AcademyRelativity
Academy
Newtonian Limits
Level 1 - Physics topic page in Relativity.
Principle
Relativistic equations reduce to Newtonian equations when all speeds are much smaller than \(c\).
Notation
\(\beta\)
speed fraction \(v/c\)
\(\gamma\)
Lorentz factor
\(p\)
momentum
\(\mathrm{kg\,m\,s^{-1}}\)
\(K\)
kinetic energy
\(\mathrm{J}\)
\(x,t\)
space and time coordinates
\(\mathrm{m,\;s}\)
Method
Derivation 1: Expand the Lorentz factor
For \(\\beta\\ll1\), the binomial approximation keeps only the leading correction.
Lorentz factor
\[\gamma=(1-\beta^2)^{-1/2}\]
Low-speed expansion
\[\gamma\approx1+\frac12\beta^2\]
Derivation 2: Recover Newtonian momentum
At low speed, \(\gamma\) is very close to one.
Relativistic momentum
\[p=\gamma mv\]
Newtonian limit
\[p\approx mv\]
Derivation 3: Recover Newtonian kinetic energy
Use the low-speed expansion of \(\gamma\) in \(K=(\gamma-1)mc^2\).
Relativistic kinetic energy
\[K=(\gamma-1)mc^2\]
Low-speed kinetic energy
\[K\approx\frac12\frac{v^2}{c^2}mc^2=\frac12mv^2\]
Rules
Gamma approximation
\[\gamma\approx1+\frac12\beta^2\]
Momentum limit
\[p\approx mv\]
Kinetic-energy limit
\[K\approx\frac12mv^2\]
Galilean position
\[x'\approx x-vt,\qquad t'\approx t\]
Examples
Question
Estimate
\[\gamma-1\]
for a car moving at \[30\,\mathrm{m\,s^{-1}}\]
Answer
\[\beta=\frac{30}{3.00\times10^8}=1.0\times10^{-7}\]
\[\gamma-1\approx\frac12\beta^2=5.0\times10^{-15}\]
Checks
- Use Newtonian formulas only when \(v/c\) is small enough for the required precision.
- The first relativistic correction usually scales as \(v^2/c^2\).
- Low-speed agreement is a limit, not a separate assumption.
- Rest energy has no Newtonian counterpart in ordinary mechanics.