<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>With a rotating black hole, what is rotating? The event horizon is not\na physical object. It\'s just defined as the set of points such and\nsuch distance from the singularity, so how can it be rotating? If you\ndefined the "effielsphere" as the set of points exactly 100 meters\nfrom the top of the Effiel Tower, and then imagined that sphere as\n"rotating", what would that mean? The event horizons are spherically\nsymmetric, and so are exactly the same whether they are "rotating" or\nnot. The spacetime curvature caused by the black hole is spherically\nsymmetric, so how would you know if it was "rotating"? The singularity\nis just a mathematical point, and this can\'t be "rotating". If you had\na rotating physical object like a planet or a neutron star, there are\nactual subatomic particles that are spinning around about the center,\nbut a black hole is not made of subatomic particles. If a physical\nobject like a planet or neutron star is rotating, you could imagine\nthat somehow spacetime is "attached" to the subatomic particles that\nmake it up, and thus somehow "dragged along", but a black hole is not\nmade of subatomic particles. It\'s just the singularity, which is a\nmathematical point, surrounded by the event horizon, which is just\ndefined as such and such distance from the mathematical point. Also,\nyou could say the coordinate system is rotating, but the coordinate\nsystem is purely a human invention, and if you wanted to, you could\nset up a rotating coordinate system for stationary object.\n\nDavid\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>With a rotating black hole, what is rotating? The event horizon is not
a physical object. It's just defined as the set of points such and
such distance from the singularity, so how can it be rotating? If you
defined the "effielsphere" as the set of points exactly 100 meters
from the top of the Effiel Tower, and then imagined that sphere as
"rotating", what would that mean? The event horizons are spherically
symmetric, and so are exactly the same whether they are "rotating" or
not. The spacetime curvature caused by the black hole is spherically
symmetric, so how would you know if it was "rotating"? The singularity
is just a mathematical point, and this can't be "rotating". If you had
a rotating physical object like a planet or a neutron star, there are
actual subatomic particles that are spinning around about the center,
but a black hole is not made of subatomic particles. If a physical
object like a planet or neutron star is rotating, you could imagine
that somehow spacetime is "attached" to the subatomic particles that
make it up, and thus somehow "dragged along", but a black hole is not
made of subatomic particles. It's just the singularity, which is a
mathematical point, surrounded by the event horizon, which is just
defined as such and such distance from the mathematical point. Also,
you could say the coordinate system is rotating, but the coordinate
system is purely a human invention, and if you wanted to, you could
set up a rotating coordinate system for stationary object.
David
tessel@tum.bot
Jul1-04, 04:47 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 29 Jun 2004, Ulmo wrote:\n\n> With a rotating black hole, what is rotating?\n\nIn a sense, the gravitational field itself.\n\nAs it happens, this is a question I thought about some years ago, so I can\nprobably say much more about this than you really want to hear. To sum up\nin advance: like a zillion other things in gravitation theory, the simple\nquestion "what is rotation?" turns out to be much more subtle in gtr than\nmost people can easily appreciate, even if they have possess a lot of the\nrequisite background. I stress that I am speaking here even of the\nrotation of ordinary objects like a spinning top, although the conceptual\nand philosophical problems are not really urgent until we consider\nsomething like a rotating neutron star.\n\nLet me try to say a few things about various ways in which many authors\nhave attempted to define notions of rotation in curved spacetimes in the\ncontext of gravitation theory, especially the current default theory,\nwhich is gtr.\n\nOne basic point is that one arrives at mathematical definitions by\ngeneralizing some definition which is familiar from flat spacetime.\nThen, because as it turns out there many confusing things come to light,\nwe must try to justify our proposed new definition by proving theorems to\nthe effect that it really does behave (in precise ways) the way that we\nexpect (but probably only after much thought about what we "should"\nexpect!).\n\nAs you work through various possible definitions, some further points\nemerge. First, there is, as usual, a crucial distinction between local\nand global definitions. Second, you see again and again that definitions\nwhich are fully equivalent in flat spacetime typically are distinct in\ncurved spacetime. Indeed, by considering examples you will probably be\nled to the conclusion that any notion of "rotation" which may be familiar\nto you from your experience of how objects move in a nominal E^3 is very\nnaive; in strongly curved spacetimes, the notion of "rotation" separates\nout into numerous distinct concepts. One then has to struggle to\nunderstand their meaning by considering examples and trying to induct\ngeneral theorems, in each case. Then one should try to understand how the\nvarious definitions are related to one another, and what when can say\nabout their physical significance in various situations.\n\nOne general approach to this problem is to try to define various notions\nof "rotation type X" by looking for various mathematical criteria for when\nan object is "nonrotating". One of the simplest examples is a local notion\ncriterion for a nonrotating spacetime (nonrotating gravitational field,\nmeaning a field typical of that produced by a nonrotating object such as a\nhypothetical nonrotating solid sphere) which is suggested in\n\nauthor = {Ignazio Ciufolini and John Archibald Wheeler},\ntitle = {Gravitation and Inertia},\npublisher = {Princeton University Press},\nyear = 1995}\n\nThe authors recall the standard interpretation of the scalar invariants\n\nF_(ab) F^(ab), *F_(ab) F^(ab)\n\nof the EM field tensor, for which see\n\nauthor = {L. D. Landau and E. M. Lifshitz},\ntitle = {The Classical Theory of Fields},\nseries = {Course of Theoretical Physics},\nvolume = 2,\nedition = {Fourth},\npublisher = {Pergamon},\nyear = 1975}\n\nNote our interpretation rests upon simple theorems to the effect that\ndepending on whether or not the invariants are zero or not in some region\nof spacetime, there may or may not exist "adapted" observers who see a\nmuch simplified EM field. See also Feynman\'s discussion in the Lectures\nof transforming away a magnetic field.