Relativistic Dynamics

Contents

Four-vectors
Four-momentum and mass–energy
The relativistic force law
Massless particles
Collisions
The twin paradox
Covariant electrodynamics in one line

Lorentz Transformations settled how coordinates transform. Dynamics must now be rebuilt on that kinematics, because Newton's second law in the form $\vect{F}=m\ddot{\vect{x}}$ is not Lorentz covariant: it singles out a frame's time as the parameter of the motion. The repair is to express everything in objects that transform under $\Lambda$ in a definite way — four-vectors — and to differentiate with respect to the invariant proper time of Definition 38.12 rather than a coordinate time.

Four-vectors

Definition 40.1 (Four-vector).

A four-vector is a quadruple $A^{\mu}$ transforming under a change of inertial frame as the coordinates do,

\begin{equation}\tag{40.1} \bar{A}^{\mu}=\Lambda^{\mu}{}_{\nu}A^{\nu}\ep \end{equation}

Its square $A^{2}:=\eta_{\mu\nu}A^{\mu}A^{\nu}$ is then invariant, by Equation (38.14), and so is the scalar product $A\cdot B=\eta_{\mu\nu}A^{\mu}B^{\nu}$ of any two four-vectors. Rests on Definition 38.6 and Equation (38.14).

Invariants are the currency of the subject: a quantity built as the square of a four-vector has the same value in every frame, so it may be computed in whichever frame makes it easiest and quoted in any other.

Definition 40.2 (Four-velocity).

For a particle with worldline $x^{\mu}(\tau)$ parametrized by its proper time, the four-velocity is

\begin{equation}\tag{40.2} u^{\mu}:=\dv{x^{\mu}}{\tau} =\gamma_{u}\left(c,\vect{u}\right)\ec \end{equation}

where $\vect{u}=\dd\vect{x}/\dd t$ is the ordinary velocity and $\gamma_{u}=\left(1-\abs{\vect{u}}^{2}/c^{2}\right)^{-1/2}$. Rests on Definitions 38.12 and 40.1.

The factor $\gamma_{u}$ appears because $\dd t=\gamma_{u}\dd\tau$ by Equation (38.26). Since $\tau$ is invariant and $x^{\mu}$ is a four-vector, $u^{\mu}$ is one too — which $\dd\vect{x}/\dd t$ is not, and that is precisely why the ordinary velocity is the wrong object.

Proposition 40.3 (Normalization of the four-velocity).

$u^{2}=c^{2}$ for every massive particle, in every frame. Rests on Equation (40.2) and Definition 38.6.

Proof.

Derives Proposition 40.3. Directly from Equation (40.2) and $\eta=\diag(1,-1,-1,-1)$,

\[ u^{2}=\gamma_{u}^{2}\left(c^{2}-\abs{\vect{u}}^{2}\right) =\frac{c^{2}-\abs{\vect{u}}^{2}} {1-\abs{\vect{u}}^{2}/c^{2}}=c^{2}\ep \]

Alternatively, divide $\dd s^{2}=c^{2}\dd\tau^{2}$ by $\dd\tau^{2}$.

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A four-velocity is therefore never zero and always timelike: a particle cannot stand still in spacetime, only redistribute its motion between the time and space directions. At rest it moves through time at rate $c$; as it speeds up, part of that is traded for spatial motion, and the trade is constrained by Proposition 40.3.

Four-momentum and mass–energy

Definition 40.4 (Four-momentum).

For a particle of rest mass $m$ — an invariant, the same in every frame — the four-momentum is

\begin{equation}\tag{40.3} p^{\mu}:=mu^{\mu} =\left(\gamma_{u}mc,\ \gamma_{u}m\vect{u}\right) =:\left(\frac{E}{c},\ \vect{p}\right)\ec \end{equation}

which defines the relativistic energy $E$ and three-momentum $\vect{p}$:

\begin{equation}\tag{40.4} E=\gamma_{u}mc^{2}\ec\qquad \vect{p}=\gamma_{u}m\vect{u}\ep \end{equation}

Rests on Definition 40.2.

Theorem 40.5 (Energy–momentum relation).

\begin{equation}\tag{40.5} E^{2}=\abs{\vect{p}}^{2}c^{2}+m^{2}c^{4}\ec \end{equation}

and equivalently $p^{2}=m^{2}c^{2}$. Rests on Definition 40.4 and Proposition 40.3.

Proof.

Derives Theorem 40.5. By Definition 40.4 and Proposition 40.3, $p^{2}=m^{2}u^{2}=m^{2}c^{2}$. Writing the same square in components, $p^{2}=E^{2}/c^{2}-\abs{\vect{p}}^{2}$, and equating the two gives Equation (40.5).

