Special Relativity

In a Galilean transform, only the time basis vector is shifted. In a Lorentz transformation, both time and space basis vectors are shifted. This is a direct consequence of the observation that the concept of "simultaneous events" depends on the reference frame. Furthermore, the lengths of the time and space basis vectors are also changed (relative to the first frame), reflecting the observations of time dialation and Lorentz contraction. The exact details behind these transformations will fall straight out of the geometric algebra for relativity.

Algebraic Framework

The fundamental thing we need for starting out is the Minkowski metric (we follow the mostly-minus convention because its cooler or something), which is given (for 3+1 spacetime) by:

This could easily be extended to any number of (spatial) dimensions but… that's a bit scary. To package this into an algebraic framework, we can work in the geometric algebra (or for general spatial dimensions, the algebra is still the same) called the Spacetime Algebra (STA).

Tensor index notation has a similar convention to what you would find in the wild: greek indices represent all possible values (0, 1, 2, 3) but latin indices represent only nonzero values (1, 2, 3).

The basis blades we have here are:

The 4 basis vectors: , where is the temporal basis vector and the others are the spatial basis vectors. Note that while .
The 6 bivectors: . These help us generate Lorentz boosts, and more generally all the Lorentz transformations. For this reason they are the generators of the Lie algebra of the Lorentz group, (insert appropriate Lorentz group's Lie algebra here (idk which of the 3 variants it is)). They also have other uses, ofcourse, but those are outside the scope of my understanding.
The 4 trivectors: . I have no idea what they do.
The pseudoscalar . Its a thing… its always a thing. You can't get rid of it. Its like the Herobrine of Clifford Algebra; you might try to patch it out, but it will always be a thing. But unlike minecraft creepypasta, this thing has some cool uses like:

With the tour of the basis blades out of the way, we can now move on to doing actual things with these now. We will first visit the properties of the algebra before turning to relativity stuff.

Timelike vs Spacelike

There is an important distinction to be made regarding the squares of certain basis blades: Basis blades squaring to are said to be spacelike, and ones squaring to are said to be timelike. So revisiting our earlier tour of the basis blades:

is timelike, are spacelike, basically by definition.
, so these are timelike. The others are spacelike: .
, so this is timelike. The others are spacelike: .
The pseudoscalar is spacelike: .

As the name suggests, timelike blades have something to do with time and spacelike blades have something to do with space. For general multivectors, timelike refers to having positive square, spacelike refers to having negative square, and lightlike or null refers to having 0 square.

Reciprocal Basis

The reciprocal basis pops up whenever you mix Geometric Algebra with calculus because… people like putting superscripts wherever possible I guess.

The reciprocal basis for STA is defined by the equation

In terms of down-to-earth tangible computable things, we have: and , ie.

Spacetime Splits

A nice property of STA is that its even subalgebra is isomorphic to the Algebra of Physical Space (APS) . The spacetime split coupled with this isomorphism allows us to separate objects in STA into their temporal part as a scalar and their spatial part as a vector in APS.

The details of the isomorphism are:

Scalars in STA are mapped to scalars in APS.
Timelike bivectors in STA are mapped to vectors in APS.
Spacelike bivectors in STA are mapped to bivectors in APS.
The pseudoscalar in STA is mapped to the pseudoscalar in APS.

The spacetime split itself is simply just right-multiplication by ¹ ; to split a spacetime vector , you simply compute . Abbreviating the STA bivectors as , we have for a general spacetime vector ,

And noting that these correspond exactly to the basis vectors of APS under the aforementioned isomorphism, this cleanly shows a vector being split into a scalar temporal part and a vector spatial part.

But these components are not absolute (there is no absolute time or space). Instead, this splits the vectors in the reference frame of an observer whose temporal basis vector is . If another observer has temporal basis vector , then performs a spacetime split in their reference frame.

Spacetime Events and Spacetime Interval

Events in spacetime are represented by vectors in STA. One can specify a temporal part (scalar), and a spatial part (vector), and by inverting spacetime splits, it can be seen that this data maps to a vector in STA.

