Lecture 8

We finished our study of the linear algebra of objects which transform covariantly under Lorentz transformations.

We reviewed the maps between four-vector and covector introduced last time, and showed that taking a vector to its covector and back reproduced the original vector.

We then considered Lorentz transformation properties. Since the matrix product of covector with vector, $v_\mu w^\mu$ , gives the Lorentz-invariant inner product $v\, \cdot\, w$ , we know how covectors must transform under Lorentz transformation. Since the four-vector $w^\mu$ gets left-multiplied by the Lorentz transformation matrix $\Lambda$ , the covector $v_\mu$ must be right-multiplied by $\Lambda^{-1}$ -- so that the product remains unchanged.

We introduced the 4-gradient operator $\partial_\mu$ , defined naturally as differentiation with respect to the coordinates $x^\mu$ . We showed that under any spacetime coordinate transformation (of which Lorentz transformations are a subgroup), this operator must transform as a covector, because the coordinates remain independent. We then talked about the Lorentz transformation properties of the metric $g_{\mu\nu}$ itself -- which is determined by requiring the inner product of two 4-vectors to remain unchanged even though each 4-vector gets matrix-multiplied by $\Lambda$ . We then introduced tensors with arbitrary numbers of upper and lower indices, determining precisely the complicated transformation properties under Lorentz transformations.

We thus now have all the elements to build Lorentz-covariant laws of physics -- grouping observables and their derivatives into 4-vectors, covectors, and tensors with known Lorentz transformation properties; then generalizing equations so that both left and right sides transform in definite, and equivalent, ways under Lorentz transformations. Note that whenever an index is contracted -- that is, repeated once as a subscript -- which transforms like a covector -- and once as a superscript -- which transforms as a four-vector -- and implicitly summed, that index contributes no transformation under Lorentz transformation, as the contraction forces the apparent transformation of the identical upper and lower indices to cancel. Thus, for example, $v_\mu v^\mu$ transforms like a scalar: since the index $\mu$ is contracted, the covector transformation of $v_\mu$ exactly compensates and cancels the 4-vector transformation of $v^\mu$ . Thus, the ``free'' uncontracted indices of a covariant expression determine entirely its Lorentz transformation properties as a particular tensor.

Your homework will explore some examples of covariant generalizations of your favorite physics equations.

About this document ...

This document was generated using the LaTeX2HTML translator Version 2K.1beta (1.47)

The command line arguments were:
latex2html -split 0 8

The translation was initiated by on 2004-02-16

2004-02-16