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A person standing on a uniformly moving car can never know (without looking outside, or at the speedometer) whether the car is at rest or in motion at a uniform nonzero velocity w.r.t earth. However, in the case of a person standing on a uniformly accelerating car, due to the effect of pseudo forces, knows that the car is accelerating.

My question is, remaining on the car, can he measure the value of the acceleration? If yes, how? Will it be identical to the acceleration measured by another person on the earth?

P.S. In this question, first I am interested in understanding the absoluteness of acceleration in Galilean relativity.

Solidification
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  • When inside "the box" there is no way to distinguish between acceleration and gravity. – ghellquist Dec 04 '21 at 19:01
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    @ghellquist - although that is correct for an arbitrary gravitational field, that is incorrect in practice. Gravity will have nonzero parallax between the ends of the box, whereas acceleration will have zero parallax. – TLW Dec 04 '21 at 20:25
  • @TLW Will that always be true? Suppose the acceleration is from a force in the center of a beam, there will be propagation along the beam towards the ends at the speed of light. If the acceleration is from three equal forces at either end of the beam and the middle it will be different. – Michael Dec 05 '21 at 00:10
  • Maybe helpful: https://math.stackexchange.com/questions/1637409/what-does-the-integral-of-position-with-respect-to-time-mean – S.H.W Dec 05 '21 at 09:49
  • @TLW It seems like a bit of a stretch to speak of distinguishing gravitational parallax between the ends of a box in practice. –  Dec 05 '21 at 23:02
  • @Michael - that effect is limited by how much the box can bend, and as such asymptotes towards zero as the measurement time period increases. – TLW Dec 30 '21 at 02:29
  • @user319419 - assuming a 1m x 1m x 1m box, the difference between local down at the two opposite faces works out to ~160nm away from 'truly' parallel, which is totally measurable. Optics would be a pain, but I suspect dropping both ends of a laser interferometer would be a workable starting point. (Of course, you'd likely just be measuring the nearest mountain, or the equipment itself, in practice instead, but meh.) – TLW Dec 30 '21 at 02:57

4 Answers4

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There are two kinds of acceleration: proper acceleration and coordinate acceleration.

Proper acceleration is “absolute” although the usual scientific term is “invariant”. This means that all observers using any reference frame will agree on the proper acceleration. Physically, proper acceleration is the acceleration that is directly measured by an accelerometer. It is also equal to the coordinate acceleration in an inertial frame. Mathematically it is given as the covariant derivative of the four velocity. This applies to both the Newton Cartan formulation of ordinary Newtonian mechanics as well as for special and general relativity.

The coordinate acceleration, on the other hand, is mathematically the second derivative of the position. Since position is defined relative to the coordinate system, this notion of acceleration is relative, not invariant. In particular, non-inertial coordinates will disagree on the amount of coordinate acceleration. Notice that proper acceleration had a physical definition and a mathematical definition, but coordinate acceleration is not physical and has only a mathematical definition.

In your example of the person on the accelerating car, the acceleration that they physically feel is the proper acceleration. They could measure it with an accelerometer, for example in their phone. If they used an inertial coordinate system then their coordinate acceleration would equal their measured proper acceleration. But if they used non-inertial coordinates, such as coordinates with respect to the car, then their coordinate acceleration would be zero.

