d/Gravity Probe B was in a
polar orbit around the earth. As
in the right panel of figure c,
the orientation of the gyroscope
changes when it is carried around
a curve and back to its starting
point. Because the effect was
small, it was necessary to let it
accumulate over the course of
5000 orbits in order to make it
detectable.e/A triangle in a space with
negative curvature has angles
that add to less than 180◦.c/Left: A 90-90-90 triangle. Its angles add up to more than 180◦.
Middle: The triangle “pops” off the page visually. We intuitively want
to visualize it as lying on a curved surface such as the earth’s. Right:
A gyroscope carried smoothly around its perimeter ends up having
changed its orientation when it gets back to its starting point.to the case where spacetime is flat.Curvature doesn’t require higher dimensions
Although we often visualize curvature by imagining embedding
a two-dimensional surface in a three-dimensional space, that’s just
an aid in visualization. There is no evidence for any additional
dimensions, nor is it necessary to hypothesize them in order to let
spacetime be curved as described in general relativity.f/Only measurements from within the plane define whether the
plane is curved. It could look curved when drawn embedded in three
dimensions, but nevertheless still be intrinsically flat.Put yourself in the shoes of a two-dimensional being living in a
two-dimensional space. Euclid’s postulates all refer to constructions
that can be performed using a compass and an unmarked straight-
edge. If this being can physically verify them all as descriptions of
the space she inhabits, then she knows that her space is Euclidean,
and that propositions such as the Pythagorean theorem are physi-
cally valid in her universe. But the diagram in f/1 illustrating the
proof of the Pythagorean theorem in Euclid’sElements (proposi-
tion I.47) is equally valid if the page is rolled onto a cylinder, 2, or
formed into a wavy corrugated shape, 3. These types of curvature,
which can be achieved without tearing or crumpling the surface,
are not real to her. They are simply side-effects of visualizing her
Section 7.4 ?General relativity 445