Centrifugal force from GR perspective

[Chi Meson]

I got this from Wikipedia (under the definition of centrifugal force):

Like all Newtonian physics, the above assumes that there is some universal frame of reference from which it can be determined whether or not an object is moving. The theory of relativity dispenses with this, and views "inertial forces" like the centrifugal and Coriolis effects as fully "real", while it is gravity that becomes "fictitious".

I need some clarification here. I had understood that the "ficticious forces" of centrifugal and coriolis were "inertial effects," and that with GR the gravitational force was also an inertial effect. The Wikipedia definition contradicts this. This messes with my head.

So, before I start digging through my old Spacetime Physics text again, is Wikipedia correct here: does Centrifugal become real while gravity becomes ficticious? [smiley:puking:]

 

[NateTG]

I got this from Wikipedia (under the definition of centrifugal force):

I need some clarification here. I had understood that the "ficticious forces" of centrifugal and coriolis were "inertial effects," and that with GR the gravitational force was also an inertial effect. The Wikipedia definition contradicts this. This messes with my head.

So, before I start digging through my old Spacetime Physics text again, is Wikipedia correct here: does Centrifugal become real while gravity becomes ficticious? [smiley:puking:]

This isn't my strong suit, so don't take this as gospel truth. Moreover, as far as I can recall, the coriolis

That said, from a GR perspective gravity bends space-time rather than directly acting as a force on objects, as a consequence, what would be considered an intertial reference with a gravitational field in Newtonian mechanics is considered an accelerated frame of reference in GR.

In that sense, both gravity in GR and the centrifugal force are 'ficticious' forces that are the result of an accelerated reference frame rather than some sort of 'force interaction'.

It's important to realize that describing forces as 'real' or 'ficticious' is primarily an interpretation issue, and is usually not an important distinction in physics. Since GR does not treat accelerated reference frames as 'unnatural' in the same way that Newtonian mechanics does, and there is no 'preferred' frame of reference GR, AFAIK, does not make a distinction between so-called real or ficticious forces.

 

[DW]

You are correct. Wikipedia is wrong. It wouldn't be the first time. In general relativity a real force i.e. four-vector force can not be transformed away. The force of gravity can be locally transformed away simply by going transforming to a free fall frame so you can tell right away that it is not a real force. The gravitational force is the same thing as an inertial force and in relativity is given by affine connections, the Cristoffel symbols of the second kind. Those connections vanish according to local free fall frames. To better understand how fictitious forces such as Centrifugal etc are gravitational consider the metric of special relativity as there is no gravitational sources for that metric. Simply transform coordinates to that of a frame with spin so that the metric becomes equation 6.3.23 at
http://www.geocities.com/zcphysicsms/chap6.htm
An exact calculation of geodesic motion for a test particle according to these coordinates results in the coordinate acceleration of equation 6.3.31. From that one can immediately read off the Coriolis, Centrifugal, and transverse forces 6.3.33. Now since the test particle is in free fall in a spacetime that has NO gravitational sources, you must realize that the forces reckoned to be acting on the particle according to the spinning frame observer absolutely must be fictitious. There is nothing in the spacetime to put a force on it at all. However, these fictitious forces are a result of geodesic motion according to the spin frame observer so he considers them to be the result of a gravitational force. You see, the gravitational force and the inertial or fictitious forces acting on a particle are completely equivalent in general relativity. They are the forces of affine connection.

 

[jcsd]

Like all Newtonian physics, the above assumes that there is some universal frame of reference from which it can be determined whether or not an object is moving. The theory of relativity dispenses with this, and views "inertial forces" like the centrifugal and Coriolis effects as fully "real", while it is gravity that becomes "fictitious".

To be honest this article isn't that great. Newytonian physics is invaraint under a Galilean transformation and as such assumes no universal reference frame, only a special set of reference frames where Newton's laws of motion apply and it givese us no way of deteriming such a frame. The theory of special relativity keeps this asusmption, but extends it so that the equations of Maxwell are also invaraint in inertial frames. The gernarl theory formulate sphysics in a frame invaraint manner.