\n\nNext, an obvious formal analogy suggests that the scalar curvature\ninvariants\n\nR_(abcd) R^(abcd), *R_(abcd) R^(abcd)\n\nmight play an analogous role for the gravitational field. In particular,\nthe analogy suggests that\n\n*R_(abcd) R^(abcd) =/=0\n\nin some region is associated with "essential magnetogravitism" in that\nregion, i.e. "magnetogravitic" effects will be in principle be measured by\n-all- observers, although of course the details will differ from one\nobserver to the next.\n\nNow invoke another chain of analogies which suggests that\n"magnetogravitism" is the "physical trace" of rotation of the source of\nthe field. In the case of a rotating object, you probably have some\ninitial intuition which is naive (as it turns out) but not entirely\nmisleading. In the case of a black hole, the magnetogravitism must\npersist in the exterior field for the same reason that the gravitational\nfield itself must persist--- even though the source of the field has\nvanished behind an event horizon which was created by the collapse, the\n"news" that the matter which was the initial source of the field has\n"vanished" (at least, vanished from that part of the universe which can\ncausally affect future events we in the exterior can someday experience),\ncannot propagate out from under the horizon. This is just like the way we\nusually explain a notorious faq, "how does gravity get out from the event\nhorizon of a black hole?", except that now we are focusing on the trace of\n"angular momentum of formerly accessible source" rather than the trace of\n"mass of formerly accessible source".\n\nI assume you know, BTW, that one major difference between Newtonian\ngravitation and gtr is that in the former, there is no difference between\nthe field of a nonrotating and rotating spherically symmetric uniform\ndensity solid ball, but in the latter, there is. This is becaues the\nsource of the gravitational field in gtr is the energy-momentum-stress\ntensor T^(ab) (with contributions from any matter and any nongravitational\nfields present), not just the matter density. (Yes, this is circular as\nstated, since I claim we don\'t yet know what "rotating" means in gtr, but\nyou can make some sense of my thought experiment by using weak-field\ngravitation, which by the way is another common and sometimes useful\napproach to trying to figure out almost any puzzle of interpretation.)\n\nI stress again that there is little difference between the gravitational\nfield of a rotating star and the field of a rotating hole having the same\nmass and angular momentum, until one reaches the surface of the star.\nWhatever difference you might measure would normally result from higher\nmoments which can be supported by a material object like a star, but not\nby a hole, which quickly settles down after any small perturbation to the\nsimplest possible field for an object with a given mass/angular momentum\n(namely Kerr)--- at least, in the exterior region outside the horizon.\n\nNext, still sticking to isolated compact objects like stars, you can\ninvoke the machinery of asymptotically flat spacetimes. Essential\nbackground is the way that moments arise in good old Newtonian\ngravitation--- or, by another useful formal analogy, Gaussian\nelectrostatics (keyword: "spherical harmonics", "Legendre functions").\nBut we also have angular momentum and stress plus higher moments for\nthese! But wait--- many of these can be transformed away. We seek\ninvariant characterizations, so we naturally want to transform away as\nmuch complexity as possible. Previously, we were "transforming away"\nneedless complications by choosing an adapted frame (a local concept).\nHere, we "transform away" needless complications by choosing an adapted\ncoordinate system (global). Previously, we dealt with scalar invariants of\ntensors (local, meaningful at each event); here, we deal with moments\nobtained by integration and we invokved in an essential way a global\nassumption (asymptotic flatness).\n\nThis is all very sketchy, but it suggests some small part of the\nbackground you must master before you can begin to understand just how\ncomplicated the situation really is.\n\nBTW, some readers may know a book with a relevant-sounding title\n\nauthor = {Jamal N. Islam},\ntitle = {Rotating Fields in General Relativity},\npublisher = {Cambridge University Press},\nyear = 1995}\n\nBut--- at least as far as I can see--- in this book, Islam doesn\'t attempt\nto seriously address the question at hand (this book is concerned with\ntechnical problems arising in pursuing certain assumptions, not to\nexamination of those assumptions from a philosopher-physicist POV).\nCiufolini & Wheeler do make a half-hearted attempt to say something about\nthe question of whether anyone who refers to "rotation" in the context of\ngtr really knows what he is talking about, but their comments are IMO only\na very modest beginning. IMO, one could write more than one book on your\nquestion, and in the future I expect such books will appear. (No, I\nmyself am not writing one, at least not yet!)\n\n> The event horizon is not a physical object. It\'s just defined as the set\n> of points such and such distance from the singularity,\n\nNo! That is -not- how it is defined at all. "Distance from the\nsingularity" doesn\'t even make sense.\n\nHmmm... I think you need to back up, way up, and get some -much- more\nelementary intuition for how the gravitational field works in gtr. Then\nyou can try to fill the gaps in what I said above. Then you can ask your\nquestion again.\n\nFor elementary intuition, see the John Baez\'s essay, "What is the EFE?",\nor something like that, on his home page. See also\n\nauthor = {Geroch, Robert},\ntitle = {General relativity from {A} to {B}},\npublisher = {University of Chicago Press},\nyear = 1978}\n\nwhich is very easy but will help a lot, I think. Then see the semipopular\nbook by Wald. Then see previous discussions here of "Vaidya models"---\nthis is by far the best way to really appreciate the local/global\ndistinction which is to critical here, as in so many places in this\nsubject.\n\n> The event horizons are spherically symmetric,\n\nNo. We have discussed this point in detail, but right now this is too\nadvanced for you, I think.\n\n> and so are exactly the same whether they are "rotating" or not.