Mach versus Einstein

[sol invictus]

Consider two identical rocket ships separated by a distance of 1 km. At t=0 the ships are at rest with respect to each other, and are pointed in opposite directions which are perpendicular to their displacement (so if one is at x=0 and the other at x=1, they are pointed in the y and -y directions respectively). They are connected by a rope, and we will posit that at t=0 the rope is extended to almost its full length, but is slack, with no tension in it.

I will ignore the effects of gravitational attraction between the rockets for now. In the real world that would be a very good approximation because the gravitational attraction between two such rockets would be extremely weak, and if they start somewhere in intergalactic space, far from any stars or matter, there are no other gravitational forces of relevance. In this thought experiment we could also set G (the gravitational constant) equal to zero, in which case there really are no such forces, but it's not necessary. We can also (in this thought experiment) just take the rest of the universe to be completely empty, so let's do that.

So then ignoring gravity, the configuration I've described is static - nothing about it will change with time. (In reality the gravitational mutual attraction would cause the rockets to fall towards each other, but very, very slowly.)

Now suppose each rocket blasts its engines for an identical (and short) amount of time. This will put the rockets into a slow rotation around their center of mass. We will assume that the rope is capable of sustaining tension up to some critical amount, at which point it will snap. We can now gradually increase the angular speed of rotation by having the ships fire short synchronized bursts of their engines every now and then. Here's a physical question - what will happen? Will the rope snap, and when?

According to Newton, that's easy. We just use standard Newtonian formulas to find the tension in the rope as a function of the inertial mass of the spaceships and their angular velocity. The answer is that T=2mv^2/r=2mrw^2, where T is the tension in the rope, r is 1km, v=rw is the velocity in km/s, and w is the angular velocity in radians per second. So as we increase w, eventually T will reach the critical value and the rope will snap.

According to Mach (at least according to the caricature of his position as it's usually presented today, which I referred to before as Straw-Mach ), since the rope was slack initially, and since there are no distant bodies to measure the rotation with respect to, there will never be any tension in the rope, and so it will not snap no matter how long the rockets fire their engines.

What about Einstein? Well, that's easy - Einstein's theories reduce to Newton's in the limit of small velocity (compared to the speed of light) and when gravitational forces are weak (as they are in this case). Therefore the answer will be just as it was for Newton, up to some tiny corrections of order (v/c)^2 (which is 10^-16 for v around a meter/second) and (Gm^2/r)/(mv^2) = Gm/(rv^2) (which is around 10^-11 if the spaceships weigh 1000kg). So if we believe Einstein, Mach was wrong and Newton was right (for this particular question at least).

Einstein also allows us to think about the problem in another frame of reference. Once the rockets are rotating, we can choose coordinates which rotate with respect to the original coordinates with exactly the same angular speed. In those new coordinates the rockets and rope will be at fixed coordinate positions. However Einstein took as a postulate that the laws of physics are invariant under coordinate transformations, so there must still be a tension in the rope. So where is it coming from?

Now it should be obvious that this procedure of changing coordinates does NOT change the space or the physics in any way. It is literally nothing more than a re-labeling of the spacetime points. It's a purely human convention, which has no more relevance to the physics than discussing the experiment in English rather than German. (That is why Einstein took general coordinate invariance as a fundamental postulate.)

So what happens in the new frame, in the language of general relativity? The answer cannot be that the space is curved, since as we have just seen it's the same space, and indeed the math tells us it's not curved. Instead, it's that the same old geodesics of the space, when expressed in the new coordinates, are no longer "straight lines" - in other words, fixed position in the new coordinates is not a geodesic. Therefore if the rockets and rope are at rest in these new coordinates they will experience a force, and it is that force which accounts for the tension in the rope.

Now it should be clear that this force has nothing to do with gravity in any ordinary usage of that term. It has nothing to do with Newtonian gravity because it arises from Newton's laws of motion, not from Newton's law of gravity. It has nothing to do with the curvature of the spacetime, because the spacetime isn't curved - we just chose to use different coordinates to describe it. And it remains non-zero and almost unchanged when the grav. constant G=0. Hence it's best to think about this force as an inertial force, a so-called "fictitious force" to use the common term, not as a fundamental force of nature.

Of course in general relativity this distinction is not so easy to make, because it general the spacetime really IS curved by the presence of stress-energy, and then it's not so easy to tell whether some forces are due to that curvature or due to some kind of "fictitious force".