It also makes sense to ask what is the Lorentz-invariant notion of distance between events. This is what the spacetime interval is, and is very straightforwardly given to us from the algebra. It is simply the squared length as computed from the inner product or squaring the vector, so for 2 events , the spacetime interval between them is . For a single event , this can be used to calculate its Lorentz-invariant squared length as . So the spacetime interval is just the squared length of the event that represents the difference of 2 events.

Note that although the quantity is being squared, the value itself can be positive, zero, or negative. What this means for the length in each case is:

Positive squared length means the event is classified to be timelike. In this case, the straightline journey from the origin to this event is slower than light. The proper time for this journey can be defined as .
Zero squared length means this event is classified to be lightlike or null. In this case, the straightline journey from the origin to this event is exactly the journey of light.
Negative squared length means the event is classified to be spacelike. In this case, the straightline journey from the origin to this event is faster than light, so this event (as hinted by its classification) actually represents a spatial direction. The proper length for this event can be defined as .

Lorentz Transformations

I will start out with a bit of a spoiler: all Lorentz transformations are just rotations (in STA). This is why I hinted earlier that the bivectors help generate all Lorentz transformations, the same formula for rotations in Geometric Algebra help produce the formulas for Lorentz transformations.

Lorentz transformations encompass 2 types of transformations: Lorentz boosts and normal euclidean rotations. Lorentz boosts are generated from timelike bivectors, and normal euclidean rotations are generated from spacelike bivectors. Everyone knows how to rotate in euclidean space, so I focus on Lorentz boosts here.

For simplicity, assume we want to boost in the direction of . The the rotor for boosting in this direction is simply given by rotation by units in the plane:

Since we know , a slight twist on the Euler identity gives us an equivalent expression in terms of hyperbolic trigonometric functions

We can now investigate the action of Lorentz boosts on events. By linearity, we can just focus on its action on the basis vectors, which is split up into 3 cases.

To have tangible results for a Lorentz boost of velocity , we set where . With by simple properties of hyperbolic trigonometric function [citation needed], it is easy to see that:

Here is called the Lorentz factor.

Action of on :
Action of on :
Action of on where :

These completely define the action of a Lorentz boost in direction on any spacetime event. By extension, you can figure out the action of a Lorentz boost in any direction on spacetime events (by simply composing with a euclidean rotation).

Squared length is Lorentz-invariant

A short proof that squared length is actually Lorentz-invariant for the doubters out there.

Let be any rotor (can include euclidean rotations along with Lorentz boosts), and a spacetime event. We have to prove that :

Speed of light is constant

Using this, we can also demonstrate that the speed of light is constant in all (inertial) reference frames. WLOG we can, as before, take the rotor to be . The worldline of light contains all vectors s.t. .

It then becomes a trivial corollary of the above result that , so this is also on the worldline of light.

So we proved the main postulate of Special Relativity using… Special Relativity? Yeah I lied, and this is where we discuss about the Minkowski metric and Spacetime Algebra. Spacetime Algebra is straightforwardly derived from the Minkowski metric. But the Minkowski metric itself is not given from thin air, it actually comes naturally from the postulate "the speed of light is constant in all inertial reference frames." So we just used the postulate to derive the postulate (it's a good sanity check though).

Time Dilation & Lorentz Contraction

Time Dialation

Time dialation is the result of measuring the temporal component of a vector with different bases from different reference frames. To illustrate this with an example, start with a purely temporal event and the rotor for our boost being . Labelling the new basis with , we can derive the following relation:

Then our purely temporal event is now measured to have components

(Note: this can also be derived from the action of on ).

Hence we see a different component in the new basis: , the effect of time dialation.

Lorentz Contraction

Lorentz contraction is a bit different. Not only are you measuring with a different basis, but you are also measuring a slightly different vector.

To see why, we start with the simple example of a plank with endpoints at time being and (so that the plank has length ). This plank's endpoints' worldlines trace out 2 parallel lines in its rest frame.

Now let's perform a Lorentz boost in the direction. Our new coordinate system has bases , and the equations to convert into these is:

And hence our very same plank's endpoints now have components

To see the problem, let's focus on the situation at the first instant, ie. put . We note the components of each endpoint in both the original and boosted coordinate frames:

The problem is that the change in frame of reference has also caused a change in the (hyper)plane of simultaneity. Since distances have to be measured in a (hyper)plane of simultaneity, we could use and as reference points for measuring the length of the plank in the original frame of reference. But in our new frame of reference, these 2 events are not simultaneous anymore, so our reference points need to be changed.

