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Tom Roberts...
Posted: Fri Jun 11, 2010 8:31 pm
 
Oh No wrote:
[quote]The method is to ascertain T for a given situation, then solve the
Einstein field equation (if we can) to determine g.
[/quote]
Except, of course, when computing T requires g. This is generally the case,
including electrodynamics, the motion of pointlike particles, and any vector or
tensor matter field(s). As I said before, this is one aspect of GR that makes it
so difficult: the geometry (g) affects the "source" (T). In general, one must
solve for both g and T in one fell swoop. Vacuum solutions (T=0) are an obvious
exception.


Tom Roberts
 
Oh No...
Posted: Sat Jun 12, 2010 1:13 am
 
Thus spake "Jonathan Thornburg [remove -animal to reply]"
<jthorn at (no spam) astro.indiana-zebra.edu>
[quote][[I think I've unwrapped the nested quoting correctly, but if I've
erred is mistakenly attributed anyone's words to someone else, my
apologies.]]

Steve Carlip wrote:
| There is a simple exact solution to the Einstein field equations
| describing a thin shell of matter collapsing to form a black hole.
| The shell starts outside its Schwarzschild radius, so there is
| initially no event horizon. The exterior is a piece of the exterior
| Schwarzschild solution. The interior is flat Minkowski space;
| as in Newtonian gravity, there is no field inside a hollow sphere.
| After a finite amount of time -- as measured, for example, by an
| observer in the flat interior

I'm going to call that finite amount of time T1.

Charles Francis commented:
I am not clear that the existence of this observer is even possible in
the situation you describe.

Why not? Our initial conditions do not contain a black hole, simply
a shell of matter with an observer inside. For all proper times of
that inside observer < T1, there is no event horizon, and thus no
obstacle to T1 sending reports out to the outside. For times >= T1,
those reports can't get out, but that doesn't stop the observer from
making the observations (the infalling matter hasn't yet reached the
observer, and spacetime near the observer remains singularity-free
(indeed *flat*).


| -- the shell reaches its Schwarzschild
| radius, and an event horizon forms. Do you contend that at the
| moment this happens, the Einstein field equations cease to hold,
| the flat interior vanishes, and the shell instantly shrinks to point?

No. from the viewpoint of an exterior observer, the shell remains a
shell at the Schwarzschild radius.

A slightly more accurate statement would be "from the viewpoint of an
exterior observer who is being fooled by a well-known optical illusion,
the shell remains at the Schwarzschild radius (with its brightness
decreasing exponentially with time)". Since we know that the optical
appearance of the shell *doesn't* match its actual position [see below]
(that's what I mean by the phrase "optical illusion"), it's silly to
use that optical appearance as a description of what's actually
happening to the shell.

But the more serious flaw with the hypothesis that "the shell remains
a shell at the Schwarzschild radius" is that the Einstein equations
say otherwise. That is, the Einstein equations let us calculate
the motion of the shell, as measured by (for example) its areal
(Schwarzschild) radius as a function of proper time (as measured by
an observer riding on the shell). When you do this calculation, you
find that the radius doesn't freeze at r=2m, but rather crosses right
through r=2m and keeps on decreasing.

[The optical illusion I mentioned earlier, that "from the viewpoint
of an exterior observer who is being fooled by a well-known optical
illusion, the shell remains at the Schwarzschild radius" is then
easily understood as a consequence of the trajectories of *photons*
emitted by the shell at various times. This is made very clear by
Misner, Thorne, & Wheeler's figure 32.1.]

[/quote]
Ok, I will try to describe a bit better what happens. This is not easy,
because we do not have a static situation. The gedanken actually shows
that we cannot describe it using static coordinates, which makes a
description particularly difficult. Undoubtedly any qualitative
description I give will not be perfect.

If it is a real shell, it must have finite thickness. As it collapses to
the position where the event horizon will be, its thickness increases,
so that some of the matter is inside this radius and some outside. The
thickness of the shell inside this radius expands to the point where the
inside observer is crushed.

To describe it more accurately I have to consider that the shell
consists of elementary particles. In this model each particle is
modelled as a small black hole. From the point of view of an observer,
each particle has a radius equal to the size of its event horizon.
However, it is not meaningful to define coordinates interior to the
particle. As the particles get closer together with the contraction of
the shell, they are affected by the gravity of neighbouring particles.
In consequence, from the point of view of an exterior observer, their
event horizons expand, and ultimately merge. Ultimately the collapse
brings all the particles to merge at a point, with a single event
horizon equal to the event horizon of the black hole. During the
process, the region inside the hole is squeezed out of existence, but
this happens in a continuous process, not in a discontinuity.

Regards

--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)

http://www.rqgravity.net
 
Tom Roberts...
Posted: Sat Jun 12, 2010 5:38 am
 
Oh No wrote:
[quote][about coordinate-free techniques]
But this is what I mean by "pulling the wool". They give the
appearance of dispensing with coordinates, and one finds people thinking
that this makes them somehow a better model of physical reality, since
coordinates are imposed by man, not by nature. Actually they include all
possible coordinates, which is a very different kind of statement.
[/quote]
You have not understood what I have been saying. This is very basic and
fundamental; it's also rather obvious once you sit down and think about it.