Gravitaional and inertial forces are treated with equality in GR, so you either say they are both 'real' or both 'ficticious'.

 

[Chi Meson]

Gravitaional and inertial forces are treated with equality in GR, so you either say they are both 'real' or both 'ficticious'.

This is how I understood it (not that I really understand it). Thank you all. Who should tell Wikepedia about their error?

 

[da_willem]

Wikipedia is a user made encyclopedia, anyone can make or edit articles. If you think an article is incorrect or incomplete you can edit it yourself. So go ahead!

 

[pmb_phy]

I got this from Wikipedia (under the definition of centrifugal force):

That article doesn't make a lot of sense.

I need some clarification here. I had understood that the "ficticious forces" of centrifugal and coriolis were "inertial effects," and that with GR the gravitational force was also an inertial effect.

You are correct. It is best to refer to the Coriolis force and the centrifugal force as an "inertial force" since it is misleading to think of them otherwise, especially in GR. To Einstein inertial forces were "real". Einstein argued that a uniform gravitational field cannot be distinguished from a uniformly accelerating frame of reference. A similar thing holds for all non-inertial frames. One is therefore in no position to claim that the Coriolis force is real and fictitious fake or gravity reak and Coriolis fictitious. It is best to simply call them "inertial forces."

Einstein showed that it was possible to generate coriolis and centripetal forces using finite distributions of matter. In particular he used a hollow spherical shell which was rotating. An observer at rest outside and far away in free-fall is in an inertial frame of reference. An observer inside the shell, where the spacetime is flat, who is in free-fall is also in an inertial frame. However the inside observer is rotating with respect to the outside observer. If he wants to be at rest relative to the outside observer then he must choose a non-inertial frame. In particular he must choose one that is rotating with respect to his inertial frame. Hence the Coriolis force can be thought of as a gravitational force.

Einstein commented on the Coriolis force in the February 17, 1921 issue of Nature

Can gravitation and inertia be identical? This question leads directly to the General Theory of Relativity. Is it not possible for me to regard the Earth as free from rotation, if I conceive of the centrifugal force, which acts on all bodies at rest relatively to the earth, as being a "real" gravitational field of gravitation, or part of such a field? If this idea can be carried out, then we shall have proved in very truth the identity of gravitation and inertia. For the same property which is regarded as inertia from the point of view of a system not taking part of the rotation can be interpreted as gravitation when considered with respect to a system that shares this rotation. According to Newton, this interpretation is impossible, because in Newton's theory there is no "real" field of the "Coriolis-field" type. But perhaps Newton's law of field could be replaced by another that fits in with the field which holds with respect to a "rotating" system of co-ordiantes? My conviction of the identity of inertial and gravitational mass aroused within me the feeling of absolute confidence in the correctness of this interpretation.

Other people have commented on inertial forces too. E.g. A.P. French did in his mechanics text, i.e. from Newtonian Mechanics, A.P. French, The M.I.T. Introductory Physics Series, W.W. Norton Pub. , (1971) , page 499.

From the standpoint of an observer in the accelerating frame, the inertial force is actually present. If one took steps to keep an object "at rest" in S', by tying it down with springs, these springs would be observed to elongate or contract in such a way as to provide a counteracting force to balance the inertial force. To describe such force as "fictitious" is therefore somewhat misleading. One would like to have some convenient label that distinguishes inertial forces from forces that arise from true physical interactions, and the term "psuedo-force" is often used. Even this, however, does not do justice to such forces experienced by someone who is actually in the accelerating frame of reference. Probably the original, strictly technical name, "inertial force," which is free of any questionable overtones, remains the best description.

It would be incorrect to think of inertial forces as "fictitious" and 4-forces as "real". It would go against the idea of GR in fact. It would imply that there are special frames of reference and there are no special frames of reference in GR.

Pete
[/quote]

 

[gptejms]

I have a question unrelated to the above discussion:-how can an 'inertial' force have a quantum?What does it really mean?

