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## 1  Executive Summary

Today we will ask a few questions such as

• What is a pseudo force?
• What is a fictitious force?
• Are pseudo forces real?
• Is gravity a pseudo force?

Alas, we will not answer these questions!

As far as I can tell, nobody knows, and nobody cares. Or at least nobody should care. If these questions have answers at all, each answer is a matter of opinion, and opinion is divided. There is no consensus. When I asked several hundred physics teachers via the phys-l mailing list, nobody seemed to have any firm opinions. I don’t have a firm opinion, either.

There is no experiment you can do that will tell you the answers to these questions.

For example, as far as I can tell, gravity can be classified as a “pseudo force” – or not. It’s six of one and half-a-dozen of the other. You can choose to do it either way, and the choice has no observable consequences.

The fact that there is no easy answer suggests that we are asking the wrong questions, and that we should ask more physically-meaningful questions instead. Examples of such questions are given in section 2.2.

## 2  Is Gravity a Pseudo Force?

### 2.1  Two Views of Gravity

In reference 1, Feynman explicitly raised the question of whether gravity was a pseudo force ... and then declined to answer the question. I am not fool enough to rush in where Feynman feared to tread. However the following discussion may something of an improvement: Feynman discussed only one side of the argument before giving up, whereas the following discusses two sides of the argument before giving up. The fact that there are two sides – i.e. two seemingly-reasonable arguments leading to different conclusions – helps you understand why there is no easy answer.

The fact that there is no easy answer makes me suspect that we are asking the wrong question, as discussed in section 2.2.

The following table summarizes a few relevant facts. The meaning of the entries will be explained below.

 F ∝ m screenable locally black-box detectable paired (action-reaction) real, not pseudo Coriolis √& — √ — — Centrifugal √ — — — — Straight-line acceleration of the frame √ — — — — Newtonian gravity (mass:mass) √ — — √ ? General relativity √& — — √ ? Electrostatics (charge:charge) — √ √ √ √ Magnetostatics (dipole:dipole) — √ √ √ √

In all cases we consider a sufficiently local measurement in a sufficiently uniform field, so that we are not sensitive to nonuniformities in the field. This idea is made precise in equation 3.

The checkmarks in the rightmost column tell us that electrostatics, magnetostatics, and Newtonian gravity have the property that every force is paired with an equal and opposite force. For example, the Newtonian law of universal gravitation:

F = G
 m1 m2 r2
(1)

is symmetric with respect to interchange of m1m2, so the force on object 1 in the field of object 2 is equal and opposite to the force on object 2 in the field of object 1. Coulomb’s law has a similar symmetry.

This stands in contrast to the Coriolis effect, the centrifugal field, and straight-line acceleration of the reference frame, where the observed accelerations are not subject to any pairing rule. For example, consider a freely moving object in a rotating reference frame. There is a Coriolis effect. This is not (in general) associated with an equal-and-opposite Coriolis effect on some other object. (This is one reason, among many, why pseudo forces cannot be considered “just like” real forces. Pseudo forces cannot freely be plugged into the third law of motion, i.e. the action/reaction law.)

General relativity is similar to Newtonian gravity, but slightly more complicated. For one thing, in GR the interaction cannot be described as a mass-to-mass interaction; instead, the first mass creates a field that acts (at some later time) on the second mass. Secondly, relativity requires there to be higher-order terms that depend on the state of motion of the mass, not just the magnitude of the mass. This is profoundly analogous to the way that relativity requires electrostatics to be accompanied by magnetic terms that depend on the state of motion of the charge, not just the magnitude of the charge.

All the known unpaired interactions have the property that they are proportional to mass. Therefore it is always possible to write the equation of motion such that these interactions appear as unforced accelerations.

There is a consensus that any unpaired interaction, if expressed in units of force, clearly must be a pseudo force, not a real force.

Similarly there is a consensus that any force that is not proportional to mass clearly must be a real force. It cannot appear in the equation of motion as an unforced acceleration.

The classification of Newtonian gravity remains unclear. It may appear in the equation of motion as an unscreenable unforced acceleration, but equally it may appear as F/m, where the force is a well-behaved paired force.

To repeat: If you use the “screenability” property, you classify Newtonian gravity one way. If you use the “pairing” property, you classify it the other way. It is unscreenable (like clearly fictitious forces) but paired (like clearly non-fictitious forces).

