This Week's Finds in Mathematical Physics (Week 37)
John Baez
John Baez
Mainly this week I have various bits of news to report from the
7th Marcel Grossman Meeting on general relativity. It was big
and had lots of talks. Bekenstein gave a nice review talk on
entropy/area relations for black holes, and Strominger gave a
talk in which he proposed a solution to the information loss
puzzle for black holes. (Recall that if one believes, as most people
seem to believe, that black holes radiate away all their mass in the
form of completely random Hawking radiation, then there's a question
about where the information has gone that you threw into the black hole
in the form of, say, old issues of Phys. Rev. Lett.. Some people
think the information goes into a new "baby universe" formed at the
heart of the black hole --- see "week31" for more. The information
would still, of course, be gone from *our* point of view in this
picture. Strominger proposed a set up in which one had a quantum theory
of gravity with annihilation and creation operators for baby universes,
and proposed that the universe (the "metauniverse"?) was in a coherent
state, that is, an eigenstate of the annihilation operator for baby
universes. This would apparently allow handle the problem, though right
now I can't remember the details.) There were also lots of talks on the
interferometric detection of gravitational radiation, other general
relativity experiments, cosmology, etc.. But I'll just try to describe
two talks in some detail here.
1) L. Lindblom, Superfluid hydrodynamics and the stability of
rotating neutron stars, talk at MG7 meeting, Monday July 5,
Stanford University.
Being fond of knots, tangles, and such, I have always liked
knowing that in superfluids, vorticity (the curl of the velocity
vector field) tends to be confined in "flux tubes", each
containing an angular momentum that's an integral multiple of
Planck's constant, and that similarly, in type II
superconductors, magnetic fields are confined to magnetic flux
tubes. And I was even more happy to find out that the cores of
neutron stars are expected to be made of neutronium that is
*both* superfluid *and* superconductive, and contain lots of flux
tubes of both types. In this talk, which was really about a
derivation of detailed equations of state for neutron stars,
Lindblom began by saying that the maximum rotation rate of a
rotating neutron star is due to some sort of "gravitational
radiation instability due to internal fluid dissipation". I
didn't quite understand the details of that, which weren't
explained, but it motivated him to study the viscosity in
neutron star cores, which are superfluid if they are cool enough
(less than a billion degrees Kelvin). There are some protons and
electrons mixed in with the neutrons in the core, and both the
protons and neutrons go superfluid, but the electrons form a
normal fluid. That means that there are actually *two* kinds of
superfluid vortices -- proton and neutron --- in addition to the
magnetic vortices. These vortices mainly line up along the axis
of rotation, and their density is about 10^6 per square
centimeter. Rather curiously, since the proton, neutron, and
electron fluids are coupled due to beta decay (and the reverse
process), even the neutron vortices have electric currents
associated to them and generate magnetic fields. This means that
the electrons scatter off the neutron vortex cores as well as the
proton vortex cores, which is one of the mechanisms that yields
viscosity.
2) Abhay Ashtekar, Mathematical developments in quantum general
relativity, a sampler, talk at MG7 meeting, Tuesday July 6,
Stanford University.
This talk, in addition to reviewing what's been done so far on
the "loop representation" of quantum gravity, presented two new
developments that I found quite exciting, so I'd like to sketch
what they are. The details will appear in future papers by
Ashtekar and collaborators.
The two developments Ashtekar presented concerned mathematically
rigorous treatments of the "reality conditions" in his approach to
quantum gravity, and the "loop states" used by Rovelli and Smolin.
First let me try to describe the issue of "reality
conditions". As I described in "week7", one trick that's important
in the loop representation is to use the "new variables" for
general relativity introduced by Ashtekar (though Sen and
Plebanski already had worked with similar ideas). In the older
Palatini approach to general relativity, the idea was to view
general relativity as something like a gauge theory with gauge
group given by the Lorentz group, SO(3,1). But to do this one
actually uses two different fields: a "frame field", also called
a "tetrad", "vierbein" or "soldering form" depending on who
you're talking to, and the gauge field itself, usually called a
"Lorentz connection" or "SO(3,1) connection". Technically, the
frame field is an isomorphism between the tangent bundle of
spacetime and some other bundle having a fixed metric of
signature +---, usually called the "internal space", and the
Lorentz connection is a metric-preserving connection on the
internal space.
