From Schwarzschild to Droste

Love's not Time's fool, though rosy lips and cheeks

Within his bending sickle's compass come:

Love alters not with his brief hours and weeks,

But bears it out even to the edge of doom.

If this be error and upon me proved,

I never writ, nor no man ever loved.

                              Shakespeare

Soon after Einstein published the field equations of general relativity in November of 1915 the exact spherically symmetrical vacuum solution of those equations was found by Karl Schwarzschild and communicated to Einstein by letter on December 22, 1915. Einstein presented it to the Prussian academy the following week and it was published in January 1916. The derivation was based on the preliminary form of the field equations contained in Einstein’s paper of November 18, which did not yet contain the trace term, but this is of no consequence for Schwarzschild’s solution, because the trace term doesn’t affect the vacuum solution. As we’ve discussed elsewhere, Schwarzschild used a system of spatial coordinates r,θ,ϕ where r is a radial coordinate and θ,ϕ are the usual angular coordinates, and he used a time coordinate t in terms of which the metric components are independent of t. His approach was to make use of the truncated field equations, which are valid only if the determinant g of the metric tensor is −1. This led him to define an auxiliary system of coordinates, in terms of which the determinant g took a simple form. The auxiliary radial coordinate he chose involved the cube of r, although this was a fairly arbitrary choice, and not the most efficient way of deriving the solution. He arrived at the line element

where

The parameters α and r₀ are constants of integration. To match Newtonian gravity in the weak slow limit we must have α = 2m (in geometrical units), where m is identified as the mass of the gravitating body. The value of the constant r₀ is not constrained by any such considerations. The peculiar function R(r) involving the cube of r was just an artifact of Schwarzschild’s arbitrary choice of auxiliary coordinates to simplify the determinant of the metric. Notice that the circumference of a spatial locus of constant r is actually 2πR(r), so the R parameter has absolute physical significance and corresponds closely to the radial coordinate in ordinary polar coordinates in a flat manifold. In contrast, the r coordinate is arbitrary, depending on what value we choose for the free constant r₀.

Schwarzschild decided to set r₀ equal to α. His motivation was that when R = α the coefficient of (dR)² in the line element is infinite and the coefficient of (dt)² vanishes. If we choose r₀ = α, so that r = 0 when R = α, the metric coefficients are well-behaved for all r > 0, which is just another way of saying the coefficients are well-behaved for all R > α. Of course, regardless of what value we choose for r₀, we are free to define quasi-Cartesian spatial coordinates x,y,z using the resulting radial coordinate r by the relations

Thus we have r² = x² + y² + z², and when r = 0 we have x = y = z = 0. But we can equally well define quasi-Cartesian coordinates based on R by the relations

Hence we have R² = X² + Y² + Z², and when R = 0 we have X = Y = Z = 0.

Incidentally, the line element for the coordinates originally used by Schwarzschild is sometimes written as

which is trivially equivalent to the previous expression in view of the identity

It’s also interesting to note that, in terms of the auxiliary coordinates that Schwarzschild actually used to show that the metric satisfies the field equations, one of the coefficients of the line element was r²/sin(θ)², which is infinite at θ = 0 and θ = π for all radial locations. This singularity is removable by the coordinate transformation, but this highlights the fact that singularities in the metric coefficients do not necessarily imply that the metrical relations of the manifold are singular, and also that coordinate transformations do not alter those intrinsic metrical relations.

In May of 1916 (the same month in which Schwarzschild died of a disease contracted on the Eastern front), Johannes Droste (a student of Lorentz) submitted a paper in which he independently derived the same solution, as he acknowledged in a footnote when the paper was finally published in 1917 after he had become aware of Schwarzschild’s paper. However, Droste did not follow Schwarzschild in his choice of radial coordinates, which he recognized as arbitrary and conceptually misleading. For example, what Schwarzschild regarded as the central “point” of the field at R = 2m and r = 0 is actually a spherical surface of area 4πR². Droste saw that the parameter Schwarzschild called R was both more convenient and more physically meaningful than the arbitrary parameter that Schwarzschild had called r. Of course, the choice of coordinates can have no effect on any of the metrical relations, since it is simply a re-labeling of events.

Later in 1917 both Weyl and Hilbert published their own derivations of the same metric, acknowledging that their results were equivalent to Schwarzschild’s. Like Droste, both Hilbert and Weyl chose what Schwarzschild called R as their radial coordinate. Hilbert comment on the fact that the metric coefficients in terms of this coordinate are not well-behaved at R = 0 and R = 2m. He wrote

For α ≠ 0 it turns out that R = 0 and (for positive α) R = α are points at which the line element is not regular. By that I mean that a line element or a gravitational field g_μν is regular at a point if it is possible to introduce by a reversible one-to-one transformation a coordinate system such that the corresponding functions g′_μν are continuous and arbitrarily differentiable at the point and in the neighborhood of the point, and the determinant g′ is different from 0.

