Re: On the controversy about the Schwarzschild radius and black holes

cafeinst@xxxxxxx wrote:

carlip-nospam@xxxxxxxxxxxxxxxxxxx wrote:

[...]
Crother's "proper radius" is easily calculated -- it is the radial distance
from the sphere r=2m. So of course it's zero at r=2m, just as the radial
distance from the surface of the Earth is zero at the surface of the Earth.
Calling it a "radius" doesn't make it a radius. The Schwarzschild metric
itself tells you that it's the distance from a *sphere* -- specifically, a
sphere of area 16pi m^2 -- and not the distance from a point.

Suppose you define a coordinate R that measures distance from the surface
of the Earth. I'm at about R = 15m now. The basement of my building is
at about R = -5m. Are you going to tell me this is impossible, because
R can't be negative? What if I decide to call R a "radius"? Does the
basement then vanish?

Read http://www.geocities.com/theometria/PhD.html. Crothers says:

"The radius of curvature is Rc(r) = r, but the proper radius is,

Rp(r) = Integral of (1 - 2m/r)^-Å? dr = [r(r - 2m)]^Å? + 2m ln|r^Å?
+ (r - 2m)^Å?| + K,

where K = const.

That's just wrong. If you make the integral on the left-hand side
a *definite* integral, between two values r+a and r=b, then it gives
the proper radial distance between those points *if they are in the
range in which the coordinate system exists.* In particular, as I
said, if you set the lower limit to r=2m, this gives the proper
distance from the surface r=2m. "Proper distance from a sphere of
finite area" is not "proper radius" -- this is a blatant abuse of
language.

"Now the lower limit of the proper radius is zero, and this occurs only
when r = 2m and K = -m ln(2m).

Sure -- because that's the range in which the coordinate system is defined..
But to conclude anything more from that is completely circular: Crothers
is saying, "I'm defining an integral in a coordinate system that breaks
down at r=2m, therefore the integral doesn't make sense for r<2m, therefore
there is no region r<2m," while what he is actually showing is merely,
"I'm defining an integral in a coordinate system that breaks down at r=2m,
therefore the integral doesn't make sense for r<2m, therefore if I want
to extend the result to r<2m I need to use a better coordinate system that
extends through r=2m."

Thus, when the proper radius Rp = 0, the
radius of curvature Rc = 2m. These are scalar invariants for Einstein's
gravitational field, and are independent of any admissible r-coordinate
system.

This is a fabrication. It is simply false.

The fictitious point-mass is always located at Rp = 0."

One more time, with feeling: r=2m IS NOT A POINT. It is a sphere. There
is *absolutely* no ambiguity about this. The Schwarzschild metric uniquely
determines the induced metric on the surface r=2m, t=const., and it is the
metric of a sphere of area 16pi m^2. Calling a sphere a point doesn't make
it a point.

Therefore, since Crothers proves that the proper radius Rp=0 when
Rc=2m, it's not just a tail that he decides to call a leg. He proves
that you can't go lower than Rc=2m, since doing such would cause the
proper radius to be nonreal.

No, he doesn't. He shows that a calculation in a particular coordinate
system of a quantity that he chooses falsely to call "proper radius"
is not defined outside the range of the coordinate system. This doesn't
mean the real, physical proper radius doesn't extend beyond r=2m; it just
means that if you want to compute it, you have to do so in a coordinate
system that extends beyond r=2m.

To extend my original analogy, what Crothers is doing is equivalent to
defining a coordinate R that is equal to the absolute value of the
distance above the surface of the Earth, pointing out that its smallest
value is at the surface of the Earth, and concluding the impossibility
of basements.

Steve Carlip

.

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