Chapter 14

The Lorentz Transformation

Theory, worked examples and 23 previous-year questions with full solutions — IIT JAM, GATE, JEST and CSIR NET-JRF, 2020 onwards.

Theory

14.1 Inertial frames and the Galilean transformation

(a) Inertial frames of reference

A frame of reference is a coordinate system, with an origin \(O\) and unit vectors \(\hat{\mathbf i},\hat{\mathbf j},\hat{\mathbf k}\) along the \(x\), \(y\) and \(z\) axes, together with a set of clocks that assign a time to every event. An event is something that happens at a definite point of space at a definite instant, and it is specified in a given frame by four numbers \((x,y,z,t)\). An inertial frame is one in which a free particle moves with constant velocity, so that Newton’s first law holds; any frame moving with constant velocity relative to an inertial frame is itself inertial, while a frame that accelerates or rotates relative to it is not (Chapter 3). The principle of relativity asserts that the laws of nature take the same form in every inertial frame, so that no experiment performed inside a closed laboratory can reveal its uniform motion.

(b) The Galilean transformation

Consider two inertial frames \(S\) and \(S'\) in the standard configuration of Figure 14.1. The frame \(S'\) moves relative to \(S\) with constant velocity \(v\) along the common \(x\)-\(x'\) axis, the \(y'\) and \(z'\) axes remain parallel to \(y\) and \(z\), and the two origins coincide at \(t=t'=0\). After a time \(t\) the origin \(O'\) is a distance \(vt\) from \(O\).

Figure 14.1 Two inertial frames in the standard configuration. The frame \(S'\) moves with velocity \(v\) along the common \(x\)-\(x'\) axis, and the event \(A\) has coordinates \((x,y,z,t)\) in \(S\) and \((x',y',z',t')\) in \(S'\).

Let a point \(A\) have position vector \(\mathbf r=x\hat{\mathbf i}+y\hat{\mathbf j}+z\hat{\mathbf k}\) in \(S\) and \(\mathbf r'=x'\hat{\mathbf i}+y'\hat{\mathbf j}+z'\hat{\mathbf k}\) in \(S'\). In Newtonian mechanics time is absolute, \(t'=t\), and the geometry of the figure gives \(x=x'+vt\), \(y=y'\) and \(z=z'\). Hence

\[\boxed{x'=x-vt,\qquad y'=y,\qquad z'=z,\qquad t'=t},\]

or in vector form \(\mathbf r'=\mathbf r-\mathbf vt\). These linear relations constitute the Galilean transformation. The inverse transformation is obtained by changing the sign of \(v\), namely \(x=x'+vt'\), \(y=y'\), \(z=z'\), \(t=t'\), as it must be, since \(S\) moves with velocity \(-v\) relative to \(S'\).

(c) Transformation of velocity and acceleration

Let the point \(A\) move with velocity components \(u_x=dx/dt\), \(u_y=dy/dt\), \(u_z=dz/dt\) in \(S\) and \(u_x'=dx'/dt'\), \(u_y'=dy'/dt'\), \(u_z'=dz'/dt'\) in \(S'\). Differentiating the Galilean transformation with respect to \(t=t'\) gives

\[u_x'=u_x-v,\qquad u_y'=u_y,\qquad u_z'=u_z,\qquad\text{that is}\qquad \mathbf u'=\mathbf u-\mathbf v.\]

This is the familiar classical law of addition of velocities. Differentiating once more, and remembering that \(\mathbf v\) is constant,

\[\frac{d^2\mathbf r'}{dt'^2}=\frac{d^2\mathbf r}{dt^2},\qquad\text{so}\qquad \mathbf a'=\mathbf a.\]

The acceleration is therefore the same in all inertial frames. Since masses and the forces between particles (which depend only on their separations and relative velocities) are also unchanged, Newton’s second law \(\mathbf F=m\mathbf a\) has the same form in every inertial frame: Newtonian mechanics is invariant under the Galilean transformation.

(d) The difficulty with light

Suppose a pulse of light leaves the origin of \(S'\) along the \(x'\) axis. An observer in \(S'\) measures its speed to be \(c\approx3\times10^8\;\text{m s}^{-1}\), so \(u_x'=c\), \(u_y'=u_z'=0\). According to the Galilean law the observer in \(S\) should find \(u_x=u_x'+v=c+v\). Maxwell’s equations, however, predict that electromagnetic waves travel in vacuum with the speed \(c=1/\sqrt{\mu_0\varepsilon_0}\), a number fixed by two constants of nature and containing no reference to any frame. Either Maxwell’s equations hold in only one privileged frame, the frame of a hypothetical medium called the ether in which light was supposed to propagate, or the Galilean transformation must be replaced. Maxwell’s equations are in fact not invariant under the Galilean transformation, and the experiment described next was designed to detect the motion of the Earth through the ether.

