# Space time Invariant

Standard Minkowski space time Invariant definition tells us $d\phi^2 = c^2 dt^2 -\sum_{n} (dx^n)^2$ $$d\phi=\textrm{Space time Invariant}$$

$"n" \textrm{refers to all possible spatial coordinates}$ $\color{#20A900}\textrm{Prove that Space time Invariant remains constant in all frame of references}$ Let's take the simplified form(only taking the one spatial dimension) for a Frame of reference $\phi^2 =c^2t^2 -x^2$ (Here $c$ = speed of light and $u$ =relative velocity between two reference frames )

For Another frame of Reference Lorentz transform says $t' = \beta({t-\frac{ux}{c^2}})$ (where $\beta= \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$ )

And $x' =\beta( {x-ut})$ Now inserting these on that equation $\phi'^2 =c^2 t'^2- x'^2 \implies \phi'^2 =c^2 \beta^2 (t-\frac{ux}{c^2})-\beta^2(x-ut)^2$ $= c^2 \frac{1}{1-\frac{u^2}{c^2}} (t^2 -2\frac{uxt}{c^2}+\frac{u^2x^2}{c^4})-\frac{1}{1-\frac{u^2}{c^2}}(x^2-2uxt+u^2t^2)$ After simplifying $\phi'^2= c^2 t^2 -x^2 = \phi^2$ Which means $\phi=\phi'$ or Space time Invariant as a constant.

But the in original equation It is stated as $d\phi^2 +dx^2 +dy^2 +dz^2 +(icdt)^2 =0$

So is it Reasonable to include all possible spatial dimensions in that Minkowski space time equation? (Shown in the first) Note by Dwaipayan Shikari
6 months, 2 weeks ago

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Hi @Dwaipayan Shikari, I can't figure out what you mean by "is it reasonable to include all possible spatial dimensions in that Minkowski space time equation?" Can you say more specifically what your question is?

- 6 months, 1 week ago

I have seen only $dx,dy,dz$ spatial dimensions in minkowski space time equation along with time dimension. It is true for 4d space . My question is , "Is it possible to add more spatial dimensions like $dx,dy,dz$ ?"

- 6 months, 1 week ago