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Gravitation and Inertia
Chapter One
Einstein Geometrodynamics
If Einstein gave us a geometric account of motion and gravity, if according to his 1915 and still-standard geometrodynamics spacetime tells mass how to move and mass tells spacetime how to curve, then his message requires mathematical tools to describe position and motion, curvature and the action of mass on curvature. The tools (see the mathematical appendix) will open the doorways to the basic ideas-equivalence principle, geometric structure, field equation, equation of motion, equation of geodesic deviation-and these ideas will open the doorways to more mathematical tools-exact solutions of Einstein's geometrodynamics field equation, equations of conservation of source, and the principle that the boundary of a boundary is zero. The final topics in this chapter-black holes, singularities, and gravitational waves-round out the interplay of mathematics and physics that is such a central feature of Einstein's geometrodynamics.
2.1 THE EQUIVALENCE PRINCIPLE
At the foundations of Einstein geometrodynamics and of its geometrical structure is one of the best-tested principles in the whole field of physics (see chap. 3): the equivalenceprinciple.
Among the various formulations of the equivalence principle (see 3.2), we give here three most important versions: the weak form, also known as the uniqueness of free fall or the Galilei equivalence principle at the base of most known viable theories of gravity; the medium strong form, at the base of metric theories of gravity; and the very strong form, a cornerstone of Einstein geometrodynamics.
Galilei in his Dialogues Concerning Two New Sciences writes: "The variation of speed in air between balls of gold, lead, copper, porphyry, and other heavy materials is so slight that in a fall of 100 cubits a ball of gold would surely not outstrip one of copper by as much as four fingers. Having observed this, I came to the conclusion that in a medium totally void of resistance all bodies would fall with the same speed."
We therefore formulate the weak equivalence principle, or Galilei equivalence principle in the following way: the motion of any freely falling test particle is independent of its composition and structure. A test particle is defined to be electrically neutral, to have negligible gravitational binding energy compared to its rest mass, to have negligible angular momentum, and to be small enough that inhomogeneities of the gravitational field within its volume have negligible effect on its motion.
The weak equivalence principle-that all test particles fall with the same acceleration-is based on the principle that the ratio of the inertial mass to the gravitational-passive-mass is the same for all bodies (see chap. 3). The principle can be reformulated by saying that in every local, nonrotating, freely falling frame the line followed by a freely falling test particle is a straight line, in agreement with special relativity.
Einstein generalized the weak equivalence principle to all the laws of special relativity. He hypothesized that in no local freely falling frame can we detect the existence of a gravitational field, either from the motion of test particles, as in the weak equivalence principle, or from any other special relativistic physical phenomenon. We therefore state the medium strong form of the equivalence principle, also called the Einstein equivalence principle, in the following way: for every pointlike event of spacetime, there exists a sufficiently small neighborhood such that in every local, freely falling frame in that neighborhood, all the nongravitational laws of physics obey the laws of special relativity. As already remarked, the medium strong form of the equivalence principle is satisfied by Einstein geometrodynamics and by the so-called metric theories of gravity, for example, Jordan-Brans-Dicke theory, etc. (see chap. 3).
If we replace all the nongravitational laws of physics with all the laws of physics we get the very strong equivalence principle, which is at the base of Einstein geometrodynamics.
The medium strong and the very strong form of the equivalence principle differ: the former applies to all phenomena except gravitation itself whereas the latter applies to all phenomena of nature. This means that according to the medium strong form, the existence of a gravitational field might be detected in a freely falling frame by the influence of the gravitational field on local gravitational phenomena. For example, the gravitational binding energy of a body might be imagined to contribute differently to the inertial mass and to the passive gravitational mass, and therefore we might have, for different objects, different ratios of inertial mass to gravitational mass, as in the Jordan-Brans-Dicke theory. This phenomenon is called the Nordtvedt effect (see chap. 3). If the very strong equivalence principle were violated, then Earth and Moon, with different gravitational binding energies, would have different ratios of inertial mass to passive gravitational mass and therefore would have different accelerations toward the Sun; this would lead to some polarization of the Moon orbit around Earth. However, the Lunar Laser Ranging28 experiment has put strong limits on the existence of any such violation of the very strong equivalence principle.
