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Relativistic energy and momentum

At speeds near light speed, the low-speed formulas p = mv and kinetic energy ½mv² no longer describe the same physical state accurately. Special relativity packages energy and momentum into one relationship with a powerful invariant check.

Think like a programmer

Represent a particle as { restMass, speed, gamma, energy, momentum }, with SI units attached in the surrounding API. Compute all derived fields from one source of truth, reject speeds at or above c, and test the invariant rather than trusting separately rounded outputs.

Model checklist

Inputs
Rest mass m in kg, signed speed v in m/s, and exact vacuum light speed c.
State
Lorentz factor γ, total energy E in J, and one-dimensional momentum p in kg·m/s.
Rule
Compute γ from speed, then compute E and p from the same γ.
Output
Energy, momentum, and invariant mass diagnostic.
Check
E² − (pc)² equals (mc²)² within numerical tolerance; γ at 0.6c is 1.25.

The shared factor is

\[\gamma=\frac{1}{\sqrt{1-v^2/c^2}}\]

For a free particle,

\[E=\gamma mc^2,\qquad p=\gamma mv,\qquad E^2=(pc)^2+(mc^2)^2\]

The final relationship is more than an equation to display. It is a test oracle: calculate E² - (p*c)² and compare it with (m*c²)². At v = 0.6c, γ = 1.25. A mass of 2 kg must produce the same invariant whether you calculate it through energy or momentum.

function lorentzFactor(speed: number) {
  if (Math.abs(speed) >= SPEED_OF_LIGHT) throw new RangeError("speed must stay below light speed");
  const beta = speed / SPEED_OF_LIGHT;
  return 1 / Math.sqrt(1 - beta ** 2);
}

Keeping the sign on v retains momentum direction, while energy remains non-negative for positive rest mass. Do not compute γ from a rounded percentage string or use a classical momentum in the invariant check: either shortcut can create a convincing but inconsistent result.

Try this experiment

Prediction: The Lorentz factor grows slowly at low speed and sharply as speed approaches c; no finite rest mass reaches c in this model.

Calculate γ at 0, 0.6c, and 0.9c. Before evaluating, predict which change is larger. Then use the invariant relation to explain why a velocity input equal to c should be rejected rather than clamped.

Where this model breaks

This is special relativity for a single free particle in an inertial frame. It does not describe gravity, accelerating coordinates, composite systems' binding energy, quantum fields, radiation reaction, or collisions without conservation equations for every participant. Floating-point arithmetic also loses relative precision when subtracting the two large squared terms in the invariant at extreme γ.

Summary

Relativistic energy and momentum are one coupled state calculation. Share the Lorentz factor, preserve momentum direction, enforce the speed domain, and validate the result with the energy–momentum invariant.

Glossary

Self-check

  1. Why should energy and momentum share one calculated γ?
  2. What exact value should γ have at 0.6c?
  3. Which quantity is a strong test oracle for the computed state?

Sources

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]
Inputs
Quantities you set or measure, with units and useful bounds.
State
Values the program must retain to reproduce the next result.
Rule
The relationship or update that turns inputs and state into a result.
Check
A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Relativistic Energy and Momentum, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Relativity, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Relativistic Energy and Momentum as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

  • Input: a measured value or chosen parameter supplied to a model.
  • State: the smallest set of values needed to continue or reproduce a model.
  • Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

  1. Which values are inputs, and which values must remain state?
  2. What observable result would tell you the model is behaving as expected?
  3. Which assumption would you test first before applying the model to a real system?

Model review: turn Relativistic Energy and Momentum into a test

Compute relativistic state with four-momentum and invariant energy–momentum consistency checks.

  1. Name the inputs and units that the relativity model needs.
  2. Separate the state you must keep from values you can calculate when needed.
  3. Write one rule that maps the current state and inputs to an observable result.
  4. Choose a limiting case, unit check, invariant, or known result before trusting an output.
  5. State one assumption you would change before using this simplified model for a real decision.
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