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)

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`\frac{2}{3}`	$\frac{2}{3}$
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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?

Tiffany Wang - 4 months, 4 weeks 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 $dx,dy,dz$ ?"

Dwaipayan Shikari - 4 months, 4 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$