The Einstein Field Equations in .NET

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The Einstein Field Equations
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However, since is the tangent vector to a geodesic u c c u a = 0. We can rearrange and then relabel dummy indices on the last term setting c u b u d R a dbc = R a dbc u b u d c = R a bcd u b u c d , and so we obtain the equation of geodesic deviation D2 a = R a bcd u b u c d D 2 We can summarize this result in the following way: Gravity exhibits itself through tidal effects that cause inertial particles to undergo a mutual acceleration. Geometrically, this is manifest via spacetime curvature. We describe the relative acceleration between two geodesics using the equation of geodesic deviation.
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In this section we introduce the Einstein equations and relate them to the equations used to describe gravity in the newtonian framework. Newtonian gravity can be described by two equations. The rst of these describes the path of a particle through space. If a particle is moving through a gravity eld with potential , then Newton s second law gives F = ma = m Canceling the mass term from both sides and writing the acceleration as the second derivative of position with respect to time, we have d2 x = dt 2 This equation is analogous to the equation of geodesic deviation, for which we found D2 a = R a bcd u b u c d D 2 And so we have one piece of the puzzle: we know how to describe the behavior of matter in response to a gravitational eld, which makes itself felt through the curvature. However, now consider the other equation used in newtonian gravity. This equation describes how mass acts as a source of gravitational eld, i.e.,
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2 = 4 G Einstein s equation will have a similar overall form. On the right-hand side, the source term in Newton s theory is the mass density in a given region of space. The lesson of special relativity is that mass and energy are equivalent. Therefore, we need to incorporate this idea into our new theory of gravity, and consider that all forms of mass-energy can be sources of gravitational elds. This is done by describing sources with the stress-energy tensor Tab . This is a more general expression than mass density because it includes energy density as well. We will discuss it in more detail in the next and following chapters. On the left side of Newton s equation, we see second derivatives of the potential. In relativity theory, the metric plays the role of gravitational potential. We have seen that through the relations
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1 ad gbc gcd gdb g + 2 xd xb xc a a e = c bd d bc + bd a ec =
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The curvature tensor encodes second derivatives of the metric. So if we are going to consider the metric to be analogous to gravitational potentials in Newton s theory, some term(s) involving the curvature tensor must appear on the left-hand side of the equations. The equation 2 = 4 G actually relates the trace of i j to the mass density; therefore, we expect that the trace of the curvature tensor, which as we learned in 4 gives the Ricci tensor, will serve as the term on the left-hand side. So the equations will be something like Rab Tab An important constraint on the form of Einstein s equations imposed by the appearance of the stress-energy tensor on the right side will be the conservation of momentum and energy, which as we will see in the next chapter is expressed by the relation b T ab = 0 This constraint means that Rab Tab will not work because b R ab = 0. The contracted Bianchi identities (see problem 1) imply that a R ab = 1 g ab a R, 2 where R is the Ricci scalar. Therefore, if we instead use the Einstein tensor on the left-hand side, we will satisfy the laws of conservation of energy and
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