The "new variables" trick is to use the fact that SO(3,1) has as
a double cover the group SL(2,C) of two-by-two complex matrices
with determinant one. (For people who've read previous posts of mine, I
should add that the Lie algebra of SL(2,C) is called sl(2,C) and is
the same as the complexification of the Lie algebra so(3), which
allows one to introduce the new variables in a different but equivalent
way, as I did in "week7".) Ignoring topological niceties for now, this
lets one reformulate *complex* general relativity (that is, general
relativity where the metric can be complex-valued) in terms of a
*complex-valued* frame field and an SL(2,C) connection that is
just the Lorentz connection in disguise. The latter is called
either the "Sen connection", the "Ashtekar connection", or the
"chiral spin connection" depending on who you're talking to. The
advantage of this shows up when one tries to canonically quantize
the theory in terms of initial data. (For a bit on this, try
"week11".) Here we assume our 4-dimensional spacetime can be split up
into "space" and "time", so that space is a 3-dimensional
manifold, and we take as our canonically conjugate fields the
restriction of the chiral spin connection to space, call it A,
and something like the restriction of the complex frame field to a
complex frame field E on space. (Restricting the complex frame field to
one on space is a wee bit subtle, especially because one doesn't really
want a frame field or "triad field", but really a "densitized
cotriad field" --- but let's not worry about this here. I
explain this in terms even a mathematician can understand in my
paper "string.tex", available by ftp along with all my "week"
files as described below.) The point is, first, that the A and E
fields are mathematically very analogous to the vector potential
and electric field in electromagnetism --- or really in SL(2,C)
Yang-Mills theory --- and secondly, that if you compute their
Poisson brackets, you really do see that they're canonically
conjugate. Third and best of all, the constraint equations in
general relativity can be written down very simply in terms of A
and E. Recall that in general relativity, 6 of Einstein's 10 equations
act as *constraints* that the metric and its time derivative must
satisfy at t = 0 in order to get a solution at later times.
In quantum gravity, these constraints are a big technical problem one
has to deal with, and the point of Ashtekar's new variables is precisely
that the constraints simplify in terms of these variables. (There's
more on these constraints in "week11".)
The price one has paid, however, is that one now seems to be
talking about *complex-valued* general relativity, which isn't
what one had started out being interested in. One needs to get
back to reality, as it were --- and this is the problem of the
so-called "reality conditions". One approach is to write down extra
constraints on the E field that say that it comes from a *real*
frame field. These are a little messy. Ashtekar,
however, has proposed another approach especially suited to
the quantum version of the theory, and in his talk he filled in
some of the crucial details.
Here, to save time, I will allow myself to become a bit more
technical. In the quantum version of the theory one expects the
space of wavefunctions to be something like L^2 functions on the
space of connections modulo gauge transformations --- actually
this is the "kinematical state space" one gets before writing the
constraints as operators and looking for wavefunctions
annihilated by these constraints. The problem had always been
that this space of L^2 functions is ill-defined, since there is no "Lebesgue
measure" on the space of connections. This problem is addressed
(it's premature to say "solved") by developing a theory of
generalized measures on the space of connections and proving the
existence of a canonical generalized measure that deserves the
name "Lebesgue measure" if anything does. One can then define
L^2 functions and work with them. For compact gauge groups,
like SU(2), this was done by Ashtekar, Lewandowski and myself;
see e.g. the papers "state.tex" and "conn.tex" available by ftp.
In the case of SU(2), Wilson loops act as self-adjoint
multiplication operators on the resulting L^2 space. But in
quantum gravity we really want to use gauge group SL(2,C), which
is not compact, and we want the adjoints of Wilson loop operators
to reflect that fact that the SL(2,C) connection A in quantum
gravity is really equal to Gamma + iK, where Gamma is the
Levi-Civita connection on space, and K is the extrinsic
curvature. Both Gamma and K are real in the classical theory, so
the adjoint of the quantum version of A should be Gamma - iK, and
this should reflect itself in the adjoints of Wilson loop
operators.
The trick, it turns out, is to use some work of Hall which
appeared in the Journal of Functional Analysis in 1994 (I don't
have a precise reference on me). The point is that SL(2,C) is
the complexification of SU(2), and can also be viewed as the
cotangent bundle of SU(2). This allows one to copy a trick
people use for the quantum mechanics of a point particle on R^n ---
a trick called the Bargmann-Segal-Fock representation.
Recall that in the ordinary Schrodinger representation of a
quantum particle on R^n, one takes as the space of states
L^2(R^n). However, the phase space for a particle in R^n, which
is the cotangent bundle of R^n, can be identified with C^n, and
in the Bargmann representation one takes as the space of states
HL^2(C^n), by which I mean the *holomorphic* functions on C^n
that are in L^2 with respect to a Gaussian measure on C^n. In
the Bargmann representation for a particle on the line, for
example, the creation operator is represented simply as
multiplication by the complex coordinate z, while the
annihilation operator is d/dz. Similarly, there is an
isomorphism between L^2(SU(2)) and a certain space HL^2(SL(2,C)).
Using this, one can obtain an isomorphism between the space of
L^2 functions on the space of SU(2) connections modulo gauge
transformations, and the space of holomorphic L^2 functions on
the space of SL(2,C) connections modulo gauge transformations.
Applying this to the loop representation, Ashtekar has found a
very natural way to take into account the fact that the chiral
spin connection A is really Gamma + iK, basically analogous to the
fact that in the Bargmann multiplication by z is really q + ip (well,
up to various factors of sqrt(2), signs and the like).
Well, that was pretty sketchy and probably not especially
comprehensible to anyone who hasn't already worried about this
issue a lot! In any event, let me turn to the other good news
Ashtekar reported: the constuction of "loop states". Briefly put
(I'm getting worn out), he and some collaborators have figured
out how to *rigorously* construct generalized measures on the
space of connections modulo gauge transformations, starting from
invariants of links. This begins to provide an inverse to the
"loop transform" (which is a construction going the other way).