Question - is there any way to make such a distinction in the general case? I don't know the answer, but it might be interesting to think about, as it might provide a way to test GR in some interesting way.

 

[Macoy]

Perhaps they spiral around their "centre of gravity" until they collide.

 

[Myriad]

Einstein also allows us to think about the problem in another frame of reference. Once the rockets are rotating, we can choose coordinates which rotate with respect to the original coordinates with exactly the same angular speed. In those new coordinates the rockets and rope will be at fixed coordinate positions. However Einstein took as a postulate that the laws of physics are invariant under coordinate transformations, so there must still be a tension in the rope. So where is it coming from?

I think you need to rethink this. A reference frame that's rotating with respect to another is not a coordinate transformation under which the laws of physics are invariant.

Respectfully,
Myriad

 

[sol invictus]

Perhaps they spiral around their "centre of gravity" until they collide.

OK, we can consider the effects of gravitational attraction. The rockets could spiral in (along an ellipse, actually) and collide only if the rotation is so slow the force of gravity overcomes it. Increasing the rotational speed a little, there is a critical point where the forces balance exactly: both ships orbit the center of mass in a stable circular orbit (with or without the rope, which would be just barely slack). For higher velocities the rope will have a tension in it, because the inertial force acting to pull the ships apart is stronger than the gravitational force acting to pull them together.

If the rockets have mass of around 1000kg and the distance is 1km, gravity will overcome inertia only if the speed of rotation is less than around .01 milimeters/second, which is extremely slow. That's why we can ignore gravity in this thought experiment. (And yes, that critical speed does depend on the mass, misunderstandings of the equivalence principle aside.)

 

[sol invictus]

I think you need to rethink this. A reference frame that's rotating with respect to another is not a coordinate transformation under which the laws of physics are invariant.

On the contrary. The laws are invariant under any coordinate transformation; that is the underlying principle of general relativity. Of course the laws may look different when written in different coordinates... but one of the consequences of the principle is that the results of any physical experiment do not depend on the coordinates some theorist chooses to use to describe it with.

If there is a tension in the rope there is a tension in the rope. If the rope snaps the rope snaps. Those are physical facts, and there cannot be any debate about them in a sensible theory. Requiring that your theory satisfy this principle is a tremendously restrictive condition, one that leads more or less inevitably to Einstein's equations for gravity. That such a simple principle takes you so far is what makes GR the most beautiful theory in physics, at least in my opinion.

The following statement corresponds to the fundamental idea of the general principle of relativity: “All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.”

We can state this general principle of relativity in still another form, which renders it yet more clearly intelligible than it is when in the form of the natural extension of the special principle of relativity. According to the special theory of relativity, the equations which express the general laws of nature pass over into equations of the same form when, by making use of the Lorentz transformation, we replace the space-time variables x, y, z, t, of a (Galileian) reference-body K by the space-time variables x', y', z', t', of a new reference-body K'. According to the general theory of relativity, on the other hand, by application of arbitrary substitutions of the Gauss variables x1, x2, x3, x4, the equations must pass over into equations of the same form; for every transformation (not only the Lorentz transformation) corresponds to the transition of one Gauss co-ordinate system into another.

 

[69dodge]

Hey, a new thread!

The thread is dead, long live the thread?

Thanks for the interesting discussion, BTW.

Now it should be clear that this force has nothing to do with gravity in any ordinary usage of that term. It has nothing to do with Newtonian gravity because it arises from Newton's laws of motion, not from Newton's law of gravity. It has nothing to do with the curvature of the spacetime, because the spacetime isn't curved - we just chose to use different coordinates to describe it. And it remains non-zero and almost unchanged when the grav. constant G=0. Hence it's best to think about this force as an inertial force, a so-called "fictitious force" to use the common term, not as a fundamental force of nature.

Of course in general relativity this distinction is not so easy to make, because it general the spacetime really IS curved by the presence of stress-energy, and then it's not so easy to tell whether some forces are due to that curvature or due to some kind of "fictitious force".

It has nothing to do with Newtonian gravity, I agree.

It looks rather like gravity in GR, though.

Clocks run faster on mountaintops than on the surface of the Earth, and the force of gravity points away from mountaintops toward the Earth. In the coordinate system in which the rockets aren't moving around each other, clocks run faster on the rope side of the rockets than on the outside, and the force "of gravity" at each rocket points away from the rope.