 

[selfAdjoint]

David, calm down. You have an excellent site, and I admire your work. But nobody likes flame wars here. Roll with the punches and help us out.

BTW, I am not the guy who deletes your posts, but I value your contributions here too much too see you banned for behavior.

 

[quantumdude]

BTW, I am not the guy who deletes your posts, but I value your contributions here too much too see you banned for behavior.

And DW has, in fact, racked up warning points for his recent outbursts. I think that he is perhaps not aware of it because he has elected not to receive private messages (which is why we are in the unfortunate situation of having to deal with this in a science thread). If he keeps it up, he most certainly will find himself banned temporarily, because there is a limit to the number of warning points a member is allowed to have.

DW, why do you carry on like this?

If I ever find out the name of the moderator here who is wrongfully deleting my posts I am going to sue him.

His name is Tom Mattson.

 

[Garth]

I have a question unrelated to the above discussion:-how can an 'inertial' force have a quantum?What does it really mean?

Being wary of jumping into a "flame fight" nevertheless I would like to point out that actually this question is right on the subject of this thread.

There are different ways of dealing with 'objects' in four dimensions, as MTW very clearly spell out in the box 3.2, page 76, in 'Gravitation'.
1. "Geometric language" favoured by MTW
2. “Coordinate language” favoured by Weinberg (G&C); and it is possible also to use
3. “Coordinate-Based Geometric Language”.

After reading these and other earlier posts, am I not correct in concluding that DW will only allow the first and Pete the second?

Geometric language is the fully 4 dimensional “spacetime” perspective that is not frame dependent. In this language the equation of four-momentum is fundamental, complete with mass and not rest mass, fictitious inertial forces and real four-forces.

Coordinate language requires a frame of reference, the observer’s, a basis tetrad, or axes, relative to which measurements are made. The mass of a body increases with its relative velocity, as it acquires ‘kinetic energy’, and therefore it may be said to have “relativistic mass”. Inertial forces can do work (fall off a cliff and you will hurt yourself!) and therefore may be thought of as real.

It might be thought more ‘pure’ to use geometric language, however as observers looking out onto the universe we are undoubtedly in a ‘preferred’ frame of reference – our own (!) – and so coordinate language is also perfectly appropriate.

It is a shame that these different perspectives are not equally recognised as being equally valid in their own right allowing a decent discussion to ensue.

Returning to gptejms question – The problem with the fully geometric language in relating GR to QM in the pursuit of a Quantum Gravity is that the most significant “fictitious” inertial force is of course gravity itself. If gravity is truly fictitious then there will be no graviton to carry it.

As the electro-magnetic, the weak and the strong forces are united in the GUT, so perhaps the situation is that all the existing 'real' forces are already united and nobody has recognised the fact!

Garth

 

[Chi Meson]

OK. I teach HS physics, and so I only barely get into SR and GR with my AP class. Bringing up tensors, transformations, and distributions is way beyond the scope of what I need to teach, and sadly also beyond what I remember from long ago.

But I want to not say anything "wrong" as I simplify the initial steps into physics for my students:

I take the adamant potsition that, from a Newtonian point of view, the centripetal force is not a force but the effect of inertia and the lack of a centripetal force (no argumants there).

I admit to my students that, when they get to GR, then they will see that gravity is also an inertial effect, so therefore if you call centrifugal a ficticious force, then gravity must also be ficticious.

What I pick up from this thread, is that ficticious vs. real is a matter of semantics, and therefore if gravity is real then centrifugal is a real force. But the point made by pmb is to refer to this group as "inertial forces" which are different from (what to call them?) "4-forces."

This still leaves the problem that when living in a Newtonian world, the inertial forces do not satisfy the definition of "force" according to Newton's Laws. For the time being, I'm going to continue calleing them "inertial effects" with the asterisk that, when the student gets to advanced physics, there will be changes in definitions, noting that the problem lies more with the language than with the physics.

I would be interested to know if anyone sees a glaring problem with this treatment of the "problem." Remember, I teach high-school, and only a tiny fraction of my students will become engineers or physicists.

