Traversable wormholes via a double trace deformation
HJE
wormholes via a double trace deformation
Cambridge
U.S.A.
Princeton
U.S.A.
Ping Gao 1
Daniel Louis Jafferis 1
Aron C. Wall 0
0 School of Natural Sciences, Institute for Advanced Study
1 Center for the Fundamental Laws of Nature, Harvard University
After turning on an interaction that couples the two boundaries of an eternal BTZ black hole, we find a quantum matter stress tensor with negative average null energy, whose gravitational backreaction renders the Einstein-Rosen bridge traversable. Such a traversable wormhole has an interesting interpretation in the context of ER=EPR, which we suggest might be related to quantum teleportation. However, it cannot be used to violate causality. We also discuss the implications for the energy and holographic entropy in the dual CFT description.
Black Holes; Gauge-gravity correspondence
-
Traversable
2
4
5
Discussion
A ´ dU TUU
1 Introduction
3 1-loop stress tensor
Modified bulk two-point function
Holographic energy and entropy
traversable wormholes [
25, 39, 52, 53
]. It states that there must be infinite null geodesics
passing through the wormhole, with tangent vector kµ and affine parameter λ, along which
ˆ +∞
−∞
Tµν kµ kν dλ < 0.
The physical picture is that by Raychaudhuri’s equation for null geodesic congruence, light
rays will defocus only when ANEC is violated. In that case, the light rays that focus in
one end of the wormhole can defocus when going out the other end.
There are reasonable arguments that the ANEC is always obeyed along infinite achronal
geodesics [22, 31, 33, 34, 55].1 This is sufficient to rule out traversable wormholes joining two
otherwise disconnected regions of spacetime [22]. Furthermore, the generalized second law
1A set of points is achronal if no two of the points can be connected by a timelike curve; otherwise it is
chronal.
– 1 –
(GSL) of causal horizons also rules out traversable wormholes connecting two disconnected
(asymptotically flat or AdS) regions, due to the fact that the future horizon of a lightray
crossing through the wormhole has divergent area at very early times, which contradicts
the increase of generalized entropy along the future horizon [57].
For small semiclassical perturbations to a stationary causal horizon, both the GSL and
the ANEC follow from lightfront quantization methods that are valid for free or
superrenormalizable field theories [56]. (There is also evidence that these results extend to more
general field theories [
13, 19, 26, 27, 32
]).
In our configuration, signals from early times on the horizon can intersect it again at
late times, by passing through the directly coupled boundaries. The causal structure of the
manifold is modified as a result, changing the commutation relations along null geodesics
through the wormhole and making them no longer achronal. For the same reason, a causal
horizon extending through the wormhole intersects itself, removing the piece with divergent
area. Hence the above impossibility results do not apply. The negative energy matter in our
configuration is similar to the Casimir effect, since the interaction between the boundaries
implies that the radial direction is effectively a compact circle.
Another problematic aspect of traversable wormholes is that they have the potential to
lead to causal inconsistencies. For example, by applying a boost to one end of a wormhole
one could attempt to create a configuration with closed time-like curves [
39
]. The direct
interaction of the boundaries that we require implies that no such paradoxes may arise (for
a more detailed discussion, see section 4).
The traversable wormhole we find is the first such solution that has been shown to be
embeddable in a UV complete theory of gravity. A phenomenological model of a static
BTZ wormhole that becomes traversable as a result of nonperturbative effects in a 1/c
expansion was proposed in [47] (c being the central charge), however it was not shown that
the metric obeys any field equations. A traversable wormhole solution of five dimensional
Einstein-Gauss-Bonnet gravity was found in [
3, 9, 50
], however that low energy effective
theory appears to lack a UV completion [14]. Another example was found [6] in a theory
with a conformally coupled scalar, in a regime in which the effective Newton’s constant
becomes negative. This suggests that this solution also cannot arise in a UV complete
model. The important fact that the boundary CFT dual of a traversable wormhole must
involve interactions between the two CFTs was noted in [3, 47].
The eternal black hole with two asymptotically AdS regions is the simplest setting to
investigate these questions [36]. We will deform the system by turning on a relevant double
trace deformation [
1
]
δS =
ˆ
dt dd−1x h(t, x)OR(t, x)OL(−t, x),
(1.2)
where O is a scalar operator of dimension less than d/2, dual to a scalar field ϕ. This
connects the boundaries with the same time orientation, since the t coordinate runs in
opposite dir (...truncated)