Lecture 7

We continued developing the mathematical structures necessary to write Lorentz covariant equations; that is, to codify the transformation properties of physical observables under Lorentz transformations.

We first introduced the notion from differential geometry of a metric $g_{ij}$, which relates coordinate components to invariant distances. We gave examples of flat space metrics in Cartesian, cylindrical, and spherical coordinates; then introduced the Minkowski metric $\eta_{\mu\nu}$ responsible for our invariant proper distances in special relativity. The signs in this metric make it nontrivial, and force us to consider how objects transform much more carefully than in the Euclidean case. In particular, 4-vectors $v^\mu$ can be written as column vectors which get left-multiplied by the Lorentz transformation matrix $\Lambda$ under Lorentz transformations, mixing their time and space components. However, there are Lorentz-invariant distances and angles, which mean that the 4-vectors have invariant norms and inner products. These invariant ways of combining vectors are encoded in the metric tensor $\eta_{\mu\nu}$, which we represent here as the diagonal ``matrix'' with entries $+1,-1,-1,-1$. (We will see later why not everything represented by a matrix can be thought of literally as a matrix, hence the quotation marks.) The inner product between 4-vectors $a^\mu$ and $b^\nu$ is then the sum $\sum_{\mu\nu} \eta_{\mu\nu} a^{\mu} b^{\nu}$. We discussed how this metric tensor is a natural extension of the metric tensors introduced on flat space, which describe how small variations in given coordinates correspond to small changes in physical position. The only new subtlety is the signature: the fact that the time eigenvalue has opposite sign from the spatial eigenvalues.

We then introduced some notation, designed to keep proper track of all the minus signs and Lorentz transformation properties while doing a minimum of writing. These notations were: 1) the Einstein implicit summation convention, where any index which is repeated, once as a subscript and once as a superscript, is understood to be summed over all its allowed values. 2) The raising and lowering conventions. These involve a deeper conceptual shift. We noted that the metric can be viewed in two ways. First and most simply, it is a map that combines two vectors into an invariant scalar, by defining the inner product. An alternative interpretation is that the metric is a map which relates vectors in a vector space to their images in a dual, or covector, space. In this picture there are two distinct vector spaces, and every vector in the vector space has exactly one associated partner in the dual space (and vice versa). The metric just tells us how to associate vectors with their partners; the pair itself (one vector, one covector) has a natural way of combining to form a scalar. This second notion -- whose structure is like kets and bras in quantum mechanics -- is more powerful, as it allows us to consider the most general allowed behaviors under transformations.

We define the operation of lowering an index as follows: $v_\mu$ is defined to be $\eta_{\mu\nu} v^\nu$ (with of course an implicit sum on the $\nu$ index). We worked out the components $v_\mu$ of the covector explicitly, and wrote them as the row vector $(v^0, -v^1, -v^2, -v^3)$. Thus the effect on components of lowering $x^\mu$ to $x_\mu$ is just to multiply all the spatial components by a minus sign. This associates with $v^\mu$, a vector, a different kind of object, called a covector. Because this notation allows us to write the norm as $v_\mu v^\mu$, we see that the product of a covector and vector is invariant. Indeed, writing $v_\mu$ as a row vector and $v^\mu$ as a column vector, their inner product is just the scalar obtained by the matrix multiplication $v_\mu v^\mu$. Thus under Lorentz transformations, when the vector is left-multiplied by the matrix $\Lambda$, the covector must be right-multiplied by $\Lambda^{-1}$ -- so that the product remains unchanged. The funny norm (with minus signs for spatial components) is here implemented entirely in the map to the covector.

We discussed how the role played by vectors and covectors is analogous to that played by column vectors and row vectors in flat space. To get the norm of a column vector, we must multiply it by its transpose, a row vector. The row vector $v^T$ is the covector associated with column vector $v$. Every column vector $v$ determines its unique partner $v^T$ among the row vectors, and the two combine by the matrix multiplication $v^Tv$ to give the vector's magnitude. Here we have the complication of the Minkowski signature. To get the magnitude of a four-vector, which we represent as a column vector, we must multiply it by its covector. However here that covector is not just its transpose, but the ``matrix'' $\eta$ multiplied times the four-vector, then transposed. The effect of this complication is to introduce a sign on every spatial component. The row covector and column four-vector then again combine by matrix multiplication to give the inner product.

We then discussed raising indices: going from a covector back to its partner vector. To do this we must use the ``matrix'' $g^{\mu\nu}$, whose row-column $\mu \nu$ entries are those of the inverse matrix $g^{-1}$ to the metric $g_{\mu\nu}$. (For flat Minkowki space with its $(+1,-1,-1,-1)$ diagonal metric, $\eta^{-1}$ is just the same as $\eta$.) We wrote our inverse map, from covector to vector, as $v^\mu = g^{\mu\nu} v_\nu$, and proved that taking a vector to its covector and back reproduced the original vector.

Important practical points in this lecture include the row-column convention, in viewing $g_{\mu\nu}$ and $g^{\mu\nu}$ as ``matrices,'' and recognizing when implicit summations correspond to matrix multiplication, which for each entry takes a Euclidean dot product of a row in the left matrix with a column of the right hand matrix.

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