I don't think the magnitude of gravitational forces are, in GR, ever due to spacetime curvature. I think maybe curvature determines tidal forces, or something like that? That is, curvature has to do with the nonuniformity of gravitational/inertial forces over a small volume, not with their absolute magnitude at a point. Their magnitude depends on the choice of coordinates.

Question - is there any way to make such a distinction in the general case? I don't know the answer, but it might be interesting to think about, as it might provide a way to test GR in some interesting way.

I'm pretty sure there isn't.

 

[sol invictus]

I don't think the magnitude of gravitational forces are, in GR, ever due to spacetime curvature. I think maybe curvature determines tidal forces, or something like that? That is, curvature has to do with the nonuniformity of gravitational/inertial forces over a small volume, not with their absolute magnitude at a point. Their magnitude depends on the choice of coordinates.

If by "magnitude of gravitational forces" you mean the results of some physical experiment, then those results cannot depend on coordinates. That's the general principle behind GR.

What people ordinarily mean by the strength of gravity somewhere is the value of coordinate invariant quantities constructed from the Riemann curvature tensor. The simplest example is the Ricci scalar, which by Einstein's equation is proportional to the trace of the stress-energy tensor. A somewhat more interesting example is the Riemann tensor squared, which can be non-zero even when there is no stress-energy present at that point. When we say gravity is weak somewhere, we mean that all those invariants are small. Of course in flat space (in any coordinates) all these quantities are zero.

 

[69dodge]

If by "magnitude of gravitational forces" you mean the results of some physical experiment, then those results cannot depend on coordinates. That's the general principle behind GR.

I just mean what a scale shows if I stand on it. A force. Something that's measured in Newtons. I thought that was the sort of thing you wanted to categorize as being "due to gravity" or "due to inertia".

What people ordinarily mean by the strength of gravity somewhere is the value of coordinate invariant quantities constructed from the Riemann curvature tensor. The simplest example is the Ricci scalar, which by Einstein's equation is proportional to the trace of the stress-energy tensor. A somewhat more interesting example is the Riemann tensor squared, which can be non-zero even when there is no stress-energy present at that point. When we say gravity is weak somewhere, we mean that all those invariants are small. Of course in flat space (in any coordinates) all these quantities are zero.

Ok. But gravity, measured in that way, isn't a force, right? If I step on a scale, it won't tell me how strong that sort of gravity is. An astronaut orbiting the Earth, and I sitting on my chair, experience nearly the same strength of gravity of that kind.

So now I'm not sure what your question is. What is the quantity whose cause we are trying to assign either to gravity or to inertia?

 

[Schneibster]

/me whistles circus calliope music.

Look there, Myriad says the laws of physics are different. You, sol, say they are the same. To Myriad, the laws of physics state that an inertial object moves in a straight line. To you, sol, the laws of physics state that an inertial object moves along a geodesic. You have totally failed to communicate clearly, because you have not accepted the basis upon which Myriad believes the "laws of physics" operate. To all appearances, you do not even understand that that is Myriad's basis. Next, you'll be telling Myriad s/he's wrong; and you'll be wrong in doing so. From Myriad's point of view, the "laws of physics" include Newton's Laws; to you, Newton's Laws must be modified. To Myriad, that modification is a modification of the laws of physics.

The upshot is, you're both right; but for Myriad to understand relativity, you must explain that the laws of physics s/he knows are not THE LAWS OF PHYSICS, they're THE LAWS OF PHYSICS IN FLAT SPACETIME. And that there are more comprehensive laws, which apply in ALL spacetimes, flat or not. And those laws of physics are THE laws of physics; the ones Myriad is used to are only a special case.

After ten or fifteen such conversations, you may begin to understand how to explain it from scratch. But first you must begin by not telling people they're wrong when they're not.

 

[The Man]

Of course in general relativity this distinction is not so easy to make, because it general the spacetime really IS curved by the presence of stress-energy, and then it's not so easy to tell whether some forces are due to that curvature or due to some kind of "fictitious force".

Question - is there any way to make such a distinction in the general case? I don't know the answer, but it might be interesting to think about, as it might provide a way to test GR in some interesting way.

