8.7 Strange Meeting

 

It seemed that out of battle I escaped

Down some profound dull tunnel...

Willfred Owen (1893-1918)

 

In the summer of 1913 Einstein accepted an offer of a professorship at the University of Berlin and membership in the Prussian Academy of Sciences. He left Zurich in the Spring of 1914, and his inagural address before the Prussian Academy took place on July 2, 1914. A month later, Germany was at war with Belgium, Russia, France, and Britian. Surprisingly, the world war did not prevent Einstein from continuing his intensive efforts to generalize the theory of relativity so as to make it consistent with gravitation - but his marriage almost did. By April of 1915 he was separated from his wife Mileva and their two young sons, who had once again taken up residence in Zurich. The marriage was not a happy one, and he later wrote to his friend Besso that if he had not kept her at a distance, he would have been worn out, physically and emotionally. Besso and Fritz Haber (Einstein's close friend and colleague) both made efforts to reconcile Albert and Mileva, but without success.

 

It was also during this period that Haber was working for the German government to develop poison gas for use in the war. On April 22, 1915 Haber directed the release of chlorine gas on the Western Front at Ypres in France. On May 23rd Italy declared war on Austria-Hungary, and subsequently against Germany itself. Meanwhile an Allied army was engaged in a disastrous campaign to take the Galipoli Peninsula from Germany's ally, the Turks. Germany shifted the weight of its armies to the Eastern Front during this period, hoping to knock Russia out of the war while fighting a holding action against the French and British in the West. In a series of huge battles from May to September the Austro-German armies drove the Russians back 300 miles, taking Poland and Lithuania and eliminating the threat to East Prussia. Despite these defeats, the Russians managed to re-form their lines and stay in the war (at least for another two years). The astronomer Karl Schwarzschild was stationed with the German Army in the East, but still kept close watch on Einstein's progress, which was chronicled like a serialized Dickens novel in almost weekly publications of the Berlin Academy.

 

Toward the end of 1915, having failed to drive Russia out of the war, the main German armies were shifted back to the Western Front. Falkenhayn (the chief of the German general staff) was now convinced that a traditional offensive breakthrough was not feasible, and that Germany's only hope of ultimately ending the war on favorable terms was to engage the French in a war of attrition. His plan was to launch a methodical and sustained assault on a position that the French would feel honor-bound to defend to the last man. The ancient fortress of Verdun ("they shall not pass") was selected, and the plan was set in motion early in 1916. Falkenhayn had calculated that only one German soldier would be killed in the operation for every three French soldiers, so they would "bleed the French white" and break up the Anglo-French alliance. However, the actual casualty ratio turned out to be four Germans for every five French. By the end of 1916 a million men had been killed at Verdun, with no decisive change in the strategic position of either side, and the offensive was called off.

 

At about the same time that Falkenhayn was formulating his plans for Verdun, on Nov 25, 1915, Einstein arrived at the final form of the field equations for general relativity. After a long and arduous series of steps (and mis-steps), he was able to announce that "finally the general theory of relativity is closed as a logical structure". Given the subtlety and complexity of the equations, one might have expected that rigorous closed-form solutions for non-trivial conditions would be difficult, if not impossible, to find. Indeed, Einstein's computations of the bending of light, the precession of Mercury's orbit, and the gravitational redshift were all based on approximate solutions in the weak field limit. However, just two months later, Schwarzschild had the exact solution for the static isotropic field of a mass point, which Einstein presented on his behalf to the Prussian Academy on January 16, 1916. Sadly, Schwarzschild lived only another four months. He became ill at the front and died on May 11 at the age of 42.

 

It's been said that Einstein was scandalized by Schwarzschild's solution, for two reasons. First, he still imagined that the general theory might be the realization of Mach's dream of a purely relational theory of motion, and Einstein realized that the fixed spherically symmetrical spacetime of a single mass point in an otherwise empty universe is highly non-Machian. That such a situation could correspond to a rigorous solution of his field equations came as something of a shock, and probably contributed to his eventual rejection of Mach's ideas and positivism in general. Second, the solution found by Schwarzschild - which was soon shown by Birkhoff to be the unique spherically symmetric solution to the field equations (barring a non-zero cosmological constant) - contained what looked like an unphysical singularity. Of course, since the source term was assumed to be an infinitesimal mass point, a singularity at r = 0 is perhaps not too surprising (noting that Newton's inverse square law is also singular at r = 0). However, the Schwarzschild solution was also (apparently) singular at r = 2m, where m is the mass of the gravitating object in geometric units.