The equivalence principle, in the medium strong form, is at the foundations of Einstein geometrodynamics and of the other metric theories of gravity, with a "locally Minkowskian" spacetime. Nevertheless, it has been the subject of many discussions and also criticisms over the years.
First, the equivalence between a gravitational field and an accelerated frame in the absence of gravity, and the equivalence between a flat region of spacetime and a freely falling frame in a gravity field, has to be considered valid only locally and not globally. However, the content of the strong equivalence principle has been criticized even "locally." It has been argued that if one puts a spherical drop of liquid in a gravity field, after some time one would observe a tidal deformation from sphericity of the drop. Of course, this deformation does not arise in a flat region of spacetime. Furthermore, let us consider a freely falling frame in a small neighborhood of a point in a gravity field, such as the cabin of a spacecraft freely falling in the field of Earth. Inside the cabin, according to the equivalence principle, we are in a local inertial frame, without any observable effect of gravity. However, let us take a gradiometer, that is, an instrument which measures the gradient of the gravity field between two nearby points with great accuracy (present room temperature gradiometers may reach a sensitivity of about [10.sup.-11] (cm/[s.sup.2])/cm per [Hz.sup.-1/2] [equivalent to] [10.sup.-2] Eötvös per [Hz.sup.-1/2] between two points separated by a few tens of cm; future superconducting gradiometers may reach about [10.sup.-5] Eötvös [Hz.sup.-1/2] at certain frequencies, see 3.2 and 6.9). No matter if we are freely falling or not, the gradiometer will eventually detect the gravity field and thus will allow us to distinguish between the freely falling cabin of a spacecraft in the gravity field of a central mass and the cabin of a spacecraft away from any mass, in a region of spacetime essentially flat. Then, may we still consider the strong equivalence principle to be valid?
From a mathematical point of view, at any point P of a pseudo-Riemannian, Lorentzian, manifold (see 2.2 and mathematical appendix), one can find coordinate systems such that, at P, the metric tensor [g.sub.[alpha][beta]] ( 2.2) is the Minkowski metric [[eta].sub.[alpha][beta]] = diag(-1, +1, +1, +1) and the first derivatives of [g.sub.[alpha][beta]], with respect to the chosen coordinates, are zero. However, one cannot in general eliminate certain combinations of second derivatives of [g.sub.[alpha][beta]] which form a tensor called the Riemann curvature tensor: [R.sup.[alpha].sub.[[beta].sub.[gamma]][delta]] (see 2.2 and mathematical appendix). The Riemann curvature tensor represents, at each point, the intrinsic curvature of the manifold, and, since it is a tensor, one cannot transform it to zero in one coordinate system if it is nonzero in some other coordinate system. For example, at any point P on the surface of a sphere one can find coordinate systems such that, at P, the metric is [g.sub.11](P) = [g.sub.22](P) = 1. However, the Gaussian curvature of the sphere (see mathematical appendix), that is, the [R.sub.1212] component of the Riemann tensor, is, at each point, an intrinsic (independent of coordinates) property of the surface and therefore cannot be eliminated with a coordinate transformation. The metric tensor can indeed be written using the Riemann tensor, in a neighborhood of a spacetime event, in a freely falling, nonrotating, local inertial frame, to second order in the separation, [delta][x.sup.[alpha]], from the origin:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.1.1)
These coordinates are called Fermi Normal Coordinates.
In section 2.5 we shall see that in general relativity, and other metric theories of gravity, there is an important equation, the geodesic deviation equation, which connects the physical effects of gravity gradients just described with the mathematical structure of a manifold, that is, which connects the physical quantities measurable, for example with a gradiometer, with the mathematical object representing the curvature: the Riemann curvature tensor. We shall see via the geodesic deviation equation that the relative, covariant, acceleration between two freely falling test particles is proportional to the Riemann curvature tensor, that is, [??][x.sup.[alpha]] ~ [R.sup.[alpha].sub.[beta][mu]v] [delta][x.sup.[mu]], where [delta][x.sup.[alpha]] is the "small" spacetime separation between the two test particles. On a two-surface, this equation is known as the Jacobi equation for the second derivative of the distance between two geodesics on the surface as a function of the Gaussian curvature.