I thought that was one of the results of general relativity or the general case, that the perceived force of gravity is simply a fictitious force like the Coriolis force or centrifugal force? Objects do not move under the force of gravity, gravity warps space-time and objects simply take the shortest path through that warped space-time. In the case of your bound rockets the force on the rope is centrifugal stretching the rope, meaning the reaction of the rope is compressive (like an extended spring) transferring a centripetal force from one rocket to the other and countering each rockets inertial tendency to continue in straight lines. Although considered a fictitious force it still has very real consequences just as the fictitious centrifugal force has on a centrifuge, the fictitious Coriolis force has on a missile trajectory or the fictitious gravitational force has on just about everything.

 

[Schneibster]

I think you need to rethink this. A reference frame that's rotating with respect to another is not a coordinate transformation under which the laws of physics are invariant.

Respectfully,
Myriad

The correct answer here is to begin by asking the question, "what laws of physics are you referring to?"

If your answer is, "Newton's Laws of Motion," then you're correct; Newton's Second Law states that an object in motion remains in motion at a constant velocity (and since velocity is a vector, that means in a straight line) and an object at rest remains at rest, unless acted upon by a force. This is obviously not true in a rotating frame of reference.

However, as it turns out, there is another answer to what the laws of physics are. This answer is the theory of relativity. What it says is that you must understand the meaning of "a straight line" (and "a vector") differently under certain circumstances. Those circumstances obtain any time you can perform an experiment that shows you are in an accelerated frame of reference. When your frame is accelerated, you can always measure how much, by performing an experiment. Once you have done so, then you can make the correct adjustment to the meaning of "straight line" (and "vector") to make the laws of motion come out right, and correctly describe what you see.

If we wish to generalize these new, more correct laws of motion, then we must define a new sort of "straight line." This new definition is called a "geodesic." You can always determine a geodesic by observing the motion of an inertial body, that is a body upon which no forces are acting. If your frame is one in which lasers make straight lines (as we ordinarily define them), and the object you are observing is in a frame where that is also true, then you will observe such a body as moving in a straight line at a constant velocity, just as Newton's Laws say. However, if either your frame, or the body's frame, or both, sees lasers making curves, then you will observe deviation from that straight line, and that deviation will be the same one that the lasers show.

One way that curvature might happen is if either you or the body you are observing or both are under the influence of a gravity field. Gravity warps spacetime, giving it an observable curvature. Laser beams in a gravity field are curved, according to an observer outside that gravity field. Inertial objects in a gravity field move in curves, according to an observer outside that gravity field. These are proof that space is curved in a gravity field.

Another way that curvature might happen is if you are under the influence of a force, and experiencing an acceleration as a result of that force.

And another way that curvature might happen is if you are rotating.

What Einstein said was, you cannot run a local experiment that will tell whether the acceleration you experience is due to gravity or not. You feel acceleration; you run an experiment, and you see inertial objects and laser beams move in curves. But you cannot do anything in that experiment to tell whether the reason for the curves is gravity, rotation, or linear acceleration. This is called the equivalence principle; it says that acceleration and gravity are indistinguishable. It is therefore correct to state that gravity and acceleration are the same thing.

Now, there are several provisos to this. First, all the gravity (defined as the curvature of space created by a massive object) that we see is based upon some massive object, and appears to radiate approximately in a sphere from the center of mass of that object. As a result, the gravity field is non-uniform; as you get closer, it gets stronger. So if you perform a non-local experiment, for example, release two objects and allow them to become inertial, then in a real gravity field, you will see them move closer together. But that is merely a matter of how the gravity field is shaped, not an intrinsic difference between gravity and acceleration. If you suppose, as I have written elsewhere recently, that you are under the influence of a gravity field that is from a sufficiently large object, then you will not be able to measure the objects moving closer together in a reasonable amount of time, and you'll conclude that you're experiencing an acceleration.

Second, as sol has pointed out recently, if you are rotating, then you will be able to distinguish this from both a real gravity field and from a linear acceleration, because of the shape of spacetime; you will see a special direction, the plane in which you are rotating, with different behavior than any other, and you will also see two other directions, the axes of your rotation, with similarly odd behavior, much closer to what you expect in flat spacetime, and a range of behaviors in the directions between those. However, this is still definable in terms of a spacetime distortion, and the equations used to do it are still derived from those of General Relativity. And although the way those equations must be varied over direction, to account for the way things vary under a rotational frame, at base they are the very same equations used to define the acceleration case, and the gravity case, and even the inertial case. Those same equations work for ALL these cases, and (physicists believe) any others you care to come up with as well.

