Nordström's theory of gravitation
In theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually "two" distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 respectively. The first was quickly dismissed, but the second became the first known example of a metric theory of gravitation, in which the effects of gravitation are treated entirely in terms of the geometry of a curved spacetime. Neither of Nordström's theories are in agreement with observation and experiment. Nonetheless, the first remains of interest insofar as it led to  the second. The second remains of interest both as an important milestone on the road to the current theory of gravitation, general relativity, and as a simple example of a self-consistent relativistic theory of gravitation. As an example, this theory is particularly useful in the context of pedagogical discussions of how to derive and test the predictions of a metric theory of gravitation. Development of the theories. Nordström's theories arose at a time when several leading physicists, including Nordström in Helsinki, Max Abraham in Milan, Gustav Mie in Greifswald, Germany, and Albert Einstein in Prague, were all trying  to create competing relativistic theories of gravitation. All of these researchers began by trying to suitably modify the existing theory, the field theory version of Newton's theory of gravitation. In this theory, the field equation is the Poisson equation formula_1, where formula_2 is the gravitational potential and formula_3 is the density of matter, augmented by an equation of motion for a test particle in an ambient gravitational field, which we can derive from Newton's force law and which states that the acceleration of the test particle is given by the gradient of the potential This theory is not  relativistic because the equation of motion refers to coordinate time rather than proper time, and because, should the matter in some isolated object suddenly be redistributed by an explosion, the field equation requires that the potential everywhere in "space" must be "updated" "instantaneously", which violates the principle that any "news" which has a physical effect (in this case, an effect on test particle motion far from the source of the field) cannot be transmitted faster than the speed of light. Einstein's former calculus professor, Hermann Minkowski had sketched a vector theory of gravitation as early as 1908, but  in 1912, Abraham pointed out that no such theory would admit stable planetary orbits. This was one reason why Nordström turned to scalar theories of gravitation (while Einstein explored tensor theories). Nordström's first attempt to propose a suitable relativistic scalar field equation of gravitation was the simplest and most natural choice imaginable: simply replace the Laplacian in the Newtonian field equation with the D'Alembertian or wave operator, which gives formula_5. This has the result of changing the vacuum field equation from the Laplace equation to the wave equation, which means that any "news" concerning redistribution of matter in one  location is transmitted at the speed of light to other locations. Correspondingly, the simplest guess for a suitable equation of motion for test particles might seem to be formula_6 where the dot signifies differentiation with respect to proper time, subscripts following the comma denote partial differentiation with respect to the indexed coordinate, and where formula_7 is the velocity four-vector of the test particle. This force law had earlier been proposed by Abraham, but it does not preserve the norm of the four-velocity as is required by the definition of proper time, so Nordström instead proposed formula_8. However, this  theory is unacceptable for a variety of reasons. Two objections are theoretical. First, this theory is not derivable from a Lagrangian, unlike the Newtonian field theory (or most metric theories of gravitation). Second, the proposed field equation is linear. But by analogy with electromagnetism, we should expect the gravitational field to carry energy, and on the basis of Einstein's work on relativity theory, we should expect this energy to be equivalent to mass and therefore, to gravitate. This implies that the field equation should be "nonlinear". Another objection is more practical: this theory disagrees drastically with observation. Einstein and  von Laue proposed that the problem might lie with the field equation, which, they suggested, should have the linear form formula_9, where F is some yet unknown function of formula_2, and where T is the trace of the stress–energy tensor describing the density, momentum, and stress of any matter present. In response to these criticisms, Nordström proposed his second theory in 1913. From the proportionality of inertial and gravitational mass, he deduced that the field equation should be formula_11, which is nonlinear. Nordström now took the equation of motion to be or formula_13. Einstein took the first opportunity to  proclaim his approval of the new theory. In a keynote address to the annual meeting of the Society of German Scientists and Physicians, given in Vienna on September 23, 1913, Einstein surveyed the state of the art, declaring that only his own work with Marcel Grossmann and the second theory of Nordström were worthy of consideration. (Mie, who was in the audience, rose to protest, but Einstein explained his criteria and Mie was forced to admit that his own theory did not meet them.) Einstein considered the special case when the only matter present is a cloud of "dust" (that  is, a perfect fluid in which the pressure is assumed to be negligible). He argued that the contribution of this matter to the stress–energy tensor should be: He then derived an expression for the stress–energy tensor of the gravitational field in Nordström's second theory, which he proposed should hold in general, and showed that the sum of the contributions to the stress–energy tensor from the gravitational field energy and from matter would be "conserved", as should be the case. Furthermore, he showed, the field equation of Nordström's second theory follows from the Lagrangian Since Nordström 's equation of motion for test particles in an