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The spacetime interval is defined as follows:

$$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$

or in tensor notation:

$$\Delta s^2 = \eta_{\mu\nu} \Delta x^\mu \Delta x^\nu$$

When I first studied introductory special relativity, I didn't even pay much attention to this quantity -- it was mostly time dilation, length contraction, and fancy paradoxes.

However, it has caught my attention now. The book I'm reading simply defines the quantity, and claims that it's invariant.

Now, just from tensor analysis and ignoring special relativity, $\eta_{\mu\nu} \Delta x^\mu \Delta x^\nu$ looks like a contracted product of a doubly covariant tensor with two contravariant tensors, mathematically proving it's an invariant. Great!

But, what I do not understand is why is the spacetime invariant defined the way it is? Why is it $-(c\Delta t)^2$, and not $(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$ ?

I want the physical motivation behind this formula.

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Leonard Susskind professor at Stanford University has en excellent explanation of why the space-time invariant is defined the way it is. I've included a video link from where he begins to talk about the subject, if you'd like to watch it.

He compares space-time to euclidean geometry where the normal Pythagorean theorem says that the square distance between two points is the sum of the square of the distance in your coordinate system. i.e $c^2= a^2+b^2$. This is a quantity that is invariant i.e. we could rotate our coordinate system and describe our new set of points in the new coordinate system with primed coordinates, then we would have the following invariant quantity between our new and old coordinates, $a^{\prime 2}+b^{\prime 2} = a^2 + b^2$. Similar in space-time we also look for an invariant quantity that all observers in different reference frames will agree upon. If we begin with the Lorentz transformation ($c=1$), we have

$$ x^\prime = \frac{(x-vt)}{\sqrt{1-v^2}} \quad t^\prime = \frac{(t-vx)}{\sqrt{1-v^2}} $$ Let us look for an invariant quantity. We could begin and try with $t^{\prime 2}+x^{\prime 2} = t^2 + x^2$, substituting $x^\prime$ and $t^\prime$ into the equation we will notice that it does not read $t^{2}+x^{2} = t^2 + x^2$, so this is not invariant property in space-time. But if we try $t^{\prime 2}-x^{\prime 2} = t^2 - x^2$ and do the same procedure we will find that this is an invariant property!

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    $\begingroup$ Great! My book just defined the space-time interval and then stated that Lorentz transformations are those which keep this interval invariant. Doing it the other way feels more intuitive to me. $\endgroup$ – SilverSlash Jan 13 '17 at 11:26
  • $\begingroup$ Yeah so did my book as well >_< $\endgroup$ – Turbotanten Jan 13 '17 at 11:50
  • $\begingroup$ Do you think you could include Susskind's arguments in the post here? Otherwise, this would be considered a "link-only" answer and would likely be deleted. $\endgroup$ – Kyle Kanos Jan 13 '17 at 12:16
  • $\begingroup$ I've included the argument. $\endgroup$ – Turbotanten Jan 13 '17 at 12:34
  • $\begingroup$ You can do it either way round. Most students prefer Lorentz transformation first; then when you get deeper in the subject, especially if you go to general relativity, then you begin to fall in love with metrics and geometry, and then you put the interval first. $\endgroup$ – Andrew Steane May 5 at 14:12
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Here are two different ways to introduce the subject of Special Relativity. Both are good ways, and each can be used to derive the other.

Approach 1: symmetry principles. We assert Relativity Postulate (physical behaviour the same relative to an inertial frame, no matter the state of relative motion of that inertial frame with others) and Speed of Light Postulate (there is a finite maximum speed for signals). From these we can derive the Lorentz transformation and hence what quantities are invariant. The spacetime interval is one such quantity.

Approach 2: geometric assertions about spacetime. We assert that spacetime is a smooth differentiable manifold with a Minkowskian metric $\eta$. The metric is itself a statement of that which is invariant; the Lorentz transformation $\Lambda$ is then defined as that class of transformations which satisfies $$ \Lambda^T \eta \Lambda = \eta $$ (Here I have used matrix notation in which $T$ is a transpose and $\eta$ has components $\eta_{ab}$.)

