6.1  An Exact Solution

 

Einstein had been so preoccupied with other studies that he had not realized such confirmation of his early theories had become an everyday affair in the physical laboratory. He grinned like a small boy, and kept saying over and over “Ist das wirklich so?”

                                                                                                                A. E. Condon

 

The special theory of relativity assumes the existence of a unique class of global coordinate systems - called inertial coordinates - with respect to which the speed of light in vacuum is everywhere equal to the constant c. It was natural, then, to express physical laws in terms of this preferred class of coordinate systems, characterized by the global invariance of the speed of light. In addition, the special theory also strongly implied the fundamental equivalence of mass and energy, according to which light (and every other form of energy) must be regarded as possessing inertia. However, it soon became clear that the global invariance of light speed together with the idea that energy has inertia (as expressed in the famous relation E2 = m2 + |p|2) were incompatible with one of the most firmly established empirical results of physics, namely, the exact proportionality of inertial and gravitational mass, which Einstein elevated to the status of a Principle. This incompatibility led Einstein, as early as 1907, to the belief that the global invariance of light speed, in the sense of the special theory, could not be maintained. Indeed, he concluded that we cannot assume, as do both Newtonian theory and special relativity, the existence of any global inertial systems of coordinates (although we can carry over the existence of a local system of inertial coordinates in a vanishingly small region of spacetime around any event).

 

Since no preferred class of global coordinate systems is assumed, the general theory essentially places all (smoothly related) systems of coordinates on an equal footing, and expresses physical laws in a way that is applicable to any of these systems. As a result, the laws of physics will hold good even with respect to coordinate systems in which the speed of light takes on values other than c. For example, the laws of general relativity are applicable to a system of coordinates that is fixed rigidly to the rotating Earth. According to these coordinates the distant galaxies are "circumnavigating" nearly the entire universe in just 24 hours, so their speed is obviously far greater than the constant c. The huge implied velocities of the celestial spheres was always problematical for the ancient conception of an immovable Earth, but it is beautifully accommodated within general relativity by the effect which the implied centrifugal acceleration field - whose strength increases in direct proportion to the distance from the Earth - has on the values of the metric components guv for this rotating system of coordinates at those locations. It's true that, when expressed in this rotating system of coordinates, those stars are moving with dx/dt values that far exceed the usual numerical value of c, but they are not moving faster than light, because the speed of light at those locations, expressed in terms of those coordinates, is correspondingly greater.

 

In general, for any given system of coordinates the velocity of light can always be inferred from the components of the metric tensor, and typically looks something like . To understand this, recall that in special relativity we have global inertial coordinate systems such that the metric tensor has the constant form

 

 

The trajectory of a light ray follows a null path, i.e., a path with dt = 0, so dividing by (dt)2 we see that the path of light everywhere satisfies the equation

 

 

Hence the velocity of light is unambiguous in terms of these preferred systems of coordinates. However, in the general theory we are no longer guaranteed the existence of a global coordinate system with a constant metric of the simple form (1). It is true that over a sufficiently small spatial and temporal region surrounding any given event in spacetime there exists a coordinate system of that simple Minkowskian form, but in the presence of a non-vanishing gravitational field ("curvature") equation (1) applies only with respect to "free-falling" inertial coordinates, which are necessarily transient and don't extend globally.

 

So, for an extended region of spacetime, instead of writing the metric in the xt plane as (dt)2 = (dt)2 - (dx)2 , we must consider the more general form

 

 

where the coefficients are functions of the coordinates. As always, the path of a light ray is null, so we have dt = 0, so the differentials dx and dt satisfy the equation

 

 

Solving this gives

 

 

If we diagonalize our metric we get gxt = 0, in which case the "velocity" of a null path in the xt plane with respect to this coordinate system is simply dx/dt = . This quantity can (and does) take on any value, depending on our choice of coordinate systems.

 