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instead, read the file preprint.info.
david dixon
In article <1994Aug10....@galois.mit.edu>,
John Baez <ba...@ucrmath.ucr.edu> wrote:
>puzzle for black holes. (Recall that if one believes, as most people
>seem to believe, that black holes radiate away all their mass in the
>form of completely random Hawking radiation, then there's a question
>about where the information has gone that you threw into the black hole
>in the form of, say, old issues of Phys. Rev. Lett.. Some people
>think the information goes into a new "baby universe" formed at the
>heart of the black hole --- see "week31" for more. The information
>would still, of course, be gone from *our* point of view in this
>picture.
I'm going to harp on this yet again, because I still haven't gotten
a very satisfying explanation of where the problem lies. Here's
my (potentially incorrect) view of the situation:
1) Black holes evaporate due to Hawking radiation. This evaporation
occurs in finite time as measured by an observer stationary w.r.t.
the black hole.
2) I drop my copy of PRL in the neighborhood of a black hole. As measured
by *my* clock (stationary w.r.t. to the black hole), the book does
not cross the event horizon in a finite time.
3) Conclusion: Given that PRL is published on quality materials, able
to withstand the rigors of tidal forces and high radiation, I get
my book back after the black hole is done evaporating. No information
loss.
Now, every time I've asked this I get responses in the form of
Penrose diagrams showing that the book hits the singularity in
finite proper time. That's great, but I would like someone to
reconcile the facts (or point out my error) that *I* don't ever
see the book fall in, but I *do* see the black hole evaporate.
Dave
Michael Clive Price
Is there a seceret code?
Dave Dixon:
> I'm going to harp on this yet again, because I still haven't gotten
> a very satisfying explanation of where the problem lies. Here's
> my (potentially incorrect) view of the situation:
>
> 1) Black holes evaporate due to Hawking radiation. This evaporation
> occurs in finite time as measured by an observer stationary w.r.t.
> the black hole.
>
> 2) I drop my copy of PRL in the neighborhood of a black hole. As
> measured by *my* clock (stationary w.r.t. to the black hole),
> the book does not cross the event horizon in a finite time.
Actually the book is *does* cross the event horizon in finite external
time. There was an article awhile back (in Phys Rev D, I think) about
what happens as a mass approaches the event horizon. The mass of the
book starts contributing (locally) to the black hole's mass (as seen by
a distant observer). Thus, get close enough and.... the event horizon
reaches up, amoeba-like, and swallows the book, in finite external time.
Munch, munch... After a short while the increase in bh radius becomes
global.
Note that this does not addresses the question of where the information
goes.
To see that it *is* lost in the radiation, note that:
1) information about the structure of the matter in a black hole is
still accessible externally, since a bh can be treated as a frozen
stars, without an event horizon, in principle.
2) all objects radiate Hawking radiation, being a property of the
gravitational field, not of the object, and so solutions like the
information disappears into a baby universe or the singularity
"eats" it, can't be not general - and probably not specific either.
> 3) Conclusion:
- no book comes back, except as radiation
- information is radiated away in photon correlations - which would
be very hard to detect
Conclusion: Hawking radiation from black holes (which is usually
calculated for a perfectly spherically symmetric black hole?) contains
data about the objects that fall in. No information loss as black hole
evaporates. No breakdown of physics etc
I've not seen anyone else reason like this, so it's probably bollocks -
but there again, I can't see the flaw either.
Mike Price pr...@price.demon.co.uk
Warren G Anderson
most important thing to remember about "evaporating black holes" is that
in the strict sense of the term as it is currently used *they are not
black holes*. How does one make sense of this seemingly contradictory
statement? Astrophysically, we typically think of a "black hole" as the
endpoint of a massive collapsed star. Mathematically, a black hole is
the region contained within the event horizon of a space-time. The event
horizon, by definition, is the last null ray that intersects observers
at asymptotically flat infinity. It happens, in classical static
black holes, to correspond with a couple of other horizons, the apparent
horizon, which is the outer most trapped surface (surface on which light
rays emitted cannot diverge spatially from one another, ie surface of
zero light ray expansion) and the static killing horizon (surface on
which observers stationary with respect to the static observers at
infinity can no longer exist). Most people think of the apparent horizon
when discussing the event horizon and collapsed stars when discussing
black holes, and this leads to confusion.
Case in point. In the current thread people are discussing the "event
horizon" of an "evaporating black hole". An "evaporating black hole" does
not have an event horizon (to the best of my knowledge, there might be one
associated with quantum gravitational effects at the end of evaporation,
but we don't know how evaporation ends, which is the whole point of the
information loss problem). It does have an apparent horizon. Thus,
investigating what happens when PRL is dropped into the "evaporating black
hole", and especially what happens at the "event horizon", one must be
careful to realise that one cannot apply all the things one "knows" about
black holes and event horizons because there are no black holes or event
horizons in the scenario. One should use what one knows about apparent
horizons and "evaporating black holes" (which I wish I had a better name
for) instead. I might add that neither Wald nor anyone else (even
Strominger) knows what the causal structure of an evaporating black hole
looks like at late times in my opinion, so the "standard evaporating
black hole Penrose diagram" should also be taken with a grain of salt.
