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Book: Classical Electrodynamics by J.D. Jackson (3rd ed.) - Chapter 12

Immediately after Eq. 12.8 he writes:

"The action (12.6) is proportional to the integral of the proper time over the path from the initial proper time $\tau_1$ to the final proper time $\tau_2$."

Eq. 12.6 reads:

$$ A=\int_{\tau_1}^{\tau_2} \gamma~L~d\tau , \tag{12.6} $$

and later in Eq. 12.7 he gives the Lagrangian of a free particle as

$$ L_{\rm free} = -m ~c^{2} \sqrt{1-\frac{u^2}{c^2}}. \tag{12.7} $$

I completely fail to understand his sentence in double quotes above (given these two equations). Please help.

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In the action you quote, $A=\int_{\tau_1}^{\tau_2} \gamma L \, \mathrm d\tau$, the Lorentz factor $$ \gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}} $$ exactly cancels out the square root in the denominator coming from the Lagrangian $$ L = -mc^2\sqrt{1-\frac{u^2}{c^2}} $$ to give simply $$ \gamma L = -mc^2, $$ and therefore an action $$ A=\int_{\tau_1}^{\tau_2} \gamma L \, \mathrm d\tau =- mc^2\int_{\tau_1}^{\tau_2} \, \mathrm d\tau =-mc^2(\tau_2-\tau_1) $$ that's proportional to the ellapsed proper time interval.

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  • $\begingroup$ Don't you think Jackson's choice of words is very confusing: 'proportional to the integral of the proper time'? I would construe 'integral of the proper time', as the definite integral of proper time over some other variable, whereas you are suggesting that he just means 'change in the proper time'. $\endgroup$ Mar 24 '17 at 15:36
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    $\begingroup$ If you insist, I guess it could be polished a bit, but I find "the integral of the proper time" to be an acceptable description of $\int_{\tau_1}^{\tau_2} \mathrm d\tau$. It's a long and complicated book - just roll with the punches and move on ;-). $\endgroup$ Mar 24 '17 at 15:59
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  1. What Jackson is trying to convey (in his quote) is the fact that the action $S\equiv A$ of a relativistic massive point particle is $$S~=~ - E_0 ~ \Delta \tau, \tag{1}$$ where $$E_0~=~m_0c^2\tag{2}$$ is the rest energy, and $$\Delta \tau~=~\int_{\tau_i}^{\tau_f}\! \mathrm{d}\tau ~=~\tau_f-\tau_i \tag{3}$$ is the change in proper time. See also this Phys.SE post.

  2. Apart from the proportionality constant $E_0$, the formula (1) has a geometrical meaning via the principle of least action:$^1$ The massive point particle chooses the path that maximizes$^2$ its change in proper time, i.e. a timelike geodesic in spacetime.

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$^1$ As always, the devil is in the details: It does not make sense to use proper time $\tau$ as parametrization for the (possibly virtual) paths in the principle of least action because the corresponding boundary conditions $$ x^{\mu}(\tau_i)~=~x^{\mu}_i\qquad\text{and}\qquad x^{\mu}(\tau_f)~=~x^{\mu}_f\qquad\qquad(\leftarrow\text{Wrong!}) \tag{4} $$ would then make all paths have the same action value (1)! Instead we should pick another parametrization $\lambda$ of the paths with corresponding boundary conditions $$ x^{\mu}(\lambda_i)~=~x^{\mu}_i\qquad\text{and}\qquad x^{\mu}(\lambda_f)~=~x^{\mu}_f. \tag{5}$$ This means that we must relate the change $\mathrm{d}\tau$ in proper time $\tau$ to the change $\mathrm{d}\lambda$ in the new parameter $\lambda$. This relationship is mediated via the spacetime metric $$ c\frac{\mathrm{d}\tau}{\mathrm{d}\lambda}~=~\sqrt{g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}} , \qquad \dot{x}^{\mu}~:=~\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\lambda}, \tag{6}$$ where we (as Jackson) have used the $(+,-,-,-)$ Minkowski sign convention. (We stress again that the parameter endpoints $\lambda_i$ and $\lambda_f$ are fixed, but that the change $\Delta\tau$ in proper time (3) is not fixed, so that the variational problem (1) is non-trivial.) In more detail, the action functional (1) therefore reads $$ S[x]~\stackrel{(1)+(2)+(3)+(6)}{=}~ -m_0c\int_{\lambda_i}^{\lambda_f}\! \mathrm{d}\lambda~\sqrt{g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}} .\tag{12.25} $$ The Euler-Lagrange (EL) equation for the action (12.25) is the geodesic equation, cf. e.g. this Phys.SE post. (The specific choice of the parametrization $\lambda$ will not matter because of parametrization invariance. Jackson's Lagrangian (12.7) corresponds to the so-called static gauge choice: $\lambda=t\equiv x^0/c$.)

$^2$ Notice the minus sign in eq. (1), which turns minima into maxima, and vice-versa.

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  • $\begingroup$ Don't you think Jackson's choice of words is very confusing: 'proportional to the integral of the proper time'? I would construe 'integral of the proper time', as the definite integral of proper time over some other variable, whereas you are suggesting that he just means 'change in the proper time'. $\endgroup$ Mar 24 '17 at 15:36
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    $\begingroup$ This is probably an instance where a formula is worth a thousand words for clarification :) $\endgroup$
    – Qmechanic
    Mar 24 '17 at 15:47

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