I had the name related to this pop in my head, as to why it is p, while letting my dog out to pee 
Before Newton/Galileo there was no momentum and impetus would be the closest term, I don’t know if ‘i’ and ‘m’ were already in use but it is probably because the Latin root of impetus is petere so ‘p’ was used.
So blame Buridan
Thanks, everybody! I am now 7% closer to understanding this musical joke I heard on Mountain Stage.
From Stewed Mulligan’s song about Albert Einstein:
“Had a frame of reference,
Set it on the fence.
Showed it relativity,
Ain’t seen the damn thing since.
(chorus)
Albert, dance around,
Albert, be profound.
Albert, let your hair stick out,
And your socks fall down.”
Ah, let me try and clarify some things.
Under Newtonian and SR you have “initial frames” which really can just be thought of as locations that don’t have external forces acting on them. So if you think of a location billions of light years away from other objects as an example. These are “privileged” frames and in Newtonian physics these privileged frames are interchangeable and you can reference a universal frame that basically ignores the complicated bits but includes all of these privileged frames.
Basically these “privileged” frames go away under GR. But this really doesn’t relate to the speed you are going in that privileged context (massive oversimplification)
To help with the speed of light question in your original question can be visualized with a spacetime diagram.
Think of it as an normal X + Y 2D graph, where up is you traveling through time and Y is any movement you make in the normal non-time dimensions.
The units here are natural units, so +1 in the X axis is the distance that a photon would travel in X time. This results in a photon’s world line being exactly 45 degrees from all observers viewpoint.
This is where it is critical to understand that the speed of light is the speed of causality, that just happens to be the speed that mass-less particles like light must travel.
You also have a “world line” and even when you are “stationary” you are still moving through time.
Here is where things get interesting, because you are still in the traditional space dimensions you are actually following the shortest path possible through time, which means your clock is moving “fast” as possible to an external observer. But that photon, which looks like it is going at a 45 degree angle will actually experience no time at all.
Lets say that you are way out in the middle of space and had no references to anything else but were moving at a significant portion of the speed of light. You wouldn’t notice anything, but because everyone views the speed of light the same, even if you were moving at 1/2c you still can’t interact with other objects in front of you faster than the speed of causality. Basically you are not following the shortest path through time, so for an external observer your clock would be running slower.
If we could violate the speed of causality and send information faster than speed of light and lets say you held a white lamp up and some how looked at the light from a position ahead of where you are. That white light would be blue shifted from that slower for a FTL front observer and red shifted fro a FTL rear observer.
The speed of causality, which really limits the ability to exchange information wouldn’t allow for that, but the energy levels of the light would change, but the user wouldn’t notice.
Thus the Einstein equivalence principle.
Tests to find violations of this principle, basically showing that a privileged frame exists are where some of the more seemingly crazy ideas like the universe is a hologram come from.
Please note this is the cliff notes version, and the above analogies won’t be exact.
The implications are kind of cool though. For example, if you ignore air resistance, if you toss a base ball it is actually following a path that is straighter in spacetime than you are experiencing, as the pressure you feel on your feet from what we call gravity is actually the earth accelerating you away from your preferred shortest path through space-time.
Or the one that makes people get riled up: The international space station (once again ignoring friction and tidal forces) is not “falling and missing” the earth, it is actually traveling in the straightest (geodesic) path through spacetime, as is the earth when orbiting the sun.
The crazy thing here is that the earth spinning, actually twists space time too, resulting in frame dragging. So light traveling in the direction of rotation of the earth will move faster than light moving against the rotation. The effect is tiny but it will induce math panic attacks for people studying black holes.
If you use the simple Schwarzschild black hole’s which don’t spin the math is hard, but once you introduce spin and have to bump up to the Kerr metric things get crazy.
This is an example of the “geodesic” shortest possible line for a body orbiting a spinning black hole
As a reminder, to the body in the above orbit, it is not undergoing acceleration!!
How cool is that.
Remember that these effects are all experimentally verified, so even if we do find a unifying theory it will still be a crazy weird world that will be a challenge and fun to work through.
TL;DR there is no privileged reference frame as we know it today in GR, all of our clocks and reference frames are just as valid yet the man or the speed of causality is still the boss.
Inertial reference frames are still just as “special” in general relativity as they are in special relativity. The lack of “privileged” reference frames means that no one inertial reference frame is in any way preferred over any other. The reference frame in which I’m at rest is no more nor less valid than one in which a cosmic ray is at rest, with a relative speed of .99999999c to each other.
In special relativity it is always possible to choose a global coordinate system where the Christoffel symbols vanish everywhere, these are the ‘privileged’ inertial frames of special relativity. In an accelerated frames of reference in SR the Christoffel symbols do not vanish and there’s not obvious procedure for extending such a frame spatially.
In general relativity the Christoffel symbols can always be made to vanish locally to construct a local free-falling frame, but, except in the special case of flat spacetime, the Christoffel symbols cannot be made to vanish globally and there’s no general procedure for constructing global frames.
Locally the inertial and/or gravitational forces felt by an observer are directly dependent on the Christoffel symbols of their frame.