\n\nNever mind holes for a moment. Back up an try to understand the surface\nof rotating star. The "too advanced" question I just referred to is\nthis: is the surface isometric to a sphere? Or to an oblate spheroid? Or\nto some still more complicated surface.\n\nCompare "equipotentials" of the gravitational potential (how are they\nrelated to the surface, in the case of a ball of gas "held together" by\nits self-gravitation, and "held up" by pressure?).\n\nNow go back to what I said above ("I assume you know.."--- I assumed\nwrongly, obviously) about rotating versus nonrotating balls in Newtonian\ngravitity versus gtr.\n\n> The spacetime curvature caused by the black hole is spherically\n> symmetric,\n\nNot if its rotating. The Kerr vacuum is -not- spherically symmetric.\n\n> so how would you know if it was "rotating"?\n\nWell, for one thing, interpretional difficulties aside, there is no doubt\nwhatever that gtr makes precise predictions about the behavior of small\nobjects orbiting a hole. It turns out that "rotation" has consequences\nwhich are in principle observable, irrespective of what kind of "rotation"\nwe mean, or what it means to say that something is rotating. OTH, there\nis not much difference here between Newtonian grav. and gtr, far from the\nobject, and we know (yes?) how to measure angular momentum in Newtonian\ngravity, although in practice this may not be easy. Interestingly enough,\nclose to an astrophysical black hole, there can be rather profound effects\nspecifically associated with rotation. An amusing example is current\nresearch aimed at literally "seeing" the horizon (or more properly a\nslighlty different "outline", a generalization of the surface of stable\ncircular photon orbits from Schwarzshild to Kerr) of the supermassive\nblack hole near the center of the Milky Way. If the hole is rotating fast\nenough, its "shadow" will look "flattened on one side".\n\n> The singularity is just a mathematical point,\n\nNo, not for Kerr. This is even more subtle, though, so forget it for now.\n\n> If you had a rotating physical object like a planet or a\n> neutron star,\n\nYes, stick to this for now.\n\n> there are actual subatomic particles that are spinning around about the\n> center, but a black hole is not made of subatomic particles. If a\n> physical object like a planet or neutron star is rotating, you could\n> imagine that somehow spacetime is "attached" to the subatomic particles\n> that make it up, and thus somehow "dragged along", but a black hole is\n> not made of subatomic particles. It\'s just the singularity, which is a\n> mathematical point, surrounded by the event horizon, which is just\n> defined as such and such distance from the mathematical point.\n\nNow you are back to the same fallacy addressed in the FAQ ("How does\nGravity Get Out of a Black Hole?"), unfortunately. So grok that, then\nthink about rotating stars, and only then should you think about rotating\nholes.\n\n> Also, you could say the coordinate system is rotating, but the\n> coordinate system is purely a human invention, and if you wanted to, you\n> could set up a rotating coordinate system for stationary object.\n\nYes, in fact you can set up a rotating coordinate system for -no- object:\nI could write down various examples of "rotating" coordinate charts for\nMinkowski spacetime. Perhaps easier to understand is the idea of\n"rotating" -frames-, corresponding to families of observers who twist\nabout one another (local notion) or who rotate about some axis (global\nnotion) in the sense you probably have in mind. This is why, even in flat\nspacetime, "invariant characterizations" of tensor fields are needed.\nHence the discussion of the EM field tensor with which I began.\n\nUnderstanding "invariant characterizations" of the metric tensor (things\nwhich describe the metric, not the coordinate chart in which we have,\nperhaps, written it out) is much more challenging--- but I\'ve already said\nway too much. Sorry if you\'re now more confused than ever--- at least\nmaybe you\'ll see that there\'s a lot to be confused -about- here! All your\nconfusions concern misconceptions which just about everyone has\ninitially--- the problem is that I started out by shooting way over your\nhead.\n\n"T. Essel" (hiding somewhere in cyberspace)\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 29 Jun 2004, Ulmo wrote:
> With a rotating black hole, what is rotating?
In a sense, the gravitational field itself.
As it happens, this is a question I thought about some years ago, so I can
probably say much more about this than you really want to hear. To sum up
in advance: like a zillion other things in gravitation theory, the simple
question "what is rotation?" turns out to be much more subtle in gtr than
most people can easily appreciate, even if they have possess a lot of the
requisite background. I stress that I am speaking here even of the
rotation of ordinary objects like a spinning top, although the conceptual
and philosophical problems are not really urgent until we consider
something like a rotating neutron star.
Let me try to say a few things about various ways in which many authors
have attempted to define notions of rotation in curved spacetimes in the
context of gravitation theory, especially the current default theory,
which is gtr.
One basic point is that one arrives at mathematical definitions by
generalizing some definition which is familiar from flat spacetime.
Then, because as it turns out there many confusing things come to light,
we must try to justify our proposed new definition by proving theorems to
the effect that it really does behave (in precise ways) the way that we
expect (but probably only after much thought about what we "should"
expect!).
As you work through various possible definitions, some further points
emerge. First, there is, as usual, a crucial distinction between local
and global definitions. Second, you see again and again that definitions
which are fully equivalent in flat spacetime typically are distinct in
curved spacetime. Indeed, by considering examples you will probably be
led to the conclusion that any notion of "rotation" which may be familiar
to you from your experience of how objects move in a nominal E^3 is very
naive; in strongly curved spacetimes, the notion of "rotation" separates
out into numerous distinct concepts. One then has to struggle to
understand their meaning by considering examples and trying to induct
general theorems, in each case. Then one should try to understand how the
various definitions are related to one another, and what when can say
about their physical significance in various situations.
