General relativity versus black holes

On Wed, 4 Aug 2004, Nick Maclaren wrote:

I think that I know the answer to this, but it may interest
some other people. Let's consider the Schwarzschild solution.
In Newtonian theory, the metric is:

ds^2 = dt^2 + dr^2 + r^2 dw^2
^
minus (very important!)

where

-infty t infty, 0 r infty, 0 u pi, -pi v pi

In Einsteinian theory, the metric is:

ds^2 = (1-2GM/r) dt^2 + (1-2GM/r)^-1 dr^2 + r^2 dw^2
^
minus (very important!)

where

dw^2 = du^2 + sin(u)^2 dv^2

-infty t infty, 2m r infty, 0 u pi, -pi v pi

Sigh...

In the interest of attempting to retard the inexorable decline in the
intellectual quality of postings to this group, from time to time I try to
remind contributors that standards of clarity in postings here are higher
than in the unmoderated groups. On behalf of all lurkers, as well as
other posters, I request that all posters here make a serious effort to
write clearly and unambiguously.

Nick, I don't want to pick nits, but unfortunately you have just given an
example of very bad writing which could -easily- have been improved before
submitting your post! IMO this is deplorable, even though in this
particular case, probably everyone knows what you presumably -meant- to
say.

In case anyone doesn't see what I am complaining about:

"The Schwarzschild vacuum" (or "solution") means a spacetime belonging to
a certain one-parameter family of static spherically symmetric exact
vacuum solutions to the EFE. This means that it is a Lorentzian manifold
(M,g), where the metric tensor g can be defined, on "the" exterior region,
wrt a "polar spherical Schwarzschild coordinate chart", by the second line
element above. Strictly speaking, to fully define the underlying smooth
manifold M we need to give some additional coordinate charts, defined on
overlapping domains, together with "transition maps" (diffeomorphisms
defined on the overlaps). Such a transition map gives the requisite
"concordance" between two competing coordinate charts, wherever their
domains of definition overlap. Physicists often omit to say all this, but
such laziness can terribly confuse beginners.

Compare Minkowski spacetime E^(1,3), a different (nonisometric) Lorentzian
manifold (M,g) whose metric tensor g is defined, on a suitable domain, wrt
a polar spherical coordinate chart, by the first line element above.
Again, strictly speaking, to fully define the underlying smooth manifold M
(usually called R^4), we need to give at least one more chart covering the
omitted line ("the world line of the observer at r = 0"), plus a
transition map. In the case of R^4, we can get away with a single global
chart (e.g. a Cartesian chart), but this is not possible for many curved
spacetimes.

BTW, a "Schwarzschild chart" is just any polar spherical chart, on some
spacetime, in which the "radial coordinate" has the obvious interpretation
in terms of the surface area of "round spheres". For example, Nick's
"exponential metric" below is a Schwarzschild chart, but the well-known
"spatially isotropic" polar spherical chart for the Schwarzschild vacuum,
and the Watt-Misner metric mentioned below, are -not- Schwarzschild
charts. So don't confuse "a Schwarzschild chart" with "the Schwarzschild
vacuum"--- they are completely different concepts!

Now, the problem with what Nick said is that writing down the E^(1,3)
metric in a polar spherical chart and calling this "the Newtonian analog
of the Schwarzschild solution", or something like that, is -very-
misleading, for a dozen reasons. To name just two:

1. In the "Newtonian" field theory of gravitation, the "field" is a
scalar function u(x,y,z), namely the gravitational potential, and the
"field equation" is just Poisson's equation

Lap(u) = u_(xx) + u_(yy) + u_(zz) = 4 pi mu

(I wrote this in a Cartesian chart, but of course it can be written in any
other chart defined on some region of R^3.) Thus, a "vacuum solution", in
this theory, is simply any harmonic function defined on R^3 (or on some
open neighborhood). The validity of euclidean geometry on "space" is an
-underlying assumption-, but no assumptions about the metric on R^3 are
needed to write down the Laplace equation! (It is true that some geometry
is secretly "preferred" here, since the symmetry group of the 3D Laplace
equation is the conformal group on E^3.)

