  I present a way to visualize the concept of curved spacetime. The result is a
curved surface with local coordinate systems (Minkowski Systems) living on it,
giving the local directions of space and time. Relative to these systems,
special relativity holds. The method can be used to visualize gravitational
time dilation, the horizon of black holes, and cosmological models. The idea
underlying the illustrations is first to specify a field of timelike
four-velocities. Then, at every point, one performs a coordinate transformation
to a local Minkowski system comoving with the given four-velocity. In the local
system, the sign of the spatial part of the metric is flipped to create a new
metric of Euclidean signature. The new positive definite metric, called the
absolute metric, can be covariantly related to the original Lorentzian metric.
For the special case of a 2-dimensional original metric, the absolute metric
may be embedded in 3-dimensional Euclidean space as a curved surface.
