Hi! Welcome to my third BeyondResearch Insight! This time again from the differential geometry side of the programme.
In a couple of previous sessions, we came across a particularly interesting topic to me as an aspiring physicist: the relationship between math and physics. In the last differential geometry session, we discussed that physics is harder than math because it is tough to build meaningful physical theories by interpreting math and relating it to the real world since modern physics is completely out of our experience. But this is also why mathematics is such a valuable tool! For example, the geometric approach to physics is extremely useful to gain intuition.
On that note, we spent the majority of the session talking about how to model spacetime, as differential geometry plays a crucial role in this. Firstly, we looked at particles in space-time and decided to represent them as parametric curves in four dimensions, three of space and one of time. This is quite intuitive but when we relate it to the physical world we encounter a question: what quantity should the parameter be if we are already using time as one of the coordinates? A mathematician would say that it can be any interval of real numbers, but a physicist needs to give the parameter some physical meaning. Both would agree, though, that the parameter should have a geometrical meaning, i.e., be independent of the coordinates we choose. For this reason, the spacetime arclength of the curve that the particle traces is a sensible choice.
How exactly do we measure distances in spacetime? Well, firstly, we might want our definition to reflect the fact that the speed of light is constant for all observers, therefore the spacetime distance of light at a given instance should be the same for all observers no matter how fast they are moving relative to each other. One way to accomplish this is to define spacetime distance is in such a way that the distance to a ray of light that started at the origin is always 0.
Special relativity also tells us that no particle can move faster than the speed of light and, therefore, be further away from the observer than a light ray that started moving away from the observer at the same time as the particle. Thus, for all a physicist cares, the distance to particles travelling faster than light can be undefined, as they are not physically meaningful. However, the distance to particles travelling slower than light should always be positive!
In the second part of this post, you will see that this will indeed be the case if we define spacetime distance with the help of the so-called Lorentz metric!
Welcome to the second part of my third BeyondResearch Insight! In the first part, I discussed how we might want to define lengths and distances in spacetime: the distance to particles travelling at the speed of light should be 0 for all observers at all times, the distance to particles travelling slower should be positive, and the distance to particles travelling faster should be undefined. But how do we achieve this?
This is what the Lorentz metric is for: it sets the length squared equal to (ct)^2-x^2-y^2-z^2. This might seem very arbitrary, but just imagine it in two dimensions. Let’s put one spatial dimension on the x-axis and time on the y-axis. Then, if we start at the origin, the points light would reach at any given moment will form two straight lines. As we go forward in time along the y-axis, the further in space light could have travelled in any direction. Therefore, the spacetime curves of light will be given by two lines creating a V shape. We can also extend these paths to the past and get a sort of X shape with the intersection at the origin. The 2D equivalent of the Lorentz metric would be (ct)^2-x^2. Because light travels at the speed of light, the following always holds: c=x/t. Therefore the distance of light from the origin, the observer, will always be ((x/t)*t)^2-x^2=x^2-x^2=0, as we wanted. For any particles with a speed greater than c, this quantity would be negative, and thus the distance is undefined. For any particles with a speed lower than c, the length would be positive.
If we instead imagined this in three dimensions, all the paths that light can reach at any given instant would create the so-called light cone, essentially our X shape rotated about the y-axis. Of course, space-time is actually four-dimensional, but since we cannot easily visualize a hypercone, let’s just work with the idea of a 3D cone. The region outside the light cone is called space-like and all particles travelling there have undefined distances. The region of space-time inside the light cone is called time-like, this is the physically meaningful part where all particles travelling slower than light are and where our Lorentz distance is well-defined. Because the distances in the time-like region are always positive, we can use the arclength of the curves particles follow as the parameter.
Interestingly, if we divide this arclength by the speed of light, we will get a measure of time. We call this the proper time, and it can be understood as the time a point particle would measure if it carried a clock with it![
All of this comes from hyperbolic geometry! When we change our frame of reference, events in space-time will move along a hyperboloid (or its 4D equivalent) to preserve the Lorentz length, as it is equal to (ct)^2-x^2-y^2-z^2.
Here we can also do a quick consistency check, that is verify if our model is reflecting our intuition. Because the hyperboloids are constrained to their respective quadrants, an event in the positive time-like region will stay in the positive time-like region, whereas an event in the positive space-like region could move to the negative space-like region. This means that an event within the future light cone will stay in the future for all observers, whereas events outside the light cone could move to the past for some! This would be bad news for our theory, but thankfully only curves inside the light cone have well-defined lengths and have any significance. So our model makes sense!
But despite this, the definition of spacetime distance might seem a little arbitrary. However, it turns out that the Lorentz metric, on which the definition stands, can actually be derived from Einstein’s relativity postulates, namely from the claims that the laws of physics are the same in all inertial reference frames and that light propagates at a constant speed no matter the speed of the observer. For instance, E. C. Zeeman has shown that just by having an order relation on spacetime, where x ≤ y means that event x can casually influence event y, implies Lorentz transformations and therefore the definition of the spacetime distance discussed before!