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So how would we know whether a force truly exists or not. I can be accelerating a car my 5 meters per second squared but another car accelerating with the same acceleration would think that my car is at rest relative to them. So is there any force on the car? Or are forces just relative and their existence just depends on our reference frame?

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    $\begingroup$ Related questions / possible duplicates: physics.stackexchange.com/q/22803/179151 , physics.stackexchange.com/q/349612/179151 , physics.stackexchange.com/q/245023/179151 $\endgroup$ May 5, 2021 at 20:38
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    $\begingroup$ “Height is relative” does not mean that slope is relative. Likewise saying that y’ is relative does not make y’’ relative. $\endgroup$ May 6, 2021 at 13:10
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    $\begingroup$ Note that forces are not defined by acceleration. physics.stackexchange.com/q/581142/195139 $\endgroup$
    – Sandejo
    May 6, 2021 at 18:30
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    $\begingroup$ Even though you are getting some quite curt replies, this is a great question. Understanding the various answers will give you some good insights. $\endgroup$
    – Andrea
    May 6, 2021 at 20:51
  • $\begingroup$ "I can be accelerating a car my 5 meters per second squared but another car accelerating with the same acceleration would think that my car is at rest relative to them." - I believe this is not true unless your velocities were identical initially. In all other cases the speed difference will be constant and visible from both cars. $\endgroup$
    – gronostaj
    May 7, 2021 at 6:05

8 Answers 8

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Velocity is relative, which means acceleration is relative

This is not correct. Velocity is relative, but (proper) acceleration is not relative. It is an invariant. Real forces lead to proper acceleration so the existence of real forces does not depend on the reference frame. In contrast, fictitious or inertial forces do not cause proper acceleration so their existence does depend on the reference frame. Reference frames where fictitious forces exist are called non-inertial reference frames.

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  • $\begingroup$ Actually without knowing about relativity some relativistic effects might appear to imply that acceleration is relative. For instance, under a constant force an object close enough to the speed of light appears to accelerate more and more slowly. I don't view this is force being relative though, more like the force carriers are travelling at a closer and closer velocity to the objects they are mediating between that they start loosing their "oomph" (to use a technical term ;) $\endgroup$
    – Michael
    May 6, 2021 at 19:34
  • $\begingroup$ @Michael gauge bosons don't physically exist, and photons (the boson of electromagnetism) always move at c regardless of the velocities of the objects they're acting on. $\endgroup$
    – OrangeDog
    May 8, 2021 at 14:30
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Within different inertial frames, velocities will be different. However, acceleration will always be the same in any inertial frame. Therefore, so will the forces.

Short proof: suppose $v(t)$ is the velocity in one inertial frame, and $v'$ is some constant shift in velocity due to choosing a different inertial reference frame. Then the velocity within that newly-chosen reference frame is $V(t)=v(t)-v'$, and upon differentiating it w.r.t. $t$, you get $A(t)=a(t)$, so the acceleration does not depend on the inertial reference frame, and neither will the force.

Now, in non-inertial reference frames, there exist what we call Fictitious forces (or not real forces), but these do not exist in inertial reference frames.

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  • $\begingroup$ Which time are you differentiating by? My point of course being that there are at least three different 'times' you could be differentiating by. $\endgroup$
    – tobi_s
    May 6, 2021 at 7:14
  • $\begingroup$ @tobi_s not sure I understand what you mean. I use the definition that $a=\dfrac{dv}{dt}$ to show that $a$ is the same in any inertial reference frame $\endgroup$
    – user256872
    May 6, 2021 at 7:22
  • $\begingroup$ There are different time variables attached to each frame and there is the proper time of the object that is subject to the forces. Thus my question is: which one is the $t$ in your denominator? To obtain an invariant definition of $v$ and $a$ it would have to be proper time. Of course this is moot when dealing with Newtonian mechanics where the three time variables coincide. Since the question was asked about a car which -- unlike trains -- is rarely used to explain concepts in relativistic mechanics I guess what I was aiming for was outside the scope of the question, so please never mind $\endgroup$
    – tobi_s
    May 6, 2021 at 8:20
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    $\begingroup$ @tobi_s There's indeed the [newtonian-mechanics] tag attached on the question. $\endgroup$
    – Blackhole
    May 6, 2021 at 10:44
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    $\begingroup$ Heh, thanks, I indeed didn't see that. All this talk about inertial frames made my imagination move near the speed of light :) $\endgroup$
    – tobi_s
    May 6, 2021 at 11:13
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People in an accelerated car know that it is accelerating due to inertial forces that can be observed inside.

