# Is acceleration an absolute quantity?

I would like to know if acceleration is an absolute quantity, and if so why?

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## 4 Answers

In standard Newtonian mechanics, acceleration is indeed considered to be an absolute quantity, in that it is not determined relative to any inertial frame of reference (constant velocity). This fact follows directly from the principal that forces are the same everywhere, independent of observer.

Of course, if you're doing classical mechanics in an accelerating reference frame, then you introduce a fictitious force, and accelerations are not absolute with respect to an "inertial frame" or other accelerating reference frames - though this is less often considered, perhaps.

Note also that the same statement applies to Einstein's Special Relativity. (I don't really understand enough General Relativity to comment, but I suspect it says no, and instead considers other more fundamental things, such as space-time geodesics.)

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By the way, in GR it depends on how you define acceleration. If you use the definition of rate of change of velocity relative to an inertial observer, no, acceleration is not absolute. But if you use a local definition e.g. based on an accelerometer, it is absolute, but you get the odd result that an observer at constant coordinates in a gravitational field (sitting on the surface of the Earth, for example) is in fact accelerating. –  David Z Nov 3 '10 at 22:21
Yes This is the correct answer. Can you please check if my proof below is correct? –  tsudot Nov 3 '10 at 23:20

Absolutely not. An observer in free fall and an observer in zero gravity both experience and observe no acceleration in their frame of relevance. One, however, is actually in an accelerating frame of reference.

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While not wrong of course, this answer brushes over too many subtleties I think. –  Noldorin Nov 3 '10 at 23:47
Why? Acceleration is only constant in inertial systems, which is the same as saying that acceleration is only constant when the system has no other accelerations. Also, what about systems with jerk? Surely newtonian mechanics must account for those too? –  Sklivvz Nov 7 '10 at 16:43

I've finally figured it out.

First, let's define precisely what it means for some quantity to be absolute or relative. In the context in question, it has to do with whether a quantity is absolute (that is, has the same value) or relative (that is, has different values) when measured by two inertial observers moving with respect to one another.

Of course, first we need to define what an inertial observer is: it's an observer for which Newton's laws are applicable without having to resort to adding fictitious forces.

Ok, so now we have two observers, Alice and Bob, both of which are inertial. They both observe the motion of some object. Let the index 1 correspond to quantities measured in A's reference frame and the index 2 correspond to quantities measured in B's reference frame. The position of the object is clearly a relative concept, since

r₂ = r₁ + u t

(where u is the velocity of Bob with respect to Alice, and is constant since they're both inertial observers). Note that the time, t, is the same for both observers, as it must be according to Newtonian Mechanics. The object position is a relative concept because r₂ ≠ r₁.

Now, take the time-derivative of both sides and we get

v₂ = v₁ + u

that is, the velocity of the object with respect to one observer is different than the velocity of the same object with respect to the other observer. Hence, velocity is a relative quantity in Newtonian Mechanics.

Next, take the time-derivative of both sides once again, and we obtain

a₂ = a₁

(since u is constant). Thus, the acceleration of the object is the same in both reference frames. Acceleration, therefore, is absolute in Newtonian Mechanics.

When we take into account the theory of relativity, then time flows at different rates for different inertial observers and the result above for the acceleration is no longer true.

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Most of what you have said is quite true. (Perhaps a little indirect, but no matter.) Your idea of relative and absolute quantities seems sound. A few things to note: in special relativity, I believe acceleration is still absolute; check the Lorentz transformations. David Zaslavsky gave an overview of the case for general relativity. Interesting, general relativity is actually required in order to rigorously define what an inertial frame is. Perhaps David can clarify this too. :) –  Noldorin Nov 3 '10 at 23:47
@Noldorin: I think you're right, acceleration is absolute in special relativity. I tried to check the Lorentz transformations but doing it properly involves more math than I have time for at the moment ;-) But here's my intuition: every object can measure its own acceleration (with an accelerometer) and can thus characterize its motion relative to a locally inertial frame. Since different inertial observers in SR will agree on what constitutes an inertial frame for the object, they will also agree that its own measurement of its acceleration corresponds to its acceleration as they observe it. –  David Z Nov 4 '10 at 0:27
@David Zaslavsky, I believe your intuition is correct - a constantly accelerating object in SR will trace out a hyperbola (asymptotically approaching $c$), so acceleration in the sense of $\partial^2 x/\partial t^2$ is dependent on the reference frame. (In particular, if you choose a frame such that the object it almost at $c$ it can't be accelerating very fast.) However, it's easy enough to define "proper acceleration" (acceleration in the object's own reference frame) which is of course absolute. –  Nathaniel Jan 20 '12 at 11:02

Acceleration will be the same in any two frames that are moving with constant speed with respect to each other (and may also be rotated and translated).

However, if you consider two frames that have relative rotation or acceleration, the acceleration of an object will be different in the two frames.

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