The Einstein Field Equations in .NET

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The Einstein Field Equations
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Fig. 6-4. A spaceship on earth, in free fall down a mineshaft. Art the astronaut releases a
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ball and nds to his astonishment that it remains at rest in front of him.
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The Weak Equivalence Principle
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The weak equivalence principle is a statement about the universal nature of the gravitational eld. Galileo found that all matter responds to the gravitational eld in exactly the same way regardless of mass or internal composition. Moreover, special relativity teaches us about the equivalence of mass and energy. Combining these two lessons of physics leads us to the weak equivalence principle. Weak Equivalence Principle. The gravitational eld couples in the same way to all mass and energy. The gravitational eld is universal. This is a formal statement of the result we obtained in the rst part of the chapter inertial and gravitational masses are equivalent.
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The Strong Equivalence Principle
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Cases 2 and 4 in our thought experiments involving the spaceship illustrate the strong equivalence principle. This principle states that the laws of physics are the
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The Einstein Field Equations
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same in an accelerated reference frame and in a uniform and static gravitational eld. Note that an experiment that takes place over a large enough time interval or large enough region of space will reveal the tidal effects of gravity. Under these conditions, the equivalence principle would no longer apply. For example, consider two balls released from rest over the surface of the earth.
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The Principle of General Covariance
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In 4 we noted that if an equation involving tensors is true in one coordinate system, it is true in all coordinate systems. This leads to the principle of general covariance, which simply states that the laws of physics, which should be invariant under a coordinate transformation, should be stated in tensorial form. Note that this principle is controversial. We merely state it here because it led Einstein in the development of his theory.
Geodesic Deviation
In ordinary at space, parallel lines always remain parallel. Now consider a more general space where the straightest possible lines are geodesics. What happens to geodesics that start off parallel in a curved space You can get a hint by taking a look at the nearest map of the earth, you can nd, showing the curved longitude lines that run from the North Pole to the South Pole. At the equator, these lines are parallel, but as you move North or South, neighboring lines begin to move together, or start off at the North Pole. Lines that emerge from the same point diverge as you move toward the equator. (see Fig. 6-5) This behavior is typical of any curved geometry. In fact, in a curved space geodesics that start off parallel will eventually cross. Since gravity is just geometry, we expect to nd this kind of behavior in the motion of particles on geodesics in spacetime. In a gravitational eld, the convergence of initially parallel geodesics is nothing more than an expression of gravitational tidal effects. Physically, this is exhibited in the shared acceleration between two particles in free fall in a gravitational eld. Release two particles from some height h above the earth. While the particles start off moving initially on parallel lines toward the ground, since they are on radial paths to the earth s center they will be seen to move toward each other if h is large enough. This is a manifestation of the tidal effects of gravity. We study this phenomenon with equation of geodesic deviation. In your studies of gravity, you will often see the term congruence. A congruence is a
Flat spaceparallel lines remain parallel
The Einstein Field Equations
Lines that were parallel converge
Lines start out parallel
Fig. 6-5. In at space, as seen on the left, lines that start off parallel remain parallel. In a curved space, however, this is not true. As shown on the right, lines that start off parallel end up converging on the sphere as you move from the equator to the North Pole. This type of deviation from being parallel is true in a general curved space or in a curved spacetime.
set of curves such that each point p in the manifold lies on a single curve. To study geodesic deviation in spacetime, we consider a congruence of timelike geodesics. If we call the tangent vector to the curve u a , then the congruence will represent a set of inertial worldlines if u a u a = 1. We de ne the connecting vector as a vector that points from one geodesic to its neighbor. More speci cally, it joins two points on neighboring curves at the same value of the af ne parameter. This is illustrated in Fig. 6-6.
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