Minkowski Space

Invariant in Lorentz Transformation

From Lorentz transformation, we know that the coordinates in changing between inertial frames are given by \begin{cases} x' = \gamma(x - vt) \\[2ex] y' = y \\[2ex] z' = z\\[2ex] t' = \gamma(t - \frac{vx}{c^2})\\[2ex] \end{cases} A small change in coordinates are given by \begin{cases} dx' = \gamma(dx - v\,dt) \\[2ex] dy' = dy \\[2ex] dz' = dz\\[2ex] dt' = \gamma(dt - \frac{v\,dx}{c^2})\\[2ex] \end{cases} The value \begin{align} &\,\,\,\,-(cd t)^2 + (d x)^2 + (d y)^2 + (d z)^2 \\ & = - \gamma^2(cd t - \frac{v}{c}d x)^2 + \gamma^2(d x - vd t)^2 + (d y')^2 + (d z')^2 \\ &= \gamma^2\left[ (d x)^2 - 2vd x d t + (vd t)^2 - (cd t)^2 + 2vd x d t - \frac{v^2}{c^2}(d x)^2 \right] + (d y)^2 + (d z)^2 \\ &= \frac{1}{1-\frac{v^2}{c^2}}\left[(dx)^2\left(1-\frac{v^2}{c^2}\right) - (c\,dt)^2\left(1- \frac{v^2}{c^2} \right) \right]+ (d y)^2 + (d z)^2 \\ &= -(c\,d t')^2 + (d x')^2 + (d y')^2 + (d z')^2 \end{align} is invariant under Lorentz Transformation.

Minkowski Space

Minkowski space is defined as a space with the metric tensor $$ds^2 = -cdt^2 + dx^2 + dy^2 + dz^2$$ Unlike the Euclidean space with metric tensor $$ds^2 = dx^2 + dy^2 + dz^2$$ Minkowski space was defined so as the quantity calculated by this metric tensor is invariant under Lorentz Transformation as shown above. The set of coordinate \((ct, x, y, z)\) is called an event.

Proper Length

Proper length is defined as the Euclidean distance between two events at the same time in an inertial frame. If the distance measured at time \(t\) in an inertial frame, {\(S\)}, is $$\Delta l = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$$ Then, in another inertial frame, {\(S'\)} moving at \(v\) relative to {\(S\)} is \begin{align} \Delta l' &= \sqrt{\Delta x'^2 + \Delta y'^2 + \Delta z'^2} \\ &= \sqrt{(\gamma(\Delta x - v\Delta t))^2 + \Delta y^2 + \Delta z^2} \\ &= \sqrt{(\gamma\Delta x)^2 + \Delta y^2 + \Delta z^2} \\ \end{align} as \(\Delta t = 0\) is {\(S\)}. If there is no \(y\) or \(z\) components, we have the length contraction expression as shown before. Note that \(\Delta t'\) is not 0 when \(\Delta x\) is non-zero. Proper length is always longer than the length measured in other inertial frame.

Proper Time

Proper time is defined as the difference of time between two events at the same position in an inertial frame. If the duration measured at postion \((x,y,z)\) in an inertial frame, {\(S\)}, is $$\Delta \tau = \int \frac{ds}{c} $$ Then, in another inertial frame, {\(S'\)} moving at \(v\) relative to {\(S\)} is \begin{align} d t' &= \gamma(d\tau - \frac{v\,dx}{c^2}) \\ &= \gamma(d\tau) \\ \end{align} or $$\tau = \int^{t_B}_{t_A}\sqrt{1-\frac{v(t)^2}{c^2}}dt,$$ as \(\Delta x = 0\) is {\(S\)}. If the events remain in the same inertial frame {\(S\)} (i.e. \(\gamma\) is constant), we have the time dilation expression as shown before. Proper time is always shorter than the duration measured in other inertial frame.

Back to the Past

Suppose the rate of change from event A to event B in an inertial frame {\(S\)} is $$ \frac{\Delta x}{\Delta t} > c$$ In another inertial frame {\(S'\)} moving at speed \(v < c\) \begin{align} \Delta t' &= \gamma\left(\Delta t - \frac{v}{c^2}\Delta x \right) \\ &= \gamma (\Delta t) \left(1 - \frac{v}{c^2}\left(\frac{\Delta x}{\Delta t}\right)\right) \\ &< \gamma (\Delta t) \left(1- \frac{v}{c} \right) \\ &< 0 \end{align} which means event A, happened before event B in {\(S\)}, is observed to happened after event B in {\(S'\)}.
If \(\frac{\Delta x}{\Delta t} > c\) between two events, they are spacelike. As their order may be reversed depending on observing inertial frame, the two events cannot have reason and consequence relationship.
If \(\frac{\Delta x}{\Delta t} <= c\) between two events, they are timelike. As their order is preserved for all observing inertial frames, the two events may have reason and consequence relationship.