Sorry for the lecture. You are now returned to you regular programming.
--
########################## _`|'_ ##############################################
## Warren G. Anderson |o o| "... for its truth does not matter, and is ##
## Dept. of Physics ( ^ ) unimaginable." -J. Ashbery, The New Spirit ##
## University of Alberta /\-/\ (ande...@fermi.phys.ualberta.ca) ##
david dixon
Emory F. Bunn <t...@physics2.berkeley.edu> wrote:
>observer proper times. The observer (neglecting the usual dimmer-switch
>effects, photon discreteness, finite mass of the object, etc.) will see
>the object fall into the hole just as the hole evaporates, though that's
>not what is actually going on-- it's a trick of the light cones.
I assume by finite mass you mean non-zero, and that the above arguments
apply for a very small test particle. But what if the thing does
have mass? It's mass contributes to the local energy density, and
changes the spacetime structure. Does, as Michael Price suggested,
the event horizon stretch out, "amoeba-like", and envelope the book,
and does this happen in finite time according to an external observer,
etc.
Also, I'm not sure what tthe phrase "that's not what is actually going
on-- it's a trick of the light cones" means. I want to make sure
I'm distinguishing between what the observer "sees" due to photons
hitting his eye, and what he calculates via relativity. So let's
put it this way: I drop my PRL into a black hole, but this time, I
tie a cord around it. Let's imagine the cord is infinitely elastic,
i.e., you can stretch it as much as you want and it won't break.
For either the case of the evaporating or non-evaporating black hole,
would I ever encounter a point (in finite time) where I can't get
my book back by pulling hard enough?
Dave
Jarle Brinchmann
In article <32o9lb$6...@galaxy.ucr.edu>, di...@galaxy.ucr.edu (david dixon) writes:
|> Also, I'm not sure what tthe phrase "that's not what is actually going
|> on-- it's a trick of the light cones" means. I want to make sure
|> I'm distinguishing between what the observer "sees" due to photons
|> hitting his eye, and what he calculates via relativity. So let's
|> put it this way: I drop my PRL into a black hole, but this time, I
|> tie a cord around it. Let's imagine the cord is infinitely elastic,
|> i.e., you can stretch it as much as you want and it won't break.
|> For either the case of the evaporating or non-evaporating black hole,
|> would I ever encounter a point (in finite time) where I can't get
|> my book back by pulling hard enough?
Yes, after the book has crossed the horizon. The things you see are just
some images left on the journey into the hole. The actual book will end
up in black hole in a finite time, this is explained by a Penrose diagram.
Jarle.
---------------------------------------------------------------------
Nuke the Whales ! | Jarle Brinchmann,
| Email: Jarle.Br...@astro.uio.no
International Krill Union. | or : jar...@astro.uio.no
david dixon
Jarle Brinchmann <jar...@leda.uio.no> wrote:
>
>In article <32o9lb$6...@galaxy.ucr.edu>, di...@galaxy.ucr.edu (david dixon) writes:
>|> For either the case of the evaporating or non-evaporating black hole,
>|> would I ever encounter a point (in finite time) where I can't get
>|> my book back by pulling hard enough?
>
>Yes, after the book has crossed the horizon. The things you see are just
>some images left on the journey into the hole. The actual book will end
>up in black hole in a finite time, this is explained by a Penrose diagram.
I realize that the proper time to hit the singularity when one is
free falling is finite. What I want to know is what does MY watch
say when the force required to extract the book goes to infinity.
The book crosses the horizon at a definite spacetime point. If
I transform from that frame to my (admittedly non-inertial) frame
at rest w.r.t. the black hole, how long would I calculate that
it takes the book to cross the event horizon.
Dave
Jarle Brinchmann
In article <32ol6l$9...@galaxy.ucr.edu>, di...@galaxy.ucr.edu (david dixon) writes:
|> I realize that the proper time to hit the singularity when one is
|> free falling is finite. What I want to know is what does MY watch
|> say when the force required to extract the book goes to infinity.
|> The book crosses the horizon at a definite spacetime point. If
|> I transform from that frame to my (admittedly non-inertial) frame
|> at rest w.r.t. the black hole, how long would I calculate that
|> it takes the book to cross the event horizon.
But how are you going to measure the force? No disturbance in the string
can travel with a speed larger than the speed of light, thus the same
problems occur with your string example. You didn't ask when you would
_feel_ the force go to infinity, merely when it would. Your setup is
actually more or less the same as if you sent a mirror into the black
hole and shone light on it.
david dixon
Jarle Brinchmann <jar...@leda.uio.no> wrote:
>
>In article <32ol6l$9...@galaxy.ucr.edu>, di...@galaxy.ucr.edu (david dixon) writes:
>|> If I transform from that frame to my (admittedly non-inertial) frame
>|> at rest w.r.t. the black hole, how long would I calculate that
>|> it takes the book to cross the event horizon.