One general approach to this problem is to try to define various notions
of "rotation type X" by looking for various mathematical criteria for when
an object is "nonrotating". One of the simplest examples is a local notion
criterion for a nonrotating spacetime (nonrotating gravitational field,
meaning a field typical of that produced by a nonrotating object such as a
hypothetical nonrotating solid sphere) which is suggested in
author = {Ignazio Ciufolini and John Archibald Wheeler},
title = {Gravitation and Inertia},
publisher = {Princeton University Press},
year = 1995}
The authors recall the standard interpretation of the scalar invariants
F_(ab) F^(ab), *F_(ab) F^(ab)
of the EM field tensor, for which see
author = {L. D. Landau and E. M. Lifshitz},
title = {The Classical Theory of Fields},
series = {Course of Theoretical Physics},
volume = 2,
edition = {Fourth},
publisher = {Pergamon},
year = 1975}
Note our interpretation rests upon simple theorems to the effect that
depending on whether or not the invariants are zero or not in some region
of spacetime, there may or may not exist "adapted" observers who see a
much simplified EM field. See also Feynman's discussion in the Lectures
of transforming away a magnetic field.
Next, an obvious formal analogy suggests that the scalar curvature
invariants
R_(abcd) R^(abcd), *R_(abcd) R^(abcd)
might play an analogous role for the gravitational field. In particular,
the analogy suggests that
*R_(abcd) R^(abcd) =/=0
in some region is associated with "essential magnetogravitism" in that
region, i.e. "magnetogravitic" effects will be in principle be measured by
-all- observers, although of course the details will differ from one
observer to the next.
Now invoke another chain of analogies which suggests that
"magnetogravitism" is the "physical trace" of rotation of the source of
the field. In the case of a rotating object, you probably have some
initial intuition which is naive (as it turns out) but not entirely
misleading. In the case of a black hole, the magnetogravitism must
persist in the exterior field for the same reason that the gravitational
field itself must persist--- even though the source of the field has
vanished behind an event horizon which was created by the collapse, the
"news" that the matter which was the initial source of the field has
"vanished" (at least, vanished from that part of the universe which can
causally affect future events we in the exterior can someday experience),
cannot propagate out from under the horizon. This is just like the way we
usually explain a notorious faq, "how does gravity get out from the event
horizon of a black hole?", except that now we are focusing on the trace of
"angular momentum of formerly accessible source" rather than the trace of
"mass of formerly accessible source".
I assume you know, BTW, that one major difference between Newtonian
gravitation and gtr is that in the former, there is no difference between
the field of a nonrotating and rotating spherically symmetric uniform
density solid ball, but in the latter, there is. This is becaues the
source of the gravitational field in gtr is the energy-momentum-stress
tensor T^(ab) (with contributions from any matter and any nongravitational
fields present), not just the matter density. (Yes, this is circular as
stated, since I claim we don't yet know what "rotating" means in gtr, but
you can make some sense of my thought experiment by using weak-field
gravitation, which by the way is another common and sometimes useful
approach to trying to figure out almost any puzzle of interpretation.)
I stress again that there is little difference between the gravitational
field of a rotating star and the field of a rotating hole having the same
mass and angular momentum, until one reaches the surface of the star.
Whatever difference you might measure would normally result from higher
moments which can be supported by a material object like a star, but not
by a hole, which quickly settles down after any small perturbation to the
simplest possible field for an object with a given mass/angular momentum
(namely Kerr)--- at least, in the exterior region outside the horizon.
Next, still sticking to isolated compact objects like stars, you can
invoke the machinery of asymptotically flat spacetimes. Essential
background is the way that moments arise in good old Newtonian
gravitation--- or, by another useful formal analogy, Gaussian
electrostatics (keyword: "spherical harmonics", "Legendre functions").
But we also have angular momentum and stress plus higher moments for
these! But wait--- many of these can be transformed away. We seek
invariant characterizations, so we naturally want to transform away as
much complexity as possible. Previously, we were "transforming away"
needless complications by choosing an adapted frame (a local concept).
Here, we "transform away" needless complications by choosing an adapted
coordinate system (global). Previously, we dealt with scalar invariants of
tensors (local, meaningful at each event); here, we deal with moments
obtained by integration and we invokved in an essential way a global
assumption (asymptotic flatness).
This is all very sketchy, but it suggests some small part of the
background you must master before you can begin to understand just how
complicated the situation really is.
BTW, some readers may know a book with a relevant-sounding title
author = {Jamal N. Islam},
title = {Rotating Fields in General Relativity},
publisher = {Cambridge University Press},
year = 1995}
But--- at least as far as I can see--- in this book, Islam doesn't attempt
to seriously address the question at hand (this book is concerned with
technical problems arising in pursuing certain assumptions, not to
examination of those assumptions from a philosopher-physicist POV).
Ciufolini & Wheeler do make a half-hearted attempt to say something about
the question of whether anyone who refers to "rotation" in the context of
gtr really knows what he is talking about, but their comments are IMO only
a very modest beginning. IMO, one could write more than one book on your
question, and in the future I expect such books will appear. (No, I
myself am not writing one, at least not yet!)
> The event horizon is not a physical object. It's just defined as the set
> of points such and such distance from the singularity,
No! That is -not- how it is defined at all. "Distance from the
singularity" doesn't even make sense.
