AcademyRelativity
Academy
Invariance of Physical Laws
Level 1 - Physics topic page in Relativity.
Principle
The laws of physics and the vacuum speed of light are the same in every inertial frame.
Notation
\(S, S'\)
inertial reference frames
\(c\)
speed of light in vacuum
\(\mathrm{m\,s^{-1}}\)
\(v\)
relative speed of two inertial frames
\(\mathrm{m\,s^{-1}}\)
\(\beta\)
speed as a fraction of light speed, \(v/c\)
\(\gamma\)
Lorentz factor
\((x,t)\)
position and time coordinates of an event
\(\mathrm{m,\;s}\)
Method
Derivation 1: Identify an inertial frame
An inertial frame is one in which a free particle moves with constant velocity. Special relativity compares such frames.
Relative speed
\[\beta=\frac{v}{c}\]
Lorentz factor
\[\gamma=\frac{1}{\sqrt{1-\beta^2}}\]
Derivation 2: Keep light speed invariant
A light pulse emitted from the shared origin must satisfy the same light-cone equation in every inertial frame.
Light in S
\[x^2+y^2+z^2=c^2t^2\]
Light in S'
\[x'^2+y'^2+z'^2=c^2t'^2\]
Derivation 3: Replace Galilean time
If \(t'=t\), a moving observer would not measure the same light speed. Space and time coordinates must mix when changing inertial frames.
Rules
Light speed
\[c=3.00\times10^8\,\mathrm{m\,s^{-1}}\]
Speed fraction
\[\beta=\frac{v}{c}\]
Lorentz factor
\[\gamma=\frac{1}{\sqrt{1-\beta^2}}\]
Light cone
\[x^2+y^2+z^2=c^2t^2\]
Examples
Question
A spacecraft moves at
\[0.600c\]
Find \(\beta\) and \(\gamma\).Answer
\[\beta=0.600\]
\[\gamma=\frac{1}{\sqrt{1-0.600^2}}=1.25\]
Checks
- Special relativity applies to inertial frames unless a problem states otherwise.
- No massive object has \(v\\ge c\).
- \(\\gamma\) is always at least \(1\).
- At low speed, \(\beta\\ll1\) and relativity approaches Newtonian mechanics.