Description
Signed +--- four-momentum norm residual in consistent caller-supplied units.
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.
Signed +--- four-momentum norm residual in consistent caller-supplied units.
When to use this tool
Use Relativistic Four Momentum Norm Residual 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 energy, momentum, mass, speedOfLight. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read invariant, residual. Use the output convention below.
For a nonempty momentum vector, nonnegative mass, and positive speedOfLight return invariant=(energy/speedOfLight)^2-dot(momentum,momentum) and residual=invariant-(mass*speedOfLight)^2.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| energy | Energy | Required. Finite number. |
| momentum | Momentum | Required. Array of finite number; maximum items 256. |
| mass | Mass in a consistent unit for the stated model. | Required. Finite number; minimum 0. |
| speedOfLight | Positive speed of light in the unit system used for energy and momentum. | Required. Finite number; strictly above 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| invariant | Invariant | Required. Finite number. |
| residual | Signed observed-minus-implied discrepancy under the stated formula. | Required. Finite number. |
{
"input": {
"energy": 10,
"momentum": [
3,
0,
0
],
"mass": 4,
"speedOfLight": 2
},
"output": {
"invariant": 16,
"residual": -48
}
}Limitations and assumptions
- Use compatible energy, momentum, mass, and speed units. The vector need not have three components and energy may be signed; a residual is not an automatic mass-shell certification or a tolerance-based pass flag.
- 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
- Momentum Energy Classical Consistency Residual
Compare supplied translational kinetic energy with the classical momentum relation p²/(2m).
- Relativistic Rapidity Velocity Addend
For positive lightSpeed and abs(velocity)<lightSpeed return combinedRapidity=rapidity+atanh(velocity/lightSpeed) and combinedVelocity=lightSpeed*tanh(combinedRapidity).