Note that the “Fm” property runs mostly – but not exactly – parallel to the “unscreenable” property. I put an ampersand (&) in the Coriolis row, because although the Coriolis pseudo force is proportional to mass, it is also proportional to velocity, so the situation is not as simple as it is for the simplest pseudo forces. Similarly general relativity contains terms that are proportional to the mass and to other factors.

Also note that when a force is unscreenable, it is often not clear what entity is exerting this force upon the object, or by what mechanism the force is being exerted. This is significant, because if there is a discernible mechanism – such as the Hookean force in extended spring – all observers can look at the mechanism and agree as to the magnitude an direction of the force. That makes a force seem “real”. In contrast, if the mechanism is not discernible, different observers might have wildly different opinions as to the magnitude and direction of the so-called force. This makes a force seem “fictitious”.

Newton ducked questions about the mechanism of gravitation with the immortal words “hypotheses non fingo” – although the idea goes back to Galileo. The point is that the laws of physics should tell us what happens ... whereas they need not tell us how it happens, and almost never tell us why it happens. For more on this, see reference 2.

The following “black-box” discussion considers a third possibility. It is included for completeness, but it doesn’t lead anywhere important, so you can skip it if you like.

If we look at the “black-box” column, we see that several of the interactions have the remarkable property that the interaction accelerates everything equally. That means that if we construct an object that contains an accelerometer inside it, if the only interactions affecting the object are of this type, the accelerometer will always read zero, because the components of the accelerometer will be accelerated along with the object as a whole.

This stands in contrast to the clearly-real forces in the bottom two rows. Electrostatic fields can easily be screened, for instance by a Faraday cage. If we build an object consisting of an accelerometer inside a Faraday cage, and act on the object with an electric field, the object can be accelerated and the accelerometer will detect this.

Therefore if we classify things according to whether they are detectable inside a small black box, then gravity must be in the same class as centrifugity and straight-line acceleration of the reference frame.

However, in this case the Coriolis effect is the odd man out, because the acceleration is proportional to velocity. In particular, a Foucault pendulum inside a black box will precess relative to the box, if the box is rotating. Similarly, a gyroscope inside a black box will precess relative to the box, if the box is rotating. Either way, we conclude that a rotation of the reference frame does not have the black-box symmetry. It might be argued that we can’t measure velocity without being slightly nonlocal, but the counterargument is that a Foucault pendulum will precess at the same angular rate, no matter how large or small the pendulum is. I say that if the effect is independent of size, it should count as a “local” effect.

I am not willing to say that the centrifugal effect gives rise to a pseudo force while the Coriolis effect gives rise to a real force ... and therefore the “black-box detectable” criterion is not an acceptable way to distinguish real forces from pseudo forces.

### 2.2  Discussion

When we ask whether gravity was a pseudo force, the fact that there is no easy answer makes me suspect that we are asking the wrong question. If there is no combination of experiments and calculations that will answer the question, we should give up on the question and ask some more physically-relevant question(s) instead.

In particular, I suggest we classify interactions according to the physically-meaningful criteria mentioned previous section:

• Is the interaction screenable?
• Is the interaction black-box detectable?
• Does the interaction give rise to pairs of forces, i.e. action/reaction pairs?

These are much better questions, much better than asking whether the force is a pseudo force or not.

Sometimes I attempt to define pseudo force, and this is how far I get: At one extreme, if the interaction is unscreenable and unpaired, then we have pseudo forces. At the other extreme, if the interaction is screenable and paired, then we have real forces. In the remaining cases, notably the case of gravity (which is paired but unscreenable), I don’t know whether they are pseudo or not.

The fact that I don’t know doesn’t bother me, because I don’t need to know. The things I need to know can be expressed in other ways.

Whether or not we resolve the gravity-related questions about pseudo forces, and whether or not we simply restrict attention to problems where gravity is not involved, there are still serious problems with the notion of pseudo forces.

The crucial question is: does a pseudo force count as a force, or not? It seems wildly inconsistent to say that a pseudo force counts as a force for the purpose of plugging into the a=F/m law, but doesn’t count as a force for the purpose of plugging into the action/reaction law. Either a pseudo force is a force, or it isn’t, and either way you’ve got problems. You’re not allowed to sacrifice one law of physics in order to save another.

## 3  Gravity and Electrostatics

Elementary texts commonly point out the analogy between Newton’s law of universal gravitation and Coulomb’s law of electrostatics:

Φgrav(r) =
G
 m1 m2 r
Φelec(r) =
kC
 q1 q2 r
(2)

We can learn something important about the equivalence principle by exploring this analogy. We will discover that the analogy breaks down sooner than you might think.

