31 Geometry of Spacetime
We begin by defining spacetime to be the set
31.1 Axiomatization
Like many things in mathematics, one way to study spacetime is to study its group of symmetries. This is analogous to how we study Euclidean space by discovering its group of isometries, and then use our newfound knowledge of translations rotations and reflections to simplify all further calculations.
But how do we go about finding the symmetries to start with? In Euclidean geometry, perhaps we start by realizing that the Euclidean plane has a particular mathematical structure on it - it has a distance function (induced by a dot product on all tangent spaces), and we can rigorously define the isometry group as the set of all transformations that preserve this dot product. But for spacetime, the situation seems much more difficult - we don’t know what sort of math structure best describes spacetime, that’s one of the things we are looking to discover! So we can’t just explicitly define spacetime symmetries to be the things that preserve this (unknown!) structure.
However, if we dig back deep enough into history, we can find a close analogy between our current predicament and geometry. The greeks after all did not know about infinitesimal tangent spaces and all that: instead, they described geometry axiomatically, by specifying properties that they observed to be true of space, and then using these as the foundations of their mathematical theory. Us moderns then could take the axioms and try to rigorously find which mathematical spaces satisfy them: we saw in Geometry class that if you take Axioms 1-4 there are two possible ways the world could be (Euclidean or Hyperbolic), but once you have all the Axioms 1-5, there is a unique structure (Euclidean dot product on every tangent space) which instantiates them.
Could we attempt an axiomatic description of spacetime? That is, could we list some rules we claim to be true (based on our observations of the world around us) and use these as constraints on our mathematical theory? Perhaps, if we choose a good set of axioms, we will be able to find a small number of possible solutions (like Euclidean / Hyperbolic space, for Euclid’s Axioms 1-4). Then, if we were left without a unique solution, we could try to find more axioms to add (based on other observations of the world around us) which further narrow down to a unique mathematical structure, which would allow us to begin a rigorous study of spacetime.
Here’s a proposal for three first axioms: in the first we take the ideas of Euclid, in the second the ideas of Galileo, and while the third observation doesn’t have a name, it certainly predates the others, going back to the first humans to imagine themselves as living in space while time passes.
Definition 31.1 (Axioms of Spacetime Symmetries)
- Symmetries of space and time are symmetries of spaceitme: If
is a Euclidean isometry, then is a symmetry of spacetime. Similarly, if is any isometry of (a translation, reflection or combination) then is a symmetry of spacetime. - Galileo’s Principle: All symmetries of spacetime must preserve the class of constant speed trajectories. And, if
and are the worldlines of any two constant speed observers, there is a symmetry of spacetime taking to . - Space is different than time: There is no symmetry of spacetime which takes the vertical axis
to a line in space .
The symmetries of spacetime form some group
31.1.1 Step 1: Interesting Symmetries Change Velocity
By Axiom (1) we know that spacetime is homogeneous, as Euclidean space is homogeneous, and the real line is homogeneous. We can use this to reduce the class of isometries we are interested in. Let
So an arbitrary symmetry of spacetime is the composition of a symmetry which fixes the origin, and a symmetry which is just a translation in space and time. Since we understand the symmetries of space and time separately (from Euclidean geometry!) this implies that if we can understand the spacetime symmetries that fix the origin, we can understand the entire group
So, what are the symmetries fixing the origin? Some of these we also already understand: if we take a Euclidean rotation
This means we are looking for isometries that fix the origin but do not fix the
Thus, the only isometries we are interested in are the ones that take
One thing to ask ourselves here;
%Note to future self: prove that things fixing the t axis pointwise must be the SAME euclidean isometry on each slice
31.1.2 Step 2: The Group is Linear
The first step is to show that the only groups that satisfy these axioms are groups of linear transformations, that is,
- Using Galileo’s axiom, we see that if
is a symmetry it must take constant speed trajectories to constant speed trajectories. Since constant speed trajectories are affine lines, this means it must preserve (at least some subset of) affine lines. also fixes the origin, so it sends affine lines through the origin to affine lines through the origin: like a linear map! (It’s more work, but in fact this is the only option, it is a linear map) - We could instead proceed and just look for the subgroup of
that is linear - and accept the fact that there could also be nonlinear symmetries out there! We will end up finding a bunch of linear symmetries, and in the end, can go back and argue that we have actually found all the symmetries: there were no nonlinear ones out there to be worried about!
