# Geometric Construction of Minkowski Space and Proper Time

In Minkowski spacetime, we define a tangent space at each point and using the metric, we calculate a real number that is the infinitesimal invariant interval between two points. This real number, based on a convention, has three possible regimes: zero for a lightlike path, positive for timelike, and negative for spacelike separated paths. We can think of it as a function or map that assigns to each point of the manifold a value on the real line.

Can we alternatively interpret this function defined over the manifold like the one described below?

At each point of the manifold, there is a light cone, where all points on it are projected to a single point on the real line.

Similarly, for each point on any timelike or spacelike path or curve, the beginning and ending points of the curve, two events, are projected to two distinct points on the real line, showing positive or negative values.

Can we use the above observation, and make sense of the spacetime manifold as a fiber bundle, the total space, over the real line as the base manifold?

Can we think of the fibers of this bundle as the lightlike paths that any movement along them will be projected to a single point on the real line?

Because we have not used any coordinate chart here, isn't the invariant interval between any two points already manifest geometrically?

• You say: "Attach a real line to each point in this base manifold, creating a 4-manifold." but you don't tell us what real line and how you attach it. Maybe you are a genius and thus imply that we fully understand your thinking on a glance, or maybe you are just learning and doing thought experiments. Either way, please add more explanation. Commented Apr 22 at 18:49
• Just a thought experiment, in the conventional Minkoswki space at each point there is a light cone, suppose that cone is deformed into or represented by a real line that passes through it, next project all points along it into another real line, representing the proper time, does the whole construct make sense, that's the question here.
– VVM
Commented Apr 22 at 19:04
• In addition to @GyroGearloose 's issue, you write: "In step 4, the first real line, or the fiber, represents the light-like path rather than the coordinate time.". Represents what lightlike path? There is a circle's worth of lightlike paths through any point in a 3-dimensional spacelike slice of spacetime. Commented Apr 22 at 19:05
• The usual formulation is that Minkowski space is a copy of $R^4$ with a metric of signature $(-1,1,1,1)$. What problem are you trying to solve with an alternative construction? Commented Apr 22 at 19:07
• Does the second real line actually have to be attached? Maybe what you want is just the 4-manifold of step 2, then a function that maps any point of that 4-manifold onto a separate real line?
– Mike
Commented Apr 22 at 19:14

There's a lot missing from this description of your idea that makes it hard to assess. For example, you talk about

the real line [and ...] the first real line [and ...] the new base manifold (the real line).

but I'm completely lost as to which space each of these refers to. Some labels and a little mathematical notation might be helpful.

Does the 4-manifold constructed in step 2 capture the essence of Minkowski space?

No. A crucial feature of Minkowski space is the spacetime interval (the metric). You don't mention anything about this.

Does the projection in step 4 correctly represent proper time, including the invariance of proper time?

No. Proper time is really a property of a (timelike) curve. You can't assign a meaningful proper time to a point in Minkowski space without reference to a particular curve passing through that point.

Does this construction successfully encode the causal structure of spacetime, including timelike, spacelike, and lightlike paths?

No. Again, causal structure requires some notion of the spacetime interval or metric, which you have not provided.

• Actually step 3, a new real line as another base manifold, could contain the same information as in the metric, because it shows the invariant proper time. No?
– VVM
Commented Apr 22 at 19:28
• Well, the metric requires two vectors as input. Equivalently, in this case, you could define a function of just one vector (called a "norm") or of a pair of points. But in any case, you need more information than just a single point of the spacetime. Also, I'll remind you again that proper time is a property of a curve, not just a single point.
– Mike
Commented Apr 22 at 19:43
• Understood, but we move along a curve in the 5-manifold, if the curve connecting two points, is along the fiber, then the curve image via the projection map, will be the same point on the real line of the base, that is the proper time, so the distance is zero.
– VVM
Commented Apr 22 at 19:48
• I can't parse this last comment. I recommend reworking the original post to explain more clearly, using symbols and being clear about which space you're talking about every time you mention a space.
– Mike
Commented Apr 22 at 20:52