# Fictitious force

1. Jul 20, 2008

### calculus_jy

can someone tell me how to distinguish fictitious and real force, together with the correlation of fictitious acceleartion and real acceleration, my research seems to fail on this subject

2. Jul 20, 2008

### Staff: Mentor

Again, I like the GR approach. Real acceleration/force is an acceleration that can be measured by an accelerometer. A fictitious force/acceleration is one which cannot be measured by an accelerometer.

If you prefer the Newtonian approach you simply make an exception to the above by considering gravity as the only real force which cannot be measured by an accelerometer.

Last edited: Jul 20, 2008
3. Jul 21, 2008

### DrGreg

In Newtonian physics and Special relativity, real forces are measured by inertial observers. A fictitious force is something that an accelerating observer would assume ought to exist in order to make Newton's second law true relative to his accelerating frame. An example is centrifugal force for a rotating observer. But, in Newtonian physics and Special relativity, Newton's second law is not true in an accelerating frame, so that is why the force is called "fictitious".

In General Relativity all forces are real -- relative to any observer -- and there are no fictitious forces. However different observers disagree on the magnitude of any real force. (The last sentence is also true in Special Relativity, but not in Newtonian physics.)

4. Jul 21, 2008

### Fredrik

Staff Emeritus
Is that really correct? It seems to me that it would make sense to define a "real" force as a non-zero value of $m\cdot d^2x^\mu/dt^2$ in a co-moving local inertial frame (it's a measure of the deviation from geodesic motion), and a "fictitious" force in an arbitrary frame as what we have to add to get Newton's second law (some four-vector version of it) to hold in those coordinates too. The first part of that definitely makes sense. I'm not sure about the second.

5. Jul 21, 2008

### yuiop

So as far as GR is concerned, centrifugal force on a turntable that produces a reading on an accelerometer IS a real force while centrifugal force acting on an orbiting body is a fictitious force?

6. Jul 21, 2008

### D H

Staff Emeritus
The accelerometer on a turntable registers an acceleration toward the center of the turntable. There is no reading for the outward fictitious centrifugal force. An accelerometer on an orbiting body registers zero acceleration. There is no reading for the inward fictitious gravitational force or for the outward fictitious centrifugal force.

7. Jul 21, 2008

### Staff: Mentor

I think you mean centripetal force. But yes, the centripetal force on a turntable is measurable by an accelerometer and is therefore real, while the centripetal force on a satellite is not measurable by an accelerometer and is therefore fictitious.

However, DrGreg disputes my interpretation of fictitious and real forces in GR, so don't take my comments as definitive.

8. Jul 22, 2008

### DrGreg

I don't claim to be a GR expert so my understanding could be wrong too. But my view is that, in GR, relative to a given frame, a force is anything that causes coordinate acceleration i.e. $d^2 \textbf{x}/dt^2$ measured in the frame's $(t, \textbf{x})$ coordinates. (Or, to be more precise, causes a change in coordinate momentum.) Thus, for an object rotating on a disk, an inertial observer would say there is a centripetal force causing the object to accelerate, and no centrifugal force, whereas an accelerating observer who is stationary on the disk would say the object is not accelerating, the net force on the object is zero consisting of equal and opposite centripetal and centrifugal forces.

For a satellite in orbit, in the satellite's own free-falling frame there are no forces acting on it -- it is moving inertially (along a geodesic). For someone hovering "stationary" above the planet at the altitude of the satellite (and therefore non-inertial), the satellite is accelerating towards the planet due to a force called "gravity".

(Nevertheless, just to confuse things and play devil's advocate, there is a case to be made for attributing proper acceleration (measured by an accelerometer and equal to acceleration in the co-moving inertial frame) to a "real" force (which I guess you could call the "proper force") and any other acceleration to a "fictitious" force. Under this interpretation, gravity is a fictitious force.)

I stand to be corrected by any GR experts reading this.

