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Each stream has its own energy-momentum tensor, and their sum T ab encodes the energy density for the whole fluid. An inertial observer with four-velocity V measures energy density ρV = T ab Va Vb , and sees an energy flow given by the spatial part of the four-vector T ab Vb . The different streams interact through collisions, but energy is conserved in the rest frame of an inertial observer, so the same energy conservation argument as before, applied to a fixed volume in an observer’s frame, gives ∇a (T ab Vb ) = 0.

So photon worldlines are null geodesics and massive particle worldlines are timelike geodesics. 8). They determine the worldlines of free particles through the geodesic equations, and so contain the same information as the ‘acceleration due to gravity’ in Newtonian theory. They vanish at the origin in local inertial coordinates, as one would expect: local inertial coordinates are the coordinates set up by an observer in free-fall at an event. In the observer’s frame, the ‘acceleration due to gravity’ is zero.

Thus it measures, in some sense, the acceleration of the new coordinates relative to the old. It should also have been anticipated, because it mirrors the acceleration term in the transformation of g when one switches to an accelerating frame in Newtonian theory. 6 Manifolds We now have one half of general relativity: we know how gravity affects matter. The gravitational field is encoded in the metric coefficients gab , and the motion of a freely falling particle is governed by the geodesic equations.

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On Some Results of the View of Characteristic Function in Optics by Hamilton W.R.

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