Rapidity Calculator

Rapidity Calculator | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Science & Education
Input types: select, number
Output type: text
Sample coverage: 4
API ready: Yes

Overview

The Rapidity Calculator computes the relativistic rapidity φ = artanh(β) from velocity, transforming non-linear Lorentz velocity additions into simple linear additions. It instantly outputs key relativistic parameters including the Lorentz factor γ = cosh(φ), proper velocity component γβ = sinh(φ), the longitudinal Doppler factor e^φ, and the doubled boost composition tanh(2φ).

When to use

When composing successive collinear relativistic boosts where rapidities add linearly instead of using Einstein's velocity addition formula.
When calculating relativistic Doppler factors (redshift/blueshift multipliers) or Lorentz gamma values directly from a particle's speed.
When analyzing high-energy particle kinematics and accelerator beam coordinates where speeds approach the speed of light.

How it works

1Select the input format: either dimensionless fraction of the speed of light (β = v/c) or speed in meters per second (m/s).
2Enter the velocity value into the corresponding field, such as a fraction of c (e.g., 0.5) or physical speed.
3The tool evaluates the hyperbolic angle φ = artanh(β) using the natural logarithm form ½·ln((1+β)/(1−β)).
4It displays the computed rapidity φ, γ = cosh(φ), γβ = sinh(φ), Doppler factor e^φ, and the composite velocity tanh(2φ).

Use cases

High-energy particle physics calculations for composing consecutive particle collisions and beam boosts.
Special relativity coursework and verification of relativistic velocity addition and hyperbolic kinematics.
Relativistic astrophysics problems involving longitudinal Doppler shifts and relativistic jets.

Examples

1. Subluminal Boost Composition at β = 0.5

Physics Student

Background

A student is solving an introductory special relativity assignment on composing two collinear reference frames moving at 0.5c.

Problem

Verify that two successive 0.5c boosts equal 0.8c using hyperbolic rapidity instead of manual algebraic substitution in Einstein's velocity addition formula.

How to use

Set 'Speed input' to 'Fraction of c (β)' and enter 0.5 in the β field.

Outcome

Yields φ = 0.549306, γ = 1.154701, Doppler factor e^φ = 1.732051, and confirms tanh(2φ) = 0.800000.

2. Cosmic Ray Muon Kinematics at β = 0.998

Astrophysics Researcher

Background

A researcher is modeling relativistic muons traveling through Earth's atmosphere at 0.998c.

Problem

Determine the Lorentz factor and longitudinal Doppler shift multiplier for a particle moving extremely close to c.

How to use

Select 'Fraction of c (β)' in 'Speed input' and enter 0.998 into the β input field.

Outcome

Computes φ = 3.453377, γ = 15.819300, and Doppler factor e^φ = 31.606961.

FAQ

What is rapidity in special relativity?

Rapidity φ is a hyperbolic angle defined as φ = artanh(v/c). Unlike velocities, collinear rapidities add linearly when chaining Lorentz boosts.

How does rapidity relate to the Lorentz factor γ?

The Lorentz factor γ is given by the hyperbolic cosine of rapidity: γ = cosh(φ), while γβ corresponds to sinh(φ).

What is the Doppler factor e^φ?

The term e^φ represents the relativistic longitudinal Doppler factor, indicating the exact frequency shift ratio for an approaching or receding source.

Can rapidity exceed 1?

Yes. While β is strictly bounded between -1 and 1 (subluminal speeds), rapidity φ ranges from -∞ to +∞ as β approaches ±1.

What does tanh(2φ) represent in the output?

tanh(2φ) represents the combined velocity (as a fraction of c) resulting from two consecutive reference frame boosts of the same rapidity φ.

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Subluminal Boost Composition at β = 0.5

2. Cosmic Ray Muon Kinematics at β = 0.998

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