I confess I understood very little of this, and frankly skimmed the
nth transformation of yet another spacetime metric. It's been
30 years since I studied general relativity (GR) and quantum field
theory (QFT); string theory hadn't even been born then. Lack of
comprehension was caused both by the inevitable decay of my
unexercised neurons, and the telegraphic style of the presentation:
this is a series of lectures written up as a book, but without much
extra explanatory material (in a real lecture you can ask questions;
in a book those questions need to be anticipated better). So this is
very heavy going unless you are a current researcher in the area. I am
not. Despite that, and the poor typography, there are snippets of some
very nice physical intuition scattered throughout, which made it
worthwhile for me.
For example, there is a nice description of how different a negative
specific heat is (and I just love the use of "well known" in
the second paragraph, too):
black holes ... are long-lived objects,
but eventually they evaporate. We can try to prevent their evaporation
by placing them in a thermal heat bath at their Hawking temperature
but that does not work. ... their specific heat is negative; their
temperature decreases as their energy or mass increases. ... suppose a
fluctuation occurs in which the black hole absorbs an extra bit of
energy from the surrounding heat bath. ... a system with negative
specific heat will lower its temperature when it absorbs energy and
will become cooler than the bath. This in turn will favor an
additional flow of energy from the bath to the black hole and a
runaway will occur. The black hole will grow indefinitely. If on the
other hand the black hole gives up some energy to the environment it
will become hotter than the bath. Again a runaway will occur that
leads the black hole to disappear.
A well known way to stabilize the
black hole is to put it in a box so that the environmental heat bath
is finite. When the black hole absorbs some energy it cools but so
does the finite heat bath. If the box is not too big the heat bath
will cool more than the black hole and the flow of heat will be back
to the bath.
And here is a great little fact about Hawking radiation I hadn't
previously been aware of:
The typical wavelength of a photon
radiated from the sun is ~ 10-5 cm, while the radius of
the surface of the sun is 1011 cm. The sun is for all
intents and purposes infinite on the scale of the emitted photon
wavelengths. The black hole on the other hand emits quanta of
wavelength ~ 1/T ~ MG, which is about equal to the
Schwarzschild radius. Observing a black hole by means of its Hawking
radiation will always produce a fuzzy image, unlike the image of the
The thrust of the book is how GR and QFT just don't work together.
For Q2 > M2G
the metric ... has a time-like singularity with no horizon to cloak
it. Such "naked singularities" indicate a break-down of
classical relativity visible to a distant observer. The question is
not whether objects with Q2 > M2G
can exist. Clearly they can. The electron is such an object. The
question is whether they can be described by classical general
relativity. Clearly they cannot.
It's not just that we need some better kind of "glue" to
patch things up at the boundaries. There's something fundamentally
wrong. Black hole thermodynamics begins to point out the problems of
reconciling the two. The entropy of a black hole is proportional to
its surface area. The Bekenstein bound says that the maximum entropy
of any region of space is proportional to its area. This is at
odds with what QFT says:
the Holographic Principle ...
says that there are vastly fewer degrees of freedom in quantum gravity
than in any QFT ...
Suppose we are dealing with a
lattice of discrete spins. Let the lattice spacing be a and
the volume ... be V. ... the maximum entropy is Smax
= (V/a3) log 2. ... it is proportional to
the volume. ...
... now consider a system that
includes gravity. ... the maximum entropy of a region of space is
proportional to its area ...
To get to this point, the authors travel through GR, information
theory, and QFT, pointing out various issues. First is the apparently
paradoxically different observations made by observers freely falling
through a black hole's event horizon (if it's a big enough black hole,
they observe ... nothing special really), compared to an
observer lurking around outside the horizon, who sees something very different.
But this is fine, because they are observing something very
different, something slowed down by red shift so much that they can
see qualitatively different things:
let us consider what happens when a
proton falls into a black hole. The baryon number is lost, and will
not be radiated back out in the Hawking radiation. ... where does the
baryon violation take place? One possible answer is that it occurs
when the freely falling proton encounters the very large curvature
invariants as the singularity is approached. From the proton's
viewpoint, there is nothing that would stimulate it to decay before
... from the vantage point of the
external observer, the proton encounters enormously high temperatures
as it approaches the horizon. ... the external observer will conclude
that baryon violation can take place at the horizon. Who is right?
... black hole complementarity
implies that they are both right. ...
... the proton is continuously
making extremely rapid transitions between baryon number states ...
Ordinary observations of the proton do not see these very rapid
... let us consider a proton
passing through a horizon ... it is not unlikely that when it passes
the horizon, its instantaneous baryon number is zero. ... from the
viewpoint of an external observer, this is not a short-lived
intermediate state. A fluctuation that is much too rapid to be seen by
a low energy observer falling with the proton appears to be a real
proton decay lasting to eternity from outside the horizon.
Along with this is the "ultraviolet/infrared connection",
where high enough energy stops being associated with small
scales, and starts being associated with large scales:
The overriding theme of 20th century
physics was the inverse relation between size and momentum/energy. ...
this trend is destined to be reversed in the physics of the 21st
century. ... Let's begin with a traditional attempt to study
interactions at length scales smaller than the Planck scale. According
to conventional thinking, what we need to do is to collide ...
particles ... We expect to discover high energy collision products
flying out at all angles. By analyzing the highest energy fragments,
we hope to reconstruct very short distance events.
