--- title: The Lorentz Transformation and Spacetime module: Foundations of Relativity moduleNumber: 1 lessonNumber: 2 order: 102 summary: > Requiring that a light sphere stay a light sphere in every inertial frame fixes the coordinate change between frames uniquely: the Lorentz transformation, with its factor gamma. Differentiating it gives relativistic velocity addition, which caps composed speeds at c. Plotting the same events on skewed spacetime axes turns the algebra into geometry, with calibration hyperbolae, an invariant interval, and a light cone that sorts events into past, future, and elsewhere. topics: [Foundations of Relativity] draft: false sources: - book: Tipler & Llewellyn ref: "Ch. 1 — Relativity I; §1-3 The Lorentz Transformation" - book: Tipler & Llewellyn ref: "§1-3 Relativistic Velocity Transformations; Calibrating the Spacetime Axes" --- The two postulates constrain the coordinate change between inertial frames enough to determine it uniquely. The [Galilean transformation](/relativity/foundations/special-relativity-postulates) $x' = x - vt$, $t' = t$ fails because it predicts a light speed $c - v$ in the moving frame. The correct transformation must reduce to the Galilean one when $v \ll c$, must be linear so that uniform motion stays uniform, and must carry a spherical light wavefront in one frame to a spherical wavefront in the other. Those requirements have one solution.[^tl-lt] ## Deriving the transformation Frames $S$ and $S'$ share parallel axes; $S'$ moves at speed $v$ along the $x$ axis and the origins coincide at $t = t' = 0$. Linearity and the requirement that the origin of $S'$ (where $x' = 0$) move as $x = vt$ force the form $$ x' = \gamma\,(x - vt), $$ with $\gamma$ a constant that may depend on $v$ and $c$ but not on the coordinates, and $\gamma \to 1$ as $v/c \to 0$. By the principle of relativity the inverse has the same form with $v \to -v$: $$ x = \gamma\,(x' + vt'). $$ Motion is only along $x$, so $y' = y$ and $z' = z$. A flash emitted from the common origin at $t = t' = 0$ expands as a sphere in each frame: $$ x^2 + y^2 + z^2 = c^2 t^2, \qquad x'^2 + y'^2 + z'^2 = c^2 t'^2. $$ Substituting the assumed forms of $x'$ and the induced $t'$ into the second sphere equation and demanding it reduce to the first fixes $\gamma$. Carrying out the algebra, the coefficient of $x^2$ must equal $1$, which gives $$ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}}, \qquad \beta \equiv \frac{v}{c}. $$ > **Definition (Lorentz factor).** The dimensionless quantity $\gamma = > (1 - \beta^2)^{-1/2}$, with $\beta = v/c$. It satisfies $\gamma \ge 1$, equals > $1$ at $v = 0$, and diverges as $v \to c$. Every relativistic correction to a > classical formula is some power of $\gamma$. The value of $\gamma$ stays near $1$ for ordinary speeds and climbs steeply only as $\beta$ approaches $1$. $$ % caption: The Lorentz factor gamma versus beta. It is indistinguishable from 1 % below about beta = 0.3 and diverges as the speed approaches c. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right] {beta}; \draw[->, black] (0,0) -- (0,4.2) node[above] {gamma}; \foreach \x/\l in {0/0, 1.5/0.25, 3/0.5, 4.5/0.75, 6/1} \draw[black] (\x,0.05) -- (\x,-0.05) node[below] {\l}; \draw[black] (-0.05,0.9) -- (0.05,0.9) node[left] {1}; \draw[black] (-0.05,2.7) -- (0.05,2.7) node[left] {3}; % gamma curve, y = 0.9/sqrt(1-(x/6)^2), sampled \draw[acc, very thick] (0,0.90) .. controls (1.8,0.95) and (3.2,1.10) .. (4.2,1.34) .. controls (4.9,1.60) and (5.4,2.05) .. (5.7,2.60) .. controls (5.85,3.10) and (5.93,3.7) .. (5.97,4.05); \draw[black, dashed] (0,0.9) -- (6,0.9); \end{tikzpicture} $$ ## The Lorentz transformation Substituting $\gamma$ back and solving for the time coordinate gives the full transformation and its inverse: $$ \begin{aligned} x' &= \gamma\,(x - vt), & \qquad x &= \gamma\,(x' + vt'), \\ y' &= y, & y &= y', \\ z' &= z, & z &= z', \\ t' &= \gamma\parens{t - \frac{vx}{c^2}}, & t &= \gamma\parens{t' + \frac{vx'}{c^2}}. \end{aligned} $$ The time equation is the decisive departure from Newton. The moving-frame time $t'$ depends on _position_ $x$, not on $t$ alone. The term $\gamma v x / c^2$ is the position-dependent offset that makes synchronized clocks in one frame appear unsynchronized in another. When $v \ll c$, $\gamma \to 1$ and $vx/c^2 \to 0$, recovering $t' = t$. > **Example (Time interval between separated events).** Two events occur at > $(x_a, t_a)$ and $(x_b, t_b)$ in $S$. Their separation in $S'$ is > $$ > t'_b - t'_a = \gamma\,(t_b - t_a) - \frac{\gamma v}{c^2}(x_b - x_a). > $$ > The interval measured in $S'$ depends not only on the interval in $S$ but on > the spatial separation of the two events. If they occur at the same place in > $S$ ($x_a = x_b$), the second term vanishes and $t'_b - t'_a = \gamma(t_b - > t_a)$: the moving frame reads a longer interval. If instead the two events are > to be simultaneous in $S'$, the spatial term must cancel the temporal one, > which requires $v/c = c(t_b - t_a)/(x_b - x_a)$. ## Velocity addition Differentiating the transformation gives the rule for combining velocities. For a particle with $x$-velocity $u_x = \d x/\d t$ in $S$, use $\d x' = \gamma(\d x - v\,\d t)$ and $\d t' = \gamma(\d t - v\,\d x/c^2)$: $$ u'_x = \frac{\d x'}{\d t'} = \frac{\d x - v\,\d t}{\d t - v\,\d x/c^2} = \frac{u_x - v}{1 - v u_x / c^2}. $$ The transverse components pick up a $\gamma$ in the denominator because $\d t'$ carries one while $\d y' = \d y$: $$ u'_y = \frac{u_y}{\gamma\,(1 - v u_x/c^2)}, \qquad u'_z = \frac{u_z}{\gamma\,(1 - v u_x/c^2)}. $$ When $v u_x \ll c^2$, the denominator is $\approx 1$ and the classical $u'_x = u_x - v$ returns. The denominator is what caps composed speeds: set $u_x = c$ and $$ u'_x = \frac{c - v}{1 - vc/c^2} = \frac{c - v}{1 - v/c} = c. $$ Light moves at $c$ in the new frame too, as the second postulate demands. No combination of sub-light speeds ever exceeds $c$. > **Example (Cosmic-ray protons, head-on).** Two protons approach Earth from > opposite directions at $0.6c$ and $0.8c$ relative to Earth. The speed of one > relative to the other is not $1.4c$. Put Earth in $S$ and one proton's rest > frame in $S'$ moving at $v = 0.6c$; the other proton has $u_x = -0.8c$ in $S$. > Then > $$ > u'_x = \frac{-0.8c - 0.6c}{1 - (0.6c)(-0.8c)/c^2} > = \frac{-1.4c}{1 + 0.48} = -0.946c. > $$ > Each proton measures the other approaching at $0.946c$, below $c$. ## Spacetime diagrams A spacetime diagram plots $ct$ vertically and $x$ horizontally, so that a light ray, $x = ct$, is a line at $45^\circ$. The path of a particle through the diagram is its **worldline**; a particle at rest traces a vertical line, and a faster particle traces a line closer to the vertical (steeper than the light line, which it can never cross). The moving frame's axes are not perpendicular on this diagram. The $ct'$ axis is the set of events with $x' = 0$, which is the worldline of the $S'$ origin, $x = vt$, a line of slope $c/v = 1/\beta$ tilted