Calculations in the Schwarzschild Metric


Motivation

At some point in your life you may (or may not – in that case you can stop reading here) want to calculate the propagation time of a photon along a path $\gamma$ in some potential field, which is given by

where $ds$ is the part corresponding to the time component of the potential field

where

  • $\tau$ is the proper time, which is constant for particles moving at the speed of light,
  • $ds=f_t(\mathbf x)dt$ for $t$ being the time coordinate,
  • $dl=f_r(\mathbf x)dr$ for $r$ being the radial spatial coordinate and
  • $dp=f_\phi(\mathbf x)d\phi$ for $\phi$ being the longitude.

In a general spherical coordinate system there is also the colatitude $\theta$, but we will ignore this one henceforth since (in the case of a radially symmetric potential field (i.e. $f_i(\mathbf x)=f_i(r), i\in\{t,r,\phi\}$ for $\mathbf x=(r,\phi)$), which will be assumed throughout this article from now on) the coordinate system can be tilted in such a way that $\theta$ is constant for geodesics without external influences like mirrors or more general any other particles interacting with the photon in question.

Preliminary results

When wanting to calculate something like $\int_\gamma ds$ several questions arise:

  1. What is $\gamma$?
  2. How does this $ds$ interact with the parameterization of $\gamma$?
  3. Is the integral path independent?

The first question will not be dealt with in detail here, but there are some things worth pointing out (at least to non-physicists): First of all, given some photon at position $\mathbf x(0)$ with initial velocity $\mathbf{\dot x}(0)$ a parameterization of its trajectory is given implicitly by the “geodesic equations” (so-called “null-geodesics” in the case of a photon), which are second order, non-linear differential equations for $\tau,t,r,\phi$ with initial conditions given by $\mathbf x(0)$ and $\mathbf{\dot x}(0)$. In particular, if you want to calculate something like $\int_A^B ds$ (which is only suggestive as it is not well-defined unless we know that this integral is path-independent!) you cannot just take $\gamma(t)$ to be $tA+(1-t)B$ in a potential field induced by a non-Euclidean metric (e.g. the Schwarzschild metric).

Concerning the second question: In general it will be hard to parameterize $t$ with a path. Since we have initial conditions for the spatial coordinates, the geodesic equations will yield solutions for those spatial coordinates, so we would like to avoid any $dt$s and rather work with $dr$ and $d\phi$. This can be achieved by noting that a particle travelling at the speed of light does not experience proper time, implying that $d\tau\equiv 0$. Thus we can (and usually do) substitute $ds$ by $\sqrt{dl^2+dp^2}$.

When it comes to path dependence we have to get a little bit more specific about the functions $f_i(\mathbf x)=f_i(\mathbf x_r), i\in\{t,r,\phi\}$. Note that, since $ds=f_tdt, \dots$, $ds$ is not an arbitrary measure, but only a (potentially not translation-invariant) multiple of the Lebesgue measure. As a result we can use the usual theory to determine if a line integral is path-independent, which amounts to checking if the 1-form $ds=\sqrt{dl^2+dp^2}$ is exact (i.e. there exists an $F$, s.t. $dF=\sum_i \frac{\partial F}{\partial x_i}dx_i = ds$). In general (i.e. for non trivial $dl,dp$ and non-trivial geodesics $\gamma$) no such $F$ exists (which can also be seen by showing that $ds$ is not closed). This brings us to the actual topic of this article:

Calculations in the Schwarzschild Metric

To do that we first characterize the Schwarzschild metric by writing out the functions $f_i(\mathbf x), i\in\{t,r,\phi\}$:

where the latter holds only under the assumption $\theta\equiv\pi/2$ and $r_s$ is a constant, called the “Schwarzschild radius”, that scales with the weight of the object inducing this metric and the gravitational constant $G$.

We already noticed that for photons we have but what does $\int_\gamma\sqrt{dl^2+dp^2}$ mean? First of all we will rewrite this integral, which is an integral over a 1-Form $da$ (mathematically speaking), as an integral with a “real” integrand by using the parameterization of the path $\gamma$ and noting that 1-Forms are functions mapping each point $x$ of the “original” space $X$ ($\mathbb R^4$ in our case, but we will only use two components and thus treat it as $\mathbb R^2$) to some dual element $da(\mathbf x)$ of the tangent space $T_x^* $, such that

for every $da$ of the form $da(x)=f_a(x)dx$.

Thus, in the Schwarzschild metric

Intuition

For $d\tau\equiv 0$ we regain a positive definite metric by writing $ds^2=dl^2+dp^2$, which looks pretty similar to the Euclidean case $dl^2=dx^2+dy^2$, where $dl$ is the “infinitesimally small length element”. The difference is that the Lebesgue measure is translation-invariant, i.e. $dx(\mathbf x)\equiv dx$ and $dy(\mathbf x)\equiv dy$ while, in general $dl(\mathbf x)\neq dl(\mathbf x’)$ and $dp(\mathbf x)\neq dp(\mathbf x’)$.

So it is reasonable to think of $f_r,f_\phi$ as non translation-invariant weighting factors in the Euclidean metric.