Double Copy Theory - From General Relativity | General Relativity Int.
The amplitudes program in quantum field theory has revealed new and unexpected connections between gauge theories and gravity.
Through the double copy relationship, in
which gravity amplitudes are closely tied to the squares of gauge theory amplitudes, it has
become possible to compute gravity amplitudes which would otherwise be prohibitively
complicated. A natural question is whether a similar relationship holds between exact solutions, in which solutions to general relativity can be generated from solutions to a
gauge theory. Indeed, using the Kerr-Schild transformation, a simple and precise relationship can be drawn between gauge fields and spacetime metrics . It is striking that
many exact solutions to the Einstein equations can be presented as the double copies of
gauge theory solutions, including all four-dimensional black hole spacetimes.
Generically, nonlinear behavior in general relativity makes a boundary-value approach
very difficult . However, in electrodynamics this is the natural way to treat a system.
As highlighted by the double copy prescription, Kerr-Schild spacetimes represent a sector
of general relativity in which the metric can be derived as the solution to a boundary-value
problem, just as in electrodynamics.
In this paper, we make use of the boundary-value character of Kerr-Schild geometries
in identifying a simple necessary condition for Maxwell sources to admit a Kerr-Schild
double copy. The corresponding field strength must admit a null geodesic eigenvector
whose differential reproduces the field strength. That is,
F
ν
µA
µ = χAν
, Fµν = ∂µAν − ∂νAµ, (1.1)
for some scalar function χ. In the case of four-dimensional Minkowski backgrounds, we
extend this condition to provide a succinct parameterization of all four-dimensional black
hole spacetimes. They are double copies of real slices of Maxwell fields sourced by point
charges moving on complex worldlines, or complex Li`enard-Wiechert fields. Different
restrictions on these complex worldlines give the double copies derived in and In higher dimensions or on curved backgrounds, it is not clear whether an analogous
classification of Kerr-Schild black hole spacetimes is possible. However, by interpreting as a necessary condition on the trajectories of particles probing a putative single copy
gauge current, this approach provides us a simple physical test for Kerr-Schild double copy
structure. This test is demonstrated to exclude the five-dimensional black ring spacetime,
furnishing a new proof that it does not admit a double copy presentation in terms of a
Kerr-Schild metric.
In section 2, we review the properties of Kerr-Schild spacetimes and the classical double copy, and present a systematic formulation of the latter. We clarify how the current
source in the gauge theory is related to the stress-energy tensor in gravity. In section 2.3,
we describe the requirements which must be satisfied by a gauge field single copy of a
Kerr-Schild metric. Referring to previous work of Newman, in section 3 we relate these
single copies to real slices of complex Li´enard-Wiechert fields [13], and use this identification to systematically construct all four-dimensional black hole spacetimes on Minkowski
backgrounds. In section 4, we present a test for Kerr-Schild structure in any number of dimensions. For sources which pass this test, we can carry the technique further to derive
their Kerr-Schild coordinates, effectively uplifting solutions of boundary value problems in
electrodynamics to solutions in general relativity. This procedure is outlined in section .
- Kerr-Schild Metrics and the Classical Double Copy
The classical double copy relates solutions in gauge theory to Kerr-Schild spacetimes in
general relativity.
In section 2.1 we review results concerning the Kerr-Schild geometries,
first introduced . In section 2.2, we review the stationary double copy discovered in , and present a generalization. In section 2.3, we derive conditions required for a Maxwell
theory solution to be related to a gravity solution by a double copy, and present the problem
of classifying Kerr-Schild spacetimes using their gauge theory counterparts.
- Kerr-Schild Spacetimes
A Kerr-Schild metric is obtained from a transformation on a fixed background metric gµν
of arbitrary dimension and curvature. Given a null vector field k
µ on this background, we
can make the Kerr-Schild transformation
gµν = gµν + φkµkν , (2.1)
where φ is some scalar function. The spacetime with metric gµν is a Kerr-Schild spacetime
and the null vector employed in the transformation is called the Kerr-Schild vector. For
reasons to be explained shortly, we will consider only geodesic Kerr-Schild vectors.
Schematically we can see the double copy structure of these spacetimes by thinking of
them as perturbations to background spacetimes in which the graviton is a tensor product
of two copies of a null vector field. We will often refer to a Kerr-Schild spacetime as the
full spacetime, in contrast to the background spacetime on which it is defined.
Contracting both sides of (2.1) by k
µk
ν
, we find that gµν k
µk
ν = gµνk
µk
ν
, so k
µ
is also
null with respect to gµν . An important consequence is the truncation of the inverse metric
to first order in the graviton,
g
µν = g
µν − φkµ
k
ν
. (2.2)
This truncation implies that k
µ
is geodesic in the background spacetime if and only if
it is geodesic in the full spacetime. It also allows us to make a simple statement of the
condition for the Kerr-Schild vector to be geodesic. Kerr-Schild transformations change
the Ricci tensor component Rµν k
µk
ν by
Rµνk
µ
k
ν − Rµν k
µ
k
ν = φ(k
µ∇µk
λ
)(k
ν∇ν kλ). (2.3)
Note that we use bars throughout to refer to quantities defined with respect to the background spacetime. Therefore, if
Rµν k
µ
k
ν = Rµνk
µ
k
ν
, (2.4)
then k
ν∇νk
µ must be a null vector. Furthermore, kµk
ν∇νk
µ = 0, so k
ν∇νk
µ
is then both
null and orthogonal to k
µ
. This implies k
ν∇ν k
µ
is proportional to k
µ
, i.e., that k
µ
is geodesic. For example, if both the background and full spacetimes saturate the null energy
condition, then k
µ
is geodesic.