from vertical toward the light line. The $x'$ axis is the set of events with $t' = 0$, which from the transformation is $ct = \beta x$, a line of slope $\beta$ tilted from horizontal toward the light line. Both axes rotate _toward_ the $45^\circ$ light line by the same angle, so the light line always bisects them, in $S'$ exactly as in $S$. $$ % caption: The moving frame S-prime drawn on the S diagram. The ct-prime axis % (worldline of the S-prime origin) tilts from vertical and the x-prime axis % tilts from horizontal, both closing on the light line at forty-five degrees, % which bisects them in every frame. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % S axes \draw[->, black] (-0.3,0) -- (5,0) node[right] {x}; \draw[->, black] (0,-0.3) -- (0,4.6) node[above] {ct}; % light line \draw[black, dashed] (0,0) -- (4.3,4.3) node[right] {x = ct}; % S' axes (beta = 0.5): ct' slope 2 (x=vt -> ct=x/beta), x' slope 0.5 \draw[->, acc, thick] (0,0) -- (2.1,4.2) node[above] {ct'}; \draw[->, acc, thick] (0,0) -- (4.2,2.1) node[right] {x'}; \node[anchor=west] at (2.2,3.4) {x' = 0}; \node[anchor=north] at (3.3,1.5) {t' = 0}; \end{tikzpicture} $$ Reading coordinates off the skewed axes needs a calibration, because a unit length along the $ct'$ axis is not the same paper distance as a unit along $ct$. ## Calibration and the invariant interval The quantity that stays fixed across frames is the **spacetime interval**. From the light-sphere condition, the combination $$ (\Delta s)^2 = (c\,\Delta t)^2 - (\Delta x)^2 $$ takes the same value in every inertial frame; the Lorentz transformation is built precisely to preserve it. A direct substitution confirms $(c\,\Delta t')^2 - (\Delta x')^2 = (c\,\Delta t)^2 - (\Delta x)^2$. > **Definition (Invariant interval).** For two events separated by $\Delta x$ in > space and $\Delta t$ in time, $(\Delta s)^2 = (c\,\Delta t)^2 - (\Delta x)^2$. > Its value is the same for all inertial observers even though $\Delta x$ and > $\Delta t$ separately are not. It plays for spacetime the role that squared > distance plays for space, but with the minus sign that distinguishes time from > space. The invariance draws the calibration curves. The locus of events at fixed interval from the origin, $(ct)^2 - x^2 = k$, is a hyperbola, and it is the same hyperbola in every frame. The hyperbola $(ct)^2 - x^2 = 1$ crosses the $ct$ axis at $ct = 1$ and crosses the $ct'$ axis at $ct' = 1$, so its intersections with the two time axes mark off equal unit ticks on each. Units on the tilted axes are read where the axes meet these hyperbolae. $$ % caption: Calibration hyperbolae. The constant-timelike-interval curve marks one % unit of time on both the ct and ct-prime axes; the constant-spacelike-interval % curve marks one unit of length on both space axes. The same invariant hyperbola % calibrates every frame. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-0.3,0) -- (4.7,0) node[right] {x}; \draw[->, black] (0,-0.3) -- (0,4.4) node[above] {ct}; \draw[black, dashed] (0,0) -- (4.2,4.2); % S' axes beta=0.5 \draw[->, thick] (0,0) -- (1.95,3.9) node[above] {ct'}; \draw[->, thick] (0,0) -- (3.9,1.95) node[right] {x'}; % timelike hyperbola (ct)^2 - x^2 = 1 : ct = sqrt(1+x^2) \draw[black!70, thick] (0,1) .. controls (0.7,1.05) and (1.4,1.35) .. (2.0,1.9) .. controls (2.5,2.35) and (3.0,2.9) .. (3.4,3.35); \node[black!70, anchor=west] at (2.7,2.55) {time unit}; % spacelike hyperbola x^2 - (ct)^2 = 1 : x = sqrt(1+ct^2) \draw[black, thick] (1,0) .. controls (1.05,0.7) and (1.35,1.4) .. (1.9,2.0) .. controls (2.35,2.5) and (2.9,3.0) .. (3.35,3.4); \node[black, anchor=north west] at (1.05,0.15) {length unit}; % unit tick on ct axis \fill[black!70] (0,1) circle (1.6pt); \fill[acc] (0.577,1.155) circle (1.6pt); \node[acc, anchor=west] at (0.72,1.28) {ct' = 1}; \end{tikzpicture} $$ ## The light cone In two space dimensions the light rays through the origin sweep out a cone, $c^2 t^2 = x^2 + y^2$. The **light cone** sorts every other event relative to the origin into three classes, and because the interval is invariant, the class is the same for all observers. - **Future.** Events inside the upper cone, with $(\Delta s)^2 > 0$ and $\Delta t > 0$: reachable from the origin by a signal slower than light. Every observer agrees they happen after the origin event. - **Past.** Events inside the lower cone, with $(\Delta s)^2 > 0$ and $\Delta t < 0$: able to send a signal to the origin. Every observer agrees they happen before it. - **Elsewhere.** Events outside the cone, with $(\Delta s)^2 < 0$: too far in space to be connected by any signal at or below $c$. Their time order relative to the origin depends on the frame, so no observer can call them cause or effect of the origin event. $$ % caption: The light cone about an event O. The forty-five-degree light lines % separate the absolute future and past (timelike, order fixed for all frames) % from the elsewhere region (spacelike, where the time order is frame-dependent). \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % cones \fill[acc!10] (0,0) -- (2.3,2.3) -- (-2.3,2.3) -- cycle; \fill[acc!10] (0,0) -- (2.3,-2.3) -- (-2.3,-2.3) -- cycle; \draw[acc, thick] (-2.6,-2.6) -- (2.6,2.6); \draw[acc, thick] (-2.6,2.6) -- (2.6,-2.6); \draw[->, black] (-3,0) -- (3,0) node[right] {x}; \draw[->, black] (0,-3) -- (0,3.2) node[above] {ct}; \fill[black] (0,0) circle (2pt); \node[anchor=north east] at (0,0) {O}; \node at (0,1.9) {future}; \node at (0,-1.9) {past}; \node[black] at (2.0,0.55) {elsewhere}; \node[black] at (-2.0,0.55) {elsewhere}; \end{tikzpicture} $$ The cone structure is the causal skeleton of relativity. The invariance of whether an event lies inside or outside the cone means the theory never lets an effect precede its cause, even though it lets different observers disagree on the timing of events that cannot influence each other. The three cases carry names and physical readings that the [next lesson](/relativity/foundations/time-dilation-length-contraction) uses to resolve the twin and pole-barn paradoxes. | Interval | Sign of $(\Delta s)^2$ | Relation | Worldline type | | --- | --- | --- | --- | | Timelike | $> 0$ | connectable by a sub-light signal; time order absolute | massive particles | | Lightlike | $= 0$ | connectable only by light | photons | | Spacelike | $< 0$ | no signal connects them; time order frame-dependent | none | The transformation and its geometry are the machinery. Applied to a clock at rest in one frame, they produce time dilation; applied to a rod, length contraction; applied to momentum and energy, the relativistic dynamics of the [fourth lesson](/relativity/foundations/relativistic-momentum-energy). [^tl-lt]: **Tipler & Llewellyn**, _Modern Physics_, §1-3 — The Lorentz Transformation: derivation of $\gamma$ from the invariance of the light sphere, the transformation and its inverse, relativistic velocity addition, calibration of the spacetime axes by invariant hyperbolae, and the light-cone causal structure.