The Riemann curvature tensor, however, cannot be eliminated with a coordinate transformation. Therefore, the relative, covariant, acceleration cannot be eliminated with a change of frame of reference. In other words, by the measurement of the second rate of change of the relative distance between two test particles, we can detect, in every frame, the gravitational field, and indeed, at least in principle, we can measure all the components of the Riemann curvature tensor and therefore completely determine the gravitational field. Furthermore, the motion of one test particle in a local freely falling frame can be described by considering the origin of the local frame to be comoving with another nearby freely falling test particle. The motion of the test particle in the local frame, described by the separation between the origin and the test particle, is then given by the geodesic deviation equation of section 2.5. This equation gives also a rigorous description of a falling drop of water and of a freely falling gradiometer, simply by considering two test particles connected by a spring, that is, by including a force term in the geodesic deviation equation (see 3.6.1).
From these examples and arguments, one might think that the strong equivalence principle does not have the content and meaning of a fundamental principle of nature. Therefore, one might think to restrict to interpreting the equivalence principle simply as the equivalence between inertial mass [M.sub.i] and gravitational mass [M.sub.g]. However, [M.sub.i] = [M.sub.g] is only a part of the medium (and strong) equivalence principle whose complete formulation is at the basis of the locally Minkowskian spacetime structure.
In general relativity, the content and meaning of the strong equivalence principle is that in a sufficiently small neighborhood of any spacetime event, in a local freely falling frame, no gravitational effects are observable. Here, neighborhood means neighborhood in space and time. Therefore, one might formulate the medium strong equivalence principle, or Einstein equivalence principle, in the following form: for every spacetime event (then excluding singularities), for any experimental apparatus, with some limiting accuracy, there exists a neighborhood, in space and time, of the event, and infinitely many local freely falling frames, such that for every nongravitational phenomenon the difference between the measurements performed (assuming that the smallness of the spacetime neighborhood does not affect the experimental accuracy) and the theoretical results predicted by special relativity (including the Minkowskian character of the geometry) is less than the limiting accuracy and therefore undetectable in the neighborhood. In other words, in the spacetime neighborhood considered, in a freely falling frame all the nongravitational laws of physics agree with the laws of special relativity (including the Minkowskian character of spacetime), apart from a small difference due to the gravitational field that is; however, unmeasurable with the given experimental accuracy. We might formulate the very strong equivalence principle in a similar way.
For a test particle in orbit around a mass M, the geodesic deviation equation gives
[??][x.sup.[alpha]] ~ [R.sup.[alpha].sub.0[beta]0] [delta][x.sup.[beta]] ~ [[omega].sup.2.sub.0] [delta][x.sup.[alpha]] (2.1.2)
where [[omega].sub.0] is the orbital frequency. Thus, one would sample large regions of the spacetime if one waited for even one period of this "oscillator." We must limit the dimensions in space and time of the domain of observation to values small compared to one period if we are to uphold the equivalence principle.
A liquid drop which has a urface tension, and which resists distortions from sphericity, supplies an additional example of how to interpret the equivalence principle. In order to detect a gravitational field, the measurable quantity-the observable-is the tidal deformation [delta]x of the drop. If a gravity field acts on the droplet and if we choose a small enough drop, we will not detect any deformation because the tidal deformations from sphericity are proportional to the size D of the small drop, and even for a self-gravitating drop of liquid in some external gravitational field, the tidal deformations [delta]x are proportional to its size D. This can be easily seen from the geodesic deviation equation with a springlike force term ( 3.6.1), in equilibrium: k / m [delta]x ~ [R.sup.i.sub.0j0]D ~ M / [R.sup.3] D, where M is the mass of an external body and [R.sup.i.sub.0j0] ~ M / [R.sup.3] are the leading components of the Riemann tensor generated by the external mass M at a distance R. Thus, in a spacetime neighborhood, with a given experimental accuracy, the deformation [delta]x, is unmeasurable for sufficiently small drops.
(Continues...)
Excerpted from Gravitation and Inertia by Ignazio Ciufolini John Archibald Wheeler Copyright © 1996 by Princeton University Press. Excerpted by permission.
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