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A while back in my Dynamics & Relativity lectures my lecturer mentioned that an object need not be accelerating relative to anything - he said it makes sense for an object to just be accelerating. Now, to me (and to my supervisor for this course), this sounds a little weird. An object's velocity is relative to the frame you're observing it in/from (right?), so where does this 'relativeness' go when we differentiate?

I am pretty sure that I'm just confused here or that I've somehow misheard/misunderstood the lecturer, so can someone please explain this to me.

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I find the phrase "acceleration need not be relative anything" to be awkward, but I can see where it comes from.

For the moment restrict our consideration the Galilean relativity (just to keep the math simple). Consider two frames of reference one ($S$) in which the body is at rest and another ($S'$) in which it moves with velocity $\vec{v'_i} = \vec{u} = u \hat{z}$.

So we have the initial velocity of the body in frame $S$ as $v_i = 0$, and $v' = v + u \hat{z}$

Now assume that the the body accelerates from time $t$ at acceleration $\vec{a} = a \hat{Z}$ resulting in a velocity in frame $S$ of $\vec{v_f} = a t \hat{z}$.

Compute the final velocity in frame $S'$ as $v'_f = v_f + u \hat{z} = (u + a t)\hat{z}$, and from that the acceleration in the primed frame as $a' = a$.

So the acceleration is the same in all frames (you can check the cases for $a \not\parallel u$ yourself), and it is reasonable to say that accelerations are not relative anything.

All of this is a consequence of the simple form of the transformation between frames:

$$ \vec{x'} = \vec{x_0} + \vec{u} t $$ $$ t' = t $$

So what about Einsteinian relativity?

Here the transformation between frames is more complicated, and the math is much more complicated resulting in observers in different frames seeing different accelerations, but they will all agree on the acceleration as measured in the body's own frame. In my opinion "the acceleration need not be relative" risks causing unnecessary confusion on these points. The magnitude and direction measured will measure in depend on the frame of the observer, which is often what is meant when people say "it's relative".

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The very simple answer to your question is that yes, acceleration is relative. Although Newtonian physics is written with respect to a preferred reference frame ("the fixed stars" as Newton said), general relativity treats all reference frames, accelerating and not, on equal footing. Newton's frame, or the "inertial frame", just means "not accelerating RELATIVE to the bulk of matter in our universe". If, locally for instance, there was enough mass moving in some direction then there would be frame dragging effects whereby the local 'inertial' reference frame (where Newton's Laws are valid) would seem to shift relative to more distant 'fixed' matter. This is basically the content of Mach's principle, which was one of Einstein's chief motives for searching for a generally covariant theory of gravitation.

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I'm not convinced this is correct. If there was nothing in the universe except an orb half-full of water, and the water was pressed primarily along one "equator", one could deduce that the orb was spinning. There is no "relative" anything in that. (There still remains to connect "spinning" to "acceleration", but I think it can be done. I'm just not sure how) –  Mooing Duck Nov 4 at 23:53
    
Aha! If there's two planets of ~equal size, and about 1% of the distance from one to the other there is a penny, gravity will pull on the penny. If acceleration was relative, then relative to the penny the close planet would accelerate rapidly toward it, and the farther planet would be rapidly accelerating away from it. But there is no mechanism for the penny to be pushing the farther planet away - gravity says the planet must accelerate toward the penny. Ergo measuring acceleration of the planet relative to the penny must be invalid. –  Mooing Duck Nov 5 at 1:23

I'll sort of cheat and neglect the relativity part--instead, I'm going to focus on your mathematical confusion regarding differentiation.

What you've forgotten is that we have a constraint. Your argument is thus, right: $$\text{if } \vec v_{b}=\vec v_{a}+\vec v_{b,a}, \text{ then } \vec a_{b}=\vec a_{a}+\vec a_{b,a} \text{ by differentiation}$$ Where $b$ and $a$ are frames of reference, and $b,a$ denotes the acceleration/velocity of the frames relative to each other.

Aah, but we have a constraint: $\vec F=\frac{\rm d\vec p}{\rm d t}=m\frac{\rm d\vec v}{\rm d t}=m\vec{a} _\text{ (for constant mass)}$.

This constraint makes little difference when we switch between frames with constant relative velocities, since the derivative of velocity stays the same. But, the moment we try to switch to an accelerating frame, things get icky. We get:

$$m\vec a_{b}=m\vec a_{a}+m\vec a_{b,a}\implies \vec F_b=\vec F_a+m\vec a_{b,a}$$

Looks OK, doesn't it? We could write $m\vec a_{b,a}\to\vec F_{b,a}$, and the equation would be handy-dandy--it would show that force is relative as well. Right?

