profile of the wave train arbitrarily. One may even
choose it to be discontinuous. The amplitude of the wave is constant on a family of wave surfaces
S = constant which correspond to a family of spacetime parallel null hyperplanes whose intersection
with any surface of constant time gives a family of parallel 2-planes in E3. Because they move at
the speed of light, wave trains cannot be brought to rest by means of a Lorentz transformation.
In non-linear theories in flat space one may, by analogy, adopt the ansatz
Fµν= F0
µν(f(S)),(2.25)
where F0
µνwill now in general depend on the arbitrary function f and where
S = n · x − v(n,S)t.(2.26)
Now we get a family of hyperplanes S = constant in E3,1but they are no-longer parallel, although
their intersections with any surface of constant time still gives a family of parallel 2-planes in E3.
The wave train therefore moves with in a constant direction but not with constant speed. They
may slow down or speed up in the sense that a hyperplane which passes a given point in space at
a later times may have a smaller or greater speed v(n,S). The hyperplanes will thus in general
intersect (see figure 3). At these locations the ansatz breaks down. Neighbouring hyperplanes will
envelope a caustic hypersurface obtained by eliminating S from the equations
S = n.x − v(S)t, 1 = −v′(S)t,(2.27)
where′indicates differentiation with respect to S. Exceptional waves are those for which
v′(S) = 0.(2.28)
If all waves are exceptional, i.e. if v′= 0 ∀S, then parallel hyperplanes are possible. If v ?= 1 these
can be brought to rest by means of a Lorentz transformation. One then has stationary solutions5
depending upon two arbitrary functions f1(z) and f2(z) of a single spatial coordinate, z say. We
shall give concrete examples in the next section for the Born-Infeld case.
To understand the the physical significance of exceptionality, in the sense of the absence of shock
waves, one should consider non-exceptional theories which do admit shock waves. As theories they
are essentially incomplete. One needs extra physical assumptions to render the evolution beyond
the shock. This typically may come from some underlying more fundamental theory. Thus the
5We say stationary rather than static because the Poynting vector may not vanish
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TIME
T0
SHOCKS
CAUSTIC
SPACE
NO SHOCKS
Figure 3: Family of hyperplanes describing the propagation of wave fronts. If the hyperplanes
intersect (right figure) the theory will be singular. Regularity (left figure) arises for exceptional
theories only, like Born-Infeld, which have no shock formation.
predictions of classical theory admitting shocks, or indeed other singularities, cannot be trusted in
situations where they arise or are about to arise. In this sense such theories “predict their own
demise”, something that is often said of classical general relativity. By contrast a classical theory,
such as Yang-Mills theory, for which the evolution of regular finite energy initial data remains
non-singular for all times [25] is certainly complete as a theory, even though, because of quantum
mechanics one does not trust every classical prediction. To check the reliability of a classical
prediction we must check to see how it might be effected by quantum effects. Generally speaking,
we expect classical Yang-Mills theory to be useful in the weak coupling limit and when dealing with
very massive excitations such as magnetic monopoles.
Classical general relativity is known to admit singularities as a consequence of gravitational
collapse. Only for weak data do we expect non-singular evolution for all time [12]. There exist
fully non-linear non-singular solutions of general relativity depending upon two arbitrary functions
propagating at unit speed. These are the pp-waves. They may be generalized to propagate in
an Anti-de-Sitter background [20]. In some ways AdS is analogous to a background B field. But
pp-waves wave-fronts in AdS are null hypersurfaces of the AdS metric. This is in contrast with Born-
Infeld theory and other non-linear electrodynamical theories, where there are plane wave solutions
traveling in some non-flat background spacetime at a slower speed. Again, by contrast with Born-
Infeld theory, the collision of two pp-waves gives rise to a spacetime singularity [21]. Such waves
definitely cannot pass through one-another. In this respect Born-Infeld theory resembles Yang-
Mills theory more than it does general relativity [25]. One is tempted to speculate that it may be
a complete classical theory. Even if this is so, any of its classical predictions is subject to quantum
correction unless there is some reason, such as supersymmetry, for believing that the quantum
corrections vanish.
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2.5Covariant Legendre Transformation
We introduce here a dual notation via the fields Pµν and Nµν ≡ ⋆Pµν. This notation has the
advantage of making Legendre self-duality of Born-Infeld theory manifest.
