foundation32 min studyLesson 10 in Physics

Special relativity: events before intuitions

Use Lorentz transformations, invariant intervals, and proper time to compare measurements without inventing a preferred frame.

Before you begin

On this page

The question for today

What you’ll understand

Specify events and frames before comparing times or lengths.
Apply a Lorentz transformation and test the invariant interval.
Identify proper time and the causal role of the light cone.

A physical event is something assigned both a place and a time: a detector click, a flash emission, or a clock reading at a location. Special relativity compares coordinates assigned to the same events by inertial observers. Its surprises become manageable when we compare those events carefully instead of asking vaguely whether time itself runs slower everywhere.

A shared light speed changes the coordinate map

Inertial observers agree on the form of physical laws and on the vacuum speed of light, but they need not agree on spatial separations, elapsed coordinate times, or simultaneity for separated events. The transformation between their coordinates must mix space and time. Simply subtracting a frame velocity from light's speed would preserve an older Newtonian transformation that is incompatible with these premises.

Define one frame moving at constant velocity along the other's x axis, with aligned axes and origins coinciding at time zero. A Lorentz transformation then maps each event, not an entire story in one step. The sign of the relative velocity depends on which frame moves in which direction. State that convention before substituting values, and transform both coordinates of each event consistently.

x'=\gamma(x-vt),\qquad ct'=\gamma(ct-\beta x),\qquad \beta=\frac vc,\qquad \gamma=\frac{1}{\sqrt{1-\beta^2}}

These are standard-configuration Lorentz transformations in flat spacetime for |v| < c. Frame S′ moves along +x at velocity v relative to S. Writing ct makes both coordinates lengths; y and z are unchanged.↔ Scroll wide equations horizontally.

The interval preserves causal structure

Although coordinate time and distance change, the spacetime interval between two events is invariant. With the sign convention used here, a positive squared interval is timelike, zero is lightlike, and negative is spacelike. Timelike-separated events can lie on a massive observer's worldline. Spacelike-separated events cannot exchange a light-speed-or-slower signal, even if another frame reverses their time ordering.

The light cone displays these possibilities geometrically. A Lorentz transformation changes coordinates while preserving the cone, rather than physically tilting a light ray away from its allowed speed. Special relativity also describes accelerating worldlines in flat spacetime; acceleration does not by itself demand general relativity. General relativity is needed when spacetime geometry and gravitation require a curved-spacetime description.

\Delta s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2=c^2\Delta\tau^2\quad\text{(timelike inertial segment)}

The (+,−,−,−) convention is used. For two events on an inertially moving clock, Δτ is that clock's proper time. Accelerated paths require integrating local proper-time increments; one cannot use a single endpoint inertial formula for every path.↔ Scroll wide equations horizontally.

Specify how a time or length is measured

A moving clock's ticks occur at different positions in a laboratory but at the same position in the clock's rest frame. For those ticks, the laboratory coordinate interval exceeds the clock's proper interval by gamma. Measuring a moving rod's length is different: the laboratory must record both endpoints simultaneously in its own frame. Mixing simultaneity conditions creates apparent contradictions.

Relativistic energy and momentum replace their Newtonian forms while agreeing at low speed to suitable accuracy. Rest energy is part of the total, not a speed-dependent change of invariant mass. Quantum field theory must respect this spacetime structure while allowing quantum superposition and particle creation. Relativity therefore supplies essential constraints for the field-theory lessons, rather than being a decorative correction to mechanics.

\Delta t=\gamma\Delta\tau,\qquad E^2=p^2c^2+m^2c^4

The time-dilation relation uses one inertially moving clock and its two ticks; Δt is the interval in the lab frame. The energy–momentum relation uses total energy E, momentum magnitude p, and invariant mass m. At rest it gives E = mc².↔ Scroll wide equations horizontally.

Work it through

Two ticks of a moving clock

A clock moves at $0.60c$ through a laboratory. Two ticks are separated by $2.0\,\mu\mathrm{s}$ on that clock. Using $c=3.0\times10^8\,\mathrm{m\,s^{-1}}$ , find the lab time and distance between the tick events.

The two ticks occur at the same location in the clock's own rest frame, so its reading is proper time: $\Delta\tau=2.0\,\mu\mathrm{s}$ . With $\beta=0.60$ , $\gamma=1/\sqrt{1-0.36}=1.25$ .
The laboratory interval is $\Delta t=\gamma\Delta\tau=(1.25)(2.0)=2.5\,\mu\mathrm{s}$ . Both observers describe the same pair of ticks, not simultaneous readings from unrelated clocks.
The laboratory displacement is $\Delta x=v\Delta t=(0.60)(3.0\times10^8)(2.5\times10^{-6})=450\,\mathrm{m}$ . Check: $c\Delta t=750\,\mathrm{m}$ and $\sqrt{750^2-450^2}=600\,\mathrm{m}=c\Delta\tau$ .

The result

The lab assigns $2.5\,\mu\mathrm{s}$ and $450\,\mathrm{m}$ to the separation. The invariant interval agrees with the clock's $2.0\,\mu\mathrm{s}$ proper time.

The living notebook

Explore the model

Interactive lab

Interactive lab · spacetime

Transform one event without changing its causal type

Change relative velocity β and event time ct while x remains fixed at 0.6 in normalized c = 1 units. Inspect the Lorentz-transformed event and light cone. This is an event-coordinate demonstration, not an animation of an accelerating observer or a simulation of curved spacetime.

Original eventTransformed eventLight cone

Lorentz factor γ: 1.12
Event x′: 0.168
Event ct′: 0.817

Relative velocity β0.45Event time ct1

t = 0 s

A fixed x = 0.6 event is Lorentz transformed with c = 1. Playback highlights the relation between coordinate descriptions; it does not move the event. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forCoordinates change under a boost, while the spacetime interval and causal classification remain fixed.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Proper time is the time recorded along the clock's worldline. For an inertially moving clock, its ticks share a position in its own rest frame. Laboratory simultaneity or equal laboratory positions do not generally describe that moving clock's readings.

Your turn

Try it yourself

In one frame, two events have $c\Delta t=5\,\mathrm{m}$ and $\Delta x=3\,\mathrm{m}$ , with no y or z separation. Classify the separation and find the proper time of the inertial clock joining them, taking the second event to be in the future.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Compute the signed squared interval before taking a square root.

Compare with the solution

The squared interval is $\Delta s^2=5^2-3^2=16\,\mathrm{m^2}>0$ , so the separation is timelike. An inertial clock can connect the events at $v=\Delta x/\Delta t=0.60c$ . Its proper interval is $\Delta\tau=(4\,\mathrm{m})/c\approx1.33\times10^{-8}\,\mathrm{s}$ . The numerical time differs by frame, but this proper interval is invariant for that inertial segment.

Keep the idea

Take it with you

Transform specified events using a stated frame and sign convention.
The invariant interval preserves causal structure even when simultaneity changes.
Relativistic field theory must combine local dynamics with quantum rules and spacetime symmetry.

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