Your question is closest in spirit to approach 2. The question then becomes, "why the Minkowski metric? Why not some other metric?" The answer goes to the heart of what kind of universe we have. One can argue that if the metric were that of a 4-dimensional Euclidean space, for example, then there wouldn't be any sense in which time is different from space, and this would amount to such a different way of things that it is hard to even describe it as a physical universe where there can be conservation laws of the type that allow one to single out and label things by their worldlines. There would be no sense of limits to causality, of past and future. Other metrics you can consider, such as diag(-1,-1,1,1) also give such a different type of spacetime that it would be totally unlike the ways things actually are in the universe as we find it.

So in so far as one can talk of a "physical motivation behind this formula" as you ask, it would be "well this is deeply and directly connected to the notion of causality and the causal structure of spacetime. It also expresses the notion that one spatial direction is as good as another in the basic structure of spacetime."

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Because it let's you quickly recognise the potential for a causal relation between the two events. So, (noting the extra $\Delta$ you added before the s is removed)

$s>0$ (space-like) more space in between than light can cross in the time => no causal relation
$s=0$ (light-like) exactly on the "light cone"
$s<0$ (time-like) less space in between than light can cross in the time => A causal relation is possible

It's the way light works "against space" or rather travelling through space that is being modelled. It's not the same as the magnitude of a vector measurement of distance.
Obviously you could give $(c\Delta t)^2$ the same sign as the distances. The resultant quantity, whilst having some uses, would just be a vector in space-time, it would not have the same (or as much, in my view) physical significance... and crucially definitely does not get to be called "space-time interval".

Further note there is a choice about whether to use -+++ or +--- signs for the terms in the equation, this choice of -+++ is just a matter of convention.

(What's really cool is that it can be shown that s is preserved under the Lorentz transform; proving that causality cannot be affected simply by changing your frame of reference. Neat.)

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Let's define an event $A$, a source of light is emanated from the origin at $t=0$. Let at time $t$ the light reaches a point $B$. Let the coordinate of space be $(x, y, z)$. If the distance travelled by the light in time $t$ is given by $$c^2t^2=x^2+y^2+z^2$$ $$c^2t^2-x^2-y^2-z^2=0.$$

Consider another frame X’ which is moving at a velocity $v$. Observing the same event in Minkowski space time gives $$c^2t'^2=x'^2+y'^2+z'^2$$ $$c^2t'^2-x'^2-y'^2-z'^2=0.$$
We called the term interval. This event shows that if the interval has zero value in one frame of reference, it should be zero in all inertial frame of reference, since light has a constant velocity in all frame of reference(Postulate of special relativity). Thus the interval between any two events of $X$ and $X'$ coordinates has a linear dependence. Suppose for an observer in $X$ frame the $X’$ frame is moving at some velocity $v$ then linear dependence of interval is given by $$c^2t^2-x^2-y^2-z^2=\alpha (c^2t'^2-x'^2-y'^2-z'^2).$$ Where $\alpha$ depends only on the magnitude of velocity. If not it will contradict the isotropic nature of space. If the observer is at $X'$ frame, then the $X$ frame will be moving at at a velocity $v$. Thus the linear dependence of interval is given by $$c^2t'^2-x'^2-y'^2-z'^2=\alpha(c^2t^2-x^2-y^2-z^2).$$ Substituting this in the above equation gives, $$c^2t^2-x^2-y^2-z^2=\alpha^2(c^2t^2-x^2-y^2-z^2).$$ Which gives $\alpha=1$.Thus the interval $$c^2t'^2-x'^2-y'^2-z'^2=c^2t^2-x^2-y^2-z^2$$ is an invariant under Lorentz transformation.

The constancy of light and isotropic nature of space defines the invariant interval.

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