The Starship Engineer's Notebook: Universal Speeding: The Lorentz Group

There are two important, well-known rules about interstellar travel:

Nothing can travel faster than light.
If you want to get anywhere interesting in a reasonable amount of time then you need to go faster than light.

This leads to three options:

You just can't go anywhere.
You can go someplace nearby, but you have to stay on your ship for a very long time.
You must break rule one.

Option 1 can be rejected as boring. Option 2 I've already looked at in the form of generation ships. Let's talk about Option 3: If the speed of light is the universal speed limit then let's start speeding. Before we can exceed the speed limit we need to know what the speed limit is, and why it's the limit. So why can't anyone go faster than the speed of light? Let's have a look.

Let's begin with some very basic concepts in Special Relativity, because we'll need them later. Typically this is introduced in introductory classes at the university and high school level by arbitrarily introducing something called the Lorentz Factor, γ. A better introduction might show how γ can be derived from Galilean Relativity and Lorentz Invariance as we did in the previous post.

While this approach is useful and instructive for discussing ideas like time dilation and length contraction, and while it does demonstrate why we shouldn't be able to exceed the speed of light, it doesn't help us understand the source of these issues much less suggest a way to solve them.

To do that, we'll need to introduce the formal language of four-vectors to discuss in detail how time and space are connected.

Spacetime And Four-Vectors

To begin with, let's consider a normal three vector, which represents a position or length in space. Picture an object being rotated in three dimensions.

Source: https://en.wikipedia.org/wiki/File:Cube_rotation.gif

Select a given point on the cube, its position relative to the origin is given by the vector:

$\vec{x}=\begin{pmatrix}x\\y\\z\end{pmatrix}$

The relationship between the vector representation of any given point before and after a rotation is given by the rotation matrix, R, which describes the rotation:

$\vec{x}_2 = \left[ R \right] \vec{x}_1 = \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix} = \begin{bmatrix} R_{11} & R_{12} & R_{13} \\ R_{12} & R_{22} & R_{23} \\ R_{13} & R_{23} & R_{33} \end{bmatrix} \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix}$

The length between two points in space is known from the Pythagorean Theorem, so the distance between our chosen point and the origin is:

$dl^2 = dx^2 + dy^2 + dz^2$

We understand, of course, that the size of an object doesn't change when it is rotated. So the rotation matrix, R, must be defined in such a way that the length is unchanged. Consider the square length above, which must be the same both before and after the rotation:

$\begin{align}(dl)^2&=\vec{x}^T\vec{x}=\vec{x}^T[R]^T[R]\vec{x}\\&\rightarrow[R]^T[R]=[I]\end{align}$

From which, we can conclude that the transpose of the rotation matrix is its inverse, the rotation matrix is orthogonal. It is a basic property of orthogonal matrices that the determinant is + 1 as demonstrated:

$(\det(R))^2=\det(R^2)=\det(R^T R)=\det(R^{-1}R)=\det(I)=1$

So it is a requirement for valid rotation matrices that the determinant be + 1. If the determinant is positive, we call it a proper rotation and if the determinant is negative, we call it an improper rotation.

A proper rotation can be broken down into a collection of smaller rotations, at the extreme end an infinite series of infinitesimal rotations starting from the identity. The identity matrix, with ones on the diagonal and zeroes for all other entries, represents no rotation at all and is a proper rotation. In the figure, a rotation of the object by 360° can be broken into two 180° rotations. An improper rotation cannot be broken down in this way. The reflection shown must have some discontinuity, there is no collection of smaller rotations that will produce a flip like this.

Proper rotations preserve chirality, or "handedness" of the object. If you look at the object above with the largest side as the bottom, the rainbow progresses from violet on the left to red on the right. This is unchanged during a rotation. Improper rotations reverse chirality. Notice that the reflected rainbow has the opposite direction, red is on the left and violet on the right.

What if we need to move in time as well as in space? If everything must travel at a finite speed, as stated above, then it must always take some amount of time to travel between points in space. How, then, do we combine these into a "spacetime" vector?