For instance, the fact that we apply coordinates to the world REQUIRES that we
be able to apply coordinates to the geometry. That is not "pulling the wool",
that is an operation parallel to what we often do in the real world. A good
model must be able to perform such parallel operations.

Physical theories are MODELS of nature and the world we inhabit. So valid
theories have a multi-level correspondence to the world:

Theory (model) <=> World
------------------------------ --------------------------
Coordinate-free description of <=> Existence of the world, (without
a manifold and its geometry any coordinates)

Coordinate-free description <=> Natural operation of physical
of physical phenomena phenomena (without any coordinates)

Application of coordinates to <=> Application of coordinates
the manifold to the world

Projection of geometry and <=> Projection of physical measurements
physical theory onto the onto the coordinates
coordinates

Coordinate-based description <=> Coordinate-based measurements of
of geometry and physical geometrical relationships and
phenomena physical phenomena


Those items are ordered in an obvious way, which is why we say that a
coordinate-free description of physical phenomena is more natural, and at a
lower level, than coordinate-based descriptions.

And I repeat: Coordinates are an arbitrary, human choice, and nature obviously
uses no coordinates, so no valid model of natural phenomena can depend in any
way on coordinates [#]. So in physics there is really no choice but to pursue
coordinate-free descriptions and techniques. This is, after all, one of the
major lessons of GR (but older authors did not realize it).

[#] But as in the world, application of coordinates must be possible.
Fortunately, the manifolds of differential geometry have precisely
these properties, as do tensors and tensor fields on such manifolds.
This is not happenstance :-)

Note that common measuring devices, such as clocks, rulers, voltmeters,
ohmmeters, ammeters, balance scales, etc. all do their thing without any
coordinates whatsoever. You can apply coordinates to many of them, but that is
not at all the same as them using the coordinates internally. Claiming that
coordinates are a necessary precursor to measurements or making physical models
is obviously wrong.


Tom Roberts
 
Gerry Quinn...
Posted: Sat Jun 12, 2010 7:42 am
 
In article <87d0krF490U1 at (no spam) mid.individual.net>, jthorn at (no spam) astro.indiana-
zebra.edu says...
[quote]Gerry Quinn <gerryq at (no spam) indigo.ie> asked:
Jonathan said:

| almost everyone who studies general relativity professionally
| (and a lot of other people as well)
| [--]
| would agree with the statement "GR provides a very good
| approximation to physical gravitation everywhere (except
| possibly very close to a singulary) inside the event
| horizon of a BH".]

And this is a statement I cannot bring myself to believe in - the most
obvious issue being the information problem. Suppose the statement is
a true statement about the physical world; then imagine a sealed
container of matter dropped into a black hole. In a short proper time
- if the statement is true - it approaches the central singularity

Up to here I think my 95% sample will agree.

It's important to realise that (at least according to the Einstein
equations) the event horizon is only *globally* defined. Locally --
and today we believe that all physics is ultimately local, not
action-at-a-distance -- there's nothing special going on at the horizon,
and in particular there's no way for our parcel of matter to even
"know" that it's crossed the horizon.
[/quote]
But of course the key words are 'according to the Einstein equations'.
If these equations are not correct at the event horizon, the premise of
the above argument is incorrect. Thus the argument shows only that the
GR solution is internally consistent. It has no strength when it comes
to ruling out alternatives.

[Of course if GR is incorrect at the Schwarzschild radius the
expression 'event horizon' may be a bit of a misnomer. But we can
agree I think that if it is approximately correct until close to that
point, there will in any case be an approximate event horizon arising
from the enormous gravitational red shift as seen from the perspective
of distant observers.]

[quote][Alice example]
In other words, in order for our parcel of matter to
"know" exactly when it had crossed the horizon today
(so that it could behave differently then), it would
have to "know" *today* what Alice is going to decide
to do *next* *week*!]

Since it doesn't "know", our parcel of matter is going to behave just
like it did before crossing the horizon, i.e., it's going to behave as
the (classical) Einstein equations say it would. That is, because
there's nothing locally special about the event horizon, then it
should be safe to extrapolate the outside-the-horizon observational
evidence that the Einstein equations are a very good approximation
to the behavior of real-world matter and spacetimes, to everywhere
inside apart from places very close to singularities.

[It's worth noting that the locally-measured
spacetime curvature near to and just inside the
event horizon need not be large, if the black hole is
massive enough. So there's no need for this region
(where I'm talking about continuing to use the
Einstein equations even though our observational
evidence only relates to outside-the-event-horizon)
to contain anything that would qualify as "extreme
conditions".]

Once we're comfortable using the (classical) Einstein equations, then
it's an easy calculation that our parcel of matter will indeed approach
the central singularity in a finite proper time.
[/quote]
But I'm not comfortable with using them, because while they do not seem
to become internally inconsistent before the central singularity, they
appear to become inconsistent with the rest of physics before they are
extrapolated so far.

