Answer by Researcher mathtalk on Sat 29 Nov 2008 - 5:26 pm UTC:

>  As far as I can tell the only justification is that the
>  absolute function allows the curve to be extended down
>  below r=2m as shown in the mathpage plot in order to
>  comply with the classical physical interpretation. This
>  has now become the single most important issue that I am
>  interested in now.

While I don't see how extending of the solution for coordinate
time t below r=2m (for a bounded trajectory with apogee R > 2m)
complies with any "classical physical interpretation," I can
can offer some mathematical observations on this extension.

The absolute value inside the logarithm is crucial to extending
the trajectory below r=2m, but its presence is sensible.  As a
quick explanation, the conventional antiderivative for 1/x is:

     d ln|x|
     ------- =  1/x        (x =/= 0)
       dx

valid on the real line except zero.  Of course a constant could
be added arbitrarily, and indeed one might use different constants
for x > 0 and x < 0, since we already have a discontinuity there.
But the above is evidently the simplest expression.

In Kevin Brown's writeup the parametric equation for t:

                                           |Q + tan(α/2)|
    t = Q*((R/2 + 2m)α + (R/2)sin(α)) + ln(|------------|)
                                           |Q - tan(α/2)|

where constant Q is:
         __________
    Q = √(R/2m) - 1     [assume R > 2m]

is to be compared with the corresponding equations:

    r = (R/2) (1 + cos(α))
               ____
    τ = (R/2) √R/2m (α + sin(α))

In the latter case we have a smooth passage of r as a function
of proper time τ across r=2m.  By smooth here we mean analytic
continuation, which implies continuous derivatives of all orders
across this boundary.

Indeed even the behavior of r(τ) as τ approaches π is actually
very nice as singularities go.  Recall from Part 1 that:
                     ____
   dr/dτ = -sin(α)/[√R/2m (1 + cos(α))]

Although this becomes infinite at odd multiples of π, those are
"simple poles" in the terminology of analytic functions, i.e.
the reciprocal dτ/dr has simple roots (zeros) at these places.

Given these mathematical facts, a physical interpretation of the
solution r(τ) as a periodic orbit seems unobjectionable to me.

However the singularity at r=2m in function r(t) of coordinate
time is much worse, an essential singularity in analytic function
theory.  Accordingly my faith in any physical interpretation of
the purely mathematical extension of r(t) across the boundary is
weak.  If anything such an interpretation would be supported best
by consistency with the smooth extension in terms of proper time.

That is, whatever metric exists within the "event horizon" r < 2m,
one could relate coordinate time there with the proper time τ and
radius r associated to the entering object.  But relating some
coordinate inside the event horizon to that outside seems quite
problematic.

For one thing as you showed above, light effectively stops at the
event horizon, so no external observational framework supports a
notion of relating spacetime coordinates across this boundary.

*  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *

If we permit speculation, a number of authors of both serious and
fictive literature have presented ideas that black holes may be
gateways to time travel.  Perhaps the simplest form of such a
theory would be an idea that time inside the event horizon runs
opposite to time outside.

But my impression is that the direction of time is mysterious
enough without dragging black holes into the picture.  An author I
like on this subject is Huw Price:

[Huw Price, Univ. of Sydney]
http://www.usyd.edu.au/time/price/

What we can say is that the parameterizations r(α) and τ(α) give
rise to a solution r(τ) in proper time that is continuous and
has period:
                      ____
   τ(π) - τ(-π) = Rπ √R/2m,

while the parameterization t(α) "blows up" at α = ±2 arctan(Q).

[Since Q involves a square root, we can consider what would be
the result if the negative root were chosen.  From the definition
of t(α) one sees that if Q and α are replaced by -Q and -α, then
exactly the same value for t is obtained.  Since r(α) is an even
function, this amounts to a different parameterization for the
same solution t(α).  So in all we would conclude as before that
the direction of coordinate time can go either way while still
satisfying the Schwarzchild metric equation.]

Evaluating t(α) between the limits ±2 arctan(Q) can be done
without applying the absolute value in the logarithm's argument.
Extension beyond these limits gets us a fraction with negative
values, so that the absolute value has to be applied before the
(real) logarithm can be taken.

If α is slightly increased above 2 arctan(Q), then the numerator
Q + tan(α/2) is little changed and the denominator Q - tan(α/2)
switches from a small positive value to a small negative one.
After applying absolute values, the argument of the logarithm
is still a large (positive) value, and taking the logarithm will
give a large positive result for t(α).  However we have now
entered a region in which t(α) decreases with increasing α, a
perhaps different sense in which one could claim time runs
backward.

Similarly an argument could be made that moving backward along
the curve t(α) until α is decreased slightly below -2 arctan(Q)
will bring us to the other branch where r < 2m, but with t(α)
increasing (as the argument to the logarithm passes thru zero).

*  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *

Is this an artifact of applying the absolute values to the
argument of logarithm?  Well, yes it is.  But there are many
things that are artifacts of following ones mathematical nose,
which is all that is being done here.  Sometimes it means a
solution to a word problem involving negative ages for family
children (which clearly has no real world meaning) and other
times it gives us negative weights for the amount of sugar to
buy (which might seem meaningless at first, but could turn
out to mean an advantage from selling sugar!).

For me the plausibility of this approach to theoretical time
travel is countermanded by two facts, one mathematical and
one physical.  The math fact has already been mentioned, and
that is the essential singularity of the logarithm we must
"pass thru" at either limit of α = ±2 arctan(Q).

The physical fact is that black holes are not static entities
in our Universe.  The best observation evidence is that they
have a definite "birth" and growth, and according to theory
advanced by Stephen Hawking, eventual dissipation.  Thus the
chronological extent of a black hole phenomenon would seem to
be finite, at least in reference to external coordinate time,
and not extended to infinity as the (necessarily simplified)
mathematical model suggests.


regards, mathtalk