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.(adsbygoogle = window.adsbygoogle || []).push({});

E^{2}= m^{2}c^{4}+ p^{2}c^{2}

m^{2}c^{4}= E^{2}- p^{2}c^{2}

m^{2}c^{4}= (γmc^{2})^{2}- (γmu)^{2}c^{2}

m^{2}c^{4}= (γmc^{2})^{2}- (γm Δx/Δt)^{2}c^{2}

Divide all terms by m^{2}γ^{2}

c^{4}/γ^{2}= (c^{2})^{2}- (Δx/Δt)^{2}c^{2}

Divide all terms by c^{2}/Δt^{2}

(cΔt)^{2}/γ^{2}= (cΔt)^{2}- (Δx)^{2}

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

(cΔt)^{2}/γ^{2}= (cΔt)^{2}- (Δx)^{2}

(Δs)^{2}= (cΔt)^{2}- (Δx)^{2}

So the spacetime interval is really (cΔt)^{2}/γ^{2}?

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!

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# What is the spacetime interval?(Δs)^2 = (ct)^2/gamma^2?

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