∎

Phenomenon 40.6 (Mass–energy equivalence).

A body at rest possesses an energy

\begin{equation}\tag{40.6} E_{0}=mc^{2}\ec \end{equation}

and any change in its internal energy changes its rest mass by $\Delta m=\Delta E/c^{2}$. Rests on Equation (40.4).

Derivation. Derives Phenomenon 40.6. Set $\vect{u}=0$ in Equation (40.4): $\gamma_{u}=1$ and $E=mc^{2}$, which is Equation (40.6). That this is a real energy and not an additive constant follows from expanding Equation (40.4) for $\abs{\vect{u}}\ll c$,

\begin{equation}\tag{40.7} E=mc^{2}\left(1-\frac{u^{2}}{c^{2}}\right)^{-1/2} =mc^{2}+\frac{1}{2}mu^{2} +\frac{3}{8}\frac{mu^{4}}{c^{2}}+O\!\left(\frac{u^{6}}{c^{4}}\right)\ec \end{equation}

whose second term is the Newtonian kinetic energy. The relativistic kinetic energy is therefore $T=E-mc^{2}=\left(\gamma_{u}-1\right)mc^{2}$, and $E_{0}$ is what remains when all motion is removed.

For the second statement, consider a body at rest emitting two equal light pulses in opposite directions with total energy $L$, so that it stays at rest. Evaluating the energy balance in a frame moving at $-v$ and expanding to second order in $\beta$, the kinetic energies before and after differ by $\left(L/c^{2}\right)v^{2}/2$, which is the kinetic energy of a body whose mass has fallen by $L/c^{2}$. This is Einstein's argument [Einstein:1905b]; he states the conclusion as a mass change $L/V^{2}$ and quotes it numerically as $L/(9\times10^{20})$ with the energy in erg and the mass in gram — that is, $c^{2}=9\times 10^{16}\,\mathrm{J}/\mathrm{kg}$ to one significant figure.

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Remark 40.7 (There is no ``relativistic mass'' in this treatise).

Some older treatments absorb $\gamma_{u}$ into the mass, writing $m_{\text{rel}}=\gamma_{u}m$ so that $\vect{p}=m_{\text{rel}}\vect{u}$ looks Newtonian. The device is consistent but costs more than it saves: the same $m_{\text{rel}}$ does not make $\vect{F}=m_{\text{rel}}\vect{a}$ true (Section 40.3), and it obscures that $m$ is a Lorentz invariant while $\gamma_{u}m$ is one component of a four-vector. Throughout this book mass means rest mass, and it never depends on speed. Rests on Equation (40.9).

Example 40.8 (Nuclear binding as a mass deficit).

Phenomenon 40.6 is not an exotic effect: it is how nuclear masses are read. The mass of a bound nucleus is less than the sum of its constituents' masses by $B/c^{2}$, with $B$ the binding energy; for $^{4}\mathrm{He}$ the deficit is about $28.3\,\mathrm{MeV}$$/c^{2}$, close to $0.8\,\mathrm{\%}$ of the total. The systematics of that deficit across the nuclide chart is what makes both fission and fusion energetically possible (Nuclear Forces and Nuclear Structure). Rests on Phenomenon 40.6.

The relativistic force law

Definition 40.9 (Four-force).

\begin{equation}\tag{40.8} f^{\mu}:=\dv{p^{\mu}}{\tau}\ec \end{equation}

whose spatial part is related to the ordinary force $\vect{F}=\dd\vect{p}/\dd t$ by $f^{i}=\gamma_{u}F^{i}$, and whose time component is $f^{0}=\gamma_{u}\left(\vect{F}\cdot\vect{u}\right)/c$, the rate at which the force does work. Rests on Definitions 38.12 and 40.4.

Newton's second law is recovered in the covariant form $f^{\mu}=m\,\dd u^{\mu}/\dd\tau$, which reduces to $\vect{F}=m\vect{a}$ when $\abs{\vect{u}}\ll c$. It does not reduce to $\vect{F}=\gamma_{u}m\vect{a}$ in general, because $\gamma_{u}$ depends on time through $\vect{u}$:

\begin{equation}\tag{40.9} \vect{F}=\dv{}{t}\left(\gamma_{u}m\vect{u}\right) =\gamma_{u}m\vect{a} +\gamma_{u}^{3}\frac{m}{c^{2}} \left(\vect{u}\cdot\vect{a}\right)\vect{u}\ep \end{equation}

From Equation (40.4) (differentiating $\vect{p}=\gamma_{u}m\vect{u}$ with respect to coordinate time, with $\gamma_{u}$ carried along). Force and acceleration are therefore not parallel unless $\vect{a}$ is parallel or perpendicular to $\vect{u}$, in which cases Equation (40.9) collapses to $F_{\parallel}=\gamma_{u}^{3}ma_{\parallel}$ and $F_{\perp}=\gamma_{u}ma_{\perp}$ respectively. The two different coefficients were once called the longitudinal and transverse masses; they are a symptom of forcing a Newtonian form onto a non-Newtonian law, and Remark 40.7 applies.

