Description
For positive lightSpeed and abs(velocity)<lightSpeed return combinedRapidity=rapidity+atanh(velocity/lightSpeed) and combinedVelocity=lightSpeed*tanh(combinedRapidity).
Relativistic quantities depend on the frame, units, and approximation being used. Rapidity supports collinear velocity composition, four-momentum has a signed invariant, and weak-field frequency shifts are approximations. A numerical term should be interpreted within its own model rather than combined indiscriminately with classical formulas.
For positive lightSpeed and abs(velocity)<lightSpeed return combinedRapidity=rapidity+atanh(velocity/lightSpeed) and combinedVelocity=lightSpeed*tanh(combinedRapidity).
When to use this tool
Use Relativistic Rapidity Velocity Addend when you need this collinear composition, invariant residual, or weak-field approximation with compatible physical units.
How to use the tool and how it works
- Enter rapidity, velocity, lightSpeed. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read combinedVelocity, combinedRapidity. Use the output convention below.
For positive lightSpeed and abs(velocity)<lightSpeed return combinedRapidity=rapidity+atanh(velocity/lightSpeed) and combinedVelocity=lightSpeed*tanh(combinedRapidity).1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| rapidity | Dimensionless collinear rapidity. | Required. Finite number. |
| velocity | Velocity | Required. Finite number. |
| lightSpeed | Positive propagation speed in compatible length per time units. | Required. Finite number; strictly above 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| combinedVelocity | Signed collinear velocity in the same unit as lightSpeed. | Required. Finite number. |
| combinedRapidity | Combined Rapidity | Required. Finite number. |
{
"input": {
"rapidity": 0.5493061443340548,
"velocity": 0,
"lightSpeed": 10
},
"output": {
"combinedVelocity": 4.999999999999999,
"combinedRapidity": 0.5493061443340548
}
}Limitations and assumptions
- This is collinear rapidity addition. Rapidity is dimensionless; floating-point saturation can return exactly the supplied light speed for extreme rapidities, and multidimensional relativistic velocity addition is not performed.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.
Alternative or Complementary methods
Use a complete frame transformation or spacetime model when collinear or weak-field assumptions do not apply. Check invariant units and approximation size before interpreting a residual or shift.
References
Similar or alternative tools
- Relativistic Four Momentum Norm Residual
Signed +--- four-momentum norm residual in consistent caller-supplied units.
- Gravitational Redshift Weak Field Term
Return only the weak-field received-minus-emitted frequency fraction -deltaPhi/c²; validity requires a small potential ratio.