Wrong. By our assumption, acceleration and velocity are relative quantities. To measure a relative quantity, one must have a reference frame. In contrast, force is something you can measure without needing a frame-- A spring balance suffices.

This equation shows that apparent force varies with your reference frame. You probably know this, but this comes from the presence of a "psuedoforce"--the $m_{b,a}$ term. It's not a real force, but it appears whenever we try to apply Newton's laws to a noninertial{*} frame. Since it's part of the apparent force, it can be measured. Since it crops up when you change frames, you will always be able to detect that a frame change occurred. This is in contrast to switching between inertial frames--unless you have a fixed object outside, you can't tell the difference.

So basically, it's not a methematical issue but an issue with what can and can't be measured inside a frame

Actually, all of this comes from the fact that Newton's laws are only applicable in an inertial frame. When asked "which of Newton's laws is the most important?" most people would say the second and/or third law. The first law is always counted as a result of the second law. This is actually false-there is no privileged law of the three-- and they are independant. The function of the first law is to define the realm of applicability of the three. In a noninertial frame, the first law does not hold(since an object at rest still gets accelerated), so the other two do not hold either. The pseudoforce is just a way of cheating the laws into working.

So, if we have a bunch of laws that only work in a non-accelerating frame, that means that "non-accelerating" has an absolute meaning. Thus all acceleration is absolute.

In short: the "relativeness" was not there in the first place. It just appears when we take a special case of $a_{frame}=0$--useful in itself, but not generalisable.

* inertial-->constant velocity no gravity, noninertial-->acceleration and/or gravity

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You missed an opportunity to mention the Coriolis force :) –  Bernhard Mar 26 '12 at 5:41
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@Bernhard: I hate the coriolis force--yes, it's another thing that can be said to pop out like a jack-in-the-box when you differentiate equations, but it's annoying. I prefer good 'ol centrifugal "force" :P –  Manishearth Mar 26 '12 at 5:58

That's because the equation of motion is a 2nd order differential equation. F=ma. If you integrate it to get r(t), you get two arbitrary integration constants. So you have two degrees of freedom, making the absolute r(t) and the absolute v(t) invariant when adding a constant.

This holds with relativistic equations of motion, and even relativistic QM. As long as your basic Differential Equation of motion is 2nd order, you get two integration constants.

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Acceleration need not be relative to any object. It is relative to space (/spacetime), which is like a universal coordinate system. Whenever you accelerate, there are signs. For example, imagine you are in an elevator in deep space. If you begin to accelerate upwards (in the direction normal to the top of the elevator car) at 9.8 m/s^2, you will begin to feel like earth's gravity is pushing you downwards. If you had a window, outside of which you saw an identical elevator car, it would appear to accelerate downwards at 9.8 m/s^2. In this sense, it is accelerating with respect to you, but a person inside of the car would be floating freely, indicating that it is not accelerating in spacetime.

Special Relativity applies to intertial reference frames, which means that the frames are not accelerating, but it is very rare to find objects that are actually not accelerating, so we make approximations, such as saying that the Earth is at rest when talking about an earth-spaceship system. For a simple example, look at the twin paradox: Al and Bob are 20-year-old identical twins. Bob flies in his spaceship at .87c (which gives a lorentz factor of 1/sqrt(1 - v^2/c^2) = 2) to a planet that is 10 lightyears away, circles around the planet, and returns to Earth. Al looks through a telescope at his brother's ship to see that Bob's clock is moving at half the speed of his clock on Earth. Bob, however, looks back at Earth and sees that Al's clock is moving at half the speed of the clock on his ship. This is because the ship is moving with a constant velocity with respect to the Earth, making the two inertial reference frames that experience Special Relativistic time dilation. When Bob returns to Earth, he is 40 years old, but Al is now 60. This is because Bob's reference frame went through accelerations (speeding up initially, then looping around the planet, and slowing down again near Earth). This example clearly shows that acceleration in a Newtonian sense must be relative to something, but in Relativity, there is a concept of absolute acceleration, which has certain effects on the fabric of spacetime.

Note: In general relativity, gravity is not a force, but a pseudoforce (just like centrifugal force - if you drive your car in circles, there is no actual force making you slide in your seat towards the outside of the circle; you are just feeling the pseudoforce that results from you accelerating towards the center of the circle). GR states that the presence of mass bends the fabric of spacetime, so that space accelerates around massive objects. The gravity that we feel on Earth is a result of the fabric of spacetime accelerating upward at 9.8 m/s^2, just the same as you would feel in an elevator car in deep space that is accelerating upward.

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