In either case, we expand the potential in a Taylor series, as follows:

 Φ(r) = Φ(r0) + Φ′(r0) Δr + Φ′′(r0) Δr2 / 2 + ⋯ potential field stress Gravity: arbitrary (gauge) arbitrary (equivalence) non-arbitrary Electrostatics: arbitrary (gauge) non-arbitrary non-arbitrary
(3)

For simplicity and clarity we have written equation 3 using one-dimensional notation; the generalization to additional dimensions is straightforward.

Let’s consider the leading term on the RHS, i.e. the zeroth-order term, i.e. the potential. In the case of gravity, we can change the potential arbitrarily, since it depends on our choice of gauge. The same is true of the electrostatic potential: the potential depends on the gauge, which we can choose arbitrarily.

Next let’s consider the first-order term. This term represents a uniform field. In the case of gravity, this term can be changed arbitrarily, via the equivalence principle, since it depends on our choice of accelerated reference frame. The interesting thing is that in the case of electrostatics, the first-order term cannot be changed arbitrarily. This is where the analogy between gravitation and electrostatics breaks down.

Note that the first order term, by definition, indeed by construction, describes a uniform field. (The nonuniformities are described by the higher-order terms.) Therefore the equivalence principle applies precisely and unreservedly to the first-order gravitational term: this particular term is indistinguishable from a uniform acceleration of the reference frame.

## 4  Are Pseudo Forces Forces?

As derived in reference 3, one of the equations of motion in a rotating frame can be expressed as:

 PM•• = − ω•  r∼(90/2) ρM   r(90/2) + ω2 ρM − 2 ω r∼(90/2) ρM•   r(90/2) + r∼(−θ/2) (1/m)FJ r(−θ/2)
(4)

Or equivalently as:

 PM•• = (1/m) { − m ω•  r∼(90/2) ρM   r(90/2) (fictitious) + m ω2 ρM (fictitious) − 2 m ω r∼(90/2) ρM•   r(90/2) (fictitious) + r∼(−θ/2) FJ r(−θ/2)  } (real)
(5)

where

 PM•• = − ω•  r∼(90/2) ρM   r(90/2) + ω2 ρM − 2 ω r∼(90/2) ρM•   r(90/2) + r∼(−θ/2) PJ•• r(−θ/2)
(6)

As pointed out by David Bowman, there are two schools of thought about the interpretation of these equations. Before discussing the details, we emphasize that the difference concerns interpretation only. There is no doubt that equation 4 and equation 5 are mathematically equivalent, and there is no doubt as to how to apply them in practice.

 The first school of thought holds that pseudo forces are not forces. Sometimes the equation of motion includes unforced accelerations. The second school of thought holds that pseudo forces are forces – at least for narrow purpose of calculating accelerations. There are no unforced accelerations.

 This school focuses attention on equation 6, which is purely a conversion between reference frames. Given the acceleration, velocity, etc. relative to one frame, we can find the acceleration, velocity, etc. relative to the other frame. There are no forces appearing in this equation, and no masses. We do not need to know the second law of motion in order to derive or use this equation. This equation applies even to objects with unknown mass, or zero mass. This school focuses attention on equation 5. Given an equation with acceleration on the LHS, the first temptation is to multiply both sides by the mass m. This is a harmless bit of mathematics, assuming m is nonzero. The second temptation is to interpret every term on the RHS as a force, so that the equation has the structure of F=ma, or rather a=(1/m){F}. This is a question of interpretation. This is a question of taste, on which opinions may differ.

 In equation 6 and equation 4, every term has dimensions of acceleration. Every term within the braces in equation 5 has dimensions of force. But remember, there is more to physics than dimensional analysis.

 In equation 4, F is of course a bona-fide force, and therefore the term involving (1/m)FJ is a force-related acceleration. The other three terms on the RHS can be interpreted as non-force-related accelerations, that is, as unforced accelerations. It is tempting to say that in equation 5, the Coriolis term should be called a Coriolis force, and the centrifugal term should be called a centrifugal force. They definitely have dimensions of force ... but whether these terms “really are” forces is a question of interpretation.

 Both schools agree that the centrifugal acceleration is real, just as real as the other terms on the RHS of equation 4. An object really will accelerate (relative to the rotating frame) in accordance with equation 4.