>
>But how are you going to measure the force? No disturbance in the string
>can travel with a speed larger than the speed of light, thus the same
>problems occur with your string example. You didn't ask when you would
>_feel_ the force go to infinity, merely when it would. Your setup is
>actually more or less the same as if you sent a mirror into the black
>hole and shone light on it.
Shit. Good point. But let's phrase it another way. Assuming I
can pull with an arbitrarily large, but finite, force, is there
ever a time when I can't pull my book out in finite time? In other
words, does information disappear from the universe in finite time?
If you've been following this thread, you'll note I've gotten
about five answers to the original question. As far as I can tell,
they don't particularly overlap, though I can always claim ignorance
on this point :-)
Suppose I did send a mirror into a black hole and shone light on
it. What would I see? Naively, one might expect some symmetry
between the blue/red shift of the photons, but on the other hand,
the mirror is receding from me ever faster, which I guess would
imply the photons come back ever more redshifted.
Dave
Jarle Brinchmann
In article <32or0c$b...@galaxy.ucr.edu>, di...@galaxy.ucr.edu (david dixon) writes:
|> Shit. Good point. But let's phrase it another way. Assuming I
|> can pull with an arbitrarily large, but finite, force, is there
|> ever a time when I can't pull my book out in finite time? In other
|> words, does information disappear from the universe in finite time
If you pull the string with a finite force you will never be able to pull
back the book. The point is more or less that as the book crosses the
horizon loosely speaking the escape velocity exceeds the speed of light
and therefore the force needed exceeds the speed of light.
|> If you've been following this thread, you'll note I've gotten
|> about five answers to the original question.
Well, actually I wrote one of them :-)
|> Suppose I did send a mirror into a black hole and shone light on
|> it. What would I see? Naively, one might expect some symmetry
|> between the blue/red shift of the photons, but on the other hand,
|> the mirror is receding from me ever faster, which I guess would
|> imply the photons come back ever more redshifted.
Yes, and in a finite time all light that you have shone on the
mirror will come back to you. This is because only a finite amount
of photons reaches the mirror before it crosses the horizon.
James A. Donald
> This question bothered me for a long time. At some point, Matt
> McIrvin posted an explanation that I understood, and fortunately, I
> saved it. The bottom line is that, as seen by an outside observer,
> your PRL does hit the singularity in finite time. In fact, the
> observer sees it hit the singularity at the exact moment the hole
> finishes evaporating.
I wrote that you see the PRL cross the "surface of the black hole"
in finite and short time.
My error. You see it cross the apparent event horizon r=2m in
finite and short time.
An infinitesimal distance inside that is the true event horizon.
A photon emitted as the PRL crosses the true event horizon
will hover just inside the apparent event horizon, slowly
moving to smaller and smaller r, as the black hole grows
smaller and smaller.
For the whole enormous time that the black hole is shrinking this
photon will suffer rapid redshift, so in reality there will
of course be no photon, but disregarding that small detail
this photon will eventually intersect the evaporation singularity,
the moment when the black hole evaporates.
Photons emitted by your copy of PRL after it crosses the true
event horizon are lost from our universe forever, as is your
copy of PRL and all the information it contained.
Photons emitted before it crosses the event horizon enter our
universe after a short time.
A photon emitted as it crosses the true event horizon
collides with the evaporation singularity after a long
finite time.
--
---------------------------------------------------------------------
We have the right to defend ourselves and our
property, because of the kind of animals that we James A. Donald
are. True law derives from this right, not from
the arbitrary power of the omnipotent state. jam...@netcom.com
GUEST-Arun Gupta(CUTS)
which showed that when one tossed a copy of PRL towards a black hole,
as the PRL approached the (apparent) event horizon, the horizon would
extend out ameoba-like and swallow the PRL.
If I understand what the no-hair theorems for black holes mean :
the black hole + unswallowed PRL system have mass multipole moments :
and the final black hole that has swallowed the PRL should settle
into one of the no-hair states, presumably by radiating away
gravitationally. So, might not the information in PRL be radiated
graviationally and not via correlations in the Hawking radiation ?
-arun gupta
Jarle Brinchmann
In article <CunCE...@nntpa.cb.att.com>, gu...@jolt.mt.att.com. (GUEST-Arun Gupta(CUTS)) writes:
|> If I understand what the no-hair theorems for black holes mean :
|>
|> the black hole + unswallowed PRL system have mass multipole moments :
|> and the final black hole that has swallowed the PRL should settle
|> into one of the no-hair states, presumably by radiating away
|> gravitationally. So, might not the information in PRL be radiated
|> graviationally and not via correlations in the Hawking radiation ?
I have thought about the same thing myself, but I'm not sure if it will
work since you would need to take account on the multipole of the book
itself due to the printing, otherwise you will only record the mass, and
possibly it's path into the hole and a few other fundamental parameters.
But I would also like to know whether this has been discussed seriously.
Matt McIrvin
Warren G Anderson <ande...@phys.ualberta.ca> wrote:
>I might add that neither Wald nor anyone else (even
>Strominger) knows what the causal structure of an evaporating black hole
>looks like at late times in my opinion, so the "standard evaporating
>black hole Penrose diagram" should also be taken with a grain of salt.
