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Ebook A first course in general relativity (2nd edition): Part 2

(BQ) Part 2 book "A first course in general relativity" has contents: Physics in a curved spacetime, the einstein field equations, gravitational radiation, spherical solutions for stars, schwarzschild geometry and black holes, cosmology. | 7 Physics in a curved spacetime 7.1 T h e t r a n s i t i o n f r o m d i f f e r e n t i a l g e o m e t r y to gravity The essence of a physical theory expressed in mathematical form is the identification of the mathematical concepts with certain physically measurable quantities. This must be our first concern when we look at the relation of the concepts of geometry we have developed to the effects of gravity in the physical world. We have already discussed this to some extent. In particular, we have assumed that spacetime is a differentiable manifold, and we have shown that there do not exist global inertial frames in the presence of nonuniform gravitational fields. Behind these statements are the two identifications: (I) Spacetime (the set of all events) is a four-dimensional manifold with a metric. (II) The metric is measurable by rods and clocks. The distance along a rod between two nearby points is |dx · dx|1/2 and the time measured by a clock that experiences two events closely separated in time is | − dx · dx|1/2 . So there do not generally exist coordinates in which dx · dx = −(dx0 )2 + (dx1 )2 + (dx2 )2 + (dx3 )2 everywhere. On the other hand, we have also argued that such frames do exist locally. This clearly suggests a curved manifold, in which coordinates can be found which make the dot product at a particular point look like it does in a Minkowski spacetime. Therefore we make a further requirement: (III) The metric of spacetime can be put in the Lorentz form ηαβ at any particular event by an appropriate choice of coordinates. Having chosen this way of representing spacetime, we must do two more things to get a complete theory. First, we must specify how physical objects (particles, electric fields, fluids) behave in a curved spacetime and, second, we need to say how the curvature is generated or determined by the objects in the spacetime. Let us consider Newtonian gravity as an example of a physical theory. For Newton, spacetime consisted of .