What is the essential difference between constant speed and acceleration? I do know what the difference is but what I am trying to understand is how an object knows its speed is changing and/or how space knows an object is accelerating. The particular thing I am interested in is the fact, as I understand it, that a charged particle that moves at changing speed generates E/M radiation whereas that same particle at rest or moving with constant speed does not.
At any given instant, the particle is either actually not moving or is moving at a constant speed -- how does a charged particle "know" what its previous speed was or how does "space" know this? Does this not imply some sort of "memory." At a macro level, a large object accelerating deforms but what about a very simple particle, like an electron which I am told lacks internal structure -- how does an electron "know" it is accelerating?
EDIT: To what extent could the following explanation be true (even if only remotely so):
An electron undergoing acceleration has a field around it and as with macro objects, pushing it causes the field to deform. This deformed field then "expresses" photons. However, if the acceleration was constant, why would you get a continuous stream of photons? I would imagine that only at the time that the field changed in shape would a photon or photons be expressed. On the other hand, if the acceleration changed so that the field kept changing shape, it would be at the time of the change that the photons would be emitted -- an electron undergoing constant acceleration in one direction would not be expected to emit a stream of photons. But I think it nonetheless does emit photons and if that is so, what is the triggering event for a photon to be emitted? Is it after a certain amount of time accelerating and if so, how does the electron measure this time.
 A: Electromagnetic radiation due to a charged particle is a classical effect, meaning we do not need special relativity or general relativity in order to derive it. This isn't to say they don't provide further insight, but I would argue they muddy the water more than necessary.
One of the fundamental ideas is that propagation of electromagnetic force occurs at a finite speed. In terms of the field (which is only different in a philosophical sense for classical electrodynamics), we can say that changes to the field due to sources (charges) propagate at a finite velocity, $c$, the speed of light.
Some other key insights: electric charge is the fundamental source and observer of the electromagnetic force (field). All changes to the EM field are due to the presence and movement of charge, and the only entities which interact with the field are, by definition, electric charges. Therefore, the presence of electromagnetic waves is a statement of the relationship between an electric charge and all other electric charges in the universe.
As a previous answer described, velocity is relative, so there's no way to tell, between two charges moving past each other at a fixed velocity, who is moving and who is stationary. Compare this with acceleration, where those continuous changes are realized in a more absolute sense. To some sense, velocity is no different than absence of motion, depending on your perspective; but the same cannot be said of acceleration. This is embedded in e.g. Newton's laws of motion ($\mathbf{F}=m\mathbf{a}$). Velocity and acceleration are not analogous concepts.
Finally, a wave in the electromagnetic field is a description of an impulse which propagates through the field; a charge moving at constant velocity does not create an impulse, but a charge which accelerates or decelerates relative to another will. See the animation of Bremsstrahlung here.
So that should put you on your way towards understanding how a charged particle "knows" it's accelerating. It doesn't. The fact is, if a charged particle accelerates relative to a fixed observer, the electromagnetic field will have "ripples" away from the particle. The particle doesn't have a say in anything. No need to consider anything quantum mechanical or anything in general relativity, just good old classical electrodynamics.
A: It's clear that an elementary particle like an electron "knows" its momentum as an intrinsic property, as that is the frequency of the wave function (at least, given a reference frame).
But that's just looking at the particle as a single point in time.  In QM, the wave function is a function over space and time.  That is, it represents the whole world line.  Where the particle is accelerating, the frequency is changing, so it's a "chirp" rather than a constant value.
So, you could say that the acceleration is also encoded in the wave function.
But... it's really the interaction with other objects that causes the acceleration.  The wave function of the entire system will encode the Newtonian movement of the particle as the path that doesn't cancel.  Acceleration is "in there" in the results, but is not a free parameter than can be set (like momentum or position), but is an effect of the interactions with other objects.  That is a description of an emergent property, in the same way that pressure and temperature of the gas laws are not really properties of single particles but emerge from the interaction of all the particles.
A: I think your "no memory" argument assumes that time is fundamentally different from space: You wouldn't be surprised that there is information exchange between "immediately adjacent" locations in space. In fact, that is the only interaction. But space and time are linked, and like with space we only "interact" — directly, that is — with immediately adjacent locations in time. Indeed, general relativity shows that we can substitute one for the other if we change our frame of reference. The general formulation is "we only interact with our immediately adjacent locations in spacetime". That we see e.g. light from many years in the past is no contradiction: We see it only when, after a long travel, it eventually alters the properties of the immediate spacetime vicinity of our retina.
You could probably say that the "memory" needed for continuity is contained in spacetime itself: It consists of the properties of each and every little spacetime "voxel" in the universe.
A: The difference is that acceleration is absolute whereas velocity is relative. In other words, the local laws of physics are exactly the same for two objects moving with a constant velocity relative to one another, so there is no local experiment that can determine whether one is moving and the other is stationary. On the other hand, if two objects are accelerating relative to one another the local laws of physics that determine their behaviour are different, so it is possible to find their absolute acceleration. How is the state of zero absolute acceleration defined ? See Mach’s principle.
So if a charged particle emits EM radiation then it is accelerating; if it does not then it has zero acceleration.