2. OTOH, Minkowski geometry is -incompatible- with the Newtonian field
theory of gravitation; an easy way to see this is to note that in
Newtonian theory, the potential responds -instantaneously- everywhere on
R^3 to changes in the distribution of matter; this obviously violates the
principle that in Minkowski spacetime, information (such as "we just
changed the distribution of matter over here!") cannot travel faster than
the speed of light. So, Newtonian gravitation is definitely not a
-relativistic- classical field theory of gravitation! (Compare the
discussion of Cartan's notion of "Newtonian spacetime" in MTW.)

OK, as I said, in this -particular- case of very bad writing, probably no
serious harm was done, but in many other cases, bandwidth is wasted
because respondents misunderstand what the original poster had in mind, or
have to request clarification, or because the question never made sense at
all, etc. So please, -everyone-, let's all try to remember that standards
have not really been lowered in this n.g., and the charter has not been
abandoned. Please bear in mind that

(1) the moderators are overworked volunteers (BTW, thanks, guys!),

(2) traffic here is exponentially increasing while the number of
moderators remains constant.

Since the moderators are far too busy to continue to suggest felicitous
improvements in wording (as John Baez often did in happier days), IMO

-All posters here need to try harder to write well in order to maintain
our standards-

TIA!

OK, enough of that, onto the rest of Nick's post:

Now, let us speculate a unification of quantum mechanics and general
relativity that produced the following metric:

ds^2 = exp(-2GM/r) dt^2 + exp(2GM/r) dr^2 + r^2 dw^2

I am, of course, not saying that there is a scrap of evidence for
such a theory. But let us assume one,

No doubt one can come up with some well-defined gravitation theory other
than gtr in which this arises as a static spherically symmetric solution,
at least if you are willing to violate various principles such as Lorentz
invariance. As for "uniting QM and gtr", I don't think this additional
demand is needed to make the point you are trying to make here, so you
should be much less ambitious, in the interests of being much more
specific about what theory you have in mind!

Warning: in the past, "exponential metrics" such as the one you propose
have been put forth by various posters here, in the absence of any
underlying theory in which it has the status of a static spherically
symmetric solution (but possibly not the -only- such solution). Such
proposals have, at best, the status of an "Ansatz" rather than a
"competing theory". Unfortunately, some past posters here could not grasp
this elementary distinction, and things got worse from there. This is
quite irrespective from the not inconsiderable issue of proposing a
gravitation theory which agrees with available evidence and which doesn't
violate too many cherished theoretical principles. But I certainly don't
want to reslay the slain!

and assume that it makes similar changes to other solutions of
Einstein's equations.

Unless you can be more specific (or can at least give some more examples),
"assume similar changes" is too vague to mean anything to me.

My question is whether we have any CURRENT data that would enable
us to distinguish these?

Well, of course this kind of question is the bread and butter of physics.
We have some theories, and we want to know: how do they compare with the
data? Ideally, we'd like to test a whole bunch of theories at one fell
swoop, and pick out the one which best explains all the available
evidence. In the context of relativistic classical field theories of
gravitation, PPN formalism provides a handy way to do this for a large
class of possible theories. (Of course, you can invent ones which don't
fit into this framework without further work; it is presumably impossible
to eliminate all competitors without making -some- assumptions about what
kind of theory you are considering.)

Here is a paper you should read:

author = {Keith Watt and Charles W. Misner},
title = {Relativistic Scalar Gravity: A Laboratory for Numerical
Relativity},
note = {gr-qc/9910032}}

This paper discusses a simple classical gravitation theory (a "stratified
conformally flat scalar theory of gravitation"). This theory requires a
preferred frame, and the authors employ a "stratified conformal chart"

ds^2 = -f(phi) dt^2 + g(phi) (dx^2 + dy^2 + dz^2),

phi(t,x,y,z) = a scalar field describing gravitation

f,g = functions to be determined

in which the distinguished family of spatial hyperslices are all
conformally flat. The exact form of phi, f, and g are to be determined by
solving their field equation. As you might expect, this problem is
simplified if you assume a lot of symmetry, e.g. you can look for static
spherically symmetric solutions.

