Spacetime Interval: Is (Δs)2 = (ct)2/γ2 a Coincidence?

[Battlemage!]

I was messing around with the relativistic energy equation and stumbled upon something that looks like the spacetime interval equation. So, I'm wondering if there is some deeper connection there, or if it's just an interesting coincidence. I'll just go through it really quickly.

E² = m²c⁴ + p²c²

m²c⁴ = E² - p²c²

m²c⁴ = (γmc²)² - (γmu)²c²

m²c⁴ = (γmc²)² - (γm Δx/Δt)²c²​

Divide all terms by m²γ²

c⁴/γ² = (c²)² - (Δx/Δt)²c²​

Divide all terms by c²/Δt²

(cΔt)²/γ² = (cΔt)² - (Δx)²​

And that looks suspiciously like the spacetime interval with the (+ - - -) sign convention if (Δs)² = (cΔt)²/γ².

(cΔt)²/γ² = (cΔt)² - (Δx)²

(Δs)² = (cΔt)² - (Δx)²​

So the spacetime interval is really (cΔt)²/γ²?

Basically it looks like rest mass corresponds to the spacetime interval, energy corresponds to time, momentum corresponds to space, and the spacetime interval is a function of the Lorentz factor: (Δs) ∝ γ^-1. (The time and space things makes sense to me since I've seen a proof that conservation of energy is related to time translations and conservation of momentum is related to space translations/rotations).

Is this a coincidence or is there a reason for it?

Thanks!

 

[Ibix]

You are basically reversing one way to derive the relativistic energy equation, the starting point for which is the interval (look up the relativistic four momentum). So yes, you've got back to the interval. Typically you would say that the interval is c times the proper time, $\Delta\tau$, which is equal to $\Delta t/\gamma$ for constant velocity (which you are assuming). It's probably better to write it using the proper time, since the way you've written it is kind of disguising the interval to look like something frame dependent when it is an invariant.

 

[Herman Trivilino]

E² = m²c⁴ + p²c²

Rewrite as

$(mc^2)^2=E^2-(pc)^2$

The quantity on the left (square of the rest energy $mc^2$) is a relativistic invariant.

$mc^2$ is the magnitude of the energy-momentum 4-vector. Energy is the temporal component, momentum is the spatial component.