Galilean and special relativity

Let frame $S'$ move at constant velocity $v$ in the positive $x$-direction relative to frame $S$, with origins together at $t=t'=0$. Galilean transformations use $x'=x-vt$, while $y'=y$, $z'=z$, and—most importantly—$t'=t$.

Differentiating the position transformation gives classical velocity addition: $u'_x=u_x-v$, or equivalently $u_x=u'_x+v$. Accelerations are the same in both frames, so Newton's second law keeps the same form for ordinary inertial motion.

The hidden assumption is universal time. If light moved at speed $c$ in $S'$, Galilean addition would predict $c+v$ or $c-v$ in $S$. Experiments instead show the same vacuum light speed in every inertial frame, so Galilean transformations are only a low-speed approximation when $|v|\ll c$.

Special relativity begins with two postulates. First, the laws of physics have the same form in all inertial frames. Second, light in vacuum has the same speed c for every inertial observer, independent of the source's or observer's motion.

A light clock sends a pulse between two parallel mirrors. In the clock's own frame, the pulse travels vertically. In a frame where the clock moves sideways, the mirrors change horizontal position during the pulse's journey, so the light follows a longer diagonal trajectory.

Take emission and return of the pulse to the same mirror as the two events. Because both observers measure the same light speed $c$, the longer ground-frame path takes more coordinate time. For uniform relative motion, geometry gives $\Delta t=\gamma\Delta\tau$, where $\Delta\tau$ is the proper round-trip time measured by a clock fixed at that mirror.

Left: signed collinear velocities in the Galilean model. Right: a transverse light clock in its own frame and in a ground frame where it moves uniformly right at speed $v\lt c$. In the ground frame both the outward and return light paths advance right; the returning pulse meets the lower mirror after it has moved. Mirror bars mark schematic initial positions, not their positions throughout the cycle; ray angles and distances are not to scale. One tick is emission and return at the same point of the clock, so $\Delta\tau=\frac{2d}{c}$ and $\Delta t=\gamma\Delta\tau$, where $d$ is the mirror separation.

Relativistic effects are organized by the Lorentz factor $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$. It depends only on the magnitude of relative speed, is dimensionless, and is always at least 1 for a massive observer moving slower than light.

For $|v|\ll c$, $\gamma\approx1$. In ordinary low-speed processes, the remaining time-coordinate correction $\frac{vx}{c^2}$ is also negligible at the required precision, and Lorentz predictions are well approximated by Galilean ones. This explains the success of Newtonian mechanics for everyday motion; a small frame speed alone does not make relativity-of-simultaneity effects identically zero over arbitrarily large separations.

As $v$ approaches $c$, the denominator approaches zero and $\gamma$ grows rapidly. The value becomes very sensitive to rounding, so enter the full ratio $\frac{v}{c}$, use parentheses around the square root, and keep guard digits.

For standard configuration, $S'$ moves at speed $v$ along positive $x$ relative to $S$, and the origins coincide when $t=t'=0$. The Lorentz transformations are $x'=\gamma(x-vt)$ and $t'=\gamma(t-\frac{vx}{c^2})$, with $y'=y$ and $z'=z$.

Space and time coordinates mix. The time assigned to an event in one frame depends on both its time and position in another frame. This mixing replaces Galilean universal time and makes the light speed invariant.

The inverse transformation is obtained by replacing $v$ with $-v$: $x=\gamma(x'+vt')$ and $t=\gamma(t'+\frac{vx'}{c^2})$. Use signed coordinates and the correct direction of frame motion rather than inserting speed magnitudes blindly.

Differentiating the Lorentz transformations gives the collinear velocity-addition rule. If an object moves at $u'$ in $S'$, while $S'$ moves at $v$ relative to $S$, then $u=\frac{(u'+v)}{(1+\frac{u'v}{c^2})}$.

The denominator prevents two subluminal velocities from combining to exceed c. It also ensures that if one observer measures light at c, every inertial observer does as well.

At low speeds, $\frac{u'v}{c^2}$ is negligible, so the denominator approaches 1 and $u\approx u'+v$. The relativistic rule therefore has the correct classical limit.

Although observers disagree on $\Delta x$ and $\Delta t$, they agree on the spacetime interval $\Delta s^2=c^2\Delta t^2-\Delta x^2$ for one-dimensional motion. Lorentz transformations preserve this quantity.

For timelike-separated events, there is an inertial frame in which both occur at the same place. An inertial clock at rest there measures the proper interval, with $c^2\Delta\tau^2=\Delta s^2$. This is the elapsed time along the straight, constant-velocity worldline joining the events. An accelerating clock can also pass through both events but need not record this same elapsed time, so the endpoint-interval formula is not a general shortcut for changing-velocity journeys.