Say we take the reference points to be and where so that . It is easy to see that , so we have the equation

By just equating the component that matters, we see that

Which now properly demonstrates Lorentz contraction.

4-Vectors

Proper Time

It's about time we give time a proper treatment. Proper time was defined before but now we explore what it actually means.

Consider some observer in uniform motion (relative to us) starting at the origin and stopping at some point . In our reference frame, the time of this journey can be extracted from the component of the event . More precisely, where is the time taken in the journey relative to us.

But ofcouse, the observer disagrees. In their reference frame the journey took time (specifically, where is the relative velocity between us and the observer). Everyone can disagree on how much time this journey took in their own reference frame (and still be equally right!), but no one can disagree on how much time the journey takes in the reference frame of the observer taking part in the journey itself. That is, the time taken in the journey from the reference frame of the journey is Lorentz-invariant, it does not depend on which reference frame you try to ask the question from.

This is proper time, the time taken in a journey from the reference frame of the journey. Since it is Lorentz-invariant, we have a straightforward Lorentz-invariant formula to calculate it too: .

Using this, we can now give a defintion to an inertial reference frame. An inertial reference frame is one such that the derivatives of its basis vectors wrt. proper time is 0:

Table of 4-Vectors

	Euclidean Vector (3-Vectors)	Minkowski Vectors (4-Vectors)
Position
Velocity
Momentum
Acceleration
Force

Specific 4-vectors have an easy relation with the corresponding 3-vectors.

4-Position:

This has the simplest relation imagineable:
4-Velocity:

This relation is a bit more involved. Following the math we get that:

Here we encounter two derivatives we don't yet know how to compute. Let's tackle them one by one. To solve , note first that by definition of proper time, we have

and thus with some rearranging and algebra, we get

Since this is physics, multiplying and dividing by differential forms is totally an allowed operation, and hence we conclude

The next derivative to solve for is a simple corollary of our obtained result:

Hence the components of the 4-velocity are , and as written out it is

There is one important thing to be noted about 4-velocity. If we compute its square, we see that

That is, the squared length of 4-velocity is always constant, equal to .
4-Momentum:

This is relatively straightforward too since we know the components of now,

Like 4-velocity, 4-momentum has a bit more to discover in its components too. For instance, note that . Expanding into a taylor series around , we see that

Thus we get the relation . But this is not the end of the story; if we analyze the squared length of 4-momentum,

On the other hand, we can directly compute the squared length from the components

Which leads us to the famous Einstein mass-energy equation

In the case of light, the relation becomes a bit more interesting. Since its speed is the speed of light (light travels at light speed, crazy), the Lorentz factor isn't well defined, so is not well defined for light and hence all of our earlier hardwork dies. However, it still has momentum (can someone explain how the hell this works), so we can do some analysis.

Its 3-velocity is still defined, and infact it gives us the starting point for defining its 4-momentum:

as a consequence, as you would probably guess from informally setting in the earlier .

Now here comes the very important bit. You can't assign a single Lorentz-invariant energy value to light². The only thing we have is some measure of energy of a photon relative to some reference frame. Let's say some object has 4-velocity relative to us. Then the energy of light wrt. to that object is measured to be
4-Acceleration:

Not much to say about the derivation other than its just boring algebra. Churning through a lot of very annoying algebra gives us that

Where the term "un-spacetime split"s the acceleration 3-vector, and the second line is derived from the identity

which is also obtained from a lot of annoying algbra.

One slightly interesting thing to note is that the spatial component of is not always parallel to .

The other thing that isn't obvious from this definition is that . This can be derived from though:

Non-Inertial Frames

Special Relativity can infact handle non-inertial reference frames. The downside is weird things occur for non-inertial observers. Notably:

Extra ficticious forces exist (although you probably already knew this from Newtonian mechanics).
Speed of a light beam depends on the frame.
Its coordinate system is curvilinear from the viewpoint of an inertial observer.

You can think of a curvilinear worldline as a series of instantaneous inertial worldlines continuously transformed by some time-dependent Lorentz transformation.

