Relativistic Energy Calculations: SI vs Natural Units¶

Overview¶

This example demonstrates the relativistic energy-momentum relation \(E^2 = (pc)^2 + (mc^2)^2\) using both SI units (with explicit c factors) and natural units (where c = 1). It showcases how mp-units supports multiple systems of quantities and how natural units simplify complex relativistic formulas while maintaining dimensional correctness.

Key Concepts¶

The Energy-Momentum Relation¶

In special relativity, the total energy of a particle is related to its momentum and rest mass through:

SI units (explicit constants):

\[E^2 = (pc)^2 + (mc^2)^2\]

\[E = \sqrt{(pc)^2 + (mc^2)^2}\]

Natural units (c = ℏ = 1):

\[E^2 = p^2 + m^2\]

\[E = \sqrt{p^2 + m^2}\]

Dual Implementation Strategy¶

The example provides two implementations of the same physics:

SI units version - requires explicit speed of light:

QuantityOf<isq::mechanical_energy> auto total_energy(QuantityOf<isq::momentum> auto p, QuantityOf<isq::mass> auto m,
                                                     QuantityOf<isq::speed> auto c)
{
  return isq::mechanical_energy(hypot(p * c, m * pow<2>(c)));
}

Natural units version - c = 1 by definition, no parameter needed:

// In natural units (ℏ = c = 1), the energy-momentum relation simplifies to E² = p² + m²
QuantityOf<natural::energy> auto total_energy(QuantityOf<natural::momentum> auto p, QuantityOf<natural::mass> auto m)
{
  return natural::energy(hypot(p, m));
}

Notice how the natural units version is simpler: no c parameter, and the formula directly matches the mathematical notation used by physicists.

SI Units: Multiple Representations¶

The SI example demonstrates the same calculation in three different unit representations:

Using GeV and c:

  const quantity p1 = isq::momentum(4. * GeV / c);
  const QuantityOf<isq::mass> auto m1 = 3. * GeV / c2;
  const quantity E = total_energy(p1, m1, c);

  std::cout << "\n*** SI units (c = " << c << " = " << c.in(si::metre / s) << ") ***\n";

  std::cout << "\n[in `GeV` and `c`]\n"
            << "p = " << p1 << "\n"
            << "m = " << m1 << "\n"
            << "E = " << E << "\n";

Converting to pure GeV:

  const quantity p2 = p1.in(GeV / (m / s));
  const quantity m2 = m1.in(GeV / pow<2>(m / s));
  const quantity E2 = total_energy(p2, m2, c).in(GeV);

  std::cout << "\n[in `GeV`]\n"
            << "p = " << p2 << "\n"
            << "m = " << m2 << "\n"
            << "E = " << E2 << "\n";

Using SI base units (kg, m, s):

  const quantity p3 = p1.in(kg * m / s);
  const quantity m3 = m1.in(kg);
  const quantity E3 = total_energy(p3, m3, c).in(J);

  std::cout << "\n[in SI base units]\n"
            << "p = " << p3 << "\n"
            << "m = " << m3 << "\n"
            << "E = " << E3 << "\n";

  std::cout << "\n[converted from SI units back to GeV]\n"
            << "E = " << E3.force_in(GeV) << "\n";

Each representation is physically equivalent, but the choice affects numerical values and readability.

Sample Output:

*** SI units (c = 1 c = 2.99792e+08 m/s) ***

[in `GeV` and `c`]
p = 4 GeV/c
m = 3 GeV/c²
E = 5 GeV

[in `GeV`]
p = 1.33426e-08 GeV s/m
m = 3.33795e-17 GeV s²/m²
E = 5 GeV

[in SI base units]
p = 2.13771e-18 kg m/s
m = 5.34799e-27 kg
E = 8.01088e-10 J

[converted from SI units back to GeV]
E = 5 GeV

Natural Units: Simplified Physics¶

The natural units example shows the elegance of setting c = 1:

  const auto p = momentum(4. * GeV);  // momentum
  const auto m = mass(3. * GeV);      // mass (rest energy: E = m when c = 1)
  const auto E = total_energy(p, m);

  std::cout << "\n*** Natural units (c = 1) ***\n"
            << "p = " << p << "\n"
            << "m = " << m << "\n"
            << "E = " << E << "\n";

Sample Output:

*** Natural units (c = 1) ***
p = 4 GeV
m = 3 GeV
E = 5 GeV

Notice:

Momentum, mass, and energy all have the same dimension (energy) in natural units
No conversion factors needed between quantities
The Pythagorean relation \(E^2 = p^2 + m^2\) is immediately visible
The formula matches exactly what appears in physics textbooks

Why This Matters¶

SI Units Advantages¶

Explicit Constants: All physical constants are visible in formulas
Intuitive Scaling: Familiar units (meters, seconds, kilograms)
Experimental Context: Direct connection to laboratory measurements
Engineering Applications: Practical for real-world calculations

Natural Units Advantages¶

Simplified Formulas: Match theoretical physics notation exactly
Dimensional Insights: Relationships like "mass IS energy" become explicit
Particle Physics: Standard in high-energy physics research
Reduced Clutter: Eliminates repetitive factors of c and ℏ

Library Support for Both¶

mp-units uniquely supports both paradigms:

SI uses ISQ quantity types with explicit dimensional relationships
Natural units use a separate quantity system based on energy dimensions
The two systems are incompatible - conversions require extracting numerical values and manually applying conversion factors (no automatic type-safe conversion)
Both provide compile-time dimensional checking within their respective systems

Practical Applications¶

This pattern is essential for:

Particle Physics: Calculating particle energies, decay rates, cross-sections
Relativistic Mechanics: Any application where \(v \approx c\)
Quantum Field Theory: Natural units are the standard
Educational Code: Teaching relativity with simplified formulas
Cross-Domain Work: Working with both experimental (SI) and theoretical (natural) contexts