9. Jul 22, 2008

### shalayka

Perhaps it can be summed up by an analogy of travel in a car:

Fictitious force - A driver inside of a car is traveling along a rectilinear path. The car turns left (accelerates, changes direction), but the driver's body attempts to continue along the rectilinear path due to inertia. The driver experiences a fictitious force, in that they feel that they are being accelerated toward the right side of the car.

GR - The car and driver in orbit around the Earth both follow the same curved path. The driver does not experience a fictitious force because the inertial path is the curved path.

10. Jul 23, 2008

### yuiop

Hi Dr Greg,
I would like to join you in the Devil's advocate game and suggest the opposite. Fictitious force is what is measured by an accelerometer while a real force is usually accompanied by tension or compression and results in real acceleration relative to an inertial observer. Some examples:

Car turning to the left while going forward.
Real centripetal force acting from right to left. (Not measured by an accelerometer) (Is accelerating from right to left according to a comoving inertial observer)
Fictititious reaction centrifugal force acting from left to right. (Is measured by an accelerometer)

Car accelerating in a straight line from right to left.
Real centripetal force acting from right to left. (Not measured by an accelerometer) (Is accelerating from right to left according to a comoving inertial observer)
Fictititious reaction force acting from left to right. (Is measured by an accelerometer)

Weight resting on a table.
Real force exerted by the table on the weight, acting upwards. (Not measured by an accelerometer) (Is accelerating upwards according to a comoving inertial observer)
Fictititious force (gravity) acting downwards. (Is measured by an accelerometer)

Satellite in orbit.
Fictitious centripetal force (gravity) acting towards the centre of orbital circle.
Fictitious centrifugal reaction force acting outwards.
In this case there is no acceleration according to a comoving inertial observer and the fictitious forces cancel out so there is no measurement on an accelerometer.

Fictitious gravity force acting downwards.
Fictitious reaction force acting upwards?
In this case there is also no acceleration according to a comoving inertial observer and the fictitious forces cancel out so there is no measurement on an accelerometer.

Person pushing against an unmoving wall.
Real force exerted by person on the wall. (Tension in muscles.) (Not measured by accelerometer)
Real reaction force exerted by the wall on the person. (Tension in the wall) (Not measured by an accelerometer)
No acceleration according to a comoving inertial observer because one real force is cancelled by another real force. The only real sign that real forces are acting in this case is tension or compression in physical objects.

From all the above, it seems real forces that are not balanced by another real force result in real acceleration relative to an inertial observer and a fictitious reaction force that is measured by an accelerometer. Note that the real force is never detected by an accelerometer

11. Jul 23, 2008

### D H

Staff Emeritus
An accelerometer essentially has a comoving inertial observer: The accelerometer's test masses. The accelerometer uses these test masses to ascertain the acceleration of the accelerometer case. That the test mass moves to the right with respect to the accelerometer case means the accelerometer case is accelerating to the left, and this is exactly what the accelerometer reports. You have the behavior of the accelerometer exactly wrong, particularly here:
The accelerometer measures the real normal force, not the fictitious gravitational force. In particular, an accelerometer resting on a table will report a 1g upward acceleration. An accelerometer does not measure the fictitious gravitational force. Attach an accelerometer to an orbiting satellite, as per your next example:
The centrifugal force only exists in the mind of an observer located at the center of the Earth and rotating with the satellite. Why this observer, rather than an observer on the surface of the Moon or in orbit around Jupiter? Moreover, this cancellation of gravitational and centrifugal force is only true for a satellite in a circular orbit in a two-body system. Centrifugal force does not cancel gravitational force for multi-body systems (e.g., a satellite whose orbit is perturbed by lunar and solar gravity), elliptical orbits or for vehicles in what some call "orthogonal orbits" (your next example):

Here your Devil's advocacy falls apart, and you know it. What fictitious reaction force acting upwards?