ambient gravitational field also follows from a Lagrangian, this shows that Nordström's second theory can be derived from an action principle and also shows that it obeys other properties we must demand from a self-consistent field theory. Meanwhile, a gifted Dutch student, Adriaan Fokker had written a Ph.D. thesis under Hendrik Lorentz in which he derived what is now called the Fokker–Planck equation. Lorentz, delighted by his former student's success, arranged for Fokker to pursue post-doctoral study with Einstein in Prague. The result was a historic  paper which appeared in 1914, in which Einstein and Fokker observed that the Lagrangian for Nordström's equation of motion for test particles, formula_17, is the geodesic Lagrangian for a curved Lorentzian manifold with metric tensor formula_18. If we adopt Cartesian coordinates with line element formula_19 with corresponding wave operator formula_20 on the flat background, or Minkowski spacetime, so that the line element of the curved spacetime is formula_21, then the Ricci scalar of this curved spacetime is just Therefore, Nordström's field equation becomes simply where on the right hand side, we have taken the trace of the stress –energy tensor (with contributions from matter plus any non-gravitational fields) using the metric tensor formula_24. This is a historic result, because here for the first time we have a field equation in which on the left hand side stands a purely geometrical quantity (the Ricci scalar is the trace of the Ricci tensor, which is itself a kind of trace of the fourth rank Riemann curvature tensor), and on the right hand stands a purely physical quantity, the trace of the stress–energy tensor. Einstein gleefully pointed out that this equation now takes the form which he had earlier  proposed with von Laue, and gives a concrete example of a class of theories which he had studied with Grossmann. Some time later, Hermann Weyl introduced the Weyl curvature tensor formula_25, which measures the deviation of a Lorentzian manifold from being "conformally flat", i.e. with metric tensor having the form of the product of some scalar function with the metric tensor of flat spacetime. This is exactly the special form of the metric proposed in Nordström's second theory, so the entire content of this theory can be summarized in the following two equations: Features of Nordström's theory. Einstein  was attracted to Nordström's second theory by its simplicity. The "vacuum" field equations in Nordström's theory are simply We can immediately write down the "general" vacuum solution in Nordström's theory: where formula_29 and formula_19 is the line element for flat spacetime in any convenient coordinate chart (such as cylindrical, polar spherical, or double null coordinates), and where formula_20 is the ordinary wave operator on flat spacetime (expressed in cylindrical, polar spherical, or double null coordinates, respectively). But the general solution of the ordinary three-dimensional wave equation is well known, and can be given rather explicit form . Specifically, for certain charts such as cylindrical or polar spherical charts on flat spacetime (which induce corresponding charts on our curved Lorentzian manifold), we can write the general solution in terms of a power series, and we can write the general solution of certain Cauchy problems in the manner familiar from the Lienard-Wiechert potentials in electromagnetism. In any solution to Nordström's field equations (vacuum or otherwise), if we consider formula_32 as controlling a "conformal perturbation from flat spacetime", then to first order in formula_32 we have Thus, in the weak field approximation, we can identify formula_32 with the  Newtonian gravitational potential, and we can regard it as controlling a "small" conformal perturbation from a "flat spacetime background". In any metric theory of gravitation, all gravitational effects arise from the curvature of the metric. In a spacetime model in Nordström's theory (but not in general relativity), this depends only on the "trace" of the stress–energy tensor. But the field energy of an electromagnetic field contributes a term to the stress–energy tensor which is "traceless", so "in Nordström's theory, electromagnetic field energy does not gravitate!" Indeed, since every solution to the field equations of this theory  is a spacetime which is among other things conformally equivalent to flat spacetime, null geodesics must agree with the null geodesics of the flat background, so "this theory can exhibit no light bending". Incidentally, the fact that the trace of the stress–energy tensor for an electrovacuum solution (a solution in which there is no matter present, nor any non-gravitational fields except for an electromagnetic field) vanishes shows that in the general "electrovacuum solution" in Nordström's theory, the metric tensor has the same form as in a vacuum solution, so we need only write down and solve the  curved spacetime Maxwell field equations. But these are "conformally invariant", so we can also write down the "general electrovacuum solution", say in terms of a power series. In any Lorentzian manifold (with appropriate tensor fields describing any matter and physical fields) which stands as a solution to Nordström's field equations, the conformal part of the Riemann tensor (i.e. the Weyl tensor) always vanishes. The Ricci scalar also vanishes identically in any vacuum region (or even, any region free of matter but containing an electromagnetic field). Are there any further restrictions on the Riemann tensor in Nordström's theory? To  find out, note that an important identity from the theory of manifolds, the Ricci decomposition, splits the Riemann tensor into three pieces, which are each fourth-rank tensors, built out of, respectively, the Ricci scalar, the trace-free Ricci tensor and the Weyl tensor. It immediately follows that Nordström's theory "leaves the trace-free Ricci tensor entirely unconstrained by algebraic relations" (other than the symmetric property, which this second rank tensor always enjoys). But taking account of the twice-contracted and detracted Bianchi identity, a differential identity which holds for the Riemann tensor