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Thermal Duality: Nemesis of
Black Holes?
Mike Hewitt
Canterbury Christ Church University
What is the thermalon?
• Condensate that forms at the Hagedorn transition.
• Euclidean single wrapping mode – related to the thermal
path integral for a single string by modular invariance.
• The thermalon mode will be interpreted as a deformation
of the vacuum, in thermal equilibrium with normal
vacuum.
• The thermalon is not a normal propagating state – rather,
a macroscopic order parameter for mixing long string
states with the field vacuum, entangling the external field
state with long strings.
Thermalon and gravity
• Why is the thermalon relevant to gravity?
• Heterotic strings: for Left moving currents and their
interaction vertex, at the dual point:
(thermalon, graviton) is equivalent to (W, 𝛾).
Gravito-thermal interaction
• The thermalon is part of the gravity sector in string theory.
• Gravity becomes part of a ‘gravito-thermal’ interaction.
• The condensate that forms at the Hagedorn transition behaves
as part of the gravity sector.
• Thermal duality implies a possibility of the thermalon
producing gravitational vacuum polarization.
• This may limit the strength of gravity, preventing the
formation of black holes – a novel macroscopic effect.
Why is this significant?
• String theory does not work in 4 dimensions without a gravity
regulator – more below.
• With a regulator, it may become like QCD, with
gravitationally free strings confined to a bag of a different
phase.
• The heterotic gravity regulator is generic in dimension, and
avoids black hole formation and related information
paradoxes.
• The heterotic regulator may generalise to other string models.
Scaling behaviour vs dimension
Holographic principle
Problem of strings in 4 dimensions
• An attractive feature of string theory is that it naturally
includes gravitons and has finite amplitudes for graviton
scattering in the critical dimension.
• High energy scattering is intermediated by an infinite
spectrum of free string states.
• However, string theory as a theory of quantum gravity
does not really work as advertised in 4 dimensions, as
high energy collisions would form black holes rather than
demonstrate string scattering amplitudes or the production
of long string states.
Free string spectrum
…operators
Average
2
𝐽
String radial scale
• It appears that most string states would collapse
behind event horizons.
• However, the fixed nature of the value for the
transition temperature from the winding mode
implies that the free spectrum is somehow
preserved when gravity is turned on.
• This supports a bag model interpretation for string
states in 4 dimensions (hadron analogy).
Thermal duality – to the rescue?
• Maximum of Z against 𝛽 from thermal duality indicates that
the greatest density of strings and information is around the
self-dual temperature.
• Heterotic string case is discussed here primarily for simplicity
as they are pure string models [Polchinski hep-th/050033] with
explicit thermal duality.
Thermalon weak field solutions
Accelerating Wall Solution
Red Shifted Source (Seen from infinity)
• Note the negative energy density close to the horizon
• Source is zero at
Area Difference across the Wall
• This can be described by a ‘warp factor’ w:
Gravity Regulator proposal
• Galileo – Einstein: The acceleration of convex surfaces with locally
constant area is limited by the Hagedorn Rindler acceleration.
• Newton: heuristic equivalent – the gravitational force between any 2
objects (from ∞ or local static c.m. frames) is limited by the string
tension.
• This Newtonian form shows the mutual interactive nature of the
effect, as a phenomenon of the gravito-thermal interaction.
• Both forms apply in all dimensions for heterotic strings.
• Not related to the magnitude of the tidal Riemannian curvature!
Adiabatic (constant area) condition
Critical acceleration
Convexity
Generalisation
• Note the generic nature of the conditions, which can be
abstracted from the heterotic context, and may apply across M
space.
• The differential geometry applies in all uncompactified
dimensions.
• The mechanism for heterotic strings relies on the self-dual
nature of the thermalon. Although the thermalon is not selfdual for non-heterotic strings, thermal duality may still apply
to the string sector (T duality in Euclidean time).
Non-heterotic case
• Mertens, Verschelde & Zakharov obtained similar results for
the Rindler solution stress tensor for thermalons in type II
string models. [arXiv:1408.7012]
• Although these models do not have explicit thermal duality,
this behaviour is related to the T duality of the torus diagram.
The conjecture also implies that (some generalisation of)
thermal duality also provides a gravitational regulator for 11
dimensional supergravity.
Gravity regulator properties
• Critical surface conditions are a string regularised version of a
Penrose trapped surface (infinite tension limit).
• Horizons are defined globally but trapped surfaces (and
thermalon traps) are defined locally by these geometric
conditions.
• Heuristically the critical gravity condition for conversion is
equivalent to the condition that the Newtonian gravity (in local
centre of mass frame) between any two objects is limited by
the string tension.
• Mutual nature of effect between two interacting bodies is
shown by the symmetry of the ‘Newtonian’ effective criterion
for the gravitational regulator – may also hold throughout M
space.
Transition
•
A thermalon shell region is stable on the outside, unstable/dynamic on the
inside.
•
Assume that thermalon polarization maintains gravitational regulation
•
Transition begins at a separation ~ 𝑀𝑚 between virtual horizons.
•
Deceleration may couple matter to thermalon production to give a stringy
bremsstrahlung process, converting kinetic energy to thermalons – a dynamic
aspect of the gravito-thermal interaction.
•
Conversion power is constant ~ T (string tension) throughout relative to local
static frames. This is consistent with a gravity sector effect.