Hmmm... I think you need to back up, way up, and get some -much- more
elementary intuition for how the gravitational field works in gtr. Then
you can try to fill the gaps in what I said above. Then you can ask your
question again.
For elementary intuition, see the John Baez's essay, "What is the EFE?",
or something like that, on his home page. See also
author = {Geroch, Robert},
title = {General relativity from {A} to {B}},
publisher = {University of Chicago Press},
year = 1978}
which is very easy but will help a lot, I think. Then see the semipopular
book by Wald. Then see previous discussions here of "Vaidya models"---
this is by far the best way to really appreciate the local/global
distinction which is to critical here, as in so many places in this
subject.
> The event horizons are spherically symmetric,
No. We have discussed this point in detail, but right now this is too
advanced for you, I think.
> and so are exactly the same whether they are "rotating" or not.
Never mind holes for a moment. Back up an try to understand the surface
of rotating star. The "too advanced" question I just referred to is
this: is the surface isometric to a sphere? Or to an oblate spheroid? Or
to some still more complicated surface.
Compare "equipotentials" of the gravitational potential (how are they
related to the surface, in the case of a ball of gas "held together" by
its self-gravitation, and "held up" by pressure?).
Now go back to what I said above ("I assume you know.."--- I assumed
wrongly, obviously) about rotating versus nonrotating balls in Newtonian
gravitity versus gtr.
> The spacetime curvature caused by the black hole is spherically
> symmetric,
Not if its rotating. The Kerr vacuum is -not- spherically symmetric.
> so how would you know if it was "rotating"?
Well, for one thing, interpretional difficulties aside, there is no doubt
whatever that gtr makes precise predictions about the behavior of small
objects orbiting a hole. It turns out that "rotation" has consequences
which are in principle observable, irrespective of what kind of "rotation"
we mean, or what it means to say that something is rotating. OTH, there
is not much difference here between Newtonian grav. and gtr, far from the
object, and we know (yes?) how to measure angular momentum in Newtonian
gravity, although in practice this may not be easy. Interestingly enough,
close to an astrophysical black hole, there can be rather profound effects
specifically associated with rotation. An amusing example is current
research aimed at literally "seeing" the horizon (or more properly a
slighlty different "outline", a generalization of the surface of stable
circular photon orbits from Schwarzshild to Kerr) of the supermassive
black hole near the center of the Milky Way. If the hole is rotating fast
enough, its "shadow" will look "flattened on one side".
> The singularity is just a mathematical point,
No, not for Kerr. This is even more subtle, though, so forget it for now.
> If you had a rotating physical object like a planet or a
> neutron star,
Yes, stick to this for now.
> there are actual subatomic particles that are spinning around about the
> center, but a black hole is not made of subatomic particles. If a
> physical object like a planet or neutron star is rotating, you could
> imagine that somehow spacetime is "attached" to the subatomic particles
> that make it up, and thus somehow "dragged along", but a black hole is
> not made of subatomic particles. It's just the singularity, which is a
> mathematical point, surrounded by the event horizon, which is just
> defined as such and such distance from the mathematical point.
Now you are back to the same fallacy addressed in the FAQ ("How does
Gravity Get Out of a Black Hole?"), unfortunately. So grok that, then
think about rotating stars, and only then should you think about rotating
holes.
> Also, you could say the coordinate system is rotating, but the
> coordinate system is purely a human invention, and if you wanted to, you
> could set up a rotating coordinate system for stationary object.
Yes, in fact you can set up a rotating coordinate system for -no- object:
I could write down various examples of "rotating" coordinate charts for
Minkowski spacetime. Perhaps easier to understand is the idea of
"rotating" -frames-, corresponding to families of observers who twist
about one another (local notion) or who rotate about some axis (global
notion) in the sense you probably have in mind. This is why, even in flat
spacetime, "invariant characterizations" of tensor fields are needed.
Hence the discussion of the EM field tensor with which I began.
Understanding "invariant characterizations" of the metric tensor (things
which describe the metric, not the coordinate chart in which we have,
perhaps, written it out) is much more challenging--- but I've already said
way too much. Sorry if you're now more confused than ever--- at least
maybe you'll see that there's a lot to be confused -about- here! All your
confusions concern misconceptions which just about everyone has
initially--- the problem is that I started out by shooting way over your
head.