Spacetime Interval

There are two big concerns with doing so. The first is a matter of the units, we measure distance in space in meters and distance in time in seconds, so we will need some conversion factor. The natural choice is the speed of light. Speed works to translate time to distance and vice-versa. This should already feel familiar, as most of us have heard of "lightyears" as a measure of interstellar distance before. For simplicity, we'll measure time in terms of the distance traveled by light in that span of time. One light-second is simply the distance traveled by light in one second, about 300,000km.

The second issue is that time does not behave like a spatial dimension, we need some means of distinguishing the two. Let's consider a simple example. You are holding a watch that is perfectly synchronized with a clock on the wall. You move away from the clock to some significant distance, say 300,000km and then compare the time you see on the clock to the time on your watch.


"The Real Quaternion Relativity" - Viktor Ariel

It will take one second for the light from the clock to reach you, and so you should read two different times on the clock and on your watch. This difference in time is, of course accounted for by the difference in distance:

$c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}=0.$

This is the "spacetime interval" which, by construction, we have set to zero, since we are using the speed of light, c, as both the conversion factor between time and distance and light is the thing separating our two events. Any two events (in this case, the time on the wall and the time on the watch) which are separated by a light signal will have an interval of zero. In general, two events closely separated in space and time should have an infinitesimal interval, ds:

Where we have used the vector arrow to signify a traditional three-vector component and set c=1, meaning that, for time in seconds, all distances are measured in light-seconds.

Lorentz Transformations

Just like with rotations, we understand that the interval between any two events must remain unchanged regardless of our vantage point. We want a four dimensional extension of the rotation matrix, known as the Lorentz transformation. In addition to simple rotation, we also need an operation that converts time to space and vice-versa. Of course, we already know what does that, velocity. Velocity acts on the four vector in a manner analogous to rotation.

By Maschen - Own work, CC0, Link

We imagine two reference frames, F and F', with an observer standing in each frame. Observer F' is moving at some velocity, v, away from observer F. Both of them measure some interval, which observer F records as ds and observer F' records as ds'. There must be some matrix, L, which allows us to describe ds' knowing only v and ds.

For the case where v is in the x direction, this general relation simplifies to:

As demonstrated in the previous post on this topic, this can be simplified to:

Where β = v/c, and γ is given by:

Lorentz Group

This is not, however, very helpful in letting us circumvent the speed limit. For that we need to dive deeper. We're not just looking for a Lorentz transformation, but the entire Lorentz group, the group of all possible transformations.

Our goal is to work out what characteristics that Lorentz transformations must have and if there are any other possible solutions beyond just the obvious. Much like the how the rotation matrix preserves size, we know that the Lorentz transformation must preserve the total interval,

For this to be the case, we can say safely that the determinant of L must be equal to +1, since we can take the determinant of both sides and determinants are distributive.

There are other ways to arrive at this result. If observer F' is moving away from F at some velocity, v, then F' will observe F moving with velocity, -v. If observer F' wanted to translate her measurements to those of observer F, she would follow the same principle as above:

But this is the same as just taking our previous boost matrix and rotating it around, and we already know that rotation matrices have a unitary determinant:

Since the transformation of ds' to ds is the inverse of the transformation of ds to ds':

Combining the two results means:

And therefore we can confidently state that:

$\det(L(v))=\det(L(-v))=\det(L^{-1}(v))=\det(L(v))^{-1}$

And the only way for 1/x = x to be true is if x = +1. So the we arrive at the conclusion that the determinant of any valid Lorentz transformation must be +1.

It is simple to find the determinant of the standard boost matrix seen above and see that it does, indeed, come to +1. This group also contains the trivial solution of the 4x4 identity matrix, reflecting the obvious fact that, if F' is the same as F then ds' = ds. We can also see that, since rotation matrices have a determinant of one, then the 3x3 rotation matrix is also a valid Lorentz transformation that has the form, where vector 0 indicates entries that are uniformly zero:

In a similar manner, a general Lorentz transformation might be described as:

Where the 3x3 matrix M describes how the space components are mapped to one another, the scalar Γ describes how time is transformed, and the vectors a and b describe how time and space are transformed into one another.

