Tidal Forces in a Schwarzschild Spacetime

The Schwarzschild solution is one of the best-known exact solutions of the Einstein equations and was derived a few months after the theory was proposed. Consider, therefore, a spherical coordinate system $(t,r,\theta,\phi)$ in vacuum. Impose the constraints that the metric is spherically symmetric and static (i.e. none of the functions $g_{\mu \nu}$ depends on $t,\theta,\phi$) and that the spacetime is asymptotically flat (i.e. $g_{\mu \nu}=1$ for $r \rightarrow \infty$). Under these conditions, the solution to the Einstein equations has a line element

$$ds^{2}=-dt^{2} \left(1 - \frac{2M}{r} \right)+ dr^{2} \left(1 - \frac{2M}{r} \right)^{-1} + r^{2}(d\theta^2 + \sin^2\theta d\phi^2 ) .$$ (37)

The spacetime described by (37) is that of a Schwarzschild black hole, where $M$ is the ``black hole mass''. Note that despite your intuition and the familiar concept of ``mass'', the Schwarzschild metric is a solution of the Einstein equations in vacuum, i.e. of the Einstein equations

$$R_{\mu \nu}=0 .$$ (38)

Indeed (37) is the only spherically symmetric and asymptotically flat solution that the equations admit (this is the thesis of Birkhoff's theorem). Because of this, the spacetime exterior (i.e. for $r \geq R_*$, where $R_*$ is the stellar radius) to a relativistic spherical (i.e. non rotating) star will be given by the line element (37) (The interior spacetime, on the other hand, will be different from that of a Schwarzschild black hole and is in general dependent on the stellar structure and equation of state.).

Let us consider what happens to an extended body located outside the black hole event horizon, is defined as the position at which the metric function $g_{tt}=0$, i.e. at $r=2M$. We will also assume that all the particles in the body move along geodesics (i.e. the body has zero internal stresses and is infinitely deformable) and monitor how the separation between two nearby geodesics varies in time. In particular, using the same notation defined above, we build the spatial separation vector

$$\eta^{\mu} \equiv h^{\mu}_{ \nu} \xi^{\nu} ,$$ (39)

where ${\boldsymbol {\it h}}$ represents the spatial projection tensor orthogonal to ${\boldsymbol {\it g}}$, i.e.

$$h_{\mu \nu} \equiv g_{\mu \nu} + u_{\mu} u_{\nu} , \qquad \qquad {\rm and} \qquad \qquad {\boldsymbol h} \cdot {\boldsymbol g} = 0 .$$ (40)

Clearly, the spatial part of $\boldsymbol{\eta}$ coincides with the 3-vector ${\vec \eta}$ introduced in Section 1.2.

The solution of the geodesic deviation equation (32) in the spacetime (37) leads to the following expressions for the spatial components of $\boldsymbol{\eta}$

$$\frac{D^2\eta^r}{D\tau^2}=\frac{2M}{r^3}\eta^r ,$$ (41)

$$\frac{D^2\eta^{\theta}}{D\tau^2}=-\frac{M}{r^3}\eta^{\theta} ,$$ (42)

$$\frac{D^2\eta^{\phi}}{D\tau^2}=-\frac{M}{r^3}\eta^{\phi} ,$$ (43)

where the positive sign indicates a stretching and a negative one a compression in that direction. A schematic view of the geodesic deviation as well as of the deformation produced on a fluid body in the presence of a strong gravitational field, produced for instance by a compact star, are shown in Fig. 4

Two comments are worth making about expressions (41). Firstly the tidal deformation is finite at $r=2M$ and thus, depending on the black hole mass, the body may well preserve its shape when crossing the event horizon (This ceases to be true for $r=0$ when the tidal stresses are divergent.). Secondly, the tidal fields at the horizon are larger for smaller black holes. This is simply because

Figure: Schematic view of the geodesic deviation as well as of the deformation produced on a fluid body in the presence of a strong gravitational field. In the case considered here the source of the gravitational field is represented by a massive body (i.e. $T_{\mu \nu }>0$) but a qualitative similar scheme would be true also in the case of a black hole.

$$\left \vert\frac{D^2\eta^a}{D\tau^2} \right\vert \sim \left\vert ... ... \left\vert \frac{1}{M^2}\eta^a \right\vert \qquad {\rm at} \qquad r \sim M .$$ (44)

For this reason, the tidal forces experienced in the vicinity of a supermassive black hole of, say, $10^8 M_{\odot}$ will be 16 orders of magnitude smaller than the corresponding ones near a stellar-mass black hole.