 

[Garth]

Keep up the good work Chi Meson! High School is where the vision is either caught or lost; behind every genius there's a teacher who inspired them.

My post about the different "languages" used in SR/GR, basically geometric or coordinate language, is rather more than just semantics. I believe it is all to do with the perspective one can adopt in order to conceptually model physical reality.

The 4-force, geometric language, is coordinate free. Therefore the 3D of space and 1D of time become a 'block' 4D spacetime. Past present and future lose their different meanings and everything seems set in 'concrete'. There are no absolute membranes of simultaneity of 'the present moment' dividing past from future; rather simultaneity is relative to each observer. The invariants of this world are the spacetime interval between two events, the mass (the norm or length of the energy-momentum vector) of a particle and the speed of light. It is the world introduced to us by Special Relativity and developed by General Relativity. In GR the inertial forces may be transformed away and are seen to be fictitious, including as we have said, gravity. Real 4-forces, such as that produced by an external force accelerating a spacecraft have the interesting property that they do not increase the norm of the 4-momentum vector but rather rotate it in energy-momentum space. Actually the total energy of the spacecraft increases, as does its normal 3-momentum, but when combined in the energy-momentum vector the effect of the signature of the metric is that one is subtracted from the other and the resultant 4-energy-momentum vector is constant in length.

However it is a world that has defied unification with QM in a Quantum Gravity. It is also a world that we manifestly do not experience, for we are set in a particular frame of reference and observe the world from it. We experience inertial forces, be it the ground supporting us against gravity or the centripetal force keeping us on the roundabout. They are real to us.

The understandings of SR and GR are useful in that they allow us to transform from one observer to another and predict the effects of gravitational fields, but their perspective may have no reality beyond this.

Personally I like to think both languages describe real viewpoints of view and it is a matter of perspective as to which one you find most convenient to use.

I hope this helps and not confuses!

Garth

 

[pmb_phy]

His name is Tom Mattson.

I'm sure you're resting easy Tom since there is nothing that you can be sued. As everyone knows, as fact, this is a private forum with a moderator. A moderator is not defined as a person who deletes posts that dw does not like.

Pete

 

[pervect]

OK. I teach HS physics, and so I only barely get into SR and GR with my AP class. Bringing up tensors, transformations, and distributions is way beyond the scope of what I need to teach, and sadly also beyond what I remember from long ago.

But I want to not say anything "wrong" as I simplify the initial steps into physics for my students:

I take the adamant potsition that, from a Newtonian point of view, the centripetal force is not a force but the effect of inertia and the lack of a centripetal force (no argumants there).

I admit to my students that, when they get to GR, then they will see that gravity is also an inertial effect, so therefore if you call centrifugal a ficticious force, then gravity must also be ficticious.

What I pick up from this thread, is that ficticious vs. real is a matter of semantics, and therefore if gravity is real then centrifugal is a real force. But the point made by pmb is to refer to this group as "inertial forces" which are different from (what to call them?) "4-forces."

This still leaves the problem that when living in a Newtonian world, the inertial forces do not satisfy the definition of "force" according to Newton's Laws. For the time being, I'm going to continue calleing them "inertial effects" with the asterisk that, when the student gets to advanced physics, there will be changes in definitions, noting that the problem lies more with the language than with the physics.

I would be interested to know if anyone sees a glaring problem with this treatment of the "problem." Remember, I teach high-school, and only a tiny fraction of my students will become engineers or physicists.

I also got it strongly hammered into me in High School that centrifugal forces weren't real. So you are doing the standard thing there. Though as I look back on the issue, I'm not sure why this concept was stressed so much and so strongly.

I've come to view a centrifugal force as a "generalized force". I'm not sure if giving that idea to HS students would be a good idea, though. The basis of this statement is the Lagrangian formulation of mechanics. (You could probably also do this with the principle of least action). Without Lagrangian mechanics, it's not clear that the students would understand the terminology.

