Lorentz Transformation Calculator

Introduction: Lorentz Transformations Between Moving Frames

This Lorentz transformation calculator describes how one physical event receives different time and position coordinates in two inertial frames moving relative to each other. At everyday speeds, it is convenient to imagine space and time as distinct entities. We assume that a pair of events separated by some time interval t and distance x will appear similarly separated to all observers. However, once velocities approach a significant fraction of the speed of light, this Newtonian intuition fails. Experiments with fast-moving particles, atomic clocks on aircraft, and the behavior of cosmic rays all reveal that time can dilate and lengths contract. These counterintuitive effects arise from Einstein’s special theory of relativity, which treats space and time as aspects of a unified spacetime. Different observers, moving relative to one another, assign coordinates differently while the underlying physical event remains the same. The mathematical rule relating coordinates in one inertial frame to those in another is the Lorentz transformation.

For the coordinate conversion used here, the speed of light $c$ has the same value for all inertial observers. Consider two reference frames: frame S, with coordinates $(t,x,y,z)$, and frame S’ moving with velocity $v$ along the positive x-axis relative to S. If an event occurs at certain spacetime coordinates in S, the Lorentz transformation determines where and when that same event is assigned coordinates in S’. Rather than treating time as universal, the equations mix time and the x-coordinate in a way that preserves the spacetime interval.

Formula: Lorentz Transformation Equations for an x-Axis Boost

This calculator’s Lorentz transformation formula uses the dimensionless speed $\beta =\frac{v}{c}$ and the Lorentz factor $\gamma =\frac{1}{\sqrt{1-{\beta }^2}}$. With frame S’ moving along the x-axis, it evaluates the following relations:

These Lorentz coordinate equations show why simultaneity depends on the frame: events with $\Delta t=0$ in S can have different times in S’ when they are separated along x. A rod at rest in S’, with proper length $L_0$ parallel to the motion, is measured in S with a length smaller by a factor of $\gamma$. Likewise, for two events at one position in S, so that $\Delta x=0$, the transformation gives ${\Delta }^t\prime =\gamma \Delta t$; this is the coordinate expression of time dilation for that setup.

Preserving the Lorentz Spacetime Interval

A defining feature of the Lorentz transformation used by this calculator is preservation of the spacetime interval $s^2=c^2{t}^2-x^2-y^2-z^2$. Unlike separate distances and times, this quantity is unchanged by the x-axis boost. It separates events into three categories: timelike ($s^2>0$), spacelike (), and lightlike ($s^2=0$). Timelike separations can connect events by a slower-than-light causal influence, whereas spacelike separations cannot. Although the displayed result reports transformed coordinates rather than an interval check, you can calculate $s^2$ from the input and output coordinates to verify the invariance.

How to use: Transforming an Event Along the x-Axis

To transform an event with this Lorentz calculator, enter the relative velocity and the event coordinates measured in frame S. The calculator uses $c=\mathrm{299,792,458}$ m/s, the exact SI value of the speed of light. After pressing Transform, it calculates the Lorentz factor and displays $(t^{\prime },x^{\prime },y^{\prime },z^{\prime })$. Enter time in seconds, positions in metres, and velocity in metres per second. The velocity magnitude must remain below $c$. This implementation handles only relative motion in the x-direction, so the y and z coordinates pass through unchanged.

Using the Lorentz transformation is particularly helpful for following the same event from different inertial viewpoints. For example, a muon moving through Earth’s atmosphere at a speed close to $0.98c$ has a Lorentz factor of approximately $\gamma \approx 5$. In an Earth-based frame, its moving clock accumulates less proper time than Earth-coordinate time; in the muon’s rest frame, distances parallel to its motion are contracted. The calculator does not model particle decay or atmospheric travel directly, but it does provide the event-coordinate conversion behind comparisons such as these.

Table of Lorentz Factors for Relativistic Speeds

This Lorentz-factor table shows how the calculator’s velocity input changes $\gamma$; the growth becomes especially sharp as the relative speed nears light speed.

For an x-axis Lorentz transformation at $0.1c$, the deviation from Newtonian coordinate conversion is small because $\gamma$ is close to one. At $0.9c$, a moving clock advances at less than half the rate assigned by the other frame, and parallel lengths contract accordingly. As $v$ approaches $c$, $\gamma$ rises without bound. This behavior is why no finite boost can carry a massive object to light speed and why relativistic coordinate transformations are essential in high-energy physics.

Beyond One Dimension in Lorentz Coordinate Transformations

This calculator applies the simplest Lorentz boost, where relative motion is confined to one spatial axis. In the full three-dimensional treatment, Lorentz transformations can be written as matrices for boosts in arbitrary directions and for rotations of spatial coordinates. In four-dimensional spacetime, a boost acts like a hyperbolic rotation: it mixes time with the coordinate parallel to the relative velocity while preserving the interval. Rapidity, relativistic velocity addition, and Minkowski diagrams extend the same ideas to broader special-relativity problems.

Experimenting with the entered velocity and event coordinates builds intuition for what this specific x-axis transformation changes. Increase the magnitude of v while holding an event fixed to see the growing difference between t and t’, or vary x to see its contribution to the transformed time. Changing y or z alone will not alter the reported transformed time or x-coordinate, because the calculator’s boost is parallel only to x. These comparisons make the coupling of space and time in special relativity more concrete without treating the output as a model of acceleration, gravity, or non-inertial motion.

Worked example: checking how x affects a Lorentz-transformed event

For a meaningful Lorentz transformation comparison, choose a subluminal velocity and hold the event time, y-coordinate, and z-coordinate fixed. First transform an event at one x-coordinate, then change only x and transform again. The time result changes through the $vx/c^2$ term, while the x result changes through $x-vt$. This isolates the relativity-of-simultaneity contribution instead of combining unrelated input units.

Limitations and assumptions of this x-Axis Lorentz Transformation

This calculator assumes two inertial frames with frame S’ moving at a constant velocity parallel to the x-axis of frame S. It applies the special-relativity equations shown above, not a treatment of acceleration, gravity, curved spacetime, or a boost in an arbitrary direction. Reliable coordinate results require seconds, metres, and metres per second to be entered consistently, and the relative velocity must have a magnitude below the speed of light. For a physical experiment or a system involving non-inertial motion, use the relevant measured data and a model appropriate to that situation.