 In this school, the centrifugal acceleration is a fine example of an unforced acceleration. If you multiply this acceleration by a mass, you get something with dimensions of a force, but it is not really a force. We call it a fictitious force or pseudo force. In this school, it is taken for granted that the centrifugal acceleration is associated with a centrifugal force. Of the four terms in braces on the RHS of equation 5, one is clearly a real force, involving F, the force exerted upon the object by its surroundings. The other three terms are called fictitious forces or pseudo forces ... but is somewhat unclear what people mean by these terms, as discussed in section 4.

 The term “pseudo” is Greek for “false” or “lying” and is meant to be very unflattering. This school shrugs off the pejorative meaning of “pseudo”.

 There is no reason to expect (let alone require) the law of motion for rotating frames to have the same structure as for nonrotating frames. Yes, the LHS of equation 5 is an acceleration, and yes, one of the terms within the braces is a force, but it does not follow that the neighboring terms are real forces. They are terms in the equation of motion, but their provenance and their meaning makes them very different from the bona-fide F term. And keep in mind that the decision to redistribute the factor of 1/m was arbitrary and somewhat unnatural. The pseudo forces contribute to the total acceleration PM•• in the same way that real forces do, so we are tempted to interpret them as forces in this narrow context.We should not entirely give in to this temptation, because in other contexts, notably the third law of motion, pseudo forces do not enter in the same way that real forces do. For details on this, see section 4.

 Since a force per unit mass is an acceleration, it is unnatural to talk about pseudo forces at all. It is more natural to just talk about the acceleration, i.e. to return to equation 4 in preference to equation 5. It is characteristic of pseudo forces that they are proportional to the mass of the object. If necessary, we can talk about these forces in terms of “force per unit mass”.

 This school of thought is in tune with modern (post-1900) physics. Since the so-called force of gravity is always proportional to mass, you might suspect that it is a pseudo force. And sure enough, in general relativity, the effect of a local and/or uniform gravitational field is treated as an unforced acceleration. See reference 4 for more on this. This school of thought is in tune with classical (pre-1900) physics. Classical gravitation is treated as a real force (not a pseudo force).

The three rotation-related terms in equation 5 (swat force, centrifugal force, and Coriolis force) are examples of what are called pseudo forces or equivalently fictitious forces. Various people seem to use these terms in various ways, but it seems that a good working definition of pseudo forces is that they appear in the equations of motion in places where you might expect a force to appear, but they do not correspond to any bona-fide force exerted upon the object by its surroundings.

## 5  Multiple Views of Centrifugity

There are several different ways of talking about centrifugal effects. Again we emphasize that the differences concern terminology and interpretation only; there is no dispute about the equations of motion, or about how to apply them.

The main points about centrifugity are summarized in the following table:

 rotating frame nonrotating frame (1) at every point centrifugal field centrifugal acceleration = ac zero centrifugal field zero centrifugal acceleration (2) acting on every object ac = unforced acceleration pseudo force is not a force ⇒ no centrifugal force ⇒ centrifugal field is an acceleration not a force ac = (1/m) Fc pseudo force is a force i.e. centrifugal force no unforced acceleration no pseudo force no centrifugal force (3) force expressed by object stationary in rotating frame Fc,exp Fc,exp (4) force on pivot Fp Fp

We use ac to represent the centrifugal acceleration, namely ac := ω2 ρ. This is just one contribution to the overall acceleration, since for any particular object, there may be other contributions to the object’s total acceleration, as detailed in equation 4.

It is generally agreed that there is such a thing as the centrifugal field, which exists in a rotating frame and not otherwise, as summarized on row (1) of the table. The field exists at every point in the rotating frame, regardless of whether there is an object at that point. At any particular point, the centrifugal field is an acceleration, called the centrifugal acceleration. This is a real acceleration, not a “pseudo” acceleration. Relative to a rotating frame, an object really will accelerate in accordance with equation 4.

Next, we turn to the topic of centrifugal force, in the context of a given object of mass m. Everyone agrees that a pseudo force can be formed by multiplying the centrifugal acceleration by m. Beyond that, there are two schools of thought, as discussed in section 4. One school holds that the pseudo force is not really a force, and that the centrifugal acceleration should be treated as an unforced acceleration. Meanwhile, the other school holds that the centrifugal pseudo force should be treated like any other force, so that the equation of motion does not have any unforced accelerations. So the question of whether the centrifugal pseudo force is “really” a force is a matter of interpretation. This is summarized row (2) of the table.

