Physics Formulae/Special Relativity Formulae

Lead Article: Tables of Physics Formulae

This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Special Relativity. There are essentially two formulations in relativity, using conventional 3-vectors with addtional time components, or the more unified formalism of four-vectors.

3-Vector Formulation

3-Vectors are conventional vectors, of dimension 3. The components are spatial corresponding to the coordinate system used. It matters not which coordinate system is used, events in space-time can only be independant of coordinates, since coordinates are simply labels to locate a point.

Einstien's Assumptions of Special Relativity

For all inertial frames of referance the following are invariant:

- Physical Laws

- Luminal Speed

Note that the second two are corolaries of the first. Since some physical laws include properties of light, often relating to the vacuum luminal speed, if the laws are to be the same for all observers in inertial referance frames, the speed of light in a vacuum must be constant.

Inertial Frames and Lorentz Transforms

Two important and very useful simplifying parameters are the lorentz factor and rapidity.

 Lorentz Factor ${\displaystyle \gamma ={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\,\!}$ Rapidity ${\displaystyle \eta \,\!}$ ${\displaystyle \tanh \eta ={\frac {v}{c}}\,\!}$

Lorentz transforms are linear transformations, relating the coordinates of an event in one frame F to another F' , where

F' moves at constant velocity v relative to F, equivalant to

F moving at constant velocity (-v) relative to F' .

The observers in each frame measure different coordinates for the same event. By this argument the inverse transforms can be deduced simply by:

- interchanging the frame coordinates

- changing the sign

 Lorentz Transformation for Space-Time ${\displaystyle t'=\gamma (t-xv/c^{2})\,\!}$  ${\displaystyle x'=\gamma (x-vt)\,\!}$  ${\displaystyle y'=y\,\!}$  ${\displaystyle z'=z\,\!}$ Lorentz Transformation for Momentum-Energy ${\displaystyle E'=\gamma \left(E-p_{x}\right)\,\!}$  ${\displaystyle p_{x}'=\gamma \left(p_{x}-{\frac {vE}{c^{2}}}\right)\,\!}$  ${\displaystyle p_{y}'=p_{y}\,\!}$  ${\displaystyle p_{z}'=p_{z}\,\!}$
 Relativistic Doppler Effect ${\displaystyle \lambda =\lambda _{0}{\sqrt {\frac {c-v}{c+v}}}\,\!}$ Doppler Shift ${\displaystyle v=|\Delta \lambda |c/\lambda _{0}\,\!}$

Often a common misonception is to obtain the incorrect sign for the relative velocity v.

Space-Time

 Proper Time ${\displaystyle \tau \,\!}$ Proper Length ${\displaystyle \chi \,\!}$ Space-Time Invariant ${\displaystyle {\frac {\chi ^{2}}{c^{2}}}=\tau ^{2}=t^{2}-{\frac {x^{2}+y^{2}+z^{2}}{c^{2}}}\,\!}$  ${\displaystyle \mathbf {r} \cdot \mathbf {r} =r^{2}=x^{2}+y^{2}+z^{2}\,\!}$ Time Dilation ${\displaystyle \Delta t=\gamma \Delta t_{0}\,\!}$ Length Contraction ${\displaystyle \Delta x={\frac {\Delta x_{0}}{\gamma }}\,\!}$

Energy-Momentum

 Rest Mass-Energy ${\displaystyle E_{0}=m_{0}c^{2}\,\!}$ Momentum-Energy Invariant ${\displaystyle E^{2}=(pc)^{2}+(m_{0}c^{2})^{2}\,\!}$  ${\displaystyle p^{2}=\mathbf {p} \cdot \mathbf {p} =p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\,\!}$ Relativitic Mass ${\displaystyle m=\gamma m_{0}\,\!}$ Momentum ${\displaystyle \mathbf {p} =\gamma m_{0}\mathbf {v} \,\!}$ Kinetic Energy ${\displaystyle T=\left(\gamma -1\right)m_{0}c^{2}\,\!}$ Total Energy ${\displaystyle E=\gamma m_{0}c^{2}\,\!}$

4-Vector Formulation

4-Vector Nomenclature Summary

 General 4-Vectors ${\displaystyle \mathbf {A} ={\begin{pmatrix}\ A_{0}\\\ A_{1}\\\ A_{2}\\\ A_{3}\end{pmatrix}}\,\!}$ General 4-Vector Lorentz Transform Transition matrix of coordinates between a frame F and another frame F' moving at velocity v in common x-x' direction relative to F. The inverse matrix is the inverse transform, from F' to F, F moving in common (-x)(-x') direction relative to F'. ${\displaystyle \mathbf {T} _{L}={\begin{pmatrix}\ \gamma &-\gamma v/c&0&0\\\ -\gamma v/c&\gamma &0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}}\,\!}$  ${\displaystyle \mathbf {T} _{L}^{-1}={\begin{pmatrix}\ \gamma &\gamma v/c&0&0\\\ \gamma v/c&\gamma &0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}}\,\!}$  Note that ${\displaystyle \mathbf {T} _{L}=\mathbf {T} _{L}^{T}\,\!}$  (symmetric matrix) ${\displaystyle \det(\mathbf {T} _{L})=1\,\!}$  (non-singular/invertible) as should be for Lorentz Invariance. Space-Time 4-Vector ${\displaystyle \mathbf {R} ={\begin{pmatrix}\ R_{0}\\\ R_{1}\\\ R_{2}\\\ R_{3}\end{pmatrix}}={\begin{pmatrix}\ ct\\\ x\\\ y\\\ z\end{pmatrix}}\,\!}$ Momentum-Energy 4-Vector ${\displaystyle \mathbf {P} ={\begin{pmatrix}\ P_{0}\\\ P_{1}\\\ P_{2}\\\ P_{3}\end{pmatrix}}={\begin{pmatrix}\ E/c\\\ p_{x}\\\ p_{y}\\\ p_{z}\end{pmatrix}}\,\!}$

Lorentz Transform

 Space-Time ${\displaystyle {\begin{pmatrix}\ ct'\\\ x'\\\ y'\\\ z'\end{pmatrix}}={\begin{pmatrix}\ \gamma &-\gamma v/c&0&0\\\ -\gamma v/c&\gamma &0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}}{\begin{pmatrix}\ ct\\\ x\\\ y\\\ z\end{pmatrix}}\,\!}$  ${\displaystyle \mathbf {R} '=\mathbf {T} _{L}\mathbf {R} \,\!}$ Energy-Momentum ${\displaystyle {\begin{pmatrix}\ E'/c\\\ p_{x}'\\\ p_{y}'\\\ p_{z}'\end{pmatrix}}={\begin{pmatrix}\ \gamma &-\gamma v/c&0&0\\\ -\gamma v/c&\gamma &0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}}{\begin{pmatrix}\ E/c\\\ p_{x}\\\ p_{y}\\\ p_{z}\end{pmatrix}}\,\!}$  ${\displaystyle \mathbf {P} '=\mathbf {T} _{L}\mathbf {P} \,\!}$