Kerr-Schild spacetimes with geodesic k
µ are most interesting, because of dramatic simplifications to their Ricci tensors. If we introduce a dimensionless perturbation parameter
λ into the Kerr-Schild transformation gµν = gµν + λφkµkν , we see that the truncation of
the inverse metric g
µν = g
µν − λφkµk
ν at first order implies that the Ricci tensor could be
at most fourth order in λ. In fact, the Ricci tensor of a general Kerr-Schild spacetime with
geodesic k
µ
truncates at second order with lowered indices,
Rαβ = Rαβ + λR(1)
αβ + λ
2R
(2)
αβ , (2.5)
R
(1)
αβ =
1
2
∇σ
∇α(φkσ
kβ) + ∇β(φkσ
kα) − ∇
σ
(φkαkβ)
, (2.6)
R
(2)
αβ = φkαk
σR
(1)
σβ . (2.7)
Because of the form of the second-order term, we can raise one index and find
R
α
β = R
α
β − λ
h
φkα
k
σRσβ − g
ασR
(1)
σβ i
, (2.8)
a first-order truncation. Fixing the perturbation parameter to unity, the explicit mixedindex Ricci tensor for Kerr-Schild spacetimes (2.1) with geodesic k
µ
is
R
α
β = R
α
β − φkα
k
σRσβ +
1
2
∇σ
∇
α
(φkσ
kβ) + ∇β(φkσ
k
α
) − ∇
σ
(φkα
kβ)
, (2.9)
where R
α
β = g
αγRγβ and ∇
σ
= g
σλ∇ .
- 2 The Classical Double Copy
It was recently discovered that stationary vacuum Kerr-Schild spacetimes are related to
vacuum solutions of Maxwell’s equations on the background spacetime. This follows
directly from the form of the Ricci tensor given in (2.9). On a flat background the terms
involving Rµν vanish, and so the vacuum Einstein equations give
∇σ
∇
α
(φkσ
kβ) + ∇β(φkσ
k
α
) − ∇
σ
(φkα
kβ)
= 0. (2.10)
With some additional assumptions and gauge choices, we can find the Maxwell equations
among these Einstein equations. We assume the spacetime is stationary and choose the
stationary coordinates, in which ∇0(φkσk
α) = 0. Furthermore, we set k0 = 1 by an
appropriate choice of φ, without changing the overall graviton. It follows that the Einstein
equations with index β = 0 are
∇σ
∇
α
(φkσ
) − ∇
σ
(φkα
)
= 0. (2.11)
We see from this equation that a Kerr-Schild spacetime naturally defines a gauge field
Aµ ≡ φkµ
. Indeed, if ηµν + φkµkν is the metric of a stationary spacetime, the vacuum
Einstein equations imply that Aµ
solves the vacuum Maxwell equations. Note the Maxwell
equations correspond only to Rµ
0 = 0, and the other Einstein equations provide additional
constraints on the Kerr-Schild graviton which are not related to the gauge field A We refer to the gauge field Aµ as the single copy of the metric gµν , or more specifically,
of the graviton φkµkν. An archetypal example of the single copy procedure is the relationship between the Schwarzschild metric and a Coulomb field. In Eddington-Finkelstein
coordinates, the Schwarzschild metric is given by
ds2 = ds2
+
rs
r
(−dt + dr)
2
, (2.12)
where ds2
is the line element of four-dimensional Minkowski space. The metric (2.12) is
manifestly in Kerr-Schild form, with φ =
rs
r
and k
µ = (∂t + ∂r)
µ
. The single copy gauge
field is Aµ = φkµ
, and it satisfies
∂µF
µν = j
ν
, (2.13)
where F
µν = ∂
µAν − ∂
νAµ and
j
µ = −qδ(3)(x)(∂t)
µ
. (2.14)
Note that we have made the replacements M → q, a charge, and κ → g, the gauge coupling,
when writing Aµ
, so that rs =
κM
4π
becomes gq
4π
.
In electrodynamics we think of a gauge field as the consequence of some configuration
of current. This view is less applicable in general relativity, owing to its nonlinear behavior. However, the double copy relationship indicates that for Kerr-Schild spacetimes, it is
instructive to think of a metric as the result of a source. Furthermore, we should think
about how the gravity source is related to its corresponding gauge source. Indeed, the
gauge current (2.14) is related to the source of the Schwarzschild metric,
T
µ
ν = Mδ(3)(x)(∂t)
µ
(dt)ν . (2.15)
If we follow the derivation of the double copy while keeping track of sources, a more general
relationship becomes clear. From (2.9), we find that a stationary Kerr-Schild solution on
flat background with k0 = 1 satisfies
R
µ
0 = −
M
2q
∇σF
σµ
. (2.16)
Using the Einstein and Maxwell equations, this implies
j
µ = −
2q
M
T
µ
0 −
T
D − 2
δ
µ
0
, (2.17)
where T = T
µ
µ. This result is also discussed in [10].
In the stationary case, this completes a web of relationships depicted in Figure 1.
Gravity, with the Einstein equations relating gµν to T
µ
ν, is shown as a layer above Maxwell
theory, which relates Aµ
to j
µ
. The classical double copy connects a Kerr-Schild metric
gµν to Aµ by the prescription Aµ ≡ φkµ
. The sources are connected by (2.17), which
we can use to construct the current j
µ
from the stress-energy tensor T
µ
ν. Note that
in both cases we are constructing elements of the gauge theory using elements of the
gravity theory. Reconstructing a gravity solution from a gauge solution requires imposing
additional constraints, which we explore in later sections.
Dr. Karam Ouharou
____
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