•
Progressively thickening solutions would be produced during conversion after
transition to the non-linear regime – limit is the ball solution.
Transition Process
• The following slide shows a possible transition processes for
formation of a thermalon region and absorption of infalling
matter.
• The infalling matter has a relativistic velocity relative to the
interior.
Encounter process
• Diagrams:
Conversion timescales
• Relativistic factors for time dilation:
• 𝛾~𝑀
• 𝛾~𝑑
for formation
for particle absorption
(d is distance from the existing boundary)
• This gives new estimates for transition times (from infinity):
~ 𝑀𝑑−3 log 𝑀
~𝑀𝑑−3 log 𝑀𝑚
for initial formation,
for particle absorption.
Conversion from Minkowski space?
• Suppose a planar thermalon trap crosses a wall of matter in
Minkowski space.
• The wall of matter cannot be converted to a thermalon
excitation because of the warp factor that would be produced.
• Agrees with intuition about conservation of energy.
• Conversion is not possible in Minkowski space.
Particle absorption
• Consider next a thermalon front geometry with a salient
(previous diagram). Could this be produced by a thermalon
trap absorbing a particle in Minkowski space?
• An expanding wave-front bounds a convex region in a
Minkowski background. This is inconsistent with a locally
constant area.
• Thus, although traps associated with Rindler frames are
everywhere in Minkowski space, these cannot gain energy by
recruiting particles.
Ball solution
• Thermal duality implies that gravitational gradients are
expelled around the self-dual point. This is similar to the
Meissner effect in superconductors.
[MH arxiv 1309.7578]
• Gravitational polarization gives a screening mechanism
which restricts the gravitational source to the boundary of
a (fully developed) thermalon region.
• Hyperbolic spatial geometry in the interior
makes the volume and information content
proportional to surface area.
• String excitations are gravitationally free in
the interior – like quarks in hadrons.
• Should resolve string state problem in 4
dimensions.
Hot objects
• The ball solution is in thermal equilibrium with the exterior
normal vacuum.
• A hot object with matching temperature and surface gravity
can be in thermal equilibrium with normal vacuum, with
𝑇𝛼𝛽 = 0 in the exterior space.
• In general, the surface temperature and gravity would only
match (i.e 2 𝜋𝑇 = 𝑔 ) by coincidence and the object would be
unstable.
Naturalness of the thermalon ball
• A thermalon deformation however naturally and stably meets
this matching condition for temperature and surface gravity as
it is tied to the location of high static gravitational acceleration.
• Here a ‘standing wave’ thermalon excitation can ride the
gravitational field.
Black holes and M theory
• To resolve short distances requires concentrating high energy
into a small volume - at increasing energy, eventually gravity
becomes strong over longer distances, and the resolution
deteriorates again.
• Making gravity consistent with quantum mechanics is
problematic for this reason, and problems with macroscopic
collapsed objects (e.g. evaporation paradoxes) will also be
present for 𝑑 > 4.
• We therefore expect that a gravitational regulator mechanism
should generalise to all regions of M theory and all
compactifications, as all will have collapsed states, and so
need to avoid paradoxes associated with black holes.
Evaporating Black Hole
• Note the causal
disconnection between the
original information and
late Hawking radiation.
• Monogamy of
entanglement prevents
reconciliation of ‘lost’ and
‘found’ information.
Black hole evaporation paradoxes
• General problem – getting information out from a photon trap
– without propagating information faster than light (outside the
light cone).
• A resort to quantum mechanics for a resolution does not seem
to work –inconsistent with the principles of quantum
information theory
• Braunstein - Pirandola theorems – the conventional picture of
black hole evaporation appears to be inconsistent with
unitarity [arXiv:1411.7195] .
Unitarity problems [B-P]
Theorem 1: A contradiction exists between:
• 1.a) completely unitarily evaporating black holes,
• 1.b) a freely falling observer notices nothing special
until they pass well within a large black hole's horizon,
and
• 1.c) the black hole interior Hilbert space dimensionality
may be well approximated as the exponential of the
Bekenstein Hawking entropy.
Theorem 2: A contradiction exists between:
• 2.a) completely unitarily evaporating black holes,
• 2.b) large black holes are described by local physics
(no signals faster than light)
and
• 2.c) externally, a large black hole should resemble its
classical theoretical counterpart (aside from its slow
evaporation).
Holographic problem
• Issue with holographic principle for stellar collapse:
• Information content of star ~ 1060 bits exceeds the
holographic limit before reaching the singularity.
• String physics apparently not relevant at this point, in the usual
interpretation.
Resolution?
• This model may resolve ‘firewall paradoxes’ by removing the
horizon, and providing a physically motivated firewall to store
the information content of a collapsing object.
• The ‘firewall’ here would be formed by string bremsstrahlung
during gravitational collapse.
• The firewall would be stable - there is no black hole, and the
interior space has been crushed so there is nowhere else for the
energy and information to go except outward by Hawking
evaporation.
Conclusions
• Because of thermal duality, string models may have a
vacuum polarization mechanism which limits the strength
of gravity.
• The criterion for this to be effective is based on the area
of accelerating surfaces, not the tidal Riemannian
curvature.
• This effect would occur at, and only at, sites of extreme
gravitational collapse.
• Collapsing objects would form hot holograms, rather
than black holes.
• This would resolve the firewall paradox: firewalls are
real, but black holes are not.
• Resolve the black hole information problem: there would
be no black holes to compromise quantum information.