In a recent paper, we discussed the formation of black holes in non-symmetric gravity. That paper was then criticized by Cornish and Moffat. In the present paper, we address the arguments raised by Cornish and Moffat. In summary, we do not see any reason to doubt the validity of our former conclusions

Burko, L M

Ori, A

CERN Document Server

Remarks on the formation of black holes in
non-symmetric gravity
Lior M. Burko and Amos Ori, Department of Physics
Technion { Israel Institute of Technology, 32000 Haifa, Israel
Abstract
In a recent paper, we discussed the formation of black holes in non-
symmetric gravity. That paper was then criticized by Cornish and Moat. In
the present paper, we address the arguments raised by Cornish and Moat.
In summary, we do not see any reason to doubt the validity of our former
conclusions.
PACS numbers: 04.50+h 04.70.Bw
1
gravitation, which incorporates General Relativity (GR) as a limiting case
[1]. Recently, Cornish and Moat (CM) [2] conjectured, based on a class
of exact static spherically-symmetric solutions, that NGT was free of black
holes. Later, Burko and Ori (BO) [3] found that there was no analogue in
NGT to the Birkho theorem of GR [4], and realized that dynamics might
have important consequences. In order to get some insight into the dynami-
cal content of NGT, BO studied in Ref. [3] the evolution of small, linearized,
anti-symmetric eld over a symmetric GR background. The behavior of the
linearized skew perturbation was found to be perfectly regular in the entire
domain of dependence, and in particular at (and beyond) the event horizon.
Based on this result, BO concluded that if, in a situation of gravitational
collapse, the initial skew perturbation is suciently small, then a black hole
is likely to form, just like in GR.
More recently, the analysis of Ref. [3] was criticized by CM [5]. In brief,
the objection of CM is based on the observation that NGT lacks a Principle
of Equivalence, and that this feature of NGT is not well captured by the
linear approximation. CM thus argue that the linear analysis is misleading,
and that the outcome of a gravitational collapse in NGT is more likely to be
a compact object which is not a black hole.
In the present paper we briefly address the objection of CM. In brief, we
nd this objection to be unjustied, and we do not see any reason to doubt
the validity of our former conclusions in Ref. [3].
The main arguments raised in Ref. [5] are the following:
1) The linear approximation to NGT is not valid, because it fails to
capture an important aspect of the fully non-linear NGT { the absence of an
Equivalence Principle.
2) The absence of the Equivalence Principle suggests that the perturba-
tive approach will break at the horizon: With no Equivalence Principle, the
evolving perturbations are allowed to \feel" the horizon, by \feeling" the
non-symmetric eld there.
3) Consider a static star surrounded by a static spherically-symmetric
NGT eld. The exterior eld is described by the static spherically-symmetric
NGT solution { the Wyman solution. On top of this background, consider
some additional small NGT perturbations. Assume now that the star under-
goes gravitational collapse. CM argue that the nal state cannot be a black
2
stable, then, upon evolution, the small initial deviation from the Wyman so-
lution must remain small. Therefore, the nal state must be given by small
perturbations on top of the Wyman solution. Such a nal state cannot be a
black hole (because the Wyman solution is not perturbatively close to any
black-hole solution).
Here is our reply to the above arguments:
1) The absence of an Equivalence Principle in NGT is certainly an inter-
esting issue. But this issue has nothing to do with the validity of the linear
approach. For, in our analysis we did not make any use of the Equivalence
Principle, or any assumption about its existence or inexistence in the theory.
Needless to say, the validity of a linear analysis has nothing to do with the
issue of whether the system in question admits an Equivalence Principle or
not. (The linear approach is used in countless number of systems in physics,
and in most of these cases, this principle is simply irrelevant.)1
In addition, let us note that the most fundamental manifestation of the
break-down of the Equivalence Principle occurs already at the linear level:
The impossibility to nd a local reference frame in which the metric tensor
is locally Minkowski, and the existence of nontrivial metric-tensor scalars
like g[]g[], are faced already at the linear level. (As we have just pointed
out, however, this issue is absolutely irrelevant to the validity of the linear
approach!)
2) We agree that (in a somewhat vague sense) the absence of the Equiva-
lence Principle allows the evolving perturbations to \feel" the non-symmetric
eld. But this has nothing special to do with the horizon: The antisymmetric
eld has a non-zero value not only at r = 2m, but also at greater (or smaller)
1Perhaps the idea behind the above argument by CM is that the usage of the linear
approach should be limited to situations where the background, or the linear eld, admits
all the qualitative aspects of the non-linear system. This is certainly not the case: In many
