So, This is the third article of the relativity series. It took me some time to write down because I was busy with other things and gave priority to the more technical articles. Furthermore I am not really sure that any of you is actually reading this stuffs… anyway, lets get started with some more basic concepts.

One of the main question that you may ask if you are new in this field is “why all this complicated stuffs? Isn’t the old good Newtonian mechanics enough to explain all the stuffs happening around us, including gravity, acceleration, distances, etc?” Well it turns out that the answers is No.

Newtonian laws are easy to understand and they work great for speeds much smaller than the speed of light. Furthermore, keeping space and time as two separate entities, they do not seems to violate our perception of world. For example suppose I am sitting in a train that is traveling at 100km/h towards a ground target. if I shoot a bullet towards that ground target and the bullet speed is 200km/h, the bullet will hit a target on the ground at 300km/h (bullet speed+ train speed). Easy to calculate and to understand. More important, in this case, the shooter, the guy sitting next to the target and the astronaut observing the scene from the space station, 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.

All this 4 points seem pretty straight forward. After all, they were all witnesses of the same event, right?. Well it turns out that, that is just an approximation and is valid only for low speeds. Even in that case they are committing a super-small error that is negligible in everyday conditions.

For understanding this we can repeat now the experiment but instead of the bullet we can shot a photon (traveling with speed c). How fast the photon will hit the target? If we just use the same formula above we get that the photon is hitting the target at speed (c+100km/h) that is wrong since the photon now exceed the speed of light… If you try to actually do the experiment, you will measure that the photon will still travel at c. [If you think that photon is a special case that you can use a bullet that travel a speed (c-10km/h).. and the result will be even stranger!…]. So, the point is the reality the universe behave in more complex manners than what described by Newton laws. In general for fast traveling bullets, the three observers notice something strange, that can be resumed with the 4 points here:

- The three observers do NOT agree when the shot was fired.
- The three observers do NOT agree when the shot hit the target.
- The three observers do NOT agree on how much time passed between the two events.
- The three observers do NOT even agree about how much space there is between things. So they cannot agree neither on how much the bullet traveled!
- All of them are right at the same time, in the meaning that all the measurements are consistent within their own reference frame.

So this seems crazy but is the way how to the universe work. I want to remark again that is not just a mathematical construct, but those bullet points are actually FACTS that can be tested experimentally (and have been tested in a million different ways!). So for how strange they can sound that is what happen in the real world… just get over it!

So back to the observers.. they do not agree on the “when”, they do not agree on “where” or even in the order of the events! event B following event A observed by the guy on the moon, could be observed with the opposite order by ET sitting on the edge of the universe… So the question is.. do observers ever agree on something?.. well luckily yes… It turn out that they do agree on something called the space-time interval. This quantity s.i. is defined as :

**s.i. = (Δx)² – (cΔt)² = (Δs)²
**

With reference to event A and B: **Δ**x is the distance between them and **Δ**t is the time between A and B measured by each observer. So s.i is the only value that observers agrees on and when they calculate it, they get the same results. So remember, that they will NOT agree on the pure **Δ**x nor they will agree on the pure **Δ**t, but they do will calculate the same s.i. value. Indeed event A can be in the past for Observer 1 and in the future for Observer 2. but both of them will calculate the same s.i.

You should notice the minus sign in between there. If the space time interval between A and B is negative means that A can influence B. If is positive, means that A will never be able to influence B. Now, since the s.i. is the same for all the observers, it is actually what define CAUSALITY between events. Indeed, since the **Δt **are relative to the observers, the only reliable quantity is s.i. !

In a bit more mathematical terms will mean that the space time interval is a quantity that is conserved for all frame or references.

Furthermore if we look at the formula, it closely resemble a distance between points in a standard Cartesian plane.. So maybe that is actually a distance calculation in some strange 4-d space? … Well that is what Minkowski found out… and the 4-d mathematical space derived was just called space-time, where time is just “another” coordinate, and the points in this 4-d space are called Events. But this is an argument for the next article, where we will discover the Minkowski space and the space-time diagrams.. Stay tuned!