"T. Essel" (hiding somewhere in cyberspace)
Jonathan Thornburg
Jul1-04, 04:48 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Ulmo <ulmo@cheerful.com> wrote:\n> With a rotating black hole, what is rotating? The event horizon is not\n> a physical object. It\'s just defined as the set of points such and\n> such distance from the singularity, so how can it be rotating?\n\nNot really.\n\nThe event horizon is actually defined as the boundary of the set S,\nwhere an event ("point in spacetime") P is in S if and only if there\'s\nan "escape route" (technically, a future-pointing null geodesic)\nfrom P to "infinity" (technically, future null infinity).\nThat is, the event horizon is the boundary between those points from\nwhich you can send a light signal out to infinity, and those points\nfrom which you can\'t.\n\n\n> If you\n> defined the "effielsphere" as the set of points exactly 100 meters\n> from the top of the Effiel Tower, and then imagined that sphere as\n> "rotating", what would that mean? The event horizons are spherically\n> symmetric\n\nNo, event horizons are in general NOT spherically symmetric.\nNotably, the event horizon of Kerr spacetime is not spherically\nsymmetric. As another example, my colleague Peter Diener has developed\na numerical code for finding event horizons in spacetimes which are\nonly known numerically, not analytically. His paper\n\nPeter Diener\n"A New General Purpose Event Horizon Finder for 3D Numerical Spacetimes"\nClass. Quantum Gravity 20 (2003), 4901-4917\nhttp://stacks.iop.org/0264-9381/20/4901\npreprint gr-qc/0305039\nhttp://arxiv.org/abs/gr-qc/0306056\n\nincludes a movie showing the event horizon in a collision of two black\nholes. This event horizon is *highly* nonspherical.\n\n\n> The spacetime curvature caused by the black hole is spherically\n> symmetric\n\nIn "interesting cases" it\'s not spherical, eg it\'s not spherical in\nKerr spacetime or in the black-hole-collision spacetime of Diener\'s paper.\n\n\n> so how would you know if it was "rotating"? The singularity\n> is just a mathematical point, and this can\'t be "rotating".\n\nA black hole is much more than just a singularity,\nit\'s a region of spacetime. The key question is, is there any\nmeasurement we can make from outside this region, which we can\nreasonably interpret as telling us something about the rotation\nof this region? The answer is yes: If we put satellites into orbit\naround the black hole, we\'ll find that for the same orbital radius,\nthe orbital periods differ depending on the angular orientation of\nthe orbit. This difference allows us to meaningfully define the\nspin (angular momentum) of the black hole.\n\nciao,\n\n--\n-- "Jonathan Thornburg (remove -animal to reply)" <jthorn@aei.mpg-zebra.de>\nMax-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),\nGolm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html\n"Washing one\'s hands of the conflict between the powerful and the\npowerless means to side with the powerful, not to be neutral."\n-- quote by Freire / poster by Oxfam\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Ulmo <ulmo@cheerful.com> wrote:
> With a rotating black hole, what is rotating? The event horizon is not
> a physical object. It's just defined as the set of points such and
> such distance from the singularity, so how can it be rotating?
Not really.
The event horizon is actually defined as the boundary of the set S,
where an event ("point in spacetime") P is in S if and only if there's
an "escape route" (technically, a future-pointing null geodesic)
from P to "infinity" (technically, future null infinity).
That is, the event horizon is the boundary between those points from
which you can send a light signal out to infinity, and those points
from which you can't.
> If you
> defined the "effielsphere" as the set of points exactly 100 meters
> from the top of the Effiel Tower, and then imagined that sphere as
> "rotating", what would that mean? The event horizons are spherically
> symmetric
No, event horizons are in general NOT spherically symmetric.
Notably, the event horizon of Kerr spacetime is not spherically
symmetric. As another example, my colleague Peter Diener has developed
a numerical code for finding event horizons in spacetimes which are
only known numerically, not analytically. His paper
Peter Diener
"A New General Purpose Event Horizon Finder for 3D Numerical Spacetimes"
Class. Quantum Gravity 20 (2003), 4901-4917
http://stacks.iop.org/0264-9381/20/4901
preprint http://www.arxiv.org/abs/gr-qc/0305039
http://arxiv.org/abs/http://www.arxiv.org/abs/gr-qc/0306056
includes a movie showing the event horizon in a collision of two black
holes. This event horizon is *highly* nonspherical.
> The spacetime curvature caused by the black hole is spherically
> symmetric
In "interesting cases" it's not spherical, eg it's not spherical in
Kerr spacetime or in the black-hole-collision spacetime of Diener's paper.
> so how would you know if it was "rotating"? The singularity
> is just a mathematical point, and this can't be "rotating".
A black hole is much more than just a singularity,
it's a region of spacetime. The key question is, is there any
measurement we can make from outside this region, which we can
reasonably interpret as telling us something about the rotation
of this region? The answer is yes: If we put satellites into orbit
around the black hole, we'll find that for the same orbital radius,
the orbital periods differ depending on the angular orientation of
the orbit. This difference allows us to meaningfully define the
spin (angular momentum) of the black hole.
ciao,
--
-- "Jonathan Thornburg (remove -animal to reply)" <jthorn@aei.mpg-zebra.de>
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\ntessel@tum.bot writes\n>On Tue, 29 Jun 2004, Ulmo wrote:\n>\n>> With a rotating black hole, what is rotating?\n>\n>In a sense, the gravitational field itself.\n\n<snip another mindblowing trip through the conceptual details, most of\nwhich goes "whoosh" over my head>\n\nA question.\n\nIt strikes me that GR is in essence a local description. Once you go too\nfar from local then you had better be very careful what question you ask\nand how you want the answer stated. I suspect that if you can properly\nprecisely state the question, and understand the answer, the bit in\nbetween is the easy bit. Anyway.....\n\nTo cut to the really simple concept that even I can more or less\ncomprehend, for an observer in a gravitational field I would make two\nstatements (which I pray are right) and one guess.\n\n1) GR in a static (locally parallel) field means that locally,\ngravitational fields are undetectable.\n\n2) GR in a nonrotating non parallel field (4D) can be detected locally\nby tidal action.\n\n3) GR in a rotating gravitational field should (I hope/guess) see a\nrotational equivalent of tidal action. That is two slightly separated\nbodies initially \'at rest\' to each other will start to rotate over time.\n\nOf course that might slightly beg the question of how a pointlike\nobserver actually rotates, but QM suggest in some sense it does. I was\n(briefly, before my hair stood on end) recently introduced to dirac\'s\nconcept that (and I probably have this wrong) ANY 4D representation\n(even of a point) requires spin (or a rotational concept) to be\nconsistent with a 4D (possibly 3+1D) description. If so I would expect\nsomething describing rotation to still remain non-zero even as the size\nof the particle tends to zero.\n\nOnce you want to extend \'local\' to \'bigger than local\', I suspect the\nmaths and the conceptual difficulties that can only be appreciated if\nyou can handle the math, become such that we leave this to people with\nthe aptitude and determination to have their braincells fried. I\'ll\nleave that to you, and others, just come and do some handwaving in our\ndirection from time to time.....\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>tessel@tum.bot writes
>On Tue, 29 Jun 2004, Ulmo wrote:
>
>> With a rotating black hole, what is rotating?