If $\Delta s^2>0$, the separation is timelike and a subluminal causal influence is possible. If $\Delta s^2=0$, it is lightlike. If $\Delta s^2<0$, it is spacelike; no signal traveling at or below $c$ can connect the events.

For two events occurring at the same position in an inertial clock's rest frame, the clock measures proper time $\Delta\tau$. A frame in uniform relative motion assigns the larger coordinate-time interval $\Delta t=\gamma\Delta\tau$, with $\gamma$ calculated from that constant relative speed.

Time dilation is reciprocal between inertial observers comparing steadily moving clocks, but a complete comparison must specify which pair of events and which clocks are used. Apparent contradictions usually come from switching event pairs or ignoring simultaneity.

Atmospheric muons provide direct evidence. Muons created high in the atmosphere have short proper lifetimes, yet many reach Earth's surface. In Earth's frame, their moving decay clocks are time-dilated; in the muon frame, the atmosphere's thickness is length-contracted. Both frames predict the same detections.

The proper length of an object is measured in the object's rest frame using the positions of its endpoints. An observer who sees the object move parallel to its length measures the contracted length.

Length contraction is $L=\frac{L_0}{\gamma}$, so $L\le L_0$. It occurs only along the direction of relative motion; transverse dimensions are unchanged.

A length measurement requires both endpoint positions at the same time in the measuring frame. Because simultaneity is frame-dependent, length contraction is inseparable from the relativity of simultaneity and is not a visual-snapshot distortion.

For two events, the time-separation transformation is $\Delta t'=\gamma(\Delta t-\frac{v\Delta x}{c^2})$. Even if $\Delta t=0$ in $S$, separated events generally have $\Delta t'\ne0$ in $S'$. Simultaneity at different places is therefore not absolute.

Clocks at different positions in one frame can be synchronized using equal outward and return light-travel times. A frame moving relative to that synchronized clock network does not judge the separated clocks to be synchronized in the same way.

For spacelike-separated events, different inertial frames can disagree on which happened first. This does not threaten causality because no signal traveling at or below c can connect those events. Timelike- and lightlike-separated distinct events keep the same causal order in every inertial frame.

A basic spacetime diagram plots position $x$ horizontally and $ct$ vertically so both axes have length units. Light rays through the origin satisfy $x=\pm ct$ and appear at $45^\circ$ when the axes use equal scales. A light cone drawn from another event is translated to that event.

A stationary object's worldline is vertical. A particle with constant velocity follows $x=x_0+vt$, where $x_0$ is its position at $t=0$; through the origin this becomes $x=(\frac{v}{c})ct$. On equal-scale $x$–$ct$ axes, its non-negative angle $\theta$ from the positive $ct$-axis satisfies $\tan\theta=\frac{|v|}{c}$. Faster subluminal motion tilts closer to a light line. The future segment remains inside the light cone drawn from any earlier event on that worldline.

The moving frame's $ct'$-axis is the worldline $x=vt$, while its $x'$-axis is the line $t'=0$, given by $ct=(\frac{v}{c})x$. Both tilt toward a light line as $|v|$ increases. Lines parallel to $x'$ represent simultaneity in $S'$; read primed coordinates by projecting parallel to the other primed axis. The tilted axes do not use the same drawn ruler scale as the unprimed axes. For a chosen positive length unit $\ell$, the curve $(ct)^2-x^2=\ell^2$ locates the $ct=\ell$ mark on the positive $ct$-axis and the $ct'=\ell$ mark on the positive $ct'$-axis; these are different events. Similarly, $x^2-(ct)^2=\ell^2$ locates the $x=\ell$ and $x'=\ell$ marks on their respective positive space axes. These constant-interval curves set the tick marks; copying unprimed tick spacing onto a tilted axis gives incorrect coordinates.

Relativity questions are usually decided before the calculator is used. Identify the two events, name the frames, state which frame moves and in what direction, and determine whether the given time or length is proper.

Choose the smallest appropriate tool: Galilean transformation for an explicitly classical low-speed model; Lorentz transformation for event coordinates; velocity addition for measured velocities; interval methods for invariant separation; or dilation/contraction after identifying proper quantities.

Solve symbolically, convert all times and distances to consistent SI units, calculate the Lorentz factor once with guard digits, and check the result against physical limits: gamma at least 1, speeds no greater than c, dilated time no smaller than proper time, and contracted length no larger than proper length.