I think the best thing to do, on the overall, is to stress that centrifugal force is not a force in Newtonian mechanics. By making the statement specific to Newtonian mechanics, you leave the door open for them being called forces in other versions of mechanics. But you don't have to go into details, in fact doing so would be outside the scope of your course.

 

[Andrew Mason]

I also got it strongly hammered into me in High School that centrifugal forces weren't real. So you are doing the standard thing there. Though as I look back on the issue, I'm not sure why this concept was stressed so much and so strongly.

This is quite an interesting discussion.

It seems to me that there is a real centrifugal force that is the 'reaction' to centripetal force. This is illustrated by a cord with a weight attached to the end through a piece of pipe and another weight at the top that is whipped around in a circle. The first (down) weight provides the centripetal force that keeps the circling mass from flying away. The force that keeps that mass from falling down through the pipe is the centrifugal force - resulting from the inertia of the circling mass. (It is not responsible for the circling mass flying away when the cord is cut. That is due to the cessation of the centripetal force).

With gravity, the centrifugal force seems to disappear for an orbiting body. While the astronaut on the training centrifuge feels a gut crushing feeling, the orbiting astronaut experiences no such feeling. Unlike the roller coaster, the orbiting object cannot flee its orbital path without adding energy: the orbiting mass does not have a natural tendency to flee the center of rotation.

At least I thought this was the case. Astro-physicist Marek Abramowicz from Sweden seems to think otherwise. See:
http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1990MNRAS.245..733A

Parts of this paper are very readable and provide several thought experiments to provide examples. I must confess, however, that I don't really understand his concept of centrifugal force in a gravitational field.

AM

 

[krab]

Keep up the good work Chi Meson! High School is where the vision is either caught or lost; behind every genius there's a teacher who inspired them.

Kudos from me too.

I also got it strongly hammered into me in High School that centrifugal forces weren't real.

Unfortunately there are HS teachers who know just enough physics to be dangerous.

This still leaves the problem that when living in a Newtonian world, the inertial forces do not satisfy the definition of "force" according to Newton's Laws.

Technically, this is correct. But keeping in mind that people already live in a "world" (not "Newtonian"), and have developed an understanding for the word "force", I find it kind of presumptuous to tell them that what they think of as a real force is in fact fictitious. That might not be clear, so let me give an example. A student says that he was cornering in a car and centrifugal effect caused his head to hit the side window. The HS physics teacher (not you, Chi Meson) says "No, this is wrong wrong wrong. There is no such thing as centrifugal force; you must observe from an inertial frame and then you will see that the car window came up and hit you on the side of the head." Do you think that the student will be impressed with the wonderful world of physics? I think not. So I think "inertial" is vastly preferable to "fictitious"; everyone knows that "fictitious" means "not real", but "inertial" would be a new adjective that a student can get a feeling for by studying physics further.

 

[Chi Meson]

A student says that he was cornering in a car and centrifugal effect caused his head to hit the side window. The HS physics teacher (not you, Chi Meson) says "No, this is wrong wrong wrong. There is no such thing as centrifugal force; you must observe from an inertial frame and then you will see that the car window came up and hit you on the side of the head."

I, uhh, I actually do say that. I do avoid the word "ficticious" though. I am adamant about this because the notion of centrifugal force confuses the idea of Newtonian force. Too many are already confused. The simple fact that there is NO object that pulls outward on a revolving object means that the 3rd law (which has just been learned) is violated.

Do you think that the student will be impressed with the wonderful world of physics? I think not.

Actually, most of my students are most fascinated by this part of my lecture. It's the presentation that makes it good.

So I think "inertial" is vastly preferable to "fictitious"; everyone knows that "fictitious" means "not real", but "inertial" would be a new adjective that a student can get a feeling for by studying physics further.

And I agree with you there. I think this is the "way out" of the conundrum.

 

[Chi Meson]

I also got it strongly hammered into me in High School that centrifugal forces weren't real. So you are doing the standard thing there. Though as I look back on the issue, I'm not sure why this concept was stressed so much and so strongly.