In cases like that, besides the inertial forces, there is the reference of the fixed environement, so it is easy for the passengers to know about their acceleration, just looking outside.

More interesting is the case of the Earth. It was not so obvious to decide if all the sky rotates around us, or if we are rotating and the stars can be considered a fixed environment. Here, inertial forces like Coriolis, that explain trade winds and hurricanes rotation are a proof that we are rotating, what means an accelerating frame.

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I can be accelerating a car my 5 meters per second squared but another car accelerating with the same acceleration would think that my car is at rest relative to them

That is not true.

When we consider just velocity, it’s simple because there are no forces involved.

But in accelerated motion we have to consider pseudo forces. You both are being pushed in this case, due to the pseudo force. That force is same for both of you, but that doesn’t mean you won’t notice a force on the other.

Say, you and your friend are moving in your cars with same acceleration. But your friend has a pendulum in his car that is hanging. Due to pseudo force, the pendulum is pushed back, and it stays like that because of the constant force on it backwards.

Like this:

enter image description here

You look over, and notice that pendulum is acting weird. That’s how you prove, that even though both are experiencing same forces, they don’t cancel out. Instead both experience it.

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In Galilean relativity, physics is unchanged for frames that are related to uniform boost. That is, we introduce an equivalence relation $\sim$ among frames and define $\mathcal{A} \sim \mathcal{B}$ if frame $\mathcal{B}$ moves with constant velocity vector as seen by the frame $\mathcal{A}$. Then Newton's first law says that there exists a particular equivalence class such that inertial motion appears as uniform linear motion. Then there comes the second law.

In fact, we can think in this way: $\vec{a}$ is Galilean invariant, and $\vec{F}=m\vec{a}$ is the simplest physical law that conforms with Galilean relativity! And this is because force, a physical quantity, should not depend on observers. Indeed, "(Physical quantity)$ = $(Geometrical invariant)" is a recurring leitmotif throughout the history of physics.

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  • $\begingroup$ "Then Newton's first law says that there exists a particular equivalence class such that inertial motion appears as uniform linear motion." Or Newton's first law tells you how to decide what "constant velocity" looks like in your reference frame: absent any forces, objects move along geodesics in spacetime (yes, we can do Newtonian mechanics on a curved spacetime, where gravity is curvature and not a force, just like we do with general relativity). Depending on how you interpret the law, it doesn't necessarily single out any particular class of frames. $\endgroup$
    – Arthur
    May 7, 2021 at 10:03
  • $\begingroup$ Dear Arthur, yes that’s a great point. There have always been controversies on the interpretation of Newton’s first law: if interpreted “literally”, it’s just a special case of F=ma and rather unnecessary. What I followed here is the interpretation that some physics texts like Marion’s classical mechanics textbook take. $\endgroup$
    – user198491
    Nov 12, 2021 at 6:23
  • $\begingroup$ I think such interpretation somewhat appreciates a bit of “philosophical content” inherent in the law, regarding the fact that not even the notion of “inertial motion,” “intertial reference frame,” or “force” was well understood before Newtonian mechanics. Yet, I wanted to be mathematically quantitative here rather than being philosophical. $\endgroup$
    – user198491
    Nov 12, 2021 at 6:23
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Or are forces just relative and their existence just depends on our reference frame?

