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  • Print publication year: 2010
  • Online publication date: June 2012

4 - Metric tensor, geodesics and covariant derivative

Summary

Introduction

We begin our study of general relativity and curved spacetime in this chapter. Chapters 4 and 5 will develop the necessary mathematical apparatus to deal with curved spacetime. As in the case of electromagnetism, the study of gravity can be divided into two separate – but interconnected – aspects. In this chapter and the next, we will study the influence of gravity on other physical systems (like particles, photons, ideal fluids, fields, etc.) without worrying about how a given gravitational field is generated – which will be discussed in Chapter 6.

All the topics introduced in this chapter will be required in the subsequent chapters and form core material for general relativity. In particular, we will start introducing index-free vector notation more liberally in the coming chapters and familiarity with the ideas and notation developed in Section 4.6.1 will be crucial. We will use units with c = 1 unless otherwise indicated.

Metric tensor and gravity

The arguments presented in the previous chapter suggest that a weak gravitational field cannot be distinguished from a modified spacetime interval as far as mechanical phenomena are concerned. We shall now generalize this result by postulating that all aspects of gravitational fields allow a geometrical description. We thus extend the tentative conclusion of the previous chapter to include arbitrarily strong gravitational fields and all physical phenomena. This leads to Einstein's theory of gravitation, which is the most beautiful of all existing physical theories.

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