
Is there a formula to determine how big of a
"dip" the spacetime fabric will curve due to
gravity given the mass/density of an object such
as the earth or the sun? If so, what is it? Could
it be used to determine the "flexibility" of the
spacetime fabric and/or how fast it moves?
  Answer 1:
There is a formula, but the math behind it is
extremely complicated  far more than my
collegelevel mathematics education is capable of.
It needs graduate school level math, and my field
is paleontology, not physics, so I never studied
it. The "flexibility" that you speak of is the
universal gravitational constant,G, which is
present even in the Newtonian approximation of
gravitational force (i.e. F
=G*m1*m2/r^{2}). That particular
formula is still a very good approximation of
gravitational interaction, but there are some
complications to the solution in the case of very
strong gravitational fields, or over very short or
very long distances.
  Answer 2:
I don't have an easy answer for this one, since
I haven't done the mathematics for advanced
general relativity.Unfortunately, the equations
that are easy to write are difficult to
understand, and vice versa. One lesson is clear:
if you'd like to understand more Physics, take
more Math!
  Answer 3:
The interaction of classical objects and
gravity is the subject ofgeneral relativity and
there are several good text books on the subject.
(My favorite is Meissner/Thorne/Wheeler  but it
is likely dated now). A suitable introduction is
the book SpaceTime Physics which needs only a bit
of calculus to read. The formula relating the
curvature of space time to the mass of an object
is the Einstein Field Equation which does exactly
that. The problem is that the curvature is in a
fourdimensional space  so the equation can be
challenging to apply. However, there are lots of
simple solutions for situations like spinning
massive bodies (like the earth).
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