Fun with the Schwarzschild metric

Approximations all the way down

The Schwarzschild metric is an approximation. That bold statement is not merely an opinion. It’s a mathematically provable fact. In the equation below, 𝜏 (tau) is proper (subjective) time. t is objective time (as seen from an observer in free space), and x is objective distance (as measured by an observer in free space).

\(dS^{2}=c^{2}d\tau ^{2} = (1-\frac{r_{s}}{r})c^{2}dt^{2} - (1-\frac{r_{s}}{r})^{-1}dx^{2} - r^{2}d\Omega ^{2}\)

With no offset motion (that’s the final term on the right), when the Schwarzschild radius is zero (empty space) or we set r to infinitely far away, we get the following.

\(c^{2}d\tau ^{2} = c^{2}dt^{2} - dx^{2}\)

When we set dt to one second and dx to one meter, we get the following.

\(89,875,517,873,681,764_{m^{2}/s^{2}} d\tau ^{2} = 89,875,517,873,681,764_{m^{2}} - 1_{m^{2}}\)

I think you see where this is going now.

\(89,875,517,873,681,764_{m^{2}/s^{2}} d\tau ^{2} = 89,875,517,873,681,763_{m^{2}}\)

Now divide, take the square roots, and you get:

\(d\tau = 0.999,999,999,999,999,994,44_{s}\)

So an observer in free space measures time going more slowly than it passes to an observer in free space. Which is clearly paradoxical, if not nonsensical.

If you go the opposite way and set r to the Schwarzschild radius (the event horizon), you also get nonsensical results. The time element goes to zero and the space element goes to infinity with a divide by zero error.

The problem here is that the Schwarzschild metric is an approximation generated from calculus. I have already shown that gravity is a force generated by the gradient of potential energy across a particle, not as the tangent at its center. In the real world, the precision of calculus occasionally leads to error.

If you want a more intuitive picture, the time (dt) element of the metric equates to the Lorentz alpha factor, which is the cosine of an angle. The space (dx) element is the Lorentz gamma factor, which is the secant (1/cos) of the same angle. In my potential energy field model, this angle is the tangent to the potential energy curve. The Schwarzschild metric uses the integral of the 1/r^2 energy curve, which is a simple 1/r hyperbolic curve, which gives the accumulated (from infinity) velocity.

By PAR - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=66751798

I must give Einstein and Schwarzschild a great deal of credit, though. Their approximation is really, really close, and they didn’t have a century of hindsight to work with. We only see so far because we stand upon the shoulders of giants.