To top it all off, you don't even have to define precisely what the equations are for any case but the inertial one, and the gravity one. All you have to do is define what motion an object will have in the inertial case, and then translate the coordinates to whatever coordinate system the observer is in; the addition of a correction for the gravitational case completes the picture. You will then get the correct answers if you do this. It's also much easier to do mathematically, and that's important, because we need to be able to define how things move under all circumstances if we're going to say that we have defined physics completely, and completely understand it.

So what is this thread about? Well, there's a question about where these laws come from. And that question involves the question of what happens when you have a totally empty universe, except for some body sitting there in space. Newton's Laws say, and GR agrees, that you cannot define whether such an object is moving in a straight line or not; in other words, there is no absolute space. But GR says something more: there may not be absolute space, but there is absolute spacetime. You can always observe laser beams, or perform local experiments, and determine whether you are accelerating or not. And if this is correct (and we have done many experiments that show that, as far as we can tell, it is), then it doesn't matter whether there is anything present in space but that single body; you will still be able to determine if it is accelerating, and you will still be able to determine if it is rotating, by performing local experiments and observing the results. If it is rotating, then it will undergo continuous acceleration at all points, varying with location; if it is accelerating, then it will undergo uniform forces. Simply by measuring these forces, you will be able to tell, and the reference you are measuring them against is absolute spacetime.

Your understanding is, therefore, correct, but incomplete. sol invictus' understanding is also correct, but more complete. It is better to have equations that describe all imaginable circumstances than only some circumstances, but that doesn't mean that the equations that describe the more limited circumstances are wrong, just that they are limited. Hopefully that helps you understand how we think spacetime, and objects that move in it, behave, more fully than you did before.

 

[sol invictus]

I just mean what a scale shows if I stand on it. A force. Something that's measured in Newtons. I thought that was the sort of thing you wanted to categorize as being "due to gravity" or "due to inertia".

Yes, that kind of measurement is what I had in mind.

Ok. But gravity, measured in that way, isn't a force, right? If I step on a scale, it won't tell me how strong that sort of gravity is. An astronaut orbiting the Earth, and I sitting on my chair, experience nearly the same strength of gravity of that kind.

Yes, that's true, and it's a good point. The issue is that forces really are dependent on the components of the Riemann tensor, metric, etc., not just on scalar invariants constructed from it.

I thought that was one of the results of general relativity or the general case, that the perceived force of gravity is simply a fictitious force like the Coriolis force or centrifugal force?

Yes, I think this is the best answer to my question. We should simply regard all forces in GR as fictitious.

Thanks.

 

[sol invictus]

Your understanding is, therefore, correct, but incomplete. sol invictus' understanding is also correct, but more complete. It is better to have equations that describe all imaginable circumstances than only some circumstances, but that doesn't mean that the equations that describe the more limited circumstances are wrong, just that they are limited.

That's a good way to say it. In my example above, it's probably pretty clear that the constraint that the laws of physics be unchanged when you change coordinates is very powerful. If we know how to compute at what point the rope breaks in the stationary frame, we'd better get the same answer in the rotating frame.

 

[Myriad]

I was responding specifically to this:

Einstein also allows us to think about the problem in another frame of reference. Once the rockets are rotating, we can choose coordinates which rotate with respect to the original coordinates with exactly the same angular speed. In those new coordinates the rockets and rope will be at fixed coordinate positions. However Einstein took as a postulate that the laws of physics are invariant under coordinate transformations, so there must still be a tension in the rope. So where is it coming from?

Never mind the tension in the rope. Angular velocity is not a parameter of a frame of reference. If one can simply transform into a rotating coordinate system at will (as opposed to a rotated one which I agree is a simple coordinate transformation), then you have a twin paradox with your two tethered rockets. In one rotating coordinate system, one rocket is stationary while the other rotates around it; in another, the other rocket is stationary. Thus in each of them, a clock in the other runs slower, and at the end of the experiment when someone winds up the rope in the center and the two rockets come together, each one's clocks will show less elapsed time than the other. That is a contradiction.

