4: Tensors
- Page ID
- 3935
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We now have enough machinery to be able to calculate quite a bit of interesting physics, and to be sure that the results are actually meaningful in a relativistic context. The strategy is to identify relativistic quantities that behave as Lorentz scalars and Lorentz vectors, and then combine them in various ways. The notion of a tensor has been introduced earlier. A Lorentz scalar is a tensor of rank 0, and a Lorentz vector is a rank 1 tensor.
- 4.1: Lorentz Scalars
- A Lorentz scalar is a quantity that remains invariant under both spatial rotations and Lorentz boosts. Mass is a Lorentz scalar. Electric charge is also a Lorentz scalar. The time measured by a clock traveling along a particular world-line from one event to another is something that all observers will agree upon; they will simply note the mismatch with their own clocks. It is therefore a Lorentz scalar.
- 4.2: Four-vectors (Part 1)
- The basic Lorentz vector is the spacetime displacement. Any other quantity that has the same behavior under rotations and boosts is also a valid Lorentz vector.
- 4.3: Four-vectors (Part 2)
- A four-vector is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.
- 4.4: The Tensor Transformation Laws
- We may wish to represent a vector in more than one coordinate system, and to convert back and forth between the two representations.
- 4.5: Experimental Tests
- The techniques developed in this chapter allow us to make a variety of new predictions that can be tested by experiment. In general, the mathematical treatment of all observables in relativity as tensors means that all observables must obey the same transformation laws. This is an extremely strict statement, because it requires that a wide variety of physical systems show identical behavior.
- 4.6: Conservation Laws
- It is natural to ask how conservation laws can be formulated in relativity. We’re used to stating conservation laws casually in terms of the amount of something in the whole universe, e.g., that classically the total amount of mass in the universe stays constant. Relativity does allow us to make physical models of the universe as a whole, so it seems as though we ought to be able to talk about conservation laws in relativity.
Thumbnail: Standard configuration of coordinate systems; for a Lorentz boost in the x-direction. (Public Domain; Gerd Kortemeyer).