3.1 Lorentz covariance for motion in electromagnetic fields, 1

A First Course in String Theory

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The Lorentz force equation (3.5) can be written relativistically as

\displaystyle{\frac{d p_\mu}{ds} = \frac{q}{c} F_{\mu \nu} \frac{d x^\nu}{ds}},

where \displaystyle{p_{\mu}} is the four-momentum.

(a) Check explicitly that this equation reproduces (3.5) when \displaystyle{\mu} is a spatial index.

(b) What does (1) gives when \displaystyle{\mu = 0}? Does it make sense?

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Eq. (3.5):

\displaystyle{\frac{d \vec p}{dt} = q \left( \vec E + \frac{\vec v}{c} \times \vec B \right)}

Eq. (2.20):

\displaystyle{ds \equiv \sqrt{ds^2}}    if    \displaystyle{ds^2 > 0}

Eq. (2.21):

\displaystyle{-ds^2 = \eta_{\mu \nu} dx^\mu dx^\nu}

The spacetime interval \displaystyle{ds^2} is Lorentz invariant. If \displaystyle{ds^2 > 0}, we have Eq. (2.27) and (2.28):

\displaystyle{\begin{aligned}     ds &= c dt_p \\     ds &= c dt \sqrt{1 - \beta^2}     \end{aligned}}

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\displaystyle{\frac{d p_\mu}{ds} \left( \frac{ds}{dt} \right) = \frac{q}{c} F_{\mu \nu} \frac{d x^\nu}{ds}} \left( \frac{ds}{dt} \right)

\displaystyle{    \begin{aligned}     \frac{d p_1}{dt}     &= \frac{q}{c} F_{1 \nu} \frac{d x^\nu}{dt} \\    &= \frac{q}{c} \left( F_{1 0} \frac{d x^0}{dt} + F_{1 1} \frac{d x^1}{dt} + F_{1 2} \frac{d x^2}{dt} + F_{1 3} \frac{d x^3}{dt} \right) \\    \frac{d p_x}{dt} &= \frac{q}{c} \left( E_x c \frac{d t}{dt} + (0) \frac{d x^1}{dt} + B_z \frac{d x^2}{dt} - B_y \frac{d x^3}{dt} \right) \\    &= q \left( E_x + \frac{1}{c} \left( \vec v \times \vec B \right)_x \right) \\    \end{aligned}}

\displaystyle{    \begin{aligned}     \frac{d p_0}{dt}     &= \frac{q}{c} \left( F_{0 0} \frac{d x^0}{dt} + F_{0 1} \frac{d x^1}{dt} + F_{0 2} \frac{d x^2}{dt} + F_{0 3} \frac{d x^3}{dt} \right) \\    &= \frac{q}{c} \left( 0 \frac{d x^0}{dt} - E_x \frac{d x^1}{dt} - E_y \frac{d x^2}{dt} - E_z \frac{d x^3}{dt} \right) \\    \frac{d }{dt} \left( \frac{-E}{c} \right)&= - \frac{q}{c} \left( \vec v \cdot \vec   E \right) \\    \frac{d E}{dt}  &= \vec v \cdot \vec F_E \\      \end{aligned}}

— Me@2022-08-04 04:17:59 PM

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2022.08.06 Saturday (c) All rights reserved by ACHK