Around 1911 Einstein proposed to incorporate gravitation into a modified version of special relativity by allowing the speed of light to vary as a scalar from place to place in Euclidean space as a function of the gravitational potential. This "scalar c field" is remarkably similar to a simple refractive medium, in which the speed of light varies as a function of the density. Fermat's principle of least time can then be applied to define the paths of light rays as geodesics in the spacetime manifold (as discussed in Section 8.4). Specifically, Einstein wrote in 1911 that the speed of light at a place with the gravitational potential j would be c0 (1 + j/c02), where c0 is the nominal speed of light in the absence of gravity. In geometrical units we define c0 = 1, so Einstein's 1911 formula can be written simply as c = 1 + j. However, this formula for the speed of light – indeed, this whole approach to gravity – turned out to be incorrect, as Einstein soon realized. In the general theory of relativity, completed in 1915, the speed of light in a gravitational field cannot necessarily be represented by a simple scalar field of c values in Euclidean space, because the speed of light at each point depends on the choice of coordinate systems, and the curvature of spacetime makes the choice of coordinates inherently ambiguous. In terms of some quite natural coordinate systems the speed of light at a given event is different in different directions, so we can't generally talk about the value of c at a given point in a non-vanishing gravitational field (although the speed if light always has the invariant value c in terms of local free-falling inertial coordinates, consistent with the equivalence principle). For example, if we consider rays of light near a spherically symmetrical (and non- rotating) mass, and we choose coordinates such that the metric coefficients are independent of t, and such that the circumference of a circular orbit of radius r is 2pr, then in terms of these coordinates the circumferential speed of light is given by the same formula as in Einstein’s 1911 paper, but the speed of radial light rays differs from the 1911 formula by a factor of 2 on the “potential” term. To explain this in detail, we must first consider how the Schwarzschild metric is derived from the field equations of general relativity.

 

To deduce the implications of the field equations for observable phenomena Einstein originally made use of approximate methods, since no exact solutions were known. These approximate methods were adequate to demonstrate that the field equations lead in the first approximation to Newton's laws, and in the second approximation to a natural explanation for the anomalous precession of Mercury (see Section 6.2). However, these results can now be directly computed from the exact solution for a spherically symmetric field, found by Karl Schwarzschild in 1916. As Schwarzschild wrote, it's always pleasant to find exact solutions, and the simple spherically symmetrical line element "let's Mr. Einstein's result shine with increased clarity". To this day, most of the empirically observable predictions of general relativity are consequences of this simple solution.

 

We will discuss Schwarzschild's original derivation in Section 8.7, but for our present purposes we will take a slightly different approach. Recall from Section 5.5 that the most general form of the metrical spacetime line element for a spherically symmetrical static field (although it is not strictly necessary to assume the field is static) can be written in polar coordinates as

 

 

where gqq = -r2, gff = -r2 sin(q)2, and gtt and grr are functions of r and the gravitating mass m. (These stipulations ensure that the circumference of a circular orbit of radius r is 2pr.) We expect that if m = 0, and/or as r increases to infinity, we will have gtt = 1 and grr = -1 in order to give the flat Minkowski metric in the absence of gravity. We saw in Section 5.5 that in this highly symmetrical context there is a fairly plausible way to derive the metric coefficients gtt and grr simply from the requirement to satisfy Kepler's third law and the inverse-square law, but with some ambiguity over the choice between proper time and coordinate time. We can now determine unambiguously the values of these metric coefficients consistent with Einstein's field equations.

 

In any region that is free of (non-gravitational) mass-energy the vacuum field equations must apply, which means the Ricci tensor

 

 

must vanish, i.e., all the components are zero. Since our metric is in diagonal form, it's easy to see that the Christoffel symbols for any three distinct indices a,b,c reduce to

 

 

with no summations implied. In two of the non-vanishing cases the Christoffel symbols are of the form qa/(2q), where q is a particular metric component and subscripts denote partial differentiation with respect

to xa. By an elementary identity these can also be written as . Hence if we define the new variable  we can write the Christoffel symbol in the form Qa with q = e2Q. Accordingly if we define the variables (functions of r)

 

 

then we have

 

and the non-vanishing Christoffel symbols (as given in Section 5.5) can be written as

 

 

We can now write down the components of the Ricci tensor, each of which must vanish in order for the field equations to be satisfied. Writing them out explicitly and expanding all the implied summations for our line element, we find that all the non-diagonal components are identically zero (which we might have expected from symmetry arguments), so the only components of interest in our case are the diagonal elements

 

 

Inserting the expressions for the Christoffel symbols gives the equations for the four diagonal components of the Ricci tensor as functions of u and v:

 

 

The necessary and sufficient condition for the field equations to be satisfied by a line element of the form (2) is that these four quantities each vanish. Combining the expressions for Rtt and Rrr we immediately have ur = -vr , which implies u = -v + k for some arbitrary constant k. Making these substitutions into the equation for Rqq we get the condition

 

 

Remembering that e2u = gtt, and that the derivative of e2u is 2ur e2u, this condition expresses the requirement

 

 

The left side is just the chain rule for the derivative of the product rgtt, and since this derivative equals the constant –e2k we immediately have rgtt = -e2kr + a for some constant a, and hence gtt = -e2k + a/r. As r increases to infinity the metric must go over to the Minkowski metric, which has gtt = 1, so we must have –e2k = 1, which implies that k = pi/2. Also, since grr = e2v where v = -u + pi/2, it follows that grr = -1/gtt, and so we have the results

 

 

To match the Newtonian limit we set a = -2m where m is classically identified with the mass of the gravitating body. These metric coefficients were derived by combining the expressions for Rtt and Rrr, but it's easy to verify that they also satisfy each of those equations separately, so this is indeed the unique spherically symmetrical static solution of Einstein's field equations.

