# Why Are Equally Spaced Events Preserved in Transformation?

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• TonyEsposito
In summary, the author in The Geometry of Spacetime by Callahan discusses the transformation from one system to another in inertial motion. He shows a spacetime diagram and explains that equally spaced events in one system will also be equally spaced in the new system due to the linearity of the transformation. This is not an assumption, but a consequence of the transformation being linear. A search for "midpoint affine transformation" can provide more information and proof on this concept.
TonyEsposito
Hi guys, I'm reading a book where the author, only form the invariance of the speed c draws conclusions about the transoformation from a system to another in inertial motion. The author shows a spacetime diagram (x,t) and the two dimensional light cone, he marks two events on the light cone (x1,t) and (-x1,t) and another event (0,t). He says since the three events are equally spaced in the first system and the transformation is linear they must be equally spaced also in the new system, why he assumes this? (the book still did not explained the invariance of the Minkwosky Norm), my brain is freezed now. Summing up the question: why he assumes that equally spaced events in a system must be equally spaced in another? thanks!

TonyEsposito said:
i'm reading a book

What book? Please give specific references.

The Geometry of Spacetime by Callahan. Page 32 "the graphical solution".

TonyEsposito said:
He says since the three events are equally spaced in the first system and the transformation is linear they must be equally spaced also in the new system, why he assumes this?

It's not an assumption, it's a consequence of linearity. A linear transformation means that the new coordinates ##x', t'## are linear functions of the old coordinates ##x, t##. In other words, we must have

$$x' = ax + bt + c$$
$$t' = dx + et + f$$

where ##a, b, c, d, e, f## are all constants (and some might be zero, we don't know at this point). You should be able to show that if we have three points equally spaced in ##x, t##, such transformations must give three points equally spaced in ##x', t'##.

I'm afraid I'm not able to find a proof of this :( can you help me?

Last edited:

## 1. Why are equally spaced events preserved in transformation?

Equally spaced events are preserved in transformation because they represent a fundamental aspect of the physical world. In physics, the concept of time is assumed to be uniform and unchanging, allowing for events to occur at regular intervals. This uniformity is a fundamental principle in the laws of physics and is therefore preserved in transformations.

## 2. How do transformations preserve equally spaced events?

Transformations, such as spatial rotations or translations, do not change the underlying structure of space and time. This means that the intervals between events remain the same before and after the transformation. In other words, the transformation does not affect the uniformity of time.

## 3. Can equally spaced events be preserved in all types of transformations?

Not all transformations preserve equally spaced events. For example, certain types of non-linear transformations, such as stretching or compressing space, can alter the uniformity of time and result in unequally spaced events. However, most commonly used transformations in physics, such as rotations and translations, do preserve equally spaced events.

## 4. Why is the preservation of equally spaced events important in physics?

The preservation of equally spaced events is important in physics because it allows us to make accurate predictions about the behavior of physical systems. By assuming that time is uniform and equally spaced intervals are preserved, we can use mathematical equations to accurately describe and understand the behavior of objects in the physical world.

## 5. Are equally spaced events preserved in all reference frames?

Yes, equally spaced events are preserved in all reference frames. This is because the concept of time and its uniformity is independent of the observer's perspective. While the measurement of time may vary between different reference frames, the underlying principle of equally spaced events remains the same.

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