Suppose we postulate instead a scenario in which spacetime really is
fundamentally flat, and objects in strong enough gravitational fields
become asymptotically 'frozen' - at least as far as ordinary low-energy
interactions are concerned - and do not reach the proper time for which
GR predicts they will come close to a central singularity. Is there
anything inconsistent with this?

There is nothing physically implausible about a stopped clock, and
theories about what will happen when a stopped clock rings the hour may
be consistent without having any physical relevance.

[We do not have to believe that the objects will remain static for
infinite time, because we will assume that gravity is a low energy
effective field, and over long enough times (from the perspective of
someone not in a high gravitational field), higher energy interactions
will take place even for matter that is seemingly 'cold', like our
packet dropped into a black hole.]

Do you see anything inconsistent in this scenario? Because it seems to
me that it can be easily reconciled with the rest of physics! Since GR
clearly cannot, there is no choice for me regarding which to believe.

[quote]And now we can certainly assume that the information it contains does
presumably become 'unlocked' by unknown processes close to the
singularity - but how can it possibly get out from this location? I
can't think of any conceivable mechanism.

[I know very little about quantum gravity; maybe someone
else who knows more give you a more helpful answer here.]

We don't know what quantum-gravity effects will do close to the
singularity. But I don't see that that's an argument that these
(unknown) quantum-gravity effects will suddenly make large modifications
to well-understood classical physics at places which are *not* close
to singularity.
[/quote]
And that's exactly the problem! If they don't, how can the information
get out? Hawking radiation with no information content can sorta kinda
be reconciled with GR, but how do you get the information from [near]
the central singularity into the Hawking radiation?

[quote] [Actually, things are a bit dicier than that:
what about places that can *see* (i.e., be causally
in the future of) a singularity? Could "seeing"
a singularity somehow influence local physics?]
[/quote]
The only actual singularities are in our models. In the real world,
some other physics has taken over.

[quote]Basically, if you want to say that quantum-gravity effects will
prevent our parcel of incoming matter from getting very close to the
singularity in finite proper time, then why stop there? Why not say
that quantum-gravity effects will cause the Brooklyn Bridge to collapse
tomorrow? What's the reason for finding one of these more or less
plausible than the other?
[/quote]
I said nothing about the Brooklyn Bridge. I was very specific about
where I think the quantum gravity effects start to become noticeable.

What I'd expect [this is my speculation], say for an astronaut who
jumps into a very large black hole, is that as he gets close to the
Schwarzschild radius, he will probably start to observe high energy
interactions that were so rare as to be unnoticeable in ordinary
gravitational fields, but whose probability is unaffected, or less
affected, by the gravitational field. These interactions can probably
- like a black hole - thermalise anything not associated with a long
range field. So I'd guess that he will see exponentially accelerating
proton decay - according to his clocks - as he approaches the horizon.
Maybe it's too simplistic, but it seems to me that combined with
gravitational red shift, this is a very straightforward and natural way
of turning an infalling object into Hawking radiation.

Anyone see any major difficulties or inconsistencies in this scenario?
There must be some problems, I guess, or surely almost everyone would
believe it!

- Gerry Quinn
 
eric gisse...
Posted: Sat Jun 12, 2010 1:19 pm
 
Gerry Quinn wrote:

[...]

[quote]It's important to realise that (at least according to the Einstein
equations) the event horizon is only *globally* defined. Locally --
and today we believe that all physics is ultimately local, not
action-at-a-distance -- there's nothing special going on at the horizon,
and in particular there's no way for our parcel of matter to even
"know" that it's crossed the horizon.

But of course the key words are 'according to the Einstein equations'.
If these equations are not correct at the event horizon, the premise of
the above argument is incorrect.
[/quote]
If the Einstein equations are not correct than the argument cannot take
place because the features of which you are arguing about exist only
_because_ of the field equations.

[quote]Thus the argument shows only that the
GR solution is internally consistent. It has no strength when it comes
to ruling out alternatives.

[Of course if GR is incorrect at the Schwarzschild radius the
expression 'event horizon' may be a bit of a misnomer.
[/quote]
Where did the notion of the 'event horizon' or 'Schwarzschild radius' come
from? Hint: GR

The concepts do not exist in a vacuum.

[...]

[quote]Once we're comfortable using the (classical) Einstein equations, then
it's an easy calculation that our parcel of matter will indeed approach
the central singularity in a finite proper time.

But I'm not comfortable with using them, because while they do not seem
to become internally inconsistent before the central singularity, they
appear to become inconsistent with the rest of physics before they are
extrapolated so far.
[/quote]
Have you looked at 'the rest of physics' recently?

Look at the self-energy of an electron in classical E&M. Ooops.
Look at renormalization in quantum mechanics, and its' lack of gravitation.
Ooops.

You can argue that GR is 'inconsistent' with 'the rest of physics' and
nobody will care because what you say is obvious and well known. However,
there exist no current observations which are relevant to both GR and
quantum theory simultaneously.

