Abstract
In this chapter, after briefly reviewing the notions of linear vector spaces, inner product of vectors and metric in (three-dimensional) Euclidean space, we shall focus on coordinate transformations, namely maps between different descriptions of the same points in space. This will allow us to introduce covariant and contravariant vectors, as well as tensors, characterized by specific transformation properties under coordinate transformations. Though we shall be mainly concerned with Cartesian coordinate transformations, which are implemented by linear relations between the old and new coordinates, the formalism is readily extended to more general transformations relating curvilinear coordinate systems, and thus also to curved spaces where Cartesian coordinates cannot be defined. We shall then study rotations in Euclidean space and show that they close an object called a Lie group, whose properties are locally captured by a Lie algebra. This will lead us to the important concept of covariance of an equation of motion with respect to rotations. The generalization of all these notions from Euclidean to Minkowski space will be straightforward.
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Notes
- 1.
Recall that this property means that \(a_1\mathbf {u}_1+a_2\mathbf {u}_2+a_3\mathbf {u}_3=0\) if and only if \(a_1=a_2=a_3=0\).
- 2.
- 3.
Such relations are, by definition, invertible, namely the Jacobian matrix \(\left( \frac{\partial x'^j}{\partial x^i}\right) \) is non-singular.
- 4.
We shall use \(\mathbf{r}=(x^i)\) to denote the collection of Cartesian coordinates. Generic coordinates will also be collectively denoted by \(x=(x^i)\).
- 5.
This remark will also apply to tensors and tensor–fields, to be introduced in next section.
- 6.
Of course Eq. (4.70) can be also written \(f^\prime (x)=f(x)\), since x is a variable.
- 7.
In what follows, when referring to Cartesian coordinate systems, the specification rectangular will be understood, unless explicitly stated, since we shall mainly restrict ourselves to coordinate systems of this kind.
- 8.
In our conventions, the rotation angle \(\theta \), on any of the three mutually orthogonal planes \(XY,\,XZ,\,YZ\), is positive if its orientation is related to that of the axis orthogonal to it (i.e. \(Z,\,Y,\,X\)) by the right-hand rule.
- 9.
Let us recall that Euclidean geometry can be fully characterized by the invariance under the corresponding congruence group.
- 10.
Complete antisymmetrization in the three indices \({\small {\mu ,\nu ,\rho }}\) on a generic tensor \(U_{\mu \nu \rho },\) is defined as follows:
$$ U_{[\mu \nu \rho ]}=\frac{1}{3!}(U_{\mu \nu \rho } + U_{\nu \rho \mu }+U_{\rho \mu \nu }-U_{\mu \rho \nu }-U_{\nu \mu \rho }-U_{\rho \nu \mu }). $$It amounts to summing over the even permutations of \(\mu ,\nu ,\rho \) with a plus sign and over the odd ones with a minus sign, the result being normalized by dividing it by the total number 6 of permutations. (see Chap. 5).
- 11.
Recall that the orthogonal group makes no difference between upper and lower indices.
- 12.
Some authors alternatively define the Lorentzian metric \(\varvec{\eta }\) as \(\mathrm{diag}(-1,+1,+1,+1)\). This notation is common in the general relativity literature and has the advantage of yielding the Euclidean metric when restricted to the spatial directions 1, 2, 3.
- 13.
If the invariant metric were diagonal with entries \((+1,+1,-1,-1)\), the corresponding group would have been \(\mathrm{SO}(2,2)\).
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D’Auria, R., Trigiante, M. (2016). The Poincaré Group. In: From Special Relativity to Feynman Diagrams. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-22014-7_4
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DOI: https://doi.org/10.1007/978-3-319-22014-7_4
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