Thirdly, we turn to the situation of an object that is stationary in the rotating frame. Everyone agrees that a centripetal (inward) force must be impressed upon this object in order to keep it in position. Correspondingly, there is a centrifugal (outward) force expressed by the object upon its surroundings. We denote this by Fc,exp, where “exp” stands for “expressed”. Beware that the laws of physics are conventionally stated in terms of impressed forces, not expressed forces, and introducing expressed forces is a notorious source of confusion. Nevertheless, there is nothing “pseudo” about the expressed force. It is undoubtedly a real, measurable force. Observers in the nonrotating frame agree with observers in the rotating frame about the existence and significance of this force. This is summarized row (3) of the table. Alas this force is not suitable for use in the conventional second law of motion, because it is an expressed force, not an impressed force, as discussed in reference 3.

Fourthly, suppose you attach a massive object to a string, and twirl it around and around in a circle at constant rate. There will be a force impressed on whatever you are using for a pivot. This is a real force, not a pseudo force. For the moment, let’s neglect air resistance, the mass of the string, and similar smallish effects. Then the force impressed on the pivot is numerically equal to the expressed force discussed in the previous paragraph. From the pivot’s point of view, this is an impressed force, impressed upon the pivot by its surroundings, i.e. by the string. This force is centrifugal, in the sense that it is directed outward, away from the center. Therefore we can call this a centrifugal force, in accordance with the plain meaning of the words.

This means we have three conflicting definitions of centrifugal force, appearing on lines (2), (3), and (4) of the table. Having multiple definitions is bad enough, but to rub salt in the wound, each definition has its own problems.

• The there are two schools of thought as to whether the centrifugal pseudo force is really a force.
• The centrifugal force expressed by an object upon its surroundings is real and has an agreed-upon meaning, but it is a trap for the unwary, because it is an expressed force, unsuitable for use in the conventional second law of motion, as discussed in reference 3. A far more serious objection to this notion of centrifugal force is that it applies only to objects that are unaccelerated relative to the rotating frame. For objects that are free to move, indeed even partially free to move (such as an Atwood machine, as shown in figure 1), the centrifugal effect is still there, as evidenced by the acceleration of the object, but the actual force expressed by the object upon its surroundings will not generally be equal to m ρ ω2.
• The force on the pivot can be expressed as Fp = m ac which is a perfectly valid equation, and can be considered a consequence of the second law of motion, but must not be considered a statement of the second law, even though it fits the general form F=ma. As discussed in reference 3, the second law speaks of the force impressed on the object, the mass of the object, and the acceleration of the object. Here we have the force on the pivot equated to the mass of something else and the acceleration of something else. The string can be considered just a trick for transporting centrifugal expressed force and converting it to an impressed force, impressed upon the pivot. This notion of centrifugal force therefore shares the objectionable property mentioned in connection with the centrifugal expressed force, namely that it doesn’t make sense for objects that are free (or even partially free) to move relative to the rotating frame.

Figure 1: Atwood Machine on a Scale

The ordinary form of the second law of motion is:

 Fimp = m  a              (7)

In the case of an object at rest in the rotating frame, we can write the remarkable equation:

 Fc,exp = m ac              (8)

which superficially resembles the second law, but actually has quite a different interpretation. The LHS of the second law of motion (equation 7) uses the impressed force Fimp, in contrast to equation 8 which uses an expressed force. Meanwhile, the RHS of the second law involves the acceleration of the object, in contrast to equation 8 which involves the centrifugal field; remember that ac has a value at each point in the field, independent of the existence of any object at the point, and certainly independent of the acceleration of any such object.

Here are the conventional and sensible ways to avoid these problems:

• If you are ever tempted to talk about the centrifugal pseudo force, you should talk about the centrifugal acceleration instead. Everybody in the rotating frame agrees about the centrifugal field and how to interpret it.
• If you are ever tempted to talk about the centrifugal expressed force Fc,exp, you should talk about the corresponding impressed force Fc,imp instead. This is the centripetal force impressed upon the object by its surroundings.

## 6  References

1.
Feynman, Leighton, and Sands, The Feynman Lectures on Physics. Volume I chapter 12: “Characteristics of Force” especially section 12-5. See also chapter 7: “The Theory of Gravitation”.

2.
John Denker, "Cause and Effect" www.av8n.com/physics/causation.htm

3.
John Denker, “Motion in a Rotating Frame” ./rotating-frame.htm

4.
John Denker, “Can You Feel Gravity?” www.av8n.com/physics/gravity-perception.htm
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