Dale
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  • The definition of acceleration that I am using is $d^2{\vec r}/dt^2$. What do you mean by proper acceleration? – Solidification Dec 04 '21 at 05:28
  • https://en.wikipedia.org/wiki/Proper_acceleration – anna v Dec 04 '21 at 08:46
  • @Solidification If a particle has 4-velocity $u^a$, then the proper acceleration $A^a=u^b\nabla_bu^a$. If $u^a$ is geodetic, it implies that $A^a=u^b\nabla_bu^a=0$ (the geodesic equation) – KP99 Dec 04 '21 at 12:27
  • @KP99 I would like to confine my question to Newtonian mechanics. – Solidification Dec 04 '21 at 12:34
  • @Solidification In Newtonian regime $u^a\approx (1,v)$, so the proper acceleration $A^0\approx 0$ and $A^i\approx \frac{\partial v^i}{\partial t}+v\cdot\nabla v^i$ (the familiar material derivative used in fluid dynamics) – KP99 Dec 04 '21 at 12:40
  • @Solidification I have expanded the answer to address your comments here. The concept of proper acceleration was originally formulated in relativistic mechanics, but through the Newton Cartan formulation it has been brought to Newtonian mechanics. – Dale Dec 04 '21 at 13:32
  • Thanks, @Dale . This is not true for the velocity, am I right? I mean, the speedometer does not measure the proper velocity but the velocity of the car w.r.t the earth. Am I right? – Solidification Dec 04 '21 at 17:28
  • @Solidification yes, that is correct. Velocity is strictly relative. There is nothing like proper velocity in the sense of something invariant like proper acceleration. (Unfortunately there is something called proper velocity, but it is not invariant, which is confusing terminology) – Dale Dec 04 '21 at 17:59
  • "coordinates wrt the car" I think that needs some clarification. Do you mean, e.g. objects inside the car? Other cars on the road? I would expect that measuring the acceleration of the car relative to road signs (say, with a video where you count the frames for which given stationary landmarks are visible) would correspond to the acceleration measured by the driver's phone, allowing for experimental error and the limitations of the measuring devices. – jpaugh Dec 04 '21 at 18:10
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    @jpaugh “coordinates with respect to the car” means coordinates where the car is stationary. Ie it is an accelerating (non inertial) reference frame which is comoving with the car. – Dale Dec 04 '21 at 18:16
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    Thanks! I interpreted that sentence in an intuitive way; and it's also fair to say I don't think physics often enough to remember this terminology. – jpaugh Dec 04 '21 at 18:36
  • I'm very curious, what do you think about tparker's answer (you can find it on this page) where they say that an accelerometer does not measure acceleration per se but rather the acceleration gradient across different parts of the device? – Maximal Ideal Jan 11 '23 at 23:45
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    @MaximalIdeal I disagree with that. With a typical proof-mass accelerometer there is a brief transient where there is an acceleration gradient, but they are designed so that transient is brief. And the reading is inaccurate during the transient. The reading is only accurate once the transient is gone and there is no more acceleration gradient. – Dale Jan 12 '23 at 00:50
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As you mention, when a vehicle is accelerating you can physically sense the amount of acceleration.

You can sense it because your body has inertia; a force is required to change your velocity.

In Newtonian mechanics, force is defined as something that causes change of velocity.


About inertia:
Inertia is in a category of its own. To categorize inertia as a force would be a self-contradiction. While it is the case that inertia acts in opposition to the change of velocity, inertia cannot be thought of as a counter-force. If inertia would be a counter-force, then the change of velocity would be impossible.

In electromagnetism, there is a phenomenon that is analogous to inertia: inductance

Imagine a current circuit where the current conductor is a superconductor. Also, let the circuit be set up to have self-inductance.

Being a superconductor, any current in the circuit will continue indefinitely. It is when you want to increase or decrease the current strength that the self-inductance of the circuit comes into play. Change of current strength elicits a magnetic field that acts in opposition to that change of current strength.

This opposition does not prevent an increase/decrease of current strength. Instead, the opposition is such that the rate of change of current strength is in some proportion to the applied voltage difference. The higher the self-inductance, the stronger the opposition to increase/decrease of current strength.


The expression 'pseudo force' is very awkward.
Inertia is not pseudo, inertia is real, and categorizing inertia as a force is a self-contradiction.


With the above in place, I turn to the basis of how an accelerometer operates.

You are in, say, a train carriage, and a weight is suspended from the roof of the carriage. It's in effect a pendulum.

When the carriage accelerates the weight lags behind; inertia.

All accelerometers use the above operating principle, in one form or another.

If you have a smartphone: today's smartphones are equipped with (tiny) accelerometers. The type of technology that is most often used is called MEMS.

The accelerometer setup consists of a (tiny) stalk, with a sensor that is sensitive enough to pick up the minute lag when the device as a whole is accelerated (just as the pendulum bob in the train carriage lags behind when the train carriage changes velocity.)


A full complement of acceleration sensing consists of three accelerometers, at right angles to each other. This is referred to as 'tri-axial arrangement'. The readings of the respective measurement axes are combined to arrive at the actual acceleration of the device.

With sufficiently accurate acceleration sensing, combined with sufficiently accurate change-of-orientation sensing, it is possible to do the acceleration counterpart of dead reckoning

With nautical dead reckoning, you keep a log of your velocity and direction at all times, and with sufficiently accurate logging you can return to exactly your starting point. That is, the dead reckoning allows you to construct your current position relative to your starting point, so you can plot a course back to that starting point.

An inertial navigation system is analogous to that, but with and inertial navigation system you are integrating acceleration readings. (More precisely: acceleration readings combined with keeping track of changes of orientation.)


I have taken the time to emphasize that an inertial navigation system is sufficient to know your current position relative to some starting point, and you can plot a course back to that starting point.

