Chapter 9 · Special relativity

中文

In Example 7.12 we saw that a Euclidean inner product space with positive definite metric tensor $g$ gives rise to a restricted tensor theory called cartesian tensors, wherein all bases ${e_{i}}$ are required to be orthonormal and basis transformations $e_{i} = A_{i}^{i^{'}} e_{i^{'}}$ are restricted to orthogonal transformations. Cartesian tensors may be written with all their indices in the lower position, $T_{ij k .}$ and it is common to adopt the summation convention for repeated indices even though both are subscripts.

In a general pseudo-Euclidean inner product space we may also restrict ourselves to orthonormal bases wherein

g_{ij} = {\pm 1 0 if i = j, if i \neq = j,

so that only pseudo-orthogonal transformation matrices $A = [A_{i}^{i^{'}}]$ are allowed. The resulting tensor theory is referred to as a restricted tensor theory. For example, in a four-dimensiona Minkowskian vector space the metric tensor in an orthonormal basis is

g_{ij} = 100001000010 000 - 1,

and the associated restricted tensors are commonly called 4-tensors. In 4-tensor theory there is a simple connection between covariant and contravariant indices, for example

U^{1} = U_{1}, U^{2} = U_{2}, U^{3} = U_{3}, U^{4} = - U_{4},

but the distinction between the two types of indices must still be maintained. In this chapter we give some applications of 4-tensor theory in Einstein’s special theory of relativity [1–3].

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Szekeres · 数学物理

章节

Chapter 9 · Special relativity

Contents

Concept index

9.1 Minkowski space-time

9.2 Relativistic kinematics

9.3 Particle dynamics

9.4 Electrodynamics

9.5 Conservation laws and energy–stress tensors

9.4 Electrodynamics

9.5 Conservation laws and energy–stress tensors

9.3 Particle dynamics

9.1 Minkowski space-time

9.2 Relativistic kinematics