>
>In a sense, the gravitational field itself.
<snip another mindblowing trip through the conceptual details, most of
which goes "whoosh" over my head>
A question.
It strikes me that GR is in essence a local description. Once you go too
far from local then you had better be very careful what question you ask
and how you want the answer stated. I suspect that if you can properly
precisely state the question, and understand the answer, the bit in
between is the easy bit. Anyway.....
To cut to the really simple concept that even I can more or less
comprehend, for an observer in a gravitational field I would make two
statements (which I pray are right) and one guess.
1) GR in a static (locally parallel) field means that locally,
gravitational fields are undetectable.
2) GR in a nonrotating non parallel field (4D) can be detected locally
by tidal action.
3) GR in a rotating gravitational field should (I hope/guess) see a
rotational equivalent of tidal action. That is two slightly separated
bodies initially 'at rest' to each other will start to rotate over time.
Of course that might slightly beg the question of how a pointlike
observer actually rotates, but QM suggest in some sense it does. I was
(briefly, before my hair stood on end) recently introduced to dirac's
concept that (and I probably have this wrong) ANY 4D representation
(even of a point) requires spin (or a rotational concept) to be
consistent with a 4D (possibly 3+1D) description. If so I would expect
something describing rotation to still remain non-zero even as the size
of the particle tends to zero.
Once you want to extend 'local' to 'bigger than local', I suspect the
maths and the conceptual difficulties that can only be appreciated if
you can handle the math, become such that we leave this to people with
the aptitude and determination to have their braincells fried. I'll
leave that to you, and others, just come and do some handwaving in our
direction from time to time.....
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<
Eric Baird
Jul16-04, 08:19 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nOn Tue, 29 Jun 2004 21:32:54 +0000 (UTC), ulmo@cheerful.com (Ulmo)\nwrote:\n\n>The event horizons are spherically\n>symmetric, and so are exactly the same whether they are "rotating" or\n>not. The spacetime curvature caused by the black hole is spherically\n>symmetric, so how would you know if it was "rotating"?\n\nThat\'s a decent argument, and yes ... one can argue that a pure\npoint-singularity should not have enough detail to support the\nproperty of rotation, since the fieldlines always have to exit\nperpendicularly from a point, so a "supposedly spinning" point and a\n"supposedly non-spinning" point might be expected to have the same\nexteriors ...\n\n.... but those "spherically-symmetrical black hole" arguments are only\nmeant to apply to /non/-rotating holes, and the rules are supposed to\nbe a bit different when a hole is spinning wrt its background.\n\nFor a rotating hole, the side profile is supposed to be more like an\nellipse, the horizon extends further because of the energy tied up in\nthe relative rotation between the hole and its environment, and the\nreceding side of the hole pulls at you more strongly than the\napproaching side.\nThe hole\'s external field has a "twist" to it that describes the\nhole\'s rotation, and the idealised singularity that we can invoke as\nbeing the minimal expression of that exterior field in a standard hole\nis no longer a central point, but a ring surrounding the central\nposition. The planar region bounded by the ring is a bit odd, so you\nmight sometimes see it referred to as a singularity, too.\n\nThere\'s still possibly a slight awkwardness in that the ring itself\ndoesn\'t tell you the /sense/ of the rotation, but the angles that the\nfieldlines splay off at gives you that.\n\nAs for the bit about fieldlines exiting the horizon at 90 degrees, the\napparent position of the horizon seems to be observer-dependent, so\nalthough an array of observers sprinkled around the hole on its\nequatorial plane might well think that all fieldlines are pointing to\nthe hole\'s apparent centre, their ideas about where that supposed\ncentre /\'is/, gleaned from their observations of the hole taken from\ntheir own individual positions, is likely to disagree. They can all\npoint back along a fieldline towards the ring, but may be pointing\ntowards different points on the ring. Methinks that each individual\nobserver probably thinks that the hole\'s centre is offset to one side\nof the actual geometrical centre, with the offset being towards the\nreceding side. Certainly by measuring the gravitational attraction of\nthe hole, the stronger attraction of the receding side would mean that\ntheir idea of the hole\'s apparent centre of gravitational mass would\nbe offset to that side, and the frame-dragging spacetime "twist" would\nmean that they\'d tend to think that the hole\'s fieldlines intersecting\ntheir position were coming from somewhere over to that side.\nYou can also try to describe an apparent visual offset in the position\nof the hole towards that side, due to differential gravitational\nlensing ... the side that pulls more strongly produces a stronger\ngravitational "magnifying lens" effect, and the hole\'s profile seems\nto extend further on that side. Or you can use all sorts of Doppler or\ntimelag or distortion arguments to argue that the observer is seeing\nmore surface on the receding side because in a sense they are actually\nseeing around a corner and their sightlines are intersecting horizon\npoints that are past the (nongravitaitonal!) horizon. There\'s a whole\nstack of these quick-and-dirty arguments and they all kinda agree on\nthe phenomenology and blur together, although the initial assumptions\nand arguments can be different.\n\nIn an old dark star type model, you even get a "porous"\nroute-dependent horizon, with radiation emitted towards the observer\nfrom the approaching edge, from a ocation that might be considered to\nbe behind the horizon for a differently-placed observer ... that\'s the\nloose classical equivalent of QM Bekenstein radiation ... but in\nmodern gravitaitonal physics (circa 2000AD), that\'s now supposed to be\nstrictly a QM effect, because indirect radiation isn\'t supposed to\nhappen in SR-based gravitational models. :(\n\n\nFor people who really like mind-twisters, you can go on to think about\nthe legal "grey area" of what would be supposed to happen if a\ndoughnut-shaped mass with major and minor axis rotations underwent\ntotal collapse! :)\nToroidal singularities ...\n\nThat\'s quite a good exercise for folk trying out wormhole geometry\nproblems, because the fieldlines get wound up around the singularity\nas they do in some wormhole descriptions, but one is dealing with\nslightly more conventional spacetime than in a "proper" wormhole\nproblem.\n\n=Erk= (Eric Baird)\n: " The principle of Nipkow\'s televisor disk is is also the principle\n: of scientific invention. The viewer ... captures and cross-references\n: flashes of illumination that appear fleetingly through chinks in an\n: otherwise impenetrable barrier, and uses these to construct a mental\n: picture of a possible world beyond.\n: The process requires persistence of vision ... "\n: -- "Seeing at a Distance: The History of Television"\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 29 Jun 2004 21:32:54 +0000 (UTC), ulmo@cheerful.com (Ulmo)
wrote:
>The event horizons are spherically
>symmetric, and so are exactly the same whether they are "rotating" or
>not. The spacetime curvature caused by the black hole is spherically
>symmetric, so how would you know if it was "rotating"?
