34 Special Relativity
positive curvature like a two-sphere the angles would add up to more than 180∘.Cor-
respondingly, the angles on a saddle surface with negative curvature would add up to
less than 180∘. This is illustrated in Figures 2.3 and 2.4. The precision in Gauss’s time
was, however, not good enough to exhibit any disagreement with the Euclidean value.
Comoving Coordinates. If the two-sphere with surface [Equation (2.18)] and con-
stant radius푅were a balloon expanding with time we replace푅by the expansion
scale푎(푡), defined in Equation (1.16). Points on the surface of the balloon would find
their mutual distances scaled by푎(푡)relative to a time푡 0 when the radius was푅 0 =1.
Figure 2.3The angles in a triangle on a surface with positive curvature add up to more than
180 ∘.
Figure 2.4 The angles in a triangle on a surface with negative curvature add up to less
than 180∘.