Accelerometers measure the acceleration due to the net real force. This example is a red herring as the net force is zero.

12. Jul 23, 2008

### DrGreg

I agree with DH's post #11.

13. Jul 23, 2008

### yuiop

Hi Dr Greg and DH.
As it so happens, I agree with both you in posts #11 and #12. As I said, I was playing Devil's advocate. It makes more sense to consider the measurement of an accelerometer as a measurement of a real force. In this interpretation, the acceleration measured by an accelerometer resting on a table is the real upward acceleration exerted by the table on the accelerometer. With this interpretation, I think we have to discard Newton's assertion that for every action there is an equal and opposite reaction especially if we only consider real forces. Acceleration is a result of a real force that has no corresponding real force acting in the opposite direction. This answers the question often raised by students being introduced to Newton physics, as to why anything accelerates if there always an equal and opposite force present. The list with real acceleration as measured by a comoving inertial observer being the same as what is indicated by an accelerometer and being the result of a real force becomes:

Car turning to the left while going forward.
Real centripetal force acting from right to left. (Measured by an accelerometer) (Is really accelerating from right to left according to an inertial observer)
Fictitious reaction centrifugal force acting from left to right. (Not measured by an accelerometer or inertial observer)

Car accelerating in a straight line from right to left.
Real centripetal force acting from right to left. (Measured by an accelerometer) (Is really accelerating from right to left according to an inertial observer)
Fictitious reaction force acting from left to right. (Not measured by an accelerometer or inertial observer)

Weight resting on a table.
Real force exerted by the table on the weight, acting upwards. (Measured by an accelerometer) (Is accelerating upwards according to an inertial observer)
Fictitious force (gravity) acting downwards. (Not measured by an accelerometer)

Satellite in orbit.
Fictitious centripetal force (gravity) acting towards the centre of orbital circle.
Fictitious centrifugal reaction force acting outwards.
In this case there is no acceleration according to a comoving inertial observer and there is no measurement on an accelerometer as there are no real forces involved.

Fictitious gravity force acting downwards.
In this case there is also no acceleration according to a comoving inertial observer and there is no measurement on an accelerometer as there are no real forces involved.

Person pushing against an unmoving wall.
Real force exerted by person on the wall. (Tension in muscles.)
Real reaction force exerted by the wall on the person. (Tension in the wall)
No acceleration according to a comoving inertial observer because one real force is cancelled by another real force. The only real sign that real forces are acting in this case is tension or compression in physical objects. There is no measurement on an accelerometer as the real forces cancel out and there is no net real force or acceleration.

When there is a non-inertial comoving observer, the real force is acting on the object and on the observer so there is no acceleration relative to the non-inertial observer. The fictitious forces (such as centrifugal force or gravity) are imaginary forces that are invented to explain the lack of acceleration relative to the non-inertial observer, but really no explanation is required beyond the fact that the non-inertial observer is also subject to the same real force and acceleration that is acting on the object he observering.

That hopefully makes more sense

Last edited: Jul 23, 2008
14. Jul 25, 2008

### DrGreg

Yes
That is certainly the view of inertial observers, and in SR and Galilean Relativity (=Newtonian mechanics) it is the only "real" view, all other views are "fictitious". But my understanding is that GR is more democratic than that, and celebrates the diversity of all observers equally. So what is fictitious to an inertial observer may be real to a non-inertial observer. (As always, I stand to be corrected by a GR expert who knows the official line on this.)

(Sorry, I will be off-line for the next week and a half, and will not be able to reply for a while.)

15. Jul 25, 2008

### Phrak

Things are moving in all directions here. The context is important. Of primary importance, in General Relativity, gravity is not a force.

In Newtonian physics, a body that is otherwise free, but is accelerated by gravity, is freely moving (and following a geodesic path) in General Relativity.

As far as ficticous forces go, where gravity is not a force, neither are there ficticous gravitational forces.