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Is there any simple way to intuitively understand spacetime interval, proper time and proper length?

Qmechanic
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Toneri Otsutsuki
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1 Answers1

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Let's start with proper length and proper time because they are probably the easiest to understand.

Let us have to inertial reference frames $S$ and $S'$ where $S'$ is moving relative to $S$. Thus, we define $S$ as "stationary" (because we are in this frame). Now a "proper" measurement on an object or event is any measurement made by an observer which is stationary with respect to that object or event.

This means that if we measure the length of an object stationary in $S$, we measure proper length. However, if we measure the length of an object in $S'$, this is not proper length because it will be contracted due to length contraction. An observer in $S'$ could measure proper length of an object in $S'$ because as seen from his frame, the object is stationary.

Now on to the invariant spacetime interval. You are probably familiar with the pythagorean theorem:

$$(\Delta {d})^{2}=(\Delta {x})^{2}+(\Delta {y})^{2}+(\Delta {z})^{2}$$

which gives the same distance for every observer in euclidean geometry. However, Minkowski spacetime is not euclidean and thus obervers at different velocities measure different distances due to length contraction. The invariant spacetime interval, however, is defined in such a way that it gives the same result for every observer. It can be written as either

$$(\Delta s)^{2}=(\Delta ct)^{2}-(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}$$ $$(\Delta s)^{2}=-(\Delta ct)^{2}+(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}$$

For more information, you may want to read Spacetime interval on Wikipedia.

jng224
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