Lorentz transformation

A Lorentz transformation is a Coordinate transformation between inertial frames in spacetime. A transformation between a frame of reference $(x,y,z,t)$ to $(x',y',z',t')$ states

\left\{\begin{align} &x'=\gamma(x-\beta ct) \\ &y'=y \\ &z'=z \\ &t'=\gamma\left( t- \frac{\beta}{c}x \right) \end{align}\right.

with parameters

\gamma\equiv \frac{1}{\sqrt{ 1-\beta ^{2} }},\qquad \beta\equiv\frac{v}{c}

In the language of proper velocity, $v$ would be the ordinary velocity. Conventionally, an object is said to be relativistic when $\beta$ is not almost-zero or $\gamma>1$ . In other words, the object's speed must be a significant fraction of the speed of light. Alternatively, in common four-vector notation $(ct,x,y,z)\equiv(x^{0},x^{1},x^{2},x^{3})$ :

\left\{\begin{align} &\bar{x}^{0}=\gamma(x^{0}-\beta x^{1}) \\ &\bar{x}^{1}=\gamma(x^{1}- \beta x^{0} ) \\ &\bar{x}^{2}=x^{2} \\ &\bar{x}^{3}=x^{3} \end{align}\right.

where the overbar now denotes the new frame. Here, time is expressed in units of $ct$ instead of $t$ , meaning that it is measured in meters and interpreted as the time it takes for light to travel that many meters.

Matrix form#

Lorentz transformations can also be written in matrix notation:

\bar{x}=\Lambda x

where

\Lambda(\beta)=\begin{pmatrix}\gamma & -\beta \gamma & 0 & 0 \\ -\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix},\qquad x=\begin{pmatrix} x^{0} \\ x^{1} \\ x^{2} \\ x^{3} \end{pmatrix},\qquad \bar{x}=\begin{pmatrix} \bar{x}^{0} \\ \bar{x}^{1} \\ \bar{x}^{2} \\ \bar{x}^{3} \end{pmatrix}

$\Lambda$ is known as the Lorentz transformation matrix. The convenience of this form lies in that, besides being more compact, it also makes it easier to handle transformations that are not on a common $x,\bar{x}$ axis, as you'd just change the matrix accordingly. Each individual component is then written

\bar{x}^{\mu}=\sum_{\nu=0}^{3} \Lambda_{\nu}^{\mu}x^{\nu}\equiv \Lambda_{\nu}^{\mu}x^{\nu}

where the second equality is using Einstein notation for implicit summation.

The matrix has a few properties:

$\gamma ^{2}-\beta ^{2}\gamma ^{2}=1$ .
Its determinant is 1.
Its inverse is equal to itself with an inverted argument: $\Lambda^{-1}(\beta)=\Lambda(-\beta)$ .

Derivation#

Say we have an event $E$ at coordinates $(x,y,z,t)$ in some inertial system $\mathcal{S}$ . We want to express this event in another inertial frame $\mathcal{S}'$ , where it'll have coordinates $(x',y',z',t')$ . We are looking for the coordinate transformation that will convert between the two. This space is four dimensional, so a complete visualization is not possible. However, think of it like this: since we have inertial frames, the origins can move at some constant velocity. Let's say that at time $t=0$ , both $\mathcal{S}$ and $\mathcal{S}'$ coincide. As time goes on, the two systems will in general move away from each other, at some constant velocity $\mathbf{v}$ . Say this velocity is on the $x$ axis (if it's not, rotate everything until it is). Then, over some time $t$ , the origin of $\mathcal{S}'$ will move a distance $vt$ away from the origin of $\mathcal{S}$ on the $x$ axis (the other coordinates don't change). The event coordinates will of course be connected by

x=x'+vt

and so the coordinate transformation is

\begin{cases} x'=x-vt \\ y'=y \\ z'=z \\ t'=t \end{cases}

These are the old Galilean transformations that applied to Galilean relativity. As you may guess from this nomenclature, these start to collapse when introducing special relativity; the cause is the postulates of special relativity and more specifically setting a maximum speed in the universe. Relativity of simultaneity and time dilation can't realistically permit something as naive as $t'=t$ and length contraction certainly invalidates the snarky "of course" regarding $x=x'+vt$ . On the other hand, $y'=y$ and $z'=z$ are safe since length contraction occurs only on the axis of motion, which we claimed to be on $x$ .

The issue in the previous derivation is in being a little too sure that $x=x'+vt$ . More generally, say we have $x=d+vt$ , where $d$ is the distance measured between the origin of $\mathcal{S}'$ and the $x$ -component of $E$ . If length contraction weren't a thing, $d$ would certainly be $x'$ and our results above would hold. But length contraction is indeed a thing, so we must take it into account:

d=\frac{x'}{\gamma}\quad\Rightarrow \quad x=d+vt=\frac{x'}{\gamma}+vt\quad\Rightarrow \quad x'=\gamma(x-vt)

The same argument could be done in reverse to go from $\mathcal{S}'$ to $\mathcal{S}$ , just this time we'll have $x'=d'-vt'$ , where $d'$ is the distance between the origin of $\mathcal{S}$ and the $x$ component of $E$ . In classical physics, this is again $d'=x$ , but in relativity, we must consider contraction, so

d'=\frac{x}{\gamma}\quad\Rightarrow \quad x'=d'-vt'=\frac{x}{\gamma}-vt'\quad\Rightarrow \quad x=\gamma(x'+vt')

which shouldn't be surprising, since relativistic effects are symmetrical between observer. We can now substitute these results into the Galilean transformations and solve for $t'$ to obtain

\boxed{\left\{\begin{align} &x'=\gamma(x-vt) \\ &y'=y \\ &z'=z \\ &t'=\gamma\left( t- \frac{v}{c^{2}}x \right) \end{align}\right.}

These are the Lorentz transformations for spacetime coordinates (or a singular Lorentz transformation if you are talking the change as a whole). The reverse transformation is just a matter of changing the primes and inverting the minuses into pluses:

\left\{\begin{align} &x=\gamma(x'+vt') \\ &y=y' \\ &z=z' \\ &t=\gamma\left( t'+ \frac{v}{c^{2}}x' \right) \end{align}\right.

Defining $\beta\equiv v/c$ , they can also be written as

\left\{\begin{align} &x'=\gamma(x-\beta ct) \\ &y'=y \\ &z'=z \\ &t'=\gamma\left( t- \frac{\beta}{c}x \right) \end{align}\right.

Note how when $v\ll c$ , then $\gamma\to 1$ and $\beta\to 0$ and these go back to being the usual Galilean transformations.

As a hyperbolic rotation#

A Lorentz transformation is a hyperbolic rotation in spacetime, the angle of which is called the rapidity $y$ :

\pmatrix{ct' \\ x' \\ y' \\ z'}=\pmatrix{\cosh y & -\sinh y & 0 & 0 \\ -\sinh y & \cosh y & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1}\pmatrix{ct \\ x \\ y \\ z}

Experimental frames#

In experimental physics that deals with special relativity, it's common to reference two well-known frames of reference.

The laboratory frame is fixed to the observer (and detectors). It shows what the detector or scientist sees when observing the phenomenon. It's particularly useful when the one of the elements of the system is at rest with respect to us, like a stationary target in a particle accelerator. In that case, its momentum vanishes and its relativistic energy is purely rest energy.
The center-of-momentum frame is the frame in which the total momentum (but not four-momentum!) of the system is zero. This provides easy access to the invariant mass of the system. The center-of-mass frame is a special case in which the center of mass is in the origin.