Black holes - Relative or Absolute?
By Wolfgang G. Gasser
00/01/04
Black holes arise in GR in situations where
classical physics would
result in escape velocities higher than the speed of light. To an
escape velocity of c corresponds a difference in gravitational
potential of 0.5 c2.
The gravitational potential decreases further if one approaches the
center of a star. If a spherical object has a constant density, then
the gravitational potential at the center is 1.5 times the one of the
surface. In the case of real stars the factor is much higher than 1.5
because of the much higher density near the center.
At least a classical consideration leads to the conclusion that a
visible star does somehow contain a black hole if the escape velocity
is higher than c at its center. But as far as I understand GR, also
this theory suggests that light cannot escape from below this
black-hole-radius of the non-black-hole.
This problem results from a more general question: are black holes
absolute or are they relative (i.e. observer-dependent)?
See: Simple Black Hole Paradox
00/01/07
: = Chris Hillman
:: = Wolfgang G. (z@z)
:: Black holes arise in GR in situations where classical physics would
:: result in escape velocities higher than the speed of light. To an
:: escape velocity of c corresponds a difference in gravitational
:: potential of 0.5 c2.
:
: That's not the right way to think of it at all. The defining feature of
: a black hole is an event horizon which acts as a "one way gate":
: objects (and light) can pass
through it in only one direction.
:: This problem results from a more general question: are black holes
:: absolute or are they relative (i.e. observer-dependent)?
:
: I think the best short answer is that according
to GTR, black holes are
: most definitely not observer dependent in the sense you mean. The
: event horizon is a real phenomenon and there is no question about
: whether any given event is inside or outside it, in any spacetime
: containing a black hole.
Assume a huge and dense globular cluster of burnt out stars. Further
assume that there is a star in the cluster whose own surface escape
velocity would be (classically calculated) lower than c, but whose
escape velocity out of the cluster is much higher than c.
Suppose that light from an observer outside the cluster moves to
this star. Doesn't GR predict an event-horizon effect for this
light? Doesn't according GR also the cluster contribute to the
space-time-curvature which ultimately can lead to a black hole?
The existence of 'absolute' black holes is inconsistent with the
very fundamental principles GT is based on. That's why Einstein
had to admit that GR cannot be valid in the range where black holes
arise. On the other hand, the argument that either black holes are
possible or GR must be fundamentally flawed is also convincing.
00/01/08
: = J. Scott Miller
:: = Wolfgang G. (z@z)
:: The existence of 'absolute' black holes is inconsistent with the
:: very fundamental principles GT is based on. That's why Einstein
:: had to admit that GR cannot be valid in the range where black
:: holes arise.
:
: I have read quite a bit on Einstein and on black holes and have never
: encountered such a statement. Care to provide
a reference?
"Aus den Gleichungen der allgemeinen Relativitätstheorie
lässt sich
die Existenz punktförmiger Singularitäten herleiten, ... Doch dem
Urheber der Graviationstheorie waren solchen Abnormitäten ein
Greuel, [the author however had a horror of such abnormalities]
und er meinte sie widerlegen zu können."
"Im Jahre 1939 veröffentlichte er in den
'Annals of Mathematics'
eine Artikel mit dem einschüchternden Titel 'On a Stationary System
with Spherical Symmetry consisting of Many Gravitating Masses'. Darin
wollte er beweisen, dass Schwarze Löcher - ... - unmöglich seien [he
wanted to prove that black holes are impossible]".
[1]
It is perfectly natural that Einstein's attempts to refute black
holes were handicapped by his instinctive reluctance to attack his
own theory. So if the description of Einstein's black-hole attack
given by Bernstein is essentially correct, then it really isn't as
effective as it could be.
I even suppose that Einstein would not have
proposed the GR equations
if he had already known that black holes are an inevitable consequence
of them. Another strong argument in favor of these equations has
been demolished by historians of science. Hilbert had not found
'independently' the same equations as Einstein but only similar ones
(which he later replaced by those of Einstein).
Unfortunately I don't remember which exact statements are the basis
of my opinion that Einstein argued that black holes are outside the
range of application of GR. Maybe someone else can clear this question
up.
[1]
Jeremy Bernstein, "Einstein und the schwarzen Löcher",
Spektrum
der Wissenschaft, August 96 (The
article can also be found in "A theory of
everything", Copernicus (Springer), 1996)
00/01/10
: = Tom Roberts
:: = Wolfgang (z@z)
:: Assume a huge and dense globular cluster of burnt out stars. Further
:: assume that there is a star in the cluster whose own surface escape
:: velocity would be (classically calculated) lower than c, but whose
:: escape velocity out of the cluster is much higher than c.
:
: That cannot happen, except for comparatively short times during the
: formation of a black hole containing the entire cluster. The condition
: that the escape velocity from the cluster is larger than c implies
: the presence of a closed trapped surface, ... also known as an event
: horizon. ...
