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9: Flux

Last updated

Feb 28, 2021
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- 8.E: Rotation (Exercises)
- 9.1: The Current Vector

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Topic hierarchy

9.1: The Current Vector
The most fundamental laws of physics are conservation laws, which tell us that we can’t create or destroy “stuff,” where “stuff” could mean quantities such as electric charge or energy-momentum. Since charge is a Lorentz invariant, it’s an easy example to start with. Because charge is invariant, we might imagine that charge density ρ was invariant. But this is not the case, because spatial (3-dimensional) volume isn’t invariant; in 3 + 1 dimensions, only four -dimensional volume is an invariant.
9.2: The Stress-Energy Tensor
A particle such as an electron has a charge, but it also has a mass. We can’t define a relativistic mass flux because flux is defined by addition, but mass isn’t additive in relativity. Mass-energy is additive, but unlike charge it isn’t an invariant. Mass-energy is part of the energy-momentum four vector. We then have sixteen different fluxes we can define.
9.3: Gauss’s Theorem
The connection between the local and global conservation laws is provided by a theorem called Gauss’s theorem. In your course on electromagnetism, you learned Gauss’s law, which relates the electric flux through a closed surface to the charge contained inside the surface. In the case where no charges are present, it says that the flux through such a surface cancels out.
9.4: The Covariant Derivative
The covariant derivative is the derivative that under a general coordinate transformation transforms covariantly, i.e., linearly via the Jacobian matrix of the coordinate transformation.
9.5: Congruences, Expansion, and Rigidity
This chapter has focused on ﬂuxes of conserved quantities in which the appearance and disappearance of world-lines would imply nonconservation of properties such as charge and mass-energy. But the mathematical techniques we’ve developed turn out to be an elegant way to approach the many diﬀerent issues.
9.6: Units of Measurement for Tensors
Analyzing units, also known as dimensional analysis, is one of the ﬁrst things we learn in freshman physics. It’s a useful way of checking our math, and it seems as though it ought to be straightforward to extend the technique to relativity. It certainly can be done, but it isn’t quite as trivial as might be imagined. We’ll see below that diﬀerent authors prefer diﬀering systems, and clashes occur between some of the notational systems in use.
9.7: Notations for Tensors
9.E: Flux (Exercises)

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