I personally get annoyed with it because it is the one thing that most people "know" about physics before they take the class, and it's wrong. even though it may be considered a force from a GR perspective, that is still not where the students are coming from. Centrifugal force, as it is misunderstood by the general public, does not satisfy any definition of force. Then they refuse to correct it. Ask any mentor how annoying that is! (See TD forum).

Understanding the centrifugal effect is one of the more satisfying moments of conceptual synthesis in HS physics. Allowing a watered down GR version of centrifugal as a force does impede the understanding of Newtionan force.

I I again agree with you all that those few who continue with physics should not have any doors closed. NEarly every day I say at one point "anyone who goes into advanced physics will find that...etc.etc." But I feel very strongly that students must learn Newtonian correctly before moving on.

 

[Chi Meson]

It seems to me that there is a real centrifugal force that is the 'reaction' to centripetal force. This is illustrated by a cord with a weight attached to the end through a piece of pipe and another weight at the top that is whipped around in a circle. The first (down) weight provides the centripetal force that keeps the circling mass from flying away. The force that keeps that mass from falling down through the pipe is the centrifugal force - resulting from the inertia of the circling mass. (It is not responsible for the circling mass flying away when the cord is cut. That is due to the cessation of the centripetal force).

Here is another wrinkle. The reaction force is actually another centripetal force. No object (from Newtonian perspective) can be the "center" of a circle while pulling on an other object that revolves around it (think earth-moon). The "central" object also moves in a smaller epi-circle. Even if the object is much larger, there will still be a theoretical radius of revolution (sort of like the Earth's theoretical acceleration toward a dropped ball).

Thanks again to everyone. I am enjoying this thread; it has been very helpful to me.

 

[Andrew Mason]

Here is another wrinkle. The reaction force is actually another centripetal force. No object (from Newtonian perspective) can be the "center" of a circle while pulling on an other object that revolves around it (think earth-moon). The "central" object also moves in a smaller epi-circle. Even if the object is much larger, there will still be a theoretical radius of revolution (sort of like the Earth's theoretical acceleration toward a dropped ball).

What about my example but with two masses on cords whirling about the center pipe - both cords tied to the central mass and both revolving in the same direction about a common center, but 180 degrees to each other? Would you say that the reaction force is the other centripetal force? or is it, at least in part, the force of gravity on the central mass to which the cords are connected?

I would be interested in hearing any coments about Abramowicz's paper. I have some difficulty with his concept of centrifugal force. His premise is that the only path in a gravitational field that does not result in a centrifugal force is the path of a photon.

He seems to conclude that just outside a black hole, the centrifugal force is toward the center of the black hole.

AM

 

[Chi Meson]

What about my example but with two masses on cords whirling about the center pipe - both cords tied to the central mass and both revolving in the same direction about a common center, but 180 degrees to each other? Would you say that the reaction force is the other centripetal force? or is it, at least in part, the force of gravity on the central mass to which the cords are connected?

In a Newtonian sense (and I'm in the wrong forum for that), yes the reaction force is the other centripetal force. The gravitational force is (assumed) downward 90 degress from the plane of the circles, so that cannot be centripetal. Even though the cords are wrapped around a central pipe, the cord-pipe-cord is still the chain of tension that carries the force from one to the other.

I can't comment on the other part of your question.

 

[Garth]

If you are in a non-inertial frame of reference inertial forces appear that otherwise would not be there.

They are real in that frame, and so hold your weights on a string and bang your head on the car window and so on, but they are "only" a response to the non-inertial frame of reference.

The understanding of GR is that standing on the ground one is in a non-inertial frame of reference, so the very-real-to-you gravitational force can be transformed away and may be called ""ficticious"", but try telling that to someone who has just fallen off a ladder!
Your frame of reference affects the way you observe the universe - I think this is called relativity?

Garth

 

[pervect]

To expand a bit on what others have said, if you have two weights connected by a string whirlling around, if you view them from a stationary inertial frame, the only force that exists is the centripetal force. This is the standard Newtonian explanation.

