# Clarification on Representing Distances and Trajectories in Minkowski Spacetime

In the context of Minkowski spacetime, where the metric has a signature of (-, +, +, +), the $$x-t$$ plane (spacetime diagram) is commonly used to visualize events and their evolution in both space and time. However, I'm seeking clarification on how to accurately represent distances and trajectories (curve or path) in such a spacetime diagram.

Specifically, considering the abstract nature of distances in Minkowski spacetime, and the fact that light-like intervals have vanishing distances, how should distances and trajectories be interpreted and represented on the $$x-t$$ plane separately from each other?

Is it accurate to interpret the trajectories of objects (including light) on the $$x-t$$ plane as curves representing their paths through spacetime? And how can we reconcile the fact that the distances between events, particularly light-like intervals, do not adhere to our intuitive notions of distance in Euclidean space?

How the notion of the curve or path in the manifold of spacetime is defined technically independent of the metric as it is an additional structure added.

[UPDATED to expand on my comment that PHY 101 position-vs-time diagrams feature a nonEuclidean geometry.]

Since points in a position-vs-time graph (a spacetime diagram) are events, the worldline of a particle is a curve through its events, with the curve restricted to have future-directed [future-causal] tangent vectors. So, we have (as you say) “trajectories of objects (including light) on the x−t plane as curves representing their paths through spacetime”.

And how can we reconcile the fact that the distances between events, particularly light-like intervals, do not adhere to our intuitive notions of distance in Euclidean space?

This is already the case in Galilean physics (whose spacetime diagram also has a nonEuclidean geometry).

In a typical PHY 101 position-vs-time graph (a spacetime diagram) below, OA and OB represent the worldlines of two particles that met at the origin at $$t=0$$ and A=(0,1) and B=(1.8,1) are the (x,t)-coordinates according to OA.

• In PHY 101, the wristwatches worn by OA and by OB both read "1.0 seconds", even though [on this diagram] the OB has a Euclidean-length that is longer than OA.

• If we complete the right-triangle, using the x-axis which is perpendicular to OA, we will find that the usual [Euclidean] Pythagorean theorem does not work on this diagram.

• Interpreting this diagram using "Galilean spacetime geometry" (i.e., using a metric-like structure where we assign wristwatch proper-times as the lengths of timelike segments), the x-axis is also [Galilean-]perpendicular to OB. In fact, one could regard the triangle OAB as a Galilean-isosceles triangle.

• So, you should have similar concerns about the PHY 101 position-vs-time graph which also "do[es] not adhere to our intuitive notions of distance in Euclidean space".

• (If we are dealing with the real world [with special relativity], and the vertical axis is in meters, the velocity of OB is 1.8 m/s. The time on OB's wristwatch is 0.999999999999999982000 sec [using WolframAlpha].
If the vertical axis is in units of $$10^8\rm\ m$$, then the velocity of OB is $$1.8\times 10^8\rm{ m/s}$$ [(3/5)c]. The time on OB's wristwatch is 0.80 sec.)

If the issue now is representing (say) square-intervals [determined by a metric], here is one that works in special relativity… at least for spacetime displacement vectors and piecewise-inertial worldlines.

From my post at https://www.physicsforums.com/threads/how-fast-does-the-particle-go-through-spacetime.1048187/post-6832310 , one can visualize the spacetime interval between two events (in a (1+1)-Minkowski spacetime diagram) by drawing a "causal diamond".

Consider events $$A=(0,0)$$ and $$Z=(10,6)$$, where I specified $$(t,x)$$-coodinates.

With $$A\ll Z$$ (Z is in the timelike future of A), the causal diamond of AZ is
the "intersection of the causal future of A and the causal past of Z",
which can be interpreted as the set of "events that can be signaled by A, and then signal Z".

By drawing "rotated graph paper" (as shown),
the area of the causal diamond on this grid represents the square-interval [in the $$(+,-)$$-convention]: $$10^2-6^2=(10+6)(10-6)=(16)(4)=64 = (8)^2,$$ which is the square of the diagonal $$AZ$$.

• (Divide AZ into eight equal segments. Construct a causal diamond for each segment. Such causal diamonds are geometrically similar to the causal diamond of AZ and have area equal to that of the diamonds in the rotated graph paper. I call the causal diamonds in the rotated grid "light-clock diamonds" since they are based on the light-signals in an inertial observer's "longitudinal light-clock".)

You can play with my Desmos visualization: https://www.desmos.com/calculator/4jg0ipstya

It's based on light-cone coordinates from the eigenvectors of the boost in (1+1)-Minkowski spacetime.

• If one of the lightlike-edges of the diamond is zero, you have a lightlike-interval.
• If the lightlike-edges have opposite sign, you have a spacelike-interval.

Let $$\Delta u=\Delta t+\Delta(x/c)$$ and $$\Delta v=\Delta t-\Delta(x/c)$$.
Then area is $$\Delta u\Delta v=(\Delta t)^2-(\Delta(x/c))^2$$.
Since the boost has determinant 1, the signed-area is invariant under boosts.

With this, one can faithfully display the elapsed-proper-time along piecewise-inertial worldlines.

Here's the Clock Effect/Twin Paradox for a traveler with outgoing and incoming speed of (3/5)c
from my slides at https://www.aapt.org/docdirectory/meetingpresentations/WM18/FG07-Salgado-RelativityRotatedGraphPaper-CalculatingWithCausalDiamonds.pdf :

The clock-diamonds in the last diagram are traced out by the light-signals in a standard-issue longitudinal light-clock carried by each piecewise-inertial observer.