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 uμ. 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 two-dimensional original metric, the absolute metric may be embedded in three-dimensional Euclidean space as a curved surface.

Topics
Four-velocity, General relativity, Special relativity, Cosmological model, Black holes, Coordinate transformations
1.
R. D’Inverno, Introducing Einstein’s Relativity (Oxford U.P., Oxford, 1998), pp. 99–101.
2.
Equation (1) defines a so-called distance function, or a metric. It can also be used considering events where dx>cdt. Then dτ2 is negative which simply means that it is related to spatial distance rather than temporal distance. A distance function like Eq. (1) corresponds to a flat spacetime, but see Eq. (10) for an example of a distance function corresponding to a curved spacetime.
3.
Steven Weinberg, Gravitation and Cosmology (Wiley, New York, 1972), pp. 337–338.
4.
Assuming that there is no black hole inside the crust.
5.
As was pointed out to me by Ingemar Bengtsson, this relation is also used by Hawking and Ellis (Ref. 18), although for completely different purposes than those of this article.
6.
If we had instead considered a metric of the form −gμν+αuμuν, where α is some general number, the inverse would have been −gμν+α/(α−1)uμuν. It is only in the case α=2 that we can simply raise the indices of the absolute metric with the original metric to get the inverse of the absolute metric.
7.
The gamma factor is defined as γ=(1−v2)−1/2 where v is the relative velocity. With this definition, it follows from Eq. (1) that dτ=dt/γ.
8.
A geometry has a Killing symmetry if there exists a vector field (called a Killing field ξμ) such that when we shift our coordinates xμ→xμ+εξμ–the metric has the same form. As an example we can consider a geometry that can be embedded as a surface of revolution. Then there exists a Killing field directed around the surface (in the azimuthal direction) with a length proportional to the embedding radius. Also, if there are coordinates where the metric is independent of one coordinate, then there is a Killing symmetry with respect to that coordinate.
9.
t′=t+φ(x) where dφ/dx=ḡtx/ḡtt gives ḡμν′=Diag(ḡtt,ḡxx−(ḡtx)2/ḡtt).
10.
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11.
Suggested to me by Sebastiano Sonego.
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14.
This may well violate the energy conditions.
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20.
The Frobenius condition in explicit form reads: uμ(∇νuρ−∇ρuν)+uρ(∇μuν−∇νuμ)+uν(∇ρuμ−∇μuρ)=0.
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© 2005 American Association of Physics Teachers.
2005
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