How physics equations transform under inverse of parameters

[JustTryingToLearn]

TL;DR

How do physics equations change under inversion of individual parameters?

It is obvious that there is a one-to-one relationship between real numbers (defined to include infinity) and their multiplicative inverses (assuming we map the inverse of zero to infinity and vice versa). Thus, one should be able to replace the distance between two points in space with it's inverse and still find reasonable results for some type of mathematical manipulation. Has there been any extensive work in theoretical physics that extends this concept and determines its limits? In other words, if time, space, mass, charge magnitude, and all other physical parameters were replaced by their inverses...how would the equations of physics change? If this has been done, can someone please tell me the name of the field? Thanks.

 

[Mark44]

Thus, one should be able to replace the distance between two points in space with it's inverse and still find reasonable results for some type of mathematical manipulation.

But why would anyone want to do this? The example here is distance between two points. What would be the meaning of "inverse distance"? The closest I can come up with for an example of what you're talking about are things like gravitation, and magnetic force, both of which obey inverse-square laws -- the closer together the two objects, the greater the forces of attraction (for gravity) or attraction/repulsion (for magnetism).

 

[Dale]

Summary: How do physics equations change under inversion of individual parameters?

Thus, one should be able to replace the distance between two points in space with it's inverse and still find reasonable results for some type of mathematical manipulation.

This is already done wherever it is found to be useful. For example, in studying waves it is common to work in “k-space” where the units are inverse distances and the relationship with physical space is through the Fourier transform.

 

[sophiecentaur]

in studying waves it is common to work in “k-space” where the units are inverse distances

Also, the period of an oscillation or the frequency are alternative ways of measurement. Is that the sort of thing the OP had in mind?

 

[Dale]

Also, the period of an oscillation or the frequency are alternative ways of measurement. Is that the sort of thing the OP had in mind?

Yes, that is the time domain of the same thing. One is in units of s and the other in units of Hz=1/s

 

[JustTryingToLearn]

I guess my question wasn't as clear as I intended. I will try to explain differently. If we let time (t) go to the "additive inverse" of time (-t), the equations of motion that we currently have remain the same. There is a symmetry there. My curiosity is this: Is there a set of parameters that can be replaced with their "multiplicative inverses" such that the equations of physics as we know them retain their original form? I'm thinking in terms of gauge transformations. Keep in mind that I am quite new to most gauge transformation ideas. So...there may be a very general reason why what I'm asking is not even possible mathematically. Either way, I'd be interested in know what that reason is.

Thanks for all the responses thus far.

 

[sophiecentaur]

But why would anyone want to do this?

Purely for convenience and depends on the actual calculations you happen to be doing.
A common example is in Electrical Measurements. There are Resistance and Reactance which go to make up the Complex quantity Impedance. Then there are Conductance and Susceptance which go to make up the quantity Admittance. If you are working with Impedances in series, you can just add them but if you have Admittances in Parallel, you just add them.
A bit boring and 'why bother' till you actually get around to using these things and then hopping from one world to the other becomes second nature and can save loads of time.