This is quite correct. My answer is based on Wald's assumption for the
causal structure, which assumes, among other things, that no remnant is
left behind. And you are on the mark when you say that things known
about classical black holes should not be applied wholesale to evaporating
ones; I think that this apparent paradox about an infalling object
*outliving* an evaporating black hole is essentially based on a
misapplication of this variety. Whatever the causal structure is, it
is going to be possible to say consistently what happens to an infalling
object in that causal structure, and it may well not resemble the fate
of an object, or the appearance of that fate, in a Schwarzschild black
hole.
--
Matt 01234567 <-- Indent-o-Meter
McIrvin ^ Harnessing tab damage for peaceful ends!
Matt McIrvin
david dixon <di...@galaxy.ucr.edu> wrote:
>I assume by finite mass you mean non-zero, and that the above arguments
>apply for a very small test particle. But what if the thing does
>have mass? It's mass contributes to the local energy density, and
>changes the spacetime structure. Does, as Michael Price suggested,
>the event horizon stretch out, "amoeba-like", and envelope the book,
Yes. This is described in _Black Holes: The Membrane Paradigm_ by
Thorne et al. Keep in mind that the event horizon is a statement about
the *future* behavior of light cones, so it behaves in a peculiar
anticausal way, responding to future rather than past events.
>and does this happen in finite time according to an external observer,
>etc.
I think this depends on whether or not the hole evaporates. For a
classical black hole I don't think you'll ever see the event horizon
finish eating the object.
>Also, I'm not sure what tthe phrase "that's not what is actually going
>on-- it's a trick of the light cones" means. I want to make sure
>I'm distinguishing between what the observer "sees" due to photons
>hitting his eye, and what he calculates via relativity.
Keep in mind that "what he calculates via relativity" has less meaning
in GR than in SR; it depends upon the choice of coordinate system, and
there's no natural non-curvilinear choice like there is in SR. Even
in SR, I can choose coordinates such that your lunch today occurs
"infinitely far in the future" by my t coordinate, though those
coordinates are not the "flat" ones normally used. For a Schwarzschild
hole there is a standard set of Schwarzschild coordinates, which do
inconvenient things near the black hole but are standard anyway. Those
coordinates are nevertheless not God-given; we could use other sets, and
often do. If the hole is evaporating, we can't use those Schwarzschild
coordinates at all.
>So let's
>put it this way: I drop my PRL into a black hole, but this time, I
>tie a cord around it. Let's imagine the cord is infinitely elastic,
>i.e., you can stretch it as much as you want and it won't break.
>For either the case of the evaporating or non-evaporating black hole,
>would I ever encounter a point (in finite time) where I can't get
>my book back by pulling hard enough?
Is it a causality-violating cord that transmits forces instantaneously
quickly?
Matt McIrvin
Michael Clive Price <pr...@price.demon.co.uk> wrote:
>Actually the book is *does* cross the event horizon in finite external
>time. There was an article awhile back (in Phys Rev D, I think) about
>what happens as a mass approaches the event horizon. The mass of the
>book starts contributing (locally) to the black hole's mass (as seen by
>a distant observer). Thus, get close enough and.... the event horizon
>reaches up, amoeba-like, and swallows the book, in finite external time.
> Munch, munch... After a short while the increase in bh radius becomes
>global.
I am a little concerned here about your use of the words "finite
external time." Were they treating this as a perturbation of the
Schwarzschild metric and using the Schwarzschild t coordinate, or
something? After all, if we ignore Hawking evaporation, then a
light ray at the *new* horizon will never escape to large distances,
so an external observer will never *see* the object reach it even
here. Even if we take Hawking radiation into account, I don't think
the phenomenon you're describing changes the situation that much
from what it would be for massless test particles.
Michael Clive Price
Date: Fri, 19 Aug 94 06:51:08 GMT
Organization: MCP plc
Reply-To: pr...@price.demon.co.uk
X-Newsreader: Simple NEWS 2.0 (ka9q DIS 1.24)
Lines: 22
Me:
> Actually the book is *does* cross the event horizon in finite
> external time. There was an article awhile back (in Phys Rev D,
> I think) about what happens as a mass approaches the event horizon.
> The mass of the book starts contributing (locally) to the black
> hole's mass (as seen by a distant observer). Thus, get close
> enough and.... the event horizon reaches up, amoeba-like, and
> swallows the book, in finite external time. Munch, munch... After
> a short while the increase in bh radius becomes global.
Matt McIrvin:
> I am a little concerned here about your use of the words "finite
> external time." Were they treating this as a perturbation of the
> Schwarzschild metric and using the Schwarzschild t coordinate, or
> something?
A perturbation, I think, but I don't recall the details.
I'll try to look up the article again, but I'm fairly sure (that the
article was definite that) the book is swallowed in finite external time
as *seen* by any external quasi-stationary observer. The way I think
it works (from memory) is that the infalling massive object approaches
the original BH's event horizon to within the object's own Schwarzschild
radius in finite external time. It takes awhile for the BH to feel the
gravitational effects "before" it responds by increasing its
Schwarzschild radius. Thus the book is lost.
