Chapter 1 Vectors and tensors 1.1 Books and links The main text for the course is AM Steane, Relativity Made Relatively Easy (OUP, 2012). Additional details can be found in JD Jackson's Classical Electrodynamics. The webpage for the course can be found at http://www-pnp.physics.ox.ac.uk/ tseng/teaching/b2/. The page includes problem sets, links to related courses and materials, and old lecture notes with detailed derivations. The old lecture notes follow the older syllabus, so in some areas contain material which is beyond the current course 1.2 Postulates of Special Relativity 1. Principle of Relativity (two versions): • Laws of physics take the same mathematical form in all inertial frames of reference. • Motion of bodies in a given space are the same among themselves, whether the space is at rest or moving uniformly. 2. Light speed postulate: there is a maximum speed for signals. 3. Space dimensions are flat (\Euclidean") 1.3 Vectors A point is used in two ways: • geometrically, it's a location, independent of how the location is described; 3 B2: Symmetry and Relativity J Tseng • a set of N variables (coordinates which describe a location, conventionally x1; x2; ··· xN The superscript is also by convention (not an exponent). A coordinate system is a mapping between sets of coordinates and the geometric locations. The sets form an N-dimensional vector space. We typically expect that the laws of physics shouldn't depend on the choice of coordinate system. Note that we are using the word \location" in a broad sense: it can be a spatial location, but if combined with a time coordinate, it's a spacetime point, sometimes called an event. A coordinate transformation is a set of N functions of xi xi = xi(x1; x2; ··· xN ) The transformation is usually invertible. A contravariant vector field is a set of N functions of xi Ai = Ai(x1; x2; ··· xN ) which transforms as follows under a coordinate transformation: N i i X @x A = Aj(x1; x2; ··· xN ) @xj j=1 A covariant vector field is a set of N functions of xi 1 2 N Bi = Bi(x ; x ; ··· x ) which transforms as follows under a coordinate transformation: N X @xj B = B (x1; x2; ··· xN ) i @xi j j=1 i The scalar product of A and Bi is N X i A Bi i=1 which is invariant. Contravariant and covariant vectors form dual linear spaces. Length (squared) is defined as the scalar product of a vector with itself N 2 X i L = A Ai i=1 4 B2: Symmetry and Relativity J Tseng 1.3.1 Summation convention 1. Sum over an index which appears as both an \upper" (contravariant) and \lower" (covariant) index, from 1 to N. 2. A differential with respect to a contravariant coordinate (@=@xi) counts as a covariant index, and vice versa. N i i i X @x i @x A = Aj ) A = Aj @xj @xj j=1 N X i i A Bi ) A Bi i=1 When we begin to work explicitly with Minkowski space for Special Relativity, we will use different index types as a shorthand for their range: 1. Latin indices will range from 1 to 3, encompassing the spatial dimensions. 2. Greek indices will range from 0 to 3, with 0 corresponding to the time dimension. 1.3.2 Coordinate transformation matrices The coordinate transformation is often written in terms of a matrix @xi Λi = j @xj i i j A = Λ jA −1 j Ai = Aj(Λ ) i (The order of the last equation is simply to write it more obviously as a matrix multiplica- tion.) 1.4 Tensors A rank-2 tensor is a set of N 2 functions with the following transformation rules: • contravariant tensor i j ij @x @x C = Cab @xa @xb • covariant tensor @xa @xb C = C ij @xi @xj ab 5 B2: Symmetry and Relativity J Tseng • mixed tensor i b i @x @x C = Ca j @xa @xj b Higher rank tensors generalize these transformations. Vectors are rank-1 tensors. Tensors are said to be of the same kind when they have the same number and order (and type) of indices. 1.4.1 Tensor algebra Tensors of the same kind form a linear space. The outer product of two tensors of rank s and r is another tensor of rank s + r: i k i k T jS = C j A tensor of rank s can be contracted by summing over a pair of upper/lower indices, producing a tensor of rank s − 2: i A ji = Dj Quotient rule: if the inner product of N p functions A (with appropriate upper and lower indices) with an arbitrary tensor B is another tensor C then A is a rank-p tensor. 1.4.2 Special tensors i i A symmetric tensor is one where two indices can be swapped, e.g., A jk = A kj if A is symmetric in the jk indices. A skew-symmetric tensor, sometimes called an anti-symmetric tensor, flips sign when the i i specified indices are swapped, e.g., A jk = −A kj. i The Kronecker delta δj is a rank-2 mixed tensor. It is also isotropic in that it doesn't change under coordinate transformations. The related Kronecker delta δij is isotropic under linear transformations in a Euclidean space, but is not a tensor in general. 1.4.3 The metric The metric or fundamental tensor gij defines the line element 2 i j ds = gijdx dx It is also isotropic under linear transformations. ij jk k jk k The reciprocal fundamental tensor is g such that gijg = δi . (In general, AijA 6= δi .) A Cartesian coordinate system is one in which the components gij of the metric tensor are constants. 6 B2: Symmetry and Relativity J Tseng A Euclidean space is a space for which there exists a coordinate transformation to a system 2 i j where ds = δijdx dx . Index manipulation The fundamental tensor allows one to raise and lower indices. abjc ab c gijT d = T i d ij ab c abic g T j d = T d One can also flip the upper/lower indices in a contraction: i i T i = Ti 1.4.4 Levi-Civita tensor The Levi-Civita symbol eijk is defined to be skew-symmetric in all pairs of its indices, with e123 = +1. It is not a tensor in general. p The Levi-Civita tensor is defined to be ijk = jgjeijk, where jgj is the determinant of the metric, considered as a matrix. For Cartesian coordinate systems in Euclidean space, jgj = 1. 1.4.5 Caveats The following caveats apply to more general coordinate systems and transformations: 1. In non-Cartesian coordinate systems, the components gij are in general not equal to the components gij. 2. Even though the gradient of a scalar function @φ/∂xi (which we can write more suc- j cinctly as @iφ) transforms as a covariant vector, the gradient of a vector @iA is not in general a tensor. However, in this course we are mostly concerned with a Cartesian coordinate system of a Minkowski space in which the metric is known 0 −1 1 B 1 C gµν = B C @ 1 A 1 and we limit ourselves to linear transformations. In this case, the components of gµν are ν identical to gµν, and @µA transforms as a rank-2 mixed tensor. 7 B2: Symmetry and Relativity J Tseng 1.5 3D Physics tensors Examples of 3D tensors in physics. Note that since these reside in Euclidean space, con- travariant and covariant components transform in the same way, so the upper and lower indices here are used here mostly to imply summation. 1.5.1 Angular momentum Angular momentum can be written as a skew-symmetric outer product of position and momentum. Lij = xipj − xjpi The angular momentum vector is related to the tensor: jk Li = ijkL : In matrix form, 0 1 0 Lz −Ly ij L = @ −Lz 0 Lx A Ly −Lx 0 1.5.2 Moment of inertia Moment of inertia for a rigid body: Z ij 3 k ij i j I = d xρ(x)[(x xk)δ − x x ] V Angular momentum: i ij L = I !j where !j is the angular velocity vector. 1.5.3 Stress The stress tensor σij can be thought of as indicating the flow of the momentum component pi across face j of a volume. \Face j" refers to the face which is normal to the j direction. Stress tensor for an ideal fluid: 0 p 1 ij σ = @ p A p Euler equation for incompressible fluids: @ui 1 + uj@ ui = @ σij @t j ρ j where ui(xj) is the velocity of each point in the fluid. 8.