Respectfully,
Myriad

 

[69dodge]

Never mind the tension in the rope. Angular velocity is not a parameter of a frame of reference. If one can simply transform into a rotating coordinate system at will (as opposed to a rotated one which I agree is a simple coordinate transformation), then you have a twin paradox with your two tethered rockets. In one rotating coordinate system, one rocket is stationary while the other rotates around it; in another, the other rocket is stationary. Thus in each of them, a clock in the other runs slower, and at the end of the experiment when someone winds up the rope in the center and the two rockets come together, each one's clocks will show less elapsed time than the other. That is a contradiction.

You're only considering time dilation due to velocity. There is another kind, due to gravity. A clock at sea level runs slower than one on a mountaintop, though their relative velocity is zero. The rockets on a rope aren't near the Earth's gravity, but they feel fictitious forces that affect clocks in a similar way.

I'm not sure of the details, but I assume they would show that the two rockets age the same amount regardless of which is considered stationary and which circling.

 

[Myriad]

Angular velocity is not a parameter of a frame of reference.

It appears I have to correct myself here. I am inappropriately equating "frame of reference" with "inertial frame of reference" (which is usually an unstated assumption in discussions of SR but not in GR). I apologize for the error.

I'm still not comfortable with calling a transformation to a rotating frame of reference a (mere) "coordinate transformation" as the actual mapping must include time as a variable. (Hmm, but in GR time is a coordinate, isn't it?... See, I do tend to catch on eventually, just takes a bit of patience is all...)

Okay, never mind my previous post, I have some catching up and rereading to do.

Respectfully,
Myriad

 

[sol invictus]

Angular velocity is not a parameter of a frame of reference.

Actually it is. I can write down the metric for you, if
a) I figure out how to use latex on this forum, and
b) you ask me.

If one can simply transform into a rotating coordinate system at will (as opposed to a rotated one which I agree is a simple coordinate transformation), then you have a twin paradox with your two tethered rockets. In one rotating coordinate system, one rocket is stationary while the other rotates around it; in another, the other rocket is stationary. Thus in each of them, a clock in the other runs slower, and at the end of the experiment when someone winds up the rope in the center and the two rockets come together, each one's clocks will show less elapsed time than the other. That is a contradiction.

What 69dodge said. In the rotating metric, the time coordinate has a coefficient that depends on radius, so there is a position-dependent time dilation as well as a velocity dependent one. I haven't checked explicitly to make sure it works out and the clocks agree, but I'll bet you any amount of money you'd like that it does.

Incidentally GPS systems must compensate for the effect of gravitational time dilation from the field of the earth.

EDIT - crossed posts. Yes, in GR time is a coordinate just like the others, but it is in SR too (think of boost transformations). The relevant thing here is that the transformation to the rotating frame is non-linear in the coordinates (something like [latex]\scriptsize $x' = x \cos(\omega t) + y\sin(\omega t)$[/latex]). That's what makes it non-inertial.

 

[69dodge]

To get this:
[latex]$E=mc^2$[/latex]

say this:
[ latex]$E=mc^2$[/latex]

(but without the space)

 

[sol invictus]

Thanks!!

[latex]$R_{\mu \nu} - {1 \over 2}g_{\mu \nu} R = 8 \pi G_N T_{\mu \nu}$ [/latex]

 

[The Man]

Yes, I think this is the best answer to my question. We should simply regard all forces in GR as fictitious.

Thanks.

I do not think we can jump to that assertion, since gravitational waves can carry energy and momentum away from certain gravitational interactions. Also escaping a gravitational field requires additional energy and momentum, both this and gravitational waves must therefore be the result of and result in what could only be considered to be non-fictitious forces. I think at this time we may need a clearer definition of what constitutes a fictitious or non- fictitious force. Generally a fictitious force is referred to as an inertial force but we may want to extrapolate on that and say that a fictitious force results in no overall net exchange of energy or momentum between particles. Certainly a temporary exchange is typical of a fictitious force but whatever is lost must be regained and whatever is gained must be lost again. An apparent gain or lose of energy and momentum of one particle without a corresponding gain or lose in some other particle is also typical of a fictitious force (Coriolis force). So we could define a fictitious force as a reciprocating exchange of energy and momentum between particles or an apparent net change in energy and momentum of one particle without a corresponding net change in energy and momentum of another particle. Perhaps it is better just to define a non- fictitious force as a non-reciprocating exchange of energy and momentum between particles.