14.2 The Michelson–Morley experiment

In 1887 A. A. Michelson and E. W. Morley used an interferometer of the kind sketched in Figure 14.2 to measure the speed of the Earth through the ether. A parallel beam of light from a single source is divided by a half-silvered mirror into two beams that travel along perpendicular arms, path \(A\) and path \(B\), to the mirrors \(A\) and \(B\). After reflection the beams recombine at the half-silvered mirror and interfere on a viewing screen. A clear glass plate in one arm ensures that both beams pass through the same thicknesses of glass and air. If the two transit times are equal the beams arrive in phase; any difference in the transit times shifts the interference fringes.

Figure 14.2 The Michelson–Morley interferometer. The hypothetical ether current \(V\) flows parallel to path \(B\).

Let each arm have length \(l\) and suppose that the ether streams past the apparatus with speed \(V\) parallel to path \(B\). Relative to the apparatus, light travels along path \(B\) with speed \(c-V\) on the outward journey and \(c+V\) on the return, so the round-trip time is

\[t_\parallel=\frac{l}{c-V}+\frac{l}{c+V}=\frac{2l}{c}\,\frac{1}{1-V^2/c^2}.\]

Along path \(A\) the light must be aimed slightly upstream so that its resultant velocity relative to the apparatus, of magnitude \(\sqrt{c^2-V^2}\), is along the arm. Hence

\[t_\perp=\frac{2l}{\sqrt{c^2-V^2}}=\frac{2l}{c}\,\frac{1}{\sqrt{1-V^2/c^2}}.\]

For \(V\ll c\) the difference is

\[\Delta t=t_\parallel-t_\perp\approx\frac{2l}{c}\left(1+\frac{V^2}{c^2}\right)-\frac{2l}{c}\left(1+\frac{V^2}{2c^2}\right)=\frac{l}{c}\,\frac{V^2}{c^2}.\]

Rotating the apparatus through \(90^\circ\) interchanges the roles of the arms and changes the path difference by \(2c\,\Delta t\), so the fringe pattern should shift by

\[\Delta N=\frac{2c\,\Delta t}{\lambda}=\frac{2l}{\lambda}\,\frac{V^2}{c^2}.\]

Taking \(V=3\times10^4\;\text{m s}^{-1}\), the orbital speed of the Earth, together with the effective arm length \(l\approx11\) m and \(\lambda\approx5.9\times10^{-7}\) m of the 1887 apparatus, gives \(V^2/c^2=10^{-8}\) and \(\Delta N\approx0.37\) of a fringe. The apparatus could detect shifts of about one hundredth of a fringe, yet no shift was found, at any time of day or season. Many repetitions with ever greater sensitivity confirmed this null result.

The negative result had two consequences. First, there is no detectable ether and therefore no state of “absolute rest”: every motion is relative to a specified frame of reference. Second, the speed of light in vacuum is the same in all inertial frames, unlike the speed of sound or of water waves, which is fixed relative to the medium. FitzGerald and Lorentz proposed to save the ether by supposing that every body moving through it with speed \(V\) contracts along its direction of motion by the factor \(\sqrt{1-V^2/c^2}\), which makes \(t_\parallel=t_\perp\) exactly. Einstein’s special theory of relativity (1905) gave a more radical resolution: the Galilean transformation itself is wrong at speeds comparable with \(c\). The contraction then appears not as a mechanical effect of an ether but as a kinematic consequence of the new transformation (Section 14.6).

14.3 The postulates of special relativity

Einstein’s special theory rests on two postulates.

The principle of relativity. The laws of physics have the same form in all inertial frames. There is no universal frame of reference and no absolute motion; only relative motion can be detected.
The constancy of the speed of light. The speed of light in vacuum has the same value \(c\) in all inertial frames, independently of the motion of the source or of the observer.

The two postulates are incompatible with the Galilean law \(u_x=u_x'+v\), and with the absolute time \(t'=t\) on which it rests. They require a new transformation between inertial frames, which must satisfy the following conditions.