 

Einstein and others argued that it wasn't physically realistic for a configuration of particles of total mass M to reside within their joint Schwarzschild radius r = 2m, and so this "singularity" cannot exist in reality. However, subsequent analyses have shown that (barring some presently unknown phenomenon) there is nothing to prevent a sufficiently massive object from collapsing to within its Schwarzschild radius, so it's worthwhile to examine the formal singularity at r = 2m to understand its physical significance. We find that the spacetime manifold at this boundary need not be considered as singular, because it can be shown that the singularity is removable, in the sense that all the invariant measures of the field smoothly approach fixed finite values as r approaches 2m from either direction. Thus we can analytically continue the solution through the singularity.

 

Now, admittedly, describing the Schwarzschild boundary as an "analytically removable singularity" is somewhat unconventional. It's customary to assert that the Schwarzschild solution is unequivocally non-singular at r = 2m, and that the intrinsic curvature and proper time of a free-falling object are finite and well-behaved at that radius. Indeed we derived these facts in Section 6.4. However, it's worth remembering that even with respect to the proper frame of an infalling test particle, we found that there remains a formal singularity at r = 2m. (See the discussion following equation 5 of Section 6.4.) The free-falling coordinate system does not remove the singularity, but it makes the singularity analytically removable. Similarly our derivation in Section 6.4 of the intrinsic curvature K of the Schwarzschild solution at r = 2m tacitly glossed over the intermediate result

 

 

Strictly speaking, the middle term on the right side is 0/0 (i.e., undefined) at r = 2m. Of course, we can divide the numerator and denominator by (r-2m), but this step is unambiguously valid only if (r-2m) is not equal to zero. If (r-2m) does equal zero, this cancelation is still possible, but it amounts to the analytic removal of a singularity. In addition, once we have removed this singularity, the resulting term is infinite, formally equal to the third term, which is also infinite, but with opposite sign. We then proceed to subtract the infinite third term from the infinite second term to arrive at the innocuous-looking finite result K = -2m/r3 at r = 2m. Granted, the form of the metric coefficients and their derivatives depends on the choice of coordinates, and in a sense we can attribute the troublesome behavior of the metric components at r = 2m to the unsuitability of the traditional Schwarzschild coordinates r,t at this location. From this we might be tempted to conclude that the Schwarzschild radius has no physical significance. This is true locally, but globally the Schwarzschild radius is physically significant, as the event horizon between two regions of the manifold. Hence it isn't surprising that, in terms of the r,t coordinates, we encounter singularities and infinities, because these coordinates are globally unique, viz., the Schwarzschild coordinate t is the essentially unique time coordinate for which the manifold is globally static.

 

Interestingly, the solution in Schwarzschild's 1916 paper was not presented in terms of what we today call Schwarzschild coordinates. Those were introduced a year later by Droste. Schwarzschild presented a line element that is formally identical to the one for which he is know, viz,

 

 

In this formula the coordinates t, q, and f have their usual meanings, and the parameter a is to be identified with 2m as usual. However, he did not regard "R" as the physically significant radial distance from the center of the field. He begins by declaring a set of rectangular space coordinates x,y,z, and then defines the radial parameter r such that

 

r2 = x2 + y2 + z2

 

Accordingly he relates these parameters to the angular coordinates q, and f by the usual polar definitions

 

 

He wishes to make use of the truncated field equations

 

 

which (as discussed in Section 5.8) requires that the determinant of the metric be constant. Remember that this was written in 1915 (formally conveyed by Einstein to the Prussian academy on 13 January 1916), and apparently Schwarzschild was operating under the influence of Einstein's conception of the condition g=-1 as a physical principle, rather than just a convenience enabling the use of the truncated field equations. In any case, this is the form that Schwarzschild set out to solve, and he realized that the metric components of the most general spherically symmetrical static polar line element