31.1.3 Step 3: We can Work with 1 Space Dimension
Let
Using this, consider the
Using the fact that we also know that such resulting transformations are linear, we can write this as a matrix:
Our goal is to fill in the missing entries! (right now, all of them!). One first thing to notice, is that with our motion in the
This implies the matrix is block diagonal: the
Now, we have a transformation which translates along the
This leaves a very manageable sized problem - everything about the symmetries of spacetime can be totally understood so long as we know how they work with one space and one time dimension!
31.1.4 Step 4: Some Matrix Calculations
This is where all the real work is!!
Now we have gotten ourselves into a sufficiently restricted situation that we can do some actual calculations. We are only interested in symmetries fixing the origin, and taking
We then did a bunch of matrix calculations that I do not feel like typing tonight (sorry!) but I emailed out a handwritten copy of. Together, this implies that there are two possibilities for the group of symmetries of
Possibility 1: The group of symmetries of spacetime consists of the following matrices
Possibility 2: The group of symmetries of spacetime consists of the following matrices, for
31.2 Implications
Our next goal, as mathematicians is to try and study these two possible worlds, and derive some properties they have. We will refer to Possibility I as the Galilean world, and Possibility II as the Lorentzian world.
Proposition 31.1 (All Lorentzian Worlds are Isomorphic) At first, it appears that there are really uncountably many different possibilities for spacetime: one possible Galilean world, but a continuum of Lorentzian worlds, one for each value of
Exercise 31.1 (Prove This:) When
- Injective
- Surjective
- A group homomorphism
31.2.1 Velocity Addition
These two worlds have very different rules for velocity addition: in the Galilean world, velocities add:
Proposition 31.2 For any
But in the Lorentzian world, velocities satisfy a rather different formula, which makes sure that the overall velocity always remains within
Proposition 31.3 For any
Exercise 31.2 Do these calculations.
Exercise 31.3 Then, using the velocity addition for
31.2.2 Existence of a Constant Speed
Show that in the Galilean world, if there is an object moving at any speed
Exercise 31.4 Say you observe an object to be moving at speed
What would happen if you speed up? Would you see its speed change at all? Show that no matter what speed you go (so, no matter which Lorentz transformation
This is an incredibly weird prediction: first, we saw that it is impossible to start out stationary and move at any speed outside of
31.2.3 Metric
The type of mathematical structure spacetime has in these two possibilities is also rather different. Perhaps surprisingly, it turns out that the Lorentzian spacetime (with the more complicated looking matrices!) actually has a nicer mathematical structure.
What should we be looking for here? Well, if Geometry taught us anything, the geometric properties of a space are usually stored in the form of a dot product on infinitesimal tangent vectors (we saw this even for strange geometries, like Minkowski space). So, now that we have two potential symmetry groups of spacetime, we should ask if these correspond to any sort of geometric structure.
Exercise 31.5 (The Galilean World) Show that if
Exercise 31.6 (The Lorentzian World) Show that the inner product
This was Hermann Minkowski’s big realization upon reading Einstein’s work: in 1908 he said in a lecture, announcing this that
“Gentlemen! The views of space and time which I wish to lay before you … They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”
Remark 31.1 (Reminder: Showing a dot product is preserved). If
This proves our main theorem: we’ve discoverd the mathematical model for spacetime in this case, and identified it with something we already understand!
Theorem 31.1 The geometry of spacetime equipped with the Lorentz symmetries is isomorphic to Minkowski space.