What is my acceleration in my own frame?

Question

This is , I suppose not a good question , but I think I am missing something which confuses me in this question.

So my question is:

Velocity is relative. So suppose a car is moving at a speed $200 \;\text{km/h}$ with respect to an inertial frame. In its frame its speed is $0$. Now suppose in a time $\text{d}t$ its speed increases by $\text{d}v$ , i.e. it has an acceleration say, $a$. We can extend the time interval but the problem will be the same. So after some time say, its speed becomes $210 \;\text{km/h}$ with respect to the same inertial frame (the acceleration being constant and continuous) but still its speed in its own frame is $0$. So its speed in its own frame has not changed. So in its own frame (which is not inertial) it has no acceleration.

Now someone sitting in the car will say that the universe accelerated past him. An inertial frame observer will say that the car accelerated with respect to him. Its true that someone sitting in the car will feel the jerk due to acceleration. But just like the concept of relative velocity and its result which ascertains the fact that no one in an inertial frame can tell whether it is he or the surroundings which are moving. Similarly an accelerated observer (though he feels the jerk due to acceleration) will say the universe accelerated past him. So who has accelerated? Is absolute acceleration real. I might have confused the question particularly the last line but that's because I am confused myself.

Actually we say that there isn't any inertial frame. But even to say this we need to compare with some basic reference (inertial) frame because a frame is Inertial when it moves at a constant velocity w.r.t to another inertial frame. So how do we say that there isn't any inertial frame ? With which frame do we compare ? I think that simply saying that Newton's laws are not valid wouldn't suffice. Because if you are in the same accelerating frame in which some forces are acting on a body, how would you know that you are in a non-inertial frame? (forgetting for the time being that there aren't any inertial frame and actually trying to prove this very fact). It might be that the frame with which you compare be the accelerating one and you could be an inertial observer w.r.t to some still unknown inertial frame. Sorry, I might have confused the last few lines, but that's because I am confused myself. And don't know exactly how should I frame my question. Would just saying that Newton's law aren't valid in the frame suffice and wouldn't have to do any comparison?

Edit In a nutshell, I mean that is there any way to know if a frame is inertial? With which frame would one compare it to show it's inertial, since different observers can measure different accelerations? I mean in the end with reference to the question above who has accelerated. It's okay that observations of all inertial observers are correct but the truth is there can't be any inertial observer.

How can one know that the frame is inertial? Does just the application of Newton's laws guarantee it since different observers can measure different accelerations?

Is this being cyclic and hence not a correct way to find an inertial frame or is it correct.

Answer 1

This is an excellent question which is much more subtle than it first appears.

On a first (or even a second) reading, you seem to answer your own question and then ignore your own answer!

What I mean is, you clearly understand that while in an accelerating vehicle you experience a 'jerk' which does not happen with uniform motion, and you also acknowledge that the rest frame of the car is not inertial.

To an experienced physicist this already answers the question of why we declare acceleration to be absolute, but it does not satisfy you. Why? Because (as I realized on a third reading) your question is more interesting than that...

You don't care whether you are in an inertial frame or not. In your frame of reference you have the ability and the natural-born right to measure your own accelerations, and who are physicists to tell you that your answers are somehow wrong! Inertial reference frame or not, different frames can give different accelerations, and therefore acceleration is relative, right?

Well, guess what? I hear ya brother! You're absolutely right, and physicists need to talk more clearly.

Let's call a kinematic quantity (position, velocity, acceleration, or any higher derivatives) a Relative Quantity if its value is measured in arbitrary frame coordinates. So we have Relative Position, Relative Velocity, Relative Acceleration, etc.

The quantity called Absolute Acceleration (which we usually abbreviate to just 'Acceleration') is introduced as an additional quantity to the relative quantities, and for a very good reason I'll get to. Your 'Relative Acceleration' still exists, and yes, you have every right to use it, but you must also be aware of a much more interesting and fundamental quantity.

You see, whereas I have no experimental reason to justify the concept of an Absolute Position or an Absolute Velocity (i.e. they have no distinguishing features), I can point to an empirical justification for a particular acceleration.

Specifically, if I define Absolute Acceleration to be the Relative Acceleration measured within any inertial reference frame (they'll all agree on that measurement by the way), then for some mysterious reason of nature, Newton's Laws will work!

In particular, a body that is sufficiently isolated from all influences will have zero acceleration. When other stuff is in the vicinity, accelerations can be accounted for by well-defined force laws that 'make sense', given the nature of the surroundings. In particular, forces come in pairs (as per the third law), rather than popping up out of nowhere.

But this wonderful scheme only works out this way if we choose to measure accelerations relative to inertial reference frames, so we declare those accelerations to be 'true' or 'absolute', even though those adjectives are not as self-explanatory as physicists would have us believe.

That much answers your question, but the next bit is really worth knowing too...

When you use the framework of spacetime, rather than space + time, the notions of position, velocity, and acceleration, disappear, and you get a more satisfying geometric way to think about all this. Basically, spacetime distinguishes straight lines from curved lines, with the straight lines corresponding to uniform/inertial motion and the curved lines corresponding to non-inertial/accelerated motion.

There are simple experiments you can do to tell whether you are following a straight line or a curved line in spacetime. If you experience a 'jerk', you're on a curved line, for example.