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Tetrads in General Relativity

VIII. Schwarzschild solution

Schwarzschild Radius

   Some important features of the Schwarzschild metric (8.1) are:

  1. Far out ($r \to \infty $ ), the metric component ${g_{tt}}$ in front of $d{t^2}$ becomes one, and the component ${g_{rr}}$ in front of $d{r^2}$ minus one. Thus, in this limit the metric becomes Minkowskian in polar coordinates.
    \[  d{s^2} \asymp d{t^2} - d{r^2} - {r^2}(d{\theta ^2} + {\sin ^2}\theta d{\phi ^2})  \]
    It implies that asymptotically $t$ and $r$ have their special-relativistic significance.
  2. Moving inward along decreasing $r$, one eventually crosses the sphere defined by $r = 2M$. Here the geometry becomes very different: ${g_{tt}}=0$ and ${g_{rr}=\infty}$; moreover, both metric components change sign there. However, nothing really singular happens at $r=2M$. As will be discussed in Section IX, the singularity turns out to be an artifact of the Schwarzschild coordinates that can be removed by a change of coordinates.
  3. Going in all the way, one encounters the only real physical singularity at $r=0$. Anything falling in would experience infinitely strong gravity and be torn apart there.

   The length ${r_{\text{S}}} := 2M$, in physical units $2GM/{c^2}$, is named the Schwarzschild radius, and the sphere with radius $r = {r_{\text{S}}}$ the Schwarzschild horizon. For almost all astrophysical objects, the Schwarzschild radius is extremely small in comparison to their physical dimensions. For example, the Schwarzschild radius of the Earth is roughly 9 mm, while the Sun, which is $3 \times {10^5}$ times as massive, has a Schwarzschild radius of approximately 3 km. The ratio becomes large only in close proximity to black holes and other ultra-dense objects such as neutron stars. In fact, a black hole is defined as a body whose mass is entirely contained within its Schwarzschild radius.

   It is worth noting that the Schwarzschild radius is the only length scale that appears in the metric. If the time and radial coordinate are rescaled to dimensionless parameters $r/{r_{\text{S}}} \to r$ and $t/{r_{\text{S}}} \to t$, the metric (8.1) appears as if $\;{r_{\text{S}}} = 1$. Thus, basically all Schwarzschild metrics are the same and for most purposes in studying the geometry of a black hole, one can simple set the Schwarzschild radius equal to one.