[quote]
Suppose we postulate instead a scenario in which spacetime really is
fundamentally flat, and objects in strong enough gravitational fields
become asymptotically 'frozen' - at least as far as ordinary low-energy
interactions are concerned - and do not reach the proper time for which
GR predicts they will come close to a central singularity. Is there
anything inconsistent with this?
[/quote]
You need to make testable predictions and have a mathematical foundation
before any discussion of 'consistency' can take place. Its' easy to make up
wild scenarios in which these things are "true" but not nearly as easy to
formulate them into a coherent theory that exists much less survives
scrutiny.

[...]

[quote]What I'd expect [this is my speculation], say for an astronaut who
jumps into a very large black hole, is that as he gets close to the
Schwarzschild radius, he will probably start to observe high energy
interactions that were so rare as to be unnoticeable in ordinary
gravitational fields, but whose probability is unaffected, or less
affected, by the gravitational field.
[/quote]
See above.

[snip rest]
 
Uncle Al...
Posted: Sat Jun 12, 2010 4:07 pm
 
eric gisse wrote:
[quote]
Gerry Quinn wrote:
[snip][/quote]

[quote]Once we're comfortable using the (classical) Einstein equations, then
it's an easy calculation that our parcel of matter will indeed approach
the central singularity in a finite proper time.

But I'm not comfortable with using them, because while they do not seem
to become internally inconsistent before the central singularity, they
appear to become inconsistent with the rest of physics before they are
extrapolated so far.

Have you looked at 'the rest of physics' recently?

Look at the self-energy of an electron in classical E&M. Ooops.
Look at renormalization in quantum mechanics, and its' lack of gravitation.
Ooops.

You can argue that GR is 'inconsistent' with 'the rest of physics' and
nobody will care because what you say is obvious and well known. However,
there exist no current observations which are relevant to both GR and
quantum theory simultaneously.
[/quote]
GR and QFT *do* overlap in one domain - string theory. String theory
is a mathematical triumph and a physical disaster. The classical and
quantum components have two assumptions in common: vacuum isotropy and
the Equivalence Principle. Photons say the spacetime geometry is
well-behaved,

http://arxiv.org/abs/0706.2031
http://arxiv.org/abs/0905.1929
http://arxiv.org/abs/0912.5057
http://arxiv.org/abs/0801.0287

When good theory fails, look for bad postulates. Physics cannot
imagine a fundamental extrinsic, extensive, emergent phenomenon.
Pharma does it as life and death: chirality. Nobody knows if left and
right shoes violate the Equivalence Principle by detecting a
left-footed chiral vacuum background. Such would reduce all physical
theory to heuristics without contradicting any prior observation,

http://www.mazepath.com/uncleal/erotor1.jpg
Somebody should look. The worst it can do is succeed.

Physics dedicates unlimited resources and personnel attempting to
discredit historically good theory. Physics should throw a sop toward
obviously vulnerable postulates where the answer can appear with no
fancy tap-dancing required before or after.

A left foot cannot be detected by socks or left shoes no matter how
large the numbers, how exotic the materials, or how fancy the styles.
A right flip-flop recyled from old tire tread is an unambiguous
identification.

Do it.

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz4.htm
 
Oh No...
Posted: Sat Jun 12, 2010 8:27 pm
 
Thus spake Tom Roberts <tjroberts137 at (no spam) sbcglobal.net>
[quote]Oh No wrote:
[about coordinate-free techniques]
But this is what I mean by "pulling the wool". They give the
appearance of dispensing with coordinates, and one finds people thinking
that this makes them somehow a better model of physical reality, since
coordinates are imposed by man, not by nature. Actually they include all
possible coordinates, which is a very different kind of statement.

You have not understood what I have been saying. This is very basic and
fundamental; it's also rather obvious once you sit down and think about it.

For instance, the fact that we apply coordinates to the world REQUIRES that we
be able to apply coordinates to the geometry. That is not "pulling the wool",
that is an operation parallel to what we often do in the real world. A good
model must be able to perform such parallel operations.

Physical theories are MODELS of nature and the world we inhabit. So valid
theories have a multi-level correspondence to the world:
[/quote]
I still cannot grasp, and you have not explained, what metaphysical
notion of "world" you have. This being so, nothing you say is obvious,
and much is plainly wrong.
[quote]
Theory (model) <=> World
------------------------------ --------------------------
Coordinate-free description of <=> Existence of the world, (without
a manifold and its geometry any coordinates)
[/quote]
This correspondence, for example, is plainly wrong. As Igor pointed out,
coordinate free notations do not apply without any coordinates. They
apply with all possible coordinate systems.
[quote]
Coordinate-free description <=> Natural operation of physical
of physical phenomena phenomena (without any coordinates)
[/quote]
ditto
[quote]
Application of coordinates to <=> Application of coordinates
the manifold to the world
[/quote]
First you need to work out what application of coordinates to the world
means. What you say below shows that you clearly don't understand it.

[quote]Projection of geometry and <=> Projection of physical measurements
physical theory onto the onto the coordinates
coordinates

Coordinate-based description <=> Coordinate-based measurements of
of geometry and physical geometrical relationships and
phenomena physical phenomena
[/quote]
Again you are assuming those geometrical relationships as some kind of
metaphysical prior. I cannot grasp that. I can only work from the
mathematical definitions, in which R^n is the prior.