Note well: the assumption of a preferred frame drastically violates the
spirit of gtr, and as the authors note, this assumption implies that the
Watt-Misner theory cannot exhibit gravitomagnetism. This phenomenon, or
rather class of phenomena, is predicted by many gravitation theories,
including gtr, but details differ, which is the point--- we can tell
theories apart by comparing their predictions in sufficient detail. Even
as we write, a dedicated satellite experiment is in progress which should
provide the first solid test of predicted gravitomagnetic effects near the
Earth.

As Watt and Misner point out, the motivation for introducing a stratified
conformally flat theory is that conformally flat gravitation theories
cannot exhibit light bending and thus are hopelessly inadequate. OTOH,
stratified conformally flat theories can exhibit light bending--- but only
at the cost of breaking global Lorentz invariance. Compare the discussion
of scalar theories vis a vis vector and tensor theories in MTW (where
Lorentz invariance is assumed).

Next, note that the static spherically symmetric solution in the
Watt-Misner theory turns out to be given (in a "polar spherical spatially
isotropic chart" or "stratified conformally flat chart") by:

ds^2 = -exp(-2m/r) dt^2 + exp(2m/r) [dr^2 + r^2 (du^2 + sin(u)^2 dv^2)],

-infty t infty, 0 r infty, 0 u pi, -pi v pi

Exercise: Try to transform your "exponential metric" above into a similar
"spatially isotropic polar spherical chart" (proceed the same way that you
transform the polar spherical Schwarzschild chart for the exterior
Schwarzschild vacuum to the well-known spatially isotropic polar spherical
chart for the exterior Schwarzschild vacuum).

Finally, to address your question, note that Watt and Misner give a very
clear discussion of how to compute the PPN parameters of their theory.
Compare this with a similar (but less realistic and more complicated!)
example in MTW. If you work through these examples, you should get a good
idea of how we can compare and test gravitation theories.

The relevance is that the speculated formulae would give many of
the properties of general relativity, but without black holes and
other singularities.

Oh dear, another "obfuscation alert": you must always distinguish clearly
between event horizons (-global- causal features; in the Schwarzschild
chart the event horizon of the Schwarzschild vacuum happens to form a
boundary where the coordinate system breaks down, i.e. a "coordinate
singularity", but in other coordinate charts the horizon is locally
unremarkable) and geometric singularities (-local- features) such as
strong scalar curvature singularities.

I think you mean here that your spacetime, an "exponential analog" of the
Schwarzschild vacuum, appears from casual inspection of the form of the
metric to lack an -event horizon-. But you should always be very careful
before jumping to any such conclusion. For example, in the Weyl canonical
chart for "the" exterior of the Schwarzschild vacuum, the horizon appears
as a -line segment- lying on the axis of axial symmetry used in writing
down the chart. (From the form of the chart, it is not evident that in
fact the spacetime is spherically symmetric.) Point being that we should
expect to work through some careful -global analysis- of the causal
structure before we can rigorously rule out the existence of an event
horizon in a given spacetime model. For, if one exists, it is a -global-
feature of the spacetime.

This, in turns, disproves the claims that the observed binary star
results are PROOFS of the existence of black holes.

Whoa, I think you just skipped over a whole lotta highly relevant
observations! Such as the orbital decay of certain binary pulsars, which
is in excellent agreement (so far) with the predictions of gtr, in which
theory the decay is attributed to gravitational radiation carrying off
energy from the binary system.

If we have any current data that DOES disprove such a theory, then the
evidence for the existence of black holes is rather stronger than I
think that it is.

Steve Carlip can probably answer this much better than I can, so I hope he
will speak up.

"T. Essel" (hiding somewhere in cyberspace)