In a message dated 2/21/01 7:16:07 PM Eastern Standard Time,
David_Bowman@GEORGETOWNCOLLEGE.EDU writes:
<< For instance, suppose we take the properly measured area A of the
sphere and calculate a predicted radius R of this sphere according to
R = sqrt(A/(4*[pi])) and then actually descend into the sphere with a
meter stick and measure the actual radius R_actual. It will be found
that R_actual > R. The actual direct distance across its interior
(i.e. 2*R_actual) is greater than we would have predicted from flat
space geometry. Similarly, suppose we predict the interior volume V of
the sphere according to the flat space formula:
V = (4*[pi]/3)*R^3 = (4*[pi]/3)*(A/(4*[pi]))^(3/2) .
>>
This is a wonderful prediction of GR and an excellent point to
highlight which really brings home the paradigm shift that GR requires when
we think of space and gravitational fields. In one of Feynman's lectures he
talks about this fact and uses some very simple mathematics to illustrate
this point.
Feynman gives us the following equation:
Rex=Rmea-(Amea/4*pi)^.5 where Rex is what he calls the excess
radius and Rmea and Amea are the measured radius and surface area of any
spherical object in our Universe. He goes to show that:
Rex=M*(G/(3*(C^2)) where G/(3*(C^2)) =2.5E-29 cm/gm
This provides a simple illustration of what happens to the surface
area of any volume which contains sufficient mass to alter the local property
of space.