 

 

where f and h are arbitrary functions of r has the determinant g = -f(r) h(r) r4sin(q)2. (Schwarzschild actually included an arbitrary function of r on the angular terms of the line element, but that was superfluous.) To simplify the determinant condition he introduces the transformation

 

 

from which we get the differentials

 

 

Substituting these into the general line element gives the transformed line element

 

 

which has the determinant g = -f(r)h(r). Schwarzschild then requires this to equal -1, so his derivation essentially assumes a priori that h(r) = 1/f(r). Interestingly, with this assumption it's easy to see that there is really only one function f(r) that can yield Kepler's laws of motion, as discussed in Section 5.5. Hence it could be argued that the field equations were superfluous to the determination of the spherically symmetrical static spacetime metric. On the other hand, the point of the exercise was to verify that this one physically viable metric is actually a solution of the field equations, thereby supporting their general applicability.

 

In any case, noting that r = (3x1)1/3 and sin(q)2 = 1 - (x2)2, and with the stipulation that h(r) = 1/f(r), and that the metric go over to the Minkowski metric as r goes to infinity, Schwarzschild essentially showed that Einstein's field equations are satisfied by the above line element if f(r) = 1 - a/r where a is a constant of integration that "depends on the value of the mass at the origin". Naturally we take a = 2m for agreement with observation in the Newtonian limit. However, in the process of integrating the conditions on f(r) there appears another constant of integration, which Schwarzschild calls r. So the general solution is actually

 

 

We ordinarily take a = 2m and r = 0 to give the usual result f(r) = 1 - a/r, but Schwarzschild was concerned to impose an additional constraint on the solution (beyond spherical symmetry, staticality, asymptotic flatness, and the field equations), which he expressed as "continuity of the [metric coefficients], except at r = 0". The metric coefficient h(r) = 1/f(r) is obviously discontinuous when f(r) vanishes, which is to say when r3 + r = a3. With the usual choice r = 0 this implies that the metric is discontinuous when r = a = 2m, which of course it is. This is the infamous Schwarzschild radius, where the usual Schwarzschild time coordinate becomes singular, representing the event horizon of a black hole. In retrospect, Schwarzschild's requirement for "continuity of the metric coefficients" is obviously questionable, since a discontinuity or singularity of a coordinate system is not generally indicative of a singularity in the manifold - the classical example being the singularity of polar coordinates at the North pole. Probably Schwarzschild meant to impose continuity on the manifold itself, rather than on the coordinates, but as Einstein remarked, "it is not so easy to free one's self from the idea that coordinates must have a direct metric significance". It's also somewhat questionable to impose continuity and absence of singularities except at the origin, because if this is a matter of principle, why should there be an exception, and why at the "origin" of the spherically symmetrical coordinate system?

 

Nevertheless, following along with Schwarzschild's thought, he obviously needs to require that the equality r3 + r = a3 be satisfied only when r = 0, which implies r = a3. Consequently he argues that the expression (r3 + r)1/3 should not be reduced to r. Instead, he defines the parameter R = (r3 + r)1/3, in terms of which the metric has the familiar form (1). Of course, if we put r = 0 then R = r and equation (1) reduces to the usual form of the Schwarzschild/Droste solution. However, with r = a3 we appear to have a physically distinct result, free of any coordinate singularity except at r = 0, which corresponds to the location R = a. The question then arises as to whether this is actually a physically distinct solution from the usual one. From the definitions of the quasi-orthogonal coordinates x,y,z we see that x = y = z = 0 when r = 0, but of course the x,y,z coordinates also take on negative values at various points of the manifold, and nothing prevents us from extending the solution to negative values of the parameter r, at least not until we arrive at the condition R = 0, which corresponds to r = -a. At this location it can be shown that we have a genuine singularity in the manifold, because the curvature scalar becomes infinite.

