Equations for gravitational field

April 30, 2015

Lines Write the equations

You told us how an almost churchlike atmosphere is pervading your desolate house now. And justifiably so, for unusual divine powers are at work in there.

Besso to Einstein,

The basis of Einstein's general theory of relativity is the audacious idea that not only do the metrical relations of spacetime deviate from perfect Euclidean flatness, but that the metric itself is a dynamical object. In every other field theory the equations describe the behavior of a physical field, such as the electric or magnetic field, within a constant and immutable arena of space and time, but the field equations of general relativity describe the behavior of space and time themselves. The spacetime metric is the field. This fact is so familiar that we may be inclined to simply accept it without reflecting on how ambitious it is, and how miraculous it is that such a theory is even possible, not to mention (somewhat) comprehensible. Spacetime plays a dual role in this theory, because it constitutes both the dynamical object and the context within which the dynamics are defined. This self-referential aspect gives general relativity certain characteristics different from any other field theory. For example, in other theories we formulate a Cauchy initial value problem by specifying the condition of the field everywhere at a given instant, and then use the field equations to determine the future evolution of the field. In contrast, because of the inherent self-referential quality of the metrical field, we are not free to specify arbitrary initial conditions, but only conditions that already satisfy certain self-consistency requirements (a system of differential relations called the Bianchi identities) imposed by the field equations themselves.

The self-referential quality of the metric field equations also manifests itself in their non-linearity. Under the laws of general relativity, every form of stress-energy gravitates, including gravitation itself. This is really unavoidable for a theory in which the metrical relations between entities determine the "positions" of those entities, and those positions in turn influence the metric. This non-linearity raises both practical and theoretical issues. From a practical standpoint, it ensures that exact analytical solutions will be very difficult to determine. More importantly, from a conceptual standpoint, non-linearity ensures that the field cannot in general be uniquely defined by the distribution of material objects, because variations in the field itself can serve as "objects".

Furthermore, after eschewing the comfortable but naive principle of inertia as a suitable foundation for physics, Einstein concluded that "in the general theory of relativity, space and time cannot be defined in such a way that differences of the spatial coordinates can be directly measured by the unit measuring rod, or differences in the time coordinate by a standard clock...this requirement ... takes away from space and time the last remnant of physical objectivity". It seems that we're completely at sea, unable to even begin to formulate a definite solution, and lacking any definite system of reference for defining even the most rudimentary quantities. It's not obvious how a viable physical theory could emerge from such an austere level of abstraction.

These difficulties no doubt explain why Einstein's route to the field equations in the years 1907 to 1915 was so convoluted, with so much confusion and backtracking. One of the principles that heuristically guided his search was what he called the principle of general covariance. This was understood to mean that the laws of physics ought to be expressible in the form of tensor equations, because such equations automatically hold with respect to any system of curvilinear coordinates (within a given diffeomorphism class, as discussed in Section 9.2). He abandoned this principle at one stage, believing that he and Grossmann had proven it could not be made consistent with the Poisson equation of Newtonian gravitation, but subsequently realized the invalidity of their arguments, and re-embraced general covariance as a fundamental principle.

It strikes many people as ironic that Einstein found the principle of general covariance to be so compelling, because, strictly speaking, it's possible to express almost any physical law, including Newton's laws, in generally covariant form (i.e., as tensor equations). This was not clear when Einstein first developed general relativity, but it was pointed out in one of the very first published critiques of Einstein's 1916 paper, and immediately acknowledged by Einstein. It's worth remembering that the generally covariant formalism had been developed only in 1901 by Ricci and Levi-Civita, and the first real use of it in physics was Einstein's formulation of general relativity. This historical accident made it natural for people (including Einstein, at first) to imagine that general relativity is distinguished from other theories by its general covariance, whereas in fact general covariance was only a new mathematical formalism, and does not connote a distinguishing physical attribute. For this reason, some people have been tempted to conclude that the requirement of general covariance is actually vacuous. However, in reply to this criticism, Einstein clarified the real meaning (for him) of this principle, pointing out that its heuristic value arises when combined with the idea that the laws of physics should not only be expressible as tensor equations...

Source: mathpages.com
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What equation is relevant to this problem - gravitational field? | Yahoo Answers

g = GM/r^2 = 28 N/kg; where G is the universal gravitational constant, M is the mass of the attractive source, and r is the distance to the center of M.
a) Find g'/g = (r/2r)^2; so that g' = (1/4)g = 7 N/kg.
b) Find g'/g = (2M/M); so that g' = 2g = 56 N/kg
c) 28 N/kg as the satellite's mass m does not affect the value of the gravitational field g.

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