By Sean B. Carroll
ISBN-10: 1292026634
ISBN-13: 9781292026633
Spacetime and Geometry: An advent to basic Relativity presents a lucid and carefully smooth advent to basic relativity for complex undergraduates and graduate students.
It introduces glossy innovations and an available and full of life writing kind to what can usually be a proper and intimidating topic. Readers are led from physics of flat spacetime (special relativity), throughout the intricacies of differential geometry and Einstein's equations, and directly to interesting purposes similar to black holes, gravitational radiation, and cosmology.
Subtle issues are illuminated during the textual content through cautious and interesting exposition.
A hassle-free and lucid process, balancing mathematical rigor and actual perception, are hallmarks of this significant textual content.
Read or Download Spacetime and Geometry: An Introduction to General Relativity PDF
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Extra info for Spacetime and Geometry: An Introduction to General Relativity
Example text
Let's pause a moment to take all this in. We introduced coordinates labeled by upper indices, which transformed in a certain way under Lorentz transformations. We then considered vector components that also were written with upper indices, which made sense since they transformed in the same way as the coordinate functions. ) The basis vectors associated with the coordinate system transformed via the inverse matrix, and were labeled by a lower index. This notation ensured that the invariant object constructed by summing over the components and basis vectors was left unchanged by the transformation, just as we would wish.
Note also that the paths defined by x' = ±t' are precisely the same as those defined by x ±t; these trajectories are left invariant under boosts along the xaxis. Of course we know that light travels at this speed; we have therefore found that the speed of light is the same in any inertial frame. 4 • VECTORS To probe the structure of Minkowski space in more detail, it is necessary to introduce the concepts of vectors and tensors. We will start with vectors, which should be familiar. Of course, in spacetime vectors are four-dimensional, and are often referred to as four-vectors.
10 • CLASSICAL FIELD THEORY When we make the transition from special relativity to general relativity, the metric r/µv will be promoted to a dynamical tensor field, gµv(x). GR is thus a particular example of a classical field theory; we can build up some feeling for how such theories work by considering classical fields defined on flat spacetime. ) Let's begin with the familiar example of the classical mechanics of a single particle in one dimension with coordinate q(t). 126) where the function L(q, q) is the Lagrangian.