Schwarzschild Metric - Flamm's Paraboloid

Flamm's Paraboloid

The spatial curvature of the Schwarzschild solution for can be visualized as the graphic shows. Consider a constant time equatorial slice through the Schwarzschild solution (θ = π/2, t = constant) and let the position of a particle moving in this plane be described with the remaining Schwarzschild coordinates (r, φ). Imagine now that there is an additional Euclidean dimension w, which has no physical reality (it is not part of spacetime). Then replace the (r, φ) plane with a surface dimpled in the w direction according to the equation (Flamm's paraboloid)

$w = 2 \sqrt{r_{s} \left( r - r_{s} \right)}.$

This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of w above,

Thus, Flamm's paraboloid is useful for visualizing the spatial curvature of the Schwarzschild metric. It should not, however, be confused with a gravity well. No ordinary (massive or massless) particle can have a worldline lying on the paraboloid, since all distances on it are spacelike (this is a cross-section at one moment of time, so all particles moving across it must have infinite velocity). Even a tachyon would not move along the path that one might naively expect from a "rubber sheet" analogy: in particular, if the dimple is drawn pointing upward rather than downward, the tachyon's path still curves toward the central mass, not away. See the gravity well article for more information.

Flamm's paraboloid may be derived as follows. The Euclidean metric in the cylindrical coordinates (r, φ, w) is written

$\mathrm{d}s^2 = \mathrm{d}w^2 + \mathrm{d}r^2 + r^2 \mathrm{d}\phi^2.\,$

Letting the surface be described by the function, the Euclidean metric can be written as

$\mathrm{d}s^2 = \left \mathrm{d}r^2 + r^2\mathrm{d}\phi^2,$

Comparing this with the Schwarzschild metric in the equatorial plane (θ = π/2) at a fixed time (t = constant, dt = 0)

$\mathrm{d}s^2 = \left(1-\frac{r_{s}}{r} \right)^{-1} \mathrm{d}r^2 + r^2\mathrm{d}\phi^2,$

yields an integral expression for w(r):

$w(r) = \int \frac{\mathrm{d}r}{\sqrt{\frac{r}{r_{s}}-1}} = 2 r_{s} \sqrt{\frac{r}{r_{s}}- 1} + \mbox{constant}$

whose solution is Flamm's paraboloid.

Read more about this topic: Schwarzschild Metric