Forces are only relative insofar as "every action has an equal and opposite reaction". So a force applied to the car must also equally-and-oppositely be applied to something else, in this case the road surface.

For the other car, then, it must also be applying some amount of force to the road surface which results in the same acceleration. (Note that the forces to achieve that acceleration won't be the same unless the masses of the two cars are the same, because of "F = ma".)

The fact that there is no net acceleration between the cars is irrelevant, because each car's forces are relative to the individual cars and the road surface.

The only way to make the other car relevant would be to connect the two together somehow, with a rope, pole or something. Now your reference frame includes both cars - but your reference frame also includes forces in the connecting device, to establish net forces on each car within that reference frame. You could for sure get the forces in the connecting device to be zero by applying the correct forces from each engine, but that only means there is no force applied via the connecting device. Other forces must be known (and be in the correct ratio) in order for the force in the connecting device to be zero.

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To understand the answer, you need the concept of an inertial or non-inertial frame. That's fundamental to this kind of question.

Basic Newtonian mechanics are applicable in an inertial frame - one where the "space" (or box, or region) that we consider, is itself not accelerating. Since movement in a circle is also acceleration, that means the frame itself is not moving at constant velocity in a curve either.

If the frame itself is accelerating, its called a non-inertial frame. In a non inertial frame you have to take into account the acceleration of the frame itself. The acceleration of the frame will make it look (to someone within the frame) like there are strange extra forces pulling things in strange ways. These are called pseudo-forces - they don't really exist, they are artifacts of the frame's own acceleration. Like if you move within a car in a circle, it will seem like some force is pulling you away from the centre of the circle (centrifugal force). That's one example - we can see that this is a pseudo-force because obviously there is nothing at a distance all the way round the circle pulling just you outward. Different non-inertial frames will have different pseudo-forces; by definition an inertial frame has zero pseudo-forces.

So the answer to your question is this:

Velocity can be relative, because a difference of velocity doesn't affect whether this forms an inertial or non inertial frame. Whether you travel at a constant 0, 20 or 50 km/h compared to another car, your frame in both cars is the same. Constant velocity doesn't affect whether a frame is inertial or not, or (for a non inertial frame) how much by.

Acceleration cannot be relative, because a different acceleration always changes the frame. It changes inertial to non-inertial frames, and changes one non-inertial frame into another (or back to an inertial frame again).

And that means....

Because a different, but constant, acceleration of the frame changes the frames inertial nature, it will always change the pseudo-forces you see in that frame, and that will mean you can tell the difference between two frames which are accelerating differently. So in an accelerating car, with blinds over the windows, you could still tell there's a difference in the cars motion between (for example) being stationary, accelerating slowly, accelerating quickly, or moving in a circle at constant velocity. You'll be pulled into the side of the car differently, or pressed into the seat differently, or a pendulum will hang or swing differently.

Because a different, but constant, velocity of the frame don't change the frames inertial nature or the pseudo-forces in it, you can't tell as an observer in the frame what velocity the frame moves at. In a car with blinds over the windows, you couldn't tell the difference between stationary, or moving in a straight line slowly, or quickly. It will all feel the same,and a pendulum will hang the same in each case.

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I think it depends on what you mean by "force is relative." Do you mean that the observed force on an object is different in different frames of references?

Then yeah, force is relative. If you are sitting in your car, and the car is accelerating. You feel/observe the inertial force (kinda like gravity) backwards. But in the ground's frame of reference, you are not acted upon by any backward force.

However, acceleration, unlike velocity, doesn't satisfy the principle of relativity. That is for different inertial frames of different velocities, the physics in all of them are exactly the same. You can not measure an absolute velocity for an inertial frame by performing any kind of experiment.

However, acceleration doesn't have principle of relativity. You can know your absolute acceleration by observing your acceleration/inertial force in a non-inertial frame.

So acceleration and force are relative in that you observe different accelerations and forces in different frames of references. But they don't satisfy principle of relativity because they need the introduction of inertial forces to fix the physics in their frames of references.

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