 

In terms of these coordinates the entire surface of the Schwarzschild horizon has the same spatial coordinates x = y = z = 0, but nothing prevents us from passing through this point into negative values of r. It may seem that by passing into negative values of x,y,z we are simply increasing r again, but this overlooks the duality of solutions to

 

 

The distinction between the regions of positive and negative r is clearly shown in terms of polar coordinates, because the point in the equatorial plane with polar coordinates r,0 need not be identified with the point -r,p. Essentially polar coordinates cover two separate planes, one with positive r and the other with negative r, and the only smooth path between them is through the boundary point r = 0. According to Schwarzschild's original conception of the coordinates, this boundary point is the event horizon, whereas the physical singularity in the manifold occurs at the surface of a sphere whose radius is r = -2m. In other words, the singularity at the "center" of the Schwarzschild solution occurs just on the other side of the boundary point r = 0 of these polar coordinates. We can shift this boundary point arbitrarily by simply shifting the "zero point" of the complete r scale, which actually extends from - to +. However, none of this changes any of the proper intervals along any physical paths, because those are invariant under arbitrary (diffeomorphic) transformations. So Schwarzschild's version of the solution is not physically distinct from the usual interpretation introduced by Droste in 1917.

 

It's interesting that as late as 1936 (two decades after Schwarzschild's death) Einstein proposed to eliminate the coordinate singularity in the (by then) conventional interpretation of the Schwarzschild solution by defining a radial coordinate r in terms of the Droste coordinate r by the relation r2 = r - 2m. In terms of this coordinate the line element is

 

 

Einstein notes that as r ranges from - to + the corresponding values of r range from + down to 2m and them back to +, so he conceives of the complete solution as two identical sheets of physical space connected by the "bridge" at the boundary r = 0, where r = 2m and the determinant of the metric vanishes. This is called the Einstein-Rosen bridge. For values of r less than 2m he argues that "there are no corresponding real values of r". On this basis he asserts that the region r < 2m has been excluded from the solution. However, this is really just another re-expression of the original Schwarzschild solution, describing the "exterior" portions of the solution, but neglecting the interior portion, where r is imaginary. However, just as we can allow Schwarzschild's r to take on negative values, we can allow Einstein's r to take on imaginary values. The maximal analytic extension of the Schwarzschild solution necessarily includes the interior region, and it can't be eliminated simply by a change of variables. Ironically, the reason the manifold seems to be well-behaved across Einstein's "bridge" between the two exterior regions while jumping over the interior region is precisely that the r coordinate is locally ill-behaved at r = 0. Birkhoff proved that the Schwarzschild solution is the unique spherically symmetrical solution of the field equations, and it has been shown that the maximal analytic extension of this solution (called the Kruskal extension) consists of two exterior regions connected by the internal region, and contains a genuine manifold singularity.

 

On the other hand, just because the maximally extended Schwarzschild solution satisfies the field equations, it doesn't necessarily follow that such a thing exists. In fact, there is no known physical process that would produce this configuration, since it requires two asymptotically flat regions of spacetime that happen to become connected at a singularity, and there is no reason to believe that such a thing would ever happen. In contrast, it's fairly plausible that some part of the complete Schwarzschild solution could be produced, such as by the collapse of a sufficiently massive star. The implausibility of the maximally extended solutions doesn't preclude the existence of black holes - although it does remind us to be cautious about assuming the actual existence of things just because they are solutions of the field equations.

 

Despite the implausibility of an Einstein-Rosen bridge connecting two distinct sheets of spacetime, this idea has recently gained widespread attention, the term "bridge" having been replaced with "wormhole". It's been speculated that under certain conditions it might be possible to actually traverse a wormhole, passing from one region of spacetime to another. As discussed above this is definitely not possible for the Schwarzschild solution, because of the unavoidable singularity, but people have recently explored the possibilities of traversable wormholes. Naturally if such direct conveyance between widely separate regions of spacetime were possible, and if those regions were also connected by (much longer) ordinary timelike paths, this raises the prospect of various kinds of "time travel", assuming a wormhole connected to the past was somehow established and maintained. However, these rather far-fetched scenarios all rely on the premise of negative energy density, which of course violates so-called "null energy condition", not to mention the weak, strong, and dominant energy conditions of classical relativity. In other words, on the basis of classical relativity and the traditional energy conditions we could rule out traversable wormholes altogether. It is only the fact that some quantum phenomena do apparently violate these energy conditions (albeit very slightly) that leaves open the remote possibility of such things.

 

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