In the dual notation, the field equations of any non-linear theory of electrodynamics are
∇µPµν= 0, (2.29)
or, in form language, d ⋆ P = 0, where the field Pµνis defined by
dL = −1
2PµνdFµν.(2.30)
Pµνcoincides with Fµνfor Maxwell’s theory. In general it reads
Pµν= −(LxFµν+ Ly⋆ Fµν).(2.31)
The components of Pµνare just D and H. Using this two-form, the energy-momentum tensor can
be cast in a form identical to TMaxwell
µν
Tµν= −PµαgαβFνβ− gµνL.(2.32)
The formulation of the theory in terms of Pµν is dual to the Fµν formulation in the sense of a
Legendre transformation. In fact if one takes the Legendre transform with respect toˆL by
ˆL = −1
2PµνFµν− L,(2.33)
one has
dˆL = −1
2FµνdPµν, (2.34)
in analogy to (2.30). For the special case of a purely electric configuration in flat space,ˆL is the
ordinary Hamiltonian. Introducing the Hodge dual field Nµν≡ ⋆Pµν, and defining s ≡1
−1
of s and t. Then we have
Fµν=ˆLsPµν+ˆLt⋆ Pµν.
4NµνNµν=
4PµνPµνand t ≡1
4⋆NµνNµν= −1
4Pµν⋆Pµνthen the theory is specified by givingˆL as a function
(2.35)
The energy momentum in tensor in terms of the dual variables follows from (2.33) and (2.35):
Tµν=ˆLsTMaxwell
µν
[P] − gµν(sˆLs+ tˆLt−ˆL),TMaxwell
µν
[P] = −PµαgαβPνβ− sgµν.(2.36)
Sufficient conditions for the dominant energy condition to hold are
ˆLs> 0,sˆLs+ tˆLt−ˆL ≥ 0. (2.37)
In the case of Born-Infeld theory one has t = −y by electric-magnetic duality invariance and
expressing also x in terms of (s,t) one gets
−ˆL = 1 −
?
1 + 2s − t2
⇔
L(Fµν) = −ˆL(Nµν). (2.38)
For Legendre self-dual theories like Born-Infeld, the equations describing propagation of pertur-
bations (2.3), (2.4), will have exactly the same form in terms of the variables (x,y) as they do in
terms of the variables (s,t).
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3Born-Infeld/String theory
3.1Open and Closed string metrics
The open string metric Gµνis usually obtained as follows [31].6One starts with the matrix g + F
whose components are gµν+ Fµν. Then one inverts to obtain a matrix with components
?
1
g + F
?µν= Gµν+ θµν,(3.1)
where Gµνis symmetric and θµνis antisymmetric. Let Gµνbe the inverse of Gµν, i.e. GαµGµβ= δβ
Calculation reveals that
α.
Gµν= (G−1)µν=?(g − F)−1g(g + F)−1?µν,(3.2)
which is conformal to the inverse of (2.11) specialized to the Born-Infeld case. Then one checks
that
Gµν= gµν− FµαgαβFβν.
A slightly more involved calculation shows that
(3.3)
θµν= −
1
1 + 2x − y2(Fµν− y ⋆ Fµν) = −
1
?1 + 2x − y2Pµν, (3.4)
where Pµνis the dual Maxwell field in the sense of (2.30). In verifying (3.4) the following four
dimensional matrix identities are useful (where 1 stands for the identity matrix):
g−1Fg−1⋆ F = −y1,
g−1Fg−1F − g−1⋆ Fg−1⋆ F = −2x1,
(3.5)
(3.6)
and hence
(g−1F − yg−1⋆ F)(g−1F +1
yg−1⋆ F) = −(1 + 2x − y2)1.(3.7)
Comparing (3.3) with (2.8) one sees that the open string metric is equal, up to a conformal factor,
to the Boillat metric governing the propagation of fluctuations around a Born-Infeld background:
Gµν=
?
1 + 2x − y2ABoillat
µν
.(3.8)
Relations (3.8) an (3.4) translate the stringy quantities Gµνand θµνinto pure non-linear electro-
dynamics language, i.e. the metric describing fluctuations around a fixed background and the dual
Maxwell field.
An essential requirement on the causal structure defined by the open string metric is to be
invariant under electric-magnetic duality rotations. To examine this we recall that the stress tensor
of Born-Infeld theory, which is known to be invariant [9], is given by
gµν− Tµν
Born−Infeld=
2
√−detg
∂?−det(g + F)
∂gµν
(3.9)
6We will always use Fµν for the gauge field, but it might represent the Kalb-Ramond potential Bµν.
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But
δ
?
−det(g + F) =1
2
?
−det(g + F)
?