Basic Newtonian mechanics can't handle anything other than an inertial frame of reference, so that is why you'll be told in a basic Newtonian physics class that there isn't any centrifugal force.

It would be a really good idea to try and understand the Newtonian viewpoint first, before going on to anything more advanced. This is why this concept is stressed so much, I suppose.

You really can't view the centrifugal force as existing in an inertial frame.

When you go from basic Newtonian physics to more advanced formulations, like Lagrangian mechanics, you gain the ability to do physics in coordinate systems that are not inertial. These cordinates are called "generalized coordinates". To go along with these "generalized coordinates" one has "generalized forces". This system of mechanics allows one to view the weights from a rotating frame of rererence. In the rotating frame one simply has two weights , not moving anymore , connected by a string. But because the frame of reference is not inertial, one also has a generalized force, the centrifugal force, pulling the weights apart. If the weights were allowed to move, one also might have anotther fictious force to contend with, the "Coriolis force".

In studying weather, because the Earth is rotating, the Coriolis force becomes important. It would be possible to insist that oen must do physics in a strictly inertial frame, but it would be inconvenient in this case. It's much more convenient to think of the Earth's surface as one's reference frame, even though it's not inertial, because the Earth is rotating.

So "generalized forces" are very useful, but they arise only when one does physics in non-inertial frames of reference.

 

[Andrew Mason]

To expand a bit on what others have said, if you have two weights connected by a string whirlling around, if you view them from a stationary inertial frame, the only force that exists is the centripetal force. This is the standard Newtonian explanation.

I agree with everything you have said.

If one thinks of a mass undergoing an acceleration (other than an acceleration due to gravity) in the frame of reference of that acclerating mass, there is an apparent force that resists acceleration. In the rest reference frame, it is viewed simply as inertia. But in the frame of the accelerating mass it appears to be a force acting in the opposite direction to the accelerating force.

The centrifugal force is the same kind of apparent force, except that the direction of the centripetal acceleration and associated centrifugal force is always changing.

A real difference occurs when the acceleration is provided by gravity. There is no inertial/centrifugal force or effect in either the inertial or non-inertial (falling) reference frames. This is where I lose Abramowicz. It seems to me that there can be no real or apparent centrifugal force associated with a gravitational force in either frame of reference.

AM

 

[Garth]

Let me reiterate:- Whether inertial forces such as centrifugal forces are "real-to-you" or not depends on your frame of reference, that is your perspective on the world around you. Looked at from a fully geometric spacetime perspective inertial forces do not exist, not even gravity, they can be transformed away and are therefore artefacts of geometry.

Garth

 

[pervect]

So far I've been busy on some of my own projects and haven't had a chance to look at the article you mentioned, but I can hopefully shed a little light on the centrifugal force "paradox" in GR.

It's well known that a massive body can orbit a black hole only outside the photon sphere, at 1.5 times the Schwarzschild radius. (3GM/c^2).

At the photon sphere, the orbital velocity is the speed of light.

Inside the photon sphere, a body can't orbit the BH, no matter how fast it moves.

Unless I've made a big error in the calculations (sadly, it's all too possible) what happens is that when you try to orbit a BH inside the photon sphere, you actually increase the acceleration (d^2 r/dtau^2) at which you fall into the black hole!

This is based on the equation for geodesic motion in geometric units

(dr/dtau)^2 = (E/m)^2 - (1-2M/r)(1+L^2/(r^2 m^2))

You can convert dr/dtau to d^2r / dtau^2 by taking

dr/dtau = f(r), the square root of the above quation

d^2r/dtau^2 = d(dr/dtau)/dr * dr/dtau = df/dr*f

When you do this you get

dr^2 / dtau^2 = -M/r^2 + L^2(3M-r)/m^2*r^4

Here M is the mass of the black hole, r is the distance away from the black hole, L is the angular momentum, and m is the mass of the small test body.

Getting away from the math, this means that inside the photon sphere, you _always_ have to thrust away from the black hole to avoid falling in. You can't just orbit it.