[quote]Those items are ordered in an obvious way, which is why we say that a
coordinate-free description of physical phenomena is more natural, and at a
lower level, than coordinate-based descriptions.
[/quote]
We would only say that if we have not grasped the fact that coordinate
free notations are true in any coordinates. The do not apply to a
structure which does not have coordinates.
[quote]
And I repeat: Coordinates are an arbitrary, human choice, and nature obviously
uses no coordinates, so no valid model of natural phenomena can depend in any
way on coordinates [#]. So in physics there is really no choice but to pursue
coordinate-free descriptions and techniques. This is, after all, one of the
major lessons of GR (but older authors did not realize it).
[/quote]
It may well be true that space is not a prior, in the sense of a
manifold everywhere locally homeomorphic to R^n. However, GR does not
teach us that. Quite the opposite, since it depends on the definition of
a manifold.. Perhaps quantum theory does, but there is no accepted
interpretation of quantum theory according to which we can say so.
[quote]
[#] But as in the world, application of coordinates must be possible.
Fortunately, the manifolds of differential geometry have precisely
these properties, as do tensors and tensor fields on such manifolds.
This is not happenstance :-)

Note that common measuring devices, such as clocks, rulers, voltmeters,
ohmmeters, ammeters, balance scales, etc. all do their thing without any
coordinates whatsoever. You can apply coordinates to many of them, but that is
not at all the same as them using the coordinates internally. Claiming that
coordinates are a necessary precursor to measurements or making
physical models
is obviously wrong.
[/quote]
I certainly did not claim such a thing, and nor did Einstein. Let us
stick, for simplicity, and definiteness with clocks and rulers. As shown
by Einstein coordinates are the results of measurements, not precursors
to them.


Regards

--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)

http://www.rqgravity.net
 
Igor Khavkine...
Posted: Sun Jun 13, 2010 12:36 am
 
On Jun 13, 8:27 am, Oh No <N... at (no spam) charlesfrancis.wanadoo.co.uk> wrote:

[quote]The notion of three as derived from, e.g., three apples has caused
mathematicians huge philosophical problems through the ages. Which is
why we have settled on von Neumann's definition based on set theory,
which divorces mathematics from such philosophical problems. One can
easily fool oneself into thinking that we can think of a point in a
manifold completely abstractly. That may even work when the manifold is
a physical 2D surface in 3 space. However, if that is how one thinks of
the spacetime manifold, it is pure metaphysics, and cannot be justified.
The actual mathematical definition of a manifold places it locally into
correspondence with R^n. One cannot escape that, unless one pulling the
wool over one's own eyes.
[/quote]
The above argument contains an excellent recipe for divorcing manifold
points from n-tuples or reals, which happens to be the same as for
divorcing threes from apples. But I wouldn't want to pull a wool over
anyone's eyes, including your own. So from now on, in our
conversations, I will only use numbers in conjunction with objects
that they count. In any case, this conversation has drifted far enough
off topic...

Igor
 
Peter...
Posted: Mon Jun 14, 2010 4:45 pm
 
On 11 Jun., 02:33, Igor Khavkine <igor... at (no spam) gmail.com> wrote:
[quote]On Jun 3, 4:14 pm, Peter <end... at (no spam) dekasges.de> wrote:
On 1 Jun., 05:59, Tom Roberts <tjroberts... at (no spam) sbcglobal.net> wrote:

Consequently, I cannot follow your arguing that, in Schwarzschild's
paper (!), r=0 is _not_ the origin, where the mass is located.

A claim that r=0 corresponds to the location of a mass (even by the
author of the solution) does not make it so. Independent checks of
this claim, in works following Schwarzschild's original, have verified
that there is indeed no mass there. As you can see, this paragraph
does not give you the details. However, it should at least make you
realize that, if you wish, you need to make a stronger argument for
the presence of mass at r=0 than "I don't know where else it could
be", because it could be nowhere.
[/quote]
Schwarzschild calculates the Mercury orbit around r=0

[quote]both solutions are different => one of them is wrong, or the solution
is not unique (in contrast to Schwarzschild's claim)

You are implicitly using a private notion of identity (or distinction)
of solutions, hence also of uniqueness and non-uniqueness. As
evidenced by your confusion, these notions seem to be different from
the standard ones of modern differential geometry. To move the
discussion forward, you need to either lookup the standard definition
and adopt it or to precisely articulate your own.
[/quote]
I'm confused by some replies to my simple questions, indeed.

If I go back to Cartesian coordinates (t,x,y,z), then, I hope, I avoid
the difficulties with spherical coordinates you have rightly
highlighted.