The problem with this thinking is
that at energies far above the Planck mass, the collision will create
a black hole ... . The interesting short distance effects that we want
to probe will be hidden behind a horizon ... and are inaccessible. ...
the products of collision will be ... low energy Hawking radiation
[with] energy ... which decreases with the incident energy. ... as the
energy increases we would be probing ever larger scales ... This is
the simplest example of the ultraviolet/infrared connection ... Very
high frequency is related to large size scale.
The UV/IR connection is deeply
connected to black hole complementarity. ... the enormous differences
in the complementary descriptions of matter falling into a black hole
are due to the very different times resolutions available to the
Having noted that QFT doesn't work with GR, the authors suggest that
string theory might be a solution. It's not guaranteed, but at least
it has the right kind of behaviour, the right number of degrees of
freedom to be compatible with the Holographic principle:
quantum field theory has too many
degrees of freedom. ... [string theory]
seems to have just the right
number of degrees of freedom
The extreme red shift between the freely
falling frame and the Schwarzschild frame may take phenomena which are
of too high frequency to be visible ordinarily and make them visible
to the outside observer. ...
This suggests that the consistency
of black hole complementarity is a deep constraint on how matter
behaves at very short times or high frequencies. Quantum field theory
gets it wrong, but string theory seems to do better.
String theory has many different kinds
of black holes ... the statistical mechanics of strings allows us to
compute the entropy up to numerical factors of order unity. In every
case the results nontrivially agree with the Bekenstein-Hawking
The whole book is rather nicely summarised in the final section,
pulling together the four key concepts covered: Black Hole
Complementarity, the IR/UV connection, the Holographic Principle, and
The views of space and time that held
sway during most of the 20th century were based on locality and field
theory ... it was assumed that all observers would agree on the usual
invariant relationships between events. ... But ... it was never
adequate to deal with the combination of quantum mechanics and general
The first sign of this was the
failure of standard quantum field theory methods when applied to the
Einstein action. For a long time it was assumed that this just meant
that the theory was incomplete at short distances ... But the dilemma
of apparent information loss in black hole physics that was uncovered
by Hawking in 1976 said otherwise. In order to reconcile the
equivalence principle with the rules of quantum mechanics the rules of
locality have to be massively modified. The problem is not a pure
ultraviolet problem but an unprecedented mix of short distance and
long distance physics. ...
The new paradigm that is gradually
emerging is based on four closely related concepts. The first is Black
Hole Complementarity. ... the location of phenomena depends on the
resolution time available to the experimenter who probes the system.
... [consider] the fate of ... Alice, falling into an enormous black
hole with Schwarzschild radius of a billion years. According to ...
Alice ... the horizon is harmless and she or her descendants can live
for a billion years before being crushed at the singularity. In
apparent complete contradiction, the high frequency observer who stays
outside the black hole finds that his description involves Alice
falling into a hellish region of extreme temperature, being
thermalized, and eventually re-emitted as Hawking radiation. ....
Obviously this has to do with more than just a modification of the
short distance physics. ... the key to black hole complementarity is
the extreme red shift of the quantum fluctuations as seen by the
The second new idea is the
Infrared/Ultraviolet connection. Very closely related to Black Hole
Complementarity, the IR/UV connection reverses one of the most
fundamental trends of 20th century physics. Throughout that century a
close connection between energy and size prevailed. If one wished to
study progressively smaller and smaller objects one had to use higher
and higher energy probes. But once gravity is involved that trend is
reversed. At energies above the Planck scale any possible short
distance physics that we might look for is shrouded behind a black
hole horizon. As we raise the energy we wind up probing larger and
larger distance scales. The ultimate implications of this, especially
for cosmology are undoubtedly profound but still unknown.
Third is the Holographic Principle.
... The non-redundant degrees of freedom that describe a region of
space are in some sense on its boundary, not its interior as they
would be in field theory. At one per Planck area, there are vastly
fewer degrees of freedom than in a field theory, cutoff at the Planck
volume. The number of degrees of freedom per unit volume becomes
arbitrarily small as the volume gets large. ...
Finally, the existence of black
hole entropy indicates the existence of microscopic degrees of freedom
which are not present in the usual Einstein theory of gravity. It does
not tell us what they are. String theory does provide a microscopic
framework for the use of statistical mechanics. In all cases the
entropy of the appropriate string system agrees with the Bekenstein
Hawking entropy. This, if nothing else, provides an existence proof
for a consistent microscopic theory of black hole entropy.
This is still an open and active research area, and I look forward
to reading about further developments (hopefully in a slightly more
readable style). And while waiting, I can enjoy little snippets like:
by bending some of the directions of
space into compact manifolds it becomes possible to generate a
cosmological constant for the resulting lower dimensional Kaluza-Klein
which make me wonder, as an extra plot point for the
as-yet-unwritten Cold Sproing:
if a new dimension uncurls, does the cosmological constant also