systems where small perturbations were successfully analyzed by the linear approach, there
are important, or even crucial, non-linear phenomena which are absent at the linear level.
As an example, consider sound waves in a perfect fluid [6]: It is the non-linear aspects which
are responsible for the formation of shock waves, as well as turbulence and other chaotic
phenomena { and yet the linear treatment is perfectly valid (in a compact neighborhood)
if the initial perturbations are small. (A similar example is the linearized Einstein theory.)
Another example: In a vibrating elastic body or string, coupling to higher harmonics
occurs at the non-linear level only, and yet the linear approach is valid for the description
of small vibrations.
3
does not alter the following simple fact: The event horizon is an absolutely
regular location at the background, which has no local signicance2.
3) This argument of CM is based on the assumption that the initial state
is, in some global sense, \close to Wyman", i.e., that it can be described by
a small perturbation over Wyman. This assumption is obviously wrong: If
we denote the initial radius of the star by rs, then the initial eld is close to
Wyman at r > rs, but is very dierent from at r < rs. Thus, while in the
Wyman solution the antisymmetric eld is typically very strong at r = 2m,
in the initial state considered here it is presumably small there. In addition,
the Wyman solution is vacuum and static, and the initial state considered
here is neither of them (at r < rs).3 Therefore, the expectation that this
conguration will evolve like Wyman with small perturbations has nothing
to base on.
Let us examine this issue more carefully: Let us denote the section r  rs
of the initial hypersurface by S. Then, since the initial data are close to
Wyman on S, the standard stability arguments indeed suggest that the evolv-
ing eld will be similar to Wyman throughout D+(S) { but not elsewhere
(here it is important to recall that there is no Birkho theorem in NGT).
The region near r = 2m is not included in D+(S), and is not even close
to it. Therefore, there is no reason to expect that the evolving eld near
r = 2m will by any means be similar to Wyman.
In fact, this last argument by CM can easily be reversed: If in the initial
conguration described in argument (3) the initial skew eld is suciently
small4 (which we assume), then the initial data can be viewed as a small
NGT perturbation on top of a GR background. The same line of reasoning
suggested by CM now implies that the evolving conguration will be just
GR plus small skew perturbation { which means that a black hole will form!
2We also point out that the results of Ref. [3] are not a consequence of the pre-
existing black hole (and event horizon). As it turns out, even if there is no black hole in
the moment (i.e., in the gravitational collapse of matter or gravitational radiation), the
subsequent formation of a black hole is to be anticipated [3].
3This strong deviation of the initial state from Wyman considered here has nothing to
do with the extra small perturbations added on top of the background; It is the background
itself which is, in overall, very dierent from Wyman.
4This would be the case if the parameter s of the exterior Wyman solution is suciently
small.
4
valid, because the initial data are close to a GR conguration in the entire
initial slice (whereas, in contrast, they are close to Wyman at r > rs only).
In summary, we nd no reason to doubt the validity of the linear approach
of Ref. [3]. Therefore, although this linear analysis does not provide a strict
mathematical proof for anything (a linear analysis almost never does), it
should be regarded as rm and trustable { just like any other situation in
physics in which linear analysis is used to study the behavior of small pertur-
bations. We therefore insist that our conclusion in Ref. [3] is justied: In the
situation of a gravitational collapse in NGT, if the initial skew perturbation
is suciently small, one should expect a black hole to form { just like in GR.
References
[1] For a review on NGT see, e.g., J.W. Moat in Gravitation { A Ban
Summer Institute, edited by R.B. Mann and P. Wesson (World Scientic,
Singapore, 1991), p. 523.
[2] N.J. Cornish and J.W. Moat, Phys. Lett. B 336, 337-342 (1994); J.
Math. Phys. 35, 6628-6643 (1994).
[3] L.M. Burko and A. Ori, \On the formation of black holes in
non-symmetric gravity," Technion Preprint TECHNION-PH-95-7, gr-
qc/9506033 (to be published).
[4] This has been shown independently (although through a completely
dierent precedure) by M.A. Clayton, \Massive NGT and Spherically
Symmetric Systems," University of Toronto preprint UTPT-95-09, gr-
qc/9505005 (unpublished), and by N.J. Cornish, J.W. Moat and D.
Tatarski, Gen. Rel. Grav., to appear (1995).
[5] N.J. Cornish and J.W. Moat, \Comment on the formation of black
holes in nonsymmetric gravity," University of Toronto preprint UTPT-
95-14, gr-qc/9506055 (unpublished).
[6] See, e.g., Landau and Lifshitz, Fluid Mechanics, second edition (Perga-
mon Press, Oxford, 1987).
5


Remarks on the formation of black holes in non-symmetric gravity

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