Thus he says the gravitational field (at a given point) represented by the metric components g_μν for a given system of coordinates is regular if these g_μν are continuous and differentiable or if there is a reversible one-to-one transformation to a different system of coordinates such that the corresponding g_μν are continuous and differentiable. Conversely, if no such coordinate transformation renders the g_μν continuous and differentiable, then Hilbert says the field is not regular (at that point). With this definition, the Schwarzschild metric is indeed not regular at R = α, but this is not a good definition, because of the unwarranted restriction to reversible one-to-one transformations. Consider, for example, the result of applying an inadmissible (according to Hilbert’s prescription) transformation to ordinary Cartesian coordinates of flat space, such that the metric components in terms of the transformed coordinates are discontinuous and/or not differentiable at some points. Had we begun with these ill-behaved coordinates, the only way to remove the bad behavior would be to apply the inverse transformation, which is also inadmissible by Hilbert’s prescription. No reversible one-to-one transformation would suffice to make the metric coefficients continuous and differentiable, so, from this standpoint, Hilbert’s definition would tell us the metrical field of the flat plane is not regular. On the other hand, if we started with the ordinary Cartesian coordinates, Hilbert’s definition would tell us that the metrical field of the flat plane obviously is regular. Clearly this definition is not self-consistent, and is not a valid criterion for discerning the underlying character of a metrical manifold in a given region, based simply on one particular metric in terms of a particular system of coordinates – coordinates that may or may not be well-behaved.

One indication of the character of Schwarzschild’s radial coordinate r can be seen by plotting the relationship between r and R, as shown below.

As the area-based radial coordinate R passes through a, the rate of change of the coordinate that Schwarzschild called r goes to infinity. However, one shouldn’t place too much significance on this particular radial transformation, because, as noted above, the r coordinate is quite arbitrary. As Hilbert remarked, it isn’t useful to transform to the origin the position R = α, and even if we wanted to do this, Schwarzschild’s transformation is not the simplest way to accomplish it. For example, instead of defining the radial coordinate r by the relation r³ = R³ – α³ as Schwarzschild did, we could just as well define the radial coordinate as r = R – α, thereby placing the event horizon at r = 0. But a coordinate transformation does not change the metrical relations of the manifold; it simply amounts to shifting the labels of events. These do not represent distinct physical solutions, they are simply different ways of expressing the same solution. It is also worth noting that, as we’ve discussed elsewhere, the spherically symmetrical solution in the form of Kruskal’s metric can be derived directly from the field equations, without ever invoking irregular coordinates at the Schwarzschild radius.

Both Droste and Schwarzschild thought we should disregard the solution inside the “Schwarzschild radius”.  Schwarzschild emphasized this by choosing a radial coordinate that is zero at the “Schwarzschild radius”, but this shift of coordinates is of no physical significance. To clarify this with a simple illustration, consider a flat Euclidean plane with polar coordinates (R,θ) and the metric line element

Now suppose that (for whatever reason) we want to deny the existence of the region 0 < R < α for some constant α. We might try to argue that the “true” radial coordinate is not R, but rather r = R – α, and hence the line element is

We might then claim that this “explains” why the region we dislike doesn’t exist, because it corresponds to r < 0, whereas (so we might argue) r cannot physically be less than zero… but of course this reasoning is absurd. The intrinsic metrical relations on the plane are totally unaffected by our change of radial coordinates. The length of every path is unchanged. The region 0 < R < α corresponds to the region –α < r < 0.  If we want to declare that this region doesn’t exist, we can just as well express this by excluding R < α or by excluding r < 0. It’s the same thing. The left hand figure below shows the plane with its actual metrical relations, and the right hand figure shows how the points would be depicted if we pretended that the locus at r = 0 was actually a single point.

The right hand figure shows that the radial coordinates in terms of r provide a double covering of the xy coordinates, which we’ve depicted by showing the two sheets as cones for clarity. The upper cone represents the region with r > 0, and the lower cone represents the region with r < 0. Every set of coordinates x,y such that x² + y² < α² actually maps to two different points on the plane, one with positive r and one with negative r. In contrast, there is a one-to-one mapping between the points of the plane and the sets of X,Y coordinates as shown in the left hand figure. Notice that all the metrical relations are identical, despite the fact that the right hand figure does not accurately depict those relations. For example, to evaluate the absolute length of the locus R = α as θ ranges from 0 to 2π, we simply integrate the original metric to give the result 2πR. We can just as well evaluate the absolute length of this path in terms of the r,θ coordinates by integrating the locus r = 0 as θ ranges from 0 to 2π. Again we get the result 2πR. The metrical manifolds described by our two metrics in terms of our two coordinate systems are equivalent, because they differ only by a choice of coordinate systems.

Likewise the spherically symmetrical solutions to the vacuum field equations described by Schwarzschild, Droste, Weyl, and Hilbert are all perfectly equivalent, differing only in the choice of coordinate systems. Of course, by Birkhoff’s theorem, we know there is only one unique spherically symmetrical solution, but we don’t need Birkhoff’s theorem to tell us that two solutions related by a coordinate transformation are equivalent. If all we knew was that Schwarzschild and Droste had both found spherically symmetrical solutions of the field equations, but we didn’t know that we can go from one to the other by a simple coordinate transformation, then Birkhoff’s theorem would assure us that we can, i.e., that there is indeed a coordinate transformation that relates them, and hence they are the same solution. But in the case of Schwarzschild and Droste we already know the solutions are related by a coordinate transformation, because we have the explicit transformation relating the different radial coordinates. This is why, even though Birkhoff’s theorem was not known in 1916, it was immediately obvious that the solutions of Droste, Weyl, and Hilbert were all equivalent to the solution originally found by Schwarzschild.

This of course is a separate question from whether the metrical relations described by the Schwarzschild solution accurately describe phenomena in nature. For example, few people think it likely that the maximal analytic extension of the Schwarzschild solution (with both a white and black hole) is ever realized in nature. If someone wants to claim that various aspects of the Schwarzschild solution are not physical, they are obviously free to do so, and to provide any rationale they think may support their claim, but it can’t be justified by trivial semantics, i.e., by simply changing the labels of events.

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