Can someone with an actual physics background clarify my understanding:
My interpretation of "dark matter" isn't that it's literally "matter" that is invisible and we are trying to find a way to detect it and interact with it.
Rather it's just a shorthand term that explains our math. We detect gravity. The only thing that we know that makes gravity is matter. We detect more gravity for the amount of matter we see. Therefore we call what we cannot detect "dark matter". But that's it. We have no other semblance that it is somehow "matter". We might as well call it "Unknown Gravity Generator".
Is that right or am I off base?
If I'm right, then could it just be as simple that gravity Works Differently at galactic scales much the same way that everything works differently at quantum scales and relativistic (ie high velocity) scales? And as such, we don't really need to "find" dark matter, but rather we just need to find out exactly under what circumstances the equations change - much as Einstein did for relativity? And it's a super hard problem because while we can make things move at the speed of light in a lab, we can't really give something galactic mass in a lab, and observing galactic-masses in the real world on a human scale is very hard as any relevant measurements would require multiple lifetimes to observe (hypothetically)
If I'm wrong, in what other ways is dark matter "matter-y" that makes it a useful term? And do scientists believe there is a chance we could actually be able to "generate" it an experimental environment?
Its characterizing bulk behaviour is that dark matter exerts a pressure on galaxies immersed in it, effectively squashing them inwards (or, if you like, preventing discoid galaxies from flying apart as they spin quickly).
Energy-density is encoded in the (symmetric, gauge-invariant, conserved, (0,2)) matter tensor of the Einstein Field Equations of General Relativity.
Among other things this preserves notions that matter/energy cannot be created or destroyed, only converted from one form to another.
It is the microscopic constitution of the bulk energy-density, and the details about how or if it can be converted into photons or some other radiation, or other types of matter, that is not known, although the bulk behaviour imposes numerous constraints.
As with all matter in General Relativity, dark matter is subject to the strong equivalence principle, just like laboratory Cavendish experiments, or binary or triple pulsar systems, or in galaxies' peculiar motions within galaxy clusters.
Finally, your "as simple that gravity Works Differently at galactic scales" is a rejection of the Strong Equivalence Principle, and also raises the question of the (unknown) microscopic details of how the inverse square law holds up so well everywhere except the edges of many known galaxies. Typical approaches involve a new long-range ("fifth") force that in bulk effectively applies a tension on the outer reaches of galaxies where MOND transitions from the familiar 1/r^2 inverse square law for gravity to the slower-decaying 1/r law. Tension has the dimensions of negative pressure, so essentially rather than outside-the-galaxy dark matter pushing the outer gas clouds and stars inwards, we have something inside the galaxy reaching out and pulling the same gas clouds and stars inwards. One is then left struggling to do anything other than to describe the microscopic behaviours of a field that sources this negative pressure, and making that field denser somewhere towards the middle of galaxies.
My interpretation of "dark matter" isn't that it's literally "matter" that is invisible and we are trying to find a way to detect it and interact with it.
Rather it's just a shorthand term that explains our math. We detect gravity. The only thing that we know that makes gravity is matter. We detect more gravity for the amount of matter we see. Therefore we call what we cannot detect "dark matter". But that's it. We have no other semblance that it is somehow "matter". We might as well call it "Unknown Gravity Generator".
Is that right or am I off base?
If I'm right, then could it just be as simple that gravity Works Differently at galactic scales much the same way that everything works differently at quantum scales and relativistic (ie high velocity) scales? And as such, we don't really need to "find" dark matter, but rather we just need to find out exactly under what circumstances the equations change - much as Einstein did for relativity? And it's a super hard problem because while we can make things move at the speed of light in a lab, we can't really give something galactic mass in a lab, and observing galactic-masses in the real world on a human scale is very hard as any relevant measurements would require multiple lifetimes to observe (hypothetically)
If I'm wrong, in what other ways is dark matter "matter-y" that makes it a useful term? And do scientists believe there is a chance we could actually be able to "generate" it an experimental environment?