The metric tensor components in these coordinates are - at least
formally - different in Schwarzschild's paper and in Wikipedia.
Unfortunately, I could not see any explanation for that in the
foregoing postings. According to Heisenberg, it should nevertheles be
possible to explain it to persons like me having obtained a basic
education in general physics, but not in differential geometry and
general relativity ;-)

[[Mod. note -- Different coordinates means different tensor bases,
so naturally the coordinate components are different. A simple example
might clarify this: Consider the 3-D wind (velocity) vector at my home
at some specified time. We can resolve this into components in many
ways, including:
(a) write the Cartesian components of this 3-D vector as (vx,vy,vz),
where I orient x=north, y=west and z=up
(b) write the polar-spherical components of this 3-D vector as
(vr,vtheta,vphi), where define (r,theta,phi) by the usual formulae
x = r sin(theta) cos(phi)
y = r sin(theta) sin(phi)
z = r cos(theta)
Does it surprise you that the 3 numbers (vx,vy,vz) aren't the same
as the 3 numbers (vr,vtheta,vphi)? Of course not.

Well, that's the same reason why the metric tensor components of
Schwarzschild spacetime at a given event will look different in
Schwarzschild's original coordinates, than in most modern treatments
(the latter including Wikipedia).
-- jt]]

<snip>

Thank you,
Peter
 
Andreas Most...
Posted: Tue Jun 15, 2010 10:38 pm
 
Peter <enders at (no spam) dekasges.de> writes:

[quote]On 11 Jun., 02:33, Igor Khavkine <igor... at (no spam) gmail.com> wrote:
On Jun 3, 4:14 pm, Peter <end... at (no spam) dekasges.de> wrote:
On 1 Jun., 05:59, Tom Roberts <tjroberts... at (no spam) sbcglobal.net> wrote:

Consequently, I cannot follow your arguing that, in Schwarzschild's
paper (!), r=0 is _not_ the origin, where the mass is located.

A claim that r=0 corresponds to the location of a mass (even by the
author of the solution) does not make it so. Independent checks of
this claim, in works following Schwarzschild's original, have verified
that there is indeed no mass there. As you can see, this paragraph
does not give you the details. However, it should at least make you
realize that, if you wish, you need to make a stronger argument for
the presence of mass at r=0 than "I don't know where else it could
be", because it could be nowhere.

Schwarzschild calculates the Mercury orbit around r=0
[/quote]
The 3 km for the Schwarzschildradius of the Sun is neglectable compared
to the 46,001,200 km Perihelion of Mercury. It wouldn't make a
measureable difference in the rotation speed of the Perihelion.

Andreas.

--
Replace animals with appropriate symbols
AndreasDogMostCatGoogleMailDogCom
 
Daryl McCullough...
Posted: Wed Jun 16, 2010 10:05 am
 
Oh No says...
[quote]
Thus spake Tom Roberts <tjroberts137 at (no spam) sbcglobal.net
Oh No wrote:
Thus spake Tom Roberts <tjroberts137 at (no spam) sbcglobal.net
Coordinate-free description of <=> Existence of the world, (without
a manifold and its geometry any coordinates)

This correspondence, for example, is plainly wrong. As Igor pointed out,
coordinate free notations do not apply without any coordinates. They
apply with all possible coordinate systems.

Not so. Coordinate-free techniques apply without any coordinates
whatsoever.

That is a fundamental misunderstanding of the mathematical definitions.
[/quote]
I assume that you have already seen the coordinate-free definitions
of vectors, 1-forms and tensors? If not, here's a quick summary:

[ Mod. note: It also helps to mention at the start that the underlying
manifold is a topological space, which can be as abstract as you
like. -ik ]

The primitive geometric notions for a differentiable manifold are
(1) smooth real scalar field (functions that assign a real number
to each point on the manifold), and (2) smooth parametrized paths
(functions that map real numbers to points on the manifold). We
certainly don't need coordinates to be able to make sense of these
concepts, except possibly to explain the notion of a "smooth" function:
a function on a differentiable manifold is smooth if the corresponding
map on n-tuples is differentiable. For much of the development of the
coordinate-free approach, you don't need to use the definition of "smooth".

The intuitive meaning of scalar field and parametrized path can be
understood without introducing a coordinate system. For example,
we can take the surface of the Earth to be our manifold. Then the function
that gives the altitude of each point can be modeled as a scalar field.
A road on the surface of the Earth, together with a starting point and
a direction, can be modeled by a parametrized path, in which the parameter
is the distance along the road.

In terms of the notions of scalar field and parametrized path, we
can define a vector to be a tangent (linear approximation) to a
parameterized path, and we can define a 1-form to be a tangent
to a scalar field. You don't need coordinates, in the sense of
assigning n-tuples to each point on the manifold, in order to make
sense of these notions.

The technical definition of a tangent to a parametrized path at a point
is in terms of the local effect of traveling along the path on the values
of scalar fields: If P(s) is a parametrized path, then the tangent to P
at point s_0 is defined as that operator V that acts on scalar fields
Phi(X) as follows:

V(Phi) == d/ds Phi(P(s)) evaluated at s=s_0

Note that you don't need coordinates to differentiate, since
the combination Phi(P(s)) is a function from reals to reals,
and so we can use ordinary calculus of real numbers.

To know V for a path P(s) all you need to know about the path
is what direction it is headed, and how fast it is headed there.