> After all, if we ignore Hawking evaporation, then a light ray at the
> *new* horizon will never escape to large distances, so an external
> observer will never *see* the object reach it even here.
This is general point, isn't it? Same's true for the *old* horizon,
which is why (in some respects) we ought not forget the frozen star
image either (as I said in my post), although I don't want to get into
the question of the "reality" of the event horizon.
Mike Price pr...@price.demon.co.uk
Archimedes Plutonium
Alright, I was looking forward to publishing TEACHING TRUE PHYSICS, 1st year College before the end of September. Those hopes were dashed because I needed to include Series and Parallel Circuits, thinking they were minor and petty to EM theory, and then I remembered, there is a counterintuitive portion to Parallel Circuits and whenever I run into counterintuitive science is a signal to me that we have not understood that science correctly. And once we do understand the science, the counterintuitive disappears. So that now dashed my hopes of end of September and places this book's publication in October. This book has been one huge challenge after another for me, because, almost everything in this book is new physics, for it is a total replacement of the Maxwell Equations and so much more.
So, well, I was thinking earlier about Kirchhoff laws and the history of those laws for much of science is understood if you understand the history. And for the life of me I could not think of a single need for a circuit, for the light bulb was not invented until the 1900s. And I knew not the age period that Kirchhoff lived. So I could not see any need for Kirchhoff's laws until Edison invented the light bulb to make circuits with resistors useful. In fact I could not think of a single resistor to use except Faraday's invention of the crude electric motor, but could not see Faraday stringing together 2 or 3 of his crude electric motors to make a series or parallel circuit, especially with his pool of mercury, a toxic element (Did Faraday know that mercury was so toxic?).
So I looked at a website for some history and a website of "howstuffworks" says in 1800 Volta invented first battery and then goes on to say "The very first circuits used a battery and electrodes immersed in a container of water. The flow of current through the water produced hydrogen and oxygen."
Alright, sounds like something Faraday would have done.
But I would have guessed the first circuit would have involved the Leyden Jar even though its current would last for a brief moment. Where people want to get shocked and so lined up and 3 people holding wires to the next person making a full circuit back to the Leyden jar. So a Series circuit shocking with people as resistors. And the Leyden Jar was invented in 1745, so they had much much time before Faraday invented the crude electric motor in 1821.
Ohm's law was in 1827. And Kirchhoff's laws were in 1845.
So what devices of a circuit did Ohm make to derive his law?
Alright, earlier today I gave a example of Parallel Circuit using Kirchhoff law, now I need the contrast with Series Circuit.
On Saturday, September 25, 2021 at 4:14:35 PM UTC-5, Archimedes Plutonium wrote:
> SAVE// AP's 151st book, Do a Problem Sample of Kirchhoff law on Parallel Circuit
>
> Alright I am looking for the terms in AP-EM Equations (replacement of the error filled Maxwell Equations), I am looking for the terms that are the two Kirchhoff's laws. And the best way to proceed is a numbers problem. Yet, usually AP finds geometry sample problems to help understand but in this case of Kirchhoff's law and series or parallel circuits, AP finds, it strange that a numbers algebra problem sample would be best to employ.
>
> Parallel Circuit Sample Problem:
>
> Voltage is 120 V
> Resistors 3 of them: 10 ohms, 20 ohms, 30 ohms
> What is the current in each resistor?
> 120/10 = 12 A
> 120/20 = 6 A
> 120/30 = 4 A
>
> The total current would be 12+6+4 = 22 A
>
> What is the overall Resistance in Old Physics of this parallel circuit?
>
> R= V/i = 120/22 = 5.45 ohms and considerably smaller than any of the three resistors.
>
> So AP has to make clear what is going on here in this counterintuitive physics.
>
> I am going to have to get a Series circuit problem sample to compare with parallel circuit.
>
>
> Looking for the terms in AP-EM Equations where the two Kirchhoff laws lie. For parallel circuit it would be a constant V with varying C, B, or E. In Series circuit it would be a constant C with varying V, B, E.
> V'BE/(BE)^2 - VB'E/(BE)^2 - VBE'/(BE)^2
> current production - Lenz law - DC, AC direction.
> V'CE/(CE)^2 - VC'E/(CE)^2 - VCE')/(CE)^2
> B production - Displacement current - parallel attract.
> V'CB/(CB)^2 - VC'B/(CB)^2 - VCB'/(CB)^2
> (E production = inverse square of distance) - synchronicity - push versus pull.
> C'BE + CB'E + CBE'
> V production + DC current of dipoles from monopoles + AC current dipoles from monopoles.
>
> What I called synchronicity and the push versus pull in the Coulomb-gravity law above, looks to me that the force of gravity has to be a connection of the Planets to the Sun as a Parallel Circuit board of astronomy. This is probably the first time any scientist ever compared the Sun and its planets to an electric circuit board, but ladies and gentlemen, that is what gravity really is-- a form of EM.