Corollary 40.10 (Nothing massive reaches $c$).

A body of nonzero rest mass cannot be accelerated to the speed of light by any finite energy. Rests on Equation (40.4).

Proof.

Derives Corollary 40.10. By Equation (40.4), $E=\gamma_{u}mc^{2}\to\infty$ as $\abs{\vect{u}}\to c$, since $\gamma_{u}\to\infty$. The kinetic energy required is $\left(\gamma_{u}-1\right)mc^{2}$, which diverges.

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This is measured directly, and simply: Bertozzi accelerated electrons through known potentials up to $15\,\mathrm{MeV}$ and timed their flight over a measured baseline, finding the speed saturating just below $c$ while the kinetic energy continued to rise in proportion to the applied potential [Bertozzi:1964]. Galilean kinematics predicts $v^{2}\propto T$ without limit; the data follow $\left(\gamma-1\right)mc^{2}=T$ instead.

Massless particles

Setting $m=0$ in Equation (40.5) gives

\begin{equation}\tag{40.10} E=\abs{\vect{p}}c\ec \end{equation}

a consistent possibility, describing a particle whose four-momentum is null, $p^{2}=0$. Such a particle has no rest frame — there is no boost carrying a null vector to a purely timelike one — and by Equation (40.4) it must travel at exactly $c$ in every frame, since $\vect{u}=\vect{p}c^{2}/E$. The photon is the instance (The Photon: Photoelectric and Compton Effects); the four-momentum $p^{\mu}=\hbar k^{\mu}$ ties Equation (40.10) to the wave description, and the Doppler and aberration formulas of Section 38.5 are then statements about how $k^{\mu}$ transforms.

Collisions

Postulate 40.11 (Conservation of four-momentum).

In any isolated process the total four-momentum is conserved:

\begin{equation}\tag{40.11} \sum_{\text{in}}p^{\mu}=\sum_{\text{out}}p^{\mu}\ep \end{equation}

Rests on Definition 40.4.

Because Equation (40.11) is an equation between four-vectors, it holds in every frame if it holds in one — which is exactly what a conservation law must do, and what separate conservation of energy and of momentum could not guarantee on its own. Its time component is conservation of energy and its spatial components conservation of momentum; the two are no longer independent statements but components of a single one.

Definition 40.12 (Invariant mass).

For a system of particles the invariant mass $M$ is defined by

\begin{equation}\tag{40.12} M^{2}c^{2}:=\left(\sum_{a}p_{a}^{\mu}\right)^{2} =\frac{1}{c^{2}}\left(\sum_{a}E_{a}\right)^{2} -\left(\sum_{a}\vect{p}_{a}\right)^{2}\ep \end{equation}

Rests on Definitions 40.1 and 40.4.

The invariant mass of a system is not the sum of its constituents' masses: kinetic energy and binding energy contribute, which is Example 40.8 again. It is, however, invariant and conserved, and it equals the total energy divided by $c^{2}$ in the frame where the total momentum vanishes — the centre-of-momentum frame.

Example 40.13 (Threshold energy for pair production).

A proton of energy $E$ strikes a proton at rest; what is the least $E$ at which $p+p\to p+p+p+\bar{p}$ can proceed? At threshold the four final particles are produced at rest in the centre-of-momentum frame, so the final invariant mass is $4m_{p}$. Evaluating Equation (40.12) for the initial state in the laboratory, where $p_{1}=(E/c,\vect{p})$ and $p_{2}=(m_{p}c,\vect{0})$,

\[ M^{2}c^{2}=\left(p_{1}+p_{2}\right)^{2} =p_{1}^{2}+p_{2}^{2}+2p_{1}\cdot p_{2} =2m_{p}^{2}c^{2}+2Em_{p}\ec \]

using $p_{i}^{2}=m_{p}^{2}c^{2}$. Setting $M=4m_{p}$ and solving,

\begin{equation}\tag{40.13} E=7m_{p}c^{2}\ec\qquad T=E-m_{p}c^{2}=6m_{p}c^{2}\approx5.6\,\mathrm{GeV}\ep \end{equation}

From Equations (40.5) and (40.12) (setting $M=4m_{p}$ at threshold and solving the laboratory invariant for $E$). Six rest masses of kinetic energy are needed to create two, because momentum conservation forbids the products from being at rest in the laboratory. This is why colliders beat fixed targets: with two beams of equal and opposite momentum the centre-of-momentum frame is the laboratory, and the whole of the energy is available (Experiment: The Higgs Boson Discovery). Rests on Postulate 40.11 and Definition 40.12.