If Phi(X) is a scalar field, then we can define the tangent to
Phi at point X_0 to be that operator W such that for all parametrized
paths P(s) that pass through X_0 at s=0, if V is the tangent to P
at s=0, then

W(V) = V(Phi)

To know W for a scalar field Phi, you just need to know how Phi
changes locally for all possible directions.

Although the mathematical model for differentiable manifolds is
indeed little pieces of R^n, I think it is an exaggeration to say
that coordinates are at the heart of this coordinate-free development.
The notion of tangent vector and 1-form don't require coordinates.
Where the patches of R^n come into play is the technical fact that
at each point, there are n different basis tangent vectors such that
every tangent vector can be expressed as a linear combination of basis
vectors. This fact *allows* us to use coordinates to describe vectors
and 1-forms, but it certainly is not needed to make sense of those
notions.

--
Daryl McCullough
Ithaca, NY
 
enders...
Posted: Fri Jun 18, 2010 9:06 pm
 
One confirmation, one correction and one renewal of questions
concerning this early posting:

On 3 Jun., 16:14, Peter <end... at (no spam) dekasges.de> wrote:

[quote]Schwarzschild uses the following coordinates.

a) x_1,2,3,4, mass in origin, x_1=x_2=x_3=0

b) r = SQRT ( x^2 + y^2 + z^2 ) = SQRT ( x_1^2 + x_2^2 + x_3^2 )

   mass in origin, r=0

c) x_1 = r^3 / 3, x_2 = -cos /theta, x_3 = /phi

   discontinuity at origin, x_1=0, see text before eq.(13)
[/quote]
Confirmation:
Several posters have written, that, in Schwarzschild's paper, r=0 is
_not_ the origin (where the mass is located). None has shown, what is
wrong in the arguing above, however.


Correction:

[quote]both solutions are different => one of them is wrong, or the solution
is not unique (in contrast to Schwarzschild's claim)
[/quote]
This statement is false, sorry!

Schwarzschild's solution (14) is correct for a wide range of values
for \alpha and \rho, in particular, for \rho=\alpha^3 he uses and for
\rho=0 yielding the cited Wikipedia metric as well as that on
http://rqgravity.net/Schwarzschild


BTW, I'm not disputing that the singularity in the latter is
coordinate related. For all values of r, the local radial speed of
light equals that in vacuo.

Thus, let me ask, again, why one has left Schwarzschild's choice \rho=
\alpha^3 in favor of \rho=0, and is there a possibility to determine
\rho experimentally? (The perihel rotation of Mercury is by far too
small, it can be explained in 2nd order, \alpha^2/r^2, see Einstein
1915, while \rho enters only in 3rd order, \rho/r^3.)


Thank you and best wishes,
Peter
 
Albert van der Horst...
Posted: Fri Jun 18, 2010 10:42 pm
 
In article <1dLDa0DCgVEMFw9B at (no spam) charlesfrancis.wanadoo.co.uk>,
Oh No <charlesDOTeDOThDOTfrancis at (no spam) googlemail.com> wrote:
[quote]Thus spake Peter <enders at (no spam) dekasges.de

Again, I'm extremely happy that you seem to share my pov. As I have
understood your web site, its T which determines g, is this correct?

Yes. The method is to ascertain T for a given situation, then solve the
Einstein field equation (if we can) to determine g.

If yes, I have to calculate T first, before I can calculate g. But can
I calculate T without a concrete coordinate system?

No, you cannot. There is a modern fashion for "coordinate free"
notation. People think this gives a deeper insight into general
relativity. Imv it is merely a way of pulling the wool over the eyes. In
practice physics starts from the definition of coordinate systems, and
cannot be done without one.
[/quote]
IMO you're far out here. There is a manifold that is amenable to
direct experimentation and allows to derive properties,
that is the space around us.
There are child's plays that reflect infinitesimal displacements
in the Lie sense. The intuitive experiments are sufficient
to e.g. establish that we are in a three dimensional world.

A child can realize that there are two directions to a rooms floor,
and that there are points there that can be identified by
e.g. placing an object there.

What is an electric field? At a certain point measure the
force on a charge. The function that relates the point to
the force is called a field. No, I don't need coordinates.
I can see the direction and extension of the spring at that
point. I can make a drawing, and there are no real numbers
there.

Far from being a "fashion", this "coordinate free" notion of
mathematical objects of a point on the floor is the deeper insight
underlying the progress not made until Descartes that the floor
can be endowed with a coordinate system.

A real number is so intricate an object that its properties
have not been understood until the late 19-th century
mathematics.
Even so it has deeply disturbing aspects, like taking on
more than countable infinite values. Only countably finite
values of those can even be characterized in any way, the
computable numbers. Also a real number is in 3 or 4 step
constructed from ordinal numbers like 3.
I have heard some one argue in this thread that numbers like
3 are too abstract and shaky a concept to base science on.

Equating triples or quadruples of real numbers to the real
world to the point of refusing to talk about what is really
out there is disturbing. Of course once you start calculating
you need numbers, but I'm convinced that progress is made by
people doing coordinate free thinking, which justifies the
fashion in the books. Physics doesn't start with introducing
coordinates. It ends with them at the point you have a
theory and you need to compare with experimental results.