The paradox arises if the escape velocity from a neutron star out
of the cluster is higher than c, but neither the escape velocity
from the cluster nor from the neutron star alone. There is no
reason at all to assume that the whole cluster should collapse.
According to standard textbooks clusters and galaxies can contain
black holes without collapsing as a whole.
The underlying problem is simple:
gravitational potential in GR
is not an absolute quantity, but the black hole concept ultimately
depends on absolute potential. (One could try to resolve the
paradox by introducing absolute potential into GR; nevertheless
such a solution would not be fully consistent with the generally
accepted black hole theory.)
If someone thinks that it is not possible to use gravitational
potential in GR, then he should read e.g. the chapters "7.5 The
Gravitational Doppler Effect" and "7.6 Metric of Static Fields"
of Wolfgang Rindler's "Essential Relativity", Springer, 1986.
If a particle travels straight to the center of a black hole and
emits radiation, a distant observer will see its light more and
more redshifted. At the event horizon its frequency would become
zero because of infinite time dilation. But time dilation depends
on the difference in potential (with respect to an observer),
doesn't it? So the assumption that the event horizon of a black
hole in a cluster is independent of the whole cluster's gravitation
cannot be correct.
00/01/11
: = Steve Carlip
:: = Wolfgang (z@z)
:: The underlying problem is simple: gravitational potential in GR
:: is not an absolute quantity, but the black hole concept ultimately
:: depends on absolute potential.
:
: The first of these statements is roughly true---the gravitational
: potential is not even a well-defined quantity in GR, much less an
: absolute one. The second is categorically false---the concept of
: a black hole in GR is completely independent of any potential,
: absolute or not.
If GR were indeed a consistent and correct theory of gravity, then
its relation to classical potential would have been cleared up in
the meanwhile. It's normal that a new theory
is counter-intuitive
at the beginning. But if a theory remains counter-intuitive, and
many simple questions cannot be consistently answered even
after decades, then something must probably be wrong. E.g. the
following questions don't seem to have a clear answer within GR:
1) What would happen if the sun suddenly disappeared?
2) Does mass depend on gravitational potential?
It cannot be true that "the concept of a black hole in GR is
completely independent of any potential", because the concept
depends on the same quantity which at least in the weak-field
approximation can be expressed as a classical potential.
: You determine whether an object is a black hole
by tracing
: light rays. If light originating in a finite region is gravitationally
: trapped in that region and can never escape, the region is in
: the interior of a black hole.
This concept makes no use of
: "gravitational potential"; it merely requires that you have a
: theory that determines the paths of light rays.
Whether light is gravitationally trapped in a finite region depends
not only on the mass inside that region. Think of a dying star
whose escape velocity has just become higher than c at its center.
It cannot be denied that the highest gravitational time dilation is
always at the center of a star. Assuming continuity, we conclude
that the black hole surface (where time dilation is infinite) must
appear at first at the center. We conclude that matter outside
the black hole surface must be involved at least during black-hole
formation.
This paradox is related to the ambiguity of "escape velocity",
which according to Tom Roberts means "the velocity required for
an inertial timelike object to escape to spatial
infinity". But does
according to this definition the escape velocity from a neutron star
in a dense cluster depend only on the star or also on the cluster?
00/01/12
: = Steve Carlip
:: = Wolfgang G. G.
:: Does mass depend on gravitational potential?
:
: Give me a precise, operational definition of
"mass" and of
: "gravitational potential", and I'll give you an unambiguous
: answer. (You can decide whether it's "clear".)
Take the definition of the mass M which is used in the formula for
the Schwarzschild radius of black holes (r_s = 2GM/c2).
Has a stone more mass on the Mount Everest than the same stone
at sea level?
:: It cannot be true that "the concept of a black hole in GR is
:: completely independent of any potential", because the concept
:: depends on the same quantity which at least in the weak-field
:: approximation can be expressed as a classical potential.
:
: What quantity, precisely, are you referring to here? What,
: precisely, do you mean when you say "the concept of a black
: hole" depends on it?
Paul A. Tipler ('Physics for Scientists and Engineers', 1991)
gives in a section on black holes the following formula for
gravitational time dilation
lambda'/lambda = 1 / sqrt(1 - v2/c2)
[42.25]
where v is the escape velocity. If it is bigger than c, then the
result is a black hole.
The quantity I was referring to is the one which is expressed
in this formula as an escape velocity.
:: Whether light is gravitationally trapped in a finite region depends
:: not only on the mass inside that region.
:
: That's certainly correct. And it's true that whether or not a
: region is a black hole depends, according to GR, not only on the
: mass in that region, but on what's going on outside. As a simple
: example, the empty region inside a collapsing spherical shell of
: matter can be a black hole, briefly, if the matter is collapsing
: fast enough.
A problem I see is that the definition of the Schwarzschild radius
does not take into consideration the absolute time dilation (or
escape velocity or potential) caused by the neighborhood of a
black hole.
My main objection: black holes depend on
absolute time dilation, but
GR is in principle incompatible with absolute time dilation.