That's a decent argument, and yes ... one can argue that a pure
point-singularity should not have enough detail to support the
property of rotation, since the fieldlines always have to exit
perpendicularly from a point, so a "supposedly spinning" point and a
"supposedly non-spinning" point might be expected to have the same
exteriors ...
.... but those "spherically-symmetrical black hole" arguments are only
meant to apply to /non/-rotating holes, and the rules are supposed to
be a bit different when a hole is spinning wrt its background.
For a rotating hole, the side profile is supposed to be more like an
ellipse, the horizon extends further because of the energy tied up in
the relative rotation between the hole and its environment, and the
receding side of the hole pulls at you more strongly than the
approaching side.
The hole's external field has a "twist" to it that describes the
hole's rotation, and the idealised singularity that we can invoke as
being the minimal expression of that exterior field in a standard hole
is no longer a central point, but a ring surrounding the central
position. The planar region bounded by the ring is a bit odd, so you
might sometimes see it referred to as a singularity, too.
There's still possibly a slight awkwardness in that the ring itself
doesn't tell you the /sense/ of the rotation, but the angles that the
fieldlines splay off at gives you that.
As for the bit about fieldlines exiting the horizon at 90 degrees, the
apparent position of the horizon seems to be observer-dependent, so
although an array of observers sprinkled around the hole on its
equatorial plane might well think that all fieldlines are pointing to
the hole's apparent centre, their ideas about where that supposed
centre /'is/, gleaned from their observations of the hole taken from
their own individual positions, is likely to disagree. They can all
point back along a fieldline towards the ring, but may be pointing
towards different points on the ring. Methinks that each individual
observer probably thinks that the hole's centre is offset to one side
of the actual geometrical centre, with the offset being towards the
receding side. Certainly by measuring the gravitational attraction of
the hole, the stronger attraction of the receding side would mean that
their idea of the hole's apparent centre of gravitational mass would
be offset to that side, and the frame-dragging spacetime "twist" would
mean that they'd tend to think that the hole's fieldlines intersecting
their position were coming from somewhere over to that side.
You can also try to describe an apparent visual offset in the position
of the hole towards that side, due to differential gravitational
lensing ... the side that pulls more strongly produces a stronger
gravitational "magnifying lens" effect, and the hole's profile seems
to extend further on that side. Or you can use all sorts of Doppler or
timelag or distortion arguments to argue that the observer is seeing
more surface on the receding side because in a sense they are actually
seeing around a corner and their sightlines are intersecting horizon
points that are past the (nongravitaitonal!) horizon. There's a whole
stack of these quick-and-dirty arguments and they all kinda agree on
the phenomenology and blur together, although the initial assumptions
and arguments can be different.
In an old dark star type model, you even get a "porous"
route-dependent horizon, with radiation emitted towards the observer
from the approaching edge, from a ocation that might be considered to
be behind the horizon for a differently-placed observer ... that's the
loose classical equivalent of QM Bekenstein radiation ... but in
modern gravitaitonal physics (circa 2000AD), that's now supposed to be
strictly a QM effect, because indirect radiation isn't supposed to
happen in SR-based gravitational models. :(
For people who really like mind-twisters, you can go on to think about
the legal "grey area" of what would be supposed to happen if a
doughnut-shaped mass with major and minor axis rotations underwent
total collapse! :)
Toroidal singularities ...
That's quite a good exercise for folk trying out wormhole geometry
problems, because the fieldlines get wound up around the singularity
as they do in some wormhole descriptions, but one is dealing with
slightly more conventional spacetime than in a "proper" wormhole
problem.
=Erk= (Eric Baird)
: " The principle of Nipkow's televisor disk is is also the principle
: of scientific invention. The viewer ... captures and cross-references
: flashes of illumination that appear fleetingly through chinks in an
: otherwise impenetrable barrier, and uses these to construct a mental
: picture of a possible world beyond.
: The process requires persistence of vision ... "
: -- "Seeing at a Distance: The History of Television"
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