(Of course, in the real world there is always some measurement drift. But this is a thought experiment, and in a thought experiment we can always assume that our measurement devices can be made so accurate that measurement error is not a factor.)


The fact that you can always plot a course back to your starting point shows that acceleration is absolute.

Now, of course: as you are at some starting point, ready to begin a journey, your acceleration measurement does not tell you where your starting point is relative to other features in the larger world. Inertia is the same everywhere, so inertia does not give you any information about your position.

The point is: once you have set yourself in motion you can always construct your current position relative to your starting point.

Inertia is the same, everywhere, and in every direction. That very uniformity allows you to use inertia as a global reference.

Conversely, imagine a universe where inertia fluctuates randomly from place to place and from orientation to orientation. Then there is no uniformity, hence no reference.

So:
If inertia is the same everywhere, then acceleration (as measured with an accelerometer) is absolute.

Adil Mohammed
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Cleonis
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The person knows he or she is accelerating because they feel compressed.

For example the car seat will push on your back in an accelerating car, but the front of your body doesn't move immediately, so a person is subject to a slight compression during the acceleration.

This defines the reference frame for an object's acceleration as the object itself. It is different to the Machian reference frame of distant matter. See also Mach's principle and a reference frame for acceleration, it expands on this proposal.

enter image description here

So the acceleration could be measured if the car seat were converted into a weighing scale. To measure the acceleration accurately it would be best to have a sliding seat with no friction and have the weighing scale $S$, behind the person and the seat. The reading on the scale could be compared to the normal weight of the person and seat and the acceleration could be calculated in terms of $g$.

John Hunter
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  • @ Solidification ...and yes the acceleration measured would be the same as that measured by an observer on earth, outside the car. – John Hunter Dec 04 '21 at 12:38
  • "The person knows he or she is accelerating because they feel compressed." Also by observing the motion of force-free objects. – Solidification Dec 04 '21 at 12:39
  • @ Solidification yes that's true, other free objects would not experience the compression and continue at constant speed, and would appear to accelerate relative to the accelerating car – John Hunter Dec 04 '21 at 12:42
  • "...yes the acceleration measured would be the same as that measured by an observer on earth, outside the car." But how? – Solidification Dec 04 '21 at 12:52
  • @Solidification: well accelerations are technically caused by forces (real or pseudo), so you could measure the pseudo-force and determine the associated acceleration; if there's no obvious source for the force (no obvious source for this mysterious additional component to gravity), you can argue you're in a non-inertial frame. – Filip Milovanović Dec 04 '21 at 13:50
  • @ Solidification the acceleration measured from inside the car is by the weighing scale method of the answer. It can me measured from outside the car by timing the car to cover a known distance, e.g. starting from rest. – John Hunter Dec 04 '21 at 14:00
  • Note that these days you don't have to convert the car seat into a scale. There are cell phone apps to measure your car's acceleration. – jamesqf Dec 04 '21 at 18:12
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Very strictly speaking, there's no way to locally measure absolute acceleration. But in practice, you always can except when it's induced by gravity. (In that case, it's debatable whether it's even useful to think of it as "absolute acceleration" at all, although you would within the traditional framework of Galilean relativity and Newtonian gravity.)

It's often stated that you can directly and locally detect absolute acceleration because you can "feel" it. However, strictly speaking this isn't true; you can only ever measure gradients in the acceleration field, regardless of its source (gravitational or otherwise).

When you literally physically feel an acceleration, you are actually feeling local compression between different parts of your body due to the fact that the forces (typically contact forces) that are accelerating you only act on certain parts of your body, which causes spatial gradients in the induced acceleration.

In principle, you can't feel (or even see) the effect of any force that imposes a sufficiently spatially uniform acceleration on objects within your range of observation, including the different parts of your own body. In an extremely idealized thought experiment, you could imagine being accelerated arbitrarily strongly by (e.g.) an electrostatic force that's extremely precisely shaped so that the local ${\bf E(x)}$ field is exactly inversely proportional to the local charge-to-mass-density ratio in your body, in which case you wouldn't be able to feel the acceleration.

In practice, this fact is only ever important for gravity, because gravity is the only force that's realistically capable of inducing reasonably uniform acceleration fields. Any other realistic source of force will always induce strong acceleration gradients - certainly at the microscopic level - which you can feel. But it's worth noting that in principle, you can only ever detect acceleration gradients and not acceleration per se.

The same is true of higher derivatives as well. In principle, you can't detect even very strong jerk, snap, crackle, or pop either, as long as it's applied in a sufficiently spatially uniform way.

tparker
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