Example 40.14 (Compton scattering).

A photon of wavelength $\lambda$ scatters off an electron at rest and emerges at angle $\vartheta$ with wavelength $\lambda'$. Conservation Equation (40.11) reads $k+p=k'+p'$; isolating the unobserved electron and squaring,

\[ p'^{2}=\left(k-k'+p\right)^{2} =-2k\cdot k'+2p\cdot\left(k-k'\right)+p^{2}\ec \]

and since $p'^{2}=p^{2}=m_{e}^{2}c^{2}$ and $k^{2}=k'^{2}=0$, the terms rearrange to $k\cdot k'=p\cdot\left(k-k'\right)$. Evaluating in the laboratory with $p=(m_{e}c,\vect{0})$ and $k^{\mu}=\left(\hbar\omega/c\right)(1,\hat{\vect{n}})$ gives the Compton relation

\begin{equation}\tag{40.14} \lambda'-\lambda=\frac{h}{m_{e}c} \left(1-\cos\vartheta\right)\ec \end{equation}

whose measurement is reported in The Photon: Photoelectric and Compton Effects. The whole derivation used only Postulate 40.11 and the invariance of the squares — no dynamics of the interaction at all. Rests on Postulate 40.11, Theorem 40.5 and Equation (40.10).

The twin paradox

Phenomenon 40.15 (Differential ageing).

Of two clocks that separate from a common event and reunite at another, the one that travelled inertially throughout records the greater proper time [Hafele:1972b] [Bailey:1977]. Rests on Definitions 38.6 and 38.12.

Derivation. Derives Phenomenon 40.15. Proper time along a worldline is $\tau=\int\dd t/\gamma_{u}$ by Equation (38.26). For the inertial twin, who may be taken at rest, $\gamma_{u}=1$ throughout and $\tau_{\text{stay}}=T$, the coordinate time between departure and reunion. For the travelling twin $\gamma_{u}>1$ wherever $\vect{u}\neq0$, so

\[ \tau_{\text{trip}}=\int_{0}^{T}\frac{\dd t}{\gamma_{u}}<T =\tau_{\text{stay}}\ec \]

strictly, since the traveller must move to return. For a trip out and back at constant speed $v$, $\tau_{\text{trip}}=T/\gamma$.

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The result is often called a paradox on the grounds that Remark 38.14 makes time dilation reciprocal, so each twin should find the other younger. The objection fails because the situation is not symmetric: the stay-at-home twin occupies one inertial frame, while the traveller occupies two different ones joined by a turnaround. Only the traveller feels an acceleration, and only the traveller's surfaces of simultaneity swing round at the turn, skipping over a long stretch of the stay-at-home's worldline.

No appeal to general relativity is needed. The proper times are path lengths in Minkowski space, and Phenomenon 40.15 is the statement that the straight path between two timelike-separated events is the longest in proper time — the reversal, forced by the minus signs in $\eta$, of the Euclidean fact that a straight line is the shortest path. The effect has been measured directly on flown atomic clocks [Hafele:1972a] [Hafele:1972b] and, far more precisely, on circulating muons [Bailey:1977]; both are reported in Experiment: Time Dilation and Relativistic Kinematics.

Covariant electrodynamics in one line

Relativistic dynamics was not invented to repair mechanics; it was extracted from electrodynamics, which had been Lorentz covariant all along without anyone noticing. The Lorentz force law $\vect{F}=q\left(\vect{E}+\vect{u}\times\vect{B}\right)$ becomes, in the covariant notation of this chapter,

\begin{equation}\tag{40.15} \dv{p^{\mu}}{\tau}=qF^{\mu}{}_{\nu}u^{\nu}\ec \end{equation}

with $F^{\mu\nu}$ the electromagnetic field tensor assembled from $\vect{E}$ and $\vect{B}$. That single equation contains both the electric and magnetic force, and the transformation properties of $F^{\mu\nu}$ explain why the split between them is frame-dependent: what one observer calls a magnetic force on a moving charge, another calls an electric force on a charge at rest. The construction belongs with Maxwell's equations and is carried out in The Maxwell Equations and Generalized Classical Field Theory.