> I never liked the Kirchhoff laws when doing them in college physics. My impression was-- who needs them, but now they come in critical importance. They fill out some of the True Physics. I am guessing no-one asked the question in Old Physics, which of the Maxwell Equations is Kirchhoff laws? Pretty sure no-one asked that question. Even Halliday and Resnick have it on page 677 before they begin to discuss the Maxwell Equations with Ampere's law on page 714. Old Physics treated the Kirchhoff's laws as some sort of periphery item, an item particular to circuits but not Maxwell theory, an item before Maxwell's Equation but not actually within Maxwell's Equation.
>
> AP suspects though that the Kirchoff's laws, not sure yet, but only a hunch, that the Kirchhoff's laws once well understood demand that Ohm's law of Old Physics as V = CR, that it is truly V = CBE where R, Resistance = Magnetic Field x Electric Field. It is my hunch but not yet proven that the bizarre case of adding more resistors actually decreases the overall resistance of a parallel circuit is explained by V= CBE. Such totally counterintuitive result should be easily explained with a V = CBE but not with a V= CR. That is, because of R=BE we have another factor than simply R alone.
>
> And in my textbook I dismiss the total Maxwell Equations as either in full error such as Gauss's law of no magnetic monopoles, when monopoles are the foundation of EM theory. Or the other laws in Maxwell Equations for they are missing many rules and laws, such as Lenz's law such as Kirchoff's laws.
>
> The trouble with Maxwell Equations, is they are not built from a sound foundation but rather Maxwell built them from "modeling experiments". When you build Physics from modeling, you capture some features of Nature, but bound to miss many. The firm sound foundation I speak of is to take New Ohm's Law Voltage = Coulomb x Magnetic Field x Electric Field take that as primal equation V= CBE and then differentiate all the permutations of V= CBE to gain all the laws of EM theory. James Clerk Maxwell should have looked to Mathematics for a foundation of EM Equations, especially to the idea that Volume in geometry encompasses all within its domain. So transfering that idea of Volume in Geometry is a completeness, then Voltage must be a volume type of formula such as V= CBE.
>
> There are 6 laws to complete EM theory, and not what Maxwell theorized to become just his 4 equations, but 6 equations.
>
> Those 6 can be written as this.
>
> 1) Magnetic Monopole has units: Magnetic Field B = kg/ A*s^2 = kg/ C*s
> Electric Field is E = kg*m^2/ A*s^2 = kg*m^2/ C*s
>
> 2) New Ohm's law V=CBE
>
> 3) C' = (V/(BE))' = V'BE/(BE)^2 - VB'E/(BE)^2 - VBE'/(BE)^2 which is Faraday's law.
> 1st term as current production -- 2nd term as Lenz law -- 3rd term as DC, AC direction
>
> 4) B' = (V/(CE))' = V'CE/(CE)^2 - VC'E/(CE)^2 - VCE')/(CE)^2 which is Ampere-Maxwell law.
> 1st term as B production -- 2nd term as Displacement current -- 3rd term as parallel attract
>
> 5) E' = (V/(CB))' = V'CB/(CB)^2 - VC'B/(CB)^2 - VCB'/(CB)^2 which is Coulomb-gravity law.
> 1st term as E production -- 2nd term as inverse square of distance -- 3rd term as synchronicity
>
> 6) V' = (CBE)' = C'BE + CB'E + CBE' which is electric generator law
> 1st term as V production -- 2nd term as DC generator -- 3rd term as AC generator
>
> Now Kirchhoff's laws one involves the Series circuit and the other involves the Parallel circuit where Voltage is constant but the resistance and current varies. So what I am looking for is one of the three terms in either V', E', B', C' looking for a term such as V/C'B or V/C'E to find Kirchoff's law for Parallel circuit. For Series circuit we have the current is constant, the Coulomb is constant and that means the V and R varies, and in AP equations R= BE. So I am looking for a term in V', E', B', C' that is this Kirchoff's law for Series in a term such as V'/CB' or V'/CE' to find Kirchhoff's law for Series circuit.
Series Circuit Problem Example.
Battery of 12 Volts
Three resistors R_1 = 1 ohm , R_2 = 2 ohm , R_3 = 3 ohm of bulbs
Current in circuit is V/R_summed = 12/6 = 2 A
Current is a constant while voltage and resistance vary. In Parallel circuit the Voltage is constant while current and resistance vary.
Voltage drop across R_1 is V_1 = iR_1 , and so 2 A x 1ohms = 2 V
Voltage drop across R_2 is V_2 = iR_2 , and so 2A x 2ohms= 4 V
Voltage drop across R_3 is V_3 = iR_3 , and so 2A x 3ohms= 6 V
In a series connection you have a Voltage drop across the circuit, whereas in parallel the voltage is a constant.
From H&R, page 677
Kirchhoff's 1st law (Junction Rule) : The sum of the currents entering any junction must be equal to the sum of the currents leaving that junction. (Conservation of electric monopoles)
Kirchhoff's 2nd law (Loop Rule) : The algebraic sum of the changes in potential encountered in a complete traversal of any closed circuit must be zero. (Conservation of energy)
So why did not the Kirchhoff laws appear in Maxwell Equations? Why are they periphery to the Maxwell Equations in Old Physics? The answer lies in the fact that Old Physics had no magnetic monopole and had their Maxwell Equations based upon modeling, rather than based on the calculus permutations of V= CBE.