Why I personally like the fashion.
I for me, each time I see a formula like T_ab I do not just
identify it with T. I go in a disturbing side path
considering in how far it can be identified. I imagine a
particular coordinate system and project 16 numbers in
there.
Then I may check whether the formula is true for this
coordinate system. Then I ask myself whether this reflects
any real properties of the manifold/space/whatever itself. A
discovery goes the opposite way. I "see" something. Then I
try to express it in a coordinate system.

If a property can be described in an arbitrary coordinate
system, it should be described without a coordinate system.

[quote]
Regards

--
Charles Francis
[/quote]
Groetjes,
Albert van der Horst

--
--
Albert van der Horst, UTRECHT,THE NETHERLANDS
Economic growth -- being exponential -- ultimately falters.
albert at (no spam) spe&ar&c.xs4all.nl &=n http://home.hccnet.nl/a.w.m.van.der.horst
 
Tom Roberts...
Posted: Fri Jun 18, 2010 10:42 pm
 
Daryl McCullough wrote:
[quote]I assume that you have already seen the coordinate-free definitions
of vectors, 1-forms and tensors? If not, here's a quick summary:
[ Mod. note: It also helps to mention at the start that the underlying
manifold is a topological space, which can be as abstract as you
like. -ik ]

The primitive geometric notions for a differentiable manifold are
(1) smooth real scalar field (functions that assign a real number
to each point on the manifold),
[/quote]
I see no need for this; the points of the manifold form a topological
space (as the moderator said), and that defines "continuous" (not
"smooth", which does not really apply to the manifold itself). You used
this field only to define "smooth", which is why it is unnecessary in
the topological approach.

BTW "smooth" is overkill, though physicists often specify it
(it means C^infinity -- continuously differentiable an
infinite number of times); continuous (C^0) is all that is
needed for a manifold, and the metric being C^2+ is all that
is needed to apply the field equation of GR [#]. See Hawking and
Ellis.

BTW "differentiable" applies to functions and maps on the
manifold, not to the manifold itself. Ditto for "smooth".
But a manifold is continuous, as defined by its topology.

[#] And one common manifold is only C^2+ -- a static universe
containing a spherically-symmetric planet of uniform mass
density surrounded by vacuum. This is a combination of the two
Schwarzschild solutions, stitched together at the surface of the
planet. (The radius of the planet must be larger than its Schw.
radius.)

[quote]and (2) smooth parametrized paths
(functions that map real numbers to points on the manifold). We
certainly don't need coordinates to be able to make sense of these
concepts, except possibly to explain the notion of a "smooth" function:
[/quote]
The topology defines "continuous" for the manifold, without any
reference to or use of coordinates or a diffeomorphism to R^N. I don't
know what "smooth" would mean for the manifold (a manifold cannot be
differentiated), but C^n and smooth (C^infinity) functions and maps can
be defined on a continuous manifold.

Once you have the topology, paths are either a) 1-d sub-manifolds with
the topology of R, or b) diffeomorphisms from R to the manifold (the
choice depends on your definitions; (a) and (b) are equivalent). While
(b) can be used to apply a path parameter (coordinate) to the path, the
locus of the submanifold of (a) can also be referenced without regard to
that diffeomorphism.

[quote]For much of the development of the
coordinate-free approach, you don't need to use the definition of "smooth".
[/quote]
IMHO it's much better to base this all on the topology. That is the usual
approach (today).

[quote]Although the mathematical model for differentiable manifolds is
indeed little pieces of R^n,
[/quote]
That is an aspect of one possible (older) definition of manifold. A more
modern definition is based on the topology, which is much better IMHO.
It is equivalent to the one you refer to, but uses no diffeomorphisms
(because those are at a higher level of structure than the topology
itself -- continuous is more basic than differentiable).

Tom Roberts
 
Dary McCullough...
Posted: Sun Jun 20, 2010 4:54 am
 
Oh No says...

[quote]GR is valid on large scales, and, it describes matter fields, not
pointlike particles. However, what we observe on small scales is that
matter consists of pointlike particles (up to quantum effects).

Instead of using the Schwarzschild R coordinate, which has an event
horizon at R=2GM, I have been using the r coordinate with r = R - 2GM
outside the event horizon. The region R<2GM does not map to r
coordinates, and I am now strictly working on a different manifold, not
just a chart on the original manifold.

The r coordinate has the event horizon at r=0. I describe a pointlike
particle at r=0 surrounded by the exterior region of a Schwarzschild
geometry.
[/quote]
That doesn't seem right at all to me. Even if you want to restrict
your manifold to the region outside the event horizon, and even if
you want to use coordinates in which r=0 is the event horizon, that
doesn't make the black hole a pointlike object. A surface enclosing
the event horizon has a nonzero area. The limit as you approach the
event horizon stays nonzero. I can't make any sense of calling it
a pointlike object.

I would think that for something to be considered pointlike, it
must be possible to enclose it with a surface of arbitrarily small
surface area. If that's not the case, then in what sense it is
pointlike?

--
Daryl McCullough
Ithaca,NY
 
 
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