It must be linear in the coordinates and the time, so that a single event in \(S\) corresponds to a single event in \(S'\) and uniform motion in one frame is uniform motion in the other. This expresses the homogeneity of space and time.
It must reduce to the Galilean transformation when \(v\ll c\), because Newtonian mechanics is well verified at low speeds.
The inverse transformation must have the same form, with \(v\) replaced by \(-v\), because by the principle of relativity neither frame is preferred.

It will appear in Sections 14.4 and 14.5 that no material body can be accelerated to the speed \(c\); the speed of light is a limiting speed for all particles with non-zero rest mass (Chapter 15).

14.4 The Lorentz transformation

(a) Derivation

We use the standard configuration of Figure 14.1, with the origins coinciding at \(t=t'=0\). The transverse coordinates are unchanged, \(y'=y\) and \(z'=z\). To see why, imagine two identical metre sticks held parallel to the \(y\) axis, one at rest in each frame, which pass each other. If the stick in \(S'\) appeared shorter in \(S\), then by the principle of relativity the stick in \(S\) would appear shorter in \(S'\), and the two conclusions contradict each other when the sticks are compared side by side at the moment of passing. Hence lengths perpendicular to the relative velocity are the same in both frames.

By linearity the most general relation between \(x'\) and \((x,t)\) that keeps the origin \(x'=0\) at \(x=vt\) is

\[x'=\gamma(x-vt),\]

where \(\gamma\) is a constant that can depend only on \(v\), and reduces to 1 when \(v\to0\). By the principle of relativity the inverse relation must have the same form with \(v\to-v\) and the same \(\gamma\),

\[x=\gamma(x'+vt').\]

Substituting the first relation into the second gives \(x=\gamma^2(x-vt)+\gamma vt'\), and solving for \(t'\),

\[t'=\gamma t+\frac{1-\gamma^2}{\gamma v}\,x.\]

The time therefore cannot be absolute: \(t'\) depends on position as well as on \(t\). The constant \(\gamma\) is fixed by the second postulate. A light pulse emitted from the common origin at \(t=t'=0\) along the \(x\) axis satisfies \(x=ct\) in \(S\) and \(x'=ct'\) in \(S'\). Multiplying the two relations \(x'=\gamma(x-vt)\) and \(x=\gamma(x'+vt')\),

\[xx'=\gamma^2(x-vt)(x'+vt'),\]

and substituting \(x=ct\), \(x'=ct'\) gives \(c^2tt'=\gamma^2(c-v)(c+v)tt'\), so that

\[\boxed{\gamma=\frac{1}{\sqrt{1-v^2/c^2}}=\frac{1}{\sqrt{1-\beta^2}},\qquad \beta=\frac vc}.\]

The same result follows by substituting \(x'=\gamma(x-vt)\) and the expression for \(t'\) into \(x'=ct'\) and requiring the result to coincide with \(x=ct\). With this value of \(\gamma\), the coefficient in the expression for \(t'\) is \((1-\gamma^2)/(\gamma v)=-\gamma^2\beta^2/(\gamma v)=-\gamma v/c^2\), so \(t'=\gamma(t-vx/c^2)\).

(b) The transformation and its inverse

Collecting the results, the Lorentz transformation from \(S\) to \(S'\) is

\[\boxed{x'=\gamma(x-vt),\qquad y'=y,\qquad z'=z,\qquad t'=\gamma\left(t-\frac{vx}{c^2}\right)},\]

and the inverse Lorentz transformation, obtained by interchanging primed and unprimed quantities and replacing \(v\) by \(-v\), is

\[\boxed{x=\gamma(x'+vt'),\qquad y=y',\qquad z=z',\qquad t=\gamma\left(t'+\frac{vx'}{c^2}\right)}.\]

When \(v\ll c\), \(\gamma\to1\) and \(vx/c^2\to0\) for all ordinary distances, so the transformation reduces to the Galilean one, as required. For \(v>c\) the factor \(\gamma\) becomes imaginary, which already suggests that no inertial frame can move faster than light relative to another. The transformation is most symmetric when time is measured in units of length, by the coordinate \(ct\):

\[ct'=\gamma(ct-\beta x),\qquad x'=\gamma(x-\beta ct).\]

