The new coordinates.
and 
where
or
alternatively 
This
set of coordinates is equally applicable above and below the event horizon.
There is the additional benefit that time remains time and distance remains
distance everywhere in the coordinate system, including above and below the
event horizon, unlike the Kruskal-Szekeres coordinates .
The interpretation of time and distance is therefore unambiguous in the new
coordinates. This considerably reduces confusion in physical interpretations.
The new metric.
The
new metric is similar to the Kruskal-Szekeres metric and differs only in the
factor f that has the same meaning as in the coordinate equations above.
This
is a graphical representation of the new metric. The animation in the
introduction illustrates the ‘morph’ from Schwarzschild
coordinates to the new coordinates. Note the similarity to an acceleration
chart in Minkowski space. In fact it can be thought of as a Minkowski diagram converted
from flat space to curved space. It is also similar to the Kruskal-Szekeres
with the North quadrant rotated 90 degrees anticlockwise. It can be seen that
the transformation to the new coordinates is geometrically more direct than the
Kruskal-Szekeres transformation and the latter can be more correctly thought of
as a 90 degree clockwise rotation of the West quadrant of the new coordinates.
Note that r is undefined in the North and South quadrants (grey regions) where
T^2 > R^2 and this suggests that there is no alternative universe or white
hole present in the new coordinates. The horizontal axis consistently
represents distance and the vertical axis consistently represents time
everywhere in the diagram (except in the grey regions where there is no
consistent definition of space and time). The left quadrant is the region below
the event horizon where r<2m while the right quadrant represents
2m<r<infinity.
Just
as in Kruskal-Szekeres coordinates, light rays follow straight diagonal paths
and it easy to see that a light ray extended continuously from the right
quadrant into the upper quadrant, never arrives in the left quadrant and light
rays emitted in the left quadrant never arrive in the right quadrant. The grey
regions represent an impassable barrier between the region below the event
horizon and the area above the event horizon. In the new coordinates the event
horizon (red lines) is a two way barrier and real particles and photons can not
pass through the event horizon in either direction. Since r is undefined in the
grey area it follows that the grey region does not correspond to any region in
Schwarzschild coordinates or by extension it does not correspond to anything in
our universe and temptations to attach physical meaning to the grey region
should probably be avoided. Although Kruskal-Szekeres coordinates can be
obtained by a simple quarter rotation of left hand quadrant in a clockwise
direction, there is no a-priori reason why the rotation should be clockwise, so
it is difficult to see why Kruskal-Szekeres coordinates should be considered as
special or unique. The grey region is best thought of as a visual reminder that
there is no causal connection in the normal sense (by particles moving at the
speed or light or less) in either direction across the event horizon. The
significance or physical implications of that interpretation is something we
have to face and is an interesting subject for further research.
A
derivation of the new metric from the new coordinates is a bit involved and is
presented for any mathematical masochists out there in the form of a maple math
worksheet here
and an alternative derivation here. If
you would like to play with the original live maple worksheets, please feel
free to contact me for the files.
©
2008 KevPegrume