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

Just doing some hobby physics while I put off working on my research. In one dimension, the function

\begin{equation}

f(a,b)=[1-\exp(-(a-b)^2)]

\end{equation}

vanishes when a=b. In Minkowski spacetime though, such a function is not so easy to find (if you require Lorentz invariance). If we attempt the same thing with the 4-vectors x and y,

\begin{equation}

f(x,y)=[1-\exp(-(x-y)^2)],

\end{equation}

the function vanishes everywhere on the lightcone (x-y)^2=0. But I'm looking for a function that vanishes only when x isequalto y. From my perspective, this hypothetical function should approach zero as x gets *near* to y. However, *near* is hard to define for what I'm looking for since the usual Minkowski metric only tells you how close you are to the lightcone. I'd like to be able to write something like

\begin{equation}

f(x,y)=[1-\exp(-(t_x-t_y)^2-(\vec{r}_x-\vec{r}_y)^2)],

\end{equation}

but that's obviously not Lorentz invariant. So I'm stuck, and would like your help. Do any of you clever people have some ideas?

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# Searching for a function that vanishes only when x^mu = y^mu

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