In matrix form,

\[\begin{pmatrix}ct'\\x'\end{pmatrix}=\Lambda(\beta)\begin{pmatrix}ct\\x\end{pmatrix},\qquad \Lambda(\beta)=\begin{pmatrix}\gamma&-\gamma\beta\\-\gamma\beta&\gamma\end{pmatrix},\qquad \det\Lambda=\gamma^2(1-\beta^2)=1.\]

The inverse matrix is \(\Lambda(-\beta)\). The product of two such matrices is again of the same form: a direct multiplication gives \(\Lambda(\beta_2)\Lambda(\beta_1)=\Lambda(\beta)\) with

\[\beta=\frac{\beta_1+\beta_2}{1+\beta_1\beta_2},\]

which is the relativistic law of addition of collinear velocities derived in Section 14.5. The Lorentz transformations along a given axis therefore form a group: the inverse of a transformation and the product of two transformations are again Lorentz transformations. The group structure is most transparent in terms of the rapidity \(\phi\), defined by \(\beta=\tanh\phi\), for which \(\gamma=\cosh\phi\), \(\gamma\beta=\sinh\phi\), and

\[\Lambda=\begin{pmatrix}\cosh\phi&-\sinh\phi\\-\sinh\phi&\cosh\phi\end{pmatrix},\qquad \phi=\phi_1+\phi_2.\]

Rapidities add like ordinary velocities, and the addition formula above is the identity \(\tanh(\phi_1+\phi_2)=(\tanh\phi_1+\tanh\phi_2)/(1+\tanh\phi_1\tanh\phi_2)\).

(c) Light-cone coordinates

The combinations \(x_\pm=x\pm ct\) transform particularly simply. From the transformation in the form above,

\[x'+ct'=\gamma(1-\beta)(x+ct),\qquad x'-ct'=\gamma(1+\beta)(x-ct),\]

and since \(\gamma(1-\beta)=\sqrt{(1-\beta)/(1+\beta)}\),

\[\boxed{x'_+=\sqrt{\frac{1-\beta}{1+\beta}}\;x_+,\qquad x'_-=\sqrt{\frac{1+\beta}{1-\beta}}\;x_-}.\]

The coordinates \(x_+\) and \(x_-\) are merely rescaled, by reciprocal factors, so their product \(x_+x_-=x^2-c^2t^2\) is unchanged. The lines \(x_-=0\) and \(x_+=0\) are the paths of light pulses through the origin moving in the \(+x\) and \(-x\) directions, and they are the same in every frame. The same factor \(\sqrt{(1-\beta)/(1+\beta)}\) reappears in the relativistic Doppler effect (Chapter 16).

(d) The invariant interval

For two events separated by \(\Delta t\), \(\Delta x\), \(\Delta y\), \(\Delta z\) in \(S\), the Lorentz transformation gives

\[c^2\Delta t'^2-\Delta x'^2=\gamma^2\left[(c\Delta t-\beta\Delta x)^2-(\Delta x-\beta c\Delta t)^2\right]=\gamma^2(1-\beta^2)\left(c^2\Delta t^2-\Delta x^2\right)=c^2\Delta t^2-\Delta x^2,\]

and \(\Delta y'=\Delta y\), \(\Delta z'=\Delta z\). Hence the interval

\[\boxed{\Delta s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2}\]

has the same value in all inertial frames. In the Galilean picture time intervals and spatial distances are separately invariant; in relativity only this combination is. For a light signal \(\Delta s^2=0\) in every frame, which is the second postulate in invariant form. The interval classifies pairs of events in a frame-independent way. If \(\Delta s^2>0\) the separation is timelike: there is a frame in which the two events occur at the same place, and in that frame the time between them is the proper time \(\Delta\tau=\Delta s/c\), which is smaller than the time between them in any other frame. If \(\Delta s^2<0\) the separation is spacelike: there is a frame in which the events are simultaneous, and their separation in that frame is the proper distance \(\sqrt{-\Delta s^2}\). If \(\Delta s^2=0\) the separation is lightlike and the events can be connected by a light signal. Since \(\Delta t'=\gamma(\Delta t-v\Delta x/c^2)\) and \(\lvert v\Delta x/c^2\rvert<\lvert\Delta t\rvert\) whenever \(c\lvert\Delta t\rvert>\lvert\Delta x\rvert\) and \(v<c\), the time order of two timelike-separated events is the same in all frames; only such events can be causally related. The geometrical representation of these ideas in space-time (Minkowski) diagrams, and the four-vector formalism built on the invariant interval, are developed in Chapter 16.

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Chapter 14 The Lorentz Transformation