The spacetime interval is a rather important thing in Special Relativity. It allows us to define the separation between any two events as spacelike, timelike or lightlike and more importantly, the Lorentz transformations can be defined as the transformations which keep the spacetime interval fixed.

In that sense, as we know: lenghts and time intervals themselves are observer dependent. They are not absolute notions. On the other hand, the spacetime interval is one absolute notion.

Now, given events $(t_1,x_1,y_1,z_1)$ and $(t_2,x_2,y_2,z_2)$, its definitios is:

$$I = -c^2 \Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2,$$

so that it is the difference between the distance that light traveled between the two events and the spatial separation of the events.

My problem here is that if we try to construct special relativity following the historical procedure there is a certain gap when introducing the spacetime interval.

We can usually go on like this: we start reviewing the problems in electrodynamics which motivated the theory as Einstein himself stated in his paper. After that, we can follow Einstein's procedure and starting with the postulates derive the relativity of simultaneity, the lengths contraction and the time dilation. From that we are able to get the Lorentz transformations.

The next natural step is to give more mathematical substance to this construction, and start labeling events with elements of $\mathbb{R}^4$ so that we finally get to the idea of spacetime. The problem is that in this point one usually just defines this formula for $I$, shows that it is preserved by the Lorentz transformations and shows that it allows us to classify the separation between events.

What I want is to be able to motivate why do we introduce the spacetime interval. It is just a certain object with certain properties, but how do we motivate its importance in the context of relativity and how do we motivate its definition?

After all, as far as I know, originally relativity is trying to solve the inconsistency between Newtonian Mechanics and Maxwell's Electrodynamics. The Lorentz transformations would seem to already to the job. How could one in this context motivate the definition of $I$?

  • Landau discusses this (with spherical electromagnetic waves) in his book. IMHO the best way to motivate $I$ is with the fact that the combination $a_0b_0-\boldsymbol a\cdot\boldsymbol b$ appears a lot in mechanics, electrodynamics, field theory, etc. (nice question by the way, I really want to read the answers) – AccidentalFourierTransform Apr 28 '16 at 15:53
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  • Because we're teaching the relativity unit of modern physics out of Tacheuchi, I've been using a geometric motivations, but I think it still needs refining. Moreover it really requires the decision to go with a geometric description of the physics from the beginning. – dmckee Apr 28 '16 at 16:32
  • As far as I know there are two main approaches to special relativity. One which is the original one would be to start from the motivation coming from Electrodynamics and using Einstein's postulates do arrive at the Lorentz transformations. The other is the more geometrical one, based proposing a spacetime $\mathbb{R}^4$ endowed with one inner product $\eta$ of signature $1,3$. In the second one, the Lorentz transformations are defined as the ones which preserve inner product. For me there seems to be a considerable gap between the approaches, what I'm trying to do is to fill that gap. – user1620696 Apr 28 '16 at 17:15

$I$ has a clear physical meaning if $I\lt 0$ – which is a significant percentage of the spacetime, so to say: $$ I = -c^2\Delta t_{\rm proper}^2 $$ where $\Delta t_{\rm proper}$ is the time measured by clock that moves by a constant velocity (without acceleration); and that visits the point $(x_1,y_1,z_1)$ at time $t_1$ and $(x_2,y_2,z_2)$ at time $t_2$.

Even though all inertial observers may observe and describe the clock above, the description is particularly simple in the reference frame of the clock itself, the rest frame. In that frame, $(x'_1,y'_1,z'_1)=(x'_2,y'_2,z'_2)$ and $I=-c^2 (t'_1-t'_2)^2$ where the primes indicate that I had to use different coordinates in that frame.

So any inertial observer must be able to calculate the total duration shown by the clock which connects the two events in the spacetime. The next question is: Why the formula in the general frame is given by the Pythagorean formula with the diverse signs?

Well, it's simple: the proper time measured on light-like trajectories must be zero, $I=0$. Why? Because under the transformation to another inertial system, the light only changes its frequency (by the Doppler shift) but it doesn't change the trajectory (world line) of the light given by $\vec r = \vec v\cdot t$ where $|\vec v|=c$.

So Einstein's postulate about the constancy of the speed of light says that if $I=0$ in one inertial system, it is $I=0$ in other inertial systems, too. But it happens exactly for the light-like trajectories that have $$ -c^2 \Delta t^2 + \Delta x^2+ \Delta y^2+\Delta z^2 = 0$$ which is equivalent to $|\Delta r|/\Delta t = c$, the right speed, as we mentioned. But that implies that $I$ has to be a function of $-c^2 \Delta t^2 + \Delta x^2+ \Delta y^2+\Delta z^2$ (a function that is zero if the argument is zero) and in the rest frame, one may figure out the power and the coefficient $c^2$.

So the invariance of $I$ is basically equivalent to the principle of the constancy of the speed of light because the latter is a special case of the constancy saying that if $I=0$, then $I'=0$, too. And for nonzero $I$, the invariance of $I$ may be seen by a symmetry – e.g. by looking at both observers from the viewpoint of a reference frame "in the middle" where both move by the same speed in opposite directions.

For $I\gt 0$, the interval is spacelike and spacelike world lines aren't really allowed. $I$ is related to the proper spatial distance between two events in the spacetime although it's hard to measure it directly (clocks aren't possible because they're not allowed to move superluminally). But because $I$ is an analytic function of $\Delta x^\mu$, it must be invariant when it's negative, too.

The comments about the "proper time" measured by the clocks is just a special example optimized for $\Delta x^\mu \Delta x_\mu$. Similar 4-products of vectors are extremely important and natural for other choices of two 4-vectors, e.g. $p^\mu \cdot \Delta x_\mu$ for the phase of a de Broglie-like wave, $p^\mu p_\mu$ for the squared mass of a particle with some momentum, and many many others. Because all the vectors such as $p^\mu$ transform like $\Delta x^\mu$, the invariance of the inner 4-products under the Lorentz transformations follows from the same algebra. But the detailed physical interpretation of all these 4-vectors and their inner products depends on what 4-vectors we consider.

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