by Michael Brunnbauer, 2021-11-04

Check out Spacetime Physics, Second Edition by Edwin F. Taylor and John Archibald Wheeler for a deep and simple overview of the principles of relativity.

Quantity | Old unit | New unit | Conversion |
---|---|---|---|

Time | \( s \) | \( m \) | \( t = t_{old} \cdot c \) |

Velocity | \( \frac{m}{s} \) | unitless | \( v = \frac{v_{old}}{c} \) |

Momentum | \( \frac{kg \cdot m}{s} \) | \( kg \) | \( p = \frac{p_{old}}{c} \) |

Energy | \( \frac{kg \cdot m^2}{s^2} \) | \( kg \) | \( E = \frac{E_{old}}{c^2} \) |

\( \tau \) is the proper time (also called wristwatch time) between the two events - as measured by a clock that is present at both events and travels uniformly between them (\(\Delta x = 0 \)).

\( \sigma \) is the ruler distance (also called proper distance) between the two events - as measured by an observer in whose frame the two events occur at the same time (\(\Delta t = 0 \)).

All inertial observers agree on whether two events have timelike, lightlike or spacelike separation and on the amount of their separation in spacetime (proper time, 0 or ruler distance).

This chart illustrates how events in spacetime are affected by the Lorentz transformation. The blue lines are the trajectories of light:

Drag the slider for v to see how

- events move along the hyperbolas defined by the spacetime metric
- every event stays timelike, lightlike or spacelike regardless of speed
- horizontal lines connect different simultaneous events at different speeds (relativity of simultaneity)
- these events had a greater spacial separation at \( v=0 \) (length contraction)
- events connected by vertical lines at \( v=0 \) have greater time separation at other speeds (time dilation)

Let's consider a mass of 1kg moving from event \( (t_1,x_1) \) to event \( (t_3,x_3) \) passing intermediary event \( (t_2,x_2) \). The velocity between each pair of events is constant:

The bar on the right is the action - calculated as momentum times distance added for the two constant velocity stages of the trip. You can drag around the intermediate event to see how the total action for the trip changes. It is easy to see that the action is at a minimum when the mass travels on a straight line between the two events. Uninfluenced objects travel at constant velocity.

The action is calculated as: $$ S = m \cdot \frac{x_2-x_1}{t_2-t_1} \cdot (x_2-x_1) + m \cdot \frac{x_3-x_2}{t_3-t_2} \cdot (x_3-x_2) $$ The trajectory is a stationary point with regard to \( t_2 \) and \( x_2 \): $$ \frac{dS}{dt_2} = 0 \quad \text{(Time translation symmetry)} $$ $$ \frac{dS}{dx_2} = 0 \quad \text{(Translational symmetry)} $$ According to Noether's theorem, two conserved quantities should arise. The quantity associated with time translation symmetry is energy and the quantity associated with translational symmetry is momentum: $$ \frac{dS}{dt_2} = 0 = - m \cdot \frac{(x_2-x_1)^2}{(t_2-t_1)^2} + m \cdot \frac{(x_3-x_2)^2}{(t_3-t_2)^2} $$ $$ m \cdot v_1^2 = m \cdot v_2^2 $$ $$ \frac{dS}{dx_2} = 0 = 2 \cdot m \cdot \frac{x_2-x_1}{t_2-t_1} - 2 \cdot m \cdot \frac{x_3-x_2}{t_3-t_2} $$ $$ m \cdot v_1 = m \cdot v_2 $$

Due to the euclidean nature of the drawing, the length of the vector does not correctly represent its magnitude \( m \) for \( v \neq 0 \). The blue lines represent possible values for the momentum-energy of light ( \( E = p, m = 0 \) ).

- \( m_1, m_2 \) mass of the objects
- \( u_1, u_2 \) velocity of the objects
- \( E = E_1 + E_2 \) energy of the objects
- \( p = p_1 + p_2 \) momentum of the objects
- \( v_c \) velocity of the "center of momentum" inertial frame
- \( u_1', u_2' \) velocity of the objects in the "center of momentum" frame

- \( v_1' = -u_1', v_2' = -u_2' \) velocity of the objects after collision in the "center of momentum" frame
- \( v_1, v_2 \) velocity of the objects after collision