Spacetime Metric

You may have noticed something odd about how we defined the interval above. How can something squared have a negative part? Properly written, the interval should be:

Where the metric, η, is the 4x4 matrix:

This construction generalizes the notion of dot products seen in regular three-vectors to account for the fact that space and time behave differently in relativity. This is specifically the flat spacetime metric, explaining the behavior of four vectors in the absence of any force. In general relativity, the metric describes the curvature of spacetime due to the presence of matter and energy.

Combine this proper definition of the interval with what we already know about its invariance under Lorentz transformation:

A general property of the matrix transpose tells us that:

Substitute and simplify to yield:

Some block matrix multiplication yields:

Which imposes a notable constraint on Γ, since $b T b$ is the squared magnitude of the vector b, which cannot be less than zero:

$\Gamma^2\rightarrow\left\{\begin{matrix}\Gamma<-1\\\Gamma>+1\end{matrix}\right.$

Symmetry also gives us:

$|\vec{a}|=|\vec{b}|$

Multiply both sides from the left by the inverse of L and from the right by the inverse of η (note that η is its own inverse) yields an expression that strongly resembles our above orthogonality relationship from rotation matrices, suggesting we're on the right track:

$[L]^{-1}=\eta^{-1}[L]^{T}\eta=\begin{bmatrix}\Gamma&-\vec{b}^T\\-\vec{a}&M^T\\\end{bmatrix}$

Lorentz Group Subsets

That was a lot of matrix math, but it gets us somewhere, finally. We have found two very important characteristics of the general Lorentz transform, based only on the simple requirement that the spacetime interval have the same magnitude regardless of observer. No matter the reference frame, the combination of distance and time between two events must be the same; this means that the transformation from one frame of reference to another, L, must have the two following characteristics:

The determinant of L must equal + 1.
The first entry of L, which describes the transformation of time, must be either greater than +1 or less than -1.

If the first entry is positive then the transformation is orthochronous and, if negative, it is antichronous. An antichronous transformation reverses the direction of time, while an orthochronous transformation preserves it.

If the determinant is positive, we call it a proper transformation and if the determinant is negative, we call it an improper transformation. The difference is most easily demonstrated using rotations. Keep in mind that rotations are a subset of Lorentz transformations with Γ = 1, where time is left unchanged between frames.

All four possible combinations are shown in the following table from Wikipedia:

Intersection, ∩	Antichronous ${\mathcal {L}}^{\downarrow }=\{\Lambda \,:\,\Gamma \leq -1\}$	Orthochronous ${\mathcal {L}}^{\uparrow }=\{\Lambda \,:\,\Gamma \geq 1\}$
Proper ${\mathcal {L}}_{+}=\{\Lambda \,:\,\det(\Lambda )=+1\}$	Proper antichronous ${\mathcal {L}}_{+}^{\downarrow }={\mathcal {L}}_{+}\cap {\mathcal {L}}^{\downarrow }$	Proper orthochronous ${\mathcal {L}}_{+}^{\uparrow }={\mathcal {L}}_{+}\cap {\mathcal {L}}^{\uparrow }$
Improper ${\mathcal {L}}_{-}=\{\Lambda \,:\,\det(\Lambda )=-1\}$	Improper antichronous ${\mathcal {L}}_{-}^{\downarrow }={\mathcal {L}}_{-}\cap {\mathcal {L}}^{\downarrow }$	Improper orthochronous ${\mathcal {L}}_{-}^{\uparrow }={\mathcal {L}}_{-}\cap {\mathcal {L}}^{\uparrow }$

Note that the identity is a proper, orthochronous transformation. Let us assume that all valid transformations must be continuous, that they can be broken down into a series of smaller, also valid, transformations and that the identity must be a valid transformation. These assumptions, which seem on the face to be perfectly reasonable, require that only proper, orthochronous transformations be allowed.

By definition, improper transformations include a reversal and cannot be described in a way that is continuous with the identity. Because we have already concluded that | Γ | > 1 we observe a gap between +1 and -1; there is no smooth path from the identity to an antichronous transformation.

Valid transforms, of course, need not only include speed. Let us return next time with how fields and forces function within a four-dimensional framework.

The Starship Engineer's Notebook

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March 18, 2025

Universal Speeding: The Lorentz Group