 

Now that we have derived the Schwarzschild metric, we can easily correct the "speed of light" formula that Einstein gave in 1911. A ray of light always travels along a null trajectory, i.e., with dt = 0, and for a radial ray we have dq and df both equal to zero, so the equation for the light ray trajectory through spacetime, in Schwarzschild coordinates (which are the only spherically symmetrical ones in which the metric is independent of t) is simply

 

 

from which we get

 

where the ± sign just indicates that the light can be going radially inward or outward. (Note that we're using geometric units, so c = 1.) In the Newtonian limit the classical gravitational potential at a distance r from mass m is j = -m/r, so if we let cr = dr/dt denote the radial speed of light in Schwarzschild coordinates, we have

 

 

which corresponds to Einstein's 1911 equation, except that we have a factor of 2 instead of 1 on the potential term. Thus, as j becomes increasingly negative (i.e., as the magnitude of the potential increases), the radial "speed of light" cr defined in terms of the Schwarzschild parameters t and r is reduced to less than the nominal value of c. The factor of 2 relative to the equation of 1911 arises because in the full theory there is gravitational length contraction as well as time dilation. Of course, the length contraction doesn’t affect the gravitational redshift, which is purely a function of the time dilation, so the redshift prediction of 1911 remains valid. Only the radial speed of light (in terms of Schwarzschild coordinates) is changed.

 

On the other hand, if we define the tangential speed of light at a distance r from a gravitating mass center in the equatorial plane (q = p/2) in terms of the Schwarzschild coordinates as ct = r(df/dt), then the metric divided by (dt)2 immediately gives

 

 

Thus, we again find that the "velocity of light" is reduced a region with a strong gravitational field, but this speed is the square root of the radial speed at the same point, and to the first order in m/r this is the same as Einstein's 1911 formula, although it is understood now to signify just the tangential speed. This illustrates the fact that the general theory doesn't lead to a simple scalar field of c values in Euclidean space. The effects of gravitation can only be accurately represented by a tensor field. (It’s possible to define so-called isotropic coordinates, as discussed in Section 8.4, in terms of which the speed of light is the same in all directions, but only by using a radial coordinate in terms of which the circumference of a circular orbit of radius r is not 2pr, which shows the non-Euclidean character of the space.)

 

As mentioned, one of the observable implications of general relativity (as well as any other theory that respects the equivalence principle) is gravitational redshift, which is a consequence of the fact that the rate of proper time at a fixed radial position in a gravitational field relative to the coordinate time (which corresponds to proper time sufficiently far from the gravitating mass) is given by

 

 

It follows that the characteristic frequency n1 of light emitted by some known physical process at a radial location r1 will represent a different frequency n2 with respect to the proper time at some other radial location r2 according to the formula

 

 

From the Schwarzschild metric we have gtt(rj) = 1+2jj where jj = -m/rj is the gravitational potential at rj, so

 

 

Neglecting the higher-order terms and rearranging, this can also be written as

 

 

Observations of the light emitted from the surface of the Sun, and from other stars, is consistent with this predicted amount of gravitational redshift (up to first order), although measurements of this slight effect are difficult. A terrestrial experiment performed by Rebka and Pound in 1960 exploited the Mossbauer effect to precisely determine the redshift between the top and bottom of a tower. The results were in good agreement with the above formula, and subsequent experiments of the same kind have improved the accuracy to within about 1 percent. (Note that if r1 and r2 are nearly equal, as, for example, at two heights near the Earth's surface, then the leading factor of the right-most expression is essentially just the acceleration of gravity a = -m/r2, and the factor in parentheses is the difference in heights Dh, so we have Dn/n = a Dh.)

 

However, it's worth noting that this amount of gravitational redshift is a feature of just about any viable theory of gravity that includes the equivalence principle, so these experimental results, although useful for validating that principle, are not very robust for distinguishing between competing theories of gravity. For this we need to consider other observations, such as the paths of light near a gravitating body, and the precise orbits of planets. These phenomena are discussed in the subsequent sections.

 

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