If you try to orbit the black hole (you have a non-zero angular momentum), you have to thrust even _harder_ inside the photon sphere than if you stayed still (no angular momentum).

Mathematically, this is because the second term above changes sign when r=3M, so L goes from helping you hold position to hurting you when you move inside the photon sphere.

I view this intuitively as the black hole's gravity becoming stronger when one tries to orbit it, rather than the direction of centrifugal force changing.

This is perhaps a somewhat problematic intepretation, so I'll say something hopefully less controversial.

If you look at the tidal force on a body orbiting a black hole, it is a stronger tidal force than a body "hovering" at the same radius. The tidal force isn't dpendent on the global coordinate system, it's something an observer can measure locally.

(Note: I've only seen this analysis performed outside the photon sphere, where orbits exist. It's fairly easy to do the analysis if you ignore frame dragging induced rotation, but it's not particularly easy if you do include this effect).

Where I saw the analysis done is

here

 

[Andrew Mason]

If you try to orbit the black hole (you have a non-zero angular momentum), you have to thrust even _harder_ inside the photon sphere than if you stayed still (no angular momentum).

Is this because orbital motion requires such enormous energy that it increases the mass of the orbiting body, and therefore the gravitational attraction?

AM

 

[pervect]

No, the best anology is with the electrostatic case of what happens to the field of a charge when it moves relative to you (or when you move relative to it). The electric field in this case increased by a factor of gamma in the transverse direction. Something roughly similar happens with the gravitational field.

One of my projects has been to work out _exactly_ what happens to the gravitational field in this situation (actually, I'm interested in the situation of a moving mass, but of course a body moving near a mass is very similar). Unfortunately, when I take different approaches to the problem, I've been coming up with different answers. Thus I don't have a result I trust yet on any but a qualitiative level. (The results at least agree qualitiatively, which is better than nothing I suppose).

So far I've learned that Maple displays the Christoffel symbols backwards, and that there are some additional terms in non-coordinate bases when calculating geodesic deviation that aren't present in a coordinate basis. But I still don't have a total resolution I'm happy with.

 

[yogi]

CHI Meson - you might want to read Feynman's short expose' on the subject in Volume 1 of his lectures on Physics. He questions whether gravity is a pseudo force like inertia since these forces are always proportional to mass. He ruminates that perhaps gravity is only a pseudo force that results from the fact we do not live in a Newtonian inertial frame.

 

[Chi Meson]

CHI Meson - you might want to read Feynman's short expose' on the subject in Volume 1 of his lectures on Physics. He questions whether gravity is a pseudo force like inertia since these forces are always proportional to mass. He ruminates that perhaps gravity is only a pseudo force that results from the fact we do not live in a Newtonian inertial frame.

Thanks, I have read it. This is what my initial viewpoint was primarily based on, and which was countered by the Wikepedia definition that has since been shown to be at fault.

I always turn to Feynman, but it has been over thirty years and a new position could have been solidified since then. I'm glad to know I'm not in the wrong tree (not very high up that tree mind you, but at least it's the right one).

 

[yogi]

I also am a Feynman fan - some years ago I wrote an article based upon Feynman's idea showing that Hubble expansion leads to a divergent volumetric acceleration of magnitude equal to the gravitational constant.

 

[Rob Woodside]

Fictious or inertial forces are dangerous. They can kill you! They differ from real forces in that they do not occur in action/reaction pairs. F=ma in an inertial frame only. If body A has an acceleration a relative to an inertial frame and body A' has a mass m and an acceleration a' relative to body A. Then for body A' in the inertial frame one has F= m(a+a') . In the accelerated frame, riding A, one has F-ma =ma'. The fictitious force -ma has no reaction partner and is in the opposite direction to the acceleration a. In relativity the acceleration is the covariant derivative of velocity and is made of two pieces: the usual coordinate partials plus a Christoffel piece which accounts for how the coordinate directions change from point to point. As others have said fictitious forces like centrifugal, coriolus, and the Newtonian mg arise from this second piece.