Space, time and the needs of a new theory

This is the third article in the relativity series. It took me some time to write down because I wanted to give priority to technology focused articles.
I also sometimes wonder if anyone is actually reading these posts, but I’ll continue regardless.
Let’s dive into some fundamental concepts.

A common question for newcomers to relativity is: why all this complexity? Isn’t Newtonian mechanics sufficient to explain the world around us, including gravity, acceleration, and distances? The answer, surprisingly, is no.

Newtonian laws are intuitive and work well at speeds much slower than the speed of light.
Treating space and time as separate entities aligns with our everyday perceptions. For example, imagine being on a train traveling at 100 km/h and firing a bullet forward at 200 km/h.
From the ground, the bullet’s speed would appear to be 300 km/h (the sum of the two speeds).
Simple and understandable. Moreover, in this scenario, observers in different locations (e.g., on the train, on the ground, in space) would agree on:

• When the bullet was fired.
• When the bullet hit the target
• The time passed between the two events
• The space traveled by the bullet.

These points seem straightforward, as all observers witnessed the same event, right? However, this is an approximation valid only at low speeds. Even then, there’s a tiny, negligible error. To understand this, consider the same experiment but with a photon (traveling at the speed of light, c) instead of a bullet. Using the same logic, the photon’s speed at the target would be c + 100 km/h, which is incorrect, as nothing can exceed the speed of light. Actual measurements would still show the photon traveling at c. (And if you tried this with a “bullet” traveling near c, the results would be even more counterintuitive.)

The universe operates in a more complex way than described by Newtonian physics. At high speeds, observers disagree on several key points:

• They disagree on when the shot was fired.
• They disagree on when the shot hit the target.
• They disagree on the time interval between the events.
• They even disagree on the distance between objects, and therefore the distance the bullet traveled.

Crucially, all of them are right within their own frame of reference. Their measurements are consistent with their individual perspectives.

This might seem bizarre, but it’s how the universe functions. These aren’t just theoretical constructs; they are observed facts supported by countless experiments. As strange as it sounds, this is reality.

So, if observers disagree on “when” and “where,” and even on the order of events (event B following event A for one observer might be reversed for another), is there anything they do agree on? Yes, they agree on the space-time interval. This quantity (s.i.) is defined as:

s.i. = (Δx)² – (cΔt)² = (Δs)²

where Δx is the spatial distance between events A and B, and Δt is the time interval between them, measured by each observer. The space-time interval is the only quantity on which all observers agree. They won’t agree on Δx or Δt individually, but they will calculate the same value for s.i. Even if the order of events differs between observers, the s.i. remains consistent.

Notice the minus sign in the equation. If the space-time interval is negative, event A can influence event B. If it’s positive, A cannot influence B. Because s.i. is invariant, it defines causality between events. Since Δt is relative, s.i. is the only reliable measure of causal relationships.
In mathematical terms, the space-time interval is a conserved quantity across all frames of reference.

The formula also resembles the distance formula in a Cartesian plane. This suggests that it might represent a distance in a four-dimensional space. This was Minkowski’s insight, leading to the concept of space-time, where time is treated as another coordinate. Points in this 4D space are called events. This will be the subject of the next article, where we’ll explore Minkowski space and space-time diagrams. Stay tuned!