in any (semi)-Riemannian manifold, we see  that in Nordström's theory, as a consequence of the field equations, we have the "first-order covariant differential equation" so the periastrion lag per orbit is and to first order in m, the long axis of the nearly elliptical orbit rotates with the rate This can be compared with the corresponding expression for the Schwarzschild vacuum solution in general relativity, which is (to first order in m) Thus, in Nordström's theory, if the nearly elliptical orbit is transversed counterclockwise, the long axis slowly rotates "clockwise", whereas in general relativity, it rotates "counterclockwise" six times faster. In the first  case we may speak of a periastrion "lag" and in the second case, a periastrion "advance". In either theory, with more work, we can derive more general expressions, but we shall be satisfied here with treating the special case of nearly circular orbits. For example, according to Nordström's theory, the perihelia of Mercury should "lag" at a rate of about 7 seconds of arc per century, whereas according to general relativity, the perihelia should "advance" at a rate of about 43 seconds of arc per century. Light delay. Null geodesics in the equatorial plane of our solution satisfy Consider  two events on a null geodesic, before and after its point of closest approach to the origin. Let these distances be formula_42 with formula_43. We wish to eliminate formula_2, so put formula_45 (the equation of a straight line in polar coordinates) and differentiate to obtain Thus Plugging this into the line element and solving for dt, we obtain Thus the coordinate time from the first event to the event of closest approach is and likewise Here the elapsed coordinate time expected from Newtonian theory is of course so the relativistic time delay, according to Nordström's theory, is To first  order in the small ratios formula_53 this is just formula_54. The corresponding result in general relativity is which depends logarithmically on the small ratios formula_53. For example, in the classic experiment in which, at a time when, as viewed from Earth, Venus is just about to pass "behind" the Sun, a radar signal emitted from Earth which grazes the limb of the Sun, bounces off Venus, and returns to Earth (once again grazing the limb of the Sun), the relativistic time delay is about 20 microseconds according to Nordström's theory and about 240 microseconds according to general relativity. Summary  of results. We can summarize the results we found above in the following table, in which the given expressions represent appropriate approximations: The last four lines in this table list the so-called "four classic solar system tests" of relativistic theories of gravitation. Of the three theories appearing in the table, only general relativity is in agreement with the results of experiments and observations in the solar system. Nordström's theory gives the correct result only for the Pound–Rebka experiment; not surprisingly, Newton's theory flunks all four relativistic tests. Vacuum gravitational plane wave. In the double null chart  for Minkowski spacetime, a simple solution of the wave equation is formula_59, where f is an "arbitrary" smooth function. This represents a plane wave traveling in the z direction. Therefore, Nordström's theory admits the "exact vacuum solution" which we can interpret in terms of the propagation of a gravitational plane wave. This Lorentzian manifold admits a "six-dimensional Lie group of isometries", or equivalently, a six-dimensional Lie algebra of Killing vector fields: For example, the Killing vector field formula_66 integrates to give the one parameter family of isometries Just as in special relativity (and general relativity), it is  always possible to change coordinates, without disturbing the form of the solution, so that the wave propagates in any direction transverse to formula_68. Note that our isometry group is transitive on the hypersurfaces formula_69. In contract, the generic gravitational plane wave in general relativity has only a "five-dimensional Lie group of isometries". (In both theories, special plane waves may have extra symmetries.) We'll say a bit more about why this is so in a moment. Adopting the frame field we find that the corresponding family of test particles are "inertial" (freely falling), since the acceleration vector vanishes Notice  that if f vanishes, this family becomes a family of mutually stationary test particles in flat (Minkowski) spacetime. With respect to the timelike geodesic congruence of world lines obtained by integrating the timelike unit vector field formula_75, the "expansion tensor" shows that our test particles are expanding or contracting "isotropically" and "transversely to the direction of propagation". This is exactly what we would expect for a transverse "spin-0 wave"; the behavior of analogous families of test particles which encounter a gravitational plane wave in general relativity is quite different, because these are "spin-2 waves". This is due to the fact  that Nordström's theory of gravitation is a "scalar theory", whereas Einstein's theory of gravitation (general relativity) is a "tensor theory". On the other hand, gravitational waves in both theories are "transverse" waves. Electromagnetic plane waves are of course also "transverse". The "tidal tensor" further exhibits the spin-0 character of the gravitational plane wave in Nordström's theory. (The tidal tensor and expansion tensor are three-dimensional tensors which "live" in the hyperplane elements orthogonal to formula_78, which in this case happens to be irrotational, so we can regard these tensors as defined on orthogonal hyperslices.) The exact solution  we are discussing here, which we interpret as a propagating gravitational plane wave, gives some basic insight into the "propagation" of gravitational radiation in Nordström's theory, but it does not yield any insight into the "generation" of gravitational radiation in this theory. At this point, it would be natural to discuss the analog for Nordström's theory of gravitation of the standard linearized gravitational wave theory in general relativity, but we shall not pursue this.