1
g + F
?µνδgµν. (3.10)
Thus
gµν− Tµν
Born−Infeld=
?−det(g + F)
√−detg
Gµν. (3.11)
Since the left hand side of (3.11) is invariant so is the right hand side. Notice that the scalar
?−det(g + F)/√−detg is not invariant but its change merely induces a conformal transformation
hand side of (3.11) coincides with the Boillat co-metric
in Gµνand hence in Gµν, preserving the causal structure. It is worthwhile noticing that the right
gµν− Tµν
Born−Infeld= Cµν
Boillat, (3.12)
which is therefore completely invariant under electric-magnetic duality rotations. It is easily seen
from (2.12) that the determinant of both sides of (3.12) equals detgµν. Thus we get the remarkable
result that
det(δµ
ν− TBorn−Infeldµ
ν) = 1.(3.13)
In the next subsections we illustrate the results above by analyzing the geometries seen by open
strings in several special cases. We use the simplest exact solutions to Born-Infeld theory: plane
wave solutions and spherically symmetric solutions.
3.2Exact plane wave solutions
Boillat [16] found an exact stationary solution to Born-Infeld theory given in terms of two arbitrary
functions f1,2of only one of the cartesian coordinates, say z, with an electric field and a magnetic
induction given by
E = coshαi + (coshαsinhβf1(z) − sinhαf2(z))k,
B = (coshαf2(z) − sinhαsinhβf1(z))i − coshβf1(z)j + sinhαk,
where α,β are arbitrary constants. The magnetic field and electric induction are easily obtained
via the constitutive relations. The two Lorentz-invariants are
(3.14)
− 2x = 1 − f2
1− f2
2,y = f2,(3.15)
so that the Born-Infeld lagrangian equals 1 − |f1|. The Poynting vector P = E × H is given by
2|f1|P =?f2
+?2f1f2sinhβ cosh2α − sinh2α(f2
One might wonder if these stationary solutions may be interpreted as domain wall solutions. That
is can one choose the asymptotic values of the arbitrary functions f1and f2so as to interpolate
between two stable “ground states”? One would then expect to have a static family of domain
walls, that is, a non trivial solution for which the Poynting vector would be zero. However, this is
1coshαsinh2β − 2f1f2sinhαcoshβ?i+
1sinh2β + f2
2+ 1)?j − 2coshαcoshβf1k.
(3.16)
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not allowed by Pzin (3.16): f1would need to be zero for which case D,H blow up. Therefore one
finds no domain walls, just as in Maxwell’s theory.
By performing a Lorentz transformation on (3.14), one gets the general fully non-linear sub-
luminal plane wave solution. It presents no shocks, in accordance to section 2.4., since it propagates
with constant speed. In general we do not expect superposition to hold in non-linear electrody-
namics and therefore such plane waves propagating on top of some background solution should
not solve the equations of motion anymore. However, the plane waves obtained by boosting (3.14)
may be superimposed to a background field and still yield a solution to Born-Infeld, as shown in
[18]. Therein a background magnetic field along the x-axis and electric field along the y-axis are
considered: B = Bi , E = Ej so that E × B = −EBk. If we set
v±=−EB ±√1 − E2+ B2
1 + B2
one checks that plane waves traveling in the z-direction can be superimposed to the background
field. The waves can do so in two polarization states, with the electric and magnetic field given by:
=
1 − E2
EB ±√1 − E2+ B2,(3.17)
• (e,b) = (v±j,−i)f?(z − v±t) for the parallel polarization state,
• (e,b) = (v±i,j)f⊥(z − v±t) for the perpendicular polarization state,
where f?and f⊥are arbitrary functions of their argument. Note that there is a net drift in the z
direction
vdrift=1
2(v++ v−) = −
in agreement with (2.19).This drift effect may be understood as a consequence of Lorentz-
invariance. If B2> E2and one performs a Lorentz boost with velocity u = E/B one may pass to a
frame in which the electric field vanishes and the magnetic field becomes equal to B0=√B2− E2.
Now the velocity v0in the this frame is symmetric with respect to reversing the z-direction and is
given by
EB
1 + B2, (3.18)
v0=
1
?1 + B2
u ± v0
1 ± uv0.
0
.(3.19)
One may check that v±, v0and u satisfy the usual relative velocity addition formula
v±=
(3.20)
If E2> B2one may reduce the magnetic field to zero. The electric field in the de-boosted frame
will be E0=
?1 − E2
(2.13)),
ds2
√E2− B2and v0=
0. In these two case the open string metrics are (using
open= dt2− dx2− (1 + B2
0)(dy2+ dz2)(3.21)
and
ds2
open= (1 − E2
0)(dt2− dy2) − dx2− dz2
(3.22)
In terms of the electric induction D0the latter is
ds2
open=
1
1 + D2
0
(dt2− dy2) − dx2− dz2. (3.23)
which illustrates invariance of the open string metric up to a conformal factor under the discrete
electric-magnetic duality transformation (B,D) → (−D,B). The general metric may be obtained
using a Lorentz transformation.
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