Take the 2-minute tour ×  If we assume space is flat the volume
of Earth is:  $$ V = \frac{4 \pi R^3}{3} = \frac{4 \pi (6378.1
km)^3}{3} = 1.086 \times 10^{21} m^3 $$  The Einstein field
equations, however, predict that the presence of matter warps
space-time. As I understand it, the hyperbolic geometry means
that the actual volume will be smaller than that predicted by
the radius with the above equation, because the space in the
center of the Earth (and around the Earth) is "dimpled". How
much will these two volumes differ by?  There would be different
ways to answer this question, depending on how you define
"radius". There is one sense, where you could drop a measuring
tape from the surface to the center of the Earth, and I think
this would be the "proper" length. This is my preferred
definition, but I think it would be the most complicated to
answer. Otherwise, you could use the universal coordinates from
someone far from Earth, which would give a smaller measure of
radius. Another, more meaningful, way m