Obviously an electron does not “know” whether or not it is accelerating - its behaviour is determined by the laws of physics. Indeed, you could say that the laws of physics are the fundamental global entities here and an electron is just a local manifestation of those laws which happens to have certain local properties such as charge, mass and spin.
A: As far as spacetime is concerned, acceleration and gravity are similar things. A charged particle that is accelerating is basically experiencing a force. Em waves are nothing but the transmission of a disturbance in the electromagnetic field of the charge.
A: If you consider velocity and acceleration from a purely kinematical point of view, there is no fundamental difference between them. One is the first time derivative of the location, the other the second. Two accelerating objects comoving with each other will be at rest relatively to each other as much as two objects with the same constant velocity.
Classically, EM radiation is purely an observer dependent effect, consisting of those components of the EM field related to the acceleration (relative to the observer). The particle itself (assuming it is a point particle) can not possibly 'know' whether it is accelerating or not.
In quantum mechanics this becomes conceptually all different of course.
A: To be perfectly honest - physics does not know the answer to your question. It seems mysterious to me, too, that the particle would have information on how fast it moved in the past. There is no strict logical inconsistency, so maybe we just have to satisfy ourselves with "that is what we have observed". But maybe there is an underlying mechanism we have not yet understood.
This reminds me a lot of a problem that bothered Newton about his theory of gravity - how does the planet orbiting a star "know" about the star's gravity, and especially changes in that gravity as the star's location shifts? This was later better understood in light of general relativity, and there were of course more underlying mechanisms to be understood.
A: I think different subfields of physics will give you different ways of thinking about what parts of the system "know" what. Different formalisms might give different answers as to where this information is stored (in the particles? in the fields?).
From the perspective of statistical mechanics, each particle carries ("knows") 6 pieces of information: the three components of its position and the three components of its velocity. Equivalently, you can say the three components of its position and the three components of momentum. For these six pieces of information, there is a kind of memory. On the other hand, in this picture, acceleration can typically be calculated from the positions of all particles in the system (and maybe velocities too if you include magnetism: $\mathbf{F}_\mathrm{EM} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B}$). So you need no memory for the acceleration.
If you code a molecular dynamics simulation (maybe simulation of ions in a plasma?), you typically have an array of 3D positions for each particle and an array of 3D velocities for each particle. Equivalently, you can have an array of the 3D positions at the current time and another array of the 3D positions at the previous time. You need to keep these arrays from one time step to the next, while the forces and accelerations do not need to be kept. You then calculate the change in velocity for this particle from the gradient of the potential energy for this particle's positions (the force). The potential energy is calculated as a function of all particle positions.
$$m \mathbf{\dot{v}}_i(t) = -\nabla_i V\left[ \mathbf{r}_1(t) \ldots \mathbf{r}_N(t) \right]$$
$$\mathbf{\dot{r}}_i(t) = \mathbf{v}_i(t)$$
This is a non-relativistic picture, because the way $V(\ldots)$ is written assumes that the motion of the other particles is communicated instantaneously. This gets more complicated if you include relativistic effects and radiation, because then you need to include the state of the fields or the state of the system at previous times (retarded time). The Feynman-Heaviside formula does seem to need to know about the position, velocity, and acceleration at the retarded time.
A: From a quantum perspective the information about the current speed of a particle is encoded in the wavefunction.
See for example the following wave packet of a one dimensional particle. The blue line shows the wavefunction; the distance to the black line shows the amplitude and the angle shows the phase. The number of turns it makes is proportional to the momentum: when the blue line is rotated more times around the black line the particle moves more quickly. To answer your question, if the universe wants to know the speed of a particle it can just look at the wave function. If this speed is increasing the particle is accelerating.

source of image: https://en.wikipedia.org/wiki/Wave_packet
A: This is a tricky concept. We learn so many times that objects have no objective sense of their own velocity that it starts to feel weird that objects do appear to have an objective (measurable) notion of their own acceleration. Your EM radiation from an accelerating particle is one example, radio emission from rotating magnetic stars is another.
I think that the cleanest answer is this. From Newton's second law we know that an acceleration is related to an external force. So the particle "knows it is accelerating" because it is subject to this external force from some other particle(s).
This is not entirely satisfying when rotating bodies are involved. Two cannon-balls that are roped together in an otherwise empty universe could be spinning around their centre, with a related tension in the rope, or they could be sill with no tension. These two situations are (in our best theories) physically distinct, but it is a bit of a puzzle exactly what the rotation is to be defined with respect to given that the example supposes their is nothing else to compare to.
See Tim Maudlin's book "Space and Time" for a very interesting account of the cannonballs.
A: I think your question goes to the heart of what we classify as epistemological or ontological, and likely a very beautiful mystery of the universe.
I'd say the answer to your question boils down to information vs dynamic. Informally, take information as one value in (multi-dimensional) phase space whereas dynamic would correspond to how these values change with time.
Velocity and position are information, whereas acceleration is dynamic. That is so most of the time in physics even though they are naturally related by time derivatives (velocity is how position varies in time and acceleration is how velocity varies in time). For some reason, nature has chosen second-order  time-derivatives to describe dynamic of systems most of the time. Maybe because with this, any given acceleration as a function of time will always yield a smooth trajectory.
So the "knowledge" that your particle and the E.M. field have about themselves is what I here called information. In terms of the dynamic that governs them, they are all one big thing.
