What is the underlying explanation behind fictitious/pseudo forces?

The popular example of the bus: Lets say you are standing in a bus and the bus is moving with a constant velocity, we can therefore agree that you are in an inertial reference frame and therefore the law of inertia applies in your reference frame.

However, as soon the bus decelerates you feel a "push" forward, moreover the frame is no longer inertial, at this point we agree that the law of inertia doesn't hold for that frame since you changed the state of your motion while no force is acting on you in your frame. Consequently, in order to make up for the "discrepancy" between the law of inertia and such situations we introduce a pseudo force (stating that this is the force that caused us to change states) in order to be able to effectively use Newtonian mechanics in a broader domain.

This is the popular explanation as to why pseudo forces are introduced, however no one really touches on to the underlying principles of the occurrence of a pseudo force, so I'm looking forward for a more in depth explanation into the nature of a pseudo force (i.e why it occurs from a physical perspective?), rather than just saying that we introduce it in non-inertial reference frames, for reasons similar to the above.

If we assume its nothing more than just a human correction used for mathematical and physical analysis, and simultaneously we can't say that inertia is the reason we tend to fall forward in the situation stated above since it is a non-inertial frame, then what would be the explanation exactly to such tendency of changing states of motion in an example like the above

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    $\begingroup$ There is no deeper explanation. If inertial systems with newton's laws exist, then in a frame of reference that is accelerated relative to an inertial one Newton's laws will not be valid, unless you introduce such pseudoforces. There is nothing physical about it, an object that moves at contant speed in one reference frame will look accelerated if seen by a reference frame that is accelerated with respect to it. Wheter or not newton's law exist (if space is galilean) $\endgroup$ – user126422 Apr 21 '17 at 1:56
  • $\begingroup$ Just to add about inertial frames existing, I thought this might be tangentially related. physics.stackexchange.com/q/78317 $\endgroup$ – Maximal Ideal Apr 21 '17 at 2:10
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    $\begingroup$ When the bus decelerates what you actually feel is you pulling yourself backward. $\endgroup$ – user253751 Apr 21 '17 at 9:34
  • $\begingroup$ You might find my earlier answer to a related question (dealing with rotating reference frames) relevant. $\endgroup$ – Ilmari Karonen Apr 21 '17 at 14:37
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    $\begingroup$ Obligatory XKCD: xkcd.com/123 $\endgroup$ – Mark Apr 21 '17 at 22:41

We actually don't introduce a pseudo-force as much as we introduce an acceleration. It is an acceleration which is experienced by all bodies in that non-inertial frame. From time to time, it can be convenient to think of it as a pseudo-force, but the deeper meaning you're looking for deals with accelerations, not forces.

In your bus example, when the bus starts decelerating, every object acquires an acceleration which corresponds to the effect of the reference frame decelerating. Thus, your human on the bus will accelerate forward unless a force generates an opposing acceleration.

A more interesting case is a rotating frame. A rotating frame is non-inertial, and the equations of motion within that frame include a centrifugal acceleration $a=\frac{v^2}{r}$ away from the center of the rotating frame. If no force pushes on the object, it will accelerate away from the center at that rate. However, in most interesting rotating frame problems, there is a force in the opposite direction as well. In the case of an orbiting body like the ISS, that force is the force of gravity, $F=mg$ towards the center of our rotating frame. This generates an acceleration of $g$, and when the acceleration $g$ from the sum of the forces is equal but opposite of the acceleration from the non-inertial reference frame $\frac{v^2}{r}$ the object appears not to move (in the rotating reference frame).

Likewise, if you are spinning a weight on the end of a string, it's the force of tension on the string which directly opposes the accelerations from the non-inertial reference frame.

The idea of a pseudo-force comes about when it is not intuitive to think about these accelerations. Consider the case where you're on a gravitron, which is the carnival ride that spins really fast and pins everyone up against the wall. In this case, it is not intuitive to think about the difference between the accelerations from your reference frame and accelerations caused by the force of the walls pressed up against your back. Every part of your body feels as though there is a force pushing you outward. In fact, if you run the math, the effect of this "centrifugal force" pushing you outward is identical to the effect of an acceleration caused by the non-inertial frame multiplied by your mass.

This is where the pseudo-force comes from. At a deeper level, its really more meaningful to treat it as an acceleration, but in practice it can be convenient to model this acceleration as a force by multiplying the acceleration by the mass of the object. When we choose to deal with these non-inertial effects as forces, we call them pseudo-forces. In particular, we like to do this when we want to say the sum of the forces on a body (that isn't accelerating) is 0. It's convenient to think in all forces instead of having to mix and match forces and accelerations. It's also convenient to think this way because the intuitive wiring in our brains is typically built to assume inertial frames (even when that isn't actually accurate). But the "meaningful" math behind them is all accelerations.

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To truly understand this thing you will have to refine your ideas about physics and coordinate systems and you have to learn Tensor Calculus and differential geometry. The main thing to understand is that Laws of nature must not depend on our coordinate system! It took Einstein a lot of time to understand that when he made General Relativity.

Here I have attempted to explain this in as simple terms as possible, using the math of General Relativity but applying it on Newtonian Spacetime.

Newton's law $$\mathfrak{F}=m\mathfrak{a}$$ is valid always, in all frames of reference. (when $\mathfrak{F}$ and $\mathfrak{a}$ are vectors). Laws of nature must not depend on our coordinate system! However $\mathfrak{a}$ is defined geometrically without refrence to coordinate system as $\mathfrak{a} =\nabla_{v} v$. ($\nabla$ is called the Covariant Derivative). In a coordinate system, the expression for $\mathfrak{a}$ (acceleration of a particle moving on a curve $ x$) is not simply $ \ddot x^{i}$ [$x^i$ are the components of $x$ in a coordinate system] but contains extra terms depending on the curvature of the coordinate system (and curvature of spacetime also) which include $ \Gamma^i_{jk}$ (called as Christoffel Symbols).

$$\mathfrak{a}^i \neq \ddot x^i$$

Now in an inertial frame, all the $ \Gamma$'s are equal to $0$, and we get $$F^i=m\ddot x^{i}\tag{Only when $\Gamma^i_{jk}=0$}.$$ If we are not in an inertial frame, we get $$\frac{F^\alpha}m=\underbrace{\ddot x^\alpha+\Gamma^\alpha_{\gamma\delta}\dot x^\gamma\dot x^\delta +2\Gamma^\alpha_{\gamma0}\dot x^\gamma+ \Gamma^\alpha_{00}}_{\mathfrak{a}^\alpha}.$$ Now this equation is still $\mathfrak{F}=m\mathfrak{a}$ but $\mathfrak{a}$ now has a complicated expression. The real force is still $\mathfrak{F}$.

However if you choose to call $\ddot x$ the "acceleration" (which it is not) then you have to deal with 3 new terms in the equation which we now call "fictitious forces". In the above equation you may be able to recognize the "centrifugal force", "Coriolis Force" etc. But they are really the parts of acceleration!

So in the end fictitious forces arise from calling something the "acceleration" which it is not.

*In fact gravity arises from the same equation. The term $\Gamma^\alpha_{00}$ is "the gravitational force".

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