To get a feel for how weird it is, we do a "short" analysis on one particular case of a non-inertial reference frame.

Rindler Coordinates

Setup

The observer we will be observing has constant proper acceleration. This takes a bit of explaining…

Consider the state of the observer at some frozen instant of time. In their reference frame with coordinate bases , their 4-velocity at this instant is

so it is purely temporal. By the earlier result that , we conclude that their 4-acceleration in this coordinate system is purely spatial

It then follows that the squared length of their 4-acceleration is . This is their (negative of squared) proper acceleration. The condition of constant proper acceleration requires that this value stays constant at all times (it is a Lorentz-invariant scalar so it doesn't matter who measures it to be constant).

Analysis

For the sake of simplicity, we assume the spatial acceleration in the frame of the observer is purely in the direction. Hence in our frame, will have nonzero components in only the basis. Applying the law that , we get that

Where in the last step, the positive branch is enforced to make acceleration in the direction. The other conversions were done using the componentwise expansion of and . A similar lengthy calculation shows that . Since we are an inertial reference frame, this leads us to a set of coupled differential equations

The general solution (for just for now) is

To fix some values of , we need to pick an initial condition and then apply a law we've derived before. The initial condition will be to enforce which gives us the restriction , and our equation reduces to

From this we derive that

Now we apply at to obtain . This simplifies both our equations to

At this point, the rest is free. Just perform simple integration to get the components of the 4-position of the observer.

The constants of integration can be arbitrary in this case, as they represent the starting point of the journey. To make things simple, we set , which amounts to .

The Rindler Horizon

We have proven an interesting fact about constant accelerated motion: that its worldline in spacetime is a hyperbola. Hence this motion is asymptotic to some straight worldline. It just so happens that this motion is asymptotic to the worldline of a light beam travelling in the direction passing through our origin:

This gives rise to an interesting phenomenon: Any information from us transmitted to this observer after will never reach the observer (unless they stop accelerating).

This line is an artifical "event horizon" called the Rindler Horizon. The observer cannot see anything cross this horizon from their perspective. In the perspective of the observer, the Rindler Horizon is always at a constant proper distance of behind them, called the Rindler Distance.

Rindler Coordinate System

Our next task would be to figure out the coordinate system this observer would be using. As noted earlier, it will be a curvilinear coordinate system:

The worldlines are series of hyperbolas with the same asymptotes placed at a general proper distance from the origin .
The lines of simultaneities connect points with… er… yeah idk yet.

This gives us the formulas to convert between components of a 4-position expressed in our basis and in the Rindler coordinate basis:

so that for a given 4-position , we have

(here the index is fixed, no Einstein summation convention applied).

Rindler Metric

(Heads Up: This subsection is in tensor algebra instead because I have no idea how to do this in Geometric Algebra.)

In our own coordinates, the line element is given by the Minkowski metric:

Our job is to convert this line element into Rindler coordinates.

Firstly we calculate where the acceleration is in the direction as always in this section. By the equation , we obtain
Then we substitute this into the line element. A bit of algebra and stuff cancelling out gives us

This completes the derivation of the Rindler metric. Idk what else to say about it; you can calculate distances and angles in the reference frame of the accelerating observer now I guess.

Speed of a Light Beam

The speed of a light beam as calculated relative to some non-inertial reference frame need not always equal . This also has the physical implication that the accelerating observer (if they have a sharp enough eye) would be able to discern some light beams going faster or slower thant .

In the case of constant proper acceleration, the metric calculated above plops out the exact formula we need for the speed of a light beam in the direction:

The second form tells us a nice relation between the Rindler Distance, the coordinate at which speed of light beam was measured, and the measured speed. The speed of a light beam is simple the fraction of the Rindler Distance from the Rindler Horizon scaled by . This means the observer at their own location sees light travel at its usual speed.

It is important to note that these measurements are only for the instant . The actual light beam over time as seen by the accelerating observer would exponentially increase in speed over time due to getting farther and farther away (or exponentially decrease if its going towards the Rindler Horizon).

¹ In most other sources, the spacetime split is usually done by left-multiplication by . That is because they use the weird convention of ordering